advances in geophysics, vol. 36

ADVANCES IN

G E O P H Y S I C S

VOLUME 36

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Advances in

G E O P H Y S I C S Edited by

RENATA DMOWSKA Division of Applied Sciences

Harvard University Cambridge. Massachusetts

BARRY SALTZMAN Department of Geology and Geophysics

Yale University New Haven, Connecticut

VOLUME 36

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CONTENTS

The Kinematics and Dynamics of Poloidal-Toroidal Coupling in Mantle Flow: The Importance of Surface Plates and Lateral Viscosity Variations

ALESSANDRO M. FORrE AND w. RICHARD PELTIER 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Analytic Description of Surface Plate Kinematics , , . , . . . . . . . . . . . . , , ,

2.3 An Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 Theory .. . . .. .. .. ... . . . .. . . .. . . . . . .. .. . .. .. . . .. .. .. ... .. .. .. ... . 3.2 Net Rotations in the Lithosphere and Mantle . . . . . . , . . . . . . . . . . . . . . . . 3.3

2. Buoyancy-Driven Plate Motions . . . . . . .......... . . . . . . . .

2.2 Buoyancy-Driven Plate Motions . . . . . . . .

3. Lateral Viscosity Variations in the Lithosphere ...................

Inverting for Lateral Viscosity Variations . . . . . . . . . . . . . . . . . . . . . 4. Mantle Dynamics with 3D Viscosity Variations . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Variational Calculation of Buoyancy-Induced Flow . . . . . . . . . . . . . . . . . . 4.3 Generalized Green Functions . . , . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . 4.4 Horizontal Divergence and Radial Vorticity of Buoyancy-Induced

Surface Flow . . . . . .

4.6 Nonhydrostatic Geoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Differential Rotation in the Mantle

5 . Conclusion . . . , . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix I. Horizontal Gradien Appendix 11. Spherical Harmonic Coupling Rules . . . . . . . . . . . . . . . . . . . . I . Appendix 111. Analytic Harmonic Decomposition of Horizontal

Divergence and Radial Vorticity , . . . . . . . . . . . . . . . , . . , . . Appendix 1V. Momentum Conservation in a Medium with

3D Viscosity Variations . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix V. Viscous Stress Acting on an Undulating Surface . . . . . . . . . . . Appendix VI. Dynamic Topography with Lateral Viscosity Variations . . , . Appendix VII. Nonhydrostatic Geoid in a Self-Gravitating Mantle . . . . . . , References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Formulation of the Variational Principle . . . .

4.5 Dynamic Surface Top y . , . . . . . . . . . . . . . . , , . . . . . . . . . . . .

i 5 6

13 27 31 31 39 41 48 48 51 58

61 72 84 89 92 94 96

101

107 109 1 1 1 115 116

V

vi CONTENTS

Seismotectonics of the Mediterranean Region

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2 . Before Plate Tectonics ( 1 885- 1970) .......................... 124

2.1 Early Tectonic Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.2 Early Seismicity and Seismotectonic Maps ......................... 130 2.3 Focal Mechanisms and Seismotectonics ............................ 136

3 . I First Interpretations Based on Plate Tectonics ....................... 142 3.2 Tectonic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.3 Further Seismotectonic Studies ................................... 156

4 . Recent Seismotectonic Studies (1986- 1993) ............................ 158 5 . Azores to Tunisia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.1 Azores-Gibraltar-Tunisia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.2 Iberia .......................................................... 169 5.3 Maghreb ....................................................... 174

6.1 Sicily-Calabria . . . . . . . . .......................... 176

6.3 Alps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.1 Hellenic Arc and Aegean ................ ...................... 186 7.2 Anatolia ........................................................ 192 7.3 Carpathians ..................................................... 195

8 . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

AGUSTiN UDiAS AND ELISA BUFORN

3 . Plate Tectonics Interpretations (1970- 1986) ............................ 142

6 . Italy and the Alps ........... .......................... 176

6.2 Apennines ...................... ........................... 180

7 . Hellenic Arc. Anatolia. and Carpathians ................................ 186

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

.

ADVANCES IN GEOPHYSICS, VOL. 36

THE KINEMATICS AND DYNAMICS OF POLOIDAL- TOROIDAL COUPLING IN MANTLE FLOW: THE IMPORTANCE OF SURFACE PLATES AND LATERAL VISCOSITY VARIATIONS

ALESSANDRO M. FORTE* W. RICHARD PELTIER Department of Eurth cind Plunetary Science

Crrinbridge. Mussuchu.sett.s 02 I38

Depnrrment of Physics Hurrrrrd Universirj Universiry of Toronto

Toronto. Onrario Cunada M5S I A I

I . INTRODUCTION

The ability of the mantle and lithosphere to creep over geologic time scales is due to the presence of naturally occurring atomic-scale defects in the lattice of crystal grains (e.g., Weertman and Weertman, 1975; Carter, 1976; Nicolas and Poirier, 1976; Weertman, 1978). The imposition of deviatoric stresses causes these defects to propagate and thus allows mantle material to creep or “flow.” If the ambient temperature is sufficiently high, this solid-state flow will persist so long as the stress is maintained and the deformation process may achieve steady state. The steady-state creep of crystalline substances may be characterized by a single parameter, the effective viscosity (e.g., Stocker and Ashby, 1973; Weertman and Weertman, 1975).

This effective viscosity 77 provides the link between imposed deviatoric stress and resulting deviatoric strain rate as follows:

( 1 )

in which E#i is the deviatoric strain-rate tensor and T,, is the deviatoric stress tensor. The deviatoric strain-rate tensor is defined in terms of the material flow velocity u as follows:

7.. = 217 E.. 11 ’

where a, = a/ilx,. Solid-state creep in crystalline media generally occurs by two independent mechanisms: dislocation glide and climb (e.g., Weertman, 1968) and diffusion of point defects through crystal grains and/or along grain boundaries (e.g., Herring, 1950; Coble, 1963; Green, 1970). The theoretical expression for

*Present address: Institut de Physique du Globe, DCparternent de Sisrnologie, 75252 Paris, France

I Copyright 0 1994 hy Academic P w s . Inc.

All rights of reproduction in uny form rcmved.

2 ALESSANDRO M . FORTE AND W. RICHARD PELTIER

the effective viscosity, derived from detailed consideration of these creep mechanisms, is

where A is a dimensional constant that depends on the details of the creep mechanism, d is the effective size of crystal grains, T = [T,,T,,]"~ is the square root of the second stress-tensor invariant (Stocker and Ashby, 1973), k is Boltzmann's constant, T is the absolute temperature, A E is the diffusion activation energy, A V is the diffusion activation volume, and P is the total pressure.

The theoretical expression (3) for the effective viscosity of crystalline media is useful for understanding the importance of viscosity variations in the Earth's mantle. If mantle creep occurred dominantly through diffusion of point defects, the effective viscosity in (3) would be independent of stress (i.e., n = 1). The grain- size dependence of diffusion creep is quite pronounced and m = 2 for Herring (bulk) diffusion and rn = 3 for Coble (grain-boundary) diffusion. If mantle creep instead occurred through propagation of dislocations, the effective viscosity will be insensitive to grain size (i.e., m = 0) and sensitive to the ambient deviatoric stress, with the stress exponent n = 3 being typical (e.g., Weertman, 1968; Carter, 1976). In either creep mechanism, steady-state creep is ultimately dependent on the creation and diffusion of point defects and is therefore thermally activated. This is manifested by the exponential temperature dependence in (3). The effective viscosity of the mantle is therefore expected to be most sensitive to variations in temperature. Over the past decade numerous studies have also indicated the importance of chemical environment (e.g., the presence of H,O and COz) on the effective viscosity of mantle rocks (e.g., Kohlstedt and Hornack, 1981; Ricoult and Kohlstedt, 1985; Karato ef al., 1986; Borch and Green, 1987). The dependence of effective viscosity on grain size, stress, pressure, temperature, and chemical environment implies that the viscosity of the mantle is expected to be very heterogeneous, owing to the lateral and depth variation of these thermodynamic state variables.

The significant mathematical difficulties arising from the treatment of arbitrary three-dimensional (3D) viscosity variations have led to an overwhelming focus on mantle flow models in which the viscosity is assumed to be constant, or to vary with depth only. Such simplifications have nonetheless led to a deep understanding of the basic physics underlying the thermal convection process responsible for the "drift" of the Earth's tectonic plates and the global variation of surface heat flux (e.g., Turcotte and Oxburgh, 1967; McKenzie et al., 1974; Peltier, 1972, 1985; Jarvis and McKenzie, 1980; Jarvis and Peltier, 1982; Solheim and Peltier, 1990, 1993, 1994; Peltier and Solheim, 1992).

Theoretical modeling of mantle flow, based on the simplifying approximation that the viscosity depends only on depth, culminated with the development of

POLOIDAL-TOROIDAL COUPLING IN MANTLE FLOW 3

models that are used to predict the 3D mantle circulation expected on the basis of seismically inferred lateral density heterogeneity (e.g., Richards and Hager, 1984; Ricard et al., 1984; Forte and Peltier, 1987, 1991a). Such flow modeling has demonstrated that the observed long-wavelength nonhydrostatic geoid may be suc- cessfully described in terms of the seismically inferred global heterogeneity in the mantle (e.g., Hager et al., 1985; Forte and Peltier, 1987, 1991a; Hager and Clay- ton, 1989; Forte et al., 1992). There therefore appears to be no evidence in the long-wavelength geoid data, or indeed the dynamic surface topography data (Forte et al., 1993a), for the presence of a significant effect due to lateral variations of mantle viscosity. Since such lateral heterogeneity must exist if the crystalline mantle is in a state of motion determined by the thermal convection process, the success of these simple models would appear to indicate that the dynamics of flow in a laterally heterogeneous mantle are such that the existence of lateral rheology variations does not significantly impact surface observables such as dynamic topography and nonhydrostatic geoid anomalies.

The inadequacy of flow models which assume a spherically symmetric viscosity distribution becomes truly apparent only by considering the observed motions of the tectonic plates (e.g., Hager and O’Connell, 1981; Forte and Peltier, 1987). In a fluid shell with spherically symmetric viscosity, buoyancy forces excite only poloidal flow (which produces a pattern of purely converging or diverg- ing flow at the surface) and thus fails completely to account for the strong toroidal (i.e. strike-slip) component of actual plate motions (e.g., Hager and O’Connell, 1981; Forte and Peltier, 1987). The observed equipartitioning of kinetic energy between poloidal and toroidal plate motions is a direct consequence of the plate- like mechanical structure of the lithosphere. This equipartitioning has also been investigated by O’Connell et al. (1991), who suggest that the present-day ratio of toroidal to poloidal energy in the plate motions appears to be nearly a minimum. A detailed consideration of the relationship between toroidal energy and the strike-slip motion of plates at transform faults has been presented by Olson and Bercovici ( 199 1 ).

The mere existence of plates, with their “weak” boundaries and relatively “strong” interiors, implies that the effective viscosity of the lithosphere exhibits extreme lateral variations. The mathematical difficulties of dealing explicitly with such extreme variations of rheology have motivated several studies that attempt to overcome these difficulties by directly employing the observed plate motions (e.g., Hager and O’Connell, 1981) or by employing the geometry of the plates as a surface boundary condition (e.g.. Ricard and Vigny, 1989; Forte and Peltier, 1991a,b; Gable et al., 1991). Such treatments of the plates are essentially kinematic. To model the plates in a dynamically consistent manner, and to understand the rheologic coupling of poloidal and toroidal surface flow, requires an explicit treatment of lateral viscosity variations.

The mathematical modeling of lateral viscosity variations in numerical simu-

4 ALESSANDRO M. FORTE AND W. RICHARD PELTIER

lations of thermal convection has been almost exclusively carried out in two- dimensional (2D) Cartesian geometry. Perhaps the most complete of such studies, in the detailed investigation of stress-, temperature-, and pressure-dependent viscosity, is that by Christensen (1984). An investigation of the coupling between distinct Fourier harmonic components of the flow field due to lateral viscosity variations in a 2D Cartesian geometry has also been carried out by Richards and Hager ( 1989). Such studies, confined to 2D geometries, are intrinsically limited because they cannot describe the excitation of toroidal flows and their coupling to poloidal flows. A treatment of poloidal-toroidal coupling requires a full 3D modeling approach.

Numerical investigations of lateral viscosity variations in 3D thermal convection simulations are relatively recent. Christensen and Harder (1991) have carried out simulations with temperature-dependent viscosity in a 3D Cartesian geometry. Recent high-Rayleigh-number simulations of thermal convection in 3D Cartesian geometry, with temperature-dependent viscosity, have also been performed by Tackley ( 1993).

Initial investigations of the effects of lateral viscosity variations in 3D spherical geometry have been rather limited in scope (e.g., Iticard et al., 1988; Stewart, 1992) and are marred by questionable assumptions. (These difficulties are described later in this chapter.) Significant progress is now being made on the basis of several dynamically consistent formulations of the effects of lateral viscosity variations in the lithosphere (e.g., Ribe, 1992; Forte, 1992) and throughout the entire volume of the mantle (e.g., Forte, 1992; Cadek ef al., 1993; Martinec et al., 1993; Zhang and Christensen, 1993).

In this chapter we present a complete description of the recent theoretical developments and results outlined in Forte ( 1992). We describe three rather different but complementary methods for investigating the effects of lateral rheology variations on mantle-lithosphere How. In Section 2 we present a formalism for calculating buoyancy-induced plate motions that explicitly satisfy the constraint that each plate moves only by a rigid-body rotation. Although this formalism has been previously employed to model plate motions in Forte and Peltier (199 la,b), a complete mathematical description has not been presented. In Section 2 we therefore provide a detailed derivation of this formalism and illustrate its implications through several calculations of buoyancy-induced plate motions. The essentially kinematic approach described in Section 2 allows us to avoid an explicit consideration of lateral viscosity variations, but we then lose any understanding of the dynamic processes that generate toroidal flow and determine its coupling to poloidal How.

In Section 3 we therefore present a dynamically consistent treatment of lithospheric How in which the effects of lateral viscosity variations are explicitly included. We therein assume that these variations are important only in the litho-


sphere and may be neglected in the underlying mantle. This assumption allows us to formulate an inverse problem that we solve for the lateral viscosity variations in the lithosphere that are consistent with the observed plate velocities.

It is not obvious, however, that lateral variations of viscosity in the deep mantle are in fact negligible. The seismically inferred lateral variations of elastic-wave speeds are significant throughout the mantle and the associated temperature variations are expected, from Eq. (3), to yield significant lateral variations of effective viscosity. In Section 4 we therefore provide a complete description of a formalism that allows us to model the effects of 3D viscosity variations in a spherical fluid shell. This theory is based on a variational formulation of the principle of momentum conservation and provides a mathematically efficient, and physically transparent, description of flow dynamics. This formalism, which was initially outlined in Forte (1992), represents the principal contribution of this chapter. In Section 5 we summarize our main conclusions.

The reader will note that the discussion in the main text makes frequent reference to the mathematical Appendixes. In these Appendixes we have assembled, and in many cases derived, the principal tools required for a mathematical analysis of flow dynamics in spherical shells with lateral viscosity variations and rigid- surface plates. While the relegation of much of the formal material to these Ap- pendixes has helped to streamline the discussion in the main text, the material is essential for the comprehension of our analysis.

2. BUOYANCY-DRIVEN PLATE MOTIONS

The existence of rigid surface plates with weak boundaries represents an extreme manifestation of lateral variations of effective viscosity in the Earth. Hager and O’Connell (198 1 ) proposed a procedure for avoiding the explicit treatment of such rheologic variations that involves matching the stresses exerted by buoyancy- driven flows, acting on a no-slip surface, with the external stresses associated with a prescribed plate velocity field. This procedure was modified somewhat in Gable et al. ( 199 I ) and Ricard and Vigny ( I989), in which the stresses exerted by buoyancy-driven flow acting on a no-slip surface are matched with the imposed surface stresses arising from some generally prescribed field of plate-like surface velocities. This matching then provided the required plate-motion parameters (e.g. the plate angular-velocity vectors). An important simplifying assumption in this approach is that the stresses acting on the plate boundaries are taken to be identically zero. As pointed out in Hager and O’Connell (1981 1, this assumption is questionable and does not account for the possibly significant collision-related stresses at subduction zones or shear stresses along transform boundaries. Such difficulties motivated the development, described in Forte and Peltier ( 199 1 a,b), of a different


approach to the problem of buoyancy-induced plate motions that does not make any u priori assumptions concerning the state of stress in the plates. The mathematical development and implementation of this approach is described fully below for the first time.

2.1. Analytic Description of Surface Plate Kinematics

The existence of effectively rigid plates constrained to the move on the surface of a sphere implies, by Euler's theorem (e.g., Goldstein, 1980), that the surface velocity field v(0, 4) of N plates may be represented by the superposition of the rigid-body rotations of each plate:

where Hi(& 4) = 1 wherever plate i is located and H,(B, 4 ) = 0 elsewhere, o' is the angular velocity vector of plate i, and r is the position vector of any location on the Earth's surface. The trivial identity x H , ( 0 , 4) = 1 may be employed to rewrite (4) as

N - I

v(e, 4) = 2 H,(B, +)(wl - W N ) x r + ON x r. ( 5 ) i= I

The last term on the right-hand side of (5) represents a net rotation of the lithosphere with angular velocity of plate N . It is clear that, apart from this net rotation, the surface velocity field is entirely given by the relative rotations of each plate relative to plate N. For example, the Nuvel-1 model (DeMets et al., 1990) speci- fies the plate velocity field by arbitrarily fixing the Pacific plate (i.e., assuming uPeclllc = 0). A convenient representation for the relative rotation rate w' - uN is given by

W' - ON = VQ, (6)

where

n, = x , (w; - WIy) + X2(W$ - w;) + x,(w\ - OC), (7)

where x,,x2,x, are the Cartesian coordinates of any position r on the Earth's surface and w; represents the j t h Cartesian component of the rotation-rate vector w'. In analogy to Eqs. (6) and (7) we also have

O N = V W , (8)

(9) n N = x , w y + x2w? + x7wIy.


Inserting Eqs. (6) and (8) into Eq. ( 5 ) , we obtain

N - I

v(0, 4 ) = - H,(B, 4 ) Alli - AllN, (10) i= I

where A = r X V is related to the angular-momentum operator (see Backus, 1958, and Appendix I). The horizontal divergence V, + v and radial vorticity P . V X v provide a complete scalar characterization of the observed plate velocities (Forte and Peltier, 1987), and using Eq. (10) we obtain

( 1 1 ) 1 N - ‘

V, . v = -- C [V, H i . Ail;], a ;=I

where r = a is the radius of the Earth’s solid surface, V, is the horizontal gradient operator on the unit sphere (see Appendix I), and A* = A A is the horizontal Laplacian operator.

In Appendix 111 we derive the following analytic expressions for the spherical harmonic coefficients of the divergence and vorticity fields in Eqs. ( 1 1) and ( 12):

(13)

N - I 3

( V H . V)T = c c ( q ) T @ ; - 4% #=I ,.=I

N - I 3 1

( P * V X v)? = c c ( R ; ) ~ ( w ; - 0;) + 2 SLINYW~”, (14) ,=I , . = I J= I

in which

(S ; )? , (R;)?, N;

are defined in Appendix I11 and a,, = 0 whenever 8 # 1. In (14) we observe that the net lithospheric rotation associated with the rotation of plate N affects only the degree 8 = 1 component of the radial vorticity field and has no effect on the horizontal divergence coefficients in (13). For degrees 8 > 1 the horizontal divergence and radial vorticity fields are sensitive to the rotation of each plate relative only to the rotation of the (arbitrarily specified) Nth plate. On the basis of the explicit expressions for the elements

(S;)? and (R;)?

derived in Appendix I11 we find


in which we emphasize that (S;)r is a (linear) function F, of only the degree t? + 2, e, e - 2 components of the plate functions H,(B, #) and (&)? is a (linear) function G, of only the degree e + 1, e - 1 components of the same plate functions. This implies that the degrec e components of the horizontal divergence are sensitive only to the degree f? + 2, 8, e - 2 components of the plate geometry while the degree e radial vorticity depends only on the degree e + 1 , e - 1 plate geometry. The plate divergence and vorticity fields thus provide an independent and complementary sampling of the geometry of the surface plates. If, for example, we have a hypothetical planet with only two hemispherical surface plates [when the boundary between the plates coincides with the equator we can easily verify that the harmonic coefficients of each plate function vanish for even degrees e thus, according to the rotational invariance expressed in Eq. (4.14) of Edmonds (1960), we know this will be true for any orientation of the boundary between the two plates] then the spherical harmonic coefficients of the horizontal divergence will vanish for even degrees 4 while the radial vorticity coefficients will vanish for odd degrees e.

To illustrate the actual use of expressions (13) and (14) let us first consider the simple example of the hypothetical planet with two hemispherical surface plates whose mutual boundary coincidences with the equator. It is straightforward to demonstrate that the harmonic coefficients of plate 1 (the northern-hemisphere plate) and plate 2 (the southern-hemisphere plate) are given by

in which = 0 whenever m f 0, 6,,, = 0 whenever e f 0, and P g ( x ) is the associated Legendre function that is normalized so that its root-mean-square (rms) amplitude is 1 . In Fig. I a we plot the amplitude spectrum of HI (8,4) in which we may observe the clear I/( amplitude dependence demonstrated in Eq. (17). The horizontal gradients of the plate functions in Eqs. ( I 1) and (12) are essentially equivalent to multiplication by e (see Appendix III), in the space of spherical harmonics, and therefore we expect the plate divergence and vorticity fields to have a flat amplitude spectrum. This expectation is confirmed in Fig. 1 b, in which we plot the amplitude spectrum of the divergence and vorticity fields, calculated using Eqs. (131, (141, and (171, for the following choice of plate rotation vectors:

W I = mi, W? = w ( - -P A - -2 fl ) ' 2 '

w = 1"iMyr (18)


0-1

0-2

- $ 1 0 - 6

a 2 10-7

3 10-8

s 1 0 - 9

u

W n

t

a

2 10-10

-I a

v)

0-RADIAL VORTlClTY b A-HORIZONTAL DIV.

0 000-

100 10’ U

DEGREE FIG. I . (a) The root-mean-square (rms) amplitude. at each spherical harmonic degree, of the hemi-

spherical plate function H ( 0 , d ) defined by Eq. (17) in the text. (b) The rms amplitude (units rad/yr), at each degree, for the horizontal divergence and radial vorticity of the two rotating hemispherical plates. with geometry defined by Eq. (17). The rotation vectors of the two plates are specified by Eq. (18) in the text.

(where I”/Myr = 1 degree per million years). In Fig. 2 we show maps of the actual two-plate divergence and vorticity fields, in which we observe the usual Gibbs side lobes that arise in any truncated series representation of discontinuous fields. An effective means for suppressing such side lobes is to multiply each harmonic coefficient of the divergence and vorticity fields by the following “1,anczos smoothing” factor (Lanczos, 196 1 ; Justice, 1978):

(19)

The values of L and M may be set equal to the maximum degree and order employed in the truncated harmonic representation of the given surface field. It should be understood that when either m = 0 or 4 = 0 the value of L+ in (19) is


FIG. 2. The horizontal divergence and radial vorticity fields due to the rotation [given by Eq. (18) in the text] of the two hemispherical plates defined by Q. (17). (a) The divergence field [units (radlyr)] is synthesized from the spherical harmonic coefficients, defined by Q. (13). corresponding to degrees k‘ = 1-32. The shaded areas indicate regions of positive divergence, and unshaded areas indicate negative divergence. (b) The radial vorticity field (units lo-’ rddlyr) is synthesized from coefficients, defined by Eq. (14), corresponding to degrees E = 2-32. The shaded areas indicate regions of negative vorticity (local clockwise circulation).

determined by the limit sinx/x + 1 as x + 0. In Fig. 3 we show the effect of applying this Lanczos smoothing (when L = M = 32) to the original (un- smoothed) divergence and vorticity fields in Fig. 2.

We now consider the application of Eqs. ( 1 3) and (14) to the observed plate motions on the Earth’s surface. In Fig. 4a we show the amplitude spectrum of the plate functions H,(O, qb) corresponding to three different tectonic plates. The very

POLOIDAL-TOROIDAL COUPLING IN MANTLE FLOW 1 1

FIG. 3. The smoothed horizontal divergence and radial vorticity fields obtained by multiplying the harmonic coefficients of the corresponding fields in Fig. 2 by the Lanczos smoothing factors in Eq. (19) of the text, with L = M = 32.

large African and Pacific plates have an amplitude spectrum that displays the 1 lt! variation characteristic of the hemispheric plate shown in Fig. la. The smaller COCOS plate instead displays the flat spectrum more characteristic of a very small disk (i.e., a 2D delta function). In Fig. 4b we show the amplitude spectrum of the plate divergence and vorticity calculated according to ( I 3) and (14) and employing the plate rotation vectors for the absolute-motion model AM1-2 of Minster and Jordan (1 978). In this calculation we treat the Pacific plate as the Nth reference plate in (1 3) and (14). In Fig. 4b we observe a relatively flat amplitude spectrum, as in Fig. lb, arising from the dominating contribution of the largest


0-PACIFIC A-AFRICA 0-COCOS

a

PI ? B P

0

b 0-RADIAL VORTlClTY A-HORIZONTAL DIV. d

mu 10-9 100 10’

DEGREE FIG. 4. (a) The rms amplitude, at each spherical harmonic degree, of the plate functions H,(H. 4)

corresponding to the Pacific. African, and Cocos tectonic plates. (b) The rnis amplitude (units rad/yr), at each degree, of the horizontal divergence and radial vorticity of the tectonic plate velocities described by the rotation vectors in model AM 1-2 of Minster and Jordan (1978).

(e.g., Pacific and African) plates. The smaller (e.g., Cocos) plates contribute to the detailed variability in the amplitude spectrum for large degrees 4. The maps of the plate divergence and vorticity synthesized from a truncated sum (up to degree e = 32) of their harmonic coefficients are very “noisy,” due to the strong Gibbs side lobes. We therefore multiplied these coefficients by the smoothing factors in (19) for L = M = 32 and the resulting smoothed divergence and vorticity fields are shown in Fig. 5. The horizontal plate divergence in Fig. 5a is clearly dominated by the fast-spreading rates along the East Pacific ridge and the associated convergence along the western Pacific plate boundary. The radial plate vorticity in Fig. 5b is similarly dominated by the strong vorticity along the edges of the Pacific plate. It is worth noting that the observed strong radial vorticity in zones of strong divergence or convergence is a striking departure from the simple model of pure convergence (or divergence) over zones of downwelling (or upwelling) in a fluid with laterally homogeneous rheology.


FIG. 5 . The smoothed horizontal divergence and radial vorticity of the tectonic plate motions given by the absolute-motion model AMl-2 of Minster and Jordan (1978). The smoothing is performed by multiplying the harmonic coefficients of the divergence and vorticity fields by the Lanczos smoothing factor i n Eq. (19), with L = M = 32. (a) The divergence field (units lo-’ rad/yr) is synthesized from the harmonic coefficients in the range 4 = 1-32. (b) The radial vorticity field (units lo-’ rad/yr) is synthesized from harmonic coefficients in the range t = 2-32.

2.2. Buoyancy-Driven Plate Motions

Theoty

The theory of buoyancy-induced plate motions employed by Forte and Peltier (1991a,b) is based on the explicit recognition of the limited class of surface plate


motions that are realizable. We begin our development of this theory by rewriting expressions (1 3) and (14) as the following matrix equations:

d = S . A o (20)

(21)

in which d and v are column vectors consisting of the spherical harmonic coefficients (V, . v)? and ( P . V X v ) ~ respectively, S and & are matrices consisting of the elements (S;)? and (ROT in (13) and (l4), Am is a column vector consisting of the Cartesian components of the relative plate rotations w; - w;”, N is the matrix consisting of the elements a,, N;” in (14), and oN is a column vector consisting of the Cartesian components of the angular velocity of plate N. The matrix S. in (20) may be represented by its Lanczos (1961) decomposition [a useful discussion of this decomposition, also called a singular-value decomposition, may be found in Aki and Richards (1980)l:

(22)

in which and 1 are orthonormal matrices (UTE = I = V‘V) and A is a diagonal matrix consisting of the singular values of 8. The columns of V constitute a subset of the totality of all vectors spanning the space of plate rotation vectors A o . If the singular values in are all nonzero, then the columns of 1 span the entire space of rotation vectors A o (i.e., 1 y T = I). The columns of u constitute the particular set of vectors that span all permissible plate-like horizontal divergence fields [it is obvious from Eqs. (20) and (22) that any field of plate divergence will be given by a linear superposition of the column vectors in u]. Since the harmonic coefficients of any realizable plate divergence field must be given by a linear superposition of the columns of u, we introduce the following projection operator p:

v = RAw + NoN,

- s = u A 1 7 ,

- P = u y , which acts on any given column vector do to yield a new column vector d, ,

di = I1 do, (24)

in which the elements of d, constitute the harmonic coefficients of a realizable field of plate divergence. If the elements of do already constitute the harmonic coefficients of a plate-like divergence field, then it is clear that Ed,) = do. Since the 5 matrix in (20) is dependent only on the geometry of the plates [see Eq. (15)], it is clear that the projection operator will also be solely dependent on the plate geometry.

In practice we inevitably work with truncated sums of harmonic coefficients, and therefore it is worth understanding the impact of this truncation on the prop-


erties of the projection operator. From expression (15) we know that all the elements in any row of the matrix in (22), which correspond to degree t? of the plate divergence field, are dependent only on the degree e + 2, 4, e - 2 coefficients of the complete harmonic expansion of the plate functions. It is clear from (22) that this dependence also applies to the matrix u in (23) and therefore an element P,, of the (in practice) finite-dimensional projection matrix p will contain a representation of the degree e + 2, t‘, e - 2 plate coefficients that is uncorrupted by the use of the truncated harmonic sum representation of the plates. This representation will obviously appear more “plate-like” as the maximum degree e in the sum is increased. This increased “plateness” is achieved by increasing the column and row dimension of the projection matrix while preserving the elements P,, that were already calculated.

Forte and Peltier ( I 987) have shown that the buoyancy-induced surface flow, in a model of the mantle that has no lateral viscosity variations, may be uniquely represented by its horizontal divergence:

(V, . u)p = - 1; Dt(r)6pT(r) dr, 7 0

in which (V, . u)y are the spherical harmonic coefficients of the horizontal divergence of the predicted surface flow, go is the gravitational acceleration (which is nearly constant in the mantle), qo is a reference (scaling value) viscosity for the mantle, r = a and b are respectively the radii of the solid surface and core-mantle boundary, spy( r ) are radially varying spherical harmonic coefficients of the internal density contrasts, and Dr( r ) are kernel functions whose behavior also depends on the radial profile of relative viscosity q(r)) /qO in the mantle. Depending on the particular field of density perturbations 6py assumed for the mantle, the predicted surface divergence in (25) will generally not be “plate-like.” A plate-like divergence field (Vl,. v): may be obtained from the flow-induced divergence (V, ’ u)a in (25) by applying the projection operator described in (23) and (24):

(V, . v): = P,f,trn(VH * U)$, (26)

in which the combination st and ern in PAf,Yrn defines a particular row and column, respectively, of the p matrix. If we now substitute (25) into (26), we obtain

in which

WS( r) = [D; I ( r)Ps,prnDt ( r)l &T( r). (28)

The density perturbations S p : ( r ) defined in (28) give rise to a surface divergence field in (27) that is perfectly plate-like (i.e., corresponds to a plate divergence


produced by rigid-body rotations of the plates). On the basis of expression (28) we define a new radially varying projection operator p.sr,,,,,( r ) :

R,.fJd = D, '(r)P,,.,,,, Dt(r), (29)

which possesses the fundamental property of all projection operators, namely, - p2 = 2 (from Eq. (23) we see that pz = E). The operator 2 will be uniquely dependent on the geometry of the surface plates and the radial viscosity profile of the mantle [the latter is implicitly contained in the divergence kernels D,( r ) and D,( r ) in (29)].

The projection operator defined in (29) allows us to partition any given field of density perturbations 6py( r ) in the mantle into two orthogonal components 6fi:(r) and @ : ( r ) as follows:

6 B X r ) = L , , ( r ) 6 p g l ( r ) , (30)

@ : ( r ) = - ~. , l , t t" (~) laPY(~) , (31)

in which (30) is simply Eq. (28) rewritten and 6.,,,l,,r is the Kronecker delta (a,,, = 0 when i # j and = 1 when i = j ) . The density perturbations Sp:(r), delivered by (30), give rise to realizable surface plate motions, while the density perturbations 6 p ; ( r ) cannot give rise to plate motions [note that p,l,,,,,(r) 6 p y ( r ) = 01. The mantle flow driven by the density perturbations &: ( r ) should be modeled with a free-slip surface boundary condition, while the flow driven by the density perturbations 6p:( r ) should be modeled with a no-slip surface boundary condition. The surface plates are effectively ''locked'' into their positions by the mantle flow driven by @:(r) . The density perturbations Sp:(r) constitute that portion of the internal mantle heterogeneity that is invisible with respect to the observable plate motions. This last point is clearly important because it shows that the interpretation of past and present plate motions (e.g., Richards and Engebretson, 1992) in terms of density perturbations in the mantle is entirely nonunique. This nonuniqueness will be illustrated in several examples below.

On the basis of the plate-like surface divergence driven by the 6f i : ( r> component of the internal density perturbations, we may determine the corresponding plate rotation vectors oi - &-The generalized inverse s' of the matrix S in (22) is given by

(32)

From Eq. (20) we see that the action of S+ on a column vector d, consisting of the harmonic coefficients of the plate-like surface divergence in (27), will yield a column vector A d :

1- - V A - l U7 - s - - - -

A d = S d, (33)

in which the elements of A@+ are the relative plate rotations w' - uN consistent with the flow-induced surface divergence. It is worth emphasizing that the hori-


zontal divergence field can only constrain the angular velocity vectors of the plates relative to that of some arbitrarily selected reference plate (i.e., plate N ) . If one or more of the singular values in the diagonal matrix A are zero, then certain com- binations of plate rotations will produce zero plate divergence. The columns of the 1 matrix in (22), which correspond to the zero singular values, define the plate rotations that produce zero plate divergence. In this situation, the plate rotations delivered by (33) will obviously describe only the restricted class of plate rotations that produce a nonvanishing plate divergence. If we now insert (33) into ( 2 1 ) we obtain

v = c d + NuN,

in which the coupling matrix

- c = RS”,

(34)

(35)

describes the radial vorticity of the plates, which arises from the buoyancy- induced plate rotations given by Eq. (33). The coupling matrix C in (35) is dependent only on the geometry of the surface plates.

At this juncture it is worth considering the assertion, by Ricard and Vigny ( 1989), that a model of buoyancy-induced plate motions cannot describe degree e = I toroidal motion (i.e., a net lithospheric rotation). The coupling matrix C in (34) does indeed give rise to a degree e = 1 toroidal motion, on the basis of the flow-induced plate-like divergence d. This predicted degree 1 toroidal plate motion is, however, incomplete. A complete specification of the degree 1 toroidal motion also requires knowing the absolute rotation uN of the Nth plate. This absolute rotation cannot be determined on the basis of the theory presented here (see, however, Section 4.7 below).

Examples

We shall now consider several applications of the theory of buoyancy-induced plate motions described in the previous section. An important ingredient in this theory is the calculation of the horizontal divergence kernels D ( ( r ) in Eq. (25). These kernels are a function of the assumed relative viscosity profile v( r)/vo of the mantle. The surface observable that is most sensitive to the depth variation of this relative viscosity is the predicted nonhydrostatic geoid. In Fig. 6 (left panel) we thus show the relative mantle viscosity inferred by Forte et al. ( I 993b) on the basis of the fits to the observed nonhydrostatic geoid provided by the seismic heterogeneity model SHS/WM13 of Woodward et al. (1993). In Fig. 6 (right panel) we also show the inferred density-velocity proportionality, d In pld In v,, which is employed to convert the Sv, /v , heterogeneity in model SH8/WM13 to an equivalent field of density heterogeneity Sp. The divergence kernels D( ( r ) which are calculated on the basis of the geoid-inferred viscosity profile are shown


0

500

7 1000

5 1500

2 2000

-Y U

P,

2500

3000 0.0 0.2 0.4

6 h p / 6 1 n v s ( r ) FIG. 6. In the left panel is shown the relative viscosity profile v ( r ) / q o inferred from the nonhydro-

static geoid by Forte ef al. (1993b). In the right panel is shown the depth-varying proportionalily between density perturbations and shear velocity perturbations in the mantle.

0

500

7 1000

g 2000

Y

I 1500 Y

+ a

2500

3000 - -0.030 -0.015 0.000

AMPLITUDE FIG. 7. The horizontal divergence kernels Df(r), defined in Eq. (2.5). calculated for the relative

viscosity profile shown in Fig. 6. The kernels were calculated for a compressible mantle according to the method described in Forte and Peltier (1991a). The location of the 670-krn seismic discontinuity is indicated by the dashed horizontal line.


in Fig, 7, where we observe that long-wavelength density heterogeneity near the bottom of the upper mantle excites surface flow most effectively.

We shall first illustrate the interaction of buoyancy-induced mantle flow with surface plates by considering the mantle flow driven only by the degree e = 2 density heterogeneity derived from model SH8/WM 13 of Woodward el a/. (1993). In Fig. 8a we show the degree 2 surface horizontal divergence that is predicted for a mantle without surface plates. In this calculation we convolve the degree 2 divergence kernel in Fig. 7 with the degree 2 density perturbation in the mantle according to Eq. (25) [in which we select qo = lo2’ Pa s (pascal seconds) on the basis of the postglacial rebound analysis of Peltier (1982, 1989)l. In Fig. 8b we show the result of the interaction of the flow in Fig. 8a with a hypothetical lithosphere consisting of two hemispherical surface plates whose mutual boundary coincides with the equator (this geometry was considered previously in Figs. 1 and 2 and in Eq. ( 1 7)). The surface divergence in Fig. 8b was calculated according to Eq. (27), in which the density perturbations Sp:( r ) were obtained using the projection operator [in Eq. (29)] calculated for the two hemispherical plates. The elements of the projection operator in Eq. (29) were calculated on the basis of a spherical harmonic description of the plate functions H,(B, 4 ) up to degree and order 32. However, for numerical convenience, we employ only the elements of the operator p3r,fm(r) that correspond to s S 15, e S 15. It is clear, from the amplitude scale in Fig. 8b, that the projection operator has almost completely annihilated the internal density perturbation and thus yields a vanishingly small surface divergence. This example illustrates the importance of the alignment between the plate-boundary geometry and the geometry of the upwellings and downwellings in underlying mantle flow. It is rather clear that the symmetry of the mantle flow in Fig. 8a is such that the hemispherical plates in Fig. 8b will be essentially “locked” into position. In Fig. 8c we now observe the result of the interaction of the mantle flow in Fig. 8a with the actual geometry of surface plates (represented by a spherical harmonic expansion up to degree and order 32). In this case we observe that the peak amplitude of the surface divergence is similar to the peak amplitude obtained in the absence of plates (Fig. 8a). The observed surface plates are apparently aligned favorably with respect to the mantle flow derived from even the longest wavelength (degree 2) seismic heterogeneity. In Fig. 9 we show that the surface plate motions in Fig. 8c agree rather well with the observed plate motions, on horizontal length scales which are much smaller than that of the internal degree 2 density perturbations.

We shall now consider the interaction of the observed surface plates with the mantle flow driven by all components (up to degree and order 8) of the mantle heterogeneity given by model SH8/WM13 of Woodward et al. (1993). The shear velocity heterogeneity in model SH8lWM 13 is converted to density heterogeneity using the a In pld In v, values in Fig. 6 (right panel). The resulting field of density perturbations is partitioned by the plate-projection operator according to (30) and

FIG. 8. (a) The degree 2 horizontal divergence predicted [according to Eq. (25)] with model SH81 WM13, using the 8 In p16 In Y , conversion in Fig. 6 and the divergence kernels in Fig. 4. Units are

rad/yr. (b) The horizontal divergence, in the range e 1 - 15, due to the interaction of the degree 2 mantle flow in (a) with a lithosphere consisting of two hemispherical plates with their boundary at the equator. This plate-like divergence field wm calculated according to Eq. (27). Units are lo ->’ rad/yr. (c) The plate-like horizontal divergence, in the range 0 = I - 15, due to the interaction of the degree 2 mantle flow in (a) with the actual plates on the Earth’s surface. The units are 10-8 rad/yr. In calculating the flow shown in (a), (b) and (c), a reference viscosity value T ~ , = 10” Pa s was employed.


a

b

g 1 0 - 9 1 I I 1 I I 1 I

E 0 4 a 12 DEGREE

FIG. 9. (a) The cross-correlation. at each harmonic degree C, between the observed plate divergence (shown in Fig. 5a) and the degree 2 divergence in Fig. 8a (0, “no plates”), the plate-like divergence in Fig. 8c (0. “with plates”). (b) The rms amplitude, at each degree, of the observed plate divergence (*, “observed”), the degree 2 divergence (0, “no plates”), and the plate-like divergence (0,

“with plates”). Units are radlyr.

(3 1) and the plate-like surface divergence is calculated according to (27) (in which we employ the divergence kernels in Fig. 7). The projection operator is calculated on the basis of a spherical harmonic description of the plate functions H,(B, 4 ) up to degree and order 32. For numerical convenience, we employ only the elements of the operator P,,fm(r) that correspond to s S 15, t S 15. In Fig. lob we show the predicted plate-like surface divergence along with the surface divergence of the observed plate velocities in Fig. 10a. The agreement between the two maps is clearly very good, and the predicted divergence in Fig. lob accounts for 70% of the variance of the observed divergence in Fig. 10a. This good match is achieved by selecting the value of the reference viscosity in Eq. (27) to be q, = lo?’ Pa s. On the basis of the relative viscosity v(r)/v0 in Fig. 6 (left panel), we thus infer an absolute viscosity of lo2’ Pa s in most of the upper mantle and the absolute viscosity at the top of the lower mantle is 2.0 X 10” Pa s (increasing linearly to a


FIG, 10. (a. c) The horizontal divergence (in the range f = I - 15) and the radial vorticity (in the range f = 2- 15) of the observed plate velocities. These two fields were previously shown (up lo e =

32) in Fig. 5. (b, d) The horizontal divergence (in the range t' = I - 15) and the radial vorticity (in the range t = 2- 15) predicted with model SHWWM13, using the 6 In p/8 In v, values in Fig. 6 and the divergence kernels in Fig. 7. In the calculation of the predicted plate-like surface motions, according to Eqs. (27), (28), and (34), the reference viscosity value qo = lo2' Pa s was employed. In all maps, the units on the scale bars are rad/( 10 Myr).

value of 20.0 X lo2' Pa s at 2500-km depth). It is interesting to note that the inferred value of the upper mantle viscosity and the viscosity at the top of the lower mantle are fully compatible with the values previously deduced on the basis of the analysis of postglacial rebound data (Peltier, 1982, 1989).

We may now obtain the buoyancy-induced radial vorticity of the plates from the predicted surface divergence, in Fig. lob, by employing the coupling matrix (calculated to degree and order 15) in Eqs. (34) and (35). The resulting plate-like


radial vorticity predictions are shown in Fig. 10d along with the radial vorticity of the observed plate motions in Fig. 1Oc. The agreement between the two maps is fairly good, and the predicted vorticity in Fig. 10d accounts for 38% of the variance of the observed radial vorticity in Figure 10c. The observed plate motions, in Figs. 10a and lOc, as well as the elements of the projection operators, in Eqs. (29) and (34), have been smoothed by the Lanczos factors, in Eq. (19) for which we select the values L = M = 32. In Fig. 11 we provide a detailed degree- by-degree comparison of the predicted plate-like surface divergence and the observed plate divergence. Here we also compare the predictions obtained for a mantle without surface plates, calculated according to Eq. (25) in which qo = 10” Pa s. It is clear from Fig. 11 that the interaction of the long wavelength mantle flow, driven by the seismically inferred density contrasts, with the surface plates


a

L L

b

10-91 I I I I I a i3 0 4 a 12 DEGREE

FIG. 11. (a) The cross-correlation, at each degree e , between the observed long-wavelength plate divergence (shown in Fig. IOa) and: the buoyancy-induced surface divergence in the absence of surface plates (0, “no plates”) (calculated with model SHUWMI3, using 6 In plS In v , i n Fig. 6, and the divergence kernels in Fig. 7), the predicted plate-like divergence in Fig. 10b (0, “with plates”). (b) The rms amplitude, at each degree e, of the observed long-wavelength plate, divergence (*, “observed”), of the buoyancy-induced surface divergence in the absence of plates (0, “no plates”), and of the plate-like surface divergence (0. “with plates”) calculated on the basis of a harmonic expansion of the plates up to degree 32. The dashed line instead represents the platelike divergence when the plate expansion is limited to degree 15. Units are radlyr.

does indeed yield a predicted surface divergence that agrees closely with the observed plate divergence. In Fig. 12 we also show a degree-by-degree comparison of the predicted plate-like radial vorticity and the radial vorticity of the observed plate motions.

A good illustration of the nonuniqueness inherent in the interpretation of surface plate motions was provided in Fig. 8. In Fig. 8b we observed that, despite the presence of substantial (degree 2 ) density heterogeneity in the mantle, the resulting plate motions were essentially zero. In Fig. 8c we also observed that very long- wavelength density heterogeneity could produce realistic plate motions on much


Z

0.8

w 0.4

0 0.0

w -0.4

-0.8

4 tY tY

0

w tY

n

a

b

~ 1 0 - 9 L I I 1 I I I I

0 4 8 12 x cc

DEGREE FtG. 12. (a) The cross-correlation. at each degree e, between the observed long-wavelength radial

vorticity of the plate velocities (shown in Fig. IOc) and the predicted plate-like radial vorticity in Fig. 10d. (b) The rnis amplitude, at each degree d , of the observed radial vorticity of the plate velocities (*, “observed”) and of the predicted plate-like radial vorticity (0. “predicted”) calculated on the basis of a harmonic expansion of the plates up to degree 32. The dashed line instead represents the plate- like vorticity when the plate expansion is limited to degree 1.5. Units are radlyr.

smaller horizontal-length scales. Such results are, of course, completely different from those expected for a mantle without surface plates.

A further illustration of the nonunique interpretation of plate motions is provided by considering two extreme situations in which the density contrasts in the mantle either exist only beneath the midocean ridges or only beneath subduction zones. In Fig. 13a we show the model of midocean ridge heterogeneity constructed from a spherical harmonic expansion (up to degree and order 32) of the four major midocean ridge systems. In Fig. 13b we show the model of slab heterogeneity constructed by Su and Dziewonski ( 1 992) from a spherical harmonic expansion (up to degree and order 50) of International Seismological Center- determined hypocenters in the depth interval 150-250 km. To each model of’ density heterogeneity in the mantle, we shall apply the plate-projection operator


FIG. 13. (a) A model of midocean ridge heterogeneity (in the degree range k' = 1-32) constructed on the basis of a 5" X 5" discretization of the Earth's surface. The heterogeneity pattern is defined by assigning a value of I to any 5" X So element that coincides with a midocean ridge and a value of 0 elsewhere. (b) A model of subducted slab heterogeneity (in the degree range k' = 1-50) constructed by Su and Dziewonski (1992) on the basis of a 1' x 1' sampling of the ISC hypocentre locations in the depth interval 150-250 km. The heterogeneity is defined by assigning a value of 1 to any I" X I" element occupied by an earthquake hypocenter and a value of 0 elsewhere.

[Eqs. (29) and (30)] employed previously for the plate-motion predictions in Figs. 8c and lob. The mantle viscosity profile we shall employ is again given by Fig. 6 (left panel), and we choose vo = lo2' Pa s. The ridge-slab heterogeneity in Fig. 13 is assumed to extend vertically downward to a depth (chosen arbitrarily) of 1500 km.

In Fig. 14b we now show the predicted plate divergence for the slab- heterogeneity model, in which we maximize the least-squares fit to the observed


plate divergence in Fig. 14a by selecting the optimum density contrast for the slabs (treating all slabs equally) in Fig. 13b. The density contrast we thus find is 0.10 Mg/m‘, and the predicted divergence field in Fig. 14b accounts for 62% of the variance in the observed divergence field in Fig. 14a. It is rather evident from these two figures that the slab-heterogeneity model by itself substantially under- predicts the rate of plate divergence along the East Pacific rise.

In Fig. 14c we show the plate divergence predicted for the ridge-heterogeneity model, in which we again maximize the least-squares fit to the observed plate divergence by assigning optimal density contrasts to each of the four ridges in Fig. 13a. The density contrasts so obtained are -0.010 Mg/m3 for the North Atlantic ridge heterogeneity, -0.01 1 Mg/m3 for the South Atlantic ridge, - 0.020 Mg/m3 for the Southeast Indian ridge, and - 0.032 Mg/m3 for the East Pacific ridge. The predicted divergence field in Fig. 14c accounts for 77% of the variance of the observed field in Fig. 14a. It is noteworthy that the ridge heterogeneity model predicts a realistic pattern of convergence in the west Pacific trench system, and in the Peru-Chile trench system, with an amplitude that is about 60% of that observed in Fig. 14a. A naive interpretation of these simulated trench convergence rates may lead to the conclusion that considerable subducted slab heterogeneity exists in the mantle when, in fact, there is none in this idealized mantle- heterogeneity model.

Finally in Fig. 14d we show the plate divergence predicted for a combined slab- ridge-heterogeneity model, in which we maximize the fit to the observed divergence by selecting optimal density contrasts for the slabs and ridges in Fig. 13. In this case we infer a slab density contrast of 0.063 Mg/m3, a North Atlantic ridge contrast of - 0.004 Mg/m3, a South Atlantic ridge contrast of - 0.006 Mg/m3, a Southeast Indian ridge contrast of - 0.012 Mg/m3, and an East-Pacific ridge contrast of -0.024 Mg/m’. I t is interesting that the inferred slab density contrast agrees closely with that independently inferred by Hager and O’Connell (198 1 ) and by Hager and Richards (1989) in separate analyses of plate motions and the geoid. The predicted divergence field in Fig. 14d now accounts for 92% of the variance of the observed field in Fig. 14a. This analysis strongly suggests that both the positive density heterogeneity beneath trenches and the negative density heterogeneity beneath ridges contribute significantly to the observed motion of the plates. The ridges therefore appear to be active regions of forcing.

2.3. An Assessment

The cornerstone of previously published treatments of buoyancy-induced plate motions (e.g., Hager and O’Connell, 1981; Ricard and Vigny, 1989; Gable et al., 199 1 ; Forte and Peltier, 199 1 a,b), including that presented above, is the assumption that the long-term behavior of tectonic plates is that of rigid bodies. Although


FIG. 14. (a) The observed long-wavelength plate divergence in the degree range P = 1 - 15. (b) The plate-like divergence (in the range t! = I - 15) predicted with a best-fitting model of subducted slab heterogeneity (see text for details). (c) The plate-like divergence predicted with a best-fitting model of niidocean ridge heterogeneity (see text for details). (d) The plate-like divergence predicted with a best- lilting model of combined slab and ridge heterogeneity. In all predictions the divergence kernels in Fig. 7 are employed and the reference viscosity value is v,, = 10” Pa s. The units on all scale bars are rad/( 10 Myr).

this assumption appears to simplify matters, so that the motion of each plate may be specified with only three parameters (the components of its angular-velocity vector), it can also lead to serious inconsistencies when matching the plate motions to the buoyancy-driven flow in the mantle. A precise mathematical description of large-scale flow in a spherically symmetric mantle may be achieved with a limited number of spherical harmonic basis functions, whereas an infinite number of harmonics are required for a precise mathematical description of rigid plate motion. This distinction presents obvious difficulties when modeling plate motions in the spectral domain.


These difficulties are clearly manifested in the spectral-domain treatment of plate motions by Hager and O’Connell(1981). Their treatment in essence consists of two separate calculations: ( 1 ) plate velocities are imposed at the surface and the surface-shear-driven flow in the mantle is calculated, and (2) the assumed density perturbations in the mantle excite a flow that is calculated with a no-slip surface boundary condition. Dynamical consistency is assumed if the surface shear stresses, generated in ( l), balance the buoyancy-induced surface shear stresses generated in (2). Problems arise because the surface shear stresses in ( 1 ) become unbounded as the number of terms retained in the harmonic description of the plates increases indefinitely (see Fig. 3 in Hager and O’Connell, 1981). Infinitely large stresses are required to move a surface layer composed of perfectly rigid contiguous plates. Such infinite stresses cannot be matched to the surface stresses in (2), which always remain finite. Hager and O’Connell (1981) suggest that this inconsistency is resolved by simply postulating that the lithosphere fails


at some critical yield stress. This failure criterion is used, however, only in a separate evaluation of net force exerted on the plates. The impact of this failure on the underlying mantle flow is never determined.

The treatment of plate motions by Ricard and Vigny (1989) is also formulated in the spectral domain and is a modification of the two-step approach proposed by Hager and O’Connell(198 1). Ricard and Vigny solve for the set of plate angular- velocity vectors that provide a balance between the surface stresses generated by the plate-driven mantle flow and by the buoyancy-driven flow acting on a no-slip surface boundary. The key assumption employed in this balancing of stresses, is that the forces acting on the plate boundaries are presumed to be identically zero. This assumption is clearly at odds, however, with the actual state of stress at the surface. The surface stresses generated by the plate-driven mantle flow must again become infinite as the number of harmonics that describe the plates increase indefinitely.

The generation of unbounded surface stresses is the inevitable consequence of a procedure that models the mantle flow driven by prescribed plate velocities. To avoid this difficulty we proposed the alternative treatment described in Section 2.2. In this treatment the observable plate motions are generated by the subset of internal buoyancy sources, described in Eq. (30), which interact with a free-slip (i.e., zero-stress) surface. The only surface stresses that are generated are those due to the buoyancy sources in Eq. (31), which interact with a no-slip surface. These surface stresses always remain finite. It is important to appreciate, however, that the fraction of internal buoyancy sources that generates the plate motions becomes increasingly smaller as the number of harmonics used to describe the plates increases. This is understood by noting that the buoyancy sources that produce observable plate motions must be located in the vicinity of the plate boundaries. As these boundaries become narrower (with the increasing resolution of the plates themselves), so does the distribution of effective buoyancy sources. In the limit in which the plate boundaries are infinitely narrow, the plate-driving buoyancy sources in Eq. (30) vanish entirely. This limit is achieved when the number of harmonics used to describe the plates is infinite.

We can illustrate this important point by calculating the plate-projection operator, in Eq. (29), using a spherical harmonic expression of the plates that is limited to degree 15. This limited representation is less plate-like, and characterized by effectively wider plate boundaries, than is a plate representation that includes harmonics up to degree 32 (as in Figs. 5 and 10). We therefore expect that the buoyancy-induced surface motions will possess significantly greater amplitudes. This is confirmed in Figs. 1 1 and 12, in which the dashed lines represent the rms amplitude of the surface divergence and vorticity predicted on the basis of a harmonic expansion of the plates that is limited to degree 15.

The maximum degree to be retained in a spherical harmonic expansion of the plates depends on the actual rigidity of the tectonic plates. Real plate boundaries


must have a finite region of weakness, or else they could not accommodate the deformations arising from their relative motions. This region of weakness along the edges of the plates may be simulated by truncating the spherical harmonic expansion of the plates. The effective width of the region of weakness, in a harmonic expansion limited to degree 32, is shown by the width of nonzero divergence rates in Fig. 5a. The choice of maximum harmonic degree is, of course, arbitrary, unless we have a priori data concerning the mechanical strength of plate boundaries. Such data would permit the determination of the actual lateral variations of rheology in the lithosphere.

3. LATERAL VISCOSITY VARIATIONS I N THE LITHOSPHERE

The technique for calculating surface plate motions, described in Section 2, is evidently rather convenient as it allows us to avoid dealing explicitly with the lateral variations of effective viscosity that characterize the plate-like structure of the lithosphere. This technique is, however, essentially kinematic because the plates are introduced only as a surface boundary constraint on the permitted surface flows. The dynamical details of the flow processes in the lithosphere that mediate the coupling of the surface poloidal and toroidal motions is never determined and remains unknown. A dynamically consistent treatment of the lithosphere requires that we return to the original problem of explicitly treating the effects of lateral variations of effective viscosity in the lithosphere. In this section we shall directly show that the observed toroidal plate motions, and their coupling to poloidal motions, may be understood to first order in terms of plausible lateral variations of viscosity in the lithosphere itself.

The term viscosiry, when applied to the lithosphere (or the mantle), must be understood in the context of a steady-state rheology of a creeping polycrystalline solid. As discussed in the Introduction (Section l) , the steady-state creep of any crystalline material may be characterized in terms of an efective viscosity. This concept of effective viscosity also applies to highly nonlinear materials characterized by stress- or strain-induced softening [see Eq. (311. The lateral variations of viscosity in the lithosphere, which will be investigated in this section, should therefore be regarded as an effective physical representation of a possibly nonlinear rheology.

3.1. Theory

The equation of momentum conservation governing quasistatic flow in a continuum with heterogeneous viscosity is given by Eq. (IV.9) in Appendix IV. The coupled scalar expressions that describe the buoyancy-induced poloidal and toroi-


dal flows in such a medium are given by Eqs. (IV.10) and (IV. 11). An inspection of these equations immediately reveals that a direct solution, for an arbitrary field of viscosity variations, will be rather difficult. This difficulty might reasonably be circumvented, in the case of the Earth's mantle, by initially assuming that the lateral variations of effective viscosity in the lithosphere are much stronger than in the underlying mantle. To the extent that this assumption is reasonable we may then treat the sublithospheric mantle as a region in which the lateral variations of viscosity may be ignored. As we show below, this assumption allows us to effectively replace the direct solution of Eq. (IV.9) by an equivalent treatment based on the matching of stresses at the two bounding surfaces that define the lithospheric layer. Since the thickness of the Earth's lithosphere (=I00 km) is much less than the radius of the solid upper surface ( r = 6368 km), we may employ a locally valid Taylor expansion for the lithospheric flow field.

By virtue of the tangent-vector fheorern (Backus, 1967) we may write the horizontal component of the lithospheric flow uH in the form

If we expand the scalars V and W as a Taylor series in r and retain the lowest- order terms, we obtain

where r = a (= 6368 km) is the radius of the solid outer surface. The application of the free-slip (zero tangential stress) boundary condition at the outer surface

implies that

v,(e, 4 ) = w,(e, 4 ) = 0. (40)

The radial component u,(r, 8, 4) of the lithospheric flow may be determined from the equation of mass conservation V - u = 0 for an incompressible lithosphere:

(41) l a I

r2u, = -- V, - uH. r2 ar r _ _

By substituting Eq. (36) into (41), employing results (37), (38), and (40), and finally integrating with respect to radius, we obtain


(a - r)3 6a3r2

A? V 2 . 1 x [ r 2 + 2 10 r(u - r) (a - r)* +

In obtaining (42) we have also made use of the condition of zero radial velocity, u,(r = a, 8, 4) = 0, at the upper bounding surface.

The scalars V, and W,, in Eqs. (37) and (38) may be directly constrained by the observed tectonic plate velocities v(8, 4). Indeed it is straightforward to verify, using (36)-(38), that

(43)

The scalars V2 and W2 in Eqs. (37) and (38) may be related to the known scalars V, and W, in (43) by substituting expressions (36) and (42) into Eqs. (IV.9) and (IV. 10) and “solving” for V, and W,, Apart from the ensuing mathematical difficulties, such an approach requires that we already know the lateral variations of effective viscosity of the lithosphere. Unfortunately we do not possess, (1 priori, a sufficiently complete or realistic description of the rheology of the lithosphere. Another method for determining the values of V, and W, involves matching the buoyancy-induced normal stress from the mantle flow to the normal stresses generated by the lithospheric flow field in Eqs. (36) and (42). This method, quite apart from requiring a knowledge of the buoyancy forces in the mantle itself, again requires that we know the lateral variations of viscosity in the lithosphere. This is unsatisfactory because we wish to discover the nature of these lateral variations in the first place.

To overcome these difficulties we shall make the following simplifying “quasirigid lithosphere” approximation:

(44)

in which r = r , ~ is the radius defining the lower bounding surface of the lithosphere and c is some as yet undetermined constant. By invoking (44) we are in effect assuming that the tangential flow velocities at the base of the lithosphere are parallel to those at the surface; hence the term yuusirigid. Equation (44) implies that

V2(& 4) = KVdB, 4) and W2(& 4) = KWdB, 41, (45)

in which K is a constant. We again emphasize that the quasirigid assumption, and the consequent appearance of the scalar K in Eq. ( 4 3 , is introduced because of our lack of knowledge concerning the lateral variations of effective viscosity in the lithosphere. The ultimate validity of the approximation in Eq. (45) may be judged by the inferences of these lateral viscosity variations, presented in Section 3.3 below.

v(& 4) = v,vn(e, 4) + AWn(8, 4) .

uH(r = r,, 8, 4) = c uH(r = a, 8, 4)


We may attempt to estimate the magnitude of the constant K by matching the dynamic surface topography, expected on the basis of the lithospheric flow in (36) and (42), with the dynamic surface topography estimated by isostatic reduction (Forte et al., 1993a) of the observed topography. From Eq. (A18) in Forte and Peltier ( 1 987) we have

in which (Sa);? is the spherical harmonic coefficient of the dynamic surface topography, AP,~,, is the density jump across the lithosphere-ocean boundary, and pi"( r ) is the poloidal generating scalar that describes the lithospheric flow field. The radial velocity in (42) is directly related to the poloidal-flow scalar by u, = A2p/r and therefore

The scalar V,,(0, 4 ) may be related to the surface horizontal divergence of the lithospheric flow field and, on the basis of Eq. (43), we have

A2

a v, . v = - V".

We similarly have the following relationship between the scalar WJB, 4 ) and the radial vorticity of the lithospheric flow field:

A2 i . v x v = - W,). U

Combining expressions (47) and (48) with Eq. (46) we finally obtain

(49)

The isostatically reduced surface topography (Forte ef al., 1993a) is most strongly correlated with the observed plate divergence at harmonic degrees 3 and 4. The rms amplitude of the isostatically reduced surface topography, at degrees 4 = 3 and 4, is respectively 0.3 and 0.5 km. The rms amplitude of the horizontal divergence of the observed plate velocities, at degrees 6 = 3 and 4, is respectively 0.46 X lo-* and 0.86 X lo-" rad/yr. We find that Eq. (50) may be satisfied at degrees 3 and 4 by the following selection of K and v,, values:


We have obviously not presented a complete list of (K , vL) values in (5 I ) , but it is nonetheless clear that a large range of K values corresponds to a much smaller variation of qL values. We do not possess reliable estimates of the absolute, spherically averaged, effective viscosity of the lithosphere and, given the very rough approximation represented by Eq. (50). we cannot use (5 1) to reliably distinguish which K value is most appropriate. We defer the selection of appropriate K values to Section 3.3 below.

The locally valid expression which describes the lithospheric flow is thus obtained by combining Eqs. (36)-(38), (40), (42), and (45) to obtain

r(a - r) + ( a - r ) ~ ) ] (r2 + 2 10

r3 (u - r)' + 6a7r? u(r, 8, #) = i -

3ar2 -

We may now consider how expression (52) may ultimately be employed, in con- junction with the condition of stress continuity across the lithosphere-mantle boundary, to constrain the possible lateral variations of the effective viscosity in the lithosphere.

As shown in Appendix V [Eq. (V.9)], the continuity of horizontal stress across the lithosphere-mantle interface implies that

in which qL(8, #) is the field of lateral viscosity variations in the lithosphere and vu is the viscosity of the sublithospheric mantle, which is assumed to be spherically symmetric. We shall find it advantageous to consider the toroidal component of the horizontal stress matching, obtained by applying the operator A - to both sides of (53):


In Eq. (54) we now substitute the lithospheric flow field given by Eq. (52) and, after some manipulation, obtain

EAY, . [V,(A2 - K)Vo - KAW,,] - ~ ( 1 + YL)KA’W~

in which E = h/a, where h is the thickness of the lithosphere, (TI [ , ) is the average lithospheric viscosity and is given by the t! = 0, rn = 0 coefficient of the spherical harmonic decomposition of ~ ~ ( 8 . +), v,-(e, 4) is the dimensionless departure of the lithospheric viscosity from its average value defined according to

(56)

and q(r, 8, 4) is the scalar field defining the sublithospheric toroidal flow as follows:

(57)

In the derivation of Eq. (55) we have neglected all terms of order c2 and smaller. The quantity EV,~ = ~(qJ(q,.) - 1 ) is analogous to the “stiffness” parameter employed by Ribe (1992). The term A2V0 that appears on the left-hand side of Eq. (55) arises from radial velocity u, in the lithosphere, given by expression (52). This term is clearly not negligible and indeed becomes increasingly dominant with decreasing horizontal wavelength (i.e., as the spherical harmonic degree e increases). We emphasize this point because the treatment of lithosphere-mantle coupling by Ricard et al. (1988), employed again in Ricard et al. (1991), assumes that the contribution to the horizontal stress by u, is negligible. Equation ( 5 5 ) demonstrates that this assumption is not valid.

The mathematical advantage of assuming a spherically symmetric viscosity in the sublithospheric mantle is most apparent when treating the dynamics of toroidal flow in this region. Indeed, if we assume for simplicity that the mantle consists of two constant viscosity layers (e.g., the upper and lower mantle) the equation governing the toroidal flow in each layer is, according to Eq. (IV. 1 1 ) in Appendix IV

%.(& 4 ) = (TL” + VL(& @)I.

u d r , 8, 4) = Aq(r, 8. 4).

V2R2q = 0, ( 5 8 )

in which we have employed Eq. (57). This simple equation demonstrates that there are no internal sources of toroidal flow in a medium with spherically symmetric viscosity. The toroidal Row in the mantle will therefore be identically zero (except at degree t! = 1 ; see below) unless there are inhomogeneous boundary conditions. In Eq. (55) we have an important example of such an inhomogeneous boundary condition. Here we observe that the presence of a lithosphere with lateral viscosity variations gives rise to horizontal shear stresses that drive toroidal flow in the sublithospheric mantle.


The solution to Eq. (58) is readily obtained if we expand q(r, 8, 4 ) in terms of spherical harmonic basis functions. We thus find that the spherical harmonic coefficients q,"l( r) satisfy the following equation:

The solution of Eq. (59) is of the form

in which the constants E y and F; are determined by the boundary conditions in our problem. In our simplified mantle, consisting of two constant-viscosity layers, the boundary condition at the core-mantle boundary r = b is

valid for a free-slip boundary. At r = d, the surface that defines the horizon between the two mantle layers, we have the following boundary conditions that express the continuity of velocity and stress:

in which q+ [ = 7)M in Eqs. (53)-(55)] is the upper-layer viscosity and 7 ) - is the lower-layer viscosity. Equation (55) provides the remaining boundary condition

in which TT is the spherical harmonic coefficient in the harmonic expansion of the left-hand side of Eq. (55 ) :

1 477

~ ; n = - / / R " ( h + . [ V , ( R 2 - K)v , , - KAW,,]

- ~ ( 1 + v,)KA'W,,)sin 0 d0 dq5. (65)

The normalization convention we employ for the Yy throughout this chapter is given by Eq. (11.3) in Appendix 11. We shall omit the algebraic details involved in determining the constants E y , F;' that satisfy conditions (61)-(64) and simply point out that the values of (E,)? and ( F , ) ; ' , which define the toroidal flow in the upper layer of the mantle, are given by


where

B~ = (e + 2 ) + y ( e - I ) + (e - i ) ( m ) 2 e + y i - y ) , (69)

in which y = v,/v+. Employing expressions (66) and (67) we then find that value of the toroidal-flow scalar, immediately below the lithosphere-mantle interface, is given by

It is straightforward to show, on the basis of expressions (68) and (69), that when the viscosity of the lower layer is much greater than that of the upper layer ( y >> l ) , Eq. (70) reduces to the following expression:

Equation (71) will be exactly true for a perfectly rigid lower mantle, which thus appears as a no-slip lower boundary (except at 8 = 1; see below) for the upper- layer toroidal flow.

The requirement that tangential shear stresses be finite at the lithosphere- mantle interface implies that the tangential flow velocity must be continuous and therefore

(A . u,),=,,+ = (A . u H ) ~ = ~ , ~ . (72)

If we now substitute Eqs. (52) and (57) into Eq. (72) and expand the flow scalars in terms of their spherical harmonic coefficients, we then obtain

( 1 - E)(Wo)? = qb"(r,), (73)

in which E is the dimensionless thickness of the lithosphere, introduced in Eq. (55). All terms of order E~ and smaller have been omitted from the left-hand side of Eq. (73). For E << 1 we may also safely ignore the term containing E in (73). Combining Eqs. (70) and (73) we thus obtain


It is clear from the defining equation [Eq. (65)] for F; that Eq. (74) describes the coupling that must exist, as a consequence of lateral viscosity variations in the lithosphere, between the toroidal and poloidal components of lithospheric flow. This coupling was obtained on the basis of the continuity of horizontal stress and horizontal velocity at the lithosphere-mantle boundary.

3.2. Net Rotations in the Lithosphere and Mantle

The matching of stresses and velocities at the lithosphere-mantle interface mer- its additional consideration when dealing with the degree e = 1 component of toroidal flow in the mantle. The toroidal-flow scalar in the upper and lower mantle layers is given by the expression (60). At 4 = 1 the free-slip condition at the core- mantle boundary (CMB), given by Eq. (61), implies that F ; = 0 in the lower layer and thus qy(r) = Eyr in the lower layer. In the upper layer Eq. (60) applies again and, owing to the matching conditions (62) and (63), we again have q;"(r) = Eyr. The degree 1 toroidal flow field throughout the mantle is thus given by the following simple expression:

+I

In Eqs. ( 5 ) and (10) we previously showed that a rigid-body rotation may be expressed as

vrigid(r, 8, 4) = o X r = - A R, (76)

where

+ ( w , + L W " ) Y I '(8, 411 (77)

is derived in Appendix 111. The vector w = w,i + w , j + w,k describes the angular velocity of the rigid-body rotation. A comparison of Eq. (75) and Eqs. (76) and (77) shows that the e = 1 toroidal flow is a rigid-body rotation of the entire mantle. The components of the angular velocity vector that describe this rotation are given by

w , = v%Re[E;], w, = -v%Im[El], w; = -v?EP. (78)

The rigid-body rotation described by Eq. (75) was derived by assuming that the


viscosity in the mantle is spherically symmetric. This motion does not produce any tangential stresses, either within the mantle or at the bounding surfaces. An arbitrary rigid-body rotation of a fluid shell, with free-slip boundaries and spherically symmetric viscosity, is always a possible mode of “flow.” We may eliminate this rigid-body motion by transferring to a frame of reference rotating with angular velocity (78) relative to the original frame. Since flow in an infinite F’randtl number fluid (like the mantle) is not influenced by inertial forces, this new frame of reference is dynamically equivalent to the original frame.

Arbitrary rigid-body rotation is a degenerate solution of the viscous flow equations in a spherically symmetric mantle. This degeneracy is eliminated by introducing lateral viscosity variations, in which case the degree l = 1 toroidal flow will not possess the simple linear dependence on radius in Eq. (75). The l = 1 toroidal flow field will in general be described by v,,, = AC,,E;’(r)Y;”(B, @), and thus each infinitesimally thick spherical shell at radius r in the mantle will rotate with a different angular velocity given by

.\/cs v3 w,(r) = -- Im[El(r)], q ( r ) = -- E:’(r). (79)

The differential mantle rotation due to lateral viscosity variations has recently been invoked as an explanation for the net rotation of the lithosphere in absolute- motion plate models based on the hotspot frame of reference (Ricard et al., 1991 ; O’Connell et al., 1991). The hotspot tracks on the Pacific plate are assumed to arise from a net rotation of the lithosphere relative to the underlying mantle in which the hotspots are presumably “anchored.” The analysis by Ricard et al. (1991) is based on their use of the following expression for the horizontal shear exerted across the lithospheric layer

(80)

in which vH(r = a ) is the horizontal flow velocity at the Earth’s solid surface, vH(r = r ! ) is the horizontal flow velocity at the lithosphere-mantle interface, h = a - r, is the thickness of the lithospheric layer, (7-l) is the depth average of the laterally varying reciprocal viscosity in the lithospheric layer, and tH is the horizontal shear stress acting at the solid surface. The lithospheric layer containing the lateral viscosity variations is assumed to be 100 km thick in Ricard etal. (1991).

Equation (80) is based on the following general expression for the tangential flow-induced stress [see Eq. (V.7) in Appendix V]:

r r

vH(r = a ) - vH(r = rL) = h ( 7 - ’ ) t H ,

t H = 7 [ V,u, . + r - R, (:)I’


It is clear from a comparison of Eqs. (80) and (81) that Ricard et al. assume that the horizontal gradient of radial velocity u, is negligible compared to the radial gradient of horizontal velocity. The significant error arising from this assumption was previously pointed out in the discussion immediately following Eq. (57). It is also clear that Ricard et al. assume, apparently without justification, that the condition

auH UH >> - d r r -

obtains throughout the lithospheric layer. The greatest problem posed by the differential rotation analysis of Ricard et al.

(1991) arises from their assumption of a nonvanishing tangential stress tH at the solid surface. The existence of such a stress arises from the fact that Ricard et ul. assume that the lithospheric plate velocities are simply imposed as a boundary condition. Clearly, the imposed plate motions are maintained against viscous dissipation by externally applied horizontal stresses. This flow calculation is physically unrealistic. In reality there are no imposed external stresses, and the plate motions must arise from internal flow driven by buoyancy forces. Since there can be no external stresses driving the plates, the boundary condition that must be employed with observable plate motions is clearly free-slip and therefore

It is evident, on the basis of Eq. (83), that the assumption in Eq. The application of Eq. (83) to the near-surface net rotation o ( r )

(82) is not valid. X r implies that

Equation (84) shows that the differential rotation across a thin lithospheric layer is entirely negligible. In other words, the observed net rotation of the lithosphere will be nearly identical to that at the top of the mantle. Finite differential rotation between the surface and the deep mantle requires that the depth interval in which lateral viscosity variations occur be considerably in excess of the thickness of the lithosphere. Such a differential rotation cannot be described by the theory described in this section; it requires a theory that treats the effects of lateral viscosity variations throughout the mantle (see Section 4 below).

3.3. Inverting for Lateral Viscosity Variations

We now show that the definition (56) of the normalized viscosity perturbations

Y L ( e r cb) = ow. ( 8 5 )

implies that


The actual viscosity v,.(B,q5) must be positive and therefore vl.(O, 4 ) > - 1. Since the horizontal average of v,(B, q5) is, by definition, equal to zero, the maximum positive values of v,(B, q5) will in general be of order unity. We may illustrate this point by considering the lithosphere. The plate boundaries may be imagined to be very weak and thus v,(B, q5) = - 1 over the small surface area occupied by plate boundaries. The value of v,(B, q5) must therefore be less than + I in the plate interiors to ensure that the horizontal average of ~ ~ ( 0 , 4 ) is identically zero.

The coefficient of the term ( WO)r on the left-hand side of Eq. (74) is O( l), and therefore, from the definition of Tp in Eq. (65) and from Eq. (85) we must have

to ensure that both sides of Eq. (74) are balanced. The viscous coupling and generation of toroidal flow in the mantle by lateral viscosity variations in the lithosphere thus implies, according to Eq. (86). that the thickness E of the lithosphere trades off inversely with its relative "stiffness" (r],)/v,,,.

We will find it useful to rewrite Eqs. (74) and (65) as

[ B , - (e + 2)(d/r,-)2e+1Ae] vu [ B P + ( e - l ) (d / rL)2c+1Ae] (vL)&

- + K)

X (Wo)? = Q?, (87) where

Q? = & /J Y?*{hv, . [Vl(h2 - K)V,, - KAW,] - v,KA2Wo)

x sin B d0 d 4 . (88)

Prior to obtaining an explicit expression for Qy"' in terms of the spherical harmonic coefficients of v,, Vo, and W, we point out that for a general field f ( B , q5) = Xfiy;" the following identities hold:


in which L = m. Equations (89)-(90) follow from Eqs. (1.8) and (1.12) in Appendix I. Expanding all quantities in (88) in terms of spherical harmonics, i.e.,

UL = -&vL)yYyr v, = z ( & , ) y Y y , w, = C(W,,)b"Yy, e nl t m 1 ,,t

and employing results (89)-(90), we may obtain (algebraic details omitted here) the following:

x ( n + 2)bi,] + Kn(n + l)(W&) + .,2L,[; v;' ."I

whereL=-,i= - ( m + u + l ) , j = - ( m + u ) , k = - ( m + v - l),and the 2 X 3 arrays enclosed in square brackets are the spherical harmonic coupling coefficients defined in Eq. (11.1) of Appendix 11. The algorithm we employ to evaluate these coefficients is also described in Appendix 11. The asterisk over Q;" in Eq. (91) denotes complex conjugation.

Equations (87) and (91) provide an explicit description of the coupling between poloidal and toroidal lithospheric flow due to lateral viscosity variations. In particular the selection rule (11.8) in Appendix I1 implies that the degree s components of the lateral viscosity variations will couple the degree It? - sI, 14 - SI + 2, - s( + 4, . . . , e + s - 2, 4? + s components of the toroidal lithospheric flow

and the degree le - sl - 1, 14 - sI + I , It? - sI + 3 , . . . , 4 + s - 1, e + s + 1 components of the poloidal lithospheric flow to the degree 4 toroidal flow in the lithosphere and underlying mantle.

Equation (87) constitutes a system of linear equations that may be rewritten as the following matrix equation:

w = Mv, (92)


in which w is a vector consisting of the complex conjugate of the left-hand side of (87) arranged in order of increasing e,m, v is a vector consisting of the spherical harmonic coefficients (v& arranged in order of increasing u,v, and M is a matrix consisting of the terms enclosed in curly brackets in Eq. (91). If we require that the lithospheric flow field correspond to the observed plate velocities, then the vector w and matrix M are known, and Eq. (92) may be inverted to determine the lateral viscosity variations that are consistent with the observed plate motions. According to Eqs. (48) and (49), we have

v)g; (93)

a (W")? = -e (e + I ) (i . v x V)?,

in which (V, . v)? and ( i . V X v ) ~ are respectively the spherical harmonic coefficients of the horizontal divergence and radial vorlicity of the observed tectonic plate velocities.

The interpretation of the lateral viscosity variations yielded by an inversion of (92) will be facilitated by considering a useful approximation to the left-hand side of Eq. (87). As pointed out above, in the derivation of Eq. (71), if the viscosity jump in the two-layer parameterization of mantle viscosity is sufficiently large (i.e., y 2 lo), then the left-hand side of (87) simplifies to

We may further simplify this expression for the case d = rfd. Writing d = r, ( 1 - a), where 8 is assumed to be small, we then obtain the following approximation to Eq. (87):

(94)

For d = rh7,), the radius corresponding to the bottom of the upper mantle, 6 = 0.09. In this case we are at the limits of the validity of expression (94), which becomes increasingly accurate as S -+ 0 ( e g , 6 = 0.01 when d = rzw-if we wish to model a 100-km-thick asthenospheric channel). In the inversion experiments described below, the system of equations (92) will be based on the approximation (94) rather than Eq. (87). We may then ensure that the normalized field of lateral viscosity variations we recover will satisfy the viscosity-positivity constraint, vL(B, 4) > - 1, by choosing the appropriate value for F in (94). We thereby constrain the magnitudes (and tradeoffs) of the various parameters appearing in the definition of E


An important concern arising from the inversion of (94) is the magnitude of the bias produced by the inevitable truncation of the spherical harmonic expansion of the plate-velocity scalars W,(O, q5), V,,(e, 4 ) and of the viscosity v,(e, 4) itself. We will examine this issue by considering inversion experiments in which we choose different levels of truncation in the spherical harmonic expansion of the lateral viscosity variations.

In the first experiment we include the harmonic coefficients of the plate-velocity scalars in (93) up to degree and order 15 and invert (94) for the harmonic coefficients of the lateral viscosity variations up to degree and order 30. This truncation level for the viscosity is based on the selection rules for the coupling coefficients, in Eq. (9 1 ), which imply that viscosity variations corresponding to degrees u > 30 will not contribute to Qin (assuming that all plate-velocity coefficients vanish for degrees n > 15). The system of equations (92) that correspond to these truncation levels thus comprises 255 equations [corresponding to 4 = 1 - 15 in Eq. (94)] and 960 unknowns [corresponding to u = 1-30 in Eq. (91)]. We may invert this underdetermined system by seeking the minimum-norm field of lateral viscosity variations which satisfies (92). The harmonic coefficients of this minimum-norm viscosity are thus given by

v = M T ( M M')-'w, (95)

in which M, v, and w are as defined previously in Eq. (92). We find, essentially by trial and error, that the field of lateral viscosity variations yielded by (95) bears no resemblance to the expected pattern of weak plate boundaries and strong plate interiors unless K < - 30.

In Fig. 15a we show the lateral viscosity variations v,(8,d) inferred, according to (99 , when K = -40. In this map it is immediately clear that the very-low- viscosity regions are almost entirely confined to the plate boundaries and the plate interiors have higher viscosities as expected. When K = - 40 the inferred lateral viscosity variations satisfy the positivity condition vL(Or 4) > - 1 when the constant F in (94) has values F < 28. The field shown in Fig. 15a assumes F = 28. If we assume a lithospheric thickness of 100 km ( E = 0.016) and an upper mantle layer defined by d = r6,() (8 = 0.09) we then infer, according to (94). that (v,)lv,+, = 10. According to the absolute lithospheric viscosity estimates in (5 1) for K = - 40, we also infer, for F = 28, that v,+, = 0.8 X lo2' Pa s.

The field of lateral viscosity variations yielded by (93 , when K = -60, is nearly identical to that shown in Fig. 15a, provided that we select F = 42 to ensure that min[v,-] = - 1. In this case we find according to (94) (for E = 0.016 and 8 = 0.09) that F = 42 implies (vL)lqM = 7. The absolute viscosity estimate in (51), for K = -60, thus implies that vM = 0.9 x lo2' Pa s. These inferences for the absolute value of the upper-mantle viscosity agree closely with those inferred, in Section 2.3, by matching the observed plate motions to the plate-like surface


FIG. 15. (a) The field of lateral viscosity variations in the lithosphere [v,(B, 4 ) defined in Eq. (56)], in the degree range e = 1-30, obtained from a minimum-norm inversion [see Eq. (95)] of Eq. (94) when K = - 40. The amplitude of the normalized viscosity variations shown here is fixed by choosing F = 28 in Eq. (94). The lateral viscosity variations shown here are constrained by the observed long- wavelength (e = 1 - IS) plate divergence and vorticity. (b) The long-wavelength component, synthesized from harmonics in the range e = I - IS, of the lateral viscosity variations in (a).

flow calculated using the density perturbations derived from seismic models of mantle heterogeneity.

The amplitude spectrum of the lateral viscosity variations in Fig. 15a is shown in Fig. 16b by the square symbols. It is clear that the root-mean-square (rms) amplitude in any harmonic degree decreases sharply beyond degree 8 = 18. This is also verified by the map in Fig. 15b in which we show the lateral viscosity variations synthesized from the harmonic coefficients up to degree and order 15. The maps in Figs. 15a and 15b agree very well in their spatial pattern and in their


a l

0 5 10 15 20 DEGREE

b

0.10 I b

p: 0.02 m B , I I 1 I

0 10 20 30 DEGREE

FIG. 16. (a) The cross-correlation, at each degree t , between the field of lithospheric lateral viscosity variations in Fig. 15a and the field of lateral viscosity variations inferred from a separate inversion of Eq. (94) (with K = - 40) in which the viscosity variations are truncated at (i.e., assumed to be zero beyond) degree 20. (b) The rms amplitude, at each degree e, of the lateral viscosity variations v,(B. 4) in Fig. 1% (0, “t = 1-30”) and the lateral viscosity variations obtained from the truncated degree 20 inversion in (a) (A, ‘*t! = 1-20”),

overall amplitude. This suggests that the coupling of the observed plate-velocity scalars synthesized from harmonic coefficients up to degree 4 = 15 may be ade- quately described by a field of lateral viscosity variations up to degree e = 18. We have directly tested this hypothesis by inverting Eq. (94) to determine the field of lateral viscosity variations, up to degree and order 20, which couples the observed plate-velocity scalars described up to degree and order 15. This inversion is again underdetermined, and thus we carry out the minimum-norm inversion described in Eq. (95). We find that when K = - 40 the condition min[v,] = - 1 requires us to select F = 29 (in close agreement with that inferred from the em,, = 30 inversion). The amplitude spectrum of the t,,, = 20 inversion is shown in Fig. 16b by the triangle symbols. In Fig. 16a we show the correlation, at each harmonic


degree, between the lateral viscosity variations obtained from the e,,,, = 20 and em,, = 30 inversions.

These viscosity inversions have demonstrated that the poloidal and toroidal components of the observed lithospheric plate motions (and the equipartitioning of energy among these two components) may be explained by physically plausible lateral variations of the effective viscosity of the lithosphere. The lateral viscosity variations we infer reveal a clear pattern of weak plate boundaries and strong plate interiors. Although this may have been expected a priori, Fig. 15 shows some interesting and unexpected complexities. Although the East Pacific ridge is a region of rapid divergence and high heat flow, it does not appear to be a prominent zone of weakness. Significant reductions of strength are manifested in western North America, the Aleutian arc, and the Pacific-Antarctic ridge. This suggests a combined thermal and nonlinear (strain-induced softening) interpretation of the lateral variations in Fig. 15. Such an interpretation must be viewed with some caution, however, because these viscosity inferences are obtained from a significantly underdetermined inversion procedure. Unfortunately the viscosity inversions described here tells us nothing concerning the possible importance of lateral viscosity variations in the deep mantle. This is the subject to which we turn in the next section.

4. MANTLE DYNAMICS WITH 3D VISCOSITY VARIATIONS

The equations describing the effects of an arbitrary three-dimensional ( 3 D ) viscosity distribution on buoyancy-induced mantle flow are derived in Appendix IV. The two coupled differential equations [Eqs. (IV.10) and (IV.l l)], governing the poloidal and toroidal flows, are evidently rather complicated and are not amenable to straightforward mathematical analysis. In this section we shall describe an alternative approach based on a variational formulation of the momentum equation. As suggested in Forte (1992), this variational treatment allows us to explicitly derive the important symmetry relations that govern the relationship between poloidal and toroidal flows and the driving density contrasts. The application of a variational approach to mantle-flow problems was also considered recently by eadek et al. ( 1993), who suggest the application of an iterative numerical scheme. The formulation that we shall present is noniterative and enables the derivation of an explicit solution for the flow velocities in a spherical shell with heterogeneous viscosity.

4.1. Formulation of the Variational Principle

The hydrodynamic field equations, in Cartesian tensor notation, that describe the conservation of mass, momentum, and the relationship between stress and strain rate are respectively, in the Boussinesq approximation:


dku, = 0, (96)

(97)

TLJ = -PIS, + 27,1E,~; E tJ = - + ' J u t > ' (98)

+ pOat41 + Pla,40 = O,

The various quantities appearing in (96)-(98) are defined in Appendix IV. In Eq. (97) we have already subtracted the hydrostatic background state in which a , P o = pOd,4" . Equations (96)-(98) describe a flow field u, driven by density perturbations p, , which are assumed to be known a priori (and thus treated as fixed). This flow is assumed to occur in a medium occupying a volume V and bounded by a surface S. The flow u, will satisfy the following boundary conditions on S:

fip, = 0, 199)

&,c~T,, = o on s,, ( 100)

u, = c, on S,, (101)

where f i , is the unit vector that is everywhere normal to S, h, is any unit vector that is tangent to any point on S(C,h, = 0), S, is the portion of S on which the free-slip condition (100) applies, S, is the remaining portion of S (i.e., S, = S - S,) on which a given tangential velocity field c, (fi,c, = 0) is prescribed.

Let us now introduce a kinematically admissible flow perturbation Su, satisfying Eq. (96), such that the flow field u, + Su, satisfies the same boundary conditions (99)-(101) [the flow u,, of course, satisfies Eqs. (96)-(lOl)]. The inner product of Su, with Eq. (97) yields

a,(Tk,au,) - Tk,(akGu,) + a,(P&iSu,) Pi6ULaL4o = 0, (102)

in which we have used d,6u, = 0. Integrating Eq. (102) over the volume V occupied by the medium, we obtain

[ P i 6 4 at40 - T A , S & , I ~ V Is [ f i~T~,au, + P o 4 ~ f i ~ ~ U k l ds = 0, (103)

in which we employed the result TA,aA6u, = Tk,SEk,, which follows from the symmetry of TA,. Both u, and u, + Su, satisfy Eq. (99) and thus fi,Su, = 0 on S. Since u, and u, + Su, satisfy Eq. (101), it is clear that the Su, must vanish on S, and, owing to condition (IOO), fikTA,8u, must vanish on S, . Clearly the surface integral in (103) must vanish, thus yielding

To obtain (104) we have also used PI Sk$ Ekr = PI 6Ekk = 0, owing to (96). As the density perturbations pI and the reference gravity a, 4o are known a priori (and are thus treated as constant), we may rewrite (104) as


s w = 0; W = 1" [@&, - ~ , u , & 4 ~ l d V . ( 105)

In writing Eq. (105) we have made the important assumption that 87 = 0, which is valid only when the viscosity is not dependent on the flow velocity u, [i.e., when the rheology is linear and thus characterized by a stress exponent n = I in Eq. (3)]. The strong temperature dependence of viscosity shown in Eq. (3) implies that the dependence on strain or stress will be relatively weak, and thus Eq. (105) is expected to hold very well in the mantle. It is possible to formulate a hybrid variational principle when the viscosity is stress-dependent; the interested reader may refer to the initial study of t'adek et al. (1993).

The quantity W i n Eq. (105) is the difference between the rate of viscous dissipation of energy (qE,,E,,) and the rate of energy released by buoyancy (pIu,d, 4,,). We have shown that a flow field satisfies the field equations (96)-(98), and the boundary conditions (99)-(101), if and only if the functional W in (105) is sta- tionary with respect to perturbations of the flow field. We may further show that W is also an absolute minimum for this flow field. Let u) be the flow which satisfies 6 W = 0 in (105) [and thus Eqs. (96)-( I O I ) ] , and let u! be any kinematically admissible flow satisfying Eqs. (96), (99), (loo), and u; = 0 on S,, For the flow u, = up + E u: the quantity W i n (105) may be written as

where El: and Ej, are the strain-rate tensors that correspond respectively to u? and ul . According to (106) we may regard Was a function of E :

where W(O), (dW/d&),, and (d2 W/d&2)o correspond respectively to the first, second, and third integrals in (106). According to Eq. (105), we have 6 W =

(dW/dE),)SE = 0 and thus (dW/dE), = 0. Expression (107) thus simplifies to

W(E) = W(0) +

Since

POLOIDAL-TOROIDAL COUPLING IN MANTLE FLOW 5 1

is a positive-definite quantity, it is clear that W(0) must be the absolute minimum of W ( E ) . This minimum principle is immediately recognizable as an extension of the minimum-dissipation rheorem of Helmholtz (e.g., Batchelor, 1967, pp. 227- 228) to fluids with internal buoyancy sources.

4.2 . Variational Calculation of Buoyancy-Induced Flow

The application of the variational principle (105) to the problem of viscous flow in a spherical shell is considerably facilitated by the use of generalized spherical harmonic basis functions. A detailed discussion of the use of generalized spherical harmonics, in problems of elastodynamics in spherical geometry, is given by Phinney and Burridge (1973) (hereafter referred to as PB for convenience). The derivations presented below will be extensively based on the covariant differentiation rules described in PB. The principal properties of generalized spherical harmonic basis functions Yfm(B, (6) are described in Appendix 11, and these properties are extensively exploited in the following derivations.

We begin by expressing EVEij in terms of the so-called contravariant canonical components that are described in PB:

(108) E..E-- B U = CiaCioEaDCiTCj8E~S = e,,eo8EUPE~s

- - ECOE(K1 + 2 E + t E - - + 2E+ - E+ - - 4EOi EO-,

in which

(10% EflP = &[uu.P + UP,"]

1 % t = - C C [uyla)pfl(r) + u ~ ~ - ) ~ ( ~ ) I Y ~ + P ~ ( o , (6).

2 (=O , " = - I

Employing the covariant differentiation rules in PB to evaluate the UylDlffi(r) in ( 109), we obtain

E(XI = C ( E , ) ~ Y ~ ~ * l , E + - = Z(E,) ; '@n, E"+ = E(E,)I 'Y)" , P . m e.m (.!?I

EO- = C ( E 4 ) i . Y F i m ,

where

E + + = Z(E,)?Y:"I , E - - = C ( E6)7 Y y ztn, t,tn l , t n Y. m

( 1 10)


in which U?m is the radially varying (generalized) spherical harmonic coefficient of the contravariant flow-velocity component uo,

uqr , 8. 4 ) = c up(r)Yy(o, 4), 1 , m

and

n: = q q e + 1y2, 11: = d(e - i)(e + 2112.

Substitution of the expressions in ( I 10) into Eq. (108), and the subsequent use of the coupling rule (11.19) in Appendix 11, yields the following result:

We now introduce the following harmonic decomposition of the 3D viscosity distribution:

v(r, 8, 4) = c T;(r )y; (o, 4). ( 1 13) 1'. I

Combining expressions (1 12) and ( 1 13), we obtain r t + ,

in which r = b defines the inner surface of the shell and r = a defines the outer surface of the shell.

We now evaluate the buoyancy integral in (105):

in which go(r ) is the radial gravitational acceleration. The radial flow u, is identi-


cal to the contravariant flow component uO, and we therefore obtain from (1 15) the following:

1" plu,d,4, dV = - c4.rr 1" (pI)?'UYmgor2 dr , (1 16)

in which (p , )? is the radially varying spherical harmonic coefficient of the density perturbation field p, (r, 8, 4).

The solenoidal velocity field u, which satisfies Eq. (96), may be written as (Backus, 1958):

u = V x A p + Aq, (1 17)

where p(r , 8,4), q(r, 8,4) are respectively the poloidal and toroidal flow scalars. The contravariant flow components u"(r, 8, 4) may be obtained from (1 17) using

u" = c:, u,, (1 18)

where Cg, is the unitary (complex rotation) matrix defined in PB. Combining ( 1 17) and ( 1 18), we may show that

c ,PI b

in which L = and p;"( r). qa( r ) are respectively the spherical harmonic coefficients of the poloidal and toroidal flow scalars.

On the basis of expression (1 19) we now obtain the explicit poloidal-toroidal dependence of the terms in ( 1 1 I ) that appear in the viscous dissipation integral (1 14):


Substitution of (120) into ( 1 14) yields

The substitution of ( 1 19) into ( I 16) also yields

The variational principle in (105) requires that we minimize the functional W with respect to the flow field u. To accomplish this minimization, in a manner that directly provides the flow solution, we expand the poloidal and toroidal flow coefficients in terms of radial basis functions:

The radial basis functions J,f;l(r) and g,!(r) must satisfy the boundary conditions (99)-(IOl). When the expressions in (123) are substituted in Eqs. (121) and (122), the function W in (105) will then vary according to the values of the coefficients ,,p$ and ,,q?. The particular values of these coefficients that minimize W will define the flow solution. When W is a minimum, the following conditions must necessarily be satisfied:


aw a - -

The set of coefficients ,,p: and ,,q: that satisfy (124) and (125) define the flow solution we seek.

The substitution of expression (123) into (121) and (122), and the evaluation of ( 1 24), yields (omitting algebraic details) the following:

2 l i ( k , e, rn; n, s , t ) APT + C C J A k , e, rn; n, s, t ) k91' h f m 4 ,,I

f t ,

J2(k, e, rn; n, s, t ) = - L [ ( 2 t + 1)(2s + 1)(2J + 1) ]1 '2 J = " - 3 l

J ) ( 1 - (;l)f+'+' ) 77 "' - ' x (' rn t - r n - t 6. 'lo

where 7,) is simply a reference viscosity employed to normalize the viscosity dis-


tribution. The symmetry of the first Wigner symbol in the integrand of ( I 27) [see Eq. (11.25) in Appendix 111, and the presence of the factor [ 1 + ( - l)y+s+J], implies that the sum in (127) extends only over the values J = - sI, (e - sI + 2, . . . , e + s - 2, e + s. Similarly, the factor [ 1 - ( - l )y+s+J] in (128) implies that the sum extends only over the values J = - sI + 3, . . . , t? + s - 3,

The substitution of (123) into Eq. (121), and the evaluation of (125), yields

- sI + 1, e + # - 1.

(again, omitting algebraic details) the following:

in which

The presence of the factor [ 1 + ( - l ) r + , + J ] in (130) implies that the sum extends only over the values J = 14 - sI, 14 - SI + 2, . . . , 4 + s - 2, e + s. Similarly, the presence of the factor [ 1 - ( - l)L+.,+J] in (1 3 1) implies that the sum extends only over the values J = - sI + 1, le - sI + 3, . . . , e + s - 3, e + s - 1.


Equations (126) and (129) constitute a coupled system of linear equations for the flow variables Lp;' and Lqi". This linearity ensures that the usual principle of superposition remains valid [i.e., if u, ( r , 8, 4) is excited by density perturbation 6p,(r , 6, #) and uz(r, 8, 4) is excited by density perturbation 6pz(r, 6 , #), then uI + u2 will be excited by 6p, + 6p2] . This linearity thus implies that the concept of Green functions (e.g. Forte and Peltier, 1987) will also be valid in a spherical fluid shell with a given 3D distribution of viscosity.

Equation (1 26) describes the flow that is directly excited by buoyancy forces, and this equation corresponds to the differential equation (IV. 10) derived in Ap- pendix IV. In Eq. (126) we observe that buoyancy forces will, in general, directly excite a turuidal flow field. This, of course, will not be true in a mantle with spherically symmetric viscosity, and indeed one may readily verify that J, (k , 4, rn; n, s, t ) = 0 when q(r, 6 , 4) = q{l(r). On the basis of the spherical harmonic coupling relations contained in expressions ( 127) and (1 28), we may observe that a degree s field of density perturbations will directly excite the following flow components:

Equation (126) also describes the viscous coupling of the buoyancy-induced flow in (132) to other flow components not directly coupled to buoyancy sources. We may understand the nature of this viscous coupling with the following example. Let us assume for simplicity that the only component of density heterogeneity is given by degree s and order f , as in (1 32). Equation (132) is valid for all degrees, including s k 1 , .s * 2, s k 3, . . . , for which the density heterogeneity is assumed to be zero. We thus see that a v;" viscosity structure will lead to the coupling of p:+y. p:_.':l. . . . , poloidal flow components to the components p;:';, pi. p:;;". in ( 1 32). Even the apparently simple interaction of (pl); density heterogeneity with I];" lateral viscosity variations will generate poloidal flow components corresponding to all harmonic degrees. In the particular case of an v;" lateral viscosity structure we expect that the flow corresponding to p : and the nearest-neighbor terms p;:";', p : ~ f , q:-" will be strongest.

The poloidal and toroidal flow components excited by buoyancy forces in Eq. (126) are not independent. Equation (129) describes the coupling that must exist between the poloidal flow components and the toroidal flow components;


this equation corresponds directly to differential Equation (IV. 1 1) derived in Ap- pendix IV. On the basis of the spherical harmonic coupling implicit in expressions ( 130) and (1 3 1) we may verify that the toroidal flow components will be coupled to poloidal flow as follows:

pi-" + q i I y , q i ; r , via qy( r ) ;

4.3. Generalized Green Functions

In the previous section we noted that the flow equations (1 26) and ( 129) are linear in the flow variables, thus ensuring the validity of the linear superposition principle. The concept of the Green function may therefore be generalized to a fluid shell with lateral viscosity variations. We begin our derivation of generalized Green functions by rewriting Eqs. (126) and (129) as follows:

2 l , ( k , e, m; n, S, - t ) + 2 J, (k , e, m; n, S, - t ) k q F A,( m k I ,n

- - 2(Ri)Z ( - 1)' i: gofn r2 dr, (134) 7 0

We now rewrite (134) and (135) as

Ah? apl' + k.L ,,I

= &(k, e, m; n, s, -t) r p ~ . (135) r , ( , ! n

cwm = ,,,) J , ( k . €, m; n, s, - t ) , D 2 = I d k , e, m; n, s, - t ) .

It is clear that Eqs. ( 1 36) and (1 37) constitute a coupled set of matrix equations in


which the rows of the individual matrices, given in ( I 38), are defined by letting (k , e, rn) vary, for a given ( n , s, t ) , and the columns are defined by letting (n , s, t ) vary for a fixed ( k , e, m).

The solution of Eqs. ( 1 36) and ( 1 37) may be formally written as

where the matrix elements &';in in (139) are obtained from

- P = [A + B c-1 121-1 , (141)

and the matrix elements Q;;; in (140) are obtained from

Q = c-1 12 r. (142)

Combining the expression for L,,,,, in (138), with expression (123) and (139) and (140), we obtain

Expressions analogous to (143) and (144) were obtained in the derivation of poloidal flow Green functions, for a spherically symmetric mantle, by Forte and Peltier (1987):

in which pt( r , r ' ) is the poloidal flow Green function. The definition of the generalized Green functions, which describe the excitation of flow by buoyancy forces in a fluid shell with lateral viscosity variations, is immediately evident from a comparison of ( 145) with ( 143) and (144):

P;:"(r> r o = c ,f,(r)P;i:n$(r')2(n1)2( - l )1r '2 , (146) I ,I

in which f&, ( r , I-') and Q&(r, r ' ) are respectively the poloidal and toroidal Green functions.

We have so far avoided making an explicit choice for the radial basis functions . f i ( r ) and g k ( r ) used to describe the poloidal and toroidal flow scalars, respectively.


Any choice for these basis functions, provided it satisfies the boundary conditions (99)-( 101), will be acceptable. In the case of free-slip, zero radial velocity, boundary conditions at r = (z, b, the poloidal and toroidal flow scalars must satisfy

Perhaps the simplest and most easily employed set of radial basis functions that satisfy (148) are the following:

A relatively straightforward illustration of the utility of the radial basis functions in (149) is provided by considering the horizontal divergence (V, . u ) ( r =

a) of surface flow in an isoviscous fluid shell. According to Eq. (A1 3) in Forte and Peltier ( 1987) we have

On the basis of expressions (143) and (146), we thus obtain

in which the generalized horizontal divergence kernel S&( r ’ ) is given by

In a fluid shell with spherically symmetric viscosity the terms P;;:,, will vanish for all t! f s and m f t. In an isoviscous fluid shell, the exact expression for the horizontal divergence kernel St(r’) , derived in Appendix A in Forte and Peltier (1987), is as follows:

‘ (e + 1 ) 2(24 + 1 )

S,(r’) = (153)

1 - ( r ’ / /7 )2 f+3 1 - ( r ’ / b ) 2 c - 1

I - (a/h)”+’ 1 - (a/b)2“


I

-0.12 -0.06 0.00

AMPLITUDE FIG. 17. Horizontal divergence kernels for an isoviscous mantle. The degree d defining each kernel

is indicated by the adjacent number. The solid lines are the kernels calculated with the exact theory embodied in Eq. (153). The dashed lines are the kernels calculated on the basis of the variational principle and are defined by Eq. (152), in which N = 10. The dashed and solid lines are essentially overlapping, indicating the accuracy of the variational procedure.

In Fig. 17 we show a selection of divergence kernels given by the exact expression ( I 53), in solid lines, along with the kernels provided by expression ( I 52), with N = 10 andf,(r) in (149). shown by dashed lines. The kernels obtained with radial basis .fi( r ) in (149) are virtually identical to the exact ones. On the basis of this excellent performance we shall truncate the sums over k and ti in (143) and (144) at k = n = 10, for all the calculations that follow.

4.4. Horizontal Divergence and Radial Vorticity of Buoyancy-Induced Surface Flow

To investigate the effects of lateral viscosity variations in a fluid shell (e.g.. the Earth's mantle) we shall once again express the viscosity variations according to Eq. (56), that is,

m, 8, 4) = V H ( ~ ) [ I + w, 6, 411, (154)

in which q X r ) is the spherical average of the field ~ ( r , 8, 4 ) at any radius r and v(r, 8, 4 ) describes the lateral variations of viscosity relative to this horizontal average (this implies that the t' = 0, m = 0 spherical harmonic component of v(r, 8, 4 ) vanishes). The parameterization in (154) allows us to describe extreme lateral variations of viscosity with relatively small amplitude variations in


v(r, 0, 4). By letting v(r, 0, 4) approach values arbitrarily close to - 1 we may easily describe viscosity reductions spanning several orders of magnitude.

The first example illustrating the effects of lateral viscosity variation will be based on the degree 1 field of lateral variations v(r, 8, 4), obtained by scaling the depth-integrated degree 1 shear-velocity heterogeneity S v,/v, in model SH8/WM 13 of Woodward et al. ( 1993):

in which the scaling factor fr is selected so that

min[u] = - x ( 156)

where x > 0. We emphasize that the degree 1 field given by Eq. (1 5 5 ) does not vary with radius. In Fig. 18 we show the degree 1 viscosity variation v(0 ,4) when f, = (i.e., min[v] = - 0.9). The region of minimum viscosity is centered near the Pacific-Antarctic spreading ridge, which, in model SH8/WMI 3, is a region of strongly reduced shear velocity extending throughout the underlying upper mantle. The choice x = 0.9 implies that, at any depth, the viscosity ~ ( r , 8, 4) will vary from a minimum value of 0.1 $(r ) to a maximum value 1.9 @ ( r ) (i-e., a factor of 19 lateral variation).

We now consider the horizontal divergence and radial vorticity of the surface flow generated in the presence of the degree 1 lateral viscosity variations in Fig. 18. The general expression for the horizontal divergence was given by Eqs. (15 l)-(152). The radial vorticity (3 a V X u)$n(r = u ) is given by

FIG. 18. The degree I field of normalized lateral variations of viscosity 1i.e.. ~ ( 0 . 4 ) in Eq. ( ISS)l, obtained from the depth-integrated degree I shear-velocity heterogeneity in model SHIlWM I 3 according to Eq. (155). The factorf, in Eq. (155) is chosen such that the minimum value of the field is - 0.9. The maxiinum value of this field is then +0.9.


( f * V x u);’(r = a) =

Combining (144) and (157), we therefore obtain

in which the radial vorticity kernel R&(r’) is given by

A consideration of some of the divergence and vorticity kernels will explicitly demonstrate the flow coupling due to the e = 1 viscosity variations in (155). We assume, for simplicity, that q$(r) = ?lo (i.e., constant). In Fig. 19 we show the divergence kernels for t = I - 6, corresponding to a Re[ Yl(0, $)I density load

1= 1

- 6000

y 5500

5000

2 4500

2 4000

3500

E U

n

y 5500 U

2 4500

2 4000

3500

n 1= 1

-0.06 0.00 - 6000

y 5500 Y

v1 5000

4500

2 4000 1 1=4 3500

-0.06 0.00

-0.06 0.00 -0.06 0.00

-0.06 0.00

AMPLITUDE -0.06 0.00

FIG. 19. The horizontal divergence kernels S;:,,(r’). defined in Eq. (152) with N = 10. corresponding to a Re[ Yi(B, 4)] density load (i.e., for s = 1. t = I ). The 3D viscosity distribution is given by the degree I field of lateral viscosity variations v(H. 4) in Fig. 18 and a spherically symmetric viscosity v{;(r ) /vo = I, such that v( r , 8, Q)/q, , = I + v ( B , d). In each of the panels are the 21 + I divergence kernels corresponding to the real and imaginary parts of S ; ; , , ( r ’ ) for m = 0. I , . . . , t . The dashed line in the panel labeled “ P = 1” is the single non-zero-divergence kernel for the spherically symmetric viscosity distribution ~ ( r , 0, d)/v(, = I [ in this case, S; ; ,? ( r ’ ) = 0 unless ( t , m ) = (s. I ) ] .

64 ALESSANDRO M. FORTE A N D W. RICHARD PELTIER

(i.e., s = 1, t = 1). According to ( I 32) we expect significant flow at degrees I and 2 that is directly driven by this density load. This is indeed the case in Fig. 19. We also observe that the divergence at € = 3 is also significant. This degree 3 field is produced by viscous coupling to the buoyancy-driven e = 1 flow. The degree 4 divergence is produced by viscous coupling to the degree 2 flow, which is itself maintained by viscous coupling to the degree 1 buoyancy forces, and thus is much weaker than the degree 3 now. Viscous coupling also yields a degree 5 divergence (coupled to degree 3 flow), a degree 6 divergence (coupled to degree 4 flow), and so on. Clearly, the strength of viscous coupling is strongly diminished beyond degree 3, owing to the very limited spectral range of the lateral viscosity field (only degree 1 in this example).

In Fig. 20 we have the radial vorticity kernels, for 8 = 1-6, corresponding to the Re[ Yl(0, 4)1 density load. According to (132), we expect a viscous coupling of the degree 1 buoyancy to degree 1 toroidal flow, and this is clearly evident in Fig. 20. The degree 2 vorticity is produced by viscous coupling to the degree 1 buoyancy-induced poloidal flow according to Eq. (133). The degree 3 vorticity is generated by viscous coupling to degree 2 and 3 poloidal flows according to

n 6000

-y 5500 E U

5000

4500

2 4000

3500 -0.03 0.00 -0.03 0.00 -0.03 0.00

- 6000

y 5500 E Y

5000

4500

2 4000

3500 I -0.03 0.00

L

-0.03 0.00

AMPLITUDE -0.03 0.00

FIG. 20. The radial vorticity kernels R;:,,(r’), defined in Eq. (10) with N = 10, corresponding LO a R e [ Y / ( O , q5)I density load ( i . e . , for s = I , t = I ) . The 3 D viscosity distribution is the same as in Fig. 19. In each panel are the 2Y + I vorticity kernels corresponding to the real and imaginary parts of R);,,(r’) for ni = 0, I , . . . , L. Observe that the vorticity kernel Re[R/ / (r ’ ) ] is identically zero.


- 6000 y 5500-

5000-

2 4500- 2 4000-

E U

n

3500

-

z -0.06 0.00 -0.06 0.00 -0.06 0.00

- 6000

y 5500

5000

4500

2 4000

3500 -0.06 0.00 -0.06 0.00 -0.06 0.00

AMPLITUDE FIG. 71. Thc horizontal divergence kcrnels S;;>,(r’) , defined i n Eq. ( 152) with N 10. correspond-

ing to an I r n [Y i (H , d)] density load 1i.e.. for ,Y = 2, I = 21. The 3D viscosity distribution is the same as in Fig. 19. In each panel are the 2P + I divergence kernels corresponding to the real and imaginary parts of S;; , , (r’) for ) ? I = 0. I . . . . , P . The dashed liiic in the panel labeled “I = 2” is the single nonzero-divergence kernel for the spherically syinrnetric viscosity distribution q(r, 0. @)/q( , = I (i.c.. for

= 2, 171 = 2 ) .

( 133). The degree 4 vorticity is generated by viscous coupling to degree 3 and 4 poloidal flows, and so on. Clearly, as in the case of the divergence field, viscous coupling yields rapidly diminishing vorticity beyond degree 3 because of the limited spectral range of the lateral viscosity variations.

As a further illustration of the mechanism of coupling, we show in Fig. 2 1 the divergence kernels corresponding to the interaction of an Im[ Gt(0, 411 density load with the degree 1 viscosity variations. According to ( 132). we expect dominant flow at t = 2 and significant flow at t‘ = 1,3 arising from the viscous coupling to the degree 2 buoyancy forces. These expectations are verified in Fig. 21. The degree 4 divergence in Fig. 2 1 arises through viscous coupling to the degree 2 flow. The degree 5 and 6 divergences are respectively coupled to the degree 3 and 4 flows and, since they are already “twice removed” from the primary buoyancy force, their amplitudes are now significantly weaker than the divergence at e 4. According to ( 132), we also expect significant buoyancy-driven degree 2 toroidal flow and this is verified by the vorticity kernels shown in Fig. 22. The degree I and 3 vorticities are generated by VISCOUS coupling to the degree 2


- 6000

.y 5500

2 4500

2 4000

3500

5000

n 1 0.00 0.03 0.00 0.03 0.00 0.03 - 6000

y 5500 E U

5000

4500

2 4000

3500 0.00 0.03 0.00 0.03 0.00 0.03

AMPLITUDE FIG. 22. The radial vorticity kernels RL(r’), defined in Q. (159) with N = 10, corresponding to

an Im[ c(8, (b)] density load [i.e., for s = 2, t = 21. The 3D viscosity distribution is the same as in Fig. 19. In each of the panels are the 26 + 1 vorticity kernels corresponding to the real and imaginary parts of R;:,,(r’) for rn = 0, I, . . . , P . Observe that the vorticity kernel Im[Rf;(r’)] is identically zero.

buoyancy-driven poloidal flow, according to ( 1 33). The degree 4 vorticity is vis- cously coupled to degree 3 poloidal flow and the degree 2 toroidal flow, according to (133). Again we see that the strength of viscous coupling to the higher degree ( e > 4) flows is considerably weaker than for the t‘ S 4 toroidal flows.

We now present predictions of buoyancy-induced surface flow, based on the degree e = 1,2 field of density perturbations derived from model SH8/WM13 (Woodward et al., 1993) and the 6 h p / 6 In v , scaling factor in Fig. 6b. In Fig. 23a we show the predicted e = 1,2 surface divergence for an isoviscous mantle [~11(r) = v,) = lo2’ Pa s ] . In Fig. 23b we show the surface divergence (for k‘ =

1 -6), produced when the e = 1,2 density perturbations interact with the depth- invariant degree 1 field of lateral viscosity variations v(O,4) = vo[ I + ~(0,411, where u(6, 4 ) is shown in Fig. 18. Comparison of Figs. 23a and 23b shows that the flow field distortion produced by the lateral viscosity variations is quite extensive. The region of much “softer” mantle, underlying the Pacific-Antarctic ridge, has strongly intensified the local vertical flow while significantly reducing its strength elsewhere. The dominant strength of the downwellings, and the relatively more diffuse upwellings, in Fig. 23a is now reversed in Fig. 23b, where the single

FIG. 23. (a) The degree E = 1-2 surface divergence in an isoviscous mantle 1i.e.. q(r , 8, q5)/q,, = I ] , calculated according to Eq. (151) with N = 10. due to the degrees 1-2 density perturbations derived from niodel SHIlWM13. The 6 In p l S In v, conversion factor is that shown in Fig. 6. (b) The degree E = 1-6 surface divergence in a mantle with 3D viscosity q(r. 8, q5)/vo = 1 + u(B,q5), where u(8,q5) is the degree 1 viscosity variation in Fig. 18. calculated according to Q. (151) with N = 10. The density perturbations are again obtained from the degree 1-2 heterogeneity in SH8lWM 13, with 6 In p/6 In v, from Fig. 6. (c) The degree E = 1-6 surface radial vorticity, with 3D viscosity as in (b), calculated according to Eq. (158) with N = 10. The density perturbations employed are us in (a) and (b). In all cases the reference viscosity value is ql, = 10” Pa s. The units on all scale bars are rdd/( 10 Myr).


a

cn a h r: 0 1 2 3 4 5 6 (1:

DEGREE FIG. 24. (a) The cross-correlation, at each degree e , between predicted divergence in Fig. 23a and

that in Fig. 2%. (b) The rms amplitude, at each degree t?, of the predicted divergence in Fig. 23a (0.

divergence, no lateral viscosity), the predicted divergence in Fig. 23b (0, divergence, with lateral viscosity), the predicted radial vorticity in Fig. 23c (A, vorticity, with lateral viscosi~y).

upwelling is clearly stronger than the adjacent downwellings. In Fig. 23c we show the corresponding radial vorticity field (for t? = 1-6). The regions of peak vorticity are situated in the zones of nearly zero surface divergence that define the transition from upwelling to downwelling. A quantitative summary of the surface flow predictions is presented in Fig. 24.

The degree 4 = 1 field of lateral viscosity variations in Fig. 18 is clearly an overly simplified representation of the actual lateral variations of strength in the Earth’s mantle. In Fig. 25 we present a more realistic representation of lateral viscosity variations obtained, as in (155), from the depth-integrated P = 1 -5 shear velocity heterogeneity in model SH8/WM13:


FIG. 25. The degree 1-5 field of normalized lateral variations of viscosity [i.e., v(0, 4) in Eq. (154)l. obtained from the depth integrated degree 1-5 shear-velocity heterogeneity in model SHI/WM13 according to Eq. (160). The factor,/; in Eq. (160) is chosen such that the minimum valuc of the field shown here is - 0.99. The maximum value is + I . 18.

In Fig. 25 we have chosen x = 0.99, such that min [v] = -0.99. The maximum value of v(8,+) is 1.18 and consequently the viscosity variations in Fig. 25 imply that, at any depth, ~ ( r , 8, 4) will vary from a minimum of 0.01 I!(.) to a maximum of 2.18 v:j( r ) (i.e.. by a factor of 2 I8 laterally).

We shall again consider the buoyancy-induced surface flow generated by the interaction of the t? = 1-2 density perturbations with the field of viscosity variations in Fig. 25. The predicted 42 = 1-2 divergence for an isoviscous mantle, with v:;(r) = yi0 = 10’’ Pa s, is shown again in Fig. 26a. In Fig. 26b we show the Y =

1 -6 surface divergence produced in the presence of the depth-invariant viscosity structure ~ ( 8 , $1 = Q,J 1 + v(0, 4) ] , where v(0, 4) is shown in Fig. 25. A comparison of Figs. 23b and 26b shows that the peak value of the divergence in the latter case is significantly smaller than in the former case. This may appear somewhat puzzling, given that the field of viscosity variations employed in Fig. 26b varies laterally by a factor 218, whereas in Fig. 23b the viscosity varied laterally by a factor of 19 only. This behavior may be understood by noting that the regular geometry of the degree 1 field of viscosity variations in Fig. 18, relative to the geometry of the mantle buoyancy sources, is ideally placed for strongly focusing the central Pacific upwelling. The relatively irregular and broken pattern of mantle “softening” in Fig. 25 is not as effectively situated, relative to the mantle buoyancy sources, and thus is less efficient at amplifying the surface flow. In Fig. 26c we show the 4 = 1-6 radial vorticity field generated in the presence of the lateral viscosity variations in Fig. 25.

FIG. 26. (a) The degree e = 1-2 surface divergence in an isoviscous mantle [i.e., v ( r , 19, r $ ) / ~ , , =

I ] , calculated according to Eq. (1.5 I ) with N = 10, due to the degree 1-2 density perturbations derived from model SH8/WM13 (using S In p/6 In Y. in Fig. 6). (b) The degree 8 = 1-6 surface divergence in a mantle with 3D viscosity q(r . 0, r$)/v,, = 1 + v(O,4) , where ~ ( 0 . 4 ) is the degree 1-5 viscosity variation in Fig. 2.5, calculated according to Eq. (151) with N = 10. The density perturbations are as in (a). (c) The degree e = 1-6 surface radial vorticity, with 3D viscosity distribution and density perturbations as in (b), calculated according to Eq. (1.58) with N = 10. In all cases the reference viscosity value is qI, = 10” Pa s. The units on all scale bars are rad/( 10 Myr).


a z

95% 80%

~ - . . . . . . - - _ _ _ s 0.8

E 0 0.0

..-.. - - .._.___ w 0.4 E

u

(A I I 1 1 I

0 1 2 3 4 5 6 DEGREE

FIG. 27. (a) The cross-correlation, at each degree P. between the predicted divergence fields in Figs. 26a and 26b. (b) The rrns amplitude, at each degree e, of the predicted divergence in Fig. 26a (0.

divergence, no lateral viscosity), the predicted divergence in Fig. 26b (0, divergence, with lateral viscosity), the predicted vorticity in Fig. 26c (A , vorticity, with lateral viscosity).

z

A quantitative summary of the buoyancy-induced surface flows in Fig. 26 is presented in Fig. 27. A comparison with Fig. 24 shows clearly that the introduction of shorter-wavelength components in the field of lateral viscosity variations yields a divergence and vorticity spectrum that is “flatter” and thus more closely resembles the observed spectrum in Fig. 4. The effect of higher-degree components in the field of lateral viscosity variations is most dramatic in the case of the radial vorticity field, which possesses a distinctly “red” spectrum in Fig. 24 and a rather “blue” spectrum in Fig. 27.

We have observed that a more realistic field of lateral viscosity variations, as in Fig. 25, acts as a “filter” that may suppress the surface expression of buoyancy- induced flow in the mantle. This behavior was evident from the comparison of Figs. 23b and 26b. The extent to which the viscosity “filters” the mantle flow field depends on the relative alignment of the mantle buoyancy sources and the zones


of weakness and strength in the mantle. This behavior is clearly analogous to the filtering affect of the surface plates, discussed in Section 2 and demonstrated in Fig. 8. We note that the analogy between plates and lateral viscosity variations was indeed confirmed in Section 3.

4.5. Dynamic Surface Topography

In the previous section we observed that lateral viscosity variations in the mantle may have a great impact on the amplitude and spatial pattern of buoyancy- induced flow. We shall now consider to what extent such viscosity variations will affect the flow-induced boundary deflections. The result of this analysis is of great importance because the gravitational field perturbations (e.g., the nonhydrostatic geoid) are known to be very sensitive to the mutually canceling gravitational contributions from flow-induced boundary deflections and the internal buoyancy sources (e.g., Richards and Hager, 1984; Ricard eta!., 1984; Forte and Peltier, 1987). It is therefore important to derive flow models that provide realistic (i.e., include all relevant physical effects) boundary deflections.

We shall begin by deriving explicit expressions for the boundary deflections in terms of the buoyancy-induced flow in a laterally heterogeneous fluid shell. We consider first the case of the outer boundary at r = a that separates the overlying inviscid fluid (e.g., the global ocean layer) from the underlying viscous medium (e.g., the mantle). The matching of normal stresses across the undulating outer surface yields [see Eq. (V.8) in Appendix V] the following expression:

in which c?a(t), 4 ) is the undulation of the boundary relative to its reference lo- cationat r = a , & = p u ( a + ) , p , ; =po(u - ) , andp , :~ , ( a ) = P,(u’) is thenonhy- drostatic pressure field in the overlying inviscid fluid arising from self-gravitation. The principal obstacle to be overcome, in the use of Eq. (161), is to obtain an expression for the nonhydrostatic pressure P , ( a - ) in a fluid with lateral viscosity variations. The usual procedure (e.g., Forte and Peltier, 1987) is to consider the horizontal component of the conservation of momentum equation [i.e., Eq. (IV.8) in Appendix IV]. In Appendix VI we present a mathematically efficient derivation of the nonhydrostatic pressure using generalized spherical harmonics and the concept of covariant differentiation, described in Phinney and Burridge ( 1973). We thus find (see Appendix VI for details) that the spherical harmonic coefficients of the flow-induced surface topography Say are given by


in which

P i ( [ , m, s, t ) = [(28 + 1)(2s + 1)(2J + I ) ]”’

x ( t k - k 0 ”)( -m t m - t ’ ” 1 and the notation &,,2 indicates summation over J = j , j + 2, j + 4, . . . . The radial derivatives of the poloidal and toroidal flow scalars in (162) are readily calculated from Eqs. (143). (144). and (149).

The continuity of normal stresses across the lower bounding surface yields, in analogy with Eq. (161), the following expression:

glblp; - p,:ISb = -Pl(h+) + W b + ) ( % ) + p i 4 1 ( h ) , (163) I , +

in which Sb(0, #) is the undulation of the lower boundary relative to its undis- turbed position at r = h, pd = po(h+), ply = p o ( b - ) , and p r $ , ( b ) = PI(& ) is the nonhydrostatic stress field in the underlying inviscid fluid (e.g., the outer core). In analogy with Eq. ( 162) we then have

in which X ! y ( b + ) is directly obtained by evaluating those terms in Eq. (162) enclosed by square brackets at r = b+ rather than r = a - .

A striking aspect of Eq. (162) is the presence of terms that correspond to toroidal flow. This may seem somewhat surprising, given that toroidal flow does not involve any radial mass flux. The toroidal-flow contribution to surface topography arises from its effect on dynamic pressure P I , as shown in Eq. (VI.7) in Appendix VI.

FIG. 28. (a) The degree e = 1-2 dynamic surface topography in an isoviscous mantle [i.e., v( r , 8, q5)/vo = I], calculated according to Eq. (162). due to the degree 1-2 density perturbations derived from model SHIlWM13 (using 6 In p / 6 In v, in Fig. 6). (b) The degree = 1-6 dynamic surface topography in a mantle with 3D viscosity v( r , 0, c$)/T,, = 1 + v(0, c$), where u(8, 4 ) is the degree 1 viscosity variation in Fig. 18, calculated according to q. (162). The density perturbations are as in (a). (c) The topography difference, in the range e = 1-6, obtained by subtracting the topography in (b) from the topography in (a). The units on all scale bars are kilometers.


In the following illustrations of flow-induced surface topography we shall again employ the degree 1-2 field of density perturbations derived from model SH8/ WM13 (Woodward et al., 1993) and 6 In p/6 In v , in Fig. 6b. The two fields of lateral viscosity variation we shall consider were defined in Eqs. (155) and ( 160), the former with x = 0.9 (thus implying a factor of 19 horizontal variation) and the latter with x = 0.99 (implying a factor of 218 horizontal variation). The degree 1 field of lateral viscosity variations is shown in Fig. 18, and the degree 1-5 field of viscosity variations is shown in Fig. 25.

The flow-induced surface topography at r = a. for an isoviscous mantle ($l(r) = ql), is shown in Fig. 28a, in which we observe the clearly dominant degree 2 pattern. The degree 1-6 dynamic topography, which is produced in the presence of the degree I field of lateral viscosity variations (again with @ ( r ) = qo), is shown in Fig. 28b. It is obvious that the topography in Figs. 28a and 28b are virtually identical, and the relative difference, obtained by subtracting the latter from the former, is indeed quite small as Fig. 28c shows. A quantitative summary of the dynamic topography predictions is presented in Fig. 29. The difference, due

a

significance level 95% ..____ ----..___-

.-__ -------.._______ 80% 0.4

b

w EJ 10-1

5 3 10-2 < v) 3 10-3

0 1 2 3 4 5 6 DEGREE

FIG. 29. (a) The cross-correlation, at each degree t , between the predicted topography fields in Figs. 28a and 28b. (b) The rms amplitude, at each degree I?. of the predicted dynamic topography in Fig. 28a (0, no lateral viscosity), the predicted dynamic topography in Fig. 2% (U, with lateral viscosity). the topography difference in Fig. 28c (A, difference).


FIG. 30. (a) The degree L = 1-6 contribution to the dynamic surface topography in Fig. 28b due only to the poloidal mantle How [calculated by setting 4: = 0 in Eq. (162)l. (b) The degree L = 1-6 toroidal How contribution to the dynamic topography in Fig. 28b lcalculated by setting p : = 0 in Eq. (162)). The superposition of the separate topography contributions in (a) and (b) yields the complete topography shown in Fig. 28b. The units on a11 scale bars are kilometers.

to the introduction of the degree 1 viscosity variations, is dominated by a degree 3 component with a relative amplitude that is quite small.

In Fig. 30 we consider the separate poloidal and toroidal flow contributions to dynamic topography shown in Fig. 28b. The poloidal contribution is almost a factor of 6 greater than the toroidal contribution. It is, however, clear that had we mistakenly assumed that the toroidal contribution was entirely negligible, then the difference between the topography predictions (with and without lateral viscosity


a z 2 0.8

W 0.4 p: lx 0 0.0 V

w -0.4 w [1: a -0.8 4!

2

U

b - fs 100 Y

w 2 lo- ' k a 10-2 2

10-3

J

0 1 2 3 4 5 6 DEGREE

E

FIG. 31. (a) The cross-correlation, at each degree C. between the dynamic surface topography in Fig. 28b and thc poloidal flow contribution in Fig. 3Oa (0, total poloidal) and the toroidal Row contribution in Fig. 30b (A, total toroidal). (b) The rins amplitude, at each degree P , of the predicted dynamic topography in Fig. 28b (0, total), the separate poloidal flow contribution in Fig. 30a (0, poloidal contribution), the separate toroidal How contribution in Fig. 30b (A , toroidal contribution).

variations) would have been much larger. A quantitative analysis of the poloidal and toroidal topography contributions is provided in Fig. 3 I .

The dynamic surface topography produced in the presence of the degree 1-5 field of viscosity variations (again with va(r) = qJ is shown in Fig. 32b. along with the isoviscous prediction in Fig. 32a. Here we again have a clear manifestation of the strong insensitivity of dynamic topography to the effects of lateral viscosity variations. The difference between the predictions (with and without lateral viscosity variations) is shown in Fig. 32c. We point out that, although the amplitude of the viscosity variations in the case of Fig. 32b are more than LO times greater than in the case of Fig. 28b, the relative difference in Fig. 28c increases only slightly from about 5% to 8% in Fig. 32c. A quantitative summary of the topography predictions in Fig. 32 may be found in Fig. 33. It is worth noting that the introduction of shorter-wavelength viscosity variations yields a difference

FIG. 32. (a) The degree e = 1-2 dynamic surface topography in an isoviscous mantle [ i t . , ~ ( r , 0, $)/~j,, = I], calculated according to Eq. (162), due to the degree 1-2 density perturbations derived from model SH8/WM13 [using 6 In pl6 In v, in Fig. 61. (b) The degree e = 1-6 dynamic surface topography in a mantle with 3D viscosity ~ ( r , 0, $)/vu = 1 + v(0, $), where v(0, 4 ) is the degree 1-5 viscosity variation in Fig. 25, calculated according to Eq. (162). The density perturbations are as in (a). (c) The topography difference, in the range e = 1-6, obtained by subtracting the topography in (b) from the topography in (a). The units on all scale bars are kilometers.


a 6

7 . .- significance level 95% 80%

.--__ ----._.____ -._ ..* --.._ .* - - -

---..__. ---.___ - - _ ---.__. *..

5 0.4

0 V o.o[-

U

b - E 100

g lo-’

5 g 10-2

-Y

w U

4

m g 10-3 0 1 2 3 4 5 6

DEGREE FIG. 33. (a) The cross-correlation, at each degree P, between the dynamic surface topography pre-

dictions in Figs. 32a and 32b. (b) The rms amplitude. at each degree e, of the predicted dynamic topography in Fig. 32a (0, no lateral viscosity), the predicted dynamic topography in Fig. 32b (0, with lateral viscosity), the topography difference in Fig. 32c (A, difference).

spectrum that is clearly “blue” and quite different from that displayed in Fig. 29. In Fig. 34 we show the separate poloidal and toroidal flow contributions to the dynamic topography. A comparison of Figs. 30 and 34 shows that the introduction of shorter-wavelength viscosity variations has evidently led to a strong reduction of the toroidal flow contribution while the poloidal flow contribution is relatively unchanged.

We have so far considered only viscosity distributions with a spherically symmetric component that is independent of depth [i.e., vS(r) = vO]. We now consider the extent to which a depth variation of the horizontally averaged viscosity, when coupled with the depth-invariant lateral viscosity variations in Eq. (160), will affect the dynamic surface topography. We shall consider the following smooth depth increase of the spherically symmetric viscosity:


FIG. 34. (a) The degree e = 1-6 poloidal flow contribution to the dynamic surface topography in Fig. 32b [calculated by setting q: = 0 in Eq. (162)j. (b) The degree e = 1 -6toroidal-flow contribution to the dynamic surface topography in Fig. 32b [calculated by setting p: = 0 in Eq. (162)]. The superposition of the separate topography contributions in (a) and (b) yields the complete topography shown in Fig. 32b. The units on all scale bars are kilometers.

The choice of 10 for the exponent in ( 165) implies the following depth increases: M r h 7 , d / ~ % a ) = 3, rlll(rlloo)/r18(a) = 8, and $KrCMB)/~1Ku) = 421. In Fig. 35a we show the degree 1-2 dynamic surface topography produced in the absence of lateral viscosity variations. A comparison with the isoviscous prediction in Fig. 32a shows that the depth increase of viscosity in (165) leads to a nearly 50% reduction in the amplitude of the surface topography. The flow-induced surface topography generated in the presence of the degree 1-5 viscosity variations is shown in Fig. 35b, and the difference, relative to Fig. 35a, is shown in Fig. 3%. It

FIG. 35. (a) The degree C = 1-2 dynamic surface topography in a spherically symmetric mantle with depth-varying viscosity q(r, 8, +)/qI, = ( d r ) ' " , calculated according to Eq. (162). The degree I -2 density perturbations employed here are derived from model SH8IWM13 [using 6 In p16 In Y, in Fig. 61. (b) The degree C = 1-6 dynamic surface topography in a mantle with 3D viscosity v(r . 0. 4)/ qI1 = ( d r P [ I + ~ ( 0 , +) I , where u(8 , 4) in the degree 1-5 viscosity variation in Fig. 25, calculated according to Eq. ( 162). The density perturbations are as in (a). (c) The topography difference, in the range E = 1-6, obtained by subtracting the topography in (b) from the topography in (a). The units on all scale bars are kilometer$.


a

b

g 10-3 0 1 2 3 4 5 6

DEGREE FIG. 36. (a) The cross-correlation, at each degree e, between the dynamic surface topography pre-

dictions in Figs. 35a and 35b. (b) The rms amplitude, at each degree e, of the predicted dynamic topography in Fig. 35a (0. no lateral viscosity), the predicted dynamic topography in Fig. 35b (0, with lateral viscosity), the topography difference in Fig. 35c (A, difference).

is important to note that, although the amplitude of the dynamic topography is strongly reduced by depth increases of viscosity, the amplitude of the difference between the predictions (with and without lateral viscosity variations) is almost unchanged. Thus the 8% relative difference in Fig. 32c is increased slightly to 12% in Fig. 3%. In Fig. 36 we provide a quantitative summary of the topography calculations in Fig. 35. We note, in comparison to Fig. 33, that the amplitude of the degree l = I ,2 differences has substantially increased.

The separate poloidal and toroidal flow contributions to the dynamic surface topography are shown in Fig. 37. Since poloidal flow involves the vertical trans- port of mass we expect that depth increases of viscosity should more strongly reduce the poloidal contribution to surface topography than the toroidal contribution. This is indeed verified by comparing Figs. 37 and 34.

We have observed that the dynamic topography appears to be remarkably insensitive to even very-large-amplitude horizontal variations of viscosity. This insen-


FIG. 37. (a) The degree P = 1-6 poloidal flow contribution to the dynamic surface topography in Fig. 3Sb [calculated by setting 9 : = 0 in Eq. (162)]. (b) The degree P = 1-6 toroidal flow contribution to the dynamic surface topography in Fig. 3Sb [calculated by setting p : = 0 in Eq. (162)l. The superposition of the separate topography contributions in (a) and (b) yields the complete topography shown in Fig. 3Sb. The units on all scale bars are kilometers.

sitivity is markedly different from the strong sensitivity of the flow field to lateral viscosity variations (see, e.g., Fig. 26). We may understand this insensitivity of the dynamic topography with the following argument. The buoyancy-induced flow field scales inversely with viscosity, and therefore regions of reduced viscosity correspond to increased flow velocities and vice versa. The normal stresses that deflect the boundaries scale as the product of viscosity and the flow field, and this leads to an effective “cancellation” of the viscosity effect. In regions of increased viscosity the normal stresses would be increased for a given flow, but the


actual flow must itself be reduced in these regions, and consequently the normal stress (and hence the dynamic topography) is relatively unaltered by the lateral viscosity variation. We may also observe this effective cancellation in direct quantitative terms. As shown in (1 32), degree 1 viscosity variations directly lead to the generation of additional degree 4 - 1, e + 1 poloidal flow and degree t? toroidal flow, by degree e density perturbations. On the other hand, the expression for dynamic topography in (162) implies that the degree 1 viscosity variations will couple the degree e - 1, e + 1 poloidal flows and degree U toroidal flow to degree e topography. Clearly, then, the effective “splitting” of the flow field by lateral viscosity variations is nullified when the “split” flow components are recombined to generate surface deflections.

4.6. Nonhydrostatic Geoid

The gravitational potential perturbations which arise over a convecting mantle are the sum of the potential perturbations due to the internal density perturbations which drive the flow and the potential perturbations due to the flow-induced boundary deflections (e.g., Pekeris, 1935; Richards and Hager, 1984; Ricard era/ . , 1984; Forte and Peltier, 1987). The surface gravity perturbations are largely controlled by the mutually canceling gravitational contributions from the flow- induced surface topography (at r = u ) and the internal density anomalies. In the previous section we have observed that while the depth variation of mantle viscosity has a strong impact on the amplitude, and hence the gravity effect, of the surface undulations, the presence of very strong lutercil variations of viscosity has a much smaller impact on the amplitude and pattern of the surface topography. We therefore expect that the nonhydrostatic geoid will show a similar insensitivity to the presence of lateral viscosity variations, and hence be dominantly sensitive only to radial viscosity variations. In the following we demonstrate this explicitly. In Appendix VII we provide explicit expressions for the nonhydrostatic geoid in a self-gravitating mantle.

In Fig. 38a we show the e = 2-3 nonhydrostatic geoid prediction for an isoviscous mantle ( q ~ i ; ( r ) = q,J, employing the e = 2-3 density heterogeneity derived from model SHS/WM13 and 6 In p/6 In v, in Fig. 6b. The degree 1-6 nonhydrostatic geoid, produced when thee = 2-3 density anomalies interact with the degree 1-5 viscosity variations in Fig. 25, again with 7 X r ) = q ~ ~ , , is shown in Fig. 38b. Apart from a small reduction in the amplitude of the geoid highs in the South Pacific and in southern Africa, Fig. 38b is nearly identical to the isoviscous prediction in Fig. 38a. A map of the difference between these two predictions is presented in Fig. 38c, where we observe that the peak difference is localized in the South Pacific. The rms amplitude of the difference field is 12 m, compared to the rms amplitude of 100 m in Fig. 38a (i.e., the relative effect of the lateral viscosity variations is only 12%). We note that the difference field in Fig. 38c is


FIG. 38. (a) The degree I' = 2-3 nonhydrostatic geoid prediction for an isoviscous mantle [i.e., V ( r . 8. ~$)/q,, = 11, calculated on the basis of the degree 2-3 density perturbations derived from model SH8/WM13 [using 8 In p/6 In v , in Fig. 61. (b) The degree e = 1-6 nonhydrostatic geoid prediction for a mantle with 3D viscosity q(r, 8, d)/q( , = 1 + u ( l ) , 4 ) , where u(8, q5) is the degree 1-5 viscosity variation in Fig. 25. The density perturbations are as in (a). (c) The geoid difference, in the range e =

1-6, obtained by subtracting the geoid in (b) from that in (a). The units on all scale bars are meters.


I a

b - E - 80 a 60

2 40 W

2

5 20

0 1 2 3 4 5 6 DEGREE

FIG. 39. (a) The cross-correlation. at each degree Y, between the nonhydrostatic geoid predictions in Figs. 38a and 38b. (b) The rms amplitude, at each degree Y , of the predicted geoid in Fig. 38a (0,

no lateral viscosity), the predicted geoid in Fig. 38b (0, with lateral viscosity), the geoid difference in Fig. 38c (A, difference).

largely reflective of the small effect of the lateral viscosity variations on the flow- induced surface topography (see Fig. 32c). A detailed summary of the geoid predictions in Fig. 38 is provided in Fig. 39.

In Fig. 40a we now show the e = 2-3 nonhydrostatic geoid prediction for a laterally homogeneous viscosity with a depth variation q~(u/r)"' given in Eq. (165). This geoid prediction has reversed sign relative to the isoviscous prediction in Fig. 38a because the strong increase of viscosity with depth has strongly reduced the amplitude of the surface topography (compare Figs. 32a and 35a), thus allowing the internal density perturbations to dominate the geoid signal. The degree 1-6 geoid prediction, calculated when the degree 1 -5 viscosity variations are introduced [with v8(r) = ~ l , ( d r ) l O ] , is shown in Fig. 40b. Again we observe that the effect of very large lateral variations of viscosity is small, and this is confirmed by the small amplitude of the difference field in Fig. 40c. The

FIG. 40. (a) The degree C = 2-3 nonhydrostatic geoid prediction for a spherically symmetric mantle with depth-varying viscosity v(r, I ) . d)/vl, = ( d r ) " ' , calculated on the basis of the degree 2-3 density perturbations derived from model SHIiWM I3 (using 6 In p16 In v. in Fig. 6). (b) The degree C = 1-6 nonhydrostatic geoid prediction for a mantle with 3D viscosity v ( r . 0. 4)/v0 = ( ( i /r) l l l [ I + o(0, c$)J, where o(0,q5) is the degree 1-5 viscosity variation in Fig. 25. The density perturbations are as in (a). (c) The geoid difference, in the range C = 1-6. obtained by subtracting the geoid in (b) from that in (a). The units on all scale hars are meters.


a

b

7 100 Y

w 60

2 40 23 =z 20

!x g o 0 1 2 3 4 5 6

DEGREE FIG. 4 I . (a) The cross-correlation, at each degree e, between the nonhydrostatic geoid predictions

in Figs. 40a and 40b. (b) The rms amplitude, at each degree e, of the predicted geoid in Fig. 40a (0,

no lateral viscosity), the predicted geoid in Fig. 40b (0, with lateral viscosity), the geoid difference in Fig. 4Oc (A, difference).

rms amplitude of this difference field is 14 m, compared to the rms amplitude of 108 m in Fig. 40a (i.e. a relative difference of only 13%). We again note that the difference field in Fig. 4Oc is a reflection of the small effect of lateral viscosity variations on the flow-induced surface topography (see Fig. 35c). A quantitative summary of the geoid predictions in Fig. 40 is provided in Fig. 4 1.

When considering the conclusions obtained in this section, and in Section 4.5, it is worth recalling that all our demonstrations have employed idealized deprh- independent lateral viscosity variations. It is our expectation that the imposition of the large-amplitude lateral variations in Fig. 23, at all depths in the mantle, probably overestimates the large-scale variations that actually exist, especially in the deep mantle (e.g., Zhang and Christensen, 1993). The final confirmation of our conclusions will be possible when we can reliably map the amplitude of lateral viscosity variations throughout the mantle. Encouraging progress is now being made thanks to recent high-pressure melting experiments (Zerr and Boehler,


1993) and also as a result of recent inferences of lateral variations of seismic Q in the deep mantle (Romanowicz and LeStunff, 1993).

4.7. Differential Rotation in the Mantle

In Section 3.2 we pointed out that a degree 1 toroidal flow field in the mantle implies that any infinitesimally thick spherical shell at radius r will rotate, as a rigid body, with angular velocity given by Eq. (79). In a mantle with spherically symmetric viscosity (and free-slip boundary conditions) the angular velocity is constant with depth, and thus the entire mantle rotates as a rigid body. This rigid- body rotation is a degenerate solution of the governing flow equations, and it may be eliminated by transferring to a new frame of reference. This degeneracy is also removed by introducing lateral viscosity variations. We have seen that lateral viscosity variations, distributed throughout the mantle, may produce buoyancy- induced toroidal flow with magnitude comparable to the poloidal flow. In particular, the degree 1 toroidal flow will imply a depth-varying net rotation within the mantle. As emphasized in Section 3.2, this differential rotation of the mantle will be significant only if the lateral viscosity variations are distributed over a sufficiently large depth interval.

The most visible manifestation of net rotation within the mantle is provided by the degree 1 toroidal-flow component of the tectonic plate motions. In Fig. 42a we show the degree 1 radial vorticity derived from the Minster and Jordan (1978) absolute plate-motion model (AM 1-2) based on the hotspot reference frame. This figure clearly illustrates the dominant east-west lithosphere rotation. In Fig. 42b we show the degree 1 component of the buoyancy-induced radial vorticity field in Fig. 26c. The predicted net rotation of the lithosphere in Fig. 42b agrees rather well with the observed net-rotation of the plates in Fig. 42a. This agreement suggests that the hotspot reference frame may indeed be dynamically plausible since it yields absolute plate motions (i.e. net rotation) that are readily explained by the interaction of large-scale buoyancy-induced flow with large-scale lateral viscosity variations (e.g., in Fig. 25).

The explicit expressions for flow throughout the mantle, given by Eq. (123), allow us to readily describe the depth-varying net rotation in the mantle. The degree 1 toroidal mantle flow, given by Eq. (144), arising from the interaction of degree 1-2 density perturbations with degree 1-5 viscosity variations (Fig. 25) is illustrated in Fig. 43. The depth variation of the net-rotation vector components w,(r) , w,.(r), w,( r ) is calculated on the basis of expression (79). The surface rotation w,(r = a) , @\( r = a), w;(r = a ) is of course identical to the degree I toroidal flow in Fig. 42b. It is clear from Fig. 43 that there is essentially no differential rotation between the lithosphere and adjacent underlying mantle, as we pointed out in Section 3.2. The net rotation within the mantle changes significantly below a depth of about 1400 km. Near a depth of 1700 km the net rotation vanishes and


FIG. 42. (a) The degree e = I radial vorticity of the tectonic plate velocities (Forte and Peltier, 1987) derived from absolute-motion model AM1 -2 of Minster and Jordan (1978), which is based on the hotspot reference frame. (b) The degree t = 1 component of the predicted buoyancy-induced surface radial vorticity shown in Fig. 26c. The units on all scale bars are rad/( 100 Myr).

changes sign, resulting in striking contra-rotation of the lower 1000 km of the mantle relative to the top 1000 km of the mantle. We also note that net rotation in the mantle appears to be dominantly in the 2 direction (i.e., along the Earth’s rotation axis).

I t is interesting to note that the net rotation in the mantle, arising from the degree 1 toroidal flow, also carries a nonvanishing angular momentum. The contribution to the Earth’s angular momentum, from flow in the mantle, is given by the general expression

(166)

in which v is the mantle flow velocity and V denotes integration over the volume of the mantle. It may be shown that the angular momentum in Eq. (166) will vanish for all poloidal flows and all toroidal flows, except for the degree 1 toroidal flow [we assume that the density distribution is essentially spherically symmetric:


-0.4 0.0 0.4 0.8

Tad/( 100 Myr) FIG. 43. The depth variation of the net rotation in the mantle, corresponding to the degree t = 1

toroidal flow in the mantle, as given by Eq. (79). The degree I toroidal flow, calculated according to Eq. (144). is due to the interaction of degree 1-2 density perturbations [derived from model SHS/WM13 using 8 In p/6 In v, in Fig. 61 with the 3D viscosity r)(r, 8, q 5 ) / ~ ( , = 1 + u(B, 4). where v(B, 4) is the degree 1-5 viscosity variation in Fig. 25. The reference viscosity value is q,, = lozi Pas.

p = p o ( r ) ] . The angular momentum associated with the degree 1 toroidal flow may be shown to be (algebraic details omitted here):

Equation (167) provides the basis for the following definition of the “effective” average rotation rate of the mantle:

On the basis of the rotation rates in Fig. 43, we thus obtain

0.15 rad -0,004 rad 100 Myr ’

(4 = F r ’ ( W J =

-0.52 rad 100 Myr ‘

(4 =


5. CONCLUSION

In this chapter we have considered three different approaches to the treatment of the effects of lateral viscosity variations in the Earth’s mantle. The first and simplest approach is based on the assumption that the tectonic plates are the most extreme, and hence, only important manifestation of lateral rheology variations. It is therefore possible to characterize the effect of surface plates only in terms of their geometry and thus circumvent on explicit treatment of lateral viscosity variations. The requirement that instantaneous plate motions be described by rigid-body rotations imposes a partitioning of the internal density perturbations 6 p ( r , 8, 4) into two orthogonal components; 6 p = S p + 6p. The component 6p(r, 8, 4) produces mantle flow whose surface pattern exactly corresponds to a permitted combination of rigid-body rotation of the individual plates. The component Sp( r, 0, 4) instead produces surface flow that is completely inconsistent with rigid-body rotations of the plates and therefore “sees” a no-slip surface boundary (i.e., the plates are ‘‘locked’’ and immobile). We emphasized that this partitioning of internal buoyancy sources renders the interpretation of observed plate motions completely nonunique. Clearly, the observed plate motions reflect only the buoyancy-induced flow arising from 6p and tell us nothing about the internal flow excited by Sp. An important illustration of the consequences of this nonuniqueness was provided in Fig. 14, where we showed that plate motions, arising from buoyancy sources only beneath midoceanic ridges or only confined to regions beneath subduction zones, provide equally good matches to the observed plate motions. This geometric treatment of the surface plates also provided us with an explicit expression [Eqs. (34)-(35)j for the coupling of the poloidal and toroidal components of the plate motions. This relationship demonstrates that the equipartitioning of kinetic energy between poloidal and toroidal plate velocities depends on the plate geometries.

Although this geometric treatment of surface plates is attractive, owing to its ease of implementation, it entirely ignores the flow dynamics and the rheological properties of the lithosphere. A lithospheric plate cannot be wholly rigid, as assumed in the geometric treatment, and it must become progressively more deformable (i.e., less viscous on average) with increasing depth. In addition, the zones of weakness at plate boundaries must have some finite horizontal extent rather than being mathematically infinitesimal as is implicit in the geometric treatment.

If we assume that the lateral variations of viscosity in the sublithospheric mantle are negligible compared to those in the lithosphere, it is then possible to formulate an inverse problem for the lateral variations of viscosity in the lithosphere that are consistent with the observed tectonic plate velocities. The inferred viscosity variations in the lithosphere (Fig. 15) clearly show an overall pattern of weak plate boundaries and strong plate interiors. Our inferences suggest, however, that not


all plate boundaries are equally weak and that a purely thermal interpretation may not be valid. The East Pacific rise is a zone of rapid divergence with very high surface heat flux, and therefore we might expect that it would be a prominent low- viscosity region. Similarly, the subduction zones in the northwest Pacific are presumably colder than average and therefore should be stiffer than average (again, assuming a purely thermal origin for viscosity variations). Instead we observed that the lithosphere is softer than average in the subduction zones and that the most prominent zone of weakness appears to be on the plate boundary in western North America. This suggests that nonlinear effects (i.e., strain-induced softening) are important for understanding lateral variations of viscosity in the lithosphere.

The lateral variations of effective viscosity may indeed be strongest in the lithosphere but the global-scale models of seismic velocity heterogeneity in the mantle suggest that very-long-wavelength variations of viscosity will be significant throughout the mantle. We have shown that a physically elegant, and mathematically efficient, procedure for modeling the effects of 3D viscosity variations in spherical fluid shells is provided by a variational treatment of the momentum conservation equation. This variational treatment is based on the result (proved in Section 4.1) that the difference between the rate of viscous dissipation of energy and the rate of energy released by buoyancy is an absolute minimum. This minimum principle yields a quasianalytic formulation of buoyancy-induced flow that explicitly describes the coupling of both poloidal and toroidal flows to internal density perturbations by lateral viscosity variations. We have employed this variational formulation to show that long-wavelength lateral viscosity variations in the mantle have a strong effect on the buoyancy-induced flow field. Toroidal flow is excited with a strength that is less than, but comparable to, the poloidal How. We find that the net rotation of the lithosphere, given by absolute plate-motion models employing the hotspot reference frame, is readily explained by the interaction of very-long-wavelength buoyancy-induced flow with lateral viscosity variations that are correlated to the seismically inferred mantle heterogeneity.

An important outcome of our investigation of the effects of 3D viscosity heterogeneity is the observation that lateral viscosity variations have a rather small effect on flow-induced boundary topography and hence the nonhydrostatic geoid. We find that even if the viscosity varies laterally by two orders of magnitude, the relative difference between the geoid predicted with and without these lateral viscosity variations is only of order 12%. We conclude from this that the nonhydrostatic geoid is overwhelmingly sensitive to radial viscosity variations and is much less sensitive to lateral viscosity variations. This result is very important because it suggests that previous geoid-derived inferences of radial viscosity variation in the mantle, based on the seismically inferred density models, will not be biased by the neglect of lateral viscosity variations. A 12% error is essentially negligible, when compared to the much larger uncertainties in the current seismic inferences of 3D mantle heterogeneity.


ACKNOWLEDGMENTS

This work represents the result of a long-term research effort spanning the period 1989-1992. A.M.F. i s grateful for the financial support provided by a Canadian NSERC postdoctoral fellowship and for the continued support through NSERC grant A9627 to W.R.P. Support at Harvard University has been provided by NSF grants EAR90-05013 and EAR92-05390. We thank two anonymous re- viewers for constructive comments and suggestions. A.M.F. also wishes to thank Adam Dziewonski, Rick O'Connell, Robert Woodward. and Wei-jia Su for useful discussions.

APPENDIX I. HORIZONTAL GRADIENTS OF SPHERICAL HARMONIC FUNCTIONS

We derive here explicit expressions for the following horizontal gradients of the scalar spherical harmonic functions Y;n(O, 4):

a i a as sin 6 a+ v,y;l'(e, 4) = B - YW, 4) + 4 - - y;n(e, 4) (1.1)

AY;(e, 4) = P x v,Y;(e, 4)

The operations V, (i.e., horizontal gradient on a unit sphere) and A [ - I A is the quantum-mechanical angular momentum operator (e.g. Backus, 1958)l appear repeatedly in formulations of elasticity and hydrodynamic theory in spherical geometry. Rather unwieldy expressions for (1.1) and (1.2) may be found in Morse and Feshbach (1953, p. 1899). We shall find it most useful to obtain expressions for (I. 1) and (1.2) in terms of a special combination of fixed Cartesian basis vectors. The expressions derived below will be employed often in the main text and in the following Appendixes.

From Edmonds (1960, Eq. [5.9.17]) we obtain the result

in which we have two special cases of general vector spherical harmonics (Ed- monds, 1960, Eq. [5.9.10]) which we write as

I

YT. = C ~ : - q ( n m - q; I q I em)&,,, (1.4)

in which 8, are the unit polarization vectors (Edmonds, 1960, Eq. [5.9.4]) defined as

q = - ,


where L = m, f, ..f, i are the unit basis vectors in a rectangular Cartesian frame of reference, and the (n m - q; 1 q 1 e m ) are the Clebsch-Gordon coefficients that define the coupling between degree e= 1 and degree e=n. Using (1.4) we then have that

yye,, = yy:,I(e + I m + I ; 1 - 1 I em)&-, + ~ : , , ( e + 1 m; 1 0 1 em)i?,

+ Yy;,l(e + 1 m - I ; I 1 I &n)&,,

(1.6)

and

Y:( , = yy+;(e - 1 m + 1 ; 1 - 1 I e m ) & - ,

+ (e - 1 m ; I o I em)&, (1.7)

+ ry:,l(e - 1 m - I ; 1 I I em)&, ,

Explicit expressions for the Clebsch-Gordon coefficients in (1.6) and (1.7) may be found in Condon and Shortley (1963, p. 76). Inserting (1.6) and (1.7) into (1.3) we obtain

where

] ' I 2 and (C + m + 2)(e + m + 1) c y = [ ( 2 e + 1)(2e + 3)

(e - m + I)([ + m + 1) by = [ ( 2 e + 1)(2e + 3 )

Since A = LL, where L is the quantum-mechanical angular momentum opera-

(1.10)

tor, we then obtain from Edmonds (1960, Eq. [5.9.14)) the result

A Y: = iv- YTf.


Employing (1.4), we write (1.10) as

AYT = L V ' ~ [ Y Y + ' ( t m + I ; 1 - 1 I em)&, (I. 1 1)

+ ~ ; ( t m ; 1 o I em);,, + Y : - ' ( t m - I ; I 1 I €rn)i?,]

Explicit expressions for the Clebsch-Gordon coefficients in (I. 11) may again be found in Condon and Shortley (1963, p. 76) and we finally write (I. 1 1) as

where

a;" = [ ( P - m ) ( e + m + l)]I'*. (1.13)

For completeness we also derive the expression for 3 Y;l(O, 4) [equivalent to the vector harmonic P,,,,,(B, +) in Morse and Feshbach (1953, p. 1898)1 where 3 is the unit vector along the radius. From Edmonds (1960, Eq. [5.9.16]) we have

(I. 14)

Using (1.6) and (1.7), and following the derivation which led to (1.8), we obtain

- e- I ( - c y y y ; ; + c - c I - I y + I i y y = - fl ( - 1 6 - 1 1

+ d,(b;'Y'C',, + b y - ' Y y - , ) (1.15)

The complex basis vectors 2, in (1.8), (I. 12) and (I. 15) have the following useful

(I. 16)

orthogonality property (Edmonds, 1960, Eq. [5.9.7]):

d $ . d', = a,,., where the asterisk denotes complex conjugation.

Expressions (1.8) and (1.12) are particularly useful when V,Y;" and AY;" are integrated or differentiated over a spatial domain because the basis vectors d , are constant and therefore unaffected by these operations.

APPENDIX 11. SPHERICAL HARMONIC COUPLING RULES

We shall summarize here the principal results, concerning the coupling of two scalar spherical harmonics, which will be often employed in the main text. We


shall also describe a technique, based on the elegant formalism of LeBlanc (1986, 1987), which permits the direct and rapid numerical computation of any given spherical harmonic coupling coefficient.

From the classical theory of angular momentum coupling in quantum mechanics we have the following basic result ( e g . Edmonds, 1960, Eq. [4.6.51):

where

(11.2)

and the asterisk denotes complex conjugation. The symbols in parentheses in (11.2) are the 3-j symbols of Wigner, which are described extensively in Edmonds (1960, pp. 45-50). The expressions in (11. I ) and (11.2) differ slightly from that in Edmonds because the spherical harmonic basis functions we employ (here and throughout this chapter) are normalized such that

1; i]: f l ; l ( O , 4)yS;'(O, +)* sin OdOd4 = 6,,.,,6 ,,,,.,,, >. (11.3)

The spherical harmonics we employ are therefore obtained from the y1' defined in Eq. 12.5.291 of Edmonds (1960) by setting 477 3 1.

The coupling coefficient in (11.2) may also be expressed in terms of the Clebsch-Gordon coefficients as follows:

477

(11.4)

x (C,0; P,0140)(e,nzl; C2nzzjtnzl + mz). The coupling coefficients in (11.2) will be identically zero unless the following conditions are satisfied:

IMII e , ; lmzl zz ( 2 ; ltnl e (11.5)

m , + nz, + in = 0, (11.6)

(11.7)

(11.8)

The condition (11.8) ensure\ that the sum in (11.1) is actually restricted to Y = el

A useful illustration of Eq. (11.1) is provided by the simple example in which

le, - ell G e s el + e2 , el + t2 + 8 = even.

+ e , , el + 4, - 2, el + 4, - 4 , . , . , le, - e,l.

we couple y;" and Yp ( m = 0, 2 I ) . According to ( ILI ) , we write


L ) + I r 1

If we employ expression (11.4) and the Clebsch-Gordon coefficients (e lm,; Im, I em, + m z ) tabulated in Condon and Shonley (1963, p. 76), we have

= - <j c~lll-,~ ytl:; + <,-,,I 2 e 1 y"'-' P I + I .

in which the quantities bp and cp are defined in Eq. (1.9) of Appendix I. Results (II.I0)-(11.12) are quite useful and they are employed in Appendix 111.

In Section 4 of the chapter we make extensive use of the generalized scalar spherical harmonics, introduced in the geophysical literature by Phinney and Bur- ridge (1973). We shall therefore summarize the symmetry properties and coupling relations needed in Section 4. The generalized associated Legendre functions PP(c9) in Phinney and Burridge (1973) are identical to the matrix elements of finite rotation d$,!,(O) in Edmonds (1960, pp. 55-58). The generalized spherical harmonic functions

Py(e, 4 ) = d $ ; ( o ) ~ + (11.13)

are related to the complete rotation-matrix elements D$,!,(4, 8, y ) in Edmonds (1960, Eq. [4.1. lo]) as follows:

Pp(e, $) = D $ ; J ~ , 8, 0). (11.14)

From the properties of d$,!,(8) (see Eqs. [4.1.25], [4.2.61, and [2.5.29] in Ed- monds) we may readily verify that


(11.15)

where the YF in (11.15) are the usual scalar spherical harmonics in (11.3). Owing to the result (11.15), we shall find it useful to employ the modified generalized spherical harmonics Y P , which are defined as

(11.16) Y;ym(e, 4 ) = ~ ' m Pp(e, 4 ) = g m D$,!,(4, 8, 0).

This definition ensures that YFm reduces, when N = 0, to the usual YF. In addition, this definition also leads to the following simplified orthonormality relation (obtained by employing (11.16) in Eq. 14.6.11 of Edmonds):

1'" [ Ytml((e, 4)* Y$n2(8, 4) sin 8d8d4 = Se,.e2Sm,.m,. (11.17)

From the symmetry relations of the functions D$A in (11.16) we have [Eq. [4.2.7] in Edmonds (1960)l the following useful result:

4T 0

YF"(8, 4)* = ( - I ) N + " ' Y C N - ' q B , 4). (11.18)

If we now employ the definition of the Y t m , in (11.16), in Eq. r4.3.21 of Edmonds, we then obtain the following important generalized coupling relation:

(11.19)

The Wigner 3-j symbols in (11.19) will be identically zero unless the following conditions are satisfied:

(11.20)

(11.21)

(11.22)

N , + N2 + N = 0, (11.23)

le, - e,l 4 e 4 e, + e2 . (11.24)

It is worth emphasizing that condition (11.8) does not apply to (11.19) and thus the sum over e must include all integer values between the limits specified in


(11.24). The Wigner 3-j symbols in (11.2) and (11.19) have two important symmetries [Eqs. [3.7.5] and [3.7.6] in Edmonds (1960)], which are exploited in Section 4:

, + ' . + l ' ( el ( 2 6 ) = ( e2 el "') = . . . - - ( e3 (2 e l ) m, m , m 3 m2 m, m3 m, m, m, ( - 1)'

(11.26)

Relation (11.26) is simply an expression of the result that an odd number of permutations of the columns is equivalent to multiplication by ( - 1 ) ( 1 + ~ 2 + ' 3 .

The practical application of the theory developed in Sections 3 and 4 obviously depends on the ability to rapidly and accurately calculate the value of any given Wigner 3- j symbol. We shall now describe an efficient computational scheme we have implemented, based on the elegant group-theoretical formulation developed by LeBlanc (1986, 1987). LeBlanc has shown that any Wigner 3- j symbol, and its associated Clebsch-Gordon coefficient, may be directly evaluated using the following closed-form expression:

in which

J = e , + e, + e , , k , = J - 2e,, s, = t , + m,, ( 1 6 i 3) d, = e, - m,, tn2 + m 2 + m, = 0,

and where the functional FA,( . . .) has a binomial-type expansion:

(11.27)

(11.28)

(11.29)

in which [xIi = (x)(x - I ) (x - 2) 9 . + (x - i + 1 ) is a lowering factorial with


[XI,, 1 . The algebraically transparent expressions (11.27)-(11.29) should be con- trasted with the corresponding expression in Eq. [3.6.11] of Edmonds (1960), which involves a complicated sum over an unspecified range. Expressions (11.27)-(11.29) may be directly implemented, in a straightforward manner, with only a few lines of computer code. The resulting numerical scheme, which directly generates the value of any given Wigner 3-j symbol, is found to be very efficient and stable. The efficiency of this scheme may be significantly improved by taking maximum advantage of the symmetry properties [i.e., Eqs. (11.25) and (II.26)] of the Wigner 3-j symbols.

As discussed in LxBlanc (1986), the Wigner 3-j symbol possesses all the symmetries of its Regge symbol counterpart defined by

I- 1

(11.30)

The Regge symbol in (11.30) is invariant under all even permutations of its rows and columns, is invariant under transposition, and is multiplied by ( - for all odd permutations of its rows and columns. Such symmetries include the important relations (11.25) and (11.26). We may search the Regge symbol in (11.30) for the element with the smallest value and, by row and/or column interchanges, we translate this element to the bottom right corner initially occupied by k , . With each row and/or column interchange we multiply by ( - l ) J , and we also redefine the values of all the symbols d,, s,, and k , in (11.27). By this procedure we obtain the smallest possible value for the upper summation limit in (11.29) and thus reduce the computational effort. Expression (11.27) now provides us with a very efficient method for calculating individual Wigner 3-j symbols and we therefore avoid the usual methods based on the recursion relations provided, for example, in Edmonds ( 1960).

APPENDIX 111. ANALYTIC HARMONIC DECOMPOSITION OF HORIZONTAL DIVERGENCE A N D RADIAL VORTICITY

We derive here analytic expressions for the spherical harmonic coefficients of the horizontal divergence and radial vorticity of the surface velocity field of N rigid plates. The starting point for the derivation is the set of equations (7), (9), ( 1 l ) , and (12), in Section 2.1 of the main text.

Introducing the spherical polar representation of the Cartesian coordinates x, in Eqs. (7) and (9), we have

x , = a sin 6, cos 4 , x? = ci sin 8 sin q5q x , = a cos 8. (111.1)


Employing (111.1) in Eqs. (7) and (9) yields the following result:

U R, = - [ W : Y ; ( e , 4 ) + w;yp(e, 4 ) + ~ L Y ; ~ ( o , 441, (111.2) v3

where

w; = w; - wg (111.3)

and

where

By virtue of Eq. (1.12) in Appendix I we obtain, on the basis of (111.2), the following:

- & , ( W & Y I ' + w:Yy)].

The last term on the right-hand side of Eq. (12) is, by virtue of (111.4),

(111.7)

We may similarly show, on the basis of (111.2), that

Employing result (1.8) of Appendix I, we have the following harmonic decomposition of the horizontal gradient of the plate function in Eq. (1 1):

POLOLDAL-TOROIDAL COUPLING IN MANTLE FLOW 103

(111.9)

If we now employ result (I. 12) in Appendix 1, we also obtain the following result:

AH,(e, 4) = A C (Hi)FY;?(8, 4) 1 .,I,

(111.10)

Combining results (111.6) and (111.9) we then obtain


+ (-mi Y i l + w ; YI)(B,)? (111.1 1)

where

Employing results (II.lO)-(II.12) in Appendix 11, we obtain from (111.1 1 ) the following:

(111.14)

From result (111.14) we immediately see that the spherical harmonic coefficients (V,, . v)/l of the plate divergence in Eq. (1 1) of the main text are given by

where


(111.16)

We may also express the harmonic coefficients of the horizontal divergence in terms of the Cartesian components of the plate rotation vectors. Employing result (111.15) and (111.16) and the definitions (111.3), we obtain

N - I

(V, . v)r = c [ (S ; ) ; l (W\ - w;") + (Si)l ' (wi - UP) ,=I (111.17)

+ (S:)?Tw: - w91,

where

L (s;);" = - [(CL)? - (C:)?'], 2

(111.18) 1

2 (sip = -- [(C'L);' + (C!+)i"],

L (p ),,, = - (C' )n,. 2 I"

3 t

We now derive analytic expressions for the radial vorticity field. Combining

1 1

results (111.6), (IIU), and (111.10) we obtain the following:

-- AH, . AR, - - H,A2CI, U U


Applying results (11.10)-(11.12) of Appendix I1 to (111.19), we obtain the following, after some manipulations:

1 1 a a

-- AH^. ASZ~ - - H,A2SZ,

where

( E l - ) ? = (H,)&?", ' , [~Fu;J~- ' + c T ( ~ - I ) ]

+ (H,)??,'(l - 8 , , , ) [ b ~ - l ar!"-' - c-"-' G I (m - ] ) I ,

(E;)? = (Hl)?+l[4br - crar+l - crma~+:1 (111.21)

+ (H,)r-l(l - 6eo)[4br-l + C F - " ~ - ~ U ? - ~ + C ~ - , ' U L - " ~ ] ,

(E: )? = (Hl)~~,'[b~~~+-l' - + l)]

+ (H,)?II'(l - 8(ll)[b~-lu~:; + cF:ll(m + l)]. Results (111.7), (111.20), and (111.21) show that the spherical harmonic coefficients (3 . V X v ) ~ of the radial vorticity in Eq. (12) of the main text are given by

(111.22)

N - I

2 + - s y I ( W y 8 , I + o:s,,, + W N 6 , . _ , ) . ~

We may also express the harmonic coefficients of the radial vorticity in terms of the Cartesian components of the plate rotation vectors. Employing result (111.22) and the definitions (111.3) and (IIIS), we obtain

N- I

(P . V X v)? = 2 [(R\)~(w; - w ? ) + (R,$)?(W$ - W Y ) I = I (111.23)

+ (RC)?(W - ~ S i ) l + C 1

N;"w,N, ,=I

where 1 2

(R;)? = - [(E'-)T - (K)?1,

(111.24)


(111.25)

2 NI;' = - 6,". v3

APPENDIX IV. MOMENTUM CONSERVATION IN A MEDIUM WITH 3D VISCOSITY VARIATIONS

The equation of momentum conservation, in Cartesian tensor form, for quasistatic deformation in a continuum is

(where Ti = T,j is the stress tensor, p is the density, and a,c#J is the body force per unit mass. The inertial force pdui/dt that normally appears on the right-hand side of (IV. 1) is neglected because the viscosity of the medium (e.g., the Earth's mantle), and hence its Prandtl number, is extremely large. The total stress Tij may be written as the sum of its deviatoric and spherical components as follows

where

and

(IV.3)

(IV.4)

As indicated in Eqs. (1) and (2) of the main text, the deviatoric stress T ~ , in a solid medium creeping with deviatoric strain rate ElI is given by

rv = 2r]KI = rl(d,u, + a p , - QakukS,,), (IV.5)

in which r] is the effective viscosity [see Eq. (3) of the text] of the solid medium. From Eqs. (IV.2) and (IV.5), we obtain

a,?, = -alp + h w a k u k ) - %~,T%%u,) + @,aIu,

+ (a,q)(aIu,) + al(uIa,q) - u ,a ,m) .


Employing vector notation, we may rewrite this last expression as

v * T = - V P + i qV(V . u) - j(V . u)Vq + r]v%l (Iv,6)

+ (Vr] . V)u + V(u . VV) - (u ' V)VV.

Employing the vector calculus identity

V(A * B) = (A . V)B + (B V)A + A X (V X B) + B X (V X A),

we may rewrite (1V.6) as

(IV.7) v . T = - V P + Q q V ( V . u) - I < V . u)Vv

+ 7 p u + 2(V?7. V)u + vq x (V x u).

Combining (IV.l) and (IV.7) now yields

-VP, + $,V(V . u) + 7 p U - a(v . u)Vq

+ 2(V7 . V)U + V~,I X (V X U) + p,V& + poV+, = 0, (IV.8)

in which we have removed the expression

- V P , + poVC#Jo = 0

for the hydrostatic equilibrium state and f l (~Pl /Pl) \ << I) , pl((pllpl)l << I ) , C#JI(I~I/&l << 1) are respectively the perturbations to the pressure, density, and gravitational potential associated with the flow u.

We may formally eliminate the nonhydrostatic pressure P I by taking the curl of (IV. 8) :

qvyv x u) + vr] x v2u + vr] x V(V ' u) + 2 v x [(Vq . V)u]

+ V X [Vr] X (V X u)] + A (P? - + PO;l) - = 0, (1V.9)

in which g,, = -d&/dr, Po = dpJdrS and A = r X V. The scalar equation describing (in the limit of no lateral viscosity variations) the poloidal flow is obtained by applying the operator A . (i.e., r . V x ) to Eq. (IV.9):

1 l a qV2 V2ru, - - - (r2V . u) + [ ( V 2 ~ ) + Vq . V][Vzru, - 2(V . u)] [ r a r

-rljV*(V . u) - A - [Vq x V(V . u)] - V(r7j) . Vzu

- Aq . V2(V X U) - 2A V X [(Vv . V)U]

- A + ( P F Po; l ) V X [Vv X (V X u)] = R2 - + - ,

(IV.10)


in which u,- = F - u, = dT/dr, A2 = A A is the horizontal Laplacian operator (Backus, 1958). In deriving Eq. (IV. 10) we have made use of the vector calculus identities

V X (A X B) = B . VA - A . VB + A V . B - BV - A ,

r . VZA = V 2 r . A - 2 V . A,

and

r . (A a VB) = A . V(r . B) - A . B.

A comparison of Eq. (IV.10) (when V . u = 0) with the corresponding Eq. (2) in Stewart (1992) shows that the latter is incomplete and incorrect. The existence of lateral viscosity variations ensures that buoyancy forces [on the right-hand side of Eq. (IV.lO)] directly excite toroidal flow, in addition to poloidal flow, and this toroidal flow is implicitly described by the last four terms on the left-hand side of Eq. (IV.10). The scalar equation describing (in the limit of no lateral viscosity variations) the toroidal flow is obtained by applying the operator r . to Eq. (IV.9):

qV2(A . U) + AT . V2u + Aq . V(V . U) + 2A * [(Vq . V)U]

+ A . [Vq x (V X u)] = 0. (IV.11)

APPENDIX V. VISCOUS STRESS ACTING ON A N UNDULATING SURFACE

We consider here a surface, in a material continuum, obtained by distorting a spherical surface r = c by a small amount S c ( 8 , 4 ) such that 16c(S, 4) lc l << 1. The equation describing this surface may be written as

(V. 1)

in whichf(r, 8, 4 ) = r - Sc(8, 4) . It is clear from (V.l) that Vfwill be normal to this surface and thus we have

.f(r, 8, 4) = c,

1 ii = Vf = 3 - - v, S C ( B , I$), W.2)

in which ii is the vector normal to the undulating surface, Pis the unit radial vector, and V, is the horizontal gradient operator (see Appendix I). The normal vector ii is, to within order (6c/c(?, a unit vector. The stress vector t acting on each element of the undulating surface is given by

C

t(r = c + 6c) = ii ' 1 (c + 6c), W.3)

(c + 6c) is the stress tensor evaluated at r = c + 6c. In an effectively in which viscous continuum we have (see Appendix IV)


- T = - P I + 2 @ , W.4)

in which P is the pressure, r] is the dynamic viscosity, and & is the deviatoric strain-rate tensor. Combining (V.2)-(V.4), we obtain

P,, t = - ( P o + P,)P + 2q(E,,P + Erg6 + E++) + - Vl S C , (V.5)

C

in which we have ignored the second-order terms P , 6c and 6 c E. Employing the expressions for the strain-rate tensor components in spherical polar coordinates (e.g., Morse and Feshbach, 1953, p. 117), we obtain from (V.5) the following expression

P,(c + SC) t ( r = c + Sc) = - 3 [ P o + Pllr=c+ac + v, 6c

C

in which Po and P , are respectively the hydrostatic and nonhydrostatic pressure, V H is the horizontal gradient operator on the spherical surface r = C, and uH = uB d + ud 4 is the horizontal flow velocity. The hydrostatic pressure gradient is df , ld r = -pogo, and thus we may write (V.6) as the following first-order accurate expression:

Po(r = C ) t ( r = c + Sc) = - i [ f , , + P I - pognSc],,r + V I S C

C

(V.7)

The continuity of stress across the undulating surface requires that t ( r = c + SC+) = t ( r = c + Sc-) , and thus, using (V.7), we obtain the following matching conditions:

POLOIDAL-TOROIDAL COUPLING IN MANTLE FLOW I l l

APPENDIX VI. DYNAMIC TOPOGRAPHY WITH LATERAL VISCOSITY VARIATIONS

We derive here explicit expressions for the spherical harmonic components of the dynamic surface topography 6n?, on the basis of Eq. (161) in the main text. The covariant equations of momentum conservation, obtained by Phinney and Burridge (1973, see their Eq. [4. lo]) applied to the problem of quasistatic deformation, yield the following expression:

r ; [ (To- )? + (TO')?] + 3[(T0+)7 + (TO-)?] - 2 f l l ( T + - ) r d

- O$[(T++)P' + (T- ) ; ' ] + 2p0Qf(4 , ) ; = 0, (VI.1)

in which the terms ( T @ ) T are the generalized spherical harmonic coefficients of the contravariant stress tensor

TUP = - P , e L @ + 2qEa0, (VI.2)

where e U p is the contravariant representation of 8, and Em@ is the contravariant strain-rate tensor defined in Eq. (109) of the main text. Substitution of Eqs. ( 1 10) and (1 1 1) of the main text into Eq. (VI.2) yields

(VI.3)

where (Up)? = ( U + ) ? + ( U - ) a . By virtue of the orthogonality property and coupling relation [see Eqs. (11.17) and (11.19) in Appendix 111 for generalized spherical harmonics Yp, we then obtain from (VI.3) the following:

( - I)"' r

( T + - ) a = (PI)? + - c [ s Z t ( U p > : - 2(u"):1

L-tr

x ( e 0 0 0 J ) ( 4 -rn f r n - f ) qJ" -


X [(2e + 1)(2s + 1)(2J + 1)l"'

77-1 - (UT): 2 - 2 0 - m t m - t x ( e ")( 5 [

1 - ( - I ) Y t r + J

J = [ E - > ( 2

(V1.5)

where (Win = (U+)T - ( U - ) ? . If we now substitute expressions (V1.4)-(VI.6) into Eq. (VLI), and we employ the free-slip (i.e., To- = To+ = 0), zero radial velocity (Le., Uo = 0) boundary conditions that obtain at r = a, we then find that


X 2 [(2e + 1)(2s + 1)(2J + 1)]1 '2 J-IF - r1.2

X [(26 + 1)(2s + 1)(2J + l)] lt2

1 + >

X 2 [(24 + 1)(2s + 1)(2J + 1)1"2 J=l( ~ r1.2

x ( - m t m - t )v7-1}, (VI.7)


in which we employ the notation 2 J = k , 2 to imply a sum over the terms with J = k, k + 2, k + 4, . . . , According to Eq. ( 1 19) in the main text, we may relate the flow scalars Up, UT, U0 to the corresponding poloidal and toroidal flow scalars as follows:

(UT)i" = LZRfqy, (VI.8)

PF (UO),.' = -2(a:)2 -, r

where L = m, pi" and qy are respectively the spherical harmonic coefficients of the poloidal and toroidal flow scalars. If we now substitute Eq. (VI.7) into Eq. (161) of the main text and employ (VI.8), we then find

go(pX - P; )SUT = @; - P C ) ( ~ ~ ) ? ( U ) - ( - I ) * C, 6 ( f l . ~ ) ~ *.I

at. a: zr

- ( - I)'" c 2fl;R;

in which

Pi( ( , m, S, t ) = [(2e + 1 ) ( 2 ~ + 1 ) ( 2 ~ + 1 ) y

x ( e k - k O J)( - m t m - t


In the limit of no lateral variations of viscosity, it is readily shown that Eq. (VI.9) reduces to

6a;n = d3py 3e(e + 1 ) dp? (4 I )?(a) + +--. -1 77: &,(PIT - Po') r dr go

APPENDIX VII. NONHYDROSTATIC GEOID IN A SELF-GRAVITATING MANTLE

We provide here a brief derivation of the nonhydrostatic geoid produced by buoyancy-induced flow in a self-gravitating mantle. The derivation closely follows that in Appendix A in Forte and Peltier (1987) and corrects the typographical errors present there.

The total gravitational potential perturbation due to internal density anomalies and boundary deflections is

in which G is the universal gravitational constant, Sa is the boundary perturbation of the outer surface and Apmn = 2.2 Mg/m3 is the corresponding density jump, Sb is the perturbation of the core-mantle boundary (CMB) and Apcm = 4.43 Mg/ m3 is density jump across the CMB. The term ( Uin,)y(r) is the gravitational potential due to internal density anomalies:

(VI1.2)

where r< = min (r, r') and r , = max (r, r'). As shown in Eqs. (162) and (164) of the main text, the surface topography ( S U ) ~ and CMB topography (Sb)? contains contributions (4 ,)?/go arising from self-gravitation. Equation (VII. 1) is therefore an implicit equation for (q51)$(r = a,b), which may be solved by evaluating (VII.1) at r = a and r = b and then solving the resulting two equations for (4 I p( a) and (4 I )y( b). In this manner we obtain

(VII.3)


in which

where ji = 5.52 Mglm’ is the Earth’s mean density, A;’ is the surface topography in the absence of self-gravitation [i.e., when ( 4 , ) ~ (a) = 0 in Eq. (162)], and B;’ is the CMB topography in the absence of self-gravitation [i.e., when (q51)f’(b) =

0 in Eq. (164)]. The term (41)y(a)/go in (VII.3) is, of course, the nonhydrostatic geoid.

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ADVANCES IN GEOPHYSICS, VOL. 36

SEISMOTECTONICS OF THE MEDITERRANEAN REGION

AGUST~N U D ~ A S A N D ELSA BUFORN ~eyurrniefil c$ Geciphysics Univertidaad Complrttenw

Madrid. Spain

1. INTRODUCTION

Seismotectonics is a term generally applied to the study of earthquake occurrence and characteristics and its relation to the tectonics of a particular region and the overall dynamics of the Earth’s crust. The word seismotectonics was first used by Sieberg in 1923 as the title of a world map in which seismic and tectonic features were represented together ( “seismisch tektonischen Uberschichkarte”). Sieberg referred also to the relations between earthquakes and tectonics as seismotectonic relations ( “seismisch-tektonischen Verhaltnisse”). The term is applied today to studies and maps in which these two types of data are analyzed together. There is no complete agreement, however, about the type of information to be used in seismotectonic studies and the form in which seismic, tectonic, and geophysical and geologic data may be integrated. Other terms are also used for the same purpose, such as geod,ynnmics and active tectonics.

For large areas, tectonic considerations, used in seismotectonic analyses, are derived from the accepted theories about the general processes active in the Earth’s crust. Historically, these theories can be divided into three types. The earliest tectonic theories were based, basically, on vertical movements of crustal blocks, finally related to the contraction of the Earth as a result of its cooling process. After Wegener’s publication of the continental-drift theory in 19 12, large horizontal motions of continental blocks were considered. However, geophysical objections to the acceptance of continental drift complicated its early use in the interpretation of seismic data. From the proposal of plate tectonics theory, about 1960, with its early use of the results from seismicity and focal mechanism of earthquakes, this theory is generally used as the guideline for seismotectonic studies.

Seismic data constitute the basic information used in seismotectonic studies. In the early times, this information was reduced to epicenter distribution and size of earthquakes to which depths of foci and seismic energy flux were later added. With the development of fault-plane solutions in the 1930s, this type of data has become a key tool for seismotectonic analysis. The orientation of the focal mechanism is given by the slip vector or the principal axes of stress (pressure and tension

121 Copppht .C 1W4 by Academic F‘ress. Inc.

All rights of reproduction 111 any form r e w v r d .

122 AGUSTfN UDfAS AND ELlSA BUFORN

AFRICA

FIG. I . Mediterranean region: geographic names.

axes). These are usually represented on maps by means of their horizontal projections. Fault-plane solutions are also represented on maps by the projection of the focal sphere with different colors or shades for compression and dilation quadrants. With further developments of source mechanism studies, more source information is included in some cases, such as seismic moments, slip rates, stress drops, and source dimensions.

The Mediterranean region (Fig. 1) is formed by the Mediterranean basin and the neighboring areas, from the strait of Gibraltar to the Anatolian peninsula and the Caucasus mountains. The Mediterranean basin is formed by a series of separate basins, which are, from west to east: Alboran, Balearic, Ligurian, Tyrrhenian, Adriatic, Ionian, and Aegean. Alpine orogenies are located on both sides of the Mediterranean basin, ranging from west to east as follows: at the western end, Riff and Tellian Atlas to the south and Betics and Pyrenees to the north; the Apennines mountains along the Italian peninsula and to its northern part the mountain belt of the Alps, which occupies the southern part of central Europe; the eastern end of the Alpine mountain chain bends southward along the coast of the Balkans by the Dinaric Alps and continues eastward by the Carpathian mountains, which bend to the south, enclosing the Transylvania-Panonian basin, and continue with the Bal- kan mountains. At the eastern end of the Mediterranean basin are located the Hel- lenic region and the Anatolian peninsula.

The Mediterranean region is seismically active, with varying degrees of activity from place to place (Fig. 2). The most active regions are, in the following order:

z 9 z 0

(C

I

.. vi

I n

I I

I

FIG. 2. Seismicity of the Mediterranean region. Period 1962- 1985. magnitudes ) 4 (US. Geological Survey Hypocenter data file). Magnitudes: (O), 4; ( 0). 5; (X), 6; ( + L 7: (0). 8.

1 24 AGUSTIN UDfAS AND ELISA BUFORN

Hellenics, Anatolian, Italy and the Alps, Carpathians and Dinarics, northern Af- rica and south Spain, and Pyrenees. Along this entire region seismic activity results in a continuous occurrence of earthquakes of low to moderate magnitude and of large (high-magnitude) earthquakes separated by longer time intervals. Large destructive earthquakes have occurred in practically all parts of the active region at different times. Most earthquakes are of shallow depth with four regions of deep and intermediate-depth foci at Hellenic, Carpathian, Sicily -Calabrian and Retics-Riff regions. Historical seismicity is well documented for the occurrence of large earthquakes, going back in some locations to the third and fourth centu- ries B.C. The “instrumental period” started at the end of the nineteenth century, and a significant number of seismographic stations was installed very early.

In this review chapter, we will proceed in chronologic order from the oldest to most recent studies. First, we will review the studies that treat the whole Mediter- ranean region, separating three periods: 1885- 1970, studies before plate tectonic theory; 1970- 1986, interpretations based on plate tectonics; and 1986- 1993, recent seismotectonic studies. Second, we will review detailed particular studies, dividing the Mediterranean region into three parts: Azores to Tunisia, Italy and the Alpine region, and Hellenics and Anatolia. There is very abundant literature on the tectonics, seismicity, and seismotectonics of the whole Mediterranean region and on each particular area. We will review more extensively the general papers on the whole Mediterranean region. We have selected studies that we consider to be most representative of each of the three particular areas. Purely tectonic studies are cited only when they have a direct bearing on seismotectonic problems.

2. BEFORE PLATE TECTONICS (1885- 1970)

2.1. Early Tectonic Interpretations

The first tectonic framework for the whole Mediterranean region was published by Suess (1885-1905) (Fig. 3). Many of the features of modern studies were already present in this early work. Two large stable blocks are recognized in Eu- rope and Africa ( “Russische Tafel” and “Afrikanische Fafelland”) and between them, a continuous region of Tertiary foldings extending from west to east and limited to the north by the mountain chains of the Betics, Pyrenees, Alps, Carpa- thians, Balkans, and Caucasus and to the south by those of the Atlas, Apennines, Dinarics, Hellenics, and Taurides. Suess emphasized the need to establish the relations between the different mountain systems that cannot be understood separately. He also recognized the presence in the orogenic belts of arc structures ( “Faltenbogen”) that governs the whole tectonics of Eurasia. Thus we see in Fig. 3 the arcs of the Alps, Carpathians, Dinarides, and especially the arc of Sicily-Calabria that links the Atlas and Apennine chains and the Hellenic and

FIG. 3. First tectonic map for the Mediterranean region by S u e s (1885- 1905).

I26 ACiUSTh UDiAS AND ELISA BUFORN

Cyprus arcs. The southern limit of the folding region coincides almost exactly with recently proposed locations for the plate boundary between Europe and Africa. Suess gave also the name of Tethys to the long seaway between Europe and Indo-Africa, bordered by the various geosynclines from which the Alpine- Himalayan mountain belts were formed. Suess’ ideas were further developed by Neumayr (Neumayr and Suess, 1920), who explained the tectonic processes underlying the formation of mountains chains in terms of vertical movements of uplifts and sinking of crustal blocks producing their over- and underthrusting. Since Neumayr and Suess adhered to the contraction theory, these vertical displacements were finally related to the cooling process in the Earth interior.

In 1912, Wegener initially proposed his theory of continental drift (Wegener, 1922) in which the crustal continental rigid Sial blocks, breaking out from one original continent named fungeu, were allowed to move horizontally along great distances through the plastic Sima material of the Earth mantle to their present locations. However, he did not explain the details of the development of the Medi- terranean region. Taylor ( I 9 lo), a precursor of Wegener in proposing horizontal motions of the continents, had advanced a general southward motion of Eurasia over the Mediterranean as cause of the Alpine orogeny. Du Toit (1937), an ardent follower of continental-drift theory, proposed the separation of the original Pangea into two continents-Laurasia to the north and Gondwana to the south-by the Tethys sea, at least from Paleozoic times. In his interpretation, the Mediterranean was formed by the closing of the Tethys sea and the Alpine-Himalayan orogenic belt by the collision of parts of Gondwana with Laurasia. Argand (1924) presented a very detailed study of the formation of the Mediterranean region in the framework of continental drift. Prior to Tertiary times, Eurasia and Gondwana (Africa) were separated by the Tethys (Fig. 4). The African coast had two promontories: Italy and Arabia (the first corresponds to the Adriatic or Apulian promontory in modern terminology). The development of the Mediterranean is due first to a motion of collision between Eurasia and Africa, which produced the formation of the mountain belts of the Alpine orogeny, and subsequent motions of distensions and counterclockwise rotations of blocks like Iberia and Italy, with the opening of basins as the Balearic and Tyrrhenian. In the present situation (Fig. 3, the limits of Gondwana (Africa) are located at the northern limit of the Betics, through Cor- sica and Sardinia and then through the northern limit of the Alps, Carpathians, and Balkans. The Alboran and Adriatic basins have Sialic nature, while the rest of the Mediterranean is underlaid by Sima or very thinned Sial. It may be of interest to compare Argand’s map with recent maps showing the plate boundary between Eurasia and Africa and to recognize some of the similarities.

A different model for the formation of the Mediterranean, in the framework of continental drift, was proposed by Carey ( 1958). He hypothesized that orogenic belts were initially straight and were bent during orogenic processes. Originally,

127 SEISMOTECTONICS OF THE MEDITERRANEAN REGION

FIG. 4. Formation of the Mediterranean region based on continental drift according to Argand (1924). Eurasia and Gondwana (Africa) are separated by the Tcthys sea. ((0 Ligurian promontory. ( h ) Western Alps. (c) Bohemia. ( d ) Carpdthians. ( e ) Getic. P and Q are Italy and Arabia promontories. (1 -4) Limits of continenral slopes. ( 5 ) Austro-Alpine nappes. (6) Axial zone of Tethys. (7- 1 I ) Embryonic stages of Helvetic, Carpathian. Dinarides, Apennines. and Atlas. (12) Axial zone.

they occupied the northern and southern margins of the Tethys (Fig. 6). The present situation is generated by what Carey calls the “Mediterranean shear system.” By this, he meant a sinisrral shear displacement of Europe with respect to Africa of the order of 1100 km, due to the faster eastward drifting of Africa in the opening motion of the Atlantic ocean. In the context of this shear motion, the rotation of Iberia is like that of a roller block between the two continents. Similar rotations of Corsica and Sardinia and of Italy resulted in the formation of the Balearic, Ligurian, and Tyrrhenian basins. Carey’s model was considered as very specula-

FIG 5 Pre\ent situation of the Mediterranean region according to Argand (1924). \howin& the houndary between Gondwand [darker ( 1 ) I and Eurava [lighter ( ? ) I

T f Z T H Y S

FIG. 6. Formation of the Mediterranean region based on continental drift proposed by Carey ( 1958). Situation o f the mountain belts before their bending and folding at both sides of the Tethys sea.

130 AGUSTfN UDfAS AND ELISA BUFORN

tive, especially in its global aspects related to a hypothetical expansion of the Earth. However, the rotation of blocks and other tectonic processes proposed by Carey have been shown to be precursors of modern ideas. Using paleomagnetic data, de Boer ( 1 965) concluded that the relative westward drift of the crustal units of Italy, Spain, and southern France and Corsica with respect to Eurasia was caused by the relative displacement of Gondwana with respect to meso-Europe. The Gondwana shield moved northwestward and the meso-Europe eastward, causing the formation of a dextral shear zone in the Tethys. This hypothesis is in contrast with the sinistral shear supposed by Carey (1958) and more in conso- nance with modern ideas. According to de Boer, the tectonics of the Mediterra- nean area is dominated by large wrench faults with dextral movement and counterclockwise rotations of small blocks such as Iberia, Corsica-Sardinia, and Italy.

Previous to plate tectonics, there was no agreement about the overall tectonic situation in the Mediterranean. Continental drift was not universally accepted, and there were several theories to explain the tectonic processes, involving horizontal and vertical movements of the crust (Goguel, 1952; Beloussov, 1981). This lack of agreement, in reference to the nature of tectonic processes, results in different interpretations given to the Mediterranean region. For example, Kober (1942) presented a tectonic map of Europe similar to that of Suess and explained tectonic processes in terms of orogenes and krarongenes, that is, active and stable areas. Most authors, at this time, considered the situation in the Mediterranean as a particularly complex example of a median area, that is, a region between two large stable blocks (Holmes, 1965).

2.2. Early Seismicity and Seismotectonic Maps

Earthquake activity in the Mediterranean has been documented since very early times. The first world map of earthquakes was published by Milne in 1886 and using epicenters determined instrumentally in 1503. In the early maps (Milne, 1939), the distribution of epicenters in the Mediterranean, shows very little detail (Fig. 7). Due to the existence of a greater number of seismologic stations, more epicenters are located in the Mediterranean than in either Central or South America. The lack of distinction between small and large earthquakes in these maps leads to erroneous interpretations of the relative seismic activity of these different regions.

Montessus de Ballore (1906, 1924) studied the relations between seismology and geology ( “le probleme stismico-gbologique”) and proposed, for the first time, the term seismicity ( “sCismicitC”) to define the distribution and characteristics of earthquakes of a particular region. In these relations, he stressed the importance of the seismogenic influences of tectonic features and the concept of geologic instability. Global seismicity was divided into two principal zones: the

FIG. 7. Early world map of seismicity by Milne (1939).

132 ACUSTIN UDIAS AND ELlSA BUFORN

Mediterranean- Himalayan and the circum-Pacific circles, which accounted for the greater part of global seismicity. Montessus de Ballore concluded that earthquakes are located at geosyncline regions, two of which are principal zones and coincide with the already mentioned Mediterranean-Himalayan or Alpine and circum-Pacific. The first extends from the Atlantic to the Indian Ocean and includes the Mediterranean region. In his work there is not an overall seismotectonic discussion of the whole Mediterranean, but regional studies divided into chapters on different areas: Carpathians and dependencies, southeast Europe, Alps and Pyrenees, Italy and western Mediterranean basin. Seismicity maps are shown for different active areas, representing epicenters by circles of different size according to their intensity, although no indication is given of this correspondence nor of the time interval used in each map. In some maps, intensity lines for large earthquakes and other types of geophysical data, such as gravity anomalies, are shown. For each region, the correlation between earthquake activity and geologic features is discussed. These observations will be mentioned later in the partial studies.

As has been mentioned, Sieberg (1923, 1932, 1933) was the first to use the word seismorectonics. He showed the close relations between epicenter distribution and tectonic features and pointed to the fact that earthquakes are distributed in narrow bands that coincide with tectonically active regions. Since Sieberg identified earthquakes with fractures in the Earth crust ( “Dislokationbeben”), he looked for the tectonic elements that are at their origin. Suess’ tectonic guidelines were used, with predominantly vertical motions of crustal blocks. The Mediterra- nean belt (“Gurtel von Mittelmeeren”) constitutes one of the two main seismic regions of the world; the other is the circum-Pacific belt. The whole Mediterra- nean was considered to be an active fractured region ( “Einbruchzone”) with related earthquake occurrence. Seismic activity and tectonic features are represented in a map that may be considered as the first seismotectonic map of Europe (Fig. 8). Regions with strong, weak, and no activity are identified, as well as epicenters of large earthquakes and lines of crustal slip. Locations of epicenters at sea are recognized, and the Hellenic arc is clearly depicted. Epicenter alignments and related fracture zones that separate the crustal blocks are shown in a second map (Fig. 9). The most important trend is NW-SE (northwest-southeast), with long lines that extend from North Africa to France, following the coasts of Italy, Greece, and Yugoslavia. A second trend is conjugate to this in NE-SW with shorter lines of only regional importance. These lines separate the large blocks in which the Earth crust is supposed to be divided and along which relative vertical motions of up- lifting and sinking take place. Despite its strange appearance, it is not difficult to recognize in this map some seismotectonic alignments that are proposed in modern studies. It is interesting to note that Sieberg never mentions continental drift and adheres to Suess’ vertical tectonics.

The seismicity maps of Europe (Fig. 10) of Gutenberg and Richter (1 949) may be considered as very simplified seismotectonic maps, since they include also the

SEISMOTECTONICS OF THE MEDITERRANEAN REGION I33

p.4 Zerstiirungen Schuden unschuuiich

hTv keine Beben a Seebeben J’eismische Wogen

- Verwerfungen o Ml‘ffel- 0 hob- * Weltbebenherd FIG. 8. First seismotectonic map of Europe according to Sieberg ( 1933). “Zersliirungen,” deslructive; “Schiiden,” damage: “unachadlich.” no damage: “keine Beben.” no earthquakcs; “Seebeben,” ma- rine earthquakes; “Seismische Wogen.” scismic waves; “Verwerfungen,” dislocations; “Mitrcl.” middle: “Gross,“ large; “Weltkbenhcrd,” world earthquake focus (very large earthquakc).

FIG. 9. Seismic alignments of epicenters and fracture zones in Europe by Sieberg ( I 932).

FIG. 10. Seismicity of Europe including trends of mountain ranges and axes of the gravity anomalies by Gutenberg and Richter ( 1949).

SEISMOTECTONICS OF THE MEDITERRANEAN REGION 135

trends of mountain ranges and the axes of gravity anomalies. Their main contributions were more accurate epicenter determinations and the importance given to the depth of the hypocenters. This last consideration made these authors aware of the importance of arc structures present in the Pacific, with dipping zones of earthquake foci from the surface to a depth of about 700 km. These zones have been also noticed by Benioff (1 949), who assigned its origin to very deep faults and are now given the name of "Benioff zones." In consequence, Gutenberg and Richter divided tectonics into two types: arc and block tectonics. In Europe, they found two clear arc structures-Hellenic and Sicily-Calabria-although not all the six elements they identified in the Pacific arcs are present. In the Hellenic arc, earthquakes are of only intermediate depth [ h (height) ( 200 km], while in Sicily- Calabria they reach 350 km. A third region of earthquakes of intermediate depth, between 100 and 150 km, is found in Rumania, in the sharp bend of the Carpathian mountains. They compare these shocks with those in the Hindu Kush region and find that, in both cases most of the arc features are not well developed. Cutenberg and Richter were thus the first to establish clearly, in the modern sense, the arc structures of the Hellenic, Sicily-Calabria and Carpathian regions with their tectonic and seismic characteristics. They noted also that seismic data support a westward continuation of the active belt of northern Africa into the Atlantic Ocean as far as the Azores, but not farther.

FIG. 10. Confinued.

136 AGUSTiN UDfAS AND ELlSA BUFORN

A seismotectonic map of Europe was published by Beloussov et al. (1966), incorporating both seismic and tectonic data. Seismic data used are epicenters for shallow earthquakes (!I ( 60 km), with intensity I,, ) VII (Mercalli), for the period 1700- 1963, with different symbols for macroseismic and instrumental determinations and for deep foci, earthquakes with M ) 4. Also, seismic activity was represented as contour lines based on the number of earthquakes of intensity greater than VI, for the period 1901 - 1955 in an area of 1000 km’. The tectonic elements were based on block tectonics with only relative vertical motions. The continental crust was divided into two regions: the Alpine geosyncline and the platform. Each of these regions contains areas separated according to their predominant uplift and subsidence. The oceanic crust is divided into regions of shelves and continental slopes and deep-sea depressions. Arc structures are not recognized as such, and seismic data are explained only in terms of block tectonics. The seismicity of Europe was studied in great detail by Karnik (1969, 1971). His catalog of earthquakes is divided into two parts, for the periods 1801 - 1900 and 190 1 - 1955. For the second period, two epicenter maps are drawn according to the magnitude ranges of 4. I -4.6 and 4.7-8.3. A map of seismic energy release and a map of maximum observed intensities are also given. Karnik does not discuss the correlation between seismicity and tectonic features, and describes seismicity only in terms of seismic belts and seismogenetic alignments.

2.3. Focal Mechanisms and Seismotectonics

A very important contribution to seismotectonics from the field of seismology is derived from focal mechanism studies. Source studies can be traced to very early times in seismology, but a working method for determination of fault-plane solutions was first established by Byerly (1928). From this time, many researchers started to work on the determinations of fault-plane solutions for earthquakes of different seismic regions, including those of Europe. One of the first applications of the results of focal mechanisms to seismotectonics was done by Scheidegger (1958). He recognized the difficulties found in this problem, because of the small number of fault-plane solutions available. Because of the ambiguity in selecting the fault plane from the two nodal planes of the solutions, he proposed the use of the null vector or B axis (axis defined by the intersection of the two planes), following a suggestion made by Willmore. He attached a tectonic meaning to the direction of the B axis, stating that it is normal to the tectonic motion direction of an area. Orientation of B axes to be used in tectonics, however, must not be derived from individual solutions but from a group of these axes in a statistical fashion. Later, Scheidegger ( I 964) investigated the tectonic relations of focal mechanism solutions for Europe and western Asia, using data from the catalog of fault-plane solutions compiled by Fara (1964). The results are summarized in a map in which horizontal projections of mean values of P, 7; and B axes for several


N

0 15 30 45 60" E FIG. 1 I . Directions of horizontal projections of 7: P, and Z(nu1l) axes derived from focal mechanisms of earthquakes. Blackened, P axis; empty, Z axis; broken, Taxis (Scheidegger, 1964).

regions are represented (Fig. 11). He attached greater importance to the orientation of B axes, showing that the results agree with his proposal that they must be normal to the tectonic motion direction. However, directions for P axes are correctly found to be normal to the Carpathian arc bend and to the trend of the Cau- casus mountains. In Italy, P axes are found to be normal to the Apennines, con- trary to modern results. He found no connection with tectonics in the results for Anatolia and Greece. Another early use of fault-plane solutions in Europe was made by Di Fillipo and Peronaci (1959), who plotted the projections of fault planes for deep earthquakes in Spain and Italy.

The first complete study of the seismotectonic implications of earthquake mechanisms in the Mediterranean- Alpine region was made by Constantinescu et al. (1966). They used 101 fault-plane solutions, 75 of which were determined by the authors and carefully examined the data, concluding that solutions were reliable. They represented the results, plotting on three maps, the strikes of the nodal planes and the horizontal projection of B and P axes, together with an outline of the Alpine mountain system taken from Kober (1942). This outline follows Suess' ideas and its southern margin resembles the boundary of the African plate. The plot of P axes shows a general trend with compression stresses transversal with respect to the main orientation of Alpine tectonic features, such as

138 AGUSTh UDfAS AND ELSA BUFORN

FIG. 12. Direction of horizontal components of pressure axes derived from focal mechanisms of earthquakes and tectonic alignments in the Mediterranean region (Constantinescu cf al.. 1966; reproduced with permission from Ccophys. J . R. Asrron. Soc.. 0 1966 Blackwell Scientific Publications).

mountain chains, folds, and shorelines (Fig. 12). Constantinescu et al. first presented this important conclusion that later other authors reformulated in terms of P axes that are nearly normal to plate boundaries. The authors conclude that for the whole Mediterranean- Alpine belt, the forces that have determined the geomorphology and tectonics of different areas have been of the same nature as those that are presently active at seismic foci. This conclusion is very important, since it establishes the relation between focal mechanisms of earthquakes and tectonics. The stress field derived from the focal mechanism of earthquakes in the Mediterranean area was studied by Shirokova (1967, 1972). The study comprises the Mediterranean- Asian seismic belt extending from the Iberian peninsula to Indonesia. From a compilation of focal mechanisms, pressure and tension axes are plotted separately in two maps. For the Mediterranean region, the maps show a considerate scattering in the direction of P axes, although a certain NW-SE trend appears in northern Africa and Anatolia and a N-S trend in the Caucasus (Fig. 13). Horizontal T axes are normal to the trend of the Apennines and in N-S direction in Greece, a result that will be found in all future studies. Along the whole seismic belt, compressive stresses are found to be horizontal and normal to the strike of surface structures, such as mountain chains and coastal lines. No further relation to tectonics is noted.

The relation between focal mechanisms of earthquakes and tectonics of the Mediterranean region was studied in great detail by Ritsema ( 1 957, 1969a,b). In his earliest paper (Ritsema, 1957), he compared the mechanisms of Pacific and Mediterranean earthquakes, and revealed an almost equal percentage of earthquakes with horizontal tension and pressure in the shallow zone of both seismic


FIG. 13. Distribution of ( a ) P axes and ( b ) Taxes obtained from the focal mechanisms of earthquakes in the Mediterranean region by Shirokova (1972).

belts. In the Pacific, there is a higher percentage of earthquakes with transcurrent fault type than in the Mediterranean, which seems to constitute a basic difference between the two seismic belts. In the Mediterranean, also, there is predominant horizontal shortening at intermediate-depth level and lengthening at deep level. According to Ritsema, this can be explained only by a combination of the Earth contraction theory together with the convection theory. Orientation of tension and pressure axes in Mediterranean earthquakes was also investigated by Ritsema

FIG. 14. Seismotectonic map of the Mediterranean region including motion of blocks and stress directions by Ritsema (1969b).


(1967), who plotted on different maps those corresponding to strike-slip and dip- slip mechanisms from only the most reliable solutions. From these data, he concluded that the transcurrent motion indicate an about east-west wrench movement of dextral type in Turkey and Iran and between Gibraltar and Azores. The strike-slip motion along the faults in these two areas is thus confirmed, In the whole region, however, the great majority of horizontal P axes is directed perpendicular to structural features and corresponds to reverse faulting.

Seismicity and focal mechanisms were used by Ritsema (1969a,b) in a seismotectonic synthesis for the whole Mediterranean (Fig. 14). In this synthesis, many modern ideas are shown, although no mention is made yet of plate tectonics. In north Africa and the Azores-Gibraltar line, earthquakes exhibit an about east- west transcurrent fault movement of dextral type. Italian shocks show horizontal tensions normal to the trend of the Apennines and Balkan earthquakes’ sinistral motion along SE-NW transcurrent faults. The tectonic situation is summarized in this fashion: the southern parts of the west Mediterranean area (from Gibraltar to Calabrian arc) move in an ESE direction toward the lonian sea, causing dextral motion in north Africa; horizontal tensions are present normal to the Apennines; the Calabrian arc overrides the Ionian sea, setting tensional motions in the wake of the arc in the Tyrrhenian sea; and sinistral transcurrent motions are active in the Balkan peninsula. This last motion produces a compression in the region of the Alps in NNW direction. The prevailing W-E stress field in western Europe is in line with the stresses generated by the drifting apart at the mid-Atlantic ridge and with the dextral movements along the Azores-Gibraltar line. In the east, dextral E-W strike-slip motion is present in the Anatolian fault. In Greece there is a convergence of the motions of the Balkan peninsula to the SE and of Anatolia to the SW, forcing the material downward from inside the arc (“Verschluckungs- zone”). In this tectonic scheme horizontal movements of blocks in the Mediter- ranean, derived from focal mechanism data, are indicated for the first time. These motions are in a WNW-ESE direction, contradicting the continenful-drift theory, which proposed a collision of Africa with Europe to generate the Alpine mountain system. Ritsema denies the N-S convergence in the Mediterranean and relates a Europe westward drifting motion to the W-E stresses that are generated at the mid-Atlantic ridge. Continental-drift ideas, such as those proposed by Argand (1924), were mentioned by Ritsema for the first time in an analysis of seismic data. The westward motion of the western Mediterranean area with respect to Africa, producing dextral transcurrent motion in northern Africa and sinistral transcurrent motion in the Balkans has been proved incorrect in more recent studies. The situation proposed for the Hellenic arc does not correspond yet to the concept of a subduction zone, but i t does imply a downward pushing of crustal material.

Before plate tectonics theory, tectonics was understood in terms of either predominant vertical motions of blocks, generally derived from the Earth contraction

142 AGUSTh UDfAS AND ELISA BUFORN

process, or predominant horizontal motions as postulated by continental drift. As has been mentioned, most seismotectonic studies of the Mediterranean region ad- here to the vertical motion of blocks. Continental-drift theory, as such, was not used as tectonic framework in the interpretation of seismic data. Only Ritsema ( 1969b) considered horizontal movements, but he may have been influenced by the already proposed ideas of sea-floor spreading, rather than by continental drift, although this is not mentioned explicitly.

3. PLATE TECTONICS INTERPRETATIONS (1970- 1986)

3. I . First Interpretations Based on Plate Tectonics

The main ideas of plate tectonics were developed during the 1960s. Among the different lines of research that contributed to this new tectonic synthesis, the following may be listed: accurate mapping of the sea-floor topography, measurements of the magnetic field above the sea floor, analyses of paleomagnetic data, and more accurate seismicity and focal mechanism determinations. The first application of seismologic observations to the new global tectonics was done by Isacks er ul. (1968). In this paper they analyzed, on a global scale, data from seismicity and focal mechanism of earthquakes, and showed their agreement with the expected motions at plate boundaries according to plate tectonic theory. The difficulty of applying these ideas to the problem of the Mediterranean region, where seismic activity occupies a very broad region, was acknowledged. It is mentioned, in this paper, that although, in principle, it seems reasonable to describe the tectonics of Eurasia by the interaction of blocks of lithosphere, it is not yet clear how successful this idea will be in practice, because of the large number of blocks involved. The complexity of this region was assigned to the fact that all the blocks are continents or pieces of continents. They hoped that in such an area as the Alpine belt, a consistent (but complex) tectonic pattern may yet emerge when the distribution of seismicity is well defined and a sufficiently large number of high-quality mechanism solutions is available. The first plate tectonic analysis of the Mediterranean area, from the point of view of geology, was presented by Dewey and Bird (1970). They recognized the complexity of the situation, which is explained in terms of a collision model. The kinds of structures developed by such a collision may depend to a large extent on the nature of the sedimentary sequences present on the borders of the colliding continents. For example, the main arcuate loop of the Alps resulted from a collision of an Atlantic-type continental margin with a Tethyan microcontinent (Carnic- Apulian), with a predominant thrust toward the north. At the eastern Mediterranean, the African plate is consumed in a trench system south of the Aegean arc. In this study, seismic data are not considered.

SEISMOTECTONICS OF THE MEDITERRANEAN REGION I43

The first plate tectonic interpretation of seismologic data for the Mediterranean region is due to McKenzie ( 1970, 1972). He objected to the idea that the simple concepts of plate tectonics could not be applied to continental boundaries and considered the Mediterranean region a suitable area to explore this problem. His analysis is based exclusively on seismicity and focal mechanism data. To improve their quality, preference was given to data after 1963 (installation of the WWSSN seismographic network). For events since that date, focal mechanisms were determined by this author using preferently long-period data from WWSSN stations that diminish inconsistencies of P-wave polarity data. Solutions were plotted on a map (Fig. 15) representing the projections of the lower hemisphere of the focal sphere with the compressional quadrants in dark. This type of representation, which has become now very popular in seismotectonic studies, was first used by Stauder (1968). To study plate motions from focal mechanisms, he gives more importance to the orientation of slip vectors than to the principal axes of stress.

McKenzie based his interpretation on the postulate of rigid rotations for the kinematics of lithospheric plates. He interpreted correctly, for the first time the triple junction in the Azores, with ridge nature in its three parts, corresponding to the relative motion of the plates of America, Eurasia, and Africa. In the Mediter- ranean, the principal plates involved are Africa, Eurasia, and Arabia. The situation on the boundary between Eurasia and Africa corresponds to relative rotation of the African plate about a pole at 22.7"N, 28.2"W with a rate of 0.273"lMyr (Fig. 16). According to this rotation, the motion along the boundary, from Gibral- tar to Sicily, varies from NW-SE to N-S direction. This direction of motion agrees with the results from focal mechanisms. Ritsema (1969b) had proposed a continuous right-lateral E- W motion along this line, which according to McKenzie was based on poor data and must be abandoned. The E- W strike-slip right-lateral motion occurs only at the center of the Azores-Gibraltar fault and transforms into underthrusting of Africa from the Gulf of Cadiz to Sicily, along north Africa. This motion is consistent with the location given to the pole of rotation between Africa and Eurasia.

The situation to the east of Sicily becomes more complicated. Mechanism solutions are consistent with normal faulting in central Italy and thrusting along the Balkan coast. Tentatively, McKenzie proposed a promontory in the African plate including the eastern part of Italy and the Adriatic, to the coast of the Balkan and extending to the north into the Alps (Fig. 17a). The intermediate-depth earthquakes at the Sicily-Calabria arc, down to a depth of 350 km, are explained in terms of a subducting slab. Since the strike of the slab is N-S and not E-W, i t cannot be produced by the northward motion of Africa. McKenzie suggested that the slab is a relic, and was produced by the motion of Italy and part at least of Sicily in an eastward direction with respect to Eurasia, with a velocity considerably greater than that between Africa and Europe. In the Aegean, the situation is explained in terms of a small aseisrnic plate bordered in the south by the active

I

P P

-40 -20 0 20 40 60

FIG. 15. Focal mechanisms of earthquakes in the Mediterranean region. Black quadrants represent compressions; white quadrants, dilations (McKenzie, 1972; reproduced with permission from Geophvs. J. R. Asrron. Soc.. 0 1972 Blackwell Scientific Publications).

61

5(

4( - R

3'

2 -40 -20 0 20 ' 40 60

FIG. 16. Sketch of plate boundaries and motions. Extensional boundaries (ridges) are shown as double lines; transform faults, as single heavy lines; and boundaries with shortenings, as il solid line with short cross-lines. Arrows show the rate and direction of the relative motion between Africa and Eurasia. Location of poles of relative rotation between Europe, Africa, and Arabia are shown (McKenzie, 1972; reproduced with permission from Geophys. J. R. Asrrotz. Soc.. G 1972 Blackwell Scientific Publications).


Hellenic arc and with a south-westward motion relative to both Eurasia and Af- rica (Fig. 176). In the southern margin, focal mechanisms correspond to thrusting, compatible with the northward direction of the motion of Africa. The intermediate shocks are explained in terms of a sinking slab following Isacks and Molnar (197 I) . The pattern of faulting in northern Greece and western Turkey is of normal faulting with E-W strike. This situation could be the result of the south- westward motion of the Aegean plate that produces a general disruption over a large area and the reactivation of old E-W structural trends.

In Turkey, the most remarkable feature is the North Anatolian fault in E-W direction with strike-slip right-lateral motion. There is in this area a very good agreement between fault-plane solutions and field observations. This fault marks the northern margin of the Turkish plate, which has a rapid westward motion with respect to Europe and Africa. The southern boundary of this plate is not so well defined mainly located at the eastern Anatolian fault with left-lateral strike-slip motion and possibly some overthrusting in the southwest (Fig. 17b). McKenzie found the geometry and motion of the Turkish and Aegean plates both complicated and surprising. He explained the proposed motions, as those that minimize the work needed to move the African plate toward Eurasia, consuming the Medi- terranean sea floor in front of the Hellenic arc. The intermediate-depth earthquakes in Rumania are explained as produced by a sunk slab of originally oceanic lithosphere now overlain by continental crust, a situation similar to that in the Hindu Kush. In the Caucasus, focal mechanisms present predominant solutions with overthrust and slip vectors in N-S direction explained in terms of the northward motion of the Arabian plate. The north part of the boundary of this plate with Africa is located along the Palestinian fault, with left-lateral strike-slip motion (Fig. 17b). The pole of rotation of Eurasia and Arabia is located at 32.2"N, 5.2"W and its rate is 0.544"/Myr, greater than the rate between Eurasia and Africa. An even greater value is found for the rate of rotation of Eurasia and Turkey: 0.65"/Myr, about a pole located at 18.S0N, 35.O"E.

The work of McKenzie (1970, 1972) lays the basis for plate tectonic interpretations of the Mediterranean region. Practically all the questions related to seismicity and focal mechanism data are addressed in his work, and the kinematics of the different plates are well established. However, details in the situation in south Spain and northern Morocco, the Italian peninsula and Aegean region are still not resolved. Lort ( 197 1 ), assuming plate tectonics, delineates the Mediterranean region into aseismic plates with seismically active boundaries. He takes the basic results from McKenzie (1970) and separates the following plates: the European plate including the Balearic, Ligurian, and Tyrrhenian seas; the African plate; and the fragmented Apulian, Aegean, and Turkish plates. Lort proposes a separate Apulian plate, comprising the Adriatic sea and the adjacent landmasses of Italy and the Balkans. He recognized that seismicity data are inadequate to permit sat- isfactory delineation of the southern boundary of the Apulian plate and, hence,


0

FIG. 17. Sketch of the plates boundaries and motions for ( ( 1 ) Italy considered as a promontory of Africa. ( b ) Plates on the eastern Mediterranean: ( I ) Eurasian, (2) African, (3) Iranian, (4) south Car- pathian, ( 5 ) Turkish, (6) Aegean, (7) Black Sea, (8) Arabian. The arrows show the directions of motion relative to Eurasia, and their lengths are proportional to the magnitude of relative motion. Boundaries are represented with the same criteria as in Fig. 16 (McKenzie, 1972; reproduced with permission from Geophys. J . R. Asfron. Soc.. 8 1972 Blackwell Scientific Publications).

148 AGUSTb4 UDIAS AND ELISA BUFORN

the relative motion between this plate and Europe and Africa. He finally concluded that the delineation of plates in the past and their relative motions can be only speculative.

McKenzie’s work stimulated the interpretation of seismicity data of the Medi- terranean region in terms of plate tectonics. Ritsema (197 1 ) reinterpreted his previous model for the motions in the Mediterranean (Ritsema, 1969b) in the light of plate tectonics. He maintains the same basic picture, introducing plate and block motions with some minor changes (Fig. 14). He argues against McKenzie (1970) that the motion of the western Mediterranean from the Azores-Gibraltar fault to Tunisia is strike-slip right-lateral, with the European plate moving toward the east with respect to Africa. This motion ends at the Calabrian arc, which plays an active overriding role in the generation of this deep earthquake zone, with the Ionian basin playing only a passive role. Ritsema postulates a united Anatolian- Aegean block that moves to the west and southwest and a separate Balkan block that moves to the northwest. He also maintains the left-lateral motion along the coast of the Balkans. Roman ( 1970, 1973) argued in favor of a separate Black Sea plate and introduced the terms subplates and bufier plates for small blocks between major plates. The motion of the Black Sea plate is related to the subducted slab that causes the intermediate-depth earthquakes at the Carpathian arc. Using the distribution of epicenters, Beuzart ( 1972) delineates the plate boundaries in the Mediterranean. Besides the two large plates of Eurasia and Africa, he separates three intermediate plates clearly independent: Adriatic, Aegean, and Turkey and secondary plates like Iberia, North Africa, Black Sea-Rumania, and Sinai peninsula.

In similar manner, Papazachos ( 1 973) determines the seismic and aseismic areas from the distribution of earthquakes. To avoid location errors he divides the data into several time intervals and differentiates between shallow, intermediate- depth, and deep activity. From the distribution of shallow earthquakes Papazachos defines I0 aseismic blocks: western Mediterranean; Apulian; Ionian; Valen- tine; Aegean; west, north, and south Anatolian; northern Greece; and Balkans. However, he did not identify these blocks with lithospheric plates or subplates. Papazachos gave great importance to the regions of intermediate-depth and deep earthquakes, especially the Calabrian and Hellenic arcs, interpreting them in terms of subducted slabs of lithosphere. In the Calabrian arc, the slab dips steeply (60”) to a maximum depth of about 500 km, while in the Hellenic arc the dip of the slab is smaller (35”) and the maximum depth is about 200 km. He assigns the tectonic situation in the Mediterranean to the approaching motion of the plates of Africa and Eurasia and compressive forces acting from east to west. Another distribution of plate boundaries based on seismicity in the Mediterranean was proposed by Udias (1975), who perceives the northern boundary of the African plate to have a continuous E- W trend from Azores to Turkey. North of this boundary a series of microplates are located from west to east: Iberian, Alboran, Italian-Tyrrhenian, Adriatic, and Aegean-Turkey. These blocks are considered as “buffer plates”


between the two large plates of Africa and Eurasia, in the sense of Roman (1973), and are not rigidly connected with them. A summary of the different plate tectonic interpretations for the Mediterranean region at this time was presented by Payo (1975).

A different approach to study the plate motion in the Mediterranean region from seismic data was proposed by North (1974). He calculated the seismic slip at 14 segments of the boundary between Eurasia and Africa, from seismic moments of earthquakes with M ) 5.5, for the period 1910- 1970, using spectral amplitudes of Rayleigh waves and a moment- magnitude relation. He uses McKenzie's scheme of five plates: Eurasia, Africa, Aegean, Turkey, and Arabia. The expected values obtained for the seismic slip vary from 0.9 to 4.2 cm/yr. Observed values, although lower, are thought to be compatible with the plate tectonic model of the region. If the situation of the region is reduced to the interaction of only the plates of Eurasia, Africa, and Arabia, the seismicity rate would be one order of magnitude lower than expected. The author concluded that a major proportion of the deformation in this region takes place in aseismic viscoelastic processes such as creep.

In the context of plate tectonics, the situation in the Mediterranean region is governed by the relative motion of the plates of Eurasia and Africa. This motion is defined by the Eulerian pole and its rate of rotation. The present-day pole of rotation of Africa with respect to Eurasia has been determined by different authors (Le Pichon, 1968; McKenzie, 1972; Morgan, 1972; Chase, 1978; Minster et a!., 1974). A very comprehensive data set including sea-tloor spreading rates derived from magnetic profile anomalies, transform fault azimuths, and slip vectors from focal mechanisms of earthquakes was used for a global study by Minster and Jordan (1978). The result for the pole of counterclockwise rotation of Africa with respect to Eurasia is 25.23"N, 2 1.19"W, and the rate of rotation 0. IW/Myr.

3.2. Tectonic Evolution

Plate tectonic theory provides a new framework to describe the tectonic evolution of continents and oceans, in terms of the dynamics and motion of lithospheric plates. Plate tectonics differs from the previous ideas of continental drift, although it preserves, in a modified way, some of its postulates. One of these is the existence before Jurassic time of a unique continent, the Pangea. Separation of the continents is achieved in plate tectonics, through the creation of new oceanic crust at midocean ridges according to the ideas of sea-floor spreading, its consumption in subduction zones, and the occurrence of lateral horizontal displacements at transform faults. The first reconstruction of the Pangea, according to plate tectonics, was done by least-squares fitting of the 1000-m-depth contour of continental shelves for continents around the Atlantic by Bullard et al. (1965) and for southern continents by Smith and Hallam (1970). The result is a grouping of the continents into a unique block, roughly corresponding to Wegener's Pangea. In

I50 AGUSTfN UDfAS AND ELlSA BUFORN

this reconstruction, a large wedge-shape ocean appears, separating the northern and southern continents that corresponds to the Tethys ocean, already postulated by Suess (1885-1905) and by some authors of the continental drift theory, as du Toit (1937).

A simplified model of surface horizontal motions at a global scale, consistent with sea-floor spreading, was first proposed by Le Pichon (1968). He calculated the pole of rotation of Africa-Eurasia at 9.3"N, 46.0"W, but he did not give many details of the situation in the Mediterranean, whereas he noted that the choice of the boundary is somewhat arbitrary, since it probably consists of several zones of compression, One of the first reconstructions of the situation of the Mediterranean region in Permo-Triassic times was presented by Smith (1971). He followed the method of Bullard et al. (1965), using geometry and a simplified spreading model of the opening of the Atlantic ocean. He did not attempt to analyze the detailed causes of deformation in the orogenic belts or in the subduction zones. One problem in the Mediterranean is the existence of fragmented blocks such as Iberia, Balearic islands, Corsica and Sardinia, Sicily and Italy. Smith considers these fragmented blocks as rigid bodies and makes the assumption that originally there were no intervening gaps between the fragments. In the reconstruction, the actual Mediterranean is closed with the southern coasts of the European blocks fitted with the northern coast of Africa (Fig. 18). The Tethys is open between the two belts of Alpine orogeny: ( 1 ) Alps, Carpathians, and Balkans to the north and (2) Apennines, Dinarides, Hellenic, and Taurides to the south. This separation was already suggested by Argand ( 1924).

Smith made the following conclusions: the Mediterranean is a Jurassic or younger ocean, probably formed as a net result of the opening of the Atlantic Ocean; the western Tethys was an ocean already existing in pre-Jurassic time that subsequently disappeared as a result of the net movements of the continental fragments around the present Mediterranean; each of these fragments and Africa itself have undergone net counterclockwise rotations relative to Eurasia since Permo- Triassic time. He considers the problem of the disappearance of the Tethys and the correlation of the rotations of the fragmented blocks with paleomagnetic data. Smith separates the process into two opening phases, the first from Upper Jurassic to Lower Cretaceous and the second from Upper Cretaceous to Late Eocene. He finally concludes that the initial motion of Africa relative to Eurasia did not eliminate the Tethys in the most efficient way; the limits of the first opening phase are close to the displacements needed to bring Africa in contact with Eurasia; the separation from North America of Eurasia and Africa has been done by independent movements about poles of rotation that are fixed with respect to each plate pair. A similar reconstruction of the situation of the Mediterranean in the Early Jurassic was done by Hsii (197 l), in which Italy and the western parts of the Balkans and Greece moved as a unit. The motion of Africa relative to Europe is done in three stages: ( I ) eastward motion from Middle Jurassic to Middle Creta- ceous, due to the opening of the Atlantic Ocean and separation with respect to


FIG. 18. Reconstruction of the situation of the Mediterranean region in Permo-Triassic times (Smith, 1971).

North America; (2) from Middle Cretaceous to Late Eocene, westward motion, as Europe was separating from North America at a greater rate; and (3) from Late Eocene to present, first eastward motion and then northward. There are two re- versals of motion from eastward to westward and back again to eastward. The evolution of the western Mediterranean basin, based mostly on geologic data, was presented by Le Pichon et al. ( 197 1) and by Auzende et al. ( 1973). The latter proposed that the triangular Provence basin was formed by drifting of Corsica and Sardinia, and the elongated north African basin is related to the building of the north African mountains.

A more detailed evolution of the Mediterranean basin and the Alpine system can be found in Dewey et af. (1973). As a starting point, the authors present an assembly of the outlines of the continents and fragmented blocks before the initial dispersion in Early Jurassic time. They recognized that the difficulty of the problem is due to the fact that the evolution of the Tethys does not involve a single simple plate boundary between Europe and Africa, but a constantly evolving mo- saic of subsiding continental margins, migrating midocean ridges, transform faults, trenches, island arcs, and marginal seas. The chapter presents a pictorial display, as time-lapse frames of the evolving Alpine system from Late Triassic to the present, through nine phases of Tethyan history. Four stages are shown in Fig. 19. The model implies that the motion of the larger plates of Eurasia and

152 AGUSTiN UDIAS AND ELlSA BUFORN

FIG. 19. Tectonic evolution of thc Mediterranean. ( u ) Situation in thc Early Jurassic, when the separation of North America from Africa started. ( b ) Early Crctaceous: North America has scparatcd from Africa but not yet from Europe. (('1 Late Cretaceous: starts the separation between North America and Europe. ( d ) From Miocene to present (Dewey ef d., 1973).


Africa dictates, by and large, the general behavior of the smaller microplates, through the particular styles of deformation, set up along the adjoining plate boundaries. It is assumed that there probably never was only a single plate boundary between Africa and Europe, but a network of compressional, extensional, and transform boundaries. The overall direction of the motion of Africa

154 A G U S T h UDfAS AND ELISA BUFORN

with respect to Europe is described as follows: from 180 to 80 Myr, SE to E and NE; from 80 to 53 Myr, NW to W; from 53 to 0 Myr, NW to N. Therefore, the direction of motion is reversed once from left-lateral to right-lateral at about 80 Myr.

A brief outline of the evolution of the Mediterranean, as proposed by Dewey et al. (1973), is discussed in the four selected stages represented in Fig. 19. The first stage (Fig. 19a) presents the situation on Early Jurassic, when the separation of North America from Africa started. The Tethys sea occupies a large area on the southern border of the Tethyan plate. Bordering north Africa are the microplates of Apulia (Italy and Adriatic), Rhodope (Rumania) and Turkey, and at the northern border of the plate are those of Carnic (southern Alps) and Moesia (Rumania). Two small plates also appear in north Africa, the Moroccan and Oran plates. In Early Cretaceous, the situation has changed considerably (Fig. 19b). North America has separated from Africa, but not yet from Europe. The Iberian plate has separated from North America and Europe and started its counterclockwise rotation. The Tethys has closed with the counterclockwise rotation of Africa with respect to Europe, and the northward movement of the plates of Apulia, Rhodope, and Turkey. In Late Cretaceous (Fig. 19c), the separation between North America and Europe started, with the formation of a triple junction in the Atlantic (Azores). Africa, which has been moving east with respect to Europe starts to move west. Iberia has already finished its rotation. Controlled by the gross Africa-Europe motion, all the internal blocks, such as Apulia, Rhodope, Turkey, and Central Iran, start to be moved in a westward course relative to Europe. From Miocene to the present, the relative motion of Africa with respect to Europe has been that of a northward collision (Fig. 194. As a result of collisions between internal blocks and their squeezing between Africa and Europe, a very complex plate margin geometry arose. The blocks of Carnic, Apulia, Rhodope, and Moesia are pressed against the southern border of Europe. The Hellenic trench fully develops and the Alboran, Provenzal, and Balearic basins open. The authors conclude that the gross effect of the Early Jurassic to present growth of the Atlantic Ocean has been to progressively reduce the Tethys sea. In this process, the extremely complex plate tectonic history of the Alpine system takes place.

Tapponnier ( 1977) stated that a kinematic description of the evolution of the Mediterranean, in terms of relative motion of rigid plates and microplates, is not sufficient and presented a description in terms of rigid-plastic behavior of continental lithosphere. First, he denied the existence of a reversal in the sense of motion of Africa with respect to Europe between 80 and 50 myr, as postulated, among others, by Hsu (1971) and Dewey et al. (1973). According to Tapponnier, there is only one change in direction from eastward to northward motion. The tectonic evolution of the Mediterranean is described in terms of interactions of rigid against plastic materials, along the boundary between Europe and Africa, instead of the relative motion of rigid microplates. He retains the existence of Italy and Arabia, as promontories of the African plate, whose impingement against the

SEISMOTECTONlCS OF THE MEDITERRANEAN REGION 155

southern boundary of Eurasia produced the Alps and Caucasus mountain belts. Explanations for the Mediterranean evolution, based on motion of rigid blocks or on regions of continuous deformations, can be considered as the two extremes of other intermediate positions. The number of microplates proposed by different authors for this region varies from 2 to 10. As their dimensions become smaller, their movements as rigid blocks are more difficult to understand and the situation approaches that of a continuous deformation. Some blocks may have moved as independent units for some time and become fixed at a later stage, as may be the case for Iberia and probably, also, for Italy. The block of Italy, Adriatic or Apulian is considered by some authors as an independent microplate and by others as a fixed promontory of Africa. A useful review of these problems is given by Smith and Woodcock ( 1982).

A more detailed reconstruction of the kinematic evolution of Eurasia and Africa is presented by Savostin et al. (1986). The change of motion, in Late Cretaceous, from left-lateral to right-lateral is suppressed, as suggested by Tapponnier ( 1977), but a similar change is introduced during the Oligocene. They conclude that the mechanical evolution of the northern Tethys boundary, where most of the deformation occurs, played a significant role in determining the motion of the major plates. This kinematic evolution is used by Dercourt et al. (1986), to explain the geologic evolution of the Tethys. They conclude that the Tethys cycle can be described as follows: migration of the accreting plate boundary from south to north across the ocean in about 30 Myr; subduction of the accreting plate boundary below Eurasia; rifting of pieces of the southern continent and formation of new accreting plate boundaries, leading to the formation of a new ocean, as the older one disappears by subduction to the north; and finally collision of the detached pieces with Eurasia and formation of new subduction zones in their southern boundary. The geologic data of the Mediterranean chains and basins have been also used by Ricou et al. (1986) as constraints for the determination of the location through time of oceanic versus continental lithosphere and the successive relations between them. Bonnin and Olivet ( 1988) describe the tectonic evolution of the Mediterranean and relate it to earthquakes occurrence, concluding that the entire region is undergoing deformation and should be regarded as a continuum rather than a set of microplates separated by clear boundaries.

In conclusion, plate tectonics has provided a powerful framework to interpret, not only the actual tectonic situation of the Mediterranean region, but also its time evolution, since the breakup of the continents to the present. Because of the interaction of plates of continental nature, complex processes take place in the boundary region, involving motions of fragmented microplates and also plastic deformations at their borders. The relative motion of the African and Eurasian plates is a consequence of the differential opening of the south and north Atlantic. This process determines dextral or sinistral horizontal E- W motion between the two plates and their N-S collision. The situation in the wide active area, between the stable parts of the two large plates, is explained in terms of the interaction of

156 AGUST~N U D ~ A S AND ELISA BUFORN

small fragmented rigid plates and of the formation of zones of continuous deformations. There is an abundant literature on the tectonic evolution of the Mediter- ranean region. We have mentioned only a few of the first and, in our opinion, more significant studies, that may show light on the seismotectonic interpretations.

3.3. Further Seismotectonic Studies

Better-defined distribution of hypocenters and a larger number of reliable focal mechanism solutions are, at this time, the instruments in new seismotectonic studies of the Mediterranean region. This type of data was analyzed by Udias (1980) for the plate boundary from Azores to Sicily -&labria. Focal mechanisms for 6 1 earthquakes were used to determine the orientation of seismic stresses (pressure and tension axes) and of seismic slip. At the Azores islands, tensions are horizontal and normal to the ridge, along the Azores-Gibraltar fault to about I2"W, both P and T axes are horizontal, corresponding to right-lateral, strike-slip motion with Eurasia moving east with respect to Africa. From 12"W to the Sicily-Cala- bria arc, P axes are horizontal and oriented NW-SE to N-S. The direction of the stresses at the arc is more dispersed, but P axes tend to be normal to the trend of the arc. This study was extended to the whole Mediterranean region (Udias, 1982, 1985). The main trends of epicenters of shallow earthquakes are used to delineate the plate boundaries, which can be followed in a fairly continuous way from the Azores islands to the Caucasus mountains. The four zones of intermediate-depth and deep earthquakes, present in the region, are related to the arc-like structures of Gibraltar, Sicily -Calabria, Hellenics, and Carpathian.

Udias (1982) found a very consistent pattern of regional stresses, derived from focal mechanisms of earthquakes along the plate boundary in the Mediterranean region (Fig. 20). From Azores to Gibraltar, the stresses change from horizontal

50"

40"

30" -30" -20" -10" 0" 10" 20" 30"

FIG. 20. Regional stress pattern in the Mediterranean derived from focal mechanisms (Udias. 1982)


tensions in NE-SW to horizontal pressure NW-SE, through a zone of right- lateral strike-slip motion in the central part of the Azores-Gibraltar fault. From Gibraltar to Calabria, stresses continue to be horizontal pressure normal to the plate boundary. Along the Italian Peninsula, there are horizontal tensions in ENE- WSW and on the coast of Yugoslavia horizontal compressions in the same direction. Along the arc of the Alps, stresses are horizontal compressions normal to the trend of the mountain range. On the Hellenic arc we have, again, normal compressional horizontal stresses. Behind the arc, in northern Greece and western Turkey, there are horizontal tensional stresses in a general N-S direction related to back- arc spreading. Along the Anatolian fault, the motion is of the same type as in the central part of the Azores-Gibraltar fault, that is, strike-slip right-lateral motion in an E- W trending fault. On the continental Carpathian arc, pressure axes are normal to its trend. The northern boundary of the African plate is considered to be continuous from Azores to the Hellenic arc, going around the Adriatic promontory. The complex behavior at the north of this boundary is considered to be the result of some parts of the Eurasian plate acting as small rigid blocks such as Anatolia and Iberia and other areas undergoing plastic deformations. Hsii (1982) introduced a small modification to this scheme, separating the Adriatic microplate from Africa. His model has only two microplates: Adriatic and Anatolia- Aegean. This model was also used by Mueller ( 1 982, 1984b) who studied, in detailed form, the collision of the Adriatic promontory or microplate in the zone of the Alps, in terms of a nearly vertical subduction zone of northern and southern lithosphere ( “Verschluckungzone”), that penetrates into the asthenosphere down to a depth of approximately 250 km.

Using the framework of plate tectonics, Armi.jo et al. (1986) published a seismotectonic map of the Mediterranean basin. In this map, data from seismicity and focal mechanism are used together with satellite images and topographic, geologic, and tectonic information. The region is understood to be at present under a collision movement, in a general N-S direction with rates increasing from west to east from 1 to 2 cm/yr. In the map, seismicity is represented in two categories: before and after 1950 and according to intensity and depth (greater or less than 70 km); lower limits of intensity I , ) , VII and IX, are used for regions of low and high activity, respectively. Only major topographic and tectonic features are represented.

Plate tectonics has provided, since its proposal, a consistent and generally accepted theoretical framework to explain present tectonics and its evolution. The situation in the Mediterranean region has been recognized to pose difficult problems, because of the complex interaction between two continental plates. Bounda- ries between two oceanic plates or between continental and oceanic plates are of simpler nature. However, the application of plate tectonic principles has shown a remarkable success in explaining the seismotectonic situation in the Mediterra- nean region. Models show that the boundary between Africa and Eurasia is rather

158 AGUSTh UDfAS AND ELlSA BUFORN

complex, involving the motion of independent microplates, like those of Adriatic, Aegean, and Anatolia, and the existence of areas of plastic continuous deformations. Seismologic data have contributed decisively to these tectonic models and, in turn, have found a coherent explanation from them.

4. RECENT SEISMOTECTONIC STUDIES (1986- 1993)

In the recent years, seismotectonic studies of the Mediterranean region have progressed with the incorporation of new types of data, such as in sifu stress measurements, geodetic observations, analysis of joint orientations, microtectonics, and other geologic field observations. The determination of the focal mechanism of earthquakes has been improved with the use of wave form analysis and seismic moment tensor inversion. Calculations of moment tensors have provided a new tool for the determination of the regional stresses. Plate tectonics is accepted as the guiding theory to interpret seismotectonic data, but its applications are somewhat modified, to account for the continent-continent collisions in the Mediter- ranean. As has been already mentioned, the situation cannot be completely explained in terms of simple rotations of a few rigid plates. The boundary between the two large plates-Africa and Eurasia-does not occupy a well-defined narrow zone, but a wide area, where a variety of deformations takes place. In this context, there are two basic approaches to the interpretation of focal mechanism data: one uses slip vectors to determine the direction of plates motion, in a kinematic sense, and the other stress axes to infer the direction of regional stresses, in order to provide a picture of the dynamic situation. These two approaches were already present in previous studies, but now they become more clearly separated.

In the first approach, slip orientations can be derived from the fault-plane solutions of earthquakes and the amount of slip in each earthquake from the value of the scalar seismic moment. From these values, averaged over a sufficiently large time interval, the rate of seismic slip can be determined for a specific area. Rates of seismic slip derived from earthquakes can be compared with those expected from large-scale relative plate motions. A comparison between plate motions and strain rates derived from seismic moments in the Mediterranean was presented by Jackson and McKenzie (1988). They show that the overall rate within a deforming area can be obtained from the sum of seismic moment tensors during a certain time interval. The stable blocks, whose motion can be considered as relative rigid- body rotations and derived from seismic slips, are Africa, Eurasia, Arabia, the Adriatic, Turkey, and the Aegean. The last one, which is intensively deforming, probably cannot be considered as relatively rigid. The relative motion of these plates is defined by the location of their corresponding Eulerian poles of rotation. The poles for Eurasia, Africa, Arabia, and Turkey have been discussed already. The Adriatic is considered as a separate block and not a promontory of Africa,


10" E 20" 30" 40" 50" 60' E 50"Nl I I I I I I I

40"

30"

=PROBABLY 15 50% SElSMl

a PROBABLY LESS THAN 15% SElS

FIG. 21. Motions predicted by the poles of rotation compared with the seismicity for the last 70 yeara in the Mediterranean region. Proportion of seismic and aseismic slip (Jackson and McKenzie, 1988; reproduced with permission from Geophy.7. J . R. Asrron. SOC., 0 1988 Blackwell Scientific Publications).

with a different pole of rotation relative to Eurasia (46.0°N, 10.2"E) (Anderson and Jackson, 1987b). The pole of rotation of Aegean with respect to Africa is 40"N, I8"E (Le Pichon and Angelier, 1979). The overall motions predicted by the poles of rotation are compared with the seismicity for the last 70 years in the belts surrounding the relatively stable blocks (Fig. 2 1). Results agree with the conclusion, already established by North ( I974), that aseismic slip plays an important role in this region. Only in the north Anatolian fault, western Turkey, and Aegean (we do not consider here the regions east of Caucasus), seismicity accounts for effectively all the expected deformations. In the Caucasus, it accounts for a considerable part, while in north Africa, Hellenic trench and Zagros account for very little. In Italy and Yugoslavia the proportion remains unknown. The nature of the aseismic creep is not known, although some possible mechanisms are suggested, such as the effect of large volumes of sediments, especially, of salt deposits.

A similar approach was used by Westaway (1990), who redetermined the pole of rotation of Africa and Eurasia (21"N, 2 1 OW), a value similar to that obtained by Searle (1980). He proposes that the southern limit of the Adriatic block must be at the Gargano peninsula and not at the southern end of Italy. Using information from older earthquakes, Westaway concludes that plate convergence in northern Africa may well be entirely coseismic. This result is extended to the whole of the Mediterranean region, in contrast with the results of Jackson and McKenzie (1988). The average convergence rate for the Mediterranean boundary is 2 mm/yr, with larger values in the Hellenic trench, due to the differential motion of the Aegean block with respect to Europe and at the north Anatolian fault (Fig. 22) .

160 AGUSTiN LJDIAS AND ELISA BUFORN

Fic. 22. Average convergence rate for the Mediterranean region in inillinleters pcr year. Cross- hatched areas represent zones of shortening; dotted, extension; single lines, strike-slip deformation. Zones where overall deformation scnse is not yet clear are unshaded. Paired arrows represent relative velocity between zones boundaries; single arrows. velocity of parts of Africa plate relative 10 stable Europe: curved arrows, rotations around vertical axes (Wcstaway, 1992).

Westaway calls the attention to the relatively inactive deforming zones in Tunisia- Libya and Gargano that take up changes in the motion direction of adjacent large rigid areas. The current motions of the African, Eurasian, and North American plates were determined by Argus et a/. (1989). The best-fitting pole for Eurasia and Africa (2 1.4"N, 20.5"W) is consistent with the rigid plate model in which the Gloria fault is a transform fault, and with the orientation of slip vectors in the Mediterranean region between Gibraltar and Sicily. In this region, the relative velocity of the plates range from 4 to 7 mm/yr. For a consistent combination of the motion of the three plates-Eurasia, Africa, and North America-the pole is located at 18.8"N, 20.3"W. A different scheme for plates kinematics in the Medi- terranean is proposed by Mantovani er al. (1993). They considered western Eu- rope as a separate plate from main Eurasia. This plate extends from the mid- Atlantic ridge to a hypothetical east boundary going through the Apennines, the Alps, the Rhine graben, and across the North Sea to Iceland. The pole of rotation between western Europe and Africa is similar to that obtained by other authors for Eurasia and Africa (22.6"N, 20.8"W). The pole for main Eurasia and Africa is at 41 SON, 17.3"W. which produces a relative motion in the eastern Mediterranean at NE-SW, instead of the generally accepted NNW-SSE to N-S direction. There is, also, an additional rotation between western Europe and main Eurasia with a pole at 53.6"N, 146.4"E. This division of the Eurasian plate seems difficult to accept in view of the seismicity data and other difficulties concerning the nature of the proposed active boundary between western Europe and main Eurasia. The NE-SW relative motion of Africa with respect to main Eurasia depends totally on the acceptance of the separation of the western Europe plate.


The use of focal mechanism solutions in order to derive the direction of regional stresses, as we have seen, has been a common practice from very early times. Plots on maps of the horizontal projections of P and Taxes were among the first uses of focal mechanism data in seismotectonics. In recent times there have been two main contributions: ( 1) the use of other types of indicators, such as in situ stress measurements and geologic and geodetic observations to complement seismic data; and (2) the improvement of focal mechanism solutions, which results in less scattering of data. The deduction of stress directions from fault-plane solutions of earthquakes, however, is not in itself exempt from ambiguity. Most earthquakes happen on preexisting faults, and slips can occur at different angles relative to the principal axes and not necessarily at 45", as was pointed out by McKenzie ( 1969).

Philip (1987, 1988) used a combination of different types of data to find the present-day stress field in the Mediterranean area and compared the results with neotectonic observations. In this way, the recent tectonic evolution during the Plio-Quaternary can also be described. From a large data set of microtectonic observations, earthquake focal mechanisms, in siru stress measurements, and the geometry of incipient tectonic structures, the orientation of the tectonic stress tensor is estimated and represented in the form of horizontal stress trajectories (Fig. 23). In the western Mediterranean, the stress field is relatively uniform with the maximum horizontal stress (compressional stress) in NNW-SSE direction. This direction is roughly the same as the relative displacement vector of the motion between Eurasia and Africa. In the central and eastern regions, numerous

Fiti. 23. Stress pattern in the Mediterranean region for the Plio-Quaternary. Derived I'rorn microtectonic observations, earthquake focal mechanisms. in siru measurements, and geometry of incipient tectonic structures; w , r?. frl respectively represent greatest, intermediate, and leiist principal stress axes (Philip, 1987).

....... .. ..

..

0

-0 : : .. .,

..

0’

-* : ,,

............ !

I

............................ ............................. ............................. ......................

..........

‘4 I:“.\ -:.K

- ..........................

.....................

i 1

.k

1

FIG. 24. Stress deviations in the Mediterranean and surrounding area. Heavy arrows show direction of motion (Reba1 et af.. 1992; reproduced with permission from Geophp. J . R. Asrron. Soc., 0 1992 Blackwell Scientific Publications).


directional changes in the stress field superpose on arc structures and reveal the complex juxtaposition of continental and oceanic blocks that are more or less deformable and mobile. Philip studies in detailed form the regions of the arc of Gibraltar and the two subduction zones of the Aegean and Tyrrhenian arcs. Rebai‘ et al. ( 1 992) used a similar approach collecting a large database of stress indicators for the Mediterranean region. They introduced the term “stress deviations” as the angular difference between local and average stress directions and discuss these stress deviations at different scales for the modern stress field in the Medi- terranean. In the western Mediterranean, the maximum horizontal stress is directed N-S to NNW-SSE, roughly parallel to the relative displacement vector of the motion between Europe and Africa. Small stress deviations are present in the areas of the Alps and Gibraltar arc. This relative simplicity contrasts with the situation in the central and eastern Mediterranean, where the stress field presents numerous deviations (Fig. 24). These directional changes are localized within collision areas, associated with large-scale faults, mountain belts, and subduction zones.

The focal mechanisms of a selected set of 83 earthquakes with M ) 6 were used by Udias and Buforn (1991) to derive the direction of the principal axes of stress along the plate boundary between Eurasia and Africa, from the Azores islands to the Caucasus mountains. Along most of the region, horizontal P axes are at an angle of 45”-90” with the trend of the plate boundary. Horizontal T axes are concentrated in central Italy and northern Greece, in NE-SW and N-S direction, respectively, associated with normal faulting. The general situation along the Mediterranean can be summarized as follows (Fig. 25). At the Azores ridge the

I

FIG. 25. Seismotectonic framework and regional stress pattern for the plate boundary between Eur- asia and Africa. Predominant focal mechanism at different areas are shown (Udias and Buforn, 1991; reproduced with permission from PAGEOPH, 0 199 I Birkhauser Verlag AG).

1 64 ACUSTiN UDIAS AND ELISA BUFORN

boundary is under horizontal normal tensions. At the Azores-Gibraltar and the north Anatolian faults P and Taxes are both horizontal with right-lateral strike- slip motion along E-W trending faults. From 10"W to Tunisia, stresses are horizontal compressions in NW-SE to N-S direction. In the Adriatic block there are horizontal tensions in central Italy, and horizontal compressions in the northern and eastern borders. Around the Hellenic arc stresses are compressional and behind the arc, the Aegean and western Turkey blocks are under N-S horizontal tensions, due to back-arc spreading. The situation in the four zones of intermediate and deep earthquakes (Gibraltar, Sicily -Calabria, Hellenic, and Carpathian regions) consists in P axes dipping toward the NW, except with the deep Spanish earthquakes whose P axes dip to the east. In the Hellenic and Carpathian regions P axes are nearly horizontal, while in Sicily-Calabria they dip more steeply. The situation at depth in the Mediterranean region is reflected not only by the presence of intermediate and deep earthquakes but also by the variations in the structure of the lithosphere and upper mantle. This imposes certain modifications in the classical plate tectonics theory, as pointed out by Panza and Suhadolc (1990a,b). These authors proposed a certain subduction of continental lithosphere, as reflected by lateral variations of the lithospheric ''lid'' thickness and lateral changes of the shear-wave velocities in the lithosphere asthenosphere system. At some areas, these heterogeneities extend to depths as great as 400 km.

A combination of several types of stress indicators-focal mechanisms; stress- induced well-bore enlargements or breakouts, in siru stress measurements, including hydraulic fracturing and stress-relief measurements; and young geologic deformational features, including both fault-slip and volcanic alignments-was also used by Miiller et al. ( 1992) to determine the regional pattern of tectonic stress in Europe. This work is part of a global project on regional lithospheric stresses (Zoback ef al., 1989; Zoback, 1992). The characteristic stress patterns are examined with respect to plate-driving forces, lithospheric properties, and tectonic settings. Their results show that a large portion of Europe is characterized by an homogeneous NW -NNW compression (Fig. 26). These stresses are controlled largely by the forces that drive plate motion. Throughout the region, the maximum principal stress is horizontal, except in the Aegean sea and western Anatolia, lower Rhine, Apennines, and western France.

Modern seismotectonic studies of the Mediterranean region integrate a great variety of data to throw light on the complex nature of the African-Eurasian plate boundary. A large number of reliable fault-plane solutions allows a more accurate determination of the orientation and rate of slip and the direction of the regional stresses. Extended areas of deformation are explained in terms of plastic processes or small rigid block rotations. Modem models for the overall situation in the Mediterranean tend to agree with each other, except for details in particular areas.


FIG. 26. Generalized stress map for Europe. Thick arrows indicate average stress direction based on numerous observations of different kinds 010); open arrows, 5- 10 obqervations; thin arrows, less than 5 observations (Miiller rid.. 1992).

5. AZORES 'To TUNISIA

The western part of the plate boundary between Eurasia and Africa extends from the Azores islands, where both plates also limit with the American plate, to Tunisia. This region can be separated in two parts: the first in the Atlantic, from Azores to Gibraltar; and the second in the Mediterranean from Gibraltar to Tuni- sia. In the first, the contact corresponds to the oceanic part of the plates; and in the second, to the continental. Although the subject of this review is the Mediterra- nean region, the continuation of the plate boundary between Eurasia and Africa to the west of Gibraltar demands the inclusion of the region up to the Azores islands in our consideration. We will first review papers that cover most of the area, especially the region from Azores to Gibraltar, and then those about the Iberian peninsula and the Maghreb.


5.1. Azores-Gibraltar-Tunisia

The first mention of the earthquakes between Gibraltar and the Azores islands was made by Gutenberg and Richter (19491, who stated that seismic data support a westward continuation of the active belt of northern Africa into the Atlantic Ocean as far as the Azores islands, but not farther. The same was noticed by RothC (195 1 , 1954) calling the attention to the importance of this line of earthquakes. The line was often referred to as the Azores-Gibraltar ridge. However, the significance of this seismic active line was not recognized until the arrival of plate tectonics. Ritsema (1969b) considered the entire boundary from Azores to Sicily as a right-lateral transform fault. The first papers on the plate tectonic interpretation of the Mediterranean region by McKenzie (1970, 1972), that we have already commented, considered this Atlantic extension of the region (Fig. 16). McKenzie studied the evolution of the Azores triple junction with ridge nature in its three parts and dominated by oblique spreading. His focal mechanism solutions for the area from Azores to Gibraltar present extensions normal to the Azores ridge, strike-slip right-lateral motion at the central part of the Azores-Gibraltar fault and reverse faulting near Gibraltar (Fig. 15). These three types of mechanisms correspond to the changes in the relative motion along the plate boundary from Azores to Gibraltar and can be explained only by the location of the pole of rotation near the boundary.

This area was more thoroughly studied by Udias and Lopez-Arroyo (1972) and Udias el al. (1976). Focal mechanisms were studied of earthquakes extending from the mid-Atlantic ridge to the Alboran sea. From the analysis of seismicity and focal mechanisms, they concluded that the Azores ridge is under horizontal tensions normal to its trend; from its eastern end to near Gibraltar runs a long transform fault with right-lateral strike-slip motion that separates the oceanic parts of the African and Eurasian plates. Large earthquakes (1941, 1975) have happened on this fracture with almost pure strike-slip motion. Near the strait of Gibraltar, the situation changes to north- south horizontal compressions and reverse faulting. This is the source region for the large Lisbon earthquake of 1755. The actual tectonics of this area is in great part based on the mechanisms of the 1969 and 1964 earthquakes. Udias et al. (1976) interpreted the motion on steep- dipping planes with the northern block going down. McKenzie (1972) selected, as fault planes, the low-angle planes with underthrusting of Africa. The same direction of motion was derived by Fukao (1973) for the focal mechanism of the 1969 earthquake with a strike in NE-SW direction. Seismicity east of the strait of Gibraltar in the Alboran and Betics region has a different character from that to the west. The independence in the past of the Iberia block with respect to Eurasia may still be preserved to some degree. These general features of the Azores- Gibraltar plate boundary are consistent with the pole of rotation near the Canary islands (Minster and Jordan, 1978). The evolution of this plate boundary was stud-


ied by Laughton and Whitmarsh (1974), who conclude that this boundary has been an essentially permanent feature since the earliest Cretaceous and has suffered several changes of differential movement, mainly of strike-slip character.

For the situation at the eastern end of the Azores-Gibraltar boundary, Purdy (1975) proposed a slow consumption of oceanic crust resulting in the formation of the Gorringen bank by an overthrust of the African crust on the surface and a downgoing of the African lithosphere in depth. The Eurasian plate at this location acts as a steeply edge chisel carving a thick slice of oceanic crust from the African plate. The Gorringen bank is lifted as a consequence of this process. Purdy bases this explanation on the focal mechanisms of earthquakes and seismic reflection and refraction profiles. He suggests the presence of a miniplate at the Gorringen bank. The western end of the region at the Azores islands is interpreted differently by Ribeiro (1982) and Madeira and Ribeiro (1990) as a right-lateral oblique transcurrent fault zone in WNW-ESE direction, linking the mid-Atlantic ridge with the Gloria fault. This is described as a leaky transform model. Faults of this type are observed on the islands of Pic0 and S. Jorge. The situation at the central part of the region was studied by Lynnes and Ruff (1985) based on the mechanisms of the large strike-slipearthquakes of 1941 and 1975. The fact that these two earthquakes are separated by 200 km perpendicular to the strike of the fault indicates that this boundary is neither well defined nor stable. These authors consider the 1975 event as an intraplate earthquake, the same as the 1969 shock near the Gorringen bank, and thus intraplate events dominate seismic energy release in this part of the Africa-Eurasia plate boundary.

The Azores-Gibraltar plate boundary has been studied by Grimison and Chen (1986, 1988; Chen and Grimison, 1989). They calculated the focal mechanisms for a number of earthquakes in the region using the inversion of waveforms of P and SH waves. The mechanisms at the Azores islands confirm the spreading motion normal to the trend of the islands. From Azores to Gibraltar there is a remarkable agreement in the P axes of the focal mechanism solutions, which are horizontal and in NNW-SSE direction, consistent with the convergence in the eastern part of the boundary. The scattered seismicity, different types of mechanisms (strike-slip and thrust) and complex bathymetry suggest that in this area a single major plate boundary does not seem to exist. The region is described as a diffuse zone of ocean-ocean convergence (Fig. 27). The diffuse zone has a width of 100- 300 km and connects with the area of continental convergence between Iberia and Africa. Typical features associated with subduction zones are not observed, and instead the deformation spreads out over a diffuse belt 100-300 km wide in the oceanic lithosphere. Westaway (1990) also considers a similar wide area of deformation from 15"W in the Atlantic to Tunisia. East of Gibraltar, this area is limited in Africa by the south Atlas and in Iberia by the Guadalquivir.

A different interpretation of the Eurasia- Africa plate boundary from Azores to Gibraltar has been proposed by Weijermars ( 1 987), based mostly on geologic and

168 A G U S T h UDfAS AND ELlSA BUFORN

Fic;. 27. Present-day deformation along Az.ores-Gibraltar. Stippled area indicates the diffuse zone of ocean-ocean convergence. Solid circles correspond to recent epicenters (m,%) and historical events since 1931 ( M ) 6) (Chen and Grimison, 1989; reproduced with perinission from Trctonophys- ics. 0 1989 Elsevier Science Publishers BV).

morphologic arguments. He locates this boundary along the parallel Azores- Crevillente fault on the north and the Hays-Atlas fault on the south, which both cross the Atlantic and continue in Iberia and in northern Africa, respectively. Both are wrench right-lateral faults, and the material between them is highly deformed. The model, however, has no seismologic support, since the Hays fault is not active and the south Atlas fault has very low seismicity. Vegas (1991) also proposed the south Atlas as the southern plate boundary limit. Between this limit and the Gua- dalquivir lineament there is a zone of pure shear deformation inside which distributed deformation is taken up by rotation of blocks. A west Moroccan block is proposed, extending from the Gloria fault to the Agadir fault, which presents a clockwise rotation. Correlation of the model with focal mechanism data is not clear.

The model proposed by Udias et al. (1976) has been further developed by Udias ct ul. (1986) and Buforn et al. (1988a), extending it to the eastern end of Tunisia. A large number of fault-plane solutions have been compiled for the region, and a new seismotectonic interpretation is given (Fig. 28). This interpretation agrees with the previous one for Azores and the central part of the Azores- Gibraltar fault. For the eastern end of this fault, the situation is now considered as underthrusting of Africa, which agrees with the general relative motion of the African plate with respect to Eurasia along the Mediterranean region. The area interpreted as a zone of diffuse seismicity by Grimison and Chen (1986) is considered here as affected by westward oceanic continuations of faults in Iberia, with predominant strike-slip right-lateral motion. The region between stable Africa,

SElSMOTECTONlCS OF THE MEDITERRANEAN REGION

POLE OF ROTATKW

I

169

50’

AO”

30’ N

FIG. 28. Srismotectonic map for the Azores-Gibraltar region showing plate boundary, secondary features, horizontal stresses, ailike-slip direction of motion, and pole of rotation of Africa with respect to Europe (Buforn et u/., 1988a; reproduced with permission from Tectonophysics, 0 1988 Elsevier Science Publishers BV).

limited by the Atlas faults, and stable Iberia forms a highly fractured zone of extended deformations, which continues into the Atlantic to approximately 1 SOW. The NNW-SSE direction of horizontal pressure axes is consistent for this whole area and agrees with the present collision motion described by the location of the pole of rotation of Africa with respect to Eurasia near Canary islands. South of‘ this region, a possible triangular subplate is proposed, limited by the Agadir- Nekor fault and by a line from Agadir to the Gloria fault. A similar model is presented by Moreira (1991), who shows a greater proportion of strike-slip mechanisms in Azores and near the strait of Gibraltar, most with right-lateral motion. However, he agrees with the general result that shows thrust to the east of the Gorringen bank and strike-slip to the west.

5.2. Iberia

The seismically active areas of the Iberian peninsula are located mainly south of the Guadalquivir lineament, in the Pyrenees and in Portugal. Montessus de Ballore ( 1906) presented the first study relating seismicity and tectonics in Iberia.


He pointed to the presence of a large fault in south Spain, from the Atlantic to the Mediterranean that separates the instable and stable parts of the peninsula in the south. The second active area considered is the Pyrenees, where greater activity is found at the western part. These general features were also recognized by other early seismicity studies (Sanchez Navarro-Neumann, 19 17; Inglada, 192 1). The first seismotectonic study and map of the peninsula was made by Rey-Pastor (1927) following the general guidelines of Sieberg. In this map, the three seismic zones are shown (south Spain, Portugal and Pyrenees). The limit of the seismic zone in south Spain is located at the Guadalquivir fault with a zone of higher activity south of the Cordillera Penibetica. In the southeast, a fault is drawn extending from Alicante in NE-SW direction, and in the south two systems of faults are represented: one in E-W and the other in NW-SE direction. These features are present in more recent studies of the area. Sieberg (1932) presents a similar seismotectonic map in which there are N-S seismic alignments in Portugal, NE- SW in southern Spain, and E-W in the Pyrenees.

Modern seismotectonic studies are based on more accurate hypocentral locations, focal mechanisms, and new tectonic models based on plate tectonics. The seismic catalog was revised by Munuera (1969) and later by Mezcua and Martinez-Solares (1983), resulting in a more accurate location of earthquakes and magnitude determinations. Seismicity in Iberia is of moderate level, with a continuing occurrence of low-magnitude earthquakes (M ( 5) and large shocks (M ) 6) separated by long time intervals (100-200 years). Last large earthquakes occurred in 1829 and 1884. Focal mechanism determinations were possible only in early times for earthquakes larger than M = 5 (Udias and Lopez Arroyo, 1969; Lopez-Arroyo and Udias, 1972; Fukao, 1973). Only very recently, with the installation of new seismographic stations in the area has it been possible to obtain solutions for small earthquakes. Andrieux er al. (197 I ) presented the earliest plate tectonic model of this region. This model explains the Betics, Rif, and arc of Gibraltar as caused by the differential movement of a subplate in the Alboran sea, with respect to Iberia and Africa. This subplate overrides the two plates, forming the nappes of the internal zones of Betics and Rif and the front structure of the arc of Gibraltar. This model was used in the seismotectonic analysis of Udias et al. (1976). Seismicity in south Spain cannot be understood without a correct knowledge of the systems of geologic faults. One of the first studies on this subject identifies the major Alhama-Palomares-Carboneras fault in southeast Spain in NE-SW direction and with left-lateral motion, between Alicante and Almeria (Bousquet and Philip, 1976; Larouzikre et al., 1988). Faults in south Spain can be grouped into three systems with strikes in N70E, N30-60W, and NlO-30E (Sanz de Galdeano, 1983, 1990; Boccaleti et al. 1987). Sanz de Galdeano calls the attention to the fault or group of faults from Cadiz to Alicante that practically coincides with the separation of the external and internal zones of the Betics and that has played an important role in the evolution of the region. These and other studies on the major faults in south Spain have helped in the interpretation of seismicity and


focal mechanisms. The geologic evolution of south Spain is very complex. New models for the tectonic evolution of the Betics-Alboran do not yet integrate earthquake data (Platt and Vissers, 1989; Banks and Warburton, 1991).

Buforn et al. (1988b) and Buforn and Udias (1991) using data from improved local seismologic networks made it possible to establish a relation between earthquakes and geologic faults in south Spain. They found that most of the seismic activity is located south of the Cadiz-Alicante fault system and limited at the east by the Alhama-Palomares-Carboneras fault. Surface shocks at this area are associated to parallel faults in Granada and Almeria of the N30-60W system. Focal mechanisms are consistent with a general regional stress pattern of approximately N-S to NW-SE horizontal compressions and E-W horizontal tensions (Fig. 29). Most mechanisms show predominant vertical motion of normal and reverse character. The general stress pattern agrees with a collision movement of Iberia against Africa. This collision forms a wide area of fractures and deformations in a wedge- like area that extends from the limit of stable Iberia to the Atlas mountains in Morocco. South of the Cadiz Alicante fault the crustal material is subjected to E-W tensional stresses that push the material against the arc of Gibraltar. These tensional stresses produce normal faulting on the faults of the Betics and Alboran sea.

A very important element in the seismicity of south Spain is the occurrence of

AFRICA / FIG. 29. Seismotectonic framework for the Ibero-Maghrebian region. Circles represent shallow

foci, triangle intermediate, and deep shocks. Small arrows indicate the directions of the horizontal component of the pressure and tension axes. Thick arrows show the direction of P axes for large earthquakes ( M ) 7). Large arrows show the inferred direction of regional stresses. Shaded areas show location of interrnediate-depth earthquakes (Buforn and Udias. 199 I ) .

I72 AGUSTh UDiAS AND ELISA BUFORN

intermediate depth (30 km ( h ( 150 km) and very deep ( h = 630 km) earthquakes (Buforn et al., 1991). The deep earthquakes are rare occurrences (1954, M = 7; 1973 and 1990, M = 5) and are located in a very small volume. The intermediate depth earthquakes are more frequent and occupy a nearly vertical section from the surface to a depth of 150 km, with epicenters in an approximate N-S trend from Granada and Malaga to the Alboran sea. Their focal mechanisms show P axes dipping about 45" to the NW direction. No seismic activity has been observed between 150 and 600 km. The mechanisms of the three deep shocks are very similar with P axes dipping 45" to the east. Their origin has been related to a detached block of lithospheric material still sufficiently cold and rigid to generate earthquakes (Udias er al., 1976; Chung and Kanamori, 1976; Grimison and Chen, 1986). This active nest of deep earthquakes is related to the large anomaly found in tomographic studies between 200 and 700 km (Blanco and Spakman, 1993). The gap between intermediate and deep earthquakes and the difference between their mechanisms argue in favor of a different origin. Buforn er al. (1991) proposed two separate processes of subduction. The most recent process responsible of the intermediate depth seismicity is formed by material being pushed under the continental crust from the east or southeast, since about 10 million years ago. In the older process responsible for the deep earthquakes, the slab has become detached and sunk to the actual depth and its origin is difficult to establish. The existence of a subduction zone in south Spain is a controversial subject and is not accepted by some authors.

The Pyrenees is a region less active than south Spain. Its seismicity stretches along the 400-km length of the mountain chain from the Atlantic to the Mediter- ranean. In a general way, seismicity is shallow ( h ( 20 km) and concentrated in a narrow zone of E-W orientation, following the trend of the north Pyreenean fault (Gallart, 1982). The first reliable mechanism for the Arette earthquake (McKenzie, 1972), shows horizontal pressure normal to the mountain range, and agrees with a movement of collision between Iberia and Eurasia. However, fault- plane solutions for small earthquakes show a variety of results, which in a very general way are compatible with horizontal compressive stresses in NW-SE direction (Gagnepain et al., 1980, 1982). These compressional stresses acting normal to the Pyrenees may be related to the collision process that have determined the formation under the mountain chain of an asymmetric cortical root dipping toward the north, as has been evidenced by deep seismic reflection data (SuriBach et al., 1992).

The seismicity of Portugal is scattered over the whole continental region, especially in the central and southern parts, and continues in adjacent Atlantic areas. Most earthquakes have small magnitudes, but large destructive shocks have occurred in the past, such as those of 1755, 1858, and 1909. Moreira (1985, 1991) has determined several focal mechanisms and related them to the main faults in Portugal. The faults of Nazare, Tagus, and Messejana are considered to continue

SEISMOTECTONICS O F THE MEDITERRANEAN REGION

P to" I

173

FIG. 30. Focal mechanism solutions in the adjacent region 10 mainland Portugal, principal faults and plate boundaries. Arrows show directions of compressional stresses (Moreird, 1991 ).

into the Atlantic along submarine valleys (Fig. 30). On the southern continental margin, the compressive seismic stresses in NNW-SSE direction, derived from focal mechanism solutions, are in agreement with the stress field expected in the area from the collision between the African plate and the Iberian subplate. Focal mechanisms for small earthquakes (M ( 4) in central Portugal have been studied


by Fonseca and Long (1991) and found to be compatible with a shear regime of sinistral motion on NNE-SSW faults. They also propose a tectonic model in which the Iberian peninsula is presently being extruded in a westward direction with a small southward component with respect to stable Africa under general NNW-SSE compressive stresses. The convergence between Iberia and the Atlan- tic in front of Portugal implied by this extrusional model is being absorbed by crustal shortening, rather than by subduction. This zone of convergence is also postulated by Ribeiro et al. (1988), who favor the hypothesis that the western continental margin of Iberia is changing from a passive to an active nature, with a northward propagating subduction zone being established, nucleating at the Gor- ringen bank and extending along the base of the continental shelf of the coast of Portugal. This is a very interesting idea, but it is doubtful that it has enough support from seismologic data. The extrusion model, however, has more difficulties, since it requires left-lateral strike-slip motion along the Pyrenees, Cadiz- Ali- cante fault, and the Atlas range, for which there is no evidence from focal mechanism data.

5.3. Maghreb

In the first seismotectonic analysis of the Maghreb region Montessus de Ballore (1906) relates earthquakes along the north coast of Morocco, Algeria, and Tunisia to areas of seismic instability related to the folds of the Tellien Atlas. Earthquakes are concentrated near Oran and Algiers in Algeria, while the area between the Tellien and Saharian Atlas is seismically stable. A tectonic model for northern Africa based on continental drift was proposed by Dubourdieu (1962) ‘In which Africa is moving to the north with respect to Europe and produces a series of parallel strike-slip faults in NE-SW direction from the coast to the interior. The westernmost of these faults is the longest and stretches from Nekor to Agadir, corresponding to an important seismic alignment, while in Algeria and Tunisia the faults are shorter. Seismicity of the Maghrebian region (RothC, 1950; Hatzfeld, 1976; Giradin et al., 1977; Ben Sari, 1978) shows three main trends, one along the coast in E-W direction (Rif and Tellien Atlas), a second in NE-SW direction along the high and middle Atlas from the Mediterranean coast to Agadir, and a third along the south or Saharian Atlas parallel to the first and with very low activity. Large earthquakes occur in the El Asnam area as sudden seismic events between periods of relative quiescence (the two most recent occurred in 1954 and 1980). Earthquakes are of predominantly shallow depth, but along the middle At- las there are also some shocks of intermediate depth (30 km < h < 150 km) similar to those in south Spain and the Alboran sea.

Focal mechanisms of earthquakes in the Maghreb region show a predominant horizontal pressure axes in N-S direction and reverse faulting, especially in Al- geria (El Asnam region) (McKenzie, 1972; Giradin et al., 1977). The occurrence


of the 1980 earthquake in El Asnam (Cisternas et d., 1982; Meghraoui et al., 1986) confirmed the thrust-faulting mechanism in the area along a fault of strike NNE-SSW and with horizontal pressure axis in nearly N-S direction. This nearly N- S direction of horizontal compression can be found along the northern coastal area from Gibraltar to Tunisia. Most faults are in NNE-SSW and motion may be of underthrust of the southeastern block or left-lateral strike-slip (Bounif et al., 1987). The situation in Morocco is more complex with seismicity associated to the Rif and the high and middle Atlas. The most important alignment of earthquakes is i n NE-SW direction from Nekor to Agadir (Medina and Cherkaoui, 1991, 1992). Two other alignments are oriented NW-SE and start at the Atlantic near the cape San Vicente toward the Atlas and in the gulf of Cadiz toward the Rif. Most focal mechanisms are compatible with the direction of compression oriented from N-S to NW-SE, also determined from other neotectonic data. Mechanisms on the Nekor fault and the high Atlas are of either strike-slip, left- lateral, or reverse motion. The situation in Tunisia is also compatible with the N- S convergence motion, and focal mechanisms yield a direction of compression from NW-SE to N-S (Kamoun and Hfaiedh, 1985). The seismotectonic situation can be summarized, following Medina and Cherkaoui (1991), as an area between the coast and the northern limit of stable Africa (high and southern Atlas), highly deformed and subjected to NW-SE compressions, resulting in reverse and strike- slip faulting (Fig. 3 1).

In conclusion, the seismically active region from Azores to Tunisia corresponds to the plate boundary between Eurasia, and Africa and its motion is the result of the counterclockwise rotation of Africa with respect to Eurasia about a pole near the Canary islands. The region can be separated into two parts. Thejrst is a west-

,/ POSSIBLE 4 TRENDOF 4 ACTIVE FAULT f P AXIS * COMPRESSIONAL STRESS

/ undetermined

# strike slip

FIG. 31. Seismotectonic sketch-map of the Maghreb region. Main faults, direction of stresses, and horizontal motion are shown (Medina and Cherkaoui, 199 I).

176 AGUSTiN UDfAS AND ELISA BUFORN

ern part of oceanic boundary, from Azores to Gibraltar, in which motion changes, from west to east, from horizontal tensions and spreading normal to the Azores ridge, to strike-slip right-lateral motion in the central part and reverse faulting and convergence near the strait of Gibraltar. At this area, the continental parts of both plates create an extended area of deformation and diffuse seismicity with continuations into the Atlantic floor of faults present inland. The second, eastern, part from Gibraltar to Tunisia is formed by a plate boundary of continental nature between Eurasia and Africa, with the Iberian peninsula acting as a somewhat independent block. The Iberian peninsula has three active borders. The northern border along the Pyrenees shows convergence between Eurasia and Iberia; the southern border shows convergence between Iberia and Africa; and the western border shows convergence along the Portuguese coast, which is not very active and not yet well understood. The situation between Iberia and Africa can be explained in terms of the Alboran subplate whose differential motion is responsible of the Betics and Rif. It is generally understood, however, that, from the southern border of stable Iberia to the northern border of stable Africa, there is a broad and complex area of deformations, subject to compressional stresses in NNW -SSE direction with a possible subduction process, responsible for intermediate-depth earthquakes. Many details of the tectonic of this region are not yet explained, such as the nature of the Gorringen bank, the active faults in the Betics and northern Morocco and Algeria, the earthquakes along the coast of Portugal, and the nature and origin of intermediate and deep earthquakes.

6. ITALY AND THE ALPS

The Italian peninsula and surrounding regions constitute the most important seismotectonic area in central Mediterranean. Montessus de Ballore ( 1906) commented that Italy is the country of the world where earthquakes are best known, and it can be said that seismology was born there. An early classic reference for earthquakes in Italy is Baratta (1901), and two recent catalogs are Guidoboni (1989) for historical earthquakes before the year 1000 and Postpischil (1985) from 1000 to 1980. Earthquakes are distributed, in a very general way, following the arc of Sicily and Calabria, along the Apennines in a NW-SE trend and further north extended over the broad area occupied by the Alps and neighboring areas. We will divide our review in these three parts: Sicily-Calabria, Apennines, and Alps.

6.1. Sicily-Calabria

Since very early studies (Montessus de Ballore, 1906), it was recognized the arc form of the tectonic alignments and earthquake distribution in Sicily-


Calabria. Large earthquakes have occurred in this area, such as those of 1783, 1883, and 1908. The significance of the seismicity at intermediate depth (most earthquakes are at less than 350 km depth, with a few shocks between 400 and 500 km) was first studied by Gutenberg and Richter (1949). Many of the features of the active arc structures of the Pacific are found in this area, but not all. As in the Pacific arcs, deeper shocks are located at the concave side and gravity anomalies present negative values on the convex part and positive in the concave. Peter- schmitt (1956) studied this area in detail and noticed that the main part of seismicity is located in Calabria and the inclination of the seismic surface is about 60" with a conical shape around a center deep in the Tyrrhenian subcrust.

An early study on the focal mechanism of earthquakes in this region is that of Di Fillip0 and Peronaci (1959). Ritsema (1969a,b, 1972) determined the fault plane solutions of I0 earthquakes and found that for intermediate-depth earthquakes, as an average, the pressure axis dips 60" toward WNW, while shallow ones have horizontal tension axes. Seismicity distribution and focal mechanism is interpreted as being caused by a nearly horizontal flow of mantle material from under the Calabrian side moving toward the basin or also by an uprising of the same material from under the Calabrian arc toward higher levels in the Tyrrhenian sea section. In his interpretation, the Calabrian arc is somehow drifting in the ESE direction over the stable basin of the Ionian sea, and in the wake of this arc motion new ocean crust is created with compensating horizontal movements within the low-velocity layer from ESE toward WNW. The process is explained by Ritsema ( 1 969b), in the context of the problem of oceanization in the Mediterranean. The active role is taken by the Calabrian arc, which overrides the Ionian sea region. Ritsema (1979) addresses the question of active or passive subduction in the Cala- brian arc. Passive subduction occurs where a continental plate or island arc ac- tively rides or thrusts over a neighboring passive oceanic plate. The passive oceanic plate is pushed down into the mantle under the edge of the moving continental plate or arc and sinks down. Evidences from the distribution of seismicity and focal mechanisms show that this is the case in the Calabrian arc. Pas- sive subduction occurs as a result of the eastward or southeastward motion of the Tyrrhenian subplate over the oceanic lithosphere of the Ionian sea. Behind the arc the Tyrrhenian subplate suffers surficial extension and oceanization with material piling up at the Calabrian arc structure. Caputo et al. (1970) proposed the existence of subduction with the oceanic plate dipping beneath the Tyrrhenian sea, along which the principal stresses are in the direction WNW. This plate may be deformed, bowed, or even fragmented, which would account for the two volcanic alignments.

McKenzie (1972), interpreting the region in terms of plate tectonics, also observed that the situation in the Calabrian arc cannot be explained by lithosphere being pushed by the northward motion of Africa under the Tyrrhenian sea, in a similar way as the processes in the formation of the Pacific arcs. This motion


would produce a slab that strikes east- west and not north-south as happens here. McKenzie proposed that the slab, which is 200 km long and penetrates to about 500-km depth, is a relic, and was produced by the motion of Italy and part, at least, of Sicily in an eastward direction with respect to Eurasia with a velocity considerably greater than that between Africa and Europe. This motion is implied in the models of rotation of Italy on the basis of paleomagnetic data. The restricted region in which intermediate shocks now occur can be explained if the rapidly moving plate was relatively small. A similar model is proposed by Alvarez ef al. ( 1974) in which Sardinia, Corsica, and Calabria rotated counterclockwise and subduction was initiated by the separation of Sardinia and Calabria about 10 million years ago. Once subduction was started, thinned crust was created by extension behind the arc in the Tyrrhenian sea. Riuscetti and Schick (1975) studied 19 shallow and 8 deep earthquakes, finding normal and strike-slip faulting in E-W faults. Cagneti et al. (1978) studied a large number (81) of fault plane solutions for Italy. They concluded that for the Calabrian arc the tension axes for shallow earthquakes are almost horizontal corresponding to an east- west extension. Intermediate-depth earthquakes have pressure axes almost vertical, dipping along the Benioff zone, which indicates that the descending slab is under internal pressure. The absence of shocks between 100 and 200 km could be an indication of the interruption of the lithosphere across this depth interval, as also suggested by McKenzie ( I 972), and would correspond to the low velocity layer found by Berry and Knopoff (1967).

Ghisetti and Vezzani (1982) proposed a seismotectonic zoning of this region in which the different structural and seismic styles of the Calabrian arc, as related to the crustal dynamics at depth, are taken into consideration. Gasparini et al. (1982, 1985) based their interpretation in a relocation of hypocenters and a larger number of fault-plane solutions (59). For crustal earthquakes the situation is quite complex. Pressure axes oriented in WNW-ESE are present along the Ionian coast of Calabria, while in Sicily strike-slip mechanisms are predominant. North of the Calabrian arc, in the southern Apennines, mechanisms show predominant dip- slip motion with tensional axes normal to the main structural trend of the chain and subordinate strike- slip faults. Intermediate-depth earthquakes show pressure axes, mostly parallel to the dip of the Benioff zone. Shocks in the northern and southern parts of the downgoing slab reveal a predominance of strike- slip motion supporting the hypothesis of lateral bending of the arc (Fig. 32). The nature of the complicated Benioff zone is interpreted as a strongly deformed remnant of a previously continuous downgoing slab extending, during Oligocene time, from northern Apennines to Gibraltar. The authors suggest that the opening of the Tyr- rhenian basin was accompanied during late Miocene by a counterclockwise rotation of the Italian peninsula, which caused additional distortion of the Calabrian arc with subsequent compression in the inner side of the arc and lateral stretching in the outer side. Thus, the classical model of arc-trench system migration must


FIG. 32. Seisrnotectonic zoning for Sicily-Calabria region based on distribution of hypocenters and a large number of fault-plane solutions. Depth contours of earthquake activity and gravity anomalies are shown (Gasparini er ul.. 1985; reproduced with permission from Tecronoph~~sics, 0 1975 Elsevier Science Publishers BV).

be rejected for this region and instead an explanation must be adopted, in terms of distortion of subducted continental lithosphere during the collision of the African and Eurasian plates. In this process, only the lower crust and upper mantle take part in the subduction, while the low-density upper crust is partly involved in nappe formation and partly nailed in the roots of the chain.

The deep structure under the Calabrian arc has been investigated by Panza ef al. ( 1980), Panza ( 1984), and Panza and Suhadolc ( 1990a) from propagation of surface and body waves. In these studies, an anomalous thin lithosphere (30 km) is found in the Tyrrhenian sea with low lid (sub-Moho layer) shear-wave velocity

180 ACUSTiN UDfAS AND ELISA BUFORN

(4.2 km/s), in contrast with the normal lithosphere thickness (90 km) in the Ionian sea. The opening of the Tyrrhenian sea and the formation of the Benioff zone is explained by a dynamic subduction occurring on a moving plate in which a gravitationally descending block is overrun by the moving plate, and as this motion progresses, it pulls down and detaches another block. In consequence, the subducting slab is broken up in blocks, and the gap in seismicity between 130 and 230 km may be due to a wide gap between blocks. Anderson and Jackson ( 1 987a,b) studied the focal mechanism of six shallow-focus earthquakes in Sicily and concluded that the slip vectors reflect the N-S-directed convergence of Africa and Eurasia predicted in this area from the Africa-Eurasia pole of rotation. The intermediate-depth earthquake distribution and focal mechanisms are also studied in detailed form. They conclude that there are no major gaps in the depth distribution of earthquakes. Activity is concentrated in the depth interval 230-300 km, but a low-level activity is present at all depths, The lack of intense shallow activity, especially of thrusting shocks, suggests that subduction is no longer occurring. There is a change of dip in the Benioff zone from 70" to 45" at a depth of about 250 kni. Fault-plane solutions show a great similarity with pressure axes aligned with the dip of the slab, indicating that there is down-dip shortening in the region of more intense seismic activity (230-300 km). The location of this intense activity at the change in dip of the Benioff zone and the possible bifurcation of the slab below this depth suggest that there is large-scale disruption of the slab at this region.

6.2. Apennines

Along the Italian peninsula the distribution of epicenters follows the trend of the Apennines mountain range. The first focal mechanisms were determined by Di Fillip0 and Peronaci (1963). Ritsema (1969a,b) and Shirokova (1972) found predominant horizontal tensional stresses normal to the trend of the Apennines and corresponding to normal faulting. Ritsema (1969b, 197 I ) interpreted the general motion of the region as the Adriatic moving in south-east direction with left- lateral strike-slip motion along the coasts of the Balkans and northern Greece. This general eastward motion is a consequence of the stress field generated at the North Atlantic ridge. The westward motion of the Balkans is associated to the compressional stresses in the Alps. McKenzie (1972) remarked that the information is not sufficient to propose any plate tectonic interpretation with any confi- dence. He considers the Adriatic as a promontory of the African plate, and the motion between Africa and Eurasia is taken up by normal faulting in Italy and thrusting in the coast of the Balkans.

Cagnetti et al. (1978), on the basis of 12 fault-plane solutions in the Apennines region, showed that normal faulting is predominant with horizontal tension axes


in an about NW-SE direction. Strike-slip is also present in the northern Apen- nines. In the southeastern part, however, they found tensions in N-S direction. They related this stress field with the spreading process of the eastern North At- lantic basin, in a west-east direction, and with the northward displacement of the African continent with respect to Europe. Gasparini et al. (1985), using a large set of focal mechanisms, concluded that tensional motion is associated with large- and moderate-magnitude earthquakes distributed along the Apennine mountain belt. The southern part is characterized by the most intense vertical movements in the whole Italian peninsula, with earthquakes that have reached 7.5 magnitudes. The occurrence of the 1980 earthquake in the southern Apennines (M = 7) confirmed the normal faulting character of the earthquakes in this region with the horizontal tension axis in NE-SW direction (Deschamps and King, 1983; Westaway and Jackson, 1987; Pantosti and Valensise, 1990). Boccaletti et al. (1985) proposed for the northern Apennines a model based on ensialic shear with crustal thickening and compressive events indicative of a shortening process.

Anderson and Jackson (1987b) addressed the question of whether the Adriatic is a promontory of Africa or an independent microplate. The idea of the Adriatic promontory goes back to the early tectonic interpretation of Suess ( I885 - 1905) and Argand (1924) and was proposed by the plate tectonic interpretations of McKenzie (1972), Tapponnier (1977), Udias (1982), and others and questioned by Giese and Reutter (1978), Hsu (1982), and others who proposed an independent microplate. To solve this problem Anderson and Jackson (1987) used a large number of well-determined fault-plane solutions in northern and central Italy and the Balkans. They found normal faulting mechanisms in the Apennines that change into thrust at the northern end of the Adriatic sea and also along the Balkan coast. Examining the directions of the slip vectors on the circum-Adriatic border, Anderson and Jackson found that they are not consistent with those predicted from the pole of motion between Africa and Eurasia. However, the direction of slip vectors agree with those expected if the Adriatic block moves relative to Eurasia about a pole at 45.8"N, 10.2"E (north Italy) (Fig. 33). They concluded that the current deformation in the Adriatic area reflects not only the N-S shortening between Africa and Eurasia but also the rotation of the relative aseismic Adriatic block. The limits of this block with Eurasia are clearly marked by active seismicity. Not so the limit with Africa, tentatively located at the south limit of Italy to the Balkan coast. This implies that the relative motion between Africa and the Adriatic block is probably very small. This model describes the present-day motion and does not preclude that in the past motions have differed from those occurring today. Westaway (1990) localized the limit of the Adriatic block with Af- rica at the Gargano seismic zone (Fig. 22). The southern part of the block moves northeastward with respect to the leading edge of the African plate. More recently, Westaway (1992) stated that the Adriatic block is neither a rigid promontory of Africa nor a rigid microplate. The absence of significant localized deformation i n


FIG. 33. Direction of slip vectors in the circurn-Adriatic border. Dashed lines, observed; continuous lines, derived from the pole of rotation for Adriatic block with respect to Europe (Anderson and Jack- son, 1987b; reproduced with permission from Geophys. J. R. Asfron. Sor., 0 1987 Blackwell Scien- tific Publications).

the Adriatic sea linking the northern ends of the extensions in the central Apen- nines and the shortening in coastal Balkans suggests that this is not a rotation of a rigid body around an Eulerian pole. He bases these results on a study of historical earthquakes in Italy. Westaway found that the Apennines rather than being a single deforming zone can be divided into four parts with different senses and rates of coseismic deformations.

In a different approach, Lavecchia (1988) linked the situation in the Apennines to the recent evolution of the central western Mediterranean as a whole. The Ligurian - Balearic Basin and the Tyrrhenian sea are interpreted as lithospheric pull-apart basins favored by an eastward component of motion of the African plate with respect to the Eurasian plate. The development of the Apennines are, then,


not due to collision and accretion at the convergent European- African boundary, but subordinate to the extensional processes that led to the progressive opening of the Balearic-Ligurian and Tyrrhenian seas. Favali ef al. (1993) presented new evidence for the existence of a roughly E-W trending tectonically active boundary across the Adriatic basin. This boundary is formed by a deformational belt of lithospheric importance that may be an active zone of decoupling between the northern and southern Adriatic blocks. The belt is continued across the Ap- ennines by a N-S right-lateral shear zone inside which blocks rotate producing left-lateral strike-slip motion. The boundary is continued in E- W direction across the Tyrrhenian sea to Sardinia. The deep structure under the northcentral Apennines has been studied by Panza et al. (1980) and Panza and Suhadolc (1990b) finding a lithospheric root overlaid by hot low-velocity material that might have been emplaced by lateral gravity sliding, connected with the opening process of the Tyrrhenian sea. This aseismic root is probably detached from the lithospheric layer above and fits quite well with the pull-arc model proposed by Laubscher (1 988). This author distinguishes between push-arcs and pull-arcs. The former are formed by frontal collision on a subduction zone, while the latter are generated at zones where the subducting plate recedes pull-down by its own weight and leaves in its wake an extensional basin. Push-arc conditions are present in the western Alps and pull-arc in northern Apennines.

6.3. Alps

Earthquake distribution in the Alps extends over a wide arcuate area. As already recognized by Sieberg (1923), seismicity is moderate and large earthquakes are very seldom occurrences. There is an abundant literature on the tectonics and geologic evolution of the Alps (Kober, 1955). Schneider (1968) presented a seismotectonic study including seismicity and focal mechanism data. Earthquakes are of shallow depth ( h ( 20 km) and occupy the arc region of the mountain belt from the southern Alps to the Jura and Swabisch Alps in the north. Focal mechanisms correspond to reverse or strike-slip faulting and show a very consistent direction of horizontal pressure axes in N-S to NW-SE directions. The horizontal tensional axes, corresponding to strike-slip mechanisms, are in E-W to NE-SW directions. These results are used in the seismotectonic stress field map of Ritsema (1969b) (Fig. 14), where the Alps are shown to be under horizontal compressive stresses in from E-W to N-S directions normal to the trend of the arc. These compressive stresses are related to the left-lateral strike-slip motion along the Balkan coast. For Ritsema (1969b), the main stress field in Europe has an E- W to ESE-WNW direction in contradiction to the N-S drift of Africa against Eu- rope, which caused the Alpine mountain system, according to the continental-drift authors (Argand, 1924). Since the stress field in the eastern Alps is clearly of


horizontal pressure in N-S direction, this is related to the northward motion in the Balkans. This idea is maintained even in his interpretation in terms of plate tectonics (Ritsema, 197 1). The presence of horizontal compressional stresses in the Alps normal to the trend of the mountain belt derived from focal mechanism determinations is documented by Ahorner et al. (1972) and Ahorner (1975). The same result is obtained by Pavoni (1975, 1980) observing a rotation of the orientation of maximum compressive stress following the rotation of the strike of the Alpine arc. The actual stress field that causes the present seismicity is very similar to the stress field of the last 5- 10 million years, which produced the neotectonic deformations. Strike-slip mechanisms present in the region have left-lateral motion. In the Friuli area, source mechanisms show predominant reverse faulting (Carulli et al., 1990).

Miller et nl. (1982) distinguish three belts of seismic activity in the eastern part of the Alps. The northern belt is associated with the overthrust of the northern Limestone Alps that override the northern foreland, the central zone starting at the Vienna basin follows the so-called “Thermen linie” with earthquakes extending all the way to the Tauern region, and the southern belt formed by a broad band of shocks in the southern Alps is caused by the underthrusting of the Adriatic microplate, viewed as a promontory of Africa. The Friuli earthquakes of 1976 are an example of increased activity in this underthrusting process (Cipar, 1980, 198 1). The seismicity of the Swiss Alps is rather diffused and that of the western Alps of moderate activity can be divided into several groups. Earthquake fault- plane solutions, as mentioned before, clearly show that the axes of maximum compression are nearly horizontal and always normal to the Alpine arc (Fig. 34). This means a gradual rotation of P axes from the NE-SW in the Ligurian Alps to NW- SE in the Swiss Alps and NNW-SSE in the eastern Alps.

The general scheme of plate tectonics in the Mediterranean with a collisional N-S motion between Africa and Europe and the consideration of Italy as a promontory of the African plate or a separate microplate leads to the idea of the Alps area as a zone of collision. Hsu (197 1, 1982) considers a block formed by Italy and western Balkans that has moved with the African plate against Europe. The Austro-Alpine nappes of the Austrian and Swiss Alps represent the original northern margin of this Italo-African block. Hsu (1982) prefers the notion of a microplate that has moved sometimes in solidarity with Africa and sometimes independently of it. To the collisional motion of the Adriatic block a residual counterclockwise rotation must be added, which may account for the strike-slip left-lateral mechanisms. The slip vectors in the Friuli area correspond to those expected from the Africa-Eurasia convergence and are consistent with the consideration of the Adriatic block as a promontory of Africa. However, they also agree with the independent rotation of the Adriatic block with respect to Eurasia proposed by Anderson and Jackson ( 1987b).

Mueller (1982, 1984a, 1989, 1993) shows that the Alpine region cannot be


I FIG. 34. Seismotectonic stres field map based on data of focal mechanisms of earthquakes in the

Alps: ( ( I ) P axes; ( b ) lines normal to the trend of mountain range (Mueller. 1984a).

completely understood unless its deep structure is considered. The Alps must, then, be considered as a deep-reaching crust-mantle structure due to the compressive push of the African plate through the Adriatic promontory against the Eura- sian plate, resulting in a continent-continent collision. The conspicuous thickening of the Alpine crust and the asymmetric shape of the crust-mantle boundary are indications of a much deeper-reaching structural anomaly beneath the Alps, involving the entire lithosphere-asthenosphere system. Results from deep seismic sounding (Miller et al., 1982) and surface wave dispersion (Panza et ~ l , , 1980) show the characteristics of the deep structure under the Alps. The Alpine crust is characterized by an asymmetric crust-mantle boundary that reaches its greatest depth of approximately 55 km south of the central region and rises rather steeply


toward the inner arc side of the Alps. Within the upper mantle a high-velocity block has been detected. This lithospheric block of higher velocity could correspond to two nearly vertical slabs of lower lithosphere subducted to the south and to the north, respectively, during the plate collision process forming the Alps. This special case of subduction is denoted by the word subjuence ( “Verschluckung”). In this process, both African and Eurasian lithosphere is subducted, penetrating into the asthenosphere to a depth of 130-200 km at the inner side of the Alpine arc. A lithospheric configuration of this type generates regionally a compressive dynamics on which, within a wider framework, rotational processes may be superimposed.

In conclusion, the Italian peninsula and part of the Adriatic form a separate plate that act at times in solidarity with the African plate. In the south, this plate is linked with the Calabrian arc, which has advanced over the Mediterranean Iitho- sphere, causing its subduction. At present, subduction is stopped and the slab is under internal pressure. Along the Apennines there are horizontal tensions that produce normal faulting. The Adriatic is considered as a separate plate causing pressure on the eastern and Dinaric Alps and the coast of the Balkan peninsula, or a deforming zone divided into several blocks. The Alpine arc is subject to compressive stresses normal to the strike of the arc. In depth there is an asymmetric root of the crust-mantle boundary corresponding to two vertical slabs subducted to the north and the south.

7. HELLENIC ARC, ANATOLIA, A N D CARPATHIANS

In the easternmost part of the Mediterranean region the Aegean Sea, the Hel- lenic arc and the Anatolian peninsula are found. To the north, of these regions are located Greece, Macedonia, the Balkans, and the Carpathians. In terms of seismicity, these are the most active regions of the whole Mediterranean. We will separate the review of these regions into three parts: Hellenic arc and Aegean, Anatolia, and Carpathians. These three areas are dominated by three different types of tectonic processes. We will not discuss the seismicity of the Balkan coast and Dinaric Alps here as these were discussed in the previous section.

7.1. Hellenic Arc and Aegean

Although the arcuate shape of the tectonic features in Greece and the Aegean islands was already recognized by Suess ( 1 885- 1905), in the modern sense the arc structure was first pointed out by Gutenberg and Richter (1949). They found that the characteristic features of the Pacific arcs are incompletely established here. Modern catalogs of earthquakes in Greece were compiled by


Galanopoulos (1960, 1961) and more recently by Markrapoulos and Burton ( 198 1). Galanopoulos (1 963) stated that earthquakes of intermediate depth (maximum depth 200 km) show a clear trend and are concentrated in the southern Ae- gean sea. The first interpretations of focal mechanisms show reverse faulting for the earthquakes on the Hellenic arc and normal and strike-slip faulting in northern Greece. Galanopoulos (1967) interpreted the fault-plane solutions available for the northern Aegean and the Cretan arc as corresponding to a conjugate fault system. However, Papazachos and Delibasis (1969) attributed the mechanisms on the arc to faults separating continental from oceanic crustal blocks in which the oceanic blocks move downward with the respect to the continental. Ritsema (1969b) related the origin of the Aegean arc with right-lateral horizontal motion in northern Turkey (north Anatolian fault) and left-lateral in the Balkans. In his opinion the Aegean basin must be regarded as a kind of subsidence zone ( “Verschluckungszone”) for material that drifts into the region from both ENE and NW directions.

McKenzie (1972) presented the first plate tectonic interpretation of the region in terms of a separate Aegean plate. This plate with an aseismic part corresponding to the central Aegean sea is surrounded by active belts on all sides. The direction of slip vectors of focal mechanisms can be explained by a south-westward motion of the Aegean plate relative to both Eurasia and Africa (Fig. 17b). On the southern limit of the Aegean plate, the Mediterranean sea floor underthrusts the Hellenic arc or Cretan arc. In the northern Aegean, the situation is more complex. There are two types of mechanisms: strike-slip right-lateral motion on planes striking N-E and normal faulting on E-W faults. The situation is described as produced by the south-westward motion of the Aegean plate that produces general disruption and seismic activity over large regions of northern Greece and Albania. The intermediate-depth earthquakes are explained in terms of a subducted Mediterranean lithosphere dipping under the Aegean plate. The length of the slab is about 350 km, which corresponds to the distance the Aegean plate has moved toward Africa in the last 10 million years at a rate of about 3.5 cm/yr. McKenzie (1978) presented a more detailed study of the Hellenic region based on new fault-plane solutions, satellite images, seismic refraction, and other geophysical data. He concluded that rapid extension is now taking place in the northern and eastern parts of the Aegean sea region. This process has produced stretching of the continental crust under the Aegean sea to half its original thickness and is still occurring in western Turkey and some parts of Greece. The stretching is not confined to a small number of faults, but occurs throughout large areas. The Hel- lenic arc itself differs little from other island arcs elsewhere, with subducted lithosphere coming from the Mediterranean sea floor and dipping under the arc. In the dynamic explanation, McKenzie suggested that the forces involved in the process result from flow associated with the sinking of the slab and from the cold lower parts of the lithosphere detaching and sinking as a block through the mantle.

188 AGUSTh UDiAS AND ELISA BUFORN

Papazachos and Comninakis (197 1) interpreted seismicity data from Greece in terms of the underthrust of the African lithosphere under the Eurasian lithosphere plate in the Aegean island arc. Papazachos (1976) studied the direction of the pressure and tension axes derived from focal mechanisms. Along the descending slab, pressure axes are horizontal. In northern Greece, he found horizontal pressure axes in E-W direction and horizontal tension axes in N-S direction. The first are associated with a collision due to the westward motion of the Anatolian block relative to Europe and the second, to the process of lithosphere accretion that takes place in the Aegean arc. Accretion instead of normal subduction occurs in areas of lithosphere convergence associated with tensions and block faulting and results in an irregular distribution of intermediate-depth earthquakes. This model is abandoned (Papazachos and Comminakis, 1977) for another involving two different Benioff zones and consequently two descending lithospheric slabs, the Mediterranean and a second to the north of the Aegean. Consequently there are also two back-arc basins: the southern and northern Aegean. This hypothesis is not accepted by McKenzie (1978) on the grounds of the absence of sufficient observational basis. Combining focal mechanisms and geologic information, Papazachos et af. (1987) studied the pattern of faulting in the Aegean area, finding thrust faults along the coast of Albania and the Hellenic arc, with slip vectors normal to the coast in the western part and parallel to it in the eastern part. Central Greece, western Turkey, southern Bulgaria, and Yugoslavia are dominated by normal faulting with slip vectors oriented in an almost N-S direction. Strike-slip is present between central Peloponesus and southwestern Turkey with slip vectors related to the southwestward motion of the aseismic southern Aegean block (Fig. 35). Distribution of stresses in the interior of the subducting slab (Kondo- poulou et al., 1985) shows that neither P nor T axes are oriented under the slab. P axes are nearly horizontal, and shocks may be related to release of stresses within the sinking slab rather than by shearing at its upper boundary. The model favored more recently by Papazachos and his coworkers (Papazachos, 1988) consists in a subducting lithospheric slab from the African plate and a back-arc broken lithosphere under the Aegean area. Between these two lithospheres there is a region of low Q, where convecting motion takes place. The convection cells exert horizontal forces at the base of the Aegean lithosphere, breaking it and forcing it to expand while hot material intrudes into it. These convective cells also drag the southern Aegean lithosphere toward the Hellenic arc. Papazachos and Kiratzi (1992) estimated the active crustal deformation in central Greece from fault-plane solutions and seismic moments and obtained that the dominant mode of deformation is extension at an azimuth of N22W and at a rate of 4.2 mm/yr.

A different approach is used by Makris (1976, 1978a,b) using a combination of geophysical data, such as seismic refraction, gravity, and magnetic anomalies and heat flow, besides seismicity distribution. Present tectonics is explained in terms of a hot plume or lithotermal system ascending from the asthenosphere through


FIG. 35. Seismotectonic framework of thc Aegean area with direction of horizontal stresses and strike-slip faulting (Papazachos el nl., 1987; reproduced with permission from PAGEOPN. 0 1987 Birkhauser Verlag AG).

the lithosphere below the Aegean sea and expanding gradually from the inner zones outward. This process causes the tensional tectonics of the expanding Ae- gean sea. The Aegean crust is forced to override part of the eastern Mediterranean lithosphere, creating the Hellenic trench and the subduction zone at the collision front. Deep seismicity is caused by crust and upper-mantle fragments subducted into the soft asthenosphere, explaining in this way the complicated Benioff zone.

Le Pichon and Angelier (1979) objected to the radial spreading suggested by Makris ( 1 978a) and explained the sinking slab by a 30" rotation of the Aegean plate around a pole at 40°N, 18"E forming a consuming boundary with respect to the eastern Mediterranean sea floor. The motion of this consuming boundary is due partly to broad extensional deformations of a nonrigid Aegean plate. The northern boundary of this plate is considered to be rigid, and the eastern is formed partly by a westward branching of the north Anatolia fault and partly by exten-


sional deformations that change the motion to the southwest. For the dynamic conditions, they suggested that the situation is governed by forces acting within the Aegean lithosphere itself due to gravitational spreading. They stated that the generalized extension within the Aegean plate cannot be explained by forces acting in the asthenosphere (McKenzie, 1978) or by the extrusion of Turkey (Tapponnier, 1977). Berckhemer (1977) argued that the difference in elevation between the Aegean sea crust and the eastern Mediterranean floor results in a large gravitational force acting outward from the Hellenic arc that overrides the African lithosphere and produces its subduction. Le Pichon and Angelier (1979) proposed besides a second gravitational force acting within the sinking slab, due to its larger density than that of the surrounding mantle. This model was revised and modified by Angelier et al. (1982), who proposed that the motion of Crete relative to central Aegean is not a rotation but a pure N-S translation and explained that the uplift of the outer arc by underplating is due to the subduction process. The tectonic evolution of the Aegean was interpreted by Dewey and Sengor (1979) in terms of complex slip patterns across the boundaries of several microplates. Continental collision involving irregular margins leads to the development of mosaics of microplates and zones of semicontinuum tectonics between converging larger plates. In this way, rapid transform motions, extensional segments, and subduction zones have been active in the region at different times. Rotstein (1985) considers the Aegean and Anatolia as parts of one complex plate that rotates in counterclockwise fashion. This motion is accommodated, at present, by subduction throughout most, but not all, of its southern boundary. The interaction at the eastern part of the arc is described as a "side-arc collision" that involves heated oceanic lithosphere with middle and lower continental lithosphere. The extension in western Turkey and throughout much of the Aegean is explained as the result of the motion of the Aegean plate when normal rigid-block motion is prevented by the various constraints.

The extensional tectonic regime in the Aegean was studied by Mercier et al. (1989) and Mercier and Carey-Gailhardis (1989) by analysis of fault kinematics resulting in an evolution of tensional direction from WNW-ESE (Upper Mio- cene) to NE-SW (Pliocene-Lower Pleistocene) and to N-S (Middle Pleistocene to present). They proposed that two major Hellenic arcs have been active in the Aegean since Middle Eocene. The present Aegean slab is not older than 16 million years. Tomographic studies by Spakman et al. (1988), however, suggest that the slab is much longer than its active part and thus may be as old as 45-50 million years. Hatzfeld et al. (1989a,b) used the results (hypocentral and focal mechanism determinations) from an extensive microseismicity survey to study the seismotectonic situation of the region. They concluded that shallow seismicity is spread over a wide area in the Peloponesus. Deeper earthquakes define a subducted slab dipping gently ( I 0") toward the northeast for the first 200 km and more steeply (45") beneath the gulf of Argolis. Tension axes of fault-plane solutions for the deeper


events are consistent with the pulling of the cold lithosphere probably toward the north. The sharp bending of the subducting slab may be due in part to its loading by the overriding deforming plate.

Jackson and McKenzie (1988) and Taymaz et al. (1991a) reexamined the problem of the connection between the westward motion of Turkey relative to Europe and the extension in the Aegean region and also the transition from localized strike-slip motion in the north Anatolia fault to the distributed deformation further west. Using newly calculated focal mechanisms of earthquakes, characteristics of active faulting and paleomagnetic, structural, and geodetic evidences of block rotations, they arrive at a new solution for the tectonics of this region. This solution is based on a kinematic model of rotation of broken slats by which strike- slip motion is accommodated. They concluded that the westward motion of Tur- key relative to Europe is accommodated not only by localized slip on the north Anatolian fault but also by distributed right-lateral shear in NW Turkey, Aegean sea, and northern Greece (Fig. 36). The resistance to the westward motion of the western margin caused by the continental collision in Albania and NW Greece does not allow the necessary rotation to accommodate all the distributed simple shear in the east. The extension in the Aegean pushes continental material over the oceanic crust of the eastern Mediterranean to form a subduction zone. More-

40

35

FIG. 36. Sketch of the horizontal motions in the Aegean region along the major strike-slip faults (Tdymaz et ul., 1991a; reproduced with permission from Grophys. J. Inr., 0 1991 Blackwell Scientific Publications).

192 AGUSTIN UDIAS AND ELISA BUFORN

over, the subducted lithosphere slab beneath the southern Aegean generates extension in the lithosphere above it as it sinks into the mantle.

7.2. Anatolia

The peninsula of Anatolia is known to have suffered frequent large earthquakes. Sieberg (1923) attributed seismic activity to the fractured nature of the regions of the western and northern coasts. Ketin (1948) presented one of the earliest studies of the activity along the long east-west fault, which is now known as the “north Anatolian fault.” For the period 1938-1948, he lists 10 earthquakes with maximum intensity between VIII and XI Mercalli with epicenters along this long fault. Most of these earthquakes produced observable surface ruptures with predominant right-lateral horizontal displacements. Ketin concluded that the alignment and the displacements of the earthquakes are due to the westward drift of thc Anatolian block with respect to its frame. He tried to explain why there is not a corresponding fault in the south, proposing that the northern limit is a weakness zone that limits with the Black Sea while the southern limit is fixed to the Taurus mountains. Ambraseys ( 1970) studied the seismicity of Turkey, finding that during 1909- 1969 over 30 shallow earthquakes with magnitudes equal to or greater than 6 occurred in the north Anatolian fault zone. Of these, at least 10 are known to have been associated with right-lateral faulting observable in the field, in most cases each new rupture overlapping or beginning where the faulting ended in the previous earthquake. Ruptures on the north Anatolian fault are mapped and studied in detailed form. Its total length is about 1000 km, and its eastern end intersects with a southwest trending “conjugate” fault (“east Anatolian fault”). Historical earthquakes show that they fall along three well-defined lines. The first on the north, along the north Anatolian fault with east-west trend. The second starts near the east end of the north Anatolian fault and runs southwest to Antiochia, joining with the Dead Sea fault system. The third starts from the Aegean coast and dis- sects the south west corner of the Asia Minor, (western Turkey) following scattered fault segments. Canitez and Uqer (1967) calculated 77 fault-plane solutions in and near Anatolia, but they did not relate the solutions with the tectonics of the region. Ritsema ( 1969b) using the solutions obtained by Canitez and coworkers pointed to their very consistent right-lateral transcurrent motion along nearly east-west striking fault planes in the north of Turkey. Fault-plane solutions are i n complete agreement with motion along the north Anatolian fault zone as derived from field studies. This motion results in a westward drift of the Anatolian block that Ritsema related with the origin of the Aegean arc.

The first plate tectonic interpretations of this region are those of McKenzie (1972, 1978). The north Anatolian fault is interpreted as the northern boundary of the Turkish plate, which moves westward with respect to both Eurasia and Africa

SEISMOTECTONICS OF THE MEDlTERRANEAN REGION 193

(Fig. 17b). The slip direction derived from focal mechanisms is very consistent, changing from 105"E at the eastern end to 90"E at the western end. At the eastern end, the mechanisms change from strike-slip to a combination of strike-slip and overthrust. The southeast border of the plate is formed by the east Anatolian fault, which can be followed from satellite photographs (McKenzie, 1976). Motion in this fault is a consequence of the relative movement between the Anatolian and Arabian plates. The union of the two faults forms a triple junction in which the Anatolian, Arabian, and Black Sea plates come together. The pole of rotation of the Anatolian plate with respect to Eurasia derived from the slip vectors is located at 18.8"N, 3S"E. Seismicity in western Turkey has a different nature and is distributed along a number of bands running approximately in E-W direction. Focal mechanisms show predominant normal faults that agree with the observed surface ruptures of large earthquakes. Slip vectors show a systematic change in trend, from ENE in the north to NW in the south. In consequence, the zone is under extension and the plate is being stretched, in a similar way as in some parts of the Aegean plate.

Seismicity on the north and east Anatolian faults is revised by Dewey (1976), who studied the interoccurrence times between shocks. The mechanism of the westward motion of Anatolia along the north and east Anatolian transform faults proposed by Sengor (1979), Sengor and Yilmaz (19811, and Sengor and Canitez (1982) is a response to the continental collision between Africa and Eurasia by which continental lithosphere (Anatolian plate) is wedged out into areas where there is easily subductable oceanic material (Hellenic arc). According to these concepts, the tectonic evolution from Miocene to the present is described. At the eastern end, the triple junction has a complex nature with two strike-slip and one thrust branch separating continental plates. On the western end, the region is under roughly N-S extension by numerous E- W striking grabens. The Anatolian plate is considered to be escaping from a N-S convergent to a N-S extensional environment (Fig. 37). Rotstein and Kafka (1982) and Rotstein ( I 984) argued that the southeast boundary of the Anatolian plate is not a transform left-lateral fault (east Anatolian fault), but a zone of convergence, a continuation of the Cyprus arc. The Cyprus arc is a case of oblique subduction, while in the southeast border the main mechanism is that of thrusting. The east Anatolian Fault is not the plate boundary but one of several faults that splay from the north Anatolian fault and deform the Anatolian block. An appropriate model, therefore, for the Anatolian plate cannot be one of rigid rotation about a single pole, but a nonuniform, counterclockwise rotation coupled with internal deformation, which includes extension in the western part and splay faults in the eastern segment.

In a more recent assessment of earthquake occurrence in Turkey and neighboring areas between 1899 and 1915, Ambraseys and Finkel (1987) concluded that although a 17-year-period is hardly sufficient to allow conclusions about the long- term seismicity, many other parts of Turkey, removed from the north Anatolian

FIG. 37. Orogenic zones and plate boundaries with direction of horizontal motions in Turkey (Sengor and Canitez, 1982).


fault zone, are capable of producing earthquakes of magnitude ) 7. According to Barka and Kadinsky-Cade (1988), fault geometry (distribution of discontinuities such as bends and stopovers along the main fault trace) in the strike-slip faults of Turkey plays an important role in the Occurrence of earthquakes. They concluded that the geometry of segmentation is not simple but occurs in complex patterns and large earthquake segments often contain restraining bends or double bends.

New focal mechanism studies combined with satellite imagery are presented by Jackson and McKenzie (1984) leading to the following conclusions. The left- lateral strike-slip east Anatolian fault may continue in south-west direction toward Cyprus. In Central Anatolia, several S-W trending faults branching off the north Anatolian fault (splays) occur west of its junction with the east Anatolian fault. These splays may record older positions of the east Anatolian fault that may have progressively jumped eastward, thereby changing the motion on E- W faults from thrusting to strike- slip and lengthening the north Anatolian fault. Another possibility is that some of the westward motion of Turkey is taken up on these splays in central Anatolia, casting doubt as to whether it can really be considered a rigid block. The situation in southern Turkey and Cyprus is not sufficiently clear, although a genuine zone of intermediate depth seismicity dipping north or northeast does exist. Its link, however, with the subduction zone in the Hellenic arc remains enigmatic. The problem of the east Anatolian fault is newly examined by Taymaz er al. (1991 b) using the focal mechanism of four large earthquakes. Their solutions include oblique normal and reverse faulting in contrast to the simple strike-slip solutions on the north Anatolian fault. This complexity of the focal mechanisms, as well as of its structure, suggests that the east Anatolian fault is less developed than the north Anatolian fault and may be changing to accommodate any instability of the kinematics of their junction. Lyberis et al. (1992) offer a different interpretation of the east Anatolian fault based on satellite and field observations. The important feature is an oblique N -S collisional thrusting between Arabia and Anatolia and sinistral strike-slip faults are second-order and local consequences of the main process. According to Peringek and Cemen (1990), the east Anatolian fault may join with the Dead Sea fault zone at three locations at the northeastern extension of the latter.

7.3. Carpathians

Earthquakes in the Carpathian region are concentrated in the sharp bending of the mountain chain known as the “Vrancea region.” The region is described as of moderate activity by Sieberg (1923). Gutenberg and Richter (1949) called the attention to the remarkable source of intermediate-depth shocks with frequent repetition from nearly the same focus at a depth of 100- 150 km. They pointed out that the arc features are not well developed. Constantinescu and Enescu (1964)

196 A G U S T h UDfAS AND ELISA BUFORN

and Constantinescu et al. (1966) determined the first focal mechanism solutions and their tectonic interpretation. Most mechanisms are of reverse dip-slip faulting with the horizontal component of pressure axes in NW-SE direction and thus transversal to the Carpathian arc. Ritsema (1969b, I97 I ) found that foci are concentrated in a more or less vertical slab in N-NE direction. The slab is about 100 km in length, 35 km in width, and 100-200 km in depth. The nearly E-W direction of compressive stresses is explained by the northward squeezing of the Bal- kan peninsula by the eastward motion of the Calabrian arc and the westward motion of the Anatolian block.

McKenzie ( 1 970, 1972), in his plate tectonic interpretation of the Mediterra- nean region, considered that earthquakes under Romania occur in a vertical slab within the mantle that closely resemble those beneath island arcs but that is now overlaid by continental crust. He assigned the origin of this slab to the rapid motion of a small plate containing the Carpathian mountains and surrounding regions in southeast direction relative to the Black Sea plate. Consumption of the oceanic crust existing there in the past could probably have produced the slab where the seismic activity occurs. Roman ( 1970, 1973) redetermined the earthquake hypocenters, finding that at depths of 30-60 km there is a region of very low activity, while the region responsible for the principal events is located between 60 and 160 km. The focal mechanisms of intermediate earthquakes show compressional axes very close to horizontal and perpendicular to the slab and tensional axes contained in the plane of the slab. This suggests that the slip is caused by the gravitational pull of the lithosphere itself. The cause of the slab is a subduction under the Carpathian arc of the Black Sea plate that moves to the WNW relative to Eurasia. There is, then, no need for a separate small plate (Transilvanian plate).

Fuchs el al. (1979) presented a tectonic model for the Vrancea region in which the continent-continent collision of the Moldavian platform with the eastern Car- pathians has brought active subduction from the NE to a standstill. The region is considered as a piece of the slab detached and separated from the lithosphere by the underthrusting of continental crust. Downward motion of the deep block pro- vokes downward creep in the low-viscosity zone, which exerts vertical suction on the lower crust. This explains the nearly horizontal fault planes of the focal mechanisms in deep earthquakes and shallow seismicity resulting from the continuing continent-continent collision. The geologic evolution of the Carpathian arc is attributed by Burchfiel(1980) to the interaction of fragments of continental crust in the eastern European Alpine system that do not act as rigid blocks, but are strongly deformed. For example, the Rhodopian fragment, which forms the orocline has been detached by thrusting from the lower crust and mantle. Studies of P-wave velocity perturbation of the deep structure by Oncescu et al. (1984) and Oncescu (1984) favor a tectonic interpretation in terms of a NE-SW subduction down to about 180-km depth, which is no longer active, except on its southeastern margin (Vrancea region), where it penetrates deeper into the mantle. Intermediate seismic


INTERMEDIATE SEISMIC ACTIVITY

FIG. 38. Sketch of the proposed model for the Vrancea seismic region (Carpathian) (Oncescu, 1984).

events occur not within the sinking slab, but at the limit of separation between the detached gravitationally sinking block and the rest of the slab. The decoupling of the now sinking slab may have been caused by the NW push of the Black Sea plate (Fig. 38). An existing low-velocity zone (40-80 km) is interpreted as a low- viscosity zone or as a thick layer of sediments subducted together with the slab.

A large set of 114 fault-plate solutions of intermediate shocks are studied by Oncescu and Trifu (1987). The general tendency of vertical tension axes and horizontal pressure axes confirms the vertical extension and SE-NW horizontal contraction of the slab corresponding to a gravitationally sinking model. At 90- 105-km depth, focal mechanisms are of different kinds, reflecting local heterogeneities in stress, and there is a tendency for Taxes to move off the vertical axis as the depth increases. Oncescu (1987) finds that in the Vrancea region, compression axes as derived from small earthquakes are in a NE-SW direction and not in a SE-NW direction. The second direction is found from all strong and most moderate earthquakes, but they may correspond to preexisting planes of weakness. The argument, however, does not seem very conclusive. Also, a clear difference between the stress tensor within the crust and in the intermediate-depth region is found, supporting the hypothesis of the existence of a gravitationally sinking slab now decoupled from the crust.

In conclusion, the Aegean plate is undergoing a N-S extension that contributes to its southward advance over the Mediterranean lithosphere at the Hellenic arc. The subducted slab dipping to the north is driven by gravitational forces and possibly also by mantle convection. The Anatolian plate has a net westward motion with respect to Europe along the north Anatolian fault. This motion is accommodated in western Turkey and northern Aegean by normal and strike-slip faults and is finally transmitted to the collision zone of northwestern Greece and Alba-


nia. The Aegean and Anatolian plates, either as one or two separate plates, are strongly deformed, and their motion can hardly be regarded as a rigid rotation. The situation in the Vrancea region, in the bend of the Carpathian mountains, is due to a remnant of a subducted slab locked, at present, in a continental area. Its origin is interpreted in different ways.

8. CONCLUSIONS

Seismotectonics of the Mediterranean region has been studied from the early dates of seismology. The distribution of hypocenters of earthquakes and their magnitude and focal mechanism constitute the basic data in these analyses that later include other information from geodetic measurements, geologic field observations, satellite images, and strain measurements. The first studies used block tectonics as guidelines to interpret seismologic data. Continental drift had little influence in seismotectonic studies, and only with the arrival of plate tectonics was a general tectonic scheme established to explain the complex situation in the Mediterranean region. This region, extending to the west up to the Azores islands, must be understood as the boundary between the Eurasian, African, and Arabian plates. From Azores to Gibraltar, the boundary separates oceanic lithosphere in both sides and is relatively simple, while along the Mediterranean, both sides have a continental nature and the situation becomes very complex with the presence of intermediate small plates. The general motion of Africa with respect to Europe can be represented by a rotation about a pole near the Canary islands that produces a N-S colliding motion of the two plates along the Mediterranean. This motion is complicated by the presence of secondary smaller plates with independent rotations with respect to the two main plates. The number of small plates vary according to different authors from 2 to 10; the most commonly accepted are those of Adriatic, Aegean, and Anatolian. The situation, however, cannot be only explained in terms of rotations of rigid plates and must also include continuous deformations. Subducted slabs of lithosphere are present at the Calabrian, Hellenic, and Carpathian arcs where deep and intermediate earthquakes occur. Behind the Calabrian and Hellenic arcs stretching of the lithosphere takes place, which contributes to its advance over the subducting material. Details of the processes in particular active seismic areas are explained differently by different authors, but there is general agreement on the overall situation.

Many questions are still open, such as the nature of the subduction processes, the limits and motion of the small plates (e.g., Adriatic, Aegean, and Anatolian), the regions of continuous deformations, and the proportion of seismic and aseismic slip. The continental nature of the plates involved in the Mediterranean and the complex evolution of this collisional boundary makes it difficult to give very definite solutions to many of these problems. A difficulty in the seismotectonic


analyses arises from the fact that seismicity has been known for only about 80 years and accurate hypocentral determinations and focal mechanisms are available for only the last 40 years. These are very short time intervals for understanding and predicting the occurrence and recurrence of earthquakes past, present, and several hundred years in the future. An important source of information to avoid this problem is historical and paleoseismicity, but these are data difficult to obtain and handle, and hence are ignored by authors who prefer to limit their studies to only very recent data. Incorporation of other sources of information such as in situ stress measurements, satellite imagery, recent geologic features, and geodetic measurements will help to understand the complicated seismotectonics of the Mediterranean. There is still much we can learn about this subject.

ACKNOWLEDGMENTS

The authors wish to thank all the colleagues that have given permission to reproduce figures from their articles. Also they thank Dr. R. Dmowska, Harvard University, who encouraged us to write this chapter, and Drs. J. Mezcua, Instituto Geogrifico Nacional, Madrid, and A. F. Espinosa de 10s Mon- teros, U.S. Geological Survey, Denver, Colorado for their help. This work was partially supported by Direccion General de Investigation Cientifica y Ttcnica, Spain, project PB92-0184 and the European Community DG XII, EPOCH, CT9 1-0042. Publication No. 354 of the Dpto. de Geofisica, Universidad Complutense, Madrid.

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INDEX

A oceanic crustal, 187 rotation, 168 tectonics, 136

Boundary, plate, see Plate boundary Buoyancy

induced flow in laterally heterogeneous fluid shell, 72 variational calculation, 5 1-58 viscous coupling, 57-58

induced plate motions, 5-31 calculation, 4 examples, 17-27 theory, 1 3 - 1 7

induced surface flow horizontal divergence and radial vorticity,

in model of mantle, 15 61 -72

internal sources, generation of plate motions, 30

Adriatic block, 159 Aegean block, 159 Aegean islands, tectonic processes, 186- 192 Africa

large stable block in, 124 southern, geoid highs, reduction in amplitude.

84-86 African plate

amplitude spectrum, 1 I - 12 boundary, 137 relative plate motion, 149- 150, 155 relative rotation, 143

Agadir-Nekor fault, 169 Alhama-Palomares-Carboneras fault, 170-

Alps, earthquake distribution, 183- 186 Amplitude

171

dynamic topography, reduction by depth in-

geoid highs, reduction, 84-86 lateral viscosity variations, mapping, 88 spectrum, 8- 12

creases of viscosity, 82-84

Anatolia, seismic activity, 192- 195 Angular momentum, contribution from mantle

Apennines, epicenter distribution, 180- 183 Arc structure, see also specijc arc

Mediterranean region, 124- 126 presence in Pacific and Europe, I35 - 136

flow, 90-91

Azores, plate boundary studies, 165- 169 Azores-Crevillente fault, 168 Azores-Gibraltar fault, 148, 157, 164, 166- 169

B

Basin, lithospheric pull-apart, 182 Benioffzone, 135, 178, 180. 188-189 Block, see also specijc block

high-velocity, in upper mantle. I86 large stable, in Europe and Africa, 124 lithospheric, interaction in Eurasia, 142

C

Cadiz-Alicante fault, 170-171, 174 Canary Islands, 175, I98 Carpathians, earthquakes in Vrancea region,

COCOS plate, amplitude spectrum, 1 I - 12 Continental drift

based model for northern Africa, 174 in early interpretations of Mediterranean re-

195- I98

gion, 126-127, 130, 141-142

poloidal-toroidal, 3D modeling, 4 scalar spherical harmonics, 96- 101 between toroidal and poloidal components of

viscous, to buoyancy-induced flow, 64-66

Coupling

lithospheric flow, 39,43,57

Coupling matrix, 17,22 Creep

aseismic, 159 mantle and lithosphere, over geologic time

in Mediterranean region, 149 scales, 1-2

21 1

212 INDEX

Creep, (continued) steady-state, characterization in terms of ef-

fective viscosity, 3 I Crust, oceanic, 136, 149, 167 Cyprus arc, 193

D

Decomposition, harmonic, horizontal divergence and radial vorticity, 101 - 107

Density heterogeneity, very long wavelength,

Density perturbation internal, 84-87

mantle, 15-16

horizontal

24-26

annihilation by projection operator, 19

Divergence

analytic harmonic decomposition, 101 - 107 buoyancy-induced surface flow, 61 -72 kernel calculation, 17-22 kernel expression, 60-61 spherical harmonic coefficients, 7- 13

surface, see Surface divergence

Pacific- Antarctic ridge, strength, 66-68 zones in fluid. 12

Downwelling

E

Earthquake, see also Seismicity Arette, 172 crustal, 178- 180 distribution in Alps, 183- I86 early maps, 130-136 Friuli, 184 intermediate-depth, I87 Maghreb region, 174- 176 region from mid-Atlantic ridge to Alboran

relation to tectonics, 121, 138 slip values, 158 Vrancea region in Carpathians, 195- 198

East Pacific ridge, heterogeneity, density con-

East Pacific rise, zone of rapid divergence, 93 Epicenter

alignments and related fracture zones, 132,

Sea, 166-167

trast, 27

134

determinations, 135 distribution along Apennines, 180- 183 for shallow earthquakes, 136

best-fitting pole of rotation, 160 general southward motion over Mediterra-

nean, 126- 128 kinematic evolution, IS5 relative plate motion, 149 tectonics, 142- 143, 145

early seismotectonic maps, 132- 136 fault-plane solutions, 136- 137 large stable block in, 124 tectonic stress, regional pattern, 164- 165 westward motion of Turkey relative to, 191

Eurasia

Europe

F

Fault, see ulso .vpeciJic fault parallel strike-slip, 174 strike-slip motion along, 141 transcurrent, 139

right-lateral oblique, 167 Fault-plane solution

Anatoiian fault zone, 192- 193 derived slip orientations, 158, 164 early application, 121 - 122 Europe, 136- 137 new seismotectonic interpretation, 168- 169 in northern and central Italy and Balkans, 181

buoyancy-induced Flow

in laterally heterogeneous fluid shell, 72 variational calculation, 5 I -58 viscous coupling, 57-58

large-scale, in spherically symmetric mantle, 28

lithospheric dynamically consistent treatment, 4-5 horizontal component, 32-33

mantle, see Mantle flow poloidal and toroidal

contributions to dynamic surface topogra

in fluid shell, 3 phy, 79-84

surface, see Surface flow toroidal

component of tectonic plate motions, 89-91 sublithospheric, 36

INDEX 213

Fluid shell buoyancy force effects, 3 isoviscous, surface flow, 60 laterally heterogeneous, buoyancy-induced

lateral viscosity variation effects. 61 -62 spherical

flow, 72

3D viscosity variations. 93 viscous flow in, 5 1-58

3D viscosity variation effects, 5 FXiil mechanism

Mediterranean region, map, 144 orientation, 121 in pattern of faulting in Aegean area, 188 related studies, and seismotectonics, 136- 142 solutions

Iberia. 172- 174 in new seismotectonic studies, 156- 158,

161, 163 Fracture zone, Europe, 132. 134

lateral density, 3 3D viscosity, 93

Hindu Kush, earthquake region, similarity to

Hotspot, frame of reference, 40, 89, 93 Rumania, 135, 146

I

Iberia, seismically active areas, 169- 174

K

Kinematics based model of rotation, 191 related evolution of Mediterranean region.

surface plate, analytic description, 6- I3 I55

L G

Geoid. see o h Mantle nonhydrostatic

effects of lateral viscosity variations. 84-

as gravitational field perturbation, 72 predicted, I7 - I8 in self-gravitating mantle, 115- I16 sensitivity to radial viscosity variations. 93

89

Gloria fault. 167- 169 Gondwana. 126- 128, 130, 150 Gorringen hank, formation, 167 Green functions, generalized, 58-61 Guadalquivir lineament, 168- 170

H

Harmonic coefficient, spherical. see Spherical

Hays-Atlas fault, I68 Hellenic arc

crustal material, 14 I earthquakes, 135 tectonic processes, 186- 192

density. very long wavelength, 24-26 internal mantle, I6

harmonic coefticient

Heterogeneity

Lanczos smoothing factor, 9- 13. 23 Laurasia, 126 Lithosphere

lateral viscosity variations in, 31 -48 and mantle, creep over geologic time scales,

net rotations, 39-41 predicted net rotation, 89 thickness and stiffness. 42

1-2

M

Maghreb. seismotectonic analyses, 174- 176 Mantle

density heterogeneity, 24-26 differential rotation, 89-91 dynamics with 3D viscosity variations, 48-92

formulation of variational principle, 48-5 I isoviscous, dynamic surface topography for,

74-75 and lithosphere, creep over geologic time

scales, 1-2 net rotations, 39-41 self-gravitating, nonhydrostatic geoid in,

softening, 69 spherically symmetric, large-scale flow, 28

115-116

214 INDEX

0 Mantle flow buoyancy-induced, interaction with surface

contribution to angular momentum, 90-91 driven by density perturbation, 15- 16 long wavelength, interaction with surface

nearly horizontal, Sicily-Calabria arc, 177 plate-driven, generated surface stresses, 30 theoretical modeling, 2-3

Azores-Gibraltar region, seismotectonics, 169 fault-plane solutions represented on, 122 Mediterranean region

focal mechanisms, 144 plate tectonics, 125 seismotectonics, 140 tectonic alignments, 138

plates, 19

plates, 23-24

Map

seismotectonic, and early seismicity, I30- 136

Mediterranean region focal mechanisms map, 144 geography, 122 proportion of seismic and aseismic slip, 159 seismicity, 122- 124 seismotectonics map, 140 tectonic alignments map, 138 tectonic evolution, 152- I56

Mid-Atlantic ridge, 166- 167 Model

density heterogeneity in mantle, 25-26 SHINM13, seismic heterogeneity, 17-24,

slab-heterogeneity, 26-27

mathematical, lateral variations in viscosity,

plate motions in spectral domain, 28-30 theoretical, mantle flow, 2-3 3D, for poloidal-toroidal coupling, 4

62,66-70,75

Modeling

3-5

Momentum, angular, see Angular momentum Morocco, seismicity, I75 Motion, plate, see Plate motion

N

North Anatolian fault, 146, 164, 189 North Atlantic ridge, heterogeneity, density con-

trast, 27

Orogenic belt, arc structures in, 124 Orogenic zone, Turkey, 193 - I94 Orogeny, Alpine, 126

P

Pacific- Antarctic ridge spreading, 62 strength of underlying mantle flow, 66-68

amplitude spectrum, 1 I - 12 hotspot tracks, 40

Pacific plate

Pangea, 126, 149 Plate. see also specijic plate

divergence field, harmonic coefficients, 14-

surface, see Surface plate velocity, scalar characterization, 7 with weak boundaries and strong interiors, 3,

17

48 Plate boundary

with finite region of weakness, 3 1 flow-induced deflections, effects of viscosity

free-slip, 37, 41, 89 free-slip and no-slip surface, modeling, 16 geometry, alignment with mantle flow geome-

Mediterranean region, 145, 147 no-slip condition, in calculation of flow, 29 no-slip lower, 38 stresses acting on, 5-6 studies, Azores to Tunisia, 165- 176

buoyancy-induced. 5-3 I

variations, 72-73

try, 19-20

Plate motion

calculation, 4 examples, 17-27 theory, 13-17

generated by internal buoyancy sources, 30 matching to plate-like surface flow, 45-46 nonunique interpretation, 25 poloidal and toroidal components, 48 smoothed by Lanczos factors, 23 strike-slip, 3

and dip-slip, 141 left lateral, 174, 180- 183 right-lateral, 148, 157

INDEX 215

from strike-slip to combination with over.

toroidal, 17

application to model of mantle density hetero-

calculation using spherical harmonic expres-

thrust, 193-195

Plate-projection operator

geneity, 25-26

sion of plates, 30 Plate tectonics, see also Tectonic plate

from 1885 to 1970, 124-142 early seismicity and seismotectonic maps,

early tectonic interpretations, 124- 130 focal mechanisms and seismotectonics,

130-136

136- 142 from 1970 to 1986, 142- 158

first interpretations, 142- 149 further seismotectonic studies, 156- 158 tectonic evolution, 149- 156

explanation in terms of lithothermal system, 188-189

interpretation of seismotectonic data, 158 theory proposal, 12 1

Portugal, seismicity, 172- 174 Projection operator

annihilation of internal density perturbation, 19 plate-, see Plate-projection operator properties, impact of truncation, 15- 16

convergence between Eurasia and Iberia, 176 seismically active area, 170- 172 strike-slip motion along, I74

Pyrenees

R

Rheology laterally homogeneous, 12 lateral variations, 3-4,92 linear, 50 lithosphere, 33 steady-state, creeping polycrystalline solid, 3 I

midocean Ridge, see also specific ridge

buoyancy sources beneath, 92 related density contrasts, 25-26

Rif, associated seismicity, I75 Rotation

in Aegean area, I90 differential, in mantle, 89-91

pole of Africa-Eurasia, I80 Mediterranean region, 159- 160

relative, African plate, 143 rigid-body. 4.92

in lithosphere. and mantle, 39-41 Rumania

earthquake region, similarity to Hindu Kush,

earthquakes under, 196 135, 146

S

Sea floor spreading, 142, 149- 150 topography mapping, 142

Seismicity, see also Earthquake Anatolia, 192- 195 deep, 189 distribution in Sicily-Calabria arc, 177- 180 early, and seismotectonic maps, 130- I36 Iberia, 170- 174 Morocco, 175 in plate tectonics theory, 121 in seismotectonic synthesis for Mediterranean

region, 141 Seismotectonics

Azores-Gibraltar region, map, 169 Maghreb, 174- I76 origin, 121 recent studies from 1986 to 1993, 158- 165 related maps, 130- I36

Shell, fluid, see Fluid shell Sicily-Calabriaarc, 176- 180

earthquakes, 135, 143, 148 Slip vector

direction, 181 for orientation of focal mechanism, 12 I , I58

South Atlantic ridge, heterogeneity, density con-

Southeast Indian ridge, heterogeneity, density

South Pacific. geoid highs, reduction in ampli-

Spherical harmonic coefficient, 37. 42

trast, 27

contrast, 27

tude, 84-85

divergence and vorticity fields, 7- 13 flow-induced surface topography, 72-73 horizontal divergence and radial vorticity, 44 plate divergence field, 14- 17

216 INDEX

Spherical harmonics basis functions, 28

generalized, 5 1 horizontal gradients, 94-96 radial, 54.59-60

components of dynamic surface topogrdphy,

coupling rules, 96- 101 description of plate functions, 19, 29 expansion of lateral viscosity variations, 45-

expansion of plates, 30-3 1

acting on plate boundaries, 5-6 compression, 137- I38 dependence of viscosity, SO deviatoric, 1-2 horizontal, continuity across lithosphere-

horizontal compressional, in Alps, 184 Mediterranean, regional pattern, 156, 158,

principle axes, for orientation of focal mechanism, 121

related field in Apennines, 18 I surface shear, 29-30 viscous, acting on undulating surface, 109-

111-115

48

Stress

mantle interface, 35-39

161-165

I l l Subduction

dynamic, occurrence on moving plate, 180 oblique, Cyprus arc, I93 passive, Calabrian arc, 177- 179

in Aegean area, 189- 192 buoyancy sources beneath, 92 consumption of oceanic crust, I49 in evolution of Tethys, 155 in Iberia, 174 related density contrasts, 2.5 -26 relationship to Hellenic arc, 141 i n south Spain, 172

Surface divergence, plate-like, 2 I -24 Surface Row

Subduction zone

buoyancy-induced horizontal divergence and radial vorticity,

in model of mantle, 15 61-72

plate-like, matching to plate motion, 45-46

geometry, 17 Surface plate

interaction with mantle flow buoyancy-induced Row, 19 long-wavelength Row, 23-24

kinematics, analytic description, 6- 13

T

Tectonic plate, see also Plate tectonics long-term behavior as rigid body, 27-28 toroidal-Row component of motions, 89-91 velocities, 44

Temperature dependence of viscosity, 50 variations

sensitivity of mantle viscosity, 2 yielding lateral variations of effective vis-

cosity, 5 Tethys sea

in early interpretations of Mediterranean re-

history, 150- 155 related microcontinent, 142

dynamic, with lateral viscosity variations,

sea floor, mapping, 142 surface

gion, 126-127, 129-130

Topography

111-115

dynamic, 72-84 isostatically reduced, 34-35

Turkey North Anatolian fault, 146 westward motion relative to Europe, 191

Tyrrhenian sea, opening, 180, 182

U

Upwelling central Pacific, 69 Pacific- Antarctic ridge, strength, 66-68 zones in fluid, 12

V

Variational principle application to problem of viscous flow in

formulation, 48-5 I spherical shell, 5 1-58

INDEX 217

Vector, slip, see Slip vector Velocity

plate poloidal and toroidal. 92 scalar characterization, 7

shear, reduced in upper mantle. 62 shear-wave, 179- 180

absolute, inference, 2 I -22 effective, 1-3 lateral variations

Viscosity

dynamic topography with, I 1 1 - 115 effects on nonhydrostatic geoid, 84-89 in lithosphere, 3 1-48

inverting for, 41 -48 theory, 3 1 - 39

mathematical modeling, 3-5 shorter wavelength variations, 77-79

temperature dependence, SO 3D variations, momentum conservation in

medium with, 107- 109 very-long-wavelength variations, 93

analytic harmonic decomposition, 101 - I07

buoyancy-induced surface flow, 61 -72 plates, 17

spherical harmonic coefficients, 7- 13

Vorticity, radial

buoyancy-induced, 22-24

Vrancea region, in Carpdthians, earthquakes, 195- 198

w

Wave, elastic, lateral speed variations, 5

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