the inﬂuences of composition-sand temperature-dependent

Physics of the Earth and Planetary Interiors 129 (2002) 43–65

The influences of composition-sand temperature-dependentrheology in thermal-chemical convection on

entrainment of the D′′-layer

B. Schotta,∗, D.A. Yuenb, A. Braunc

a Theoretical Geophysics, Utrecht University, P.O. Box 80021, 3508 TA Utrecht, The Netherlandsb Department of Geology and Geophysics, Minnesota Supercomputing Institute, University of Minnesota,

1200 Washington Avenue South, Minneapolis, MN 55415-1227, USAc GeoForschungsZentrum Potsdam, Section 1.2, Division 1, Potsdam, c/o DLR Oberpfaffenhofen, 82234 Wessling, Germany

Received 7 November 2000; received in revised form 3 May 2001; accepted 8 May 2001

Abstract

The entrainment dynamics in the D′′-layer are influenced by multitudinous factors, such as thermal and compositionalbuoyancy, and temperature- and composition-dependent viscosity. Here, we are focusing on the effect of compositionallydependent viscosity on the mixing dynamics of the D′′-layer, arising from the less viscous but denser D′′-material. The markermethod, with one million markers, is used for portraying the fine scale features of the compositional components, D′′-layerand lower-mantle. The D′′-layer has a higher density but a lower viscosity than the ambient lower-mantle, as suggestedby melting point systematics. Results from a two-dimensional finite-difference numerical model including the extendedBoussinesq approximation with dissipation number Di= 0.3, show that a D′′-layer, less viscous than the ambient mantleby 1.5 orders of magnitude, cannot efficiently mix with the lower-mantle, even though the buoyancy parameter is as low asRρ = 0.6. However, very small-scale schlieren structures of D′′-layer material are entrained into the lower-mantle. Thesesmall-scale lower-mantle heterogeneities have been imaged with one-dimensional wavelets in order to delineate quantitativelythe multiscale features. They may offer an explanation for small-scale seismic heterogeneity inferred by seismic scattering inthe lower-mantle. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: D′′-layer; Entrainment; Seismic scattering; Thermochemical convection

1. Introduction

Almost 30 years ago, lower-mantle seismic hetero-geneity close to the core mantle boundary (CMB) wasdetected from travel time anomalies due to the seis-mic phase precursor PKP (Cleary and Haddon, 1972;

∗ Corresponding author. Tel.:+31-30-253-5076;fax: +31-30-253-5030.E-mail addresses: [email protected] (B. Schott),[email protected] (D.A. Yuen), [email protected](A. Braun).

Cleary, 1974). Since then D′′-layer and lower-mantleheterogeneity was found on all length scales. Modelsof seismic wave propagation have shown, that thesePKP precursors can be explained by either small-scaleCMB topography undulations or by deep lower-mantleheterogeneity (e.g. Bataille and Flatte, 1988). How-ever, recent seismological studies using PKP andPKKP precursors indicate a uniform distribution ofsmall-scale low-amplitude scatterers throughout themantle (Hedlin et al., 1997; Shearer et al., 1998).Moreover, the P-wave reflectivity of the D′′-layer was

0031-9201/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0031-9201(01)00234-5

44 B. Schott et al. / Physics of the Earth and Planetary Interiors 129 (2002) 43–65

explained by heterogeneities with scale lengths inthe range of 10–100 km (Reasoner and Revenaugh,1999), and recent investigations by (Cormier, 1999;Cormier, 2000) on seismic scattering further under-scored the multiscale nature of the heterogeneities inthe deep mantle. These observed small-scale seismicvelocity perturbations can be due to small-scale com-positional heterogeneity resulting from incompleteconvective mantle mixing, because of long mixingtime-scales for fluids having a large lateral viscositycontrast (Spence et al., 1987; Manga, 1996). Seismicscattering in the lowermost mantle is also attributedto scattering at partial melt, when the scatterersfall into the ultralow velocity zone (Garnero et al.,1993; Garnero and Helmberger, 1995; Garnero andHelmberger, 1998). Therefore, the question is, wheredo these small-scale compositional heterogeneitiescome from. In an attempt to address this question,we will study with a marker–convection model theplume driven mixing behavior of the Earth’s D′′-layer(Dahm, 1934; Dahm, 1936) with the lower-mantle.

It has long been suggested (Davies and Gurnis,1986; Hansen and Yuen, 1988), that the D′′-layer ischemically distinct from the overlying mantle, butits origin is still debatable. Possible mechanismsseem to be: (1) chemical reactions between ironcore and silicate mantle (Knittle and Jeanloz, 1986;Song and Ahrens, 1994), (2) the breakdown of (Mg,Fe)SiO3-perovskite (Saxena et al., 1996; Mao et al.,1997), (3) the accumulation of (parts of) subductedslabs (Grand et al., 1997; van der Hilst et al., 1997) atthe base of the mantle, and possibly any combinationof these mechanisms, which finally would lead to theformation of an oxide-rich D′′-layer (Anderson, 1998;Ruff and Anderson, 1980).

Formation of the D′′-layer by the segregation ofFeO from the ambient lower-mantle has been mod-eled by double-diffusive thermochemical convection(Nauheimer et al., 1996), and by the segregation ofeclogite from deeply subducted slabs, using a markertechnique for tracing the different components, withinthe framework of the Boussinesq approximation(Christensen and Hofmann, 1994). An additional neg-ative compositional buoyancy equivalent toRρ = 1.5was needed to stabilize the forming D′′-layer at theCMB.

The use of markers to portray mixing processes(Olson et al., 1984; Hoffman and McKenzie, 1985)

has been around for a long time and extension hasbeen made to three-dimensional space (Schmalzl et al.,1996), where it was shown that due to both geometryof the flow and the dynamics, mixing might be less ef-ficient than in two dimensions. Up to now, mixing pro-cesses in mantle convection have often been restrictedto thermal convection, even with variable viscositystudies (Ten et al., 1997; van Keken and Ballantine,1999). In this study, we will investigate mixing effi-ciency in the presence of both thermal-chemical con-vection and temperature- and composition-dependentviscosity.

Our intention is not to cover a large parameter space,but to draw attention to the complicated phenomenol-ogy coming along with the mixing dynamics of differ-ent mantle materials. We are presenting results from atwo-component numerical model of thermal-chemicalconvection in two-dimensional Cartesian geometry tomonitor the structural evolution of an initially strat-ified D′′-layer. An ultrahigh resolution of ‘chemical’markers is used to resolve the entrainment structuresarising from the interaction of compositionally denserD′′-layer material with lower-mantle convection, tomonitor the evolution of mixing and finally to inves-tigate the mixing efficiency from chemical buoyancyand composition-dependent rheology of the D′′-layer.

The dominant mode of mantle convection to-day consists of plate-scale flow interacting with astrongly time-dependent mantle circulation. The ap-parent lag between the surface plate configuration andthe lower-mantle was explained by a large viscos-ity increase in the lower mantle or by some layeredmantle convection (Chase, 1979). Slabs subductingin the upper mantle may not be blocked temporar-ily by endothermic phase transition (Christensen andYuen, 1984). This would seriously exert a dramaticeffect on the flow structures and cause a triggeringof lower-mantle plumes (Honda et al., 1993), likethe one causing the Pacific superswell (Cazenave andThoraval, 1994; Maruyama, 1994).

Another prominent lower-mantle upwelling, the hotplume under Africa (Ritsema et al., 1998), may belocated there, because of higher mantle temperaturesdue to the continent’s blanketing effect, amplified byits immobility with respect to the hotspot referenceframe since the opening of the Atlantic about 100 Maago, as suggested by the regular appearance of theseafloor magnetization (Muller et al., 1997).

B. Schott et al. / Physics of the Earth and Planetary Interiors 129 (2002) 43–65 45

Because of the manifold interactions between platesand global mantle circulation, we have not includedplates or subducting slabs in our model, to avoid un-necessary complications and to focus on the dynamicsin the deep mantle.

The entrainment structure modeled, here, may welldescribe those plumes in the deep mantle produced byvarious forms of focussing processes (Hansen et al.,1993), which may be responsbile for the megaplumeslike under the Pacific (Cazenave and Thoraval, 1994)or under Africa (Ritsema et al., 1998).

In Section 2 we give a model description of thethermal-chemical convection employed. This then isfollowed by a comparison between the style of plumeconvection near the CMB for constant and vari-able viscosity, in which the composition-dependentviscosity will be emphasized. We then apply thewavelet method to analyse the multiscale natureof the thermal-chemical plumes developed in thelower-mantle. Section 6 gives a summary and conclu-sions drawn from these simulations.

2. Thermal-chemical convection—modeldescription

The problem to be studied is the mixing and entrain-ment characteristics of a dense D′′-layer material withthe ambient lower-mantle. Our lower-mantle modelof thermochemical convection in two-dimensionalCartesian geometry is covering a depth of 2000 km.The aspect ratio is one, because of the high resolutionrequired to resolve the fine structures presented in thefollowing sections. We are using the finite-differenceprogram FDCON, developed by Schmeling (Wein-berg and Schmeling, 1992), which combines afinite-difference and marker technique to solve thebelow mentioned partial differential equations (PDE)governing multi-component thermochemical convec-tion. FDCON was successfully checked in benchmarktests (e.g. Blanckenbach et al., 1989; van Kekenet al., 1997) and applied to several multi-componentproblems (e.g. Schott and Schmeling, 1998; Walzerand Hendel, 1997). Our approach, based on afinite-difference and marker technique, calls for thesolution of three conservation equations for

1. mass and momentum, combined together as an el-liptic partial differential equation for the stream-

functionΨ with either constant or variable viscos-ity η and both thermally and compositionally in-duced buoyancy forces (Eq. (1));

2. energy, resulting in an equation for the heat trans-port, and including the effects of viscous and adi-abatic heating (Eq. (2));

3. composition, by applying a marker approach tokeep track of the high- and low-density componentsbeing advected by the convective velocity field.

The elliptic PDE for the streamfunctionΨ is solvedover an equidistant 121× 121 FD-grid in the limit ofPr → ∞ by the Cholesky decomposition method. Theextended-Boussinesq approximation (e.g. Christensenand Yuen, 1985) is used. Its non-dimensional form is

(∂2

∂z2− ∂2

∂x2

)η

(∂2Ψ

∂z2− ∂2Ψ

∂x2

)+ 4

∂2η

∂x∂z

∂2Ψ

∂x∂z

= −Ra∂T

∂x− Rc

∂C

∂x(1)

where the viscosityη = η(C, T , z) or η = constant.The equation for the heat transport becomes

∂T

∂t+ (v · ∇)T = κ �T + Di

(Φ

Ra− vz(T + T0)

)

(2)

whereT denotes the temperature, which is non-dimen-sionalised by the scaling temperatureTscale= 1000 K.Eq. (2) is solved on a grid with four times higherspatial resolution (481×481 grid points). The surfacetemperatureT0 = 0.273 is included and the ther-mal diffusivity κ is constant.v is the velocity fieldand vz is the vertical velocity component.C is thechemical field representing a binary system consist-ing of D′′-layer material (high density) and ambientlower-mantle (low density) substance.x andz are thehorizontal position and the depth.

In Eq. (1) the thermal and the chemical Rayleighnumbers are defined asRa = ρ0gα �T h3/κη0 andRc = �ρ gh3/κη0, respectively, and Di= αgh/cp =0.3 is the dissipation number. The dissipation functionΦ is defined by the relationΦ = 2ηeijeij (van denBerg and Yuen, 1997).

As an equation of state we useρ(T , C) = ρ0(1 −αT + C �ρ/ρ0), whereρ0 is the reference density,�ρ is the density difference due to the chemical com-position andα is the thermal expansivity.�T is the


temperature difference between the bottom and the topsurface,h is the distance between bottom and top, re-spectively andη0 = 1021 Pa s is the scaling viscosityof the mantle, the acceleration of gravityg points inthez-direction.

FDCON is using a marker approach with afourth-order Runge–Kutta integration in time com-bined with a predictor-corrector method. The markersare carrying the valuesC = 0 or 1, representing thedifferent materials by their densities and rheologies,to minimize numerical diffusion. The markers areadvected with the flow, and for each time-step theeffective compositionC is calculated from the sur-rounding markers at each finite difference grid point.The density and rheology is then chosen according tothe actualC value (Weinberg and Schmeling, 1992;van Keken et al., 1997). The exact percentage of man-tle to D′′-material is given by the value of theC-field,which comes from measuring the relative fraction ofthe two types of markers within a local grid area. Apure mantle material is given byC = 0, whileC = 1represents an area occupied only by D′′-material.

The C-field, which exists on a grid with spatialresolution dxdz, is interpolated from the irregularlydistributed markers by counting the markers within asquare of dimensions dxdz centered at each grid point(i, j). When the resolution of the grid, on which theC-field is interpolated, becomes too fine, then it canhappen, that there are marker-free squares within aC = 1 domain, getting the valueC = 0 by default.In this case, the interpolation produces a few singu-

Fig. 1. Initial conditions illustrated by the marker distribution with the streamlines overlaid (left panel) and the isothermal temperaturefield (right panel). D′′-markers appear black in the left panel. Ambient lower-mantle markers are not shown.

lar C = 0 points within theC = 1 domain, whichare purely numerical. This type of interpolation of theC-field by counting markers within grid squares workswell as long as the grid spacing dx is larger than ap-proximately 2

√2dm, where dm is the average marker

spacing. TheC-field can therefore easily be interpo-lated on a 361×361 grid from 1000×1000 irregularlybut ‘almost uniformly’ distributed markers.

3. Boundary and initial conditions

The square computational domain has free-slipboundaries at the top and bottom boundaries, includingreflecting boundary conditions at the vertical side-walls. The thermal boundary conditions are isother-mal top (T = 0) and bottom (T = 1), developing athermal boundary layer at the CMB over which thetemperature increases by≈1000 K. Heat flux throughthe side boundaries is set to zero.

Initially the model area is isothermal (T = 0) with ahot bottom boundary (T = 1) and a thermal perturba-tion in the left lower corner. Therefore, cold-lookingtemperatures likeT = 0.1 are representing material,that is 100 K hotter than the surrounding material andthe area, whereT ≥ 0.1 will be used to define thermalplumes and boundary layers.

The D′′-layer material is initially stratified along thebottom boundary, the CMB, with a constant thicknessof 200 km (Hansen and Yuen, 1988; Sidorin et al.,1999) (Fig. 1).


Table 1List of the modelsa

Model η �η Rρ Ra Di

C0 Constant – 0 106 0.3C1 Constant – 0.4 106 0.3C2 Constant – 0.6 106 0.3C3 Constant – 0.8 106 0.3V3.1 Variable 40 0.4 106 0.3V3 Variable 30 0.4 106 0.3V2 Variable 30 0.6 106 0.3V1 Variable 30 0.8 106 0.3

a Symbols are explained in the text.

4. Results

The runs we have conducted are based on either con-stant viscosity (Section 4.1) or variable viscosity (Sec-tion 4.2), which depends on composition, temperature,and depth. We have varied the buoyancy parameterRρ = Rc/Ra = �ρ/(ρ0α �T ) from no chemicalbuoyancy,Rρ = 0, to strongly chemically buoyant,Rρ = 0.8. An overview of the models is given inTable 1. Here, we want to draw the reader’s attentionto the technical point, that it is very difficult to displayall of the markers, because there are so many of them.We cannot see individual markers any longer becauseof their extreme dense spacing. In Fig. 1, left panel,the D′′-markers are shown in black, resulting in a darkD′′-layer, while the ambient lower-mantle markers arenot at all shown.

Fig. 2. In a constant viscosity model without any compositional buoyancy (Rρ = 0) the material boundary between D′′-layer andlower-mantle is almost isothermal in a developing plume (Ra = 106, model C0).

In Fig. 2, left panel, single markers are visible in thecompositional plume head (at least in a zoom-in) dueto a change in marker spacing from the internal fluiddeformation. In this case, it would not help maters byincreasing the size of the markers. They would simplyoverlap each other, producing anything but a blackunstructured area.

4.1. Constant viscosity

We first show how a constant viscosity plume with-out any chemical buoyancy,Rρ = 0, would behave asa reference model with other more complicated plumesto follow. We also show the development of the mark-ers, originating in the D′′-layer. This starting plume isthermally driven and is entirely composed of pristineD′′-material. The material boundary between D′′-layerand the ambient lower-mantle is nearly parallel to theT = 0.1 isotherm, in the starting plume scenario, asdepicted in Fig. 2. As already stated, theT = 0.1isotherm is used for defining the thermal plume.

In Fig. 3, we show the effects from a small amountof chemical buoyancy,Rρ = 0.4, due to the denserD′′-material. The thermal plume (area in whichT ≥0.1) becomes now larger than the compositional plumefilled with the D′′-material. The compositional buoy-ancy decelerates the rising plume and thermal diffu-sion is more effective in removing heat from the plumedue to the longer time of plume evolution.

This effect becomes more prominent with a largerchemical buoyancy ofRρ = 0.6 (Fig. 4). In this case,


Fig. 3. In a constant viscosity model with an increased compositional buoyancy ofRρ = 0.4 of the D′′-layer material, the thermal plumefills a larger area than the compositional plume (Ra = 106, model C1).

the thermal plume speeds away upward from the com-positional plume, giving rise to a situation in whichthe thermal plume has a bigger head than that of thecompositional plume upon reaching the surface.

With an additional influx of denser material atRρ = 0.8, hill-like structures are formed (Hansen andYuen, 1988, 1989), since the really dense materialcannot be entrained into the thermal plume any longer.This scenario of little entrainment is displayed inFig. 5. The D′′-layer now thickens, giving rise to theinteresting possibility for internal convection to occurin the D′′-layer, which can be clearly discerned insidethe D′′-layer in the right panel of Fig. 5. Hansen andYuen, 1989 also found some evidence for such a type

Fig. 4. With a compositional buoyancy of the D′′-layer material ofRρ = 0.6 the thermal plume is ascending faster than the compositionalplume, leading to a larger thermal plume head than the compositional one (Ra = 106, model C2).

of internal convection inside the D′′-layer but theirresolution was not as dense as the one used here. Thistype of small-scale secondary convection cells hasbeen suggested for the foot of the D′′-layer under theHawaiian hotspot from seismic studies (Russell et al.,1998). These constant viscosity results are shown herein order to compare with the results from a more com-plicated rheology involving a large viscosity contrastbetween the D′′-layer and the lower-mantle.

4.2. Variable viscosity

We now introduce a more realistic rheologi-cal model, where the viscosityη(C, T , z) depends


Fig. 5. A compositional buoyancy ofRρ = 0.8 of the D′′-layer material does not allow that material to enter the thermal plume any more.However, there is little entrainment (Ra = 106, model C3).

on composition, temperature, and depth, with thedenser material having a weaker rheology be-cause of a lower melting point (Weertman, 1970;Karato, 1997). This viscosity takes the followingform:

η(C, T , z)

= η0 exp

((γ a + λbz)C + (a + bz)(1 − C)

T + T0

)(3)

wherea = 1, b = 0.3, andT0 = 0.273, resulting ina viscosity contrast of�η = η(C = 0, T , z)/η(C =1, T , z) ≈ 30 for γ = λ = 0.3 and�η ≈ 40 forγ = 0.1, λ = 0.3, respectively; with the D′′-layerbeing the weaker material (η(C = 1, T , z)) than theambient lower-mantle (η(C = 0, T , z)). Almost halfof the viscosity drop is due to the temperature in-crease in the hot thermal boundary layer and plumes,and the viscosity is increasing by a factor of 4.5 overthe depth of the mantle due to increasing pressure.This range of viscosity contrast between lower-mantleand D′′-material is on the conservative side of theestimates of Yamazaki and Karato, 2001, who sug-gest a viscosity drop of 1–3 orders of magnitude inthe D′′-layer. Nearly a quarter of a century ago, itwas recognized, that a constant Newtonian viscos-ity is too much an oversimplification in this stronglytime-dependent problem (Jones, 1977).

We then increase incrementally the strength of thechemical buoyancy fromRρ = 0.4–0.6 and finally

to 0.8. Small-scale secondary convection is devel-oped ever more with an increase ofRρ , as we cansee from the sequence of time-frames from Figs. 6–8(Rρ = 0.4) to Fig. 9 (Rρ = 0.6). The latter portrays aflat D′′-layer scenario due to the localized strength ofchemical buoyancy. The effect of the low-viscosity inthe D′′-layer, caused by the combined efforts of ther-mal and compositional dependence of the rheology isto inhibit mixing in the D′′-layer by virtue of the largelateral variations in the viscosity field (Manga, 1996).Therefore, from Fig. 6 even for a modest chemi-cal buoyancy strength ofRρ = 0.4, very much lessD′′-layer material is entrained into the lower-mantle,as compared to the isoviscous case, shown inFig. 3.

In the case of a moderately strong chemical buoy-ancy (Rρ = 0.4, Fig. 6), this dense layer can be sweptaside by the strong lower-mantle flow to form lumps orhills, which can reach a height of around 500 km abovethe CMB, when they are hot enough. This resemblesthe giant plume-like structures in the lower-mantleunder the Pacific Ocean and Africa (Cadek et al.,1984).

Subsequently, when they have lost sufficient amountof heat, these hills would collapse. This kind of mo-tion in the deep lower-mantle, in which the lump isalways connected to the CMB, produces extremelysmall-scale entrainment of the D′′-material intothe lower-mantle, which leads to the production ofschlieren structures above the CMB. The expression


Fig. 6. D′′-layer dynamics with variable viscosity visualized by the chemical composition of the two-material system—the lower-mantle andthe D′′-layer (black markers)—with stream-lines overlaid. A not too high compositional buoyancy ofRρ = 0.4 allows the D′′-material torise up to 500 km above the CMB (t = 346 Ma). However, these D′′-hills collapse, when having lost sufficient heat, producing small-scaleschlieren structures due to the compositional viscosity contrast of�η = 30 (t = 366 Ma, model V3).


Fig. 7. Thermal evolution of the initially horizontally stratified isothermal lower-mantle—D′′-layer system withRρ = 0.4 and�η = 30(model V3).


Fig. 8. Evolution of the D′′-layer with a viscosity contrast of�η = 40,Rρ = 0.4 (model V3.1). Again only the markers of the D′′-materialare shown, not to obfuscate the tiny schlieren structures developing within≈200 Ma after an initial transient period of D′′-layer internalconvection.


Fig. 9. D′′-layer internal convection is dominant at higher compositional buoyancy ofRρ = 0.6 and a viscosity contrast of�η = 30(model V2). Thermal plumes starting from the D′′-hills are entraining little D′′-material into the ambient mantle. Composition is shownby D′′-markers in the left, and temperature in the right panels.

‘schlieren’ is the one used in, e.g. schlieren photog-raphy, a common technique to visualize flow patternsin experimental fluid dynamics (van Dyke, 1982). Wewill analyze these schlieren structures with waveletsin Section 5. A wavelet analysis of a whole time se-ries would lie beyond the scope of this paper, wherewe analyze two representative snapshots, to illustratetypical multiscale features of the evolved structures.With increasing value ofRρ in the D′′-layer of upto Rρ = 0.6 (Fig. 9), the dense D′′-layer becomesmore and more stably stratified with thermal plumesissuing from the top of the hill, but entraining verylittle D′′-material (Figs. 8–10). This scenario of largevalues ofRρ has already been studied by Hansen andYuen (1989), Hansen and Yuen (1990) and Tackley(1998b).

5. Analysis of the schlieren structure usingwavelets

The structures in thermal-chemical convection havea distinct multiscale character because of the dif-ferences in the scales of the chemical and thermalforcings and in the thermal and chemical viscositygradients. Wavelets represent a recently developedtool in mathematics, which allows one to analyzemultiscale structures in a readily understandable for-mat (Mallat, 1998). These schlieren or tendril-likestructures have been analyzed by interpolating a con-tinuous compositional fieldC either on a 121× 121grid or on a 361× 361 grid from 1000× 1000 mark-ers, thus yielding a spatial resolution of 16.5 km or5.5 km, respectively. These continuousC-fields are


Fig. 10. A high compositional buoyancy ofRρ = 0.8 is causing a flat D′′-layer (�η = 30, model V1). The layer internal convection cellshave aspect ratios between 1.4 and 2 with an average of 1.6. Very little D′′-material is entrained into the ambient mantle. Composition isshown by D′′-markers in the left, and temperature in the right panels.

displayed in Figs. 11 and 12, they have been takenfrom the last time-steps in Figs. 6 and 8. We areanalyzing two different, however typical, schlierenstructures at two different spatial resolutions in orderto demonstrate the sensitivity of the analysis to spatialresolution.

First, we will focus on the high resolution grid(Fig. 12). For extracting the relevant scale informa-tion from the grid, a suitable way is to transformone-dimensional spatial profiles, using both Fourierand wavelet methods. The mixing schlieren structureshown in Fig. 12 is now analyzed along the nine hori-zontal profiles X1 to X9. Analysis was done along thehorizontal and vertical profiles, however, the resultsare the same for both sets of profiles. Therefore, weare focusing on the horizontal profiles only.

Due to computational efficiency, all profiles wereextended by zero leading to 512 intervals with a spac-ing of 5.5 km (Fig. 13. This particular spacing allowsa minimum resolvable wavelength of 11 km, whichis comparable to the length-scale of heterogeneities,as observed by scattered waves in PKIKP/PKP stud-ies (Cormier, 2000). The maximum wavelength is5632 km corresponding to twice the extended pro-file length. Before transforming the profiles, a lineartrend was removed for avoiding aliasing effects. Thehorizontal profiles are located in depths ranging from600 km in the lower-mantle (profile X1) down to1900 km near the CMB (profile X9). The normalizedFourier amplitude spectra are shown in Fig. 14.

Obviously, the spectral energy is transferred fromthe short wavelengths in profiles crossing the top of


Fig. 11. Chemical field of the last time-step shown in Fig. 6 interpolated on a 121× 121 grid, yielding a spatial resolution of 16.5 km.The locations of the examined horizontal profiles across the composital schlieren structure are indicated by lines.

the mixing structure (profiles X2 and X3) to longerwavelengths near the CMB (profile X9). The shortestwavelength is associated with the entrainment of theD′′-material.

For multiscale features, such as shown in the en-trainment process, a wavelet analysis is a more suit-able tool for analyzing the spatial profiles. Fourier

transformation cannot identify precisely the sourceof the energy, it can only determine the sum of thepower, while losing the phase information needed forlocating the position of the heterogeneities. Addition-ally, the advantage of a wavelet analysis as comparedto a windowed Fourier transform comes from its be-ing scale-independent, because wavelets with short


Fig. 12. Chemical field of the last time-step shown in Fig. 8 interpolated on a 361× 361 grid, yielding a spatial resolution of 5.5 km. Thelocations of the examined profiles, X1 to X9, across the composital schlieren structure are shown by dashed lines.

windows are used for short wavelengths and longwindows for long wavelengths. Hence, there are noaliasing effects due to high and low frequency contri-butions. We will take again one-dimensional profilesand determine the dominant modes of variabilityand how these modes would change along profile(Fig. 13). This technique has been used by Simonsand Hager (1997) to examine post-glacial rebound.We have used the continuous wavelet transform(CWT) (Daubechies, 1992) and a one-dimensionalMorlet wavelet (Goupillaud et al., 1984) for analyz-ing the 9 profiles shown in Fig. 11 and the 18 profilesshown in Fig. 12, respectively. Generally, a CWTcan be written likeWψ(s, χ) = ∫

w(x)ψs,χ (x)dx,where ψs,χ (x) represents the waveletψs,χ (x) =(1/

√s)ψ((x − χ)/s) (s > 0, χ ∈ R). A set of

waveletsψs,χ (x) is created from a so-called mother

waveletψx by shifting and scaling. The shifting pa-rameterχ is responsible for the location, ands is thescaling parameter, stretching (s > 1) or compressing(s < 1) the mother wavelet. We have chosen a com-plex Morlet wavelet (Morlet et al., 1982) for the anal-ysis to minimize the space–wavenumber uncertaintyof �χ �k ≥ 1/4π . Here, the scaling parameters canbe regarded as a wavelength, although for most otherwavelets,s has a more general meaning and is called‘scale’. This kind of Morlet wavelet has been usedrecently in analyzing time-series in mantle convec-tion (Vecsey and Matyska, 2001). A complex Morletwavelet is a plane wave modulated by a Gaussianfunction and takes the form

ψs,χ (x)= π−1/4(sl)−1/2e−i2π(1/s)(x−χ)

×e−1/2((x−χ)/sl)2 (4)


Fig. 13. Horizontal profiles of the chemical field. Their location is shown in Fig. 12.


Fig. 14. Fourier amplitude spectra of the nine horizontal profiles, for location see Fig. 12. Each spectrum is normalized to its maximum.The shaded lines indicate how the energy is moving from short to longer wavelength. The profiles starting at the top of the mixing structurein 600 km depth (top) down to the CMB (bottom).

�

Fig. 15. The nine selected horizontal profiles of Fig. 13 were analyzed using a continuous wavelet transform with a Morlet wavelet of lengthsix. Each of the nine panels is divided into four diagrams, showing in the upper left—the wavelet-scalogram; upper right—the normalizedFourier amplitude spectrum; lower left—the analyzed profile; lower right—the real part of the Morlet wavelet. The wavelet-scalogramshows the color-coded space–wavelength distribution of the analyzed profile. Red colors indicate high-amplitudes, blue colors indicatelow-amplitudes, values below 15% of the maximum are left white. The color-coded plots allow to identify both the wavelength of theindividual signal as well as its location in the spatial profile, since the spatial and wavelength axes of the original profile, the Fourierspectrum and the wavelet transform are identical, respectively. The upper left panel is representing the wavelet transform of a singleschlieren structure of thickness 5.5 km, here, the advantages of the wavelet transform become obvious, since the Fourier transform is notable to detect neither the wavelength nor the location of the signal. The upper three panels crossing the top of the schlieren structureshow wavelengths from 11 to 40 km on the one hand, and from 200 to 400 km on the other. The middle three panels show wavelengthsranging from 100 to 700 km. Finally, the lower three panels indicate wavelengths around 1000 km representing the thickness of the baseof the evolving D′′-structure. Thus, we identify three different levels of schlieren thickness, 6–20 km at the top, 50–350 km in the middleand round 500 km at the base of the model.


Fig. 15.

An additional parameterl was introduced to allow foran arbitrary shift of the wavelet transform resolution infavor of spatial or wavenumber resolution for a fixedscales without changing the central wavenumber. Wehave determinedl, the length of the wavelet, by thespatial and wavenumber resolution, thus,l representsthe number of oscillations of the wavelet. From the

uncertainty principle, the greater the number of oscil-lations, the lower would be the spatial resolution andvice versa. The sampling in space and wavenumber isthen determined by

�χ = sl√2, �k =

√2

4πsl,

�s

s=

√2

4πl(5)


Fig. 16. The nine selected horizontal profiles of Fig. 11 were analyzed using a continuous wavelet transform with a Morlet wavelet oflength six. Grouping of the panels is analog to that in Fig. 15, where the panels are explained in the figure caption. The upper left panelis representing the wavelet transform of a single schlieren structure of thickness 16.5 km. The upper three panels crossing the top of theschlieren structure show a maximum amplitude for wavelengths around 100 km, with a significant contribution for wavelengths down tosome 10 km. The middle three panels show a maximum amplitude for wavelengths around 1000 km, with a significant contribution forwavelengths down to 100 km. Finally, the lower three panels indicate wavelengths around 1000 km representing the thickness of the baseof the evolving D′′-structure. The lower right panel indicating an amplitude shift towards shorter wavelength. Thus, we identify threedifferent levels of schlieren thickness, 10–50 km at the top, 50–500 km in the middle and round 500 km at the base of the model.


By inspecting at the third equation in Eq. (5), we cansee that a logarithmic equidistant sampling inχ is ap-propriate. For maintaining a balance between the spa-tial and the wavenumber resolution, a length ofl =6 was chosen. A linear fit was again removed fromthe profiles and each profile was then operated on bythe Morlet wavelet. The numerical implementation ofthe CWT is a discrete convolution between the spa-tial profile and the discretized wavelet at all scales.For interpretation, the wavelet transform was interpo-lated in the space–wavenumber domain and the am-plitudes were color-coded. Figs. 15 and 16 present thewavelet-scalograms of the nine respective horizontalprofiles (Figs. 11 and 12) together with the spatial pro-file, its Fourier spectrum and the real part of the Morletwavelet used. The L2-normalized wavelet-scalogramin the upper left box of each panel of Figs. 15 and 16reveal clearly the high-amplitudes in red colors andthe low-amplitudes in blue colors. A cut-off at 15% ofthe maximum is introduced for suppressing the lowestamplitudes in order to enhance the clarity.

Three different levels of tendril wavelength/thicknesscan be identified in both schlieren structures(Figs. 15 and 16): the upper three panels containhigh-amplitudes for wavelengths between 11 and40 km at 5.5 km resolution (Fig. 15, 361× 361 grid,corresponding profiles X1 to X3 in Fig. 12) andwavelengths between some 10 and 100 km at 16.5 kmresolution (Fig. 16, 121× 121 grid, correspondingprofiles in Fig. 11). Panels in the middle indicatewavelength between 100 and 700 km at 5.5 km reso-lution (Fig. 15, 361×361 grid, corresponding profilesX4 to X6 in Fig. 12) and wavelengths between 100and 1000 km at 16.5 km resolution (Fig. 16, 121×121grid, corresponding profiles in Fig. 11). The lowerthree panels show highest values at wavelengths ofabout 1000 km on both grids (Figs. 15 and 16).

Thus, the corresponding schlieren thickness is5–20 km at the top of the mixing structure, 50–500 kmin the middle, and 500 km at the CMB. This resultremains the same for both schlieren structures andholds for both spatial resolutions examined. Such en-trainment pattern typically evolve in thermochemicalconvection, when a large viscosity difference betweenthe fluids is included. It also confirms the usefulnessof the wavelet analysis as a tool for examining andcomparing the characteristics of multiscale structuresat different spatial resolution. We note especially that

the small-scale schlieren at the top of the mixingstructure may explain the existence of small-scalescatterers, found in seismic wave inversions (Frey-bourger et al., 1999; Cormier, 2000).

6. Conclusions and discussion

The purpose of our work on compositionally de-pendent rheology in thermal-chemical convectionis to draw attention to the complicated D′′-layerentrainment structures most likely existing in thelower-mantle, which have the potential of explainingthe observed seismic scattering. Here, we are notconcerned with subduction and the overall mantlemixing processed (e.g. Gurnis, 1986a; Olson et al.,1984; Christensen and Hofmann, 1994).

The results of our numerical thermal-compositionalconvection model show, that a low viscous D′′-layerwith a rather small density increased of 5% maynot be entrained into thermal plumes starting fromthe D′′-layer. A similar behavior has been found formulti-layer Rayleigh–Taylor instabilities occurring inmagmatic diapirs (Cruden et al., 1995). However, thepartially entrained D′′-layer material can form distinctlumps, which do not need to have a hill-like shapewith a smooth surface (Tackley, 1998a), but can have acomplex tree- or polyp-like structure. Similar complexstructures have been modeled by Hansen and Yuen(2000), however, their model could not resolve thesmall-scale structure of branches or tentacles reachingout into the lower-mantle (Fig. 12). The heterogeneityis developing the shape of a large ‘blob’ connected tolong thin tendrils, which lengthen exponentially withtime, while the ‘blob’ deforms much slower (Gurnis,1986b). Similar ‘blobs’ may, therefore, explain someaspects of heterogeneity in the Earth’s mantle (Beckeret al., 1999). For these tiny schlieren structures wefind scaling diameters in the order of a few to 10 km,which is too small to be resolved in numerical mod-els of double-diffusive thermal-chemical convection(Hansen and Yuen, 2000). The double-diffusive con-vection approach also inhibits the development ofsmall-scale mixing (Montague and Kellogg, 2000).

Schlieren structures of diameters down to about5 km are robust features in our 2000 km× 2000 kmlarge model and subsequent analysis, when 1000×1000 markers are used, because a 5 km thick ‘tentacle’


is defined by about three markers along its diame-ter. Additional schlieren structures with even smallerdiameters could be made visible by an increasingnumber of markers, while already detected structureswould become smoother, but would not disappear.

We suggest, that these tiny schlieren structures canexplain the recently proposed uniform (Hedlin et al.,1997) or slightly depth dependent (Shearer et al.,1998; Cormier, 2000) distribution of small-scalelow-amplitude scatterers throughout the mantle withtypical scale-lengths around 10 km. Due to thermaldiffusion pronounced small-scale thermal perturba-tions cannot survive on the long time-scale our mod-els are covering, but seismic velocity perturbationson length-scales of 200–1000 km may be of thermalorigin (Fig. 7).

The observed ‘non-decaying’ power spectrum ofscattered seismic energy attributed to heterogeneitysignificantly departs from that one expected from ther-mal variations in a convecting mantle. The entrain-ment of small-scale (10–100 km) schlieren structuresis a likely explanation for such a ‘non-decaying’ powerspectrum (Cormier, 2000).

A maximum of about 40% of the initially strati-fied D′′-layer material can be ‘mixed’ into the am-bient lower-mantle within a model time of 475 Ma(Fig. 6), if we are counting theC = 1 markers (rep-resenting the D′′-material) having a depth shallowerthan 500 km above the CMB. However, most of theD′′-material is forming into large lumps, which are allthe time staying close to the CMB. We, therefore, con-clude, that these lumps can still serve today as a dis-tinct chemical reservoir in the Earth’s mantle, similarto the deep-mantle blobs (Becker et al., 1999).

While the small-scale schlieren cause seismic scat-tering (Cormier, 2000), the bulk of the D′′-layer showsa strong longer–wavelength topographical undulation,which is reflected by the short-length scale horizontalvariations in the topography, due to a P-wave reflec-tor attributed to the top portion of the D′′-layer. Suchvariations of the depth of the D′′-reflector between 150and 200 km have been found beneath the southwest-ern Pacific (Yamada and Nakanishi, 1998) and north-ern Siberia (Freybourger et al., 1999). Under northernSiberia PdP travel times have also been explained witha 7 km thick 3% low velocity lamella lying approx-imately 300 km above the CMB (Freybourger et al.,1999).

Models of mantle mixing should principally be stud-ied in three dimensions (Ferrachat and Ricard, 1998),because it is difficult to scale the mixing behavior from2D to 3D, and mixing in 2D is more efficient thanin 3D (Schmalzl et al., 1996). However, the need ofvery high spatial resolution and many markers to trackthe materials with high accuracy over long time spansis not easily achievable in 3D today. Computationallyfaster 2D models of thermal-chemical convection re-main still a versatile tool to study basic mixing prop-erties.

For the choice Di= 0 (incompressible convec-tion) an isothermal mantle is the only reasonable ini-tial condition, that is gravitationally stable. For thechoice Di> 0 (‘compressible’ convection) all initialconditions between an isothermal mantle and an adi-abatic temperature gradient are gravitationally stable.While the choice of an adiabatic temperature gradi-ent is essential from the viewpoint of thermodynam-ics, we have chosen the isothermal initial condition tomake the different results easier to compare. We aretherefore overestimating the thermal buoyancy forcesfor Di > 0. However, preliminary results for startingwith an adiabatic temperature gradient show, that evenvery little additional compositional buoyancy of theD′′-layer, as low asRρ ≈ 0.2, are enough for stabiliz-ing the D′′-layer at the CMB. This is in full agreementwith the results of Tackley (1998b).

Values ofRρ greater than one are typically neededto stabilize the D′′-layer at the CMB when the Boussi-nesq approximation is used together with constant ma-terial properties, such as the thermal expansivity andviscosity (Christensen and Hofmann, 1994; Davaille,1999; Montague and Kellogg, 2000). The Boussinesqapproximation is, therefore, not appropriate at all forasessing the stability of the D′′-layer.

We have not considered the additional stirring ofthe D′′-layer by other processes such as by subduct-ing slabs, because recent numerical models show, thatcold dense downwellings sinking into the D′′-layerdo not necessarily trigger instability (thermochemicalplumes) in their surrounding, by either sweeping theD′′-layer aside or by increasing the temperature gra-dient, but may lead to the formation of highly compli-cated structures of layered thermochemical convection(Hansen, 2000). Moreover, subducting slabs may notreach the CMB, but hover in the mid-lower-mantle,due to the decreasing thermal expansivity with in-


creasing pressure (Chopelas and Boehler, 1992;Kesson et al., 1998; Kaneshima and Helffrich, 1999).However, this picture may change from slab to slab(Cizkova et al., 1998) because of the rheological de-pendence on the subducting velocity (Karato et al.,2001). Such additional complications are beyond thisstudy, where we are focusing on the style of entrain-ment of D′′-material into the lower-mantle. Oncethe D′′-material is entrained by the plumes, it willcertainly be further mixed into the mantle by fluiddynamical processes stirring the mantle.

Acknowledgements

We thank discussions with U. Christensen, U.Hansen, S. Karato, H. Paulssen, P. Tackley, W. Wang,and D. Yamazaki, and particularly H. Schmeling andan anonymous reviewer for their excellent reviews,that helped us improving the manuscript. Specialthanks to H. Schmeling for kindly providing his codeFDCON, and to Swedish NFR for supporting BertramSchott with a Post-Doc stipend, which made this re-search possible. Support for this research also comesfrom the geophysics program of NSF. All calculationshave been done on the SGI-Origin2000 computers ofthe Minnesota Supercomputing Institute (MSI).

References

Anderson, D.L., 1998. The EDGES of the mantle. In: Gurnis,M., Wysession, M.E., Knittle, E., Buffett, B.A. (Eds.),The Core-Mantle Boundary Region: Geodynamics, Vol. 28.American Geophysical Union, pp. 255–271.

Bataille, K., Flatte, S.M., 1988. Inhomogeneities near thecore-mantle boundary inferred from short-period scattered PKPwaves recorded at the global digital seismograph network. J.Geophys. Res. 93, 15057–15064.

Becker, T.W., Kellogg, L.H., O’Connell, R.J., 1999. Thermalcontraints on the survival of primitive blobs in the lower mantle.Earth Planet. Sci. Lett. 171, 351–365.

Blanckenbach, B., Busse, F., Christensen, U., Cserepes, L., Gunkel,D., Hansen, U., Harder, H., Jarvis, G., Koch, M., Marquart,G., Moore, D., Olson, P., Schmeling, H., Schnaubelt, T., 1989.A benchmark comparison for mantle convection codes. J.Geophys. Int. 98 (1), 23–38.

Cadek, O., Yuen, D.A., Steinbach, V., Chopelas, A., Matyska, C.,1984. Lower mantle thermal structure deduced from seismictomography, mineral physics and numerical modeling. EarthPlanet. Sci. Lett. 121 (3/4), 385–402.

Cazenave, A., Thoraval, C., 1994. Mantle dynamics constrainedby degree 6 surface topography, seismic tomography and geoid:inference on the origin of the South Pacific superswell. EarthPlanet. Sci. Lett. 122 (1/2), 207–219.

Chase, C.G., 1979. Subduction, the geoid, and lower mantleconvection. Nature 282, 464–468.

Chopelas, A., Boehler, R., 1992. Thermal expansivity of the lowermantle. Geophys. Res. Lett. 19 (19), 1983–1986.

Christensen, U.R., Hofmann, A.W., 1994. Segregation of subductedoceanic-crust in the convective mantle. J. Geophys. Res. SolidEarth 99 (B10), 19867–19884.

Christensen, U.R., Yuen, D.A., 1984. The interaction of asubducting lithospheric slab with a chemical or phase boundary.J. Geophys. Res. 89, 4389–4402.

Christensen, U.R., Yuen, D.A., 1985. Layered convection inducedby phase transitions. J. Geophys. Res. 90, 10291–10300.

Cizkova, H., Cadek, O., Slancova, A., 1998. Regional correlationanalysis between seismic heterogeneity in the lower mantle andsubduction in the last 180 million years: implications for mantledynamics and rheology. Pure Appl. Geophys. 151 (2–4), 527–537.

Cleary, J.R., 1974. The D′′-region. Phys. Earth Planet. Inter. 9,13–27.

Cleary, J.R., Haddon, R.A.W., 1972. Seismic wave scattering nearthe core-mantle boundary: a new interpretation of precursorsto PKP. Nature 240, 549–551.

Cormier, V.F., 1999. Anisotropy of heterogeneity scale lengths inthe lower mantle from PKIKP precursors. Geophys. J. Int.136 (2), 373–384.

Cormier, V.F., 2000. D′′ as a transition in the heterogeneityspectrum of the lowermost mantle. J. Geophys. Res. 105 (B7),16193–16205.

Cruden, A.R., Koji, H., Schmeling, H., 1995. Diapiric basalentrainment of mafic into felsic magma. Earth Planet. Sci.Lett. 131 (3/4), 321–340.

Dahm, C.G., 1934. A study of dilational wave velocity in theEarth, as a function of depth, based on a comparison of the P,P′′ and PcP phases. Ph.D. dissertation, St. Louis Univesity, St.Louis.

Dahm, C.G., 1936. Velocity of P and S waves. Bull. Seism. Soc.Am. 26, 157–171.

Daubechies, I., 1992. Ten lectures on wavelets. In: Proceedingsof the CBMS–NSF Regional Conference Series in AppliedMathematics, Vol. 61. Philadelphia, SIAM Press, Philadelphia,p. 357.

Davaille, A., 1999. Simultaneous generation of hotspots andsuperswells by convection in a heterogeneous planetary mantle.Nature 402, 756–760.

Davies, G.F., Gurnis, M., 1986. Interaction of mantle dregs withconvection: lateral heterogeneity at the core-mantle boundary.Geophys. Res. Lett. 13, 1517–1520.

Ferrachat, S., Ricard, Y., 1998. Regular vs. chaotic mantle mixing.Earth Planet. Sci. Lett. 155 (1/2), 75–86.

Freybourger, M., Krüger, F., Achauer, U., 1999. A 22◦ long seismicprofile for the study of the top of D′′. Geophys. Res. Lett.26 (22), 3409–3412.


Garnero, E.J., Grand, S.P., Helmberger, D.V., 1993. Low P-wavevelocity at the base of the mantle. Geophys. Res. Lett. 20 (17),1843–1846.

Garnero, E.J., Helmberger, D.V., 1995. A very slow basal layerunderlying large-scale low-velocity anomalies in the lowermantle beneath the Pacific-evidence from core phases. Phys.Earth Planet. Inter. 91 (1–3), 161–176.

Garnero, E.J., Helmberger, D.V., 1998. Further structuralconstraints and uncertainties of a thin laterally varyingultralow-velocity layer at the base of the mantle. J. Geophys.Res. 103 (B6), 12495–12509.

Goupillaud, P., Grossmann, A., Morlet, J., 1984. Cycle-octave andrelated transforms in seismic signal analysis. Geoexploration23, 85–102.

Grand, S.P., van der Hilst, R.D., Widiyantoro, S., 1997. Highresolution global tomography: a snapshot of convection in theEarth. Geol. Soc. Am. Today 7, 1–7.

Gurnis, M., 1986a. The effects of chemical density differenceson convective mixing in the Earth’s mantle. J. Geophys. Res.91 (B11), 11407–11419.

Gurnis, M., 1986b. Stirring and mixing in the mantle by plate-scaleflow: large persistent blobs and long tendrils coexist. Geophys.Res. Lett. 13 (13), 1474–1477.

Hansen, U., 2000. Evolution of structures by double-diffusiveconvection at the core-mantle boundary. EOS Trans. AGU81 (48), F907.

Hansen, U., Yuen, D.A., 1988. Numerical simulations ofthermal-chemical instabilities and lateral heterogeneities at thecore-mantle boundary. Nature 334, 237–240.

Hansen, U., Yuen, D.A., 1989. Dynamical influences fromthermal-chemical instabilities at the core-mantle boundary.Geophys. Res. Lett. 16 (7), 629–632.

Hansen, U., Yuen, D.A., 1990. Nonlinear physics ofdouble-diffusive convection in geological systems. Earth Sci.Rev. 29 (1–4), 385–399.

Hansen, U., Yuen, D.A., 2000. Extended-Boussinesq thermal-chemical convection with moving heat sources and variableviscosity. Earth Planet. Sci. Lett. 176 (3/4), 401–411.

Hansen, U., Yuen, D.A., Kroening, S.E., Larsen, T.B.,1993. Dynamical consequences of depth-dependent thermalexpansivity and viscosity on mantle circulations and thermalstructure. Phys. Earth Planet. Inter. 77 (3/4), 205–223.

Hedlin, M., Shearer, P., Earle, P., 1997. Seismic evidence forsmall-scale heterogeneity throughout the earth’s mantle. Nature387 (6629), 145–150.

Hoffman, N.R.M., McKenzie, D.P., 1985. The destruction ofgeochemical heterogeneities by differential fluid motions duringconvection. Geophys. J. R. Astron. Soc. 82, 163–206.

Honda, S., Balachandar, S., Yuen, D.A., Reuteler, D., 1993.Three-dimensional mantle dynamics with an endothermic phasetransition. Geophys. Res. Lett. 20 (3), 221–224.

Jones, G.M., 1977. Thermal interaction of the core and the mantleand long-term behavior of the geomagnetic field. J. Geophys.Res. 82 (11), 1703–1709.

Kaneshima, S., Helffrich, G., 1999. Dipping low-velocity layer inthe mid-lower mantle: evidence for geochemical heterogeneity.Science 283 (5409), 1888–1891.

Karato, S., 1997. Phase transformation and rheological propertiesof mantle minerals. In: Crossley, D., Soward, A.M. (Eds.),Earth’s Deep Interior. Gordon and Breach, New York.

Karato, S., Riedel, M.R., Yuen, D.A., 2001. Rheological structureand deformation of subducted slabs in the mantle transitionzone: implications for mantle circulation and deep earthquakes.Phys. Earth Planet. Inter., in press.

Kesson, S.E., Gerald, J.D.F., Shelley, J.M., 1998. Mineralogy anddynamics of a pyrolite lower mantle. Nature 393, 252–254.

Knittle, E., Jeanloz, R., 1986. High-pressure metallization of FeOand implications for the Earth’s core. Geophys. Res. Lett. 13,1541–1544.

Mallat, S., 1998. A Wavelet Tour of Signal Processing. AcademicPress, Boston.

Manga, M., 1996. Mixing of heterogeneities in the mantle: effectof viscosity differences. Geophys. Res. Lett. 23, 403–406.

Mao, H.K., Shen, G.Y., Hemley, R.J., 1997. Multivariabledependence of Fe–Mg partitioning in the lower mantle. Science278 (5346), 2098–2100.

Maruyama, S., 1994. Plume tectonics. J. Geol. Soc. Jpn. 100 (1),25–49.

Montague, N.L., Kellogg, L.H., 2000. Numerical models of adense layer at the base of the mantle and implications for thegeodynamics of D′′. J. Geophys. Res. 105 (B5), 11101–11114.

Morlet, J., Arens, G., Fourgeau, E., Giard, D., 1982. Wavepropagation and sampling theory. Part II. Sampling theory andcomplex waves. J. Geophys. 47 (2), 222–236.

Muller, R.D., Roest, W.R., Royer, J., Gahagan, L.M., Sclater, J.G.,1997. Digital isochrons of the world’s ocean floor. J. Geophys.Res. 102 (B2), 3211–3214.

Nauheimer, G., Fradkov, A.S., Neugebauer, H.J., 1996. Mantleconvection behavior with segregation in the core-mantleboundary. Geophys. Res. Lett. 23 (16), 2061–2064.

Olson, P.L., Yuen, D.A., Balsiger, D.S., 1984. Mixing of passiveheterogenieties by mantle convection. J. Geophys. Res. 89,425–436.

Reasoner, C., Revenaugh, J., 1999. Short-period P wave constraintson D′′ reflectivity. J. Geophys. Res. 104 (B1), 955–961.

Ritsema, J., Ni, S., Helmberger, D.V., Cortwell, H.P., 1998.Evidence for strong shear velocity reduction and velocitygradients in the lower mantle beneath Africa. Geophys. Res.Lett. 25 (23), 4245–4248.

Ruff, L.J., Anderson, D.L., 1980. Core formation, evolution, andconvection: a geophysical model. Phys. Earth Planet. Inter. 21,181–202.

Russell, S.A., Lay, T., Garnero, E.J., 1998. Seismic evidence forsmall-scale dynamics in the lowermost mantle at the root ofthe Hawaiian hotspot. Nature 396 (6708), 255–258.

Saxena, S.K., Dubrovinsky, L.S., Lazor, P., Cerenius, Y., Haggkvist,P., Hanfland, M., Hu, J., 1996. Stability of perovskite (MgSiO3)in the Earth’s mantle. Science 274, 1357–1359.

Schmalzl, J., Houseman, G.A., Hansen, U., 1996. Mixing invigorous, time-dependent three-dimensional convection andapplication to Earth’s mantle. J. Geophys. Res. 101, 21847–21858.

Schott, B., Schmeling, H., 1998. Delamination and detachment ofa lithospheric root. Tectonophysics 296 (3/4), 225–247.


Shearer, P.M., Hedlin, M.A.H., Earle, P., 1998. PKP and PKKPprecursor observations: implications for small-scale structureof the deep mantle and core. In: Gurnis, M., Wysession, M.E.,Knittle, E., Buffett, B.A. (Eds.), The Core-Mantle BoundaryRegion: Geodynamics, Vol. 28. American Geophysical Union,pp. 37–55.

Sidorin, I., Gurnis, M., Helmberger, D.V., 1999. Evidence fora ubiquitous seismic discontinuity at the base of the mantle.Science 286, 1326–1331.

Simons, M., Hager, B.H., 1997. Localization of the gravity fieldand the signature of glacial rebound. Nature 390, 500–504.

Song, X., Ahrens, T.J., 1994. Pressure-temperature range ofreactions between liquid iron in the outer core and mantlesilicates. Geophys. Res. Lett. 21, 153–156.

Spence, D.A., Sharp, P.W., Turcotte, D.L., 1987. Buoyancy-drivencrack propagation: a mechanism for magma migration. J. FluidMech. 174, 135–153.

Tackley, P.J., 1998a. Self-consistent generation of tectonic platesin three-dimensional mantle convection. Earth Planet. Sci. Lett.157, 9–22.

Tackley, P.J., 1998b. Three-dimensional simulations of mantleconvection with a thermochemical CMB boundary layer: D′′.In: Gurnis, M., Wysession, M.E., Knittle, E., Buffett, B.A.(Eds.), The Core-Mantle Boundary Region: Geodynamics, Vol.28. American Geophysical Union, pp. 231–253.

Ten, A., Yuen, D.A., Podladchikov, Yu., Larsen, T.B., Pachepsky,E., Malvesky, A.V., 1997. Fractal features in mixing ofnon-Newtonian and Newtonian mantle convection. EarthPlanet. Sci. Lett. 146, 401–414.

van den Berg, A.P., Yuen, D.A., 1997. The role of shear heating

in lubricating mantle flow. Earth Planet. Sci. Lett. 151 (1/2),33–42.

van der Hilst, R., Widiyantaro, S., Engdahl, E.R., 1997. Evidencefor deep mantle circulation from global tomography. Nature386, 578–584.

van Dyke, M., 1982. An Album of Fluid Motion. Parabolic Press,Stanford, CA.

van Keken, P.E., Ballantine, C.J., 1999. Dynamical models ofmantle volatile evolution and the role of phase transitions andtemperature-dependent rheology. J. Geophys. Res. 104 (B4),7137–7152.

van Keken, P.E., King, S.D., Schmeling, H., Christensen, U.R.,Neumeister, D., Doin, M., 1997. A comparison of methods forthe modeling of thermochemical convection. J. Geophys. Res.102 (B10), 22477–22495.

Vecsey, L., Matyska, C., 2001. Wavelet spectra and chaos inthermal convection modelling. Geophys. Res. Lett. 28 (2),395–398.

Walzer, U., Hendel, R., 1997. Tectonic episodicity and convectivefeedback mechanism. Phys. Earth Planet. Inter. 100 (1–4),167–188.

Weertman, J., 1970. The creep strength of the Earth’s mantle.Rev. Geophys. Space Phys. 8, 145–168.

Weinberg, R.F., Schmeling, H., 1992. Polydiapirs—multi-wavelength gravity structures. J. Struct. Geol. 14 (4), 425–436.

Yamada, A., Nakanishi, I., 1998. Short-wavelength lateral variationof a D′′ P-wave reflector beneath the southwestern Pacific.Geophys. Res. Lett. 25 (24), 4545–4548.

Yamazaki, D., Karato, S., 2001. Some mineral physics constraintson the rheology and geothermal structure of Earth’s lowermantle. Am. Mineral. 86, 385–391.

the inﬂuences of composition-sand temperature-dependent

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