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INTRODUCTION

During the past decades, geophysicists have pondered overthe causes and consequences of surface hotspots within theframework of plate tectonics. Morgan (1971) developed theseminal idea of thin plumes emanating from the core-mantleboundary and their rapid passage to the surface. Such a mecha-

nism corresponds to some burgeoning fluid-dynamical ideas ofboundary-layer instabilities and their finite-amplitude growth todiapirs and thin vertical upwellings (Howard, 1964; Goldsteinet al., 1990). The catalogs of surface hotspots (e.g., Burke andWilson, 1976) were compiled in the 1970s, and since then it hasbeen recognized that some of the hotspots might have origins inthe transition zone (Allègre and Turcotte, 1985) or in continen-

The Geological Society of AmericaSpecial Paper 430

2007

Lower-mantle material properties and convection models of multiscale plumes

Ctirad MatyskaDepartment of Geophysics, Faculty of Mathematics and Physics, Charles University,

V Holesovièkách 2, 180 00 Prague 8, Czech RepublicDavid A. Yuen

Department of Geology and Geophysics and Minnesota Supercomputing Institute,University of Minnesota, 117 Pleasant Street, SE, 599 Walter Library,

Minneapolis, Minnesota 55455-0219, USA

ABSTRACT

We present the results of numerical mantle convection models demonstrating thatdynamical effects induced by variable mantle viscosity, depth-dependent thermal ex-pansivity, radiative thermal conductivity at the base of the mantle, the spinel to per-ovskite phase change and the perovskite to post-perovskite phase transition in the deepmantle can result in multiscale mantle plumes: stable lower-mantle superplumes are fol-lowed by groups of small upper-mantle plumes. Both radiative thermal conductivity atthe base of the lower mantle and a strongly decreasing thermal expansivity of perovskitein the lower mantle can help induce partially layered convection with intense shear heat-ing under the transition zone, which creates a low-viscosity zone and allows for the pro-duction of secondary mantle plumes emanating from this zone. Large-scale upwellingsin the lower mantle, which are induced mainly by both the style of lower-mantle viscos-ity stratification and decrease of thermal expansivity, control position of central upper-mantle plumes of each group as well as the upper-mantle plume-plume interactions.

Keywords: partially layered convection, shear heating, temperature- and pressure-depend-ent viscosity, depth-dependent thermal expansivity, post-perovskite phase transition, radia-tive heat transfer in D″

137

*e-mail: [email protected]

Matyska, C., and Yuen, D.A., 2007, Lower mantle material properties and convection models of multiscale plumes, in Foulger, G.R., and Jurdy, D.M., eds., Plates,plumes, and planetary processes: Geological Society of America Special Paper 430, p. 137–163, doi: 10.1130/2007.2430(08). For permission to copy, contact [email protected]. ©2007 The Geological Society of America. All rights reserved.

tal edges via small-scale convection in the upper mantle (Kingand Anderson, 1995). The secondary instabilities from the tran-sition zone were modeled successfully in the 1990s by Honda et al. (1993), Steinbach and Yuen (1994), and Cserepes and Yuen(1997), who showed the importance of sufficiently high resolu-tion in the transition zone in capturing these thinner plumes witha width of about 50 km. Courtillot et al. (2003) also consider thatthe population of hotspots over the Earth’s surface should havetheir origins at different depths. This idea is also supported byrecent seismic imaging by Zhao (2004) and Montelli et al. (2004).Implicit in this type of reasoning is the assumption of a relativelystationary state of mantle convection occurring today.

In the lower mantle, large structures with low seismic velocities were found under Africa and the central Pacific byDziewonski’s pioneering work (Dziewonski, 1984; Su and Dzie-wonski, 1991) exhibiting axial symmetry (Matyska, 1995) andcorresponding to the long-wavelengths geoid anomalies and twogeographical groups of surface hotspots (e.g., Crough and Jurdy,1980; Stefanick and Jurdy, 1984; Richards and Hager, 1988;Matyska et al., 1998). These lower-mantle structures were called“superplumes” and recognized by Maruyama (1994) as a late-stage development of mantle evolution. They should be joinedwith the secondary smaller plumes generated at the boundarybetween the lower- and the upper-mantle at a depth of 670 km.

The basic physics of heat transfer in the Earth’s mantle canbe critically dependent on various combinations of mantle prop-erties. We emphasize that this physics is nonlinear and thus onecannot estimate at all the effects of some material changes a pri-ori without carrying out numerical experiments. The main purpose of our paper is to point out that the multiscale nature of mantle plumes in numerical models is a consequence of therichness and complexity of mantle physical properties andprocesses, such as phase transitions in both the upper and deepmantle. We demonstrate this point by showing some illustrativeexamples, drawn from both simple and complex models in thelong-time regime of mantle convection. One cannot obtain ascenario of multiscale plumes using a simple physical model.Simple steady-state models without phase transitions may workto some degree in the upper mantle, but still they would haveproblems explaining volcanoes associated with subducting slabs,which require thermal-chemical Rayleigh-Taylor instabilities(Gerya and Yuen, 2003) or secondary convective instabilities(Honda et al., 2002).

Although thermal-chemical convection has long been rec-ognized as being an integral component of mantle convection(Hansen and Yuen, 1989, 1994; Tackley, 1998; Nakagawa andTackley, 2005; Tan and Gurnis, 2005), we restrict ourselves tothe use of thermal convection. Our purely thermal models havethe following geophysical attributes:

1. Phase transitions both at the depth of 670 km and in thelower mantle (Iitaka et al., 2004; Murakami et al., 2004;Oganov and Ono, 2004; Tsuchiya et al., 2004);

2. Radiative thermal conductivity in the lower mantle (Lubi-

mova, 1958; Matyska et al., 1994; van den Berg et al., 2002;Matyska and Yuen, 2005, 2006);

3. Temperature- and depth-dependent viscosity, where thedepth-dependent viscosity has a peak in the mid-lower-mantle (Mitrovica and Forte, 2004);

4. Depth-dependent thermal coefficient of expansion (Chopelasand Boehler, 1992; Katsura et al., 2005); and

5. Nonlinear feedback effects, such as viscous dissipation andits coupling to temperature-dependent viscosity.

Other effects, not considered here, which are nonetheless veryimportant, are non-Newtonian rheology (e.g., Larsen and Yuen,1997), grain-size–dependent rheology (Solomatov, 1996, 2001;Korenaga, 2005) and grain-size–dependent thermal conductiv-ity (Hofmeister, 2005).

MODEL DESCRIPTION AND THEORETICAL BACKGROUND

The fundamental laws of conservation describing transferof heat in a dynamic Earth under the fluid approximation to-gether with rheological behavior and an equation of state are described in this section (for details, see the electronic lecturenotes displayed on the web page http://geo.mff.cuni.cz/~cm/geoterm.pdf ; see also the monographs, e.g., by Ranalli, 1995,and Schubert et al., 2001).

Equation of Continuity (Conservation of Mass)

The conservation of mass is written as:

∂ρ—— + ∇ ⋅ (ρv) = 0, (1)∂t

where ρ is the density, v is the velocity of motion and t is thetime. The symbol ∇ denotes the nabla operator and the dot ⋅ isthe scalar product.

Momentum Equation

The momentum equation in a nonrotating earth model is:

∂v∇ ⋅ τ + ρg = ρ —— + ρv ⋅ ∇v, (2)∂t

where τ is the Cauchy stress tensor and g is the gravity acceler-ation.

Conservation of Moment of Momentum

Conservation of the moment of momentum is given by:

τ = (τ)T, (3)

where T denotes transpose of a matrix.

138 Matyska and Yuen

Rheological Relationship

The stress tensor is considerd in the form:

τ = –pI + σ(v), limσ(v) = 0, (4)v→0

where p is the pressure, I is the identity tensor, and σ is a non-pressure part of the stress tensor. We will apply this system tothe Newtonian fluid, that is:

2 1σ = η(∇v + (∇v)T – — ∇ ⋅ vI) ≡ 2η(e – — ∇ ⋅ vI), (5)3 3

where —h is the dynamic viscosity and e the strain-rate tensor.

Heat Equation

For the heat equation, we have:

∂sρT (— + v ⋅ ∇s) = ∇ ⋅ (k∇T) + σ : ∇v + Q, (6)∂t

where T is the absolute temperature, s is the entropy per unitmass, k is the thermal conductivity and Q are the volumetric heatsources; the symbol : denotes the total scalar product of the sec-ond-order tensors. The first term on the right-hand side of equa-tion 6 describes conduction of heat and the second term thedissipation of heat.

If we assume that there is a reference hydrostatic state char-acterized by v = 0 in which the hydrostatic pressure p0, hydro-static density ρ0, and hydrostatic gravity acceleration g0 arerelated by

∇p0 = ρ0g0, (7)

and, moreover, that pressure deviations Π = p – p0 are negligi-ble in the heat equation, the transfer of heat in a homogeneousmaterial (i.e., entropy may be considered as a function of onlyp and T) is then described by the well-known equation:

∂Tρcp — = ∇ ⋅ (k∇T) – ρcpv ⋅ ∇T – ρvrαTg + σ : ∇v + Q, (8)∂t

where cp is the isobaric specific heat, α is the thermal expansioncoefficient and vr denotes the radial component of velocity. Theleft-hand side of equation 8 represents local changes of heat bal-ance; the second (third) term on the right-hand side describes ad-vection of heat (adiabatic heating and/or cooling).

Equation of State

The equation of state gives the density as a function of thepressure and temperature:

ρ = ρ(p, T). (9)

Today, with the advances made in computational quantum me-chanics (Tsuchiya et al., 2005) ρ(p,T) can be constructed read-ily as a look-up table and be part of the physical setup formodeling (Jacobs et al., 2006).

Approximations

The widely used Boussinesq approximation linearizes thesebasic laws near the reference hydrostatic state (e.g., Spiegel andVeronis, 1960; Ogura and Phillips, 1962; Schubert et al., 2001).If we neglect density changes caused by the pressure deviationsΠ = p – p0, we may linearize the state equation with respect tothe temperature deviations T – T0, where T0 is a referencetemperature, and write:

ρ = ρ0(1 – α(T – T0)). (10)

This approximation thus means that the influence of hydro-static pressure (as well as temperature T0) on density is hiddenin a spatial dependence of the reference density ρ0. For exam-ple, Monnereau and Yuen (2002) used in the role of depth-vari-able density model ρ0 the PREM model (Dziewonski andAnderson, 1981).

The reference density ρ0 is assumed to be a time-inde-pendent function. Considering only the largest term in the equa-tion of continuity, that is, neglecting thermal expansion, wearrive at the simplified equation:

∇ ⋅ (ρ0v) = 0 (11)

(see also Jarvis and McKenzie, 1980). After putting equations 7and 10 into the momentum equation 2, we get:

∂v–∇Π + ∇ ⋅ σ – ρ0α(T – T0)g0 + ρ0(g – g0) = ρ0 (— + v ⋅ ∇v),∂t(12)

where we have neglected the quadratic term –ρ0α(T – T0)(g –g0) on the left-hand side and the thermal expansion on the right-hand side, that is, the changes of the inertial force caused by thethermal expansion. Note that the deviation of the gravity accel-eration g – g0 is due to the self-gravitation of the Earth. The mag-nitude of the term ρ0(g – g0) in the mantle is much lower than that of the buoyancy term –ρ0α(T – T0)g0 except for thelongest wavelengths (Ricard et al., 1984); therefore, it does notinfluence substantially the basic physics of mantle thermal con-vection. For this reason, we omit the self-gravitation termthroughout the rest of this study.

Simplification of the heat equation consists of replacing ρby ρ0:

∂Tρ0cp — = ∇ ⋅ (k∇T) – ρ0cpv ⋅ ∇T – ρ0vrαTg0 + σ : ∇v + Q.∂t

(13)

Lower-mantle material properties and convection models of multiscale plumes 139

The system of equations 11–13 is referred to as the anelas-tic liquid approximation of the basic laws of conservation. How-ever, it is common to neglect compressibility in the equation ofcontinuity (11) and to replace it simply by:

∇ ⋅ v = 0. (14)

The obtained system of equations 12–14 is then usually calledthe extended Boussinesq approximation (e.g., Christensen andYuen, 1985), which is suitable for general mantle convectionstudies because shear heating and adiabatic heating and/or cool-ing are the substantial physical mechanisms influencing temper-ature and velocity patterns. However, the role played by com-pressibility in equation 11 is minor except for regions of phasetransitions.

The classical Boussinesq approximation, although an over-simplification for mantle convection studies, is suitable formany fluids in laboratory conditions and represents a furthersubstantial simplification of the studied system of equations.The reference density ρ0, the reference gravity acceleration g0,the thermal expansion coefficient α, the isobaric specific heat cp,and the thermal conductivity k are constant and the abovemen-tioned system is applied again to the Newtonian fluid with a con-stant dynamic viscosity η. Moreover, both dissipation σ: ∇v andadiabatic heating and/or cooling –ρ0vrαTg0 are not taken intoaccount. We thus get the system:

∇ ⋅ v = 0, (15)

∂v–∇Π + η∇2v – ρ0α(T – T0)g0 = ρ0 (— + v ⋅ ∇v), (16)∂t

∂T Q— = κ∇2T – v ⋅ ∇T + —— , (17)∂t ρ0cp

where κ = k/ρ0 cp is the thermal diffusivity. Two reasons whythe extended Boussinesq approximation has gained popularityis that it is easy to convert a Boussinesq code to extended-Boussinesq code, and computationally, this model is muchfaster than the anelastic approximation (Jarvis and McKenzie,1980).

We now introduce new dimensionless variables (denotedby *) by means of the relations:

d2 κs ηsκsr = dr*, t = — t*, v = — v*, Π = —— Π*, T = Ts + (Tb – Ts)T*,σs d d2

(18)

where r is the position vector and d is the characteristic dimen-sion of the system—for example, the thickness of the mantle inmantle convection problems or the vertical dimension of thefluid layer in problems in Cartesian geometry—and the sub-script s denotes surface values of corresponding quantities,whereas b denotes their bottom values. The system of equations12–14 in dimensionless variables thus reads:

∇* ⋅ v* = 0, (19)

η–∇*Π* + ∇* ⋅ (— (∇*v* + (∇*v*)T ))ηs

α ∂v*+ Ras — (T* – T*0)er = Prs–1 (—— + v* ⋅ ∇*v*),αs ∂t*

(20)

with er being the radial unit vector,

∂T* k Raqs—— = ∇* ⋅ ( —— ∇*T*) – v* ⋅ ∇*T* + ——— ∂τ∗ κs Ras

α Ts Dis η– Dis — (T* + ———) v*r + —— —— (∇*v* + (∇*v*)T ) : ∇*v*,αs Tb – Ts Ras ηs

(21)

where we introduced the (surface) Prandtl number Prs = ηs/ρ0κs,the (surface) Rayleigh number Ras = ρ0αs(Tb – Ts)g0d3/ηsκs, the(surface) Rayleigh number for heat sources Raqs = ρ0αsg0Qd5/ηsκsks and the (surface) dissipation number Dis = αsg0d/cp. Notethat the dimensionless heating term R = Raqs/Ras = Q d2/ks(Tb – Ts).

As the Prandtl number is extremely high (more than 1020,something like1022 at least) for mantle convection applications,we may use the infinite Prandtl number approximation; that is,we may replace equation 20 by:

η α–∇*Π + ∇* ⋅ (— (∇*v* + (∇*v*)T)) + Ras — (T* – T*0)er = 0.ηs as

(22)

From the physical point of view, this approximation means thatthe inertial force is negligible. As for the other numbers, in mostof the models of this study we have used Ras = 107, Dis = 0.5,and R = 3, which are standard values for mantle material prop-erties (e.g., Turcotte and Schubert, 2002). Enhanced Joule heat-ing (Braginskii and Meitlis, 1987) caused by an increase ofelectrical conductivity at the base of the mantle from electronictransitions (Badro et al., 2004; Li et al., 2005) in the D″ layercan also influence the thermal-electrical coupling between thecore and the lower mantle. We, however, neglect Joule heatingin this study, as it should play a remarkable role only if it reachesvery high magnitudes (Matyska and Moser, 1994).

Cartesian Geometry

In this section, we describe the equations that were used toobtain numerical models presented in this study. They are writ-ten in Cartesian coordinates (x,z), where x is a dimensionlesshorizontal coordinate and z denotes the dimensionless depth,that is, z = 0 at the surface and z = 1 at the bottom of a convect-ing layer. We can now obtain velocity field satisfying the equa-tion of continuity 19 by expressing velocity in the form:


∂ψ ∂ψv* ≡ (v*x , v*z ) = (—— , – ——). (23)

∂z ∂x

where ψ = ψ(x,z) is the stream function. As

v* ⋅ ∇*ψ = 0, (24)

it is clear that the isolines of ψ are the streamlines of velocity.The momentum equation 22 can now be rewritten as:

∂Π* ∂ η ∂2ψ ∂ η ∂2ψ ∂ψ– —— + —— [2 — ——]+ —— [— (—— – —— )] = 0, (25)

∂x ∂x ηs ∂x∂z ∂z ηs ∂z2 ∂x2

∂Π* ∂ η ∂2ψ ∂ η ∂2ψ ∂2ψ– —— + —— [2 — ——]+ —— [— (—— – —— )] =

∂z ∂z ηs ∂x∂z ∂x ηs ∂z2 ∂x2

αRas —— (T* – T*0). (26)αs

After applying the operator ∂/∂z to equation 25, the operator∂/∂x to equation 26, and subtracting both equations, we obtainthe final form of the momentum equation:

∂2 ∂2 η ∂2ψ ∂2ψ ∂2 η ∂2ψ(—— – ——)[— (—— – ——)] + 4 —— [— ——] = ∂z2 ∂x2 ηs ∂z2 ∂x2 ∂x∂z ηs ∂x∂z

∂ α–Ras — [—— (T* – T*0)]. (27)

∂x αs

The heat equation 21 now reads:

∂T* ∂ k ∂T* ∂ k ∂T* ∂ψ ∂T* ∂ψ ∂T*—— = —— (— ——) + —— (— ——) – —— —— + —— —— +R–∂t* ∂x ks ∂x ∂z ks ∂z ∂z ∂x ∂x ∂z

α Ts ∂ψ Dis η ∂2ψ ∂2ψ 2 ∂2ψ 2

–Dis —— (T* + ———) —— + —— — [(—— – ——) + 4 (——)]=αs Tb – Ts ∂x Ras ηs ∂z2 ∂z2 ∂x∂z

k 1 ∂k ∂T* 1 ∂k ∂ψ ∂T*= — ∇*2T* + — —— —— + — —— (∇∗Τ∗ ⋅ ∇T*) – —— —— + ks ks ∂z ∂z ks ∂T* ∂z ∂x

∂ψ ∂T*—— —— + R –∂x ∂z

α Ts ∂ψ Dis η ∂2ψ ∂2ψ 2 ∂2ψ 2

– Dis —— (T* + ———) —— + —— — [(—— – ——) + 4 (——) ],αs Tb – Ts ∂x Ras ηs ∂z2 ∂x2 ∂x∂z

(28)

as we consider k = k(z,T*) (see equation 42 below). We can seethat this heat equation is strongly nonlinear, as it contains thenonlinear terms describing the following effects: nonlinear dif-fusion of heat caused by the temperature dependence of thermalconductivity, horizontal and vertical advection of heat, adiabaticheating and/or cooling, and dissipation of heat. Additional nonlinearity, which appears in equation 27, is caused by thetemperature-dependence of viscosity (see equations 40 and 41).In principle, nonlinear creep mechanisms, such as dislocation

creep characterized by a dependence of viscosity on the strain-rate tensor, can also be taken into account, but dislocation creepdominates near cold slabs and not in hot lower-mantle regions(McNamara et al., 2001); nevertheless it can still be importantin olivine in the upper mantle.

The two scalar equations 27 and 28 for the two scalar un-knowns ψ and T* thus describe thermal convection in a 2-DCartesian box <0,a> × <0,1>, where a is the aspect ratio of thebox. To complete the equations, we need to add boundary con-ditions. We consider all boundaries to be impermeable with afree-slip:

∂2ψ ∂2ψψ = —— = 0 for x = 0, a and ψ = —— = 0 for z = 0, 1.∂x2 ∂x2

(29)

We assume no horizontal heat flow at the sidewalls

∂T*—— = 0 for x = 0, a and T* = 0 for z =, T* = 1 for z = 1.∂x

(30)

In other words, the boundary conditions at the sidewalls are re-flecting.

The nonlinearities of the dynamical system equations27–30 are the reason why both the time evolution of the systemand its spatial properties are crucially dependent on values of thephysical properties. For example, it is well known that theRayleigh number is the key parameter controlling the chaos in the system (e.g., Turcotte, 1992; Schubert et al., 2001). In this study, we also demonstrate that depth- and temperature-dependence of physical properties, such as viscosity, thermalexpansivity, and/or conductivity, are important factors for thelength scales of convecting fluid and its time-behavior as well.

Phase Changes

Phase changes in multicomponent systems, such as mantleminerals, are characterized by the existence of zones in whichtwo or more phases coexist. For simplicity, we assume that thedepth span of these zones is negligible and that we may describethem as the phase interfaces with jumps of density and entropy.Lateral variations of temperature generate undulations of phase-interface topography, which represent substantial additionalbuoyancy force caused by the density jump ∆ρ. Moreover, thejump in the entropy results in the release or consumption of la-tent heat when material flows through the phase-change inter-faces. We neglect the density jumps in the continuity equation,as it can only change the velocity by a few percent.

If Γ is the Clapeyron slope of such an interface, its undula-tion h (measured downward) caused by the temperature differ-ence T – T0 is approximately

h = Γ(T – T0)/ρog0. (31)


The buoyancy of the undulation can thus be described by meansof the additional pressure –∆ρg0h, which has the character of anexternal force in the momentum equation. In the first-order ap-proximation, we may thus add the force term

∆ρf = —— Γ(T – T0)δ(r – rp )er (32)ρ0

into the left-hand side of equation 12 or the force term

∆ρf * = ———— ΓRas(T* – T*0)δ(r* – r*p )er (33)

αsρ20gd

into the left-hand side of equation 20. Here δ is the Dirac δ-func-tion and rp is the radial distance of the phase interface withtemperature T0.

From a formal point of view, we only replace the thermalexpansivity α by

∆ρα′ = α + ——— Γδ(r – rp ) (34)ρ2

0g0

or α/αs by

α′ α ∆ρ α— = — + ———— Γδ (r* – r*p ) ≡ — + Pδ (r* – r*p ), (35)αs αs αsρ

20g0d αs

where P = ∆ρΓ/αsρ02g0d is called the phase buoyancy parame-

ter (see also Christensen and Yuen, 1985). It is clear that thesame replacement of the thermal expansivity in the adiabaticheating and/or cooling term of the heat equation then includesthe latent heat release or consumption.

Thermal Expansivity, Viscosity, and Radiative Heat Transfer

We have used the two profiles of the depth-dependent ther-mal expansivity. The first model has been parameterized to

α 8— = ——— , (36)αs (2 + z)3

where the thermal expansivity decreases by 8/27 across thewhole mantle (see, e.g., Zhao and Yuen, 1987). Quite recently,Katsura et al. (2005) announced that Anderson-Grüneisen pa-rameter of perovskite is close to 10, and thus the thermal ex-pansivity decrease over the lower mantle should be higher. Byapproximating a linear depth dependence of density in both theupper and the lower mantles, and assuming that Anderson-Grüneisen parameter of the upper-mantle minerals is close to 5,which corresponds to the estimates for olivine (Chopelas andBoehler, 1992), we arrive at the following relations (see Fig. 1):

α— = (1 + 0.78z)–5 if 0 ≤ z ≤ 0.23, (37)αs


1

0.5

0

0 0.5 1 1.5 2 2.5 3

dim

ensi

onle

ss d

epth

old thermal expansivitynew thermal expansivity

log_10 of viscosity

Figure 1. The depth dependence of dimensionless thermal expansivity(see equation 36) for the old thermal expansivity profile and for the newthermal expansivity profile (equations 37 and 38). The remaining curvedenotes the decadic logarithm of the depth-dependent part of the di-mensionless viscosity (see equation 39).

α— = 0.44(1 + 0.35(z 0.23))–10 if 0.23 < z ≤ 1. (38)αs<Figure 8-01 near here>Note that the magnitude of thermal expansivity at the top of thelower mantle should be ~3.5 × 10–5 K–1 according to Katsura etal. (2005).

We have considered the depth dependence of viscosity inthe form used by Hanyk et al. (1995; see also Figure 1):

η0(z)—— = 1 + 214.3z exp(–16.7(0.7 – z)2), (39)ηs

which gives rise to the lower-mantle viscosity maximum ataround 1800 km. Such a maximum is consistent with dynamicgeoid and postglacial rebound modeling (Ricard and Wuming,1991; Forte and Mitrovica, 2001; Mitrovica and Forte, 2004).

To take into account the temperature dependence of vis-cosity as well, we include the temperature-dependent part:

0.6f(T*) = exp (10.0 (———— – 1)) (40)0.2 + T*

in the Arrhenius form of a thermally activated process (e.g.,Davies, 1999; see also Matyska and Yuen, 2006). The compos-ite temperature- and depth-dependent viscosity, which has beenconsidered in complex models, is given by

η(z, T*) η0(z)——— = —— min {100, max {0.001, f (–T*)}}, (41)ηs ηi

that is, the temperature dependence of dimensionless viscosityis confined by the limits 0.01 and 100, so that the momentumequation may be solved by a conjugate-gradient iterative solver.

As one of the main aims of this study is to demonstrate po-tential creation of the lower-mantle superplumes by means ofthe radiative transfer of heat, we have neglected the depth andtemperature dependence of phonon thermal conductivity (Hof-meister, 1999) and considered the total thermal conductivity inthe form:

k(z, T*) Ts3

———— = 1 + g(z)(———— + T*) , (42)ks Tb – Ts

where the prefactor g(z) enables inclusion of the depth depend-ence of radiative heat transfer caused by the changes of compo-sition, opacity, and the like.

The set of equations 27–30, which describes our numericalmodels, thus consists of the fourth-order linear elliptic equation27 for the streamfunction ψ with the boundary conditions ofequation 29 (with both laterally and vertically varying coeffi-cients because of the presence of η(z,T*)/ηs) and the non-linear time-evolutionary advection-diffusion equation 28 for thetemperature T* with the boundary conditions described by equa-tion 30. The depth changes of thermal expansivity result in thechanges of the buoyancy forcing term in the elliptic equation aswell as in the changes of adiabatic heating and/or cooling in thetime-evolutionary equation. The inclusion of radiative transferof heat by means of the temperature-dependent thermal con-ductivity then changes the ratio between diffusion and advectionof heat.

Computations have been carried out in a wide box with anaspect ratio of 10 to avoid the influence of side boundaries. Wehave used 1281 × 129 equally distributed nodal points for a second-order finite difference scheme in space. The elliptic equa-tion has been solved by the conjugate-gradient iterative scheme,and a second-order Runge-Kutta scheme has been applied forthe time-stepping in the time-evolutionary equation.

RESULTS

Basic Physics of Simple Models: Influence of the Depth Dependence of Mantle Properties

We begin in Figure 2 by illustrating the style of mantle con-vection without the presence of major phase transitions but withthe extended Boussinesq approximation. We vary a whole gamutof depth-dependent properties in the thermal expansion co-efficient and viscosity. The top panel is a typical snapshot oftemperature field in the long-time regime for constant materialproperties and Ras = 106. The dissipation number is 0.5. Con-vection is rather chaotic with many cold downwellings and hotupwellings. We can see that the role played by adiabatic heatingand/or cooling is substantial in such a model; plume heads dis-appear before reaching the top boundary layer of convection be-cause of the rapid cooling due to constant thermal expansivity(Zhao and Yuen, 1987). This extended Boussinesq model thusdoes not correspond to the classical Boussinesq idea of plumes

generating locally hot material, which then rises and interactswith the cold upper boundary layer (lithosphere). This main fea-ture of convection is not substantially changed when theRayleigh number is increased to 107 in the second panel. Theonly remarkable difference from the previous case is that influ-ence of the internal heating is minor, and thus we get more sym-metry between the cold and hot anomalies.

The middle panel shows the case when the depth-dependentviscosity (see equation 39) is taken into account. Because an increase of viscosity corresponds to a relative decrease of buoy-ancy in the momentum equation, convection is then character-ized by fewer big stable plumes, emerging from the lowerthermal boundary layer (see also Hansen et al., 1993, for 2-Dmodels and Cserepes, 1993, for 3-D modeling). It is of greatgeophysical interest that there is a high lateral temperature con-trast between the plume and a very cold ambient mantle in thelower part of the convecting layer, but this contrast becomessmall at the top of the model. Such a large thermal contrast inthe lower mantle may induce seismic velocity contrasts of a cou-ple of percentage points (e.g., Ni and Helmberger, 2003). Thecold thermal boundary layer at the top of the model is unstableand acts as a source of cold “lumps,” which slowly fall to the“viscosity hill” (e.g., Ricard and Wuming, 1991; Mitrovica andForte, 2004) in the mid-lower mantle.

The bottom two panels in Figure 2 represent cases with de-creasing thermal expansivity in the lower mantle. They revealthat a similar stabilization of convection cells, creation of bigplumes, and lowering of the average temperature also can beproduced by a decrease of thermal expansivity in the lower man-tle, as it causes a decrease of buoyancy. There is, however, onesubstantial difference: low thermal expansivity at the bottom ofthe model also means a low adiabatic cooling of the bottom partof (super)plumes. It results in higher plume temperatures, andthus very hot plume heads (see also Zhao and Yuen, 1987) areable to reach and interact with the cold upper boundary layer.

To deal with more realistic models, where the critical de-pendence of mantle heat transfer on combinations of mantleproperties is demonstrated, we incorporate the effects of vari-able viscosity, radiative thermal conductivity, and the two ma-jor phase transitions in the upper and lower mantles, which areillustrated in Figure 3. These more realistic models allow for agreater variety of dynamical possibilities.

Complex Models

First, we show the results for the old thermal expansivityprofile, decreasing by 8/27 across the mantle (see equation 36),depth- and temperature-dependent viscosity, and the two phasetransitions characterized by the buoyancy parameters P670 =–0.15, PD″ = 0.10. Typical snapshots of temperature are shownin Figure 4, thermal anomalies obtained by subtracting out thehorizontally averaged temperature are shown in Figure 5, andcorresponding stream functions are shown in Figure 6. The re-sults in the top panel were obtained for the constant thermal con-



Figure 2. Typical snapshots of the temper-ature field for a long time scale (on theorder of 109 years). No phase changeswere included. The red color representsthe maximum temperature, whereas thedark blue color denotes the minimumtemperature. The medium temperature isgiven by the cold-to-warm transitionfrom green to yellow. The panels showthe effect of a change of the surfaceRayleigh number from 106 to 107 in thetwo top panels and inclusion of thedepth-dependence of viscosity or ther-mal expansivity in the remaining panels.See Figure 1 and equation 36 for the oldthermal expansivity, equations 37 and 38 for the new thermal expansivity, and equation 39 for the depth-dependentviscosity.

Figure 3. The major geophysical features of the complex models usedfor generating multiscale plumes. CMB—core-mantle boundary; PPV—postperovskite; PV—perovskite.

ductivity k = 1 considered for the entire mantle. We can see thatconvection is partially layered and the consequence of thetemperature dependence of viscosity is the formation of largecold anomalies that are able to penetrate the phase boundary atthe 670 km. The highest velocity of downwellings is reached inthe lower mantle. This downward flow is balanced at 670 km bysmall plumes originating below this interface, from which hotmaterial is ejected into the upper mantle. The counterpart of thedownwellings in the lower mantle are the plumes, which are big-ger than those in the upper mantle and are mainly visible in thebottom half of the lower mantle.

The next three panels show the influence of increasing ra-diative transfer of heat through the D″ layer, which is parame-terized by the coefficient g adjacent to the cube of absolutetemperature in the radiative thermal conductivity (see equation42). An important consequence of this phenomenon is an in-

crease of average temperature in the lower mantle and, subse-quently, a much lower temperature contrast between the lower-mantle (super)plumes and the ambient mantle. The vigor ofconvection is increased together with an enhancement of partiallayering.

An outstanding result of these models is the creation of avery hot, thin layer just below 670 km, which is again the sourceof small upper-mantle plumes. Jetting of hot material into theupper mantle is periodically repeated (inverse flushing events)in several places, thus resulting in the appearance of many upper-mantle plumes, which are then drifted horizontally by an upper-mantle mean flow toward the central location of the main upper-mantle plume. Positions of the central upper-mantleplumes are, however, controlled by the lower-mantle super-plumes. Stability of the lower-mantle superplumes, the attrac-tion of the upper-mantle plumes to one place, and subsequentplume-plume interactions may then explain why the upper-mantle plumes are able to act as a “fixed” sublithospheric heat

source for a long time. Figure 7 illustrates that this hot layer be-low the 670-km discontinuity and the upper-mantle plumes to-gether form the regions of the lowest viscosities in our models,which correspond to the second asthenosphere, as revealed byKido and Cadek (1997) under oceanic regions (see also Mitro-vica and Forte, 2004).

We also computed the model in which the radiative heatingterm is taken into account in the whole mantle. Such an overallincrease of thermal conductivity further boosted the convectivevelocities, where cold downwellings in the lower mantle are inthe form of huge blobs and the average temperature of the lowermantle decreases. However, lateral temperature contrast betweenthe upper-mantle plumes and the ambient upper mantle becomessmaller.

Figure 8 illustrates that the shear heating generated insidethe convecting layers is maximal in downwellings because ofthe combination of high viscosities together with high rates of deformation; the secondary maxima are usually reached in-


Figure 4. Typical snapshots of the temper-ature field for a long time scale. The olddepth-dependent thermal expansivityaccording to equation 36 was considered.Viscosity was both depth and temper-ature dependent (see equations 39–41).An endothermic phase change with P =–0.15 at the depth of 670 km and anexothermic phase change with P = 0.10at the depth of 2650 km were included.Thermal conductivity was considered inthe form of equation 42, where g(z) = 0(no radiative heat transfer) in the toppanel and g(z) = 10 (strong global radia-tive heat transfer) in the last panel. In theremaining panels, radiative heat trans-fer is considered only below the depth2650 km, with g equal to 2, 5, and 10, respectively.

Figure 5. Residual temperature (the difference between the local temperature and horizontally averaged temperature) forthe cases shown in Figure 4.

k= 1

k=1+2( T 0+T ) 3 in D� k= 1 above D�

k=1+5( T 0+T ) 3 in D� k= 1 above D�

k=1+10( T 0+T ) 3 in D� k= 1 above D�

Figure 6. Streamlines of the model flow for the cases shown in Figure 4. The streamfunction is equal to zero on all sidesof the box, and the contour interval for streamlines is 50 in dimensionless units. The contour interval is the same in allpanels. The solid lines display the zero and negative values and the dashed lines show the positive values of the stream-function.

k= 1

k=1+2( T 0+T ) 3 in D� k= 1 above D�

k=1+5( T 0+T ) 3 in D� k= 1 above D�

k=1+10( T 0+T ) 3 in D� k= 1 above D�

Figure 7. Decadic logarithm of dimensionless viscosity for the cases shown in Figure 4. Dashed lines show negative val-ues and solid lines mark zero and positive values. The contour interval is 0.5 in all panels.

Figure 8. Decadic logarithm of dimensionless shear heating for the cases shown in Figure 4.

side the upper-mantle plume regions, thus generating the hotlayer under the transition zone, as mentioned above. Thus wecan see clearly in Figure 8 that shear heating can be locally muchhigher than the averaged bulk heating, and so it should not beneglected in physical descriptions of complex models wherethere exist many regions with stagnation points at which flowsundergo severe deformation.

Recent measurements on perovskite (Katsura et al., 2005)indicate that decrease of thermal expansivity with increasingpressure is much higher than previously estimated for olivine(Chopelas and Boehler, 1992). For this new thermal expansiv-ity profile in the lower mantle—with a decrease of thermal ex-pansivity in the lower mantle by a factor of eleven—we haveconsidered various models with different types of mantle ther-mal conductivities, ranging from constant thermal conductivityto prevailing radiative thermal conductivity. As for the old ther-mal expansivity profile, we have also studied a suite of inter-mediate cases, in which the radiative conductivity is presentonly in the D″ layer and is varied from twice to ten times theconstant thermal conductivity value. Figures 9 and 10 show thetemperature fields and the residual temperature fields. Becausethe decrease of thermal coefficient of expansion results in a de-crease in the average temperature, the lower-mantle plumes canbe discerned much more easily than in the previous cases shownin Figures 7 and 8. It is clear that the influence of radiative ther-mal conductivity and depth-dependent thermal expansivity canbe quite profound on the development of the lower-mantle up-wellings, because the reference model with constant thermalconductivity results in clusters of smaller, unstable lower-mantle plumes. In contrast, a substantial increase of thermalconductivity in the D″ layer gives rise to the production oflower-mantle superplumes. Broad upwellings are much moreprominent, with a thin radiative thermal conductive layer in theD″ layer than with the radiative thermal conductivity distributedthroughout the mantle.

In these models, we also obtained flow reversals (see thestreamlines shown in Fig. 11) at 670 km boundary and partiallayered convection. Flow reversals can generate a considerableamount of mechanical heating. There is a low-viscosity regionin the vicinity of the transition zone (see Fig. 12), due to the in-jection of hot lower-mantle material. Shear heating (Fig. 13) ishigher than in the previous models, with a smaller decrease inthe thermal expansivity. There are even places located below the670-km boundary that are overheated by viscous heating to veryhigh temperatures, exceeding the dimensionless temperature ofunity, which is the temperature at the core-mantle boundary (wehave used a periodic color scale and thus these sites are por-trayed by dark blue inside dark red regions). The upper-mantleplumes are again small but very hot.

Note that the convection pattern at 670 km is sensitive tothe magnitude of the phase buoyancy parameter P670 , buoyancyof the lower-mantle superplumes, and viscosity stratification,because a low-viscosity zone below the interface between themantles facilitates horizontal flow. We used lower “new” ther-

mal expansivity (see equations 37 and 38) in the second set ofmodels and obtained partial layering for P670 = –0.08, which isin good agreement with the values estimated for the spinel toperovskite phase change. However, in the first set of modelswith the old thermal expansivity profile (see equation 36), buoy-ancy of the lower-mantle superplumes is higher, and we demon-strated that similar partial layering of convection can beobtained for P670 = –0.15, that is, with slightly overestimatedmagnitude of the buoyancy parameter.

The change of the model behavior in the case in which theradiative thermal conductivity is considered in the whole man-tle (see the bottom panels of Figures 9–13) is now much moreremarkable. The combined effect of the small value of thermalexpansivity (i.e., small adiabatic gradient) with higher velocitiesof convection results in a well-developed lower thermal bound-ary layer. Subsequently, the average lower-mantle temperatureis lower, temperature contrast between the superplume and theambient mantle is higher, and the superplumes are thinner.Moreover, the amount of material passing through the 670-kminterface is higher because of higher buoyancy. This combina-tion finally results in thicker upper-mantle plumes. The conse-quence is that the surface Nusselt number (i.e., the surface heatflow) is rather high, as it varies between 20 and 45. However,typical surface Nusselt numbers of the models with high ther-mal conductivity confined to D″ layer are lower and usually os-cillate between 10 and 15. It is interesting that the model withradiative transfer of heat in the whole mantle yields similar bimodal plumes to those obtained by van Keken et al. (1992) for an ad hoc stratification of rheology and constant thermal conductivity.

To understand better the small-scale features of upper-man-tle plumes, we zoomed into the plumes in the upper mantle inFigures 14 and 15 for both thermal expansivity models. It is ob-vious that many upper-mantle plumes come from the transitionzone. The overall morphology of the upper-mantle upwellingsis similar in both models, with the upper mantle being slightlyhotter for the more steeply decreasing thermal expansivity. Onlyin the case for which the radiative thermal conductivity prevailsthroughout the mantle is the upper-mantle plume directly fedfrom the deep lower mantle (see bottom of Fig. 15). There is also evidence of overheating (see the dark blue color) in somelocal areas, corresponding to the roots of upper-mantle plumes.Although such overheating need not be fully realistic for theEarth, it points to the importance of shear heating in convectiondynamics.

To demonstrate the time scale of upper-mantle plume dy-namics, we present several snapshots of temperature showingthe temporal evolution of the upper-mantle plumes in the casefor which the thermal conductivity in the D″ layer is dominatedby the radiative term, whereas radiative heat transfer is negligi-ble above the D″ layer (see Fig. 16). Time steps between twosubsequent panels correspond to ~15 m.y. in dimensional time.We note that non-Newtonian rheology can result in shorter timescales of about a few million years in the upper mantle (Larsen


Figure 9. Typical snapshots of the temperature field for a long time scale. The new depth-dependent thermal expansivityaccording to equations 37 and 38 was considered. Viscosity was both depth and temperature dependent (see equations39–41). An endothermic phase change with P = –0.08 at the depth of 670 km and an exothermic phase change with P =0.05 at the depth of 2650 km were included. Thermal conductivity was considered in the form of equation 42, where g(z)= 0 (no radiative heat transfer) in the top panel and g(z) = 10 (strong global radiative heat transfer) in the last panel. In theremaining panels, radiative heat transfer is considered only below the depth 2650 km with g equal to 2, 5 and 10, respec-tively. We used a periodic color scale, that is, dark blue below the transition zone shows where dimensionless tempera-tures are slightly higher than 1, which corresponds to the temperature at the core-mantle boundary.

Figure 10. Residual temperature (the difference between the local temperature and horizontally averaged temperature) forthe cases shown in Figure 9.

k= 1

k=1+2(T 0+T ) 3 in D� k= 1 above D�

k=1+5(T 0+T ) 3 in D� k= 1 above D�

k=1+10(T 0+T ) 3 in D� k= 1 above D�

k=1+10(T 0+T) 3 in the whole mantle

Figure 11. Streamlines of the model flow for the cases shown in Figure 9. The streamfunction is equal to zero on all sidesof the box and the contour interval of streamlines is 50 in dimensionless units. The contour interval is the same in all panels. The solid lines display the zero and negative values and the dashed lines show the positive values of the stream-function.

k= 1

k=1+2(T 0+T ) 3 in D� k= 1 above D�

k=1+5(T 0+T) 3 in D� k= 1 above D�

k=1+10(T 0+T ) 3 in D� k= 1 above D�

Figure 12. Decadic logarithm of dimensionless viscosity for the cases shown in Figure 9. Dashed lines show negative val-ues and solid lines mark zero and positive values. The contour interval is 0.5 in all panels.

Figure 13. Decadic logarithm of dimensionless shear heating for the cases shown in Figure 9. Note that the amount of radiogenic heating due to chondritic abundance is ~10.

Figure 14. Zoom of the upper-mantle plumes. Left (right) column is for the cases shown in the left (right) part of Figure4. Vertical range is ~1200 km, and the horizontal length corresponds to ~7200 km. The difference between the localtemperature and horizontally averaged temperature is displayed.

Figure 15. Zoom of the upper-mantle plumes. Left (right) column is for the cases shown in the left (right) part of Figure9. Vertical range is ~1200 km, and the horizontal length corresponds to ~7200 km. The difference between the localtemperature and horizontally averaged temperature is displayed.

Figure 16. Time series of snapshots ofthe upper-mantle plumes, in which thedepth-dependent thermal expansivityaccording to equations 37 and 38 was considered. An endothermic phasechange with P = –0.08 at the depth of670 km and an exothermic phase changewith P = 0.05 at the depth of 2650 kmwere included. Thermal conductivitywas considered in the form of equation42, where radiative heat transfer is con-sidered only below the depth 2650 kmwith g = 10. Vertical range is ~1100 km,and the horizontal length corresponds to~5300 km. Dimensionless time differ-ences between two subsequent snap-shots is 0.00005, corresponding to ~15m.y. We used a periodic color scale, thatis, dark blue below the transition zoneshows where the dimensionless temper-atures are slightly higher than 1.

and Yuen, 1997). In our model, the temperature of the plume rootsbelow the 670-km interface is very high—it can even be locallyslightly higher than the temperature of the core-mantle boundary,which results in faster plume-plume interactions and more turbu-lent plume evolution in comparison to places where the upper-mantle plume roots are colder below the transition zone.

CONCLUSIONS

In the past decade, seismic imaging of the mantle has im-proved immensely because of the great advances in data acqui-sition (Grand et al., 1997; Ritsema et al., 1999), theoreticaldevelopments in finite-frequency effects (Montelli et al., 2004;Zhou et al., 2005), and computational hardware and numericaltechniques (Komatitsch and Tromp, 2002a,b). Both super-plumes in the lower mantle and smaller upper-mantle plumeshave now been unveiled by body waves (Zhao, 2001, 2004; Leiand Zhao, 2005; Pilidou et al., 2005) and surface waves (Zhouet al., 2004). These images undoubtedly have provoked a revi-sion in the traditional concept of how mantle upwellings shouldappear, because we have been so ingrained by ideas inculcated

in the 1980s from steady-state laboratory experiments, usingsimple fluids and steady point sources of heating (e.g., White-head and Luther, 1975; Olson and Singer, 1985). These recenttomographic images have revealed clearly the multiscale natureof mantle plumes, which we have portrayed schematically inFigure 17. Implicit in this drawing is the multiscale nature inboth time and space, as mantle flow is intrinsically very time-dependent because of the nonlinear physics in the transportproperties and the phase transitions, and the feedback loops be-tween the various processes. In Figure 17, we see the differentorigins of the different spatial scales of mantle plumes. Such achange of spatial scales between the upper and lower mantles isalso consistent with corresponding change of spatial tomo-graphic spectra (Dziewonski, 2000).

In numerical models of mantle convection, the superplumesin the lower mantle can be created and stabilized in both timeand space by the viscosity stratification (e.g., Hansen et al., 1993)and increased thermal conductivity (e.g., Matyska et al., 1994;van den Berg et al., 2001, 2002; Dubuffet et al., 2002; Matyskaand Yuen, 2006; Naliboff and Kellogg, 2006), which should bepresent due to radiative heat transfer. However, radiative trans-



Figure 17. Sketch showing the multi-scale nature of mantle plumes, which in-volves the creation of the lower-mantlesuperplume from the D″ layer and gen-eration of the upper-mantle plumes fromthe low-viscosity layer below 670 km.

fer of heat in the deep mantle is still a matter of controversy. Itshould be important or even dominant in olivine and perovskite(Hofmeister 1999, 2005; Badro et al., 2004; Gibert et al., 2005)as well as in post-perovskite (Mao et al., 2005), but a decreaseof radiative heat transfer with increasing pressure was observedfor magnesiowüstite (Goncharov et al., 2006). Stabilization ofsuperplumes enables forward studies of their properties, such as the adiabaticity of the superplume mode of heat transfer(Matyska and Yuen, 2001). Here we showed that a decrease ofthermal expansivity with depth is another factor facilitating theexistence and stability of the lower-mantle superplumes. More-over, increased thermal conductivity at the base of the mantletogether with intensive shear heating at the 670-km depth is ableto create the low-viscosity zone (Kido and Cadek, 1997; Mitro-vica and Forte, 2004) acting as the source of the upper-mantlesmaller plumes, which are stabilized by interaction with thelower-mantle superplumes. Note that there also can be chemicalheterogeneities associated with the seismically observed super-plumes (Ishii and Tromp, 1999; Trampert et al., 2004), whichcan be modeled in the framework of thermal-chemical con-vection (e.g., Tan and Gurnis, 2005). Together with iron-richpatches in post-perovskite (Mao et al., 2004, 2006), these chem-ical heterogeneities could also stabilize lower-mantle convec-tion. It is still a problem to distinguish between thermal andchemical heterogeneities; for example, the magnitude of densityheterogeneities inferred directly from tomographic models issimilar to that obtained from thermal convection models(Matyska and Yuen, 2002).

Smaller plumes with a greater propensity for time depend-ence because of the lower viscosity and thermal conductivity

caused by water content have shorter lifetimes (e.g., Davailleand Vatteville, 2005) and emerge rapidly from the transitionzone, which is further aided by non-Newtonian rheology in theupper mantle. In other words, the mantle plumes not only havea multiscale spatial nature, but also richness in the temporalspectrum, which can span over several orders of magnitude.From the Taylor hypothesis (Zaman and Hussain, 1981; Mene-veau and Sreenivasan, 1991) in fluid mechanics, we would ex-pect that in a strongly nonlinear regime, larger-scale coherentstructures would have longer lifetimes than would the smaller-scale features. In the interpretation of tomographic images, onemust then keep in mind that the classical picture of a mantleplume with a long conduit connecting to its source at a thermalboundary layer is only valid for a short time before the detach-ment process occurs. This criterion is especially pertinent in theupper mantle, where the lifetime of plumes may be only of theorder of 10 m.y., from boundary-layer stability estimates basedon the physical properties of the transition zone (e.g., Howard,1964). In this connection, Maruyama (1994) has offered com-pelling geological arguments concerning the long lifetime of thesuperplume under the central Pacific.

Our aim in this article is to demonstrate that by using real-istic physics, such as the depth-dependent thermal expansivity,variable viscosity, radiative thermal conductivity, and phasetransitions, one can also produce a rich spectrum in both the spa-tial and temporal scales of mantle upwellings for realistic man-tle conditions without going into the hard turbulent thermalconvection regime (Yuen et al., 1993). Although our modelingwas in two dimensions, the recent 3-D findings in a Cartesianbox with an aspect-ratio of 6 × 6 × 1 by Kameyama and Yuen

(2006) show that convection behavior in the presence of thepost-perovskite phase transition in three dimensions is similarto the results obtained in two dimensions. Therefore, we main-tain that our results may be representative of convection in theEarth’s mantle. Indeed, one is hard pressed to do the same in lab-oratory experiments with simple fluids and in a heated-from-below configuration, even at very high Rayleigh numbers. Wehope that this article will stimulate more realistic numericalmodeling of mantle plumes in the future, with a resurgent focuson the transient nature of mantle plumes (King and Ritsema,2000; Davaille and Vatteville, 2005) and their interactions withthe lithosphere (Thoraval et al., 2006). With the relentless drivetoward petascale computing (Cohen, 2005), geodynamicists canovercome the current numerical limitations in speed, memory,and data storage and can address in due course some of the is-sues raised above for both thermal and thermal-chemical con-vection, which is a much more difficult computational problemthat has the potential to broaden the complexity of plume mod-eling (Farnetani and Samuel, 2005).

ACKNOWLEDGMENTS

We highly appreciate constructive reviews and comments byGillian Foulger, Donna Jurdy, Arie van den Berg, Garrett Ito,and Scott King. We thank Cesar da Silva, Renata Wentzcovitch,Charley Kameyama, Shige Maruyama, and Marc Monnereaufor their discussions with us. We also thank Donna Jurdy for herencouragement in writing this article and Ying-Chun Liu for her drawings. This research has been supported by National Sci-ence Foundation Cooperative Studies of the Earth’s Deep Inte-rior and Information Technology Research grants, by the CzechScience Foundation grant 205/06/0580, and by the researchproject MSM 0021620800 of the Czech Ministry of Education.

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Manuscript Accepted by the Society 31 JANUARY 2007



DISCUSSION

25 December 2006, Gillian R. Foulger

It has been remarked that the heating mode in the article byMatyska and Yuen (this volume) is not clear, the figures give theimpression that the mantle is assumed to have no radioactivity,and the 2-D models are driven entirely by heating from below.Would the authors care to clarify the thermal boundary condi-tions, and the bearing any simplifying assumptions have on thecontribution of core heat and the resulting Rayleigh number?

6 January 2007, Ctirad Matyska and David A. Yuen

Our convection models are partly heated from within, as wehave used the dimensionless heat source term R = 3, which isequivalent to about 25% of the chondritic value. Moreover, withradiative thermal conductivity, there is a tendency for over-heating in the lower mantle for a chondritic mantle. The bot-tom boundary is isothermal, and thus a substantial amount ofheat comes from below. This heating mode could be caused by a decrease of the core gravitational potential energy due to inner-core formation, ohmic dissipation, and perhaps someradioactivity from potassium. Moreover, the core was heatedduring Earth’s formation, and its cooling represents another

source of heat, which flows from the core to the mantle. How-ever, we did not compute cooling of the core, and thus thetemperature of the bottom boundary does not change with timein our computations.

The surface Rayleigh number was set to 107; we note, how-ever, the concept of Rayleigh number becomes vague in thepresence of depth-dependent thermal expansivity, depth- andtemperature-dependent viscosity, and radiative thermal conduc-tivity. An “effective” Rayleigh number is much lower than thesurface value, especially at the bottom of the lower mantle—something like 105. The combined effect of the depth-depend-ent mantle material properties resulted in the creation of thelower-mantle superplumes with very low lateral temperaturevariations, followed by classical thin plumes emerging from the670-km boundary.

REFERENCE

Matyska, C., and Yuen, D.A., 2007 (this volume), Lower mantle material prop-erties and convection models of multiscale plumes, in Foulger, G.R., andJurdy, D.M., eds., Plates, plumes, and planetary processes: Boulder, Col-orado, Geological Society of America Special Paper 430, doi: 10.1130/2007.2430(08).

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