JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 95, NO. B13, PAGES 21,329-21,344, DECEMBER 10, 1990

Mantle Flow Tectonics' The Influence of a Ductile Lower Crust

and Implications for the Formation of Topographic Uplands on Venus

DUANE L. BINDSCHADLER 1 AND E. MARC PARMENTIER

Department of Geological Sciences, Brown University, Providence, Rhode Island

Convective flow within the interiors of terrestrial planets is a primary source of large-scale topographic and tectonic features. We treat the crust and mantle as a layered viscous half-space and find solutions for flow driven by a buoyancy force distribution within the mantle and by relief at the surface and crust-mantle boundary. The crust initially responds to mantle upwelling with uplift, but continued flow causes significant crustal thinning and subsidence. Downwelling leads to initial subsidence and subsequent crustal thickening and surface uplift. Changes in crustal thickness are driven by (1) vertical normal stresses due to mantle flow and (2) shear coupling of horizontal mantle flow into the crust. The degree and time scale of crustal deformation are sensitive to crustal thickness and viscosity and the presence of elastic layers in the upper crust and mantle. Deformation can occur on time scales less than several hundred million years for reasonable values of these parameters. Strong or elastic upper crustal layers enhance crustal thinning or thickening, while large viscosity contrasts between the lower crust and upper mantle or elastic mantle layers tend to diminish deformation. For time-dependent mantle flow due to a rising or sinking diapiric body, crustal deformation depends upon the ratio of diapir radius to crustal thickness, and the ratio of crustal to mantle viscosity. A sinking diapir can lead to crustal thickening, elevated topography, and a specific sequence of time-varying deformation, While regions such as Beta Regio have long been suggested to result from convective upwelling in the Venus mantle, we suggest that a mantle downwelling origin is consistent with many of the characteristics of the compressional mountain ranges of western Ishtar Terra.

INTRODUCTION

Convective flow in the interior of a planet is the primary driving force for planetary-scale tectonics and the creation of large-scale surface features. It is therefore important to establish the surface expression of convective flow. On Earth, surface plate boundaries and motions, and hotspot swells are two particular expressions of convection. On Venus, the clearest expressions of convective flow appear to be swell-like features such as Beta, Bell, and Atla Regiones. Such large-scale topographic features display highly correlated long- wavelength gravity and topography [Sjogren et al., 1983] and great depths of apparent Airy compensation. These characteristics, as well as observations indicating volcanism and rifting [e.g., McGill et al., 1981; Campbell et al., 1984], have led to the suggestion that most large topographic features on Venus are supported by mantle upwelling in the form of hot, rising plumes [Phillips and Malin, 1983, 1984; Morgan and Phillips, 1983; Kiefer et al., 1986]. It has also been suggested that the high surface temperature of Venus tends to suppress crustal and lithospheric subduction by depressing the basalt- eclogite transition to great depths [Anderson, 1981] and by rendering the lithosphere positively buoyant [Phillips and Malin, 1983]. This may mean that Venus is unlike Earth, where ridge push and slab pull forces are primary driving forces for

1Now at Department of Earth and Space Sciences, UCLA, Los Angeles, California.

Copyright 1990 by the American Geophysical Union.

Paper number 90JB01602 0148-0277/90/9 0JB-01602505.00

plate motions and surface deformation [Forsyth and Uyeda, 1975; Chapple and Tullis, 1977].

Although there are numerous quantitative estimates of topography and heat flow due to mantle hotspots, less consideration has been given to how mantle flow might produce tectonic features and terrains on Venus. We treat the crust and mantle as a layered viscous half-space and consider the effects of flow driven by relief at the surface, relief at the crust-mantle boundary, and by density contrasts within the interior [e.g., Ramberg, 1968; Morgan, 1971; Hager and O'Connell, 1981; Parsons and Daly, 1983]. Phillips [1986] examined the effect of mantle flow on the lithosphere without considering a crustal layer, while Grimm and Solomon [1988] and Smrekar and Phillips [1988] treated crustal deformation without considering mantle flow as a driving force. This study specifically addresses the coupling between mantle flow and crustal deformation.

The high surface temperatures of Venus imply that ductile deformation is likely to occur at shallow depths within the crust [Weertman, 1979; Zuber, 1987]. On Earth, the presence of a ductile layer in the continental crust is well established [e.g., Brace and Kohlstedt, 1980; Chen and Molnar, 1983] and is thought to be partly responsible for differences in the styles of deformation observed in continental and oceanic lithosphere. On Venus, a ductile lower crust would have significant consequences for the surface expressions of mantle flow. In particular, this ductile layer would allow crustal thinning or thickening to occur in response to long-term mantle flow. Crustal thinning/thickening is driven by the interaction of gravitational forces and normal stresses associated with mantle flow, and by shear-coupling of horizontal flow into the lower crust. It is also expected to be accompanied by tectonic deformation. Thus the effects of mantle flow are not necessarily restricted to uplift in areas such as Beta Regio, but can also

21,329

21,330 BINDSCHADLER AND PARMEN'IIER: VENUS MANTLE FLOW TECTONICS

include crustal thinning, perhaps manifested as the observed rift, Devana Chasma [McGill et al., 1981; Campbell et al., 1984; Stofan et al., 1989]. Downwelling mantle flow can cause crustal thickening, and by implication, the formation of large- scale compressional tectonic features, perhaps like those in western Ishtar Terra.

An idealized layered structure, stratified in both density and viscosity, that represents the crust and mantle of Venus consists of (1) a brittle-elastic upper crustal layer, (2) a ductile, weaker lower crustal layer, (3) a strong upper mantle layer, approximately 10% denser than the crust, and (4) a weaker substrate, representing the portion of the mantle in which convective flow occurs. It is the interactions between mantle

flow driven by thermal or compositional variations and these four layers, representing the basic stratification in strength, rheology, and density of a planetary lithosphere and sublithospheric mantle, that we wish to examine.

MODEL FORlVlULATION

Uniform Density and Viscosity

Consider the simple case of a harmonic buoyancy force

F(x) = og sin (kx) (1)

due to a mass sheet at depth d within a half-space of uniform Newtonian viscosity and uniform density (Figure 1), representing, for example, temperature variations within the mantle. The term o represents the amplitude of the harmonic mass sheet (mass per unit area), g is gravitational acceleration, and k = 2rc/•, is the wavenumber of the harmonic density anomaly at depth. An initially flat surface moves vertically with velocity W s in response to buoyancy forces at depth. Topographic relief at the surface causes gravitational relaxation flow w r opposing w s (Figure 1). After sufficient surface topography has accumulated, the relaxation velocity Wr matches the source velocity W s and the surface stops moving. Relaxation velocity W r is calculated by matching the vertical normal stress at the surface due to topography to the su'ess due to relaxation flow. As long as surface slopes are small (essentially kh << 1), this stress matching condition can be

z w r h(t) sinkx

!:i:i:!:i:i:i:!:i:i:i:i:i:!:! W ':':':':':':':':':':':':':':':':':':':':':':':':':':

Fig. 1. Sketch showing the basic components of the no-crust model. Flow is driven by a harmonic mass sheet at depth d within a uniform viscosity half-space. The steady state height of topography depends upon the ratio of velocities at the surface due to the source and due to relaxation.

evaluated at z = O. Following Turcotte and Schubert's [1982] treatment of postglacial rebound but with a shear-stress-free surface boundary condition, we find that for harmonic topography of amplitude h on a uniform viscous half-space:

xzzlz=o = pgh sin (ka:) = 2txkw r sin (kx) (2) Here Xzz is the vertical normal component of the stress tensor, p is the density of the half-space, and [.t is viscosity. It is convenient to introduce the notation

^ Wr Wr= (3)

pgh

where the velocity response function Wr represents the vertical relaxation velocity per unit of topographic stress. Similarly, Ws represents vertical velocity per unit stress difference due to buoyancy forces. To find •s, a discontinuous vertical normal stress representing a buoyancy force is applied at depth d. Solutions to the viscous flow equation in regions above and below the discontinuity are found from boundary conditions on the half-space and matching conditions at the discontinuity or interface. At the interface, in addition to the imposed jump in vertical normal stress, shear stress and velocity components must be continuous. At the surface of the half-space, shear and vertical normal stresses vanish. Both components of velocity vanish as z --> oo. The velocity response function •Vr, discussed above, can be defined in an analogous way except that the discontinuity in vertical normal stress is placed at the surface, representing topographic stresses.

The vertical velocity of the surface may then be expressed as a function of time,

dh ^ pg•vrh (4) d-• = Ogws +

Taking the surface to be flat at t = 0, when flow at depth begins, the solution is

^

h(t) = øws [exp(pg•vrt) - 1] (5)

The first part of this solution is time-dependent and describes

the growth of individual harmonics of to•pography at the surface with a characteristic time constant p gw r. Note that this time constant is dependent only on W r. The second part represents a steady state; after sufficiently long times the amplitude of surface topography approaches the value

^ A A

-OWs/pW r. This ratio is positive since w s and •v r always are opposite in sign. Thus the velocity response functions w s and w r, which depend on wavelength, both determine the steady state topography, but only •v r determines the time required to reach that steady state.

Stratification in Density

The "no-crust" model discussed above could be applied to terrestrial oceanic regions, where a brittle/elastic lithosphere sits atop ductile mantle. Because oceanic crust is embedded within the elastic lithosphere, the density difference between crust and mantle material does not drive flow within the crust or

mantle. However, the lower crust of Venus is likely to be ductile if the crust is more than a few kilometers thick

[Weertman, 1979; Zuber, 1987]. A ductile lower crust permits decoupling of the crust-mantle

BINDSCHADLER AND PARMENTIER: VENUS MAN'I• FLOW TECTONICS 21,331

boundary from the surface; that is, even if the crust is initially of uniform thickness, it does not necessarily remain so. This requires that we consider flow driven by topography at the crust-mantle boundary (Figure 2) in addition to flow driven by the buoyancy force and by surface topography (Figure 1). As in the simpler no-crust model, velocity response functions are found from boundary conditions on the half-space and matching conditions across interfaces. Vertical motions of the surface and crust-mantle boundary are given by

dhl(t) A A A

dt = øgwls + pgwllhi + Apgw12h2 (6a)

dh2(t) A

dt = ogw2• + Apg½22h2 + pg½21hl (6b)

respectively, where the first subscript to each velocity response function represents a given interface and the second represents the location of a density contrast driving flow. Subscripts 1, 2, and s represent the surface, crust-mantle boundary, and buoyancy force interfaces, respectively. For

A .

example, w2s represents the velocity response function of the crust-mantle boundary due to a unit buoyancy force at depth and w12 represents the response of the surface interface to a unit of topography at the crust-mantle boundary. The quantities p and Ap represent the density contrasts at the surface and at the crust-mantle boundary, respectively.

initially present at both interfaces. That is,

hi(0) = H1 h2(0) = H2 (7)

where H 1 and H2 represent initial topographic amplitudes at the surface and crust-mantle boundary. These conditions result in the following solution to (6)'

(•1-•2) hi(t) = e½lt [ (Pg •11 - •2)C1 + APg•12C2 ] + e½2t [ (•1 - Pg½11)C1 - APg½12C2 ] + (H1 - C1) (½1-•2)

(½1-•2) h2(t) = e*lt [ (Apg •22- ½2)C2 + Pg•'21C1 ] + e*2t [ (½1- Apg •,22)C 2 - Pg•'21C1 ] + (H2- C2)

(8a)

(8b)

where qbl and •2 are the eigenvalues

pg 6p ½1,2 = -1/Xl,2 = -•- ( •11 + •½22 ) p

i Ap )2 Ap +•[(•11 +•22 -4• I•ijl ] p p

1/2 (9)

and x 1 and x 2 are time constants describing the time-dependent relief at the surface and crust-mantle boundary. Constants C1 and C 2 are found using initial conditions (7):

elastic layer h 1 (x,t)

!i:!.m..an! ................. :.:.:.. ...... ::::::::::::::::::::::::::::::: (}:,'•5::i': i!iii!iii!iiiiiiiii!i!i!i!i!i!iii' viscous or elastic i:i:!:!:i:i:!:!:!:i:i:i:i:i:!:i:i mantle layer :i:i:i:i:i:!:i:: ........

Fig. 2. The basic components of the model. A crust of density p and consisting of a strong elastic upper layer and a weaker, ductile lower layer overlies a mantle of density p + Ap and similarly stratified in viscosity. The surface and crust-mantle boundary are initially level but are deflected by flow from a source at depth (og).

Solutions to the coupled differential equations (6) consist of a solution to the homogeneous part of the equations plus a particular solution which depends only upon initial conditions. Homogeneous solutions are of the form A e{ t where A is a constant and • is one of the two eigenvalues of a linear system of equations. Although we typically assume that the surface and crust-mantle interfaces are initially flat, we present here general solutions to include cases in which topography is

w22- w2sw12) C1 = +H 1 (10a)

p I½01

A A A A

0 (W2sW 11 - WlsW21) C 2 -

Ap and I v•ij I denotes the determinant.

+H 2 (10b)

Stratification in Strength and Viscosity A

The velocity response functions Wsj and v•ij depend upon the choice of viscosity structure and the presence or absence of elastic layers. Viscosity stratification is treated by including interfaces which separate layers of uniform viscosity and across which viscosity changes discontinuously. Stress and velocity matching conditions are imposed across these viscosity interfaces and, along with boundary conditions, determine solutions to the equations of flow [e.g., Ramberg, 1968; Hager and O'Connell, 1981; Fleitout and Froidevaux, 1982].

Brittle portions of the crust and mantle can either be represented as high-viscosity layers [e.g., Grimm and Solomon, 1988] or as elastic layers. In the former representation, frictional sliding on faults is parameterized as ductile flow and the layer may be deflected vertically, thinned, or thickened. In the latter, the layer may be deflected vertically, but horizontal deformation is limited to elastic strain and

occurs rapidly on time scales for viscous flow [Phillips, 1986]. This allows the influence of an elastic layer on ductile portions of the crust and mantle to be treated as a matching or boundary condition. An elastic layer causes horizontal velocity to vanish along the elastic-viscous interface and provides flexural support of vertical normal stress (Figure 3). Flexure of a thin elastic plate can support a vertical harmonic load q [Turcotte

21,332 BINDSCHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTONICS

and Schubert, 1982],

q = Dk4h sin (kx) (11)

where k = 2•;/)•, h is the amplitude of the vertical deflection, and D is flexural rigidity [Turcotte and Schubert, 1982]:

3

Et e

D= 12(1_v2) (12) Here E is Young's modulus, v is Poisson's ratio, and t e is the thickness of the elastic layer. If flexural effects are included, the boundary condition on vertical normal stress at the top of the viscous portion of the crust (Figure 3) becomes

Xzz = (D1 k4 + pg)hl (13)

where D 1 is the flexural rigidity of the upper crustal layer. An elastic layer in the uppermost mantle alters matching conditions at the crust-mantle boundary such that horizontal velocity vanishes and the change in vertical normal stress across the layer is analogous to (13). The load that can be supported by an elastic layer at the surface depends on the ratio of its wavelength to a flexural wavelength

•cf= 2• -- (14) Pg

At wavelengths •<<•cf the flexural resistance of the elastic layer effectively prevents any topographic expression of mantle flow. At long wavelengths (•>>•cf), the resistance to flexure is negligible and topography must be created in order to balance vertical normal stresses.

A brittle elastic layer is also subject to horizontal normal stresses due to mantle flow and may fail in either compression or extension. We examine an example of such failure by treating an upper crustal layer (thickness te) as a material in which failure is governed by frictional sliding on faults [e.g., Brace and Kohlstedt, 1980], so that the least (03) and most compressive stresses (Ol) are proportional to one another

ch -= 503 (15)

for 03 < 110 MPa [Brace and Kohlstedt, 1980]. Assume that the layer overlies a uniformly viscous crust and

mantle in which a buoyancy force o at depth d drives flow. Shear and normal stresses vanish at the top of the elastic layer. The average horizontal normal stress within the layer is found from shear stresses imposed by flow at its base. This horizontal normal stress varies with the wavelength of mantle flow. Taking one principal stress to be vertical and lithostatic and the other to be horizontal and due to mantle flow, we find

that the maximum brittle strength of the layer in extension is exceeded for wavelengths of flow greater than a critical wavelength

= [ 15-•• ]'• •xt 2rid In (16)

Pte • The layer fails in compression for wavelengths greater than

•cmp = 2rid In • (17)

For example, consider d = 100 km, tJ = 106 kg m -2 (equivalent to approximately 300 m of dynamically supported topography) and p = 3000 kg m -3. A 5-km-thick brittle upper crust is predicted to completely fail in extension for wavelengths of flow greater than ~300 km, but does not in compression. To cause complete failure in compression, either the magnitude of the buoyancy force or its depth [Fleitout and Froidevaux, 1982; Phillips, 1986] must be increased. A brittle layer within the upper mantle is expected to be more difficult to disrupt in this manner, since higher lithostatic pressures inhibit frictional sliding.

Results such as these are difficult to generalize, since the magnitude of horizontal normal forces within brittle layers depends upon such factors as the strength and depth of the buoyancy force, the thickness of brittle layers, and the strengths of ductile portions of the crust and mantle. Phillips [1990] has examined plastic failure of the Venus lithosphere in some detail. Within the context of specific models, the failure of brittle layers requires more thorough investigation. For the general purposes of this study, we consider both the case in which elastic layers are present and the case in which they are absent in order to understand their effects on crustal

deformation due to mantle flow.

RESLILTS

We examine simple structures that can be used to gain insight into model behavior and the particular parameters that control it. We begin with a uniform viscosity model in order to assess the effects of elastic layers, then consider more complex viscosity distributions. The buoyancy force which drives flow is in the form of a harmonic mass sheet. More complex models can be created by superimposing different wavelengths of flow and mass sheets at different depths.

Uniform Viscosity

Consider a layer (representing the crust) of thickness L, density p, and viscosity •t overlying a substrate (representing the mantle) of density p+Ap and of equal, uniform viscosity, as shown in Figure 2. For the cases examined here, depth of the buoyancy force (d) is held constant at five crustal thicknesses. As noted above, the density contrast between crust and manfie is taken to be 10%. Three models are represented by different boundary/matching conditions (Figure 3). The first consists of a viscous mantle and crust and represents a reference model (VV). The second model (El) includes an elastic surface layer representing a brittle upper crust (12). The third model (E2) adds a second elastic layer to represent a brittle layer at the crust-mantle boundary. Models E1 and E2 represent end members in which only elastic strains accumulate in near surface layers. Our principal interest is in large-scale topographic and tectonic features, so the flexural rigidity in models E1 and E2 has been chosen such that •cf<< L.

The behavior of these models is illustrated in Figures 4-6, which show topography at the surface and on the crust-mantle boundary in the steady state (t ---> oo) limit (Figure 4), surface topography as a function of time (Figure 5), and the difference between surface and crust-mantle boundary topography (i.e., crustal thinning) as a function of time (Figure 6). Plots were calculated for an upwelling buoyancy force, but apply equally well to downwelling if the values on the vertical axes are negated. Models VV and E1 behave similarly and are discussed first. We then discuss the distinct behavior of model E2.

BINDSCHADLER AND PARMENllER: VENUS MANTI• FI_DW TECrONICS 21,333

Model VV Model E1

4

Xzz = pgh 1 + Dlk h 1 u=0

Model E2

u=0

[w] =0

u=O

Fig. 3. Boundary and matching conditions for the three models VV, El, and E2. VV represents a uniform viscous crust and mantle. Model E1 adds a surface elastic layer (diagonal bars), representing a brittle upper crust. Model E2 adds an elastic layer at the crust-mantle boundary, to represent a brittle upper mantle layer. Square brackets indicate the change in a quantity across a viscous interface or an elastic layer. For example [u] = 0 at the crust-mantle boundary in model VV indicates that there is no change in horizontal velocity across that interface.

As t --> o% time-dependent terms in (6) decay exponentially, and surface and crust-mantle boundary topography approach steady state values. In the absence of a ductile crust, upwelling (downwelling) mantle flow will result in uplift (subsidence) of the surface. However, above upwelling (downwelling) mantle flow, models VV and E1 are characterized at long wavelengths by steady state subsidence (uplift) of the surface and uplift (subsidence) of the crust-mantle boundary (Figure 4). Examining time-dependent behavior, we find that the surface initially undergoes uplift due to mantle upwelling for model E1 (Figure 5a), but later begins to subside. The amount of subsidence is significantly greater than the amount of earlier uplift. There is no change in crustal thickness during this initial uplift (Figure 6a), indicating that the entire crust is raised as a unit by vertical normal stresses associated with upwelling. Surface subsidence is accompanied by significant crustal thinning (Figure 6a). Results for model W (not shown in Figures 5 and 6) are qualitatively the same.

In the steady state limit, model E2 differs from VV and E1 in that it shows no relief at the surface and less relief at the crust-

mantle boundary (Figure 4). Initial surface uplift occurs (Figure 5b) and is identical to the uplift undergone by model El. However, in model E2 the surface simply subsides back to its original level (Figure 5b) and the amount of crustal thinning is much less that in model E1 (Figure 6b).

hip

2

0

-2

-4

-6

-8

-1 0 1 2 3 4

log( •./L )

Fig. 4a. Steady state surface topography as a function of wavelength for d=5L, and models VV, El, E2. Heights are scaled by the ratio of crustal density (p) to the mass anomaly per unit area within the mantle (•). Note that surface subsidence is predicted over upward mantle flow for models VV and El.

h2P

80] -• VV '-'- E1

60

40

20

0•s .•-.•-.1 •.-• i ß i ß i ß i

-1 0 1 2 3 4

log( •./L )

Fig. 4b. Steady state relief at the crust-mantle boundary for the same set of calculations. At the crust-mantle interface, model E2 displays less relief than models E1 and VV. The magnitude of uplift here is much greater than subsidence at the surface.

The differences between models E1 and E2 occur because two

distinct effects cause crustal thinning. The first is flow driven by a horizontal gradient in vertical normal stress due to surface topography. Thinning occurs when ductile crust flows away from high topography. Lowering of surface elevations forces the crust-mantle boundary to rise so that balance of vertical forces is preserved, resulting in net crustal thinning. The second effect is shear-coupling of horizontal mantle flow into horizontal flow of crustal material away from a region of upwelling. This effect does not occur in model E2 because the continuous elastic layer between the crust and mantle prevents such coupling. Thus subsidence and thinning in model E2 reflect only the decoupling of normal stresses at the crust- mantle boundary from topography at the surface as provided by the ductile lower crustal layer. Subsidence and thinning in models E1 and VV reflect both this decoupling and shear coupling of horizontal flow from the mantle into the crust.

It is also important to determine the time required for crustal uplift (x2) and deformation (thickening or thinning, x l)to occur. Figure 7 illustrates the effect of elastic layers on these scale times. At short wavelengths ()•/L<10), the flexural resistance of an upper crustal elastic layer (model El) reduces

21,334 BINDSCHAD• AND PARMENTIER: VENUS MAN'ILE FLOW TECTONICS

2

-2 --- )dL=100

-o- )dL=320

ALp

8O

4O

2O

-•- ML=32

---- )dL=100

-o- ML=320

-4 -2 0 2 4 6 8 -z -2 0 2

log( t/x ) log( t/x )

4 6 8

Fig. 5a. Surface height as a function of time for three wavelengths for a crust capped by an elastic layer (model El). Times are sealed by x = (21.tm/pgL), a convenient time scale. Heights are sealed to the ratio of crustal density to the amplitude of the buoyancy force. Over a region of upwelling, the surface is first uplifted, and later subsides. Downwelling results in initial subsidence and later uplift, but the absolute relief would be the same as for upwelling at a given time. Long-term changes in elevation are greater than the initial response of the surface to manfie flow. Results for model VV are qualitatively similar. 10'

8

1.0 --• ),/L=32

0.8 f it -x•-• -'- )dL=100 ALp 64 • )dL=32 0.6

h

0.4

0.2 ................

0.0 -0.2 ' Fig. 6b. Crustal thinning (hl-h2) as a function of time for three

-4 -2 0 2 4 6 8

log( t/x )

Fig 5b. Surface heights as before, but with the addition of an elastic layer at the crust-mantle boundary (model E2). The amount of early- time uplift (or subsidence) is nearly identical to that of model El, but in the steady state, topography is reduced to its original level.

the characteristic time of crustal uplift (x2) considerably and also suppresses surface topography. At longer wavelengths, the elastic layer increases Xl by slowing the flow of crust away from a region of upwelling or toward a region of downwelling. Adding an elastic layer at the crust-mantle boundary (E2) reduces X l significantly for short wavelengths and increases it for long wavelengths. This increase in X l occurs because horizontal velocities vanish at the elastic-viscous layer interfaces, restricting horizontal flow to a relatively narrow region in the lower crust. Comparison of predicted surface height and crustal thinning for model VV with the same quantities for models E1 and E2 (Figures 5-7) indicates that relatively little topography or deformation is expected at wavelengths )• < 2d (i.e.,)•/L < 10 in Figures 5-8), where d is the depth of the buoyancy force. This suggests that if brittle layers do not fail, their most important effect is the restriction of horizontal flow in the lower crust, not resistance to vertical deflections.

Fig. 6a. Crustal thinning (hl-h2)as a function of time for three wavelengths, model El. Time and height scales are the same as Figure 5. Early-time changes in surface elevation are not accompanied by crustal thickness changes. Long-term changes are accompanied by substantial thinning or thickening of the crust.

Ag = 1 - exp(-kZade) (18)

Agtop which assumes columnwise isostatic balance of surface

topography by the density anomaly in the mantle [e.g., Esposito et al., 1982]. The terms in the exponential are the wave number (k) and apparent depth of compensation (Zadc).

Admittance values are large and negative at very early times, until enough surface topography is created to balance the buoyancy force at depth. Values then reach a plateau until crustal thinning or thickening begins. Admittance values in

Topography at the surface and on the crust-mantle boundary in these models leads to gravity anomalies that can vary greatly as functions of time. To illustrate this variation, we plot admittance (Figure 8) and apparent depths of compensation (ADC, Figure 9) for model E1 (Figure 6a). Here, admittance is the ratio of total gravity anomaly (Ag) to the anomaly due only to surface topography (Agtop). Anomaly values are calculated by treating topographic variations as mass sheets. Apparent depth of compensation is calculated from

wavelengths, model E2. Thinning is much less than for model E1 (note ddferent vertical scale) and occurs at later times.

BINDSCHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTONICS 21,335

3O

2O

--• •o

o

-lO

-20

-1

-a- xl (vv)

--a- xl (El)

0 1 • 3 4 5 6 log( •/L )

Fig. 7a. Time constants for models VV and E 1, illustrating the effect of an elastic layer in the upper crest on the response time of the crest to mantle flow. The sharp bend in the curves of •:2 for model E 1 occurs at the critical flexural wavelength, reflecting the resistance of elastic layers to flexure (s•e text).

-•- )•/L = 32

I • )•/L= 100 -o- )•/L = 320

!

-1 0 1 2 3 4 5 6 7

log ( tR )

Fig. 8. Admittance (total gravity anomaly over anomaly due to topography) as a function of time for three wavelengths for model E 1. Dashed lines indicate discontinuity that occurs when surface elevations pass through zero (Figure 5a). Steady state values are small, but still greater than zero.

30' --a- xl (VV) • 20 -a- x• (E2) •

--E- ;o

o

-lO

-20 ß , ß , , ß , ß ,

-1 0 1 • • 4 5 6 log(•./L)

Fig. 7b. Time constants for models VV and E2, illustrating the effect of an elastic layers in the upper crust and uppermost mantle on the response time of the crust to manfie flow. The sharp bend in the curves of x 2 for model E2 occurs at the critical flexural wavelength, Xl is also greatly affected at short wavelengths.

ADC

10 I • )•/L = 32 --*- )•/L= 100

5 I -n- )•/L = 320

-1 0 1 2 3 4 5 6 7

log ( t/c )

0.5'

the steady state (large values of t/x) are less than those calculated, while the surface and crust-mantle boundary are both uplifted. More important, however, are the large negative and positive values that occur at different times as surface elevations pass through zero. Crustal thinning or thickening due to mantle flow could lead to admittance curves which are not

well fit by conventional models such as Airy isostasy or flexural support of topography. We note that admittances for the region surrounding Beta Regio [Reasenberg et al., 1982] do not fit either of these conventional models and might be explained if subsidence and crustal thinning are currently occurring due to mantle upwelling.

Similar behavior is exhibited by ADC's as a function of time (Figure 9). At relatively long wavelengths and while both surface and crust-mantle boundary are uplifted, ADC values correspond to the depth of the mantle density anomaly (d = 5L). As with admittance, ADC values vary greatly as the crust is thinned. Once the steady state is attained (Figure 9, lower portion), ADC's are small, in this case less than half the crustal thickness (L). This occurs because relief at the crust-mantle boundary overcompensates for the topography at the surface in order to also balance normal and shear stresses due to the

buoyancy force at depth.

ADC

0.4

0.3

0.2

0.1

0.0

• )•/L = 32

• [ [ -4- •/L: 100

4 5 6 7

log ( t/c )

Fig. 9. Apparent depth of compensation (ADC) as a function of time for three wavelengths for model El. ADC is normalized to crustal thickness (L). Dashed lines indicate surface elevations passing though zero, as in Figure 8. Boxed region is shown in detail in lower plot. Note that steady state ADC's are less than one-half the thickness of the crust.

Variable Viscosity

The strength stratification of the crust and mantle is represented by three layers (upper and lower crust, and upper

2 ! ,336 BINDSCHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTONICS

18

9.0 8.5

8.0 • 7.5 •

24

18 10

10 20 30 40 50 60 70 80 90 100

Crustal Thickness

(km)

16

10 20 30 40 50 60 70 80 90 100

Crustal Thickness

(km)

Fig. 10. Contour plots of the characteristic crustal deformation time scale (x 1) as a function of crustal thickness (L) and viscosity (•Lc). Contour intervals are marked in log(years); e.g., the label 8.0 indicates 100 m.y. Plots were calculated for •trn = 1021 Pa s and a wavelength of 2000 km. (a) Contours of x 1 for model VV. Sharp bends in some contour lines at small values of L indicate a transition between rate-

limiting of flow by mantle viscosity to rate-limiting by crustal viscosity. The location of the bends is dependent upon the ratio of viscosities in the crust and mantle and the ratio of wavelength to crustal thickness. (b) Contours of x 1 for a model with an upper elastic layer (e.g., model El). Similarity of contours indicates that flow is strongly influenced by the presence of an elastic layer. Contour lines for model E2 (not shown) are very similar in shape, but a given value of crustal viscosity results in larger x 1 because the additional elastic layer further inhibits horizontal flow.

mantle) over a ductile interior. Here we examine a number of simple viscosity distributions to gain insight into how individual layers influence topography, deformation, and time scales for crustal thinning/thickening. These time scales are strongly influenced by crustal viscosity and are also affected by crustal thickness and the presence or absence of elastic layers.

To understand the effects of crustal viscosity and thickness on the characteristic time scale for crustal thinning/thick- ening, we have calculated x 1 as a function of these variables for models with (El) and without (VV) an elastic upper crustal layer. Results in Figure 10 were based on a characteristic wavelength of 2000 km for the buoyancy force and a mantle viscosity of 1021 Pa s and are independent of the depth of the buoyancy force (d). Contours in x 1 are labeled in log(years) and represent the time required for topography and crustal thinning/thickening to reach approximately 60% of its steady state value. With no crustal elastic layer present (Figure 10a), this degree of crustal thickness variation can occur in less than

several hundred million years as long as crustal viscosity is less than approximately 1022 Pa s and crustal thickness is greater than approximately 15 km. The addition of an elastic upper crustal layer (model El) requires crustal viscosities to be smaller and crustal thicknesses larger (Figure 10b) if significant thickness variations are to occur. The addition of second elastic layer, in the uppermost mantle, further slows thinning/thickening, although to a lesser extent than the fixst such layer.

We have also examined two pairs of parameters relevant to the time scales and wavelengths at which crustal thinning or thickening first becomes significant using five viscosity distributions (Figure 11) and holding the depth of the buoyancy force constant at d = 5L (Table I). The first pair is Xss, the wavelength at which steady state surface topography reaches one-half the amplitude in the limit of long wavelengths, and Xss, the crustal deformation time (x l) at this wavelength. The second pair (•def, q:def) consists of the wavelength at which uplift at the crust-mantle boundary first becomes significant (h2 = 5•J/p) and the corresponding crustal deformation time. Clearly, the choice of this particular value of h2 is somewhat arbitrary. However, we are primarily interested in relatively long wavelengths (X>L, Figures 4-6), and have also chosen a crust-mantle density contrast of 10%. For these conditions, h2 = 5g/p means that vertical normal stress due to relief at the crust-mantle boundary balances one half of the vertical normal stress due to the buoyancy force at depth. Moreover, during early-time crustal uplift, h 1 never exceeds •/p. Therefore h 2 = 5c•/p requires that crustal thinning has occurred in order to preserve balance of forces. For example, if early-time crustal uplift was 1 km, the crust has been thinned by at least 5 km.

The five models in Figure 11 each consist of three layers over a substrate in which viscosities vary up to 3 orders of magnitude from the reference mantle viscosity. By varying the viscosity of layers individually and in combination, we can observe their effect on the characteristic wavelengths and time scales of crustal deformation. Values of viscosity are dimensionless, normalized to the viscosity of the mantle

ß .

ß

.. ..

..

VV WK LC ST UC

ST UM WS

Fig. 11. Illustrations of simple viscosity distributions discussed in the text. VV, uniform viscous half-space; WK LC, weak lower crustal layer, ST UC, strong upper crustal layer, ST UM, strong upper mantle layer; WK ST, weak lower crust + strong upper mantle.

BINDSCHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTONICS 21.337

substrate. Results shown in Table I may be summarized as follows:

1. Variations in viscosity produce large variations in Xss, and Xdef (up to 2 orders of magnitude over the range of parameters considered here), while •,def is ~10L and varies only by a factor of ~2. Crustal thinning at wavelengths > 10L can thus be expected to occur over a wide range of conditions. In addition, comparison of Xss with Xdef indicates that crustal deformation can occur long before the steady state is reached (nearly 3 orders of magnitude in time for model WK ST) and over a broader range of crustal viscosities and thicknesses than might be inferred from Figure 10.

2. The ratio of lower crust and upper mantle viscosities appears to be the primary determinant of •,ss and Xss. If the lower crust is relatively weak, the upper mantle layer suppresses shear coupling in the same fashion as an elastic upper mantle layer (i.e., model E2) except at long wavelengths.

3. Variations in Xdef appear to be affected by the overall viscosity structure. However, as might be expected for the moderate wavelengths of •,def, near-surface layers contribute most strongly to changes in Xdef (e.g., compare results of model ST UC with ST UM).

Some of these characteristic wavelengths and time scales of deformation might also be expected to vary with the depth of the buoyancy force. We find that increasing depth results in approximately proportional increases (within a factor of ~2) in •def for all models, while •,ss exhibits similar behavior for models VV and ST UC. The remaining three models (Figure 11) have large viscosity contrasts at the crust-mantle boundary and exhibit little or no change in •'ss with changing buoyancy force depth. In such cases, the large viscosity contrast suppresses shear coupling of horizontal flow into the crust, so that extensive crustal thickening or thinning can occur only at very long wavelengths. Deformation time scales are affected only to the extent that •,ss and •'def increase or decrease. Time constants for a given wavelength of flow (9) are functions only of the viscosity structure and the presence or absence of elastic layers and are not altered by changes in the depth of the buoyancy force.

Model

_

w

WKLC

ST UC

ST UM

TABLE 1. Effects of Varying Viscosity

g2 l-t3 •s• '•m •def 'gdef

1 1 1 40 170 10 91

1 0.001 1 3,200 5,100 15 14

1,000 1 1 16 2,100 8 2,100

1 1 1,000 200 13,000 16 240

1 0.001 1,000 6,300 20,000 15 23

APPLICATION TO LARGE-SCALE

FEATURES ON VENUS

The presence of swells such as Beta Regio, and the high degree of correlation of long wavelength gravity and topography have led a number of workers to suggest that convection on Venus is dominated by upwelling plumes

[Phillips and Malin, 1983; Kiefer et al., 1986; Bills et al., 1987]. However, there has been little consideration of the surface manifestations of convective downwelling. Here we wish to examine the effects of convection on the crust for two

simple cases: time-invariant flow and time-dependent flow driven by a single diapiric body (either rising or sinking).

Time-Invariant Mantle Flow

Above a region of upwelling mantle flow, the crust is first uplifted with little or no thinning (Figures 6 and 7). Tectonic features formed at this time would be caused by flexure of the brittle upper crustal and/or mantle layers, similar to deformational features predicted for doming [Withjack and Scheiner, 1982]. Continued upwelling leads to thinning of the crust (Figure 6), and the surface subsides as the crust-mantle boundary is uplifted. If an elastic mantle layer is present at the crust-mantle boundary (model E2), shear coupling of horizontal mantle flow into the crust is prevented and the surface simply relaxes to its original level. Vertical normal stresses due to the mantle flow are then balanced solely by relief at the crust- mantle boundary, and crustal thinning of ~10 times the early time uplift occurs (for Ap/p = 0.1). One predicted effect of such a steady state is a gravity anomaly that results from the mass anomaly at depth and relief at the crust-mantle boundary, but without associated topography. The highly positive correlation of low-degree harmonics of gravity and topography [Bills et al., 1987] suggests that such a situation does not dominate large-scale features on Venus.

In the absence of an elastic mantle layer (model E1 or VV), crustal thirming is accentuated (Figure 6), resulting in greater subsidence of the surface. Enhanced crustal thinning occurs because shear coupling of horizontal mantle flow into the crust is allowed by the absence of an elastic layer at the crust-mantle boundary. The amount of crustal thinning and subsidence is proportional to the amplitude of the buoyancy force ({J), and also depends on its depth (d), its characteristic wavelength, and the viscosity distribution with depth, particularly in near- surface layers. For example, a strong upper crustal layer (ST UC, Figure 11) tends to enhance crustal thinning due to upwelling at moderate wavelengths, but a strong upper mantle layer (ST UM) has the opposite effect (Figure 12). In the limit that these layers are extremely viscous, their effects are similar to elastic layers at the surface (El) and the crust-mantle boundary (E2). Strong upper mantle layers decrease crustal thinning by inhibiting shear coupling of horizontal flow into the crust. Strong upper crustal layers force horizontal velocities to approach zero near the surface, creating large vertical gradients of horizontal velocity, and thereby increasing the magnitude of shear stresses. Thus the presence of strong near-surface layers increases shear stresses within the lower crust and leads to additional crustal thinning.

To understand the effects of long-lived mantle downwelling, we simply reverse the sign on the vertical axes in Figures 4, 5, and 6. The initial reaction of the crust is subsidence. Tectonic

features may include ridges due to thrust faulting or folding. In the longer term, the surface rises toward its original elevation, while the crust-mantle boundary deepens as the crust is thickened. If an elastic layer is present in the upper mantle, the surface simply returns to its original level, and crustal thickening is limited to ~10 times the maximum early-time subsidence (for A p/p = 0.1). If no elastic mantle layer is present, crustal thickening is enhanced and topographic highs

21,338 BINDSCHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTONICS

70

60

ALp (• •0

30

20

10

0

0 1 2 3 4 5

log( X/L )

Fig. 12a. Crustal thinning as a function of wavelength in the steady state limit for three models of Figure 11: VV, ST UC, and ST UM. For mantle downwelling, these curves represent crustal thickening. A strong upper crustal layer enhances long-term changes in crustal thickness by enhancing shear coupling of horizontal flow into the crust. A strong upper mantle (or equivalently, a weak lower crust) tends to reduce such coupling.

0

-1

h•p -2 -3

-4

-x- vv

--- ST UC

ß i ß i ß ! ß i ß i ß

-1 0 1 2 3 4

log( X/L )

Fig. 12b. Surface topography as a function of wavelength in the steady state limit for the models in Figure 12a. Enhanced shear coupling in model ST UC is manifested as increased surface subsidence (over upwellings).

with a vertical velocity

2Apdga 2 w =• (19)

9lXm

where Apd is the density contrast between the body and the surrounding mantle, a is the radius of the body, and !Xm is mantle viscosity. For large-scale features, we consider diapiric bodies for which a > L. When such a body is more than several diameters from the surface, its effects at the surface are strongly attenuated. Thus a characteristic time scale over which the

diapir affects the surface is given by the diameter of the body divided by its vertical velocity:

9lXm •:o = • (20)

APdga

An upper limit on the time scale for crustal deformation can be obtained if we approximate model E2 using one-dimensional channel flow [Turcotte and Schubert, 1982, p. 234]. Horizontal velocities are required to vanish at both the upper and lower boundaries of the ductile lower crust, approximating the effect of upper crustal and mantle elastic layers and maximizing the time required for crustal deformation. For flow driven only by crustal thickness variations, the average horizontal velocity in the ductile lower crust is

u=- 12•tc

Incompressibility requires that

• •H

•x (uH) = •t (22)

where u is the average horizontal velocity, !x c is the viscosity of the ductile lower crust, and H the lower crustal thickness. For

flow driven only by topographic gradients, and in the limit of small slopes (small variations in H), equation (22) takes the form of a diffusion equation with a time constant Xc that is an upper limit on the time scale of crustal deformation:

12gc

XC- pgH3k2 (23) are created at the surface, characterized by compressional deformational features. The mechanisms by which crustal thickening occurs are the same as described above for crustal thinning over mantle upwellings.

Diapiric Flow

Convective flow in planetary mantles occurs at a variety of spatial and temporal scales. One such scale could be determined by thermal or compositional instabilities, in the form of rising or sinking diapiric bodies. The effects of a rising or sinking body can be considered in two limits: (1) the characteristic rising/sinking time of the diapir much less than a characteristic crustal deformation time (Xc), and (2) characteristic rising/sinking time on the order of or greater than xc. We formulate expressions for these time scales in order to understand the conditions under which diapiric manfie flow may cause crustal deformation.

Consider the rising/sinking diapir as a Stokes-law body

where k = 2•/3,. As an example, consider a region with a characteristic wavelength of 1000 km, and a 20 km thick lower crust with an effective linear viscosity of 1021 Pa s. The corresponding value of x c is ~50 m.y., compared to estimates of surface age for northern Venus of 250 m.y. to 1.5 b.y. [Ivanov et al., 1986; Schaber et al., 1987]. Although values of viscosity are sensitive to assumptions of temperature, composition, and the applicability of experimental flow laws, such a time scale illustrates the potential significance of mantle flow for producing crustal thickness variations and associated large-scale topographic and tectonic features on Venus.

For a body at depths less than several diapir diameters, the characteristic diameter of the affected surface region is approximately 2a. Thus the principal wavelengths of flow at which a rising or sinking diapiric body affect the crust are all less than or equal to twi•e the diameter of the diapir (4a). An estimate of the ratio of crustal deformation time to diapiric

BINDSCHADLER AND PARMI•I7•R: VENUS MANTLE FLOW TECTONICS 21,339

rising/sinking time is then

Xc 16Apd Rc a3 T- - L3 (24) x D 3•2p •1. m

For cases in which T >> 1, the rising or sinking of a diapiric body is rapid enough that little or no change in crustal thickness is expected. If T < 1, significant changes in crustal thickness are anticipated. We take p = 3.0 gcm '3 and Apd = 0.015 gcm '3 is found for a diapiric temperature difference of 250 K and a coefficient of thermal expansion o[ = 2 x 10 '5 K '1. Crustal thinning/thickening and accompanying deformation are then expected when

g_.•_c < 370 (25) I.tm

This formulation for T assumes a conservative upper limit on x c, in which no horizontal deformation (flow) is allowed in the upper crust or uppermost mantle. If deformation occurs in these regions, diapiric rising or sinking of material will result in crustal deformation over a broader range of conditions.

To illustrate the effects of time-dependent flow within the manfie, we model the diapirie sinking of an axisymmetric body within the manfie. The diapiric body is treated as a small region of higher density at a single depth within a viscous half-space. For the illustrated results a point source was used; however, the general results discussed here are not sensitive to the particular shape of the sinking body. A two-dimensional fast Fourier transform (128 x 128 elements) was used to represent the density distribution in terms of harmonic components, from which velocity response functions w ij and ultimately topo- graphy and strains, are calculated. Solutions are parameterized in terms of T, which determines the downward velocity of the diapir. Continuous movement of the diapir is simulated by placing the diapir at depth d for a time interval (At<<Xc) and calculating the accumulated topography at the surface and crust- mantle boundary and the hoop (e00) and radial (err) strains at the surface. The diapir is then moved a distance Az = wAt, where w is determined from (19) and the resulting Az is much less than crustal thickness (L). At each new location, additional topography and strains are added to those which have already accumulated. Results were tested for convergence by halving the time interval and comparing topography and strains. These were found to be identical within ~1%.

For this example, we consider the sinking of a body characterized by T = 1.0 and At = 0.5% with Rc = Rm. These parameters correspond to a diapir radius a -• 10L. The diapir was started at a depth of 10L and allowed to sink to a depth of-•22L. Beyond this depth, additional accumulation of topography was negligible. Results are summarized in terms of surface heights (Figure 13a) and horizontal surface strains (Figures 13b and 13c). The surface initially subsides, but by the time the sinking body has reached a depth of-•13L, a central elevated region has developed (Figure 13). The height and width of this topographic high broaden as the diapir continues to sink. Plots of accumulated strain (Figures 13b and 13c) show that during the early part of the deformation, hoop strains are compressional throughout the region, while radial strains are extensional near the edges of the high topography. By the time the diapir has reached a depth of ~13L, instantaneous strain rates within the high have become extensional and accumulated strains begin to decrease. This occurs because the depth of the diapir continues to increase, resulting in weaker coupling of

0.02

0.01

h•p o o.oo

-0.01

-35 -25 -15 ;5 '5 15 25 ;5

Fig. 13a. Surface topography due to the sinking of a diapiric body. As in previous figures, heights are dimensionless, scaled by p/c•. The horizontal axis is radial distance from the diapir in units of crustal thickness (L). Topography is shown at four times, corresponding to locations of the sinking body at depths of 10L, 11.5L, 13L, and 22L.

0.002 ] t

-0.002 t \k// -• •=•_•'•

-0.004 ] kx•,j -o- d/L= 13 -0.•6 1 .......

-35 -;5 -;5 -'5 • 1'5 2'5

Fig. 13b. Horizontal radial strains (l•rr) at the surface at the same times as in Figure 13a.

0.002

-0.000

-0.002

-0.004

-0.006 ....... - , ß

R/L

Fig. 13c. Hoop strains at surface (EOO) for the same times as Figure 13a. Although the topographically high region initially undergoes compression and total accumulated strains are net compressional, at later times, strain rates become extensional, leading to a reduction in accumulated strains.

flow into the crust at relatively short wavelengths, and allowing short wavelength topography to relax. At longer wavelengths, response is slower (Figure 7), and the topographic high continues to grow and broaden.

The changes in instantaneous strain rates at the surface suggest that deformational features due to diapir-driven mantle flow may be complex. Both radial and concentric features can be expected, although hoop strains might also be taken up by strike-slip deformation. Compressional deformation is

21,340 BIND$CHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTONICS

expected to propagate outward, horizontally away from the sinking body and may be followed by extension within a central region.

The effects of a rising diapiric body were also calculated (Figure 14) using the same parameters as in the case of the sinker. In this case, an initial broad uplift is created, which then undergoes subsidence due to diapir-driven crustal thin- ning. However, as the rising body approaches the surface, the vertical forces associated with the diapix outweigh the effects of crustal thinning, leading to the formation of a progressively higher and narrower central region (Figure 14a). Radial and hoop strains within the topographic high are increasingly extensional as the diapiric body nears the surface, and a zone of radial compression develops on the flanks of the high (Figures 14b and 14c). Such radial compression may explain the origin of the ridges which deCree the circular corona structures, which are suggested to be related to diapiric rise of mantle material [Stofan and Head, 1990].

DISCUSSION

Results of the previous section show how both upwelling and downwelling mantle flow might result in the formation of topographic uplands on Venus. Here we specifically consider several large-scale features on Venus as possible manifestations of mantle convection. Regions such as Atla, Bell, and Beta Regio are likely to be manifestations of mantle upwelling [Esposito et al., 1982; Phillips and Malin, 1983]. We suggest that mantle downwelling may be linked to the formation of western Ishtar Terra and some regions of tessera, while both upwelling and downwelling models may be considered for Ovda and Thetis Regiones in western Aphrodite Terra.

Topographic swells such as Aria, Beta, and Bell Regiones are 1000-3000 km diameter, 1-2 km high domes characterized by large line-of-sight (LOS) gravity anomalies [Sjogren et al., 1983]. Examination of gravity anomalies and topography of Beta Regio led Esposito et al. [1982] to conclude that the high topography of Beta Regio is supported at depths of-300 km, while Srnrekar and Phillips [1989] find apparent depths of compensation of ~160 km beneath Bell Regio. The surfaces of these swells are dominated by plains units, and large volcanic constructs are also present [McGill et al., 1981; Campbell et al., 1984; Janle et al., 1987; Stofan et al., 1989; Senske and Head, 1989], consistent with the presence of melting and thermal anomalies at depth. These characteristics strongly suggest at least partial support of topography by thermal anomalies within the mantle.

Although regions such as Beta have been uplifted by mantle upwelling [Stofan et al., 1989], little crustal thinning appears to have occurred. A rift zone (Devana Chasma) is present in Beta Regio, but appears to indicate only limited extension [McGill et al., 1981; Stofan et al., 1989]. Venera 15/16 images reveal that Bell Regio has undergone minor extensional deformation at most, manifested as subparallel graben striking approximately north-south across the center of the topographic high [Janle et al., 1987]. Chasmata interpreted as rifts intersect Atla Regio [Schaber, 1982], but available Pioneer Venus SAR data have insufficient resolution (~30 km) to reveal morphologic details indicative of the degree of extensional deformation.

The apparent lack of significant crustal thinning in these regions implies that T is greater than one; i.e., that the

h•p

0.01

0.008

0.006

0.004

0.002

-0.002

-35

+ d/L = 20.0

--•-- d/L = 19.2 o

i ' i ß i ' i ' i ' i

-25 -15 -5 5 15 25

0.001

0.0008

0.0006

0.0002

0

-35

--x--d/L = 20.0

+d/L = 19.2 A '-0--•; 1610 / \

' i ' i ' i i ' ; ' i . i -25 -15 -5 5 5 25 35

R/L

0.001 -

0.0008

.• o.ooo4

õ o.ooo2

-•x---d/L = 20.0

---•--d/L = 19.2 --o--d/L = 16.0

' I ' I ' I ' i ' i ß i i

-35 -25 -15 -5 5 15 25 35

Fig. 14. (a) Surface topography, (b) radial strain, and (c) hoop strain due to a rising diapiric body. All parameters are the same as in Figure 13, except for the initial depth (20L) and the sign of the buoyancy force. Note the zone of radial compression (Figure 14b) surrounding the topographic high created by the riser.

characteristic rise time of the mantle density anomaly (•D) is small compared to the characteristic time for crustal deformation (•:c). We investigate the constraint this second possibility places on lower crustal viscosity. Consider Beta Regio, using a characteristic wavelength of 3000 km, a corresponding mantle diapir radius a = 750 km, Apa = 0.05 g cm -3, and a crust with p = 3.0 gcm '3 and lower crust thickness

BINDSCHADLER AND PARMENTIER: VENUS MANTLE FLOW TECTOl•CS 21,341

of 20 km. For T > 1, we find that tXc/lXrn > 2 x 10 -3. If, for example, Venus' mantle viscosity is 1021 Pa s, then the effective viscosity of the lower crust must be at least 2 x 1018 Pa s. Bell Regio is smaller than Beta by a factor of 3, which leads to txc/lXrn > 5 x 10 -2.

Mantle downwelling, if it persists for long enough, can also create high topography. Although the initial response of the crust to mantle downwelling is subsidence, continued downwelling results in compressional thickening of the crust and elevated topography. Downwelling may be driven by the sinking of a portion of the relatively cold, dense lithosphere, or may simply result from larger scale, longer term patterns of convective flow within the Venus mantle.

Compressional deformation is perhaps most clearly manifested on Venus in western Ishtar Terra [Campbell et al., 1983; Barsukov et al., 1986; Crurnpler et al., 1986; Pronin, 1986]. This region contains the highest elevations and steepest regional slopes [Sharpton and Head, 1985] on Venus. Compressional mountain ranges surround Lakshmi Planum, a large (~l.3x106 km 2) elevated (>2.5 km above mean planetary radius) plateau covered by smooth, volcanic plains [Barsukov et al., 1986]. To first order, this geometry can be represented as axially symmetric. Finally, western Ishtar also lies within a larger region characterized by the highest crater densities on Venus thus far observed [Plaut and Arvidson, 1988], suggesting that western Ishtar is relatively old compared to the rest of the region mapped by the Venera orbiters.

For western Ishtar to be both relatively old and characterized by high elevations and steep slopes, some dynamic process must support the region. Otherwise, relaxation of high topography would occur. Pronin [1986] suggested that Lakshmi Planum is a locus of mantle upwelling, and that the surrounding mountain ranges were created and are supported by shear stresses due to flow outward from the center of Lakshmi.

However, as this study demonstrates, if the Montes represent crustal thickening due to mantle flow, then crustal thinning and surface extension of equal (or greater) magnitude is expected to occur within Lakshmi (e.g., Figure 14b). However, extensional features are largely absent on the Planum surface, aside from a few small (~10 km wide) graben.

Within the context of mantle flow tectonics, we consider whether mantle downwelling might explain the high topography and compressional deformation which characterize western Ishtar Terra. Long-term downwelling causes crustal thickening and the formation of high topography. Conductive heating of a thickened crust may lead to basal melting and surface volcanism, as suggested for the formation of Lakshmi Planum [Magee and Head, 1988]. Downwelling also tends to support the extreme elevations of Maxwell, Akna, and Freyja Montes against gravitational relaxation. Horizontal convergence due to downwelling is expected to produce compressional radial and hoop strains (Figure 13). The approximately concentric Danu, Akna, Freyja, and Maxwell Montes may have formed in response to radial compression. Radial features that might be characteristic of compressional hoop strains (Figure 13c) are not commonly observed. However, numerous large-scale strike-slip features that have been mapped within Akna, Freyja, and Maxwell Montes [Crurnpler et al., 1986; Vorder Bruegge et al., 1990] may accommodate hoop strains. Finally, this study predicts that elevated regions due to mantle upwelling should be younger than those due to downwelling. This is consistent with the observed difference in impact crater density in the regions

surrounding western Ishtar Terra and Beta Regio [Plaut and Arvidson, 1988; Head et al., 1988].

Pioneer Venus gravity data for Ishtar Terra have yielded an ADC of approximately 150 km [Sjogren et al., 1984]. Our results suggest that for crustal thickening in the steady state, ADC's are likely to be less than the actual crustal thickness (Figure 9). There are a number of possible implications of this ADC for a mantle flow tectonic model for western Ishtar.

Among the more plausible possibilities are the following: (1) Long-term, steady mantle flow has created extreme crustal thicknesses (>150 km) which are supported dynamically. (2) The measured ADC reflects the actual crustal thickness; the

region is no longer dynamically supported, and is in a state of gravitational collapse. (3) Some process or processes other than (or in addition to) the one suggested here is responsible for the high topography and deformation observed in western Ishtar Terra.

Cessation of downwelling allows relaxation of elevated regions to occur, resulting in extensional deformation. This postorogenic deformation may be related to the formation of tessera [Bindschadler and Head, 1988a, 1989a] and we note that regions of tessera bound all of the mountains of western Ishtar. Moreover, as demonstrated by the deformation due to a sinking diapiric body (Figure 13), deformational histories at the surface may be complex, with extension, compression, and strike slip faulting all occurring at different times within the same region. Mantle downwelling thus represents a useful working hypothesis for the formation of regions of tessera and is one of several currently being evaluated [Bindschadler and Head, 1989a].

The largest topographic upland region on Venus is Aphrodite Terra. Both mantle hotspots [Kiefer and Hager, 1988] and crustal divergence combined with hotspots [Head and Crurnpler, 1987; Sotin et al., 1989] and have been suggested to explain the origin of Ovda and Thetis Regiones, in western Aphrodite. We wish to introduce two additional possibilities. First, consider the case of a long-lived upwelling, in which crustal attenuation ceases, while crust still has a finite

thickness. Partial melting may then lead to production of melts which underplate, intrude, and thicken the crust. This forces further extension in order to preserve balance of vertical forces, and may lead to additional intrusion. The subsurface nature of such a "spreading" center may have significant consequences for its basic morphology in comparison to terrestrial spreading centers.

A second possibility is raised by the steep topographic slopes along the outer portions of Ovda and Thetis Regiones and the high values of small-scale (5 cm to 10 m) surface roughness of these regions [Bindschadler and Head, 1988b; Pettengill et al., 1980, 1988]. These steep topographic slopes are not associated with chasmata and may be more characteristic of compressional deformation, similar to slopes observed in Ishtar Terra. Enhanced small-scale roughness and steep outer topographic slopes are characteristic of regions of tessera rather than Beta-like regions [Bindschadler and Head, 1989b]. And unlike Beta, no large-scale volcanic constructs in western Aphrodite have been unambiguously identified, although this may be a factor of the relatively low resolution of topographic and radar image data available for the region. Such characteristics allow for the possibility that the high topography, steep slopes, and rough surfaces of Ovda and Thetis Regiones were formed by compression and crustal thickening due to mantle downwelling. The large chasmata

21,342 BINDSCHADLER AND PARMENTIER: VENUS MANTLE FIX)W TECTONICS

[Masursky et al., 1980; Schaber, 1982] are potentially explained as late stage features, formed as the upland relaxed subsequent to the cessation of mantle downwelling. Clearly, such a model is speculative and does not explain proposed characteristics of the region [Crumpler et al., 1987; Crumpler and Head, 1988]. On the other hand, available data do not rule out such a hypothesis. The range of plausible hypotheses is likely to be narrowed considerably by high-resolution data collected by the Magellan spacecraft.

CONCLUSIONS

The presence of a ductile lower crust within the lithosphere of Venus has important implications for the effects of mantle convection at the surface. Mantle flow causes thinning or thickening of the crust in two ways. First, vertical forces uplift or downwarp the crust-mantle boundary and the surface. Opposing normal stresses due to gravity and upward or downward flow at the crust-mantle boundary create a horizontal pressure gradient within the crust, leading to crustal thinning or thickening. Second, horizontal flow is coupled into the crust, driving material away from upwellings and toward downwellings. Shear stresses applied at the base of the crust primarily determine the amplitude of surface topography and crustal thickness changes that can occur due to mantle flow.

The amount of crustal thinning or thickening is sensitive to the presence or absence of elastic layers in the uppermost mantle and upper crust. An elastic mantle layer greatly reduces the maximum amount (long wavelength, steady state limit) of crustal thinning or thickening by preventing coupling of horizontal flow into the crust. An elastic upper crust has the opposite effect, increasing shear stresses and thus enhancing crustal thinning or thickening. Elastic layers also increase the characteristic time required for crustal deformation. Our results suggest that significant crustal deformation due to mantle flow can occur on time scales less than a few hundred million years over a wide range of crustal thicknesses, as long'as mantle viscosities are on the order of 1021 Pa s or less, and crustal viscosities are no more than 2 or 3 orders of magnitude greater. Variations in viscosity with depth also affect the time scales for deformation and the spatial wavelengths at which mantle flow efficiently changes crustal thickness. The ratio of lower crustal to upper mantle viscosity is particularly important, since this ratio strongly influences the coupling of horizontal mantle flow into the crust.

Manfie flow tectonics can be driven by either large-scale convective motions in the mantle, or by the sinking or rising of more discrete, diapiric bodies within the mantle. For time- varying mantle flow driven by the diapiric rising or sinking of material, the potential for crustal deformation depends most critically upon the ratio of diapir radius to crustal thickness (a/L), and the ratio of lower crustal to mantle viscosity (gc/grn). For large a/L, a significant portion of the lower crust must be weaker than the mantle for discernible crustal thinning or thickening to occur. Characteristic time and length scales for crustal deformation depend strongly on such variables as the characteristic depth of buoyancy forces, the crustal thickness, viscosities as a function of depth, and the presence or absence of elastic layers. Modeling the effects of a sinking diapiric body, we confirm the prediction that high topography can result from downwelling and find that diapiric sinking can result in complex, time-varying surface deformation. A specific

sequence of deformation can be identified and warrants further study.

Topographic uplands due to mantle flow can be created by both upwelling and downwelling within the mantle. Over a region of mantle upwelling, the crust is first uplifted and then thinned. The relatively minor amounts of extension, and thus crustal thinning, inferred for Venusian swells constrain the ratio of characteristic time scales for crustal deformation and

diapiric rise. We find that this constrains the ratio of lower crustal to mantle viscosity to values on the order of or greater than 10 -3.

Downwelling flow, if it persists for long enough, results in surface uplift due to crustal thickening. The compressional mountain ranges and high topography of western Ishtar Terra may be the result of such downwelling. If so, the region is predicted to be older than Beta Regio, since more time is required to thicken the crust than to uplift it. Such an age relationship is consistent with the observed impact crater densities. Comparison of the observed sequence of deformation and gravity with predictions of specific models is needed to more completely test a manfie downwelling model for western Ishtar Terra.

a

d

D

g h

k

L

wij x

gc

gm

P

Ap Apa

Xl,2 Xc

XD T

NOTATION

radius of diapiric riser or sinker. characteristic depth of buoyancy force. flexural rigidity. gravitational acceleration. relief at surface (h 1) or crust-mantle boundary (h2). wave number (2•/•). crustal thickness.

change in crustal thickness due to mantle flow. vertical velocity response function (see text). horizontal coordinate.

vertical coordinate.

critical flexural wavelength. crustal viscosity. mantle viscosity. density of crustal material. density contrast at crust-mantle boundary. density contrast between diapir and mantle. mass anomaly per unit area driving mantle flow. characteristic time constants for crustal deformation.

upper limit on x 1. scale time for rising/sinking of a diapiric body. ratio of Xc to XD.

Acknowledgments. This research was supported by NASA under grant NSG-7605 (now NAGW-1928) to E.M.P. Support for computing and for D.L.B. was provided by NASA grants NGR 40-002-116 and NAGW-713 to James W. Head. Thanks to Roger Phillips for his comments on an earlier version of this manuscript, and to Robert Grimm for his careful and thorough review.

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Duane L. Bindschadler and E. Marc Parmentier, Dept. of Geological Sciences, Box 1846, Brown University, Providence, RI 02912.

(Received July 10, 1989; revised June 28, 1990;

accepted July 11, 1990.)