ORIGINAL PAPER

Catherine A. Hier Majumder Æ David A. Yuen

Erik O. Sevre Æ John M. Boggs Æ Stephen Y. Bergeron

Finite Prandtl number 2-D convection at high Rayleigh numbers

Received: 15 November 2001 /Revised: 20 April 2002 /Accepted: 22 April 2002 / Published online: 16 July 2002� Springer-Verlag 2002

Abstract Finite Prandtl number thermal convection isimportant to the dynamics of planetary bodies in thesolar system. For example, the complex geology on thesurface of the Jovian moon Europa is caused by a con-vecting, brine-rich global ocean that deforms the over-lying icy ‘‘lithosphere’’. We have conducted a systematicstudy on the variations of the convection style, asPrandtl numbers are varied from 7 to 100 at Rayleighnumbers 106 and 108. Numerical simulations show thatchanges in the Prandtl number could exert significanteffects on the shear flow, the number of convection cells,the degree of layering in the convection, and the numberand size of the plumes in the convecting fluid. We foundthat for a given Rayleigh number, the convection stylecan change from single cell to layered convection, forincreasing Prandtl number from 7 to 100. These resultsare important for determining the surface deformationon the Jovian moon Europa. They also have importantimplications for surface heat flow on Europa, and forthe interior heat transfer of the early Earth during itsmagma ocean phase. Electronic Supplementary Material

is available if you access this article at http://dx.doi.org/10.1007/s10069-002-0004-4. On that page (frame on theleft side), a link takes you directly to the supplementarymaterial.

Keywords Rayleigh-Benard convection Æ Finite Prandtlconvection Æ E-max Æ k-max Æ Mexican Hat wavelet Æwavelets

Introduction

The Prandtl number is the ratio of the kinematic vis-cosity to the thermal diffusivity of a fluid and is an im-portant control variable in thermal convection.Although infinite Prandtl number approximations areappropriate for the Earth’s crust and mantle, finitePrandtl number fluids are also ubiquitous throughoutour Earth. They include many of the fluids associatedwith our everyday lives, such as air, water, alcohols, andoils. Convection in air, which has a Prandtl number ofabout 1, is responsible for the weather. Convection inwater, with a Prandtl number of 7, is important foroceanic currents. The liquid iron of the Earth’s outercore is a fluid with a finite Prandtl number between 0.1and 5. Convection in this electrically conducting fluidgenerates the Earth’s magnetic field (Glatzmaier andRoberts 1995). Finite Prandtl convection is important tocrystallization processes in magma chambers (Bergantz1992). Convection in a magma ocean under highRayleigh numbers at Prandtl numbers of around 106

(Spera 2000) was likely to have occurred early in Earth’shistory. A convecting lunar magma ocean early in themoon’s history would had a lower Prandtl number thanthat of the Earth because of the higher MgO and TiO2

content (Spera 2000).A knowledge of convection styles in this type of fluid

may give important insights into the heat flux andchemical differentiation of the early Earth. FinitePrandtl number flows are especially important to thedynamics of the outer planets and their moons. Con-vection occurs inside Jupiter, which is a fluid with aPrandtl number of around 0.01 (Zhang and Schubert2000). The Galileo missions have confirmed the exis-tence of convecting oceans on the Jovian moons Europaand Callisto (Carr et al. 1998; Khurana et al. 1998).

Electronic Geosciences (2002) 7: 11–30DOI 10.1007/s10069-002-0004-4

Electronic Supplementary Material is available if you access thisarticle at http://dx.doi.org/10.1007/s10069-002-0004-4. On thatpage (frame on the left side), a link takes you directly to the sup-plementary material.

C.A. Hier Majumder (&) Æ D.A. Yuen Æ E.O. SevreDepartment of Geology and Geophysics,University of Minnesota, Minneapolis, MN 55455, USAE-mail: cathy@msi.umn.edu

C.A. Hier Majumder Æ D.A. Yuen Æ E.O. Sevre Æ J.M. BoggsMinnesota Supercomputing Institute,University of Minnesota, Minneapolis, MN 55455, USA

S.Y. BergeronBTI Inc., 1092A Rue du Convent, BP 58,St. Albert, OC, J0A-1E0, Canada

Thomson and Delaney (2001) have used numericalmodels to show that a seafloor spreading and hydro-thermal vent system similar to the Earth’s mid-oceanridges could produce convection in Europa’s globalocean. A magnetic field is induced as the salty, convec-ting, global oceans on the moons Europa and Callistoare dragged through Jupiter’s magnetosphere (Khuranaet al. 1998). Europa’s icy lithosphere has a wealth ofsurface geology created by convection in its global ocean(Carr et al. 1998). Evidence from the Voyager missionsindicates that the surface features of Ganymede may becaused by convection in a water–ice mantle, which isoverlain by an icey lithosphere (Kirk and Stevenson1987). A better understanding of finite Prandtl numberconvection at high Rayleigh numbers will allow us tobetter model heat flux and surface deformation on theJovian moons.

Numerical simulations of Rayleigh–Benard convec-tion at both high Rayleigh and Prandtl numbers aredifficult. As the Rayleigh number increases, strong in-ertial forces generate complex structures in the temper-ature and velocity fields that require increased gridpoints and smaller time steps. Increasing the Prandtlnumber leads to sharp temperature gradients and strongshear flows. These effects also require increasing gridpoints and decreasing time steps. The problem is com-pounded by the fact that as the Prandtl number is in-creased, the fluid becomes stiffer and more time isrequired for movement to develop from an initial tem-perature and velocity perturbation. Vincent and Yuen(1999) have conducted numerical simulations for aPrandtl number of unity (air) at Rayleigh numbers from108 to 1014. Numerical simulations have been conductedfor Prandtl number 7 (water) up to Rayleigh numbers of1013 (Yuen et al. 2000). These two-dimensional simula-tions revealed the complex wavelike structures generatedby finite Prandtl number convection, with Reynoldsnumbers reaching over 104 at these ultra-high Rayleighnumbers (Yuen et al. 2000). Wavelets are shown to be anideal tool for studying these structures (Yuen et al.2000).

Heat flux scaling laws tend to show that the Nusseltnumber is proportional to (RaPr)n where n is 1/3 forplume theories with a single length scale and 2/7 forplume theories with several length scales (Zaleski 2000).Grossmann and Lohse (2000) suggest that a singlepower law may not appropriately explain the depen-dence of the Nusselt number on the Rayleigh andPrandtl numbers because more than one heat transportregime may be operating in a given experiment. Insteadthey propose that linear combinations of several powerlaws may be necessary. They also indicate that for Pr > 7formation of the wind of turbulence, on which classicscaling theories are based, becomes less likely. The windof turbulence is associated with shearing in the boundarylayers that moves plumes in the horizontal direction andresults in a bulk stirring of the fluid. Verzicco andCamussi (1999) conducted numerical experiments forPrandtl numbers ranging from 0.022 (mercury) to 15 at

Rayleigh number 6·105. They found that, below Pr =0.35, heat is transferred by large-scale recirculation cells;above Pr = 0.35, the heat is transferred by thermalplumes. The aspect-ratio (A) may also play an importantrole in the heat transport, as suggested by the Shraimanand Siggia (1990) theory, in which the Nusselt number ispredicted to be proportional to Ra2/7Pr–1/7A–3/7.

The Prandtl and Rayleigh numbers of some convec-ting systems are shown in Table 1. Previously, fluids withPrandtl numbers as different as the Earth’s mantle withPrandtl number 1025 and mushy ice with Prandtl number104, such as found in Europa’s lithosphere, can both beapproximated as infinite Prandtl number fluids, thoughdifferences may still remain at high Rayleigh numbers,such as 108. Significant differences may exist in the con-vection styles of fluids because of changes in Prandtlnumber in the intermediate range from Pr = 1 to 100.

This study focuses on fluids with Prandtl numbersranging from 7 to 100. Fluids in this range of Prandtlnumbers have important industrial applications. Forexample, ethanol has a Prandtl number of 17. A solutionof 60% glycerin and 40% water has a Prandtl number of100. Shevtsova et al. (2001) studied the onset of ther-mocapillary convection in liquid bridges for fluids withPrandtl numbers up to 35. This type of convection cancause undesired compositional variations in crystalsgrown industrially.

We simulated convection at Rayleigh numbers of 106

and 108. The Prandtl numbers in this study are stillsignificantly below those of some important geologicfluids, such as the mushy ice that forms diapirs onEuropa or basaltic magma. Both of these fluids havePrandtl numbers of order 104. This study represents afirst step in the process of simulating systems such asEuropa. It allows us to begin to assess the importance ofPrandtl number in the behavior of Rayleigh–Benardconvection at higher Prandtl numbers. It represents thebeginning of the development of techniques that willallow us to conduct simulations at even higher Prandtland Rayleigh numbers in the future.

We have conducted simulations in an aspect-ratio 3box for Prandtl numbers of 7, 17, 40, 80, and 100 at aRayleigh number of 106 and for Prandtl numbers of 7,17, 40, 60, 80, and 100 at a Rayleigh number of 108. Theconvection simulations have a complex time evolution,which is discussed in detail for the simulations of Prandtlnumber 7 and Rayleigh number 106. We have illustratedthe time evolution through the use of animations. Wehave conducted wavelet analysis on the long-term solu-tions of the temperature deviation and vorticity fields for

Table 1. Rayleigh and Prandtl numbers of some convective sys-tems

System Rayleigh number Prandtl number

Pot of water 1·107 7Europa’s ocean 1·1025 7Europa’s icey crust 1·1018 1·104Earth’s mantle 5·106 1·1024

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each of the simulations to help characterize the con-vection style.

Wavelets have been used as a tool to better under-stand the convection patterns captured by seismic to-mography of the Earth’s mantle (Bergeron et al. 1999,2000; Yuen et al. 2000). One-dimensional wavelettransforms have been used for many years in the analysisof time series, such as interseasonal Earth rotationvariations (Chao and Naito 1995). Malamud and Tur-cotte (2001) analyzed one-dimensional tracks of Martiantopography using wavelets. Vecsey and Matyska (2001)used wavelets to analysis temperature, kinetic energy,and Nusselt number time series in mantle convectionsimulations. Castillo et al. (2001) used wavelet trans-forms of seismograms to characterize the transition zonestructure below California.

Wavelets are also a powerful tool for processing 2-Dconvection simulation data, such as temperature andvorticity fields (Hier et al. 2001). Temperature and vor-ticity maps from convection simulations can be quitecomplicated and difficult to interpret at higher Rayleighand Prandtl numbers. The wavelet transform allows us toseparate the large-scale features, such as convection cells,from smaller scale features, such as shear layers. This leadsto a better understanding of the different processes thatproduce the temperature and vorticity maps.

In order to take advantage of theWeb format for paperpublishing, we have used figures extensively in this paper.We focus on the temperature deviation and vorticity fieldsgenerated by the simulations along with their wavelettransforms. The finite Prandtl simulations resulted in richpatterns in the temperature deviation and vorticity fieldsthat is most easily conveyed by figures. The time evolutionof the simulations was also very rich. TheWeb format hasallowed us to feature in this communication animations ofhow the complex dynamics develop with time (see Elec-tronic Supplementary Material).

Numerical model

The non-dimensional conservation equations for mass,momentum, and energy equations of finite Prandtlnumber convection in the Boussinesq approximationare:

r � v ¼ 0 ð1Þ

@v

@tþ v � rv ¼ Prr2v �rP þ PrT ez ð2Þ

@T@t

þr � ðvT Þ ¼ r2T þ Ravzez ð3Þ

where t is the time non-dimensionalized by the thermaldiffusivity and the depth of the layer, T is the tempera-ture deviation from the conductive profile, v is the ve-locity vector, P is the pressure, and ez is the unit vectorfrom the top to the bottom of the fluid layer. Ra is theRayleigh number for Rayleigh–Benard convection in abasely heated box, and Pr is the Prandtl number.

The numerical method used is a spectral method for2-D convection with finite Prandtl number (Vincent andYuen 1999) with sine and cosine series expansion as thebasis functions. The top and bottom of the box havefree-slip boundary conditions. The time-marching uses amixed leap frog-Cranck–Nicholson two-step pressurecorrection scheme. The grid size for each simulation isshown in Table 2. The aspect-ratio of the box is definedas the width of the box over the layer depth.

Wavelet transform

The wavelet transform (e.g., Resnikoff and Weiss 1998)allows one to study the scales within a dataset withoutlosing the spatial information. Whereas the Fouriertransform or spherical harmonics provides a global de-scription of a wavelength over a dataset; the wavelettransform provides a localized description of a particu-lar length scale at each grid point in the multidimen-sional dataset (van den Berg 1999). The 2-D wavelettransform is:

~ff ða; bÞ ¼ 1

a

ZLx0

ZLy0

f ðxÞw x � b

ad2x ð4Þ

where vector b is the two-dimensional position param-eter, scalar a is the scaling parameter, and Lx and Ly arethe lengths of the periodic rectangle in Cartesian space

Table 2. Grid size forsimulations

Rayleighnumber

Prandtlnumber

Aspect-ratio(width · depth)

Grid size(width · depth)

106 7 3·1 768·256 points106 17 3·1 768·256 points106 40 3·1 768·256 points106 80 3·1 768·256 points106 100 3·1 1,080·360 points108 7 3·1 1,080·360 points108 17 3·1 1,080·360 points108 40 3·1 1,080·360 points108 60 3·1 1,080·360 points108 80 3·1 1,080·360 points108 100 3·1 1,080·360 points

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(Bergeron et al. 2000). We have used as the motherwavelet the Mexican Hat, or the second-derivative of theGaussian function (Daubechies 1992):

wðxÞ ¼ 2� xj j2� �

exp� xj j2

2

!ð5Þ

The wavelet transform is done in Fourier space tosimplify and speed-up the calculation (Bergeron et al.2000).

Scalogram

The result of a Fourier transform is a coefficient de-scribing the strength of a given wavelength over thewhole dataset. The specific location or the phase infor-mation is not explicitly retained in the course of theFourier transform. Therefore, you cannot compare thestrength of a given wavelength at two different points.

The wavelet transform produces a coefficient de-scribing the strength of a given length scale at each datapoint. The wavelet transform of a 2-D dataset is a 3-Dfunction known as a scalogram. The scalogram is afunction of both the position vector and the scale. Thespatial information is explicitly preserved in the scalo-gram. This allows one to analyze how the length scalevaries with position. One can say that large lengthscales are more important at point a, and small lengthscales are more important at point b.

Because a scalogram is a function of both scale andposition, the dimension of the original dataset is in-creased by one. Scalograms of our two-dimensionalvorticity fields are three-dimensional boxes. The extradimensionality makes the scalograms difficult to storeand visualize because of the large memory required. The3-D boxes contain vast amounts of data that cannot beeasily digested by the eye when displayed on a computerscreen. Therefore, we took slices of the box at a givenscale. For each scalogram we took three slices: one at alarge-scale, one at a medium-scale, and one at a small-scale. These individual slices are a function of positiononly. They show the importance of features at a givenscale at different positions in space (see Fig. 11).

E-max and k-max

The scale parameter in the wavelet transform [Eq. (4)]results in a dimensionality of one degree higher than thephysical space analyzed. This can make the scalogramsdifficult to visualize and interpret. Analysis of E-maxand k-max distributions is used to synthesize the scalo-gram (Bergeron et al. 2000; Yuen et al. 2000). E-maxrepresents the energy of the scale that has the largestenergy value at a given position. An E-max map plotsthe energy value of the scale with the highest energy at agiven point in space. It, therefore, selects the energyvalue from the scale that is most important at that point.

We define the energy of the wavelet transform as the L2

norm:

E ¼ sign f bð Þð Þ ~ff a; bð Þ�� ��2 ð6Þ

Because the sign of both the thermal anomaly and thevorticity is important in convection studies, we havemultiplied the L2-norm by the original sign of the ther-mal anomaly or vorticity function.

The k-max is the wavenumber at which maximumenergy occurs for a given position vector b. A k-maxmap plots the scale that is most important at a givenpoint. The scale, a, is related to the wave number, k, by:

a ¼ exp �bkð Þ ð7Þ

where b is a tuning parameter, which we have chosen as0.22. High k-max values represent small-scale features.These features are associated with areas of strong gra-dients. Therefore, the k-max map is an excellent tool forpicking out areas of strong gradients in a dataset. Fortemperature deviation and vorticity fields from convec-tion simulations, the areas of sharp gradients are asso-ciated with fluid movement. We will use the k-max mapsto emphasize areas of relative movement within thefluid.

The E-max and k-max analysis results in a lower di-mensional approximation of the data. It allows us tolook at the data over a given range of scales. For ex-ample, to study small-scale features in the dataset wewould find E-max and k-max maps for scales varyingfrom k = 15 to k = 20.

Temporal evolution

The temporal evolution of the simulations is best illus-trated by an animation. Animation 1 shows the tempo-ral evolution for the Ra = 106 and Pr = 7 simulation.An example of a fluid with Pr = 7 is water at 20 �C and1 atm (Weast 1987). The animation shows the temper-ature deviation, which is the total temperature minus theconductive profile, and the vorticity (see ElectronicSupplementary Material).

The simulations are begun from an initial tempera-ture deviation and vorticity perturbations in the form ofa sine wave with fundamental wavelength Lx/2 (Fig. 1).The number of grid points for the simulation is specifiedin Table 2.

For Ra = 106 and Pr = 7, plumes and circulationcells develop from these perturbations. The plumes havethin stems with thicker plume-heads that spread hori-zontally along the thermal boundary layers (Fig. 2). Thevorticity shows that the circulation cells have pro-nounced shear layers along the edges with relativelyweak rotation in their cores (Fig. 2). This indicates thatthe majority of the movement is along the edge of theconvection cells. Hot material rises vertically in plumesalong the edge of a convection cell, and cold material

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descends vertically in downwellings along the oppositeedge. There is only weak movement in the central area ofthe circulation cell.

As the Ra = 106, Pr = 7 simulation advances to t =0.410, the plume-heads spread horizontally across thethermal boundary layer and begin to impinge onneighboring plume-heads. They then travel downwardsforming regions of hot material below the upper thermalboundary layer (Fig. 3). This creates a new thermalboundary layer. New plumes begin to develop from theoriginal stem region below the hot regions. In the vor-

ticity field, the center of the circulation cells reverse flowdirections (Fig. 3). The sharp shear layers along theedges of the circulation cells remain in the same orien-tation as in Fig. 2. These shear layers are associated withthe new plumes arising from the original stem area.

In the next frame of the temperature deviation we cansee new plumes and downwellings originating from thenew central thermal boundary layers (Fig. 4). The ver-tical symmetry, which originates from the symmetricalinitial condition, is still maintained. The plumes and

Fig. 1. Initial temperature deviation and vorticity fields at dimen-sionless time, t = 0.000 for Ra = 106, Pr = 7

Fig. 2. Temperature deviation and vorticity fields for initial plumegrowth at dimensionless time, t = 0.405 for Ra = 106, Pr = 7

Fig. 3. Temperature deviation and vorticity fields for secondaryplumes rising from boundary layers at dimensionless time, t =0.410 for Ra = 106, Pr = 7

Fig. 4. Temperature deviation and vorticity fields of plumebranching at dimensionless time, t = 0.415 for Ra = 106, Pr = 7

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downwellings growing from the original lower thermalboundary layer begin to form caps below the lowerthermal boundary layer. The original vorticity cells be-gin to divide, thus forming two separate layers of cir-culation cells. This results in layered convection (Fig. 4).

At time t = 0.420, the plumes growing from thecenter become connected with those growing from thebottom of the box to form a branched plume structure(Fig. 5). New plumes begin at the junction of the twooriginal plumes.

At t = 0.430, the layering structure still persists. Theplume branching results in a vorticity pattern that is acombination of a whole box circulation with a layeredstructure (Fig. 6). We can still see the original four cir-culation cells that cover the whole height of the box.These four cells are now each divided into three separatelayers with the strongest circulation in the middle layer.

At t = 0.473, the layered circulation cells divide toform vortex pairs (Fig. 7). Small plumes begin to risefrom both the lower and middle thermal boundary lay-ers, and downwellings form at both the upper andmiddle thermal boundary layers. The new plumes anddownwellings are evenly spaced along the length of theboundary layers. The symmetry originating from theinitial condition is preserved.

As convection progresses to t = 0.490, however, thelayering that has developed disappears (Fig. 8). As canbe seen in the vorticity field, the four original circulationcells are no longer divided horizontally (Fig. 8). Insteadthey are each divided vertically into three long narrowcells. At t = 0.490, the symmetry that had been carriedalong from the initial condition is lost.

At t = 0.525, the layered mode is destroyed, and thecirculation cells take up the whole box height (Fig. 9).

Although the four circulation cells are still visible, thecirculation is reversed from that of the original cells.There is a strong horizontal shear component resultingin a wind that sweeps across the box. The temperaturedeviation shows that the plumes move up from the lowerthermal boundary layer along thin stems, which aretilted by the horizontal shear (Fig. 9). The plumes anddownwellings spread out horizontally once they reach

Fig. 5. Temperature deviation and vorticity fields of fully devel-oped plume branching at dimensionless time, t = 0.420 for Ra =106, Pr = 7

Fig. 6. Temperature deviation and vorticity fields showing thedevelopment of a layered convection style at dimensionless time, t= 0.430 for Ra = 106, Pr = 7

Fig. 7. Temperature deviation and vorticity fields showing divisionof plumes and development of smaller eddies at dimensionlesstime, t = 0.473 for Ra = 106, Pr = 7

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the top and bottom with relatively thick thermalboundary layers.

The long-term solution is reached at t = 0.534. In thelong-term solution there are still four convection cells,but two of the cells have become wider at the expense oftwo cells that have narrowed (Fig. 10). The horizontalshear remains in the long-term solution. This results inshear layers on the edge of the wide convection cells.

Wavelet analysis

The use of wavelets allows us to separate the structuresseen in the temperature deviation and vorticity datasetsby the scale of the features. This helps us to interpret theprocesses operating at different scales in the convectionpattern. The long-term solution of the temperature de-viation and vorticity fields for each simulation wastransformed using the continuous wavelet transform[Eq. (4)] with the Mexican Hat as the mother wavelet[Eq. (5)]. The scalogram was computed from localwavenumber k = 1–20, which corresponds to a resolu-tion varying from 65 to 1.2% of the horizontal boxwidth. The resulting scalogram for Ra = 106 and Pr= 7is a 16-MB file, which consists of a 3-D box with768·256·20 points. The scalogram is difficult to visual-ize because of the large memory required and the factthat it contains more information then can be easilydigested by the eye. Therefore, for better interpretationand to minimize data storage, we have taken slices of thescalogram at three scales: k = 5, k = 11, and k = 20,which corresponds to resolutions of 27, 7.3, and 1.2% ofthe box width (Figs. 11 and 12).

For the simulation Ra = 106 and Pr = 7, the wavelettransform was performed on the temperature deviationand vorticity fields shown in Fig. 10. The resulting sca-logram slices are shown in Figs. 11 and 12. In the large-scale slice of the temperature deviation (Fig. 11), theonly visible structure is the partitioning of the field into acold bottom layer and a top hot layer. In the medium-scale we can clearly discern the shapes of the individualplume heads expanding horizontally across the bound-ary layers. The small-scale structures are dominated bythe plume tails. For the hot plumes, the plume tails in

Fig. 8. Temperature deviation and vorticity fields showing de-struction of layered convection and development of tall aspect-ratioplumes at dimensionless time, t = 0.490 for Ra = 106, Pr = 7

Fig. 9. Temperature deviation and vorticity fields showing devel-opment of wind structure at dimensionless time, t = 0.525 for Ra= 106, Pr = 7

Fig. 10. Long-term solution, at the dimensionless time, t = 0.534for Ra = 106, Pr = 7

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the small-scale are surrounded by cold signatures. Theopposite occurs for the cold plumes. These false signalssurrounding the plume are caused by the attempt to fitthe structure to the edges of the Mexican Hat wavelet.The true sign of the signal is associated with the centralcolor.

For the large-scale of the vorticity scalogram(Fig. 12), the wavelets pick out the two wide and twonarrow convection cells. In the medium-scale we can seethe four convection cells along with stronger areas ofcirculation created on their edges by the horizontalshear. The small-scale picks up the strong shear layerassociated with the edges of the convection cells.

We have used E-max and k-max maps to synthesizedata over several scales. This allows us to study the effectof a range of scales with datasets that are easier to storeand visualize. Figure 13 shows E-max and k-max mapsfor the vorticity field shown in Fig. 10. The maximumenergy is computed over scales of k = 15–20, whichcorresponds to a resolution varying from 3.0 to 1.2% ofthe horizontal box width. The E-max map is very similarto the small-scale scalogram slice in Fig. 12. However,only the highest values near the shear layers are revealed.This simplifies the pattern and makes it easier for us topick out the most significant features, the shear layers.

The lower energy features within the convection cells arenot shown on the E-max map. In the Ra = 106 and Pr= 7 simulation shown here, the pattern is relatively

Fig. 11. Scalogram of the Ra = 106 and Pr = 7, associated withthe long-term solution temperature deviation field in Fig. 10

Fig. 12. Scalogram of the Ra = 106 and Pr = 7, associated withthe long-term solution vorticity field in Fig. 10

Fig. 13. E-max and k-max maps for the vorticity computed from k= 15 to k = 20 for Ra = 106 and Pr = 7 for the temperaturedeviation field shown in Fig. 10

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simple and many of the features highlighted in the E-max map can be easily picked out on the small-scalescalogram slice. The ability of the E-max to reveal thehighest energy features and simplify the pattern willbecome increasingly important as the convection pat-terns become more complicated with increasing Ray-leigh and Prandtl numbers.

Extremely large values of k-max (Fig. 13) occur alongthe edges of sharp gradients. These areas are associatedwith differential movement of fluid. Identifying andunderstanding areas of fluid movement is essential togrowing metallic alloys from a crystal mush becausefluid movement within the mush can cause areas ofanomalous composition or freckles. The movement offluid in a crystal mush is also important to crystallizationin geologic settings such as magma chambers, ice crys-tallization, and the inner/outer core boundary (Becker-mann 2000). Material scientists have previously relied onstudies of the local Rayleigh number to identify regionsof strong fluid movement (Beckermann 2000). Thek-max has the added advantage of showing movementboth in the horizontal and vertical. The k-max not onlypicks out areas of current movement, but also delineateswhere movement has occurred recently. It marks thepath of the fluid throughout the box.

Effect of increasing Prandtl number up to 100

For the Ra = 106 simulations we increased the Prandtlnumber from 7 to 17, 40, 80, and, finally, 100. An ex-ample of a fluid with Pr = 17 is ethyl alcohol at 15 �Cand 1 atm (Turcotte and Schubert 1982). Fluids athigher Prandtl numbers can be seen as a solution ofwater with glycerin at 20 �C and 1 atm (Weast 1987).For example, Pr = 40 is a solution of water and48 vol% glycerin. Pr = 80 is a solution of waterand 60 vol% glycerin. Pr = 100 is a solution of waterand 64 vol% glycerin. As the Prandtl number is in-creased, Eq. (2) becomes stiffer for time integration. Thesolution becomes more difficult to solve because manymore time steps are needed before the initial temperatureand velocity perturbations cause convective movement.The overall time development of the solution evolvesslower. We also found that, as the Prandtl number wasincreased from Pr = 7, sharper temperature gradientsand stronger shear flows, i.e., velocity boundary layersdeveloped. This required using more grid points andsmaller time steps. This further increased the computa-tion time required for the simulations.

The most noticeable effect of even a small increase inPrandtl number from 7 to 17 at Rayleigh number 106 isthe increase in horizontal shear flow (Fig. 14). This re-sults in a ‘‘wind’’ driving the plumes. This shearing ef-fect, caused by inertia, could be substantial in a systemsuch as Europa where there is an icy ‘‘lithosphere’’ beingdeformed by a convecting, brine-rich, global ocean. Ifthe dissolved species in the ocean and the formation ofice increase the Prandtl number of the system, estimates

on the behavior of the system based on studies atPrandtl number 7 may not be appropriate, and we mustuse higher Prandtl numbers of around 50 or 100.

The smaller scale structures resulting from the stronghorizontal shear flow makes it difficult to pick out thenumber of convection cells from the vorticity field inFig. 14. This a situation where the wavelet transformproves especially valuable. In the large-scale vorticityscalogram slice (Fig. 15), we can easily see that the samepattern of two wide convection cells separated by twonarrow ones that we saw in simulations for Ra = 106

and Pr = 7 (Fig. 12). The features related to the shearare clearly discerned in the medium and small-scales.

The E-max map for the smaller scales of the vorticity(Fig. 16) gives a similar picture to the small-scale scalo-gram slice. The E-max, however, shows that the strongestsignatures are coming from two layers of circulation cells,one in the top of the fluid and one in the bottom. Thehorizontal shear in this fluid appears to be creating alayered convection style in the smaller scale. Vortical cellswith a positive sense of circulation have much strongersignatures than those with a negative sense of circulation.The original vorticity field (Fig. 14) also shows strongerpositive vortices than negative ones. The k-max map(Fig. 15) emphasizes both the vertical shearingmovementof the fluid along with its significant horizontal compo-nent. The ability to pick out both the horizontal andvertical movement within the time-dependent fluid mo-tions provides an advantage over the more traditionallocal Rayleigh number approach, which only emphasizesthe vertical movement (Beckermann 2000).

As the Prandtl number is increased to 100, the lateraltemperature distribution shows more layered structures.

Fig. 14. Long-term solution of the temperature deviation andvorticity fields for Ra = 106 and Pr = 17

19

This indicates that there is better homogenization of thetotal temperature field within each layer (Fig. 17). Boththe hot and cold thermal boundary layers decrease inthickness. The plume heads still spread laterally across thehot and cold boundary layers, but they have less verticalthickness. This indicates the possibility that the Prandtlnumber may have an important control over the verticalheat flux in finite, but large Prandtl number convection.

The vorticity field undergoes noticeable changes withincreasing Prandtl number (Fig. 17). Individual circu-lation cells are weakly delineated by strong vertical shearlayers along their edges. The large-scale slice of thevorticity scalogram shows that there are only two con-vection cells (Fig. 18) rather than the four seen forsmaller Prandtl numbers. One of the convection cells is1.5 times wider than the other. Although at first glance itmay appear that the scalogram shows three convectioncells, this is because the continuous convection field hasbeen cut along the box edges. We see clearly that thereare actually two convection cells in the medium scalo-gram slice, where one can see the shear layers along thevertical edges of the cells. The medium- and small-scalescalogram slices show sharp vertical shear layers alongthe larger convection cell boundaries along with smallerconvection cells embedded within the larger ones.

The E-max map (Fig. 19) for the smaller scales por-trays areas of sharp vertical shear layers, which coincidewith long narrow circulation cells seen in the medium-scale scalogram slice. These long narrow cells are foundin between the larger scale circulation cells. The k-max(Fig. 19) emphasizes the primarily vertical nature offluid motions; although it shows that there is also somedegree of horizontal movement.

Fig. 17. Long-term solution of the temperature deviation andvorticity fields for Ra = 106 and Pr = 100

Fig. 16. E-max and k-max maps for the vorticity computed from k= 15 to k = 20 for Ra = 106 and Pr = 17 for the temperatureperturbation field shown in Fig. 14

Fig. 15. Scalogram of the Ra = 106 and Pr = 17, associated withthe long-term solution vorticity field in Fig. 14

20

Effect of increasing Rayleigh number

Simulations were run at a Rayleigh number of 108 forPrandtl numbers of 7, 17, 40, 60, 80, and 100. The in-crease in Rayleigh number leads to development ofsmaller scale structures (Animation 2, SupplementaryElectronic Material). The convection evolves throughthe branching of plumes and the division and re-co-alescence of vorticity cells. The plumes become thinnerthan for Ra = 106. The temperature deviation is morelayered, thus indicating a greater separation of theconvective field. The shear layers along the edges of theconvection zones are much sharper (see the Supple-mentary Electronic Material).

ForRa=108 andPr=7, the convection pattern in thelong time regime (Fig. 20) is similar to the pattern for Ra=106 andPr=100 (Fig. 17). There is one upwelling andone downwelling, both of which travel across the com-plete vertical distance between the horizontal boundarylayers. The vorticity is characterized by two circulationcells with one cell 1.5 times larger than the other. Thevorticity field is also characterized by many small-scalefeatures deriving from the larger convection cells.

By comparing the temperature deviation scalogram forconvection in a Prandtl number 7 fluid at Ra = 108

(Fig. 21) to that in a fluid withRa=106 (Fig. 11), we cansee the variations of the scales associated with the plumes.ForRa=106, the general shape of the plumes is visible inthe medium-scale. In the medium-scale for Ra = 108, wecan see hot regions in the boundary layer where theplumes impinge, but there are only faint traces of theplume stems in the medium-scale. The full details of theplumes are visible only at the smaller scale for Ra= 108.

Fig. 18. Scalogram of the Ra= 106 and Pr = 100, associated withthe long-term solution of the vorticity field in Fig. 17

Fig. 19. E-max and k-max maps for the vorticity computed from k= 15 to k = 20 for Ra = 106 and Pr = 100 for the temperaturedeviation field shown in Fig. 14

Fig. 20. Long-term solution of the temperature deviation andvorticity fields for Ra = 108 and Pr = 7

21

Wavelet analysis becomes invaluable in analyzing thecomplicated vorticity fields produced in higher Rayleighnumber convection (Ra = 108) (Fig. 22). In the large-scale scalogram slice we can see the structure of the twolarge convection cells. The medium-scale slice gives us thesmaller convection cellswithin the larger cells; whereas thesmall-scale slice shows the small-scale shear layers lyingalong the edge of the large-scale circulation cells.

Because a large portion of the activity in this vorticitypattern occurs at the smallest scales, it is useful to ex-amine the E-max over the small-scales, from mode k =15–20 (Fig. 23). The plume signatures that were alsovisible in the small-scale scalogram slice are highlightedalong with the vortices within the larger convection cells.These vortices occur over a range of smaller scales. Thek-max map (Fig. 23) shows a finer pattern than found inthe k-max maps for the simulations at Ra = 106. Thevertical and horizontal movements associated with theplume and the downwelling coexist with the movementassociated with the smaller vortices within the two largercirculation cells.

The temperature and vorticity fields at Ra = 108, Pr= 17 and 40 are somewhat similar to those for Ra =108, Pr = 7. The differences with high Prandtl number

still persist, though, at high Rayleigh number. When thePrandtl number is increased to 60 at Ra = 108 however,the convection style changes dramatically (Fig. 24). Thetemperature deviation field becomes more layered, in-dicating better homogenization of the temperaturewithin each layer. The vorticity field has become quitecomplicated, and it is difficult to interpret these fieldswithout the use of wavelets. The higher Rayleigh num-ber brings out increased detail and makes it more likelythat subtle differences between Prandtl numbers 100 and1,000 will be found in future simulations.

The change in the convection style when the Prandtlnumber is increased from 7 to 60 becomes clearly evidentin the temperature deviation scalogram (Fig. 25). We donot see any signatures fromplumes or downwellings in thelarge-scale. In the medium-scale, we only see the bound-ary layers. The boundary layer signatures are broken bynumerous small upwellings and downwellings. The shapeof these upwellings and downwellings is only apparent inthe small scale. We can see that these features do nottraverse vertically through the center of the box.A layeredconvection style has developedwith an additional internalboundary layer across the center of the box.

The large-scale slice of the vorticity scalogram showsthat large-scale circulation cells still exist at this medi-umly large Prandtl number (Fig. 26). There are four

Fig. 21. Scalogram of the temperature deviation for the Ra = 108

and Pr = 7, associated with the long-term solution of thetemperature deviation field in Fig. 20

Fig. 22. Scalogram of the Ra = 108 and Pr = 7, associated withthe long-term solution of the vorticity field in Fig. 20

22

circulation cells, which are much narrower than thelarge-scale circulation cells at lower Prandtl numbers.The medium-scale scalogram slice shows a complexstructure of circulation cells within the large-scale cir-culation cells. The small-scale slice of the vorticity sca-logram is almost identical to the small-scale slice of thetemperature deviation scalogram (Fig. 25). There are

areas of downwellings and upwellings associated withsharp shear layers that travel vertically to the center ofthe box. Large plumes that travel through the verticaldistance of the box no longer exist. We note that thischange in convection style is brought about by increas-ing the Prandtl number, while keeping the Rayleighnumber constant. This indicates that the Prandtl numberplays a significant role in determining the convectionstyle. The relative increase in viscosity to thermal diffu-sivity as the Prandtl number increases makes themovement of large-scale plumes and downwelling moredifficult. More of the heat appears to be transported bysmaller scale plumes at higher Prandtl numbers.

The E-max of the vorticity field shows the location ofthe shear layers of the stronger plumes and downwel-lings (Fig. 27). The heads of the plumes and downwel-lings have especially strong signatures. We also see thelayered structure of the convection with no plumescrossing the entire box. The k-max map for Ra = 108

and Pr= 60 is quite different from that of Ra= 108 andPr = 7 (Fig. 23). We can see evidence of a large amountof horizontal movement, but only limited verticalmovement. Although many of the structures do not

Fig. 23. E-max and k-max maps for the vorticity computed from k= 15 to k = 20 for Ra = 108 and Pr = 100 for the temperaturedeviation field shown in Fig. 20

Fig. 24. Long-term solution of the temperature deviation andvorticity fields for Ra = 108 and Pr = 60

Fig. 25. Scalogram of the Ra = 108 and Pr = 60, associated withthe long-term solution of the temperature deviation field in Fig. 24

23

cross the center of the box, it appears that there is stillsome movement across the center. This indicates that theconvection style is not completely layered.

The convection style becomes more distinctively lay-ered for Ra = 108 and Pr = 80. The scale of thestructures also becomes smaller, and the convectionappears to have a more chaotic character. The featuresjust discussed become quite well defined for Ra = 108

and Pr = 100 (Fig. 28). The temperature deviation andvorticity fields are quite complicated, and the scalogramtechnique is required for their interpretation.

The slice showing the vorticity scalogram for thelarge-scale (Fig. 29) displays that there are still somelarge circulation cells. There are basically two layers oflarge-scale cells; the convection style has become indis-putably layered. The convection cells are tilted, indi-cating the presence of horizontal shear. The medium-scale scalogram slice shows a complicated pattern ofnarrow, tilted convection cells. This confirms the pres-ence of a horizontal shear. The small-scale slice showsthe small shear layers created by individual plumes. Wecan see that the plumes fade away when they reachabout halfway through the box height. A new boundary

layer has been formed across the middle of the box bythe presence of layered convection.

The small-scale scalogram slice is complicated forthis simulation. Although it only looks at k = 20, theamount of small-scale features in this vorticity field in-dicates that other small-scales should play an important

Fig. 26. Scalogram of the Ra = 108 and Pr = 60, associated withthe long-term solution vorticity field in Fig. 24

Fig. 27. E-max and k-max maps for the vorticity computed from k= 15–20 for Ra = 108 and Pr = 60 for the vorticity field shown inFig. 24

Fig. 28. Long-term solution temperature deviation and vorticityfields for Ra = 108 and Pr = 100

24

role. We have examined the E-max over scales rangingfrom k = 15 to 20 (Fig. 30). In this image we can seeboth the long, narrow shear layers along the edges oflarger convection cells and the smaller scale vortices thatare further subdividing the vortices observed in themedium-scale. The vortices with scales ranging frommedium to small indicate that although the convectionstill has some large-scale organization, the smaller scalesare becoming increasingly important. The k-max(Fig. 30) emphasizes the chaotic movement throughoutthe box. There is less movement across the center of thebox than for Ra = 108 and Pr = 60. This indicates thatthe convection style is reaching a truly layered state forthis intermediately high Prandtl number situation.

Effect of aspect-ratio

In order to study the effect of aspect-ratio on theconvection we conducted one simulation for Ra = 106,Pr = 7 in a box with a height three times taller than itswidth. The simulations are begun from an initial

temperature deviation and vorticity perturbations in theform of a sine wave with fundamental wavelength Lx/2where Lx is the box width. The convection evolvesthrough time in a fashion very similar to the time evo-lution seen in the wide aspect-ratio boxes (see Supple-mentary Electronic Material).

The plumes develop a branching structure from theinitial perturbations (Fig. 31). The original vorticity per-turbations that are the full box height divide to form acombination of whole box and layered convection. This isvery similar to the evolution seen in the simulation forRa= 106, Pr = 7 with an aspect-ratio of 3 (see Fig. 5). Thestructures are thinner and taller for the narrow aspect-ratio case than for the wide aspect-ratio one, but they areotherwise strikingly similar. The narrow aspect-ratioconvection simulation also develops about ten times fasterthan the wide aspect-ratio simulation.

There are four circulation cells in the final long-termvorticity solution (Fig. 32). There are two plumes andtwo downwellings. The plumes and downwellings aremoved by a horizontal wind, which deflects them first tothe right and then to the left. This wind has a period of t= 0.025 where t is the dimensionless time.

Overall, the narrow aspect-ratio results in a convec-tion style strikingly similar to that seen in the largeraspect-ratio. The convection develops in time throughplume branching and division of vorticity cells. Theplumes and vorticity cells adjust to the narrow aspect-ratio by stretching in the vertical direction relative to thefeatures in the larger aspect-ratio simulations. The long-term solution for the Ra= 106, Pr = 7, aspect-ratio 1·3case has the same general structure as the Ra = 106, Pr

Fig. 29. Scalogram of the Ra= 108 and Pr = 100, associated withthe long-term solution of the vorticity field in Fig. 28

Fig. 30. E-max and k-max of Ra = 108 and Pr = 100 for k = 15–20 of the long-term solution of the vorticity field in Fig. 28

25

= 7, aspect-ratio 3·1 case of four convection cells withtwo upwellings and two downwellings.

Reynolds number

The turbulent Reynolds number was calculated a pos-teriori (Table 3) by post-processing the data from therun. The Reynolds number was not calculated from anassumed Rayleigh and Prandtl number relationship.

A Reynolds number is defined as:

Re ¼ vlm

ð8Þ

where v is the velocity, l is the length-scale, and m is thekinematic viscosity.

We calculated the turbulent Reynolds number aposteriori as:

Re ¼ v0l0Pr

ð9Þ

v0 is the non-dimensional root-mean-square or quadraticmean velocity. It represents the velocity of the largesteddies. l0 is the non-dimensional integral length scale orautocorrelation length. It represents the size of the largesteddies. v0 and l0 are local quantities that are calculated inthe course of the run. Because v0 and l0 are non-dimen-

Fig. 31. Temperature deviation and vorticity for Ra = 106 and Pr= 7 with an aspect-ratio of 1·3 at dimensionless time t = 0.075.For the temperature deviation red represents a positive deviationand blue a negative one. For the vorticity red representscounterclockwise circulation and blue clockwise

Fig. 32. Long-term solution for Ra = 106 and Pr = 7 with anaspect-ratio of 1·3. For the temperature deviation red represents apositive deviation and blue a negative one. For the vorticity redrepresents counterclockwise circulation and blue clockwise

Table 3. Reynolds numbers

Rayleighnumber

Prandtlnumber

Aspect ratio Reynoldsnumbera

106 7 3 19.2106 17 3 6.0106 40 3 2.2106 80 3 0.63106 100 3 0.45108 7 3 193108 17 3 122108 40 3 58.6108 60 3 27.4108 80 3 20.5

aReynolds numbers were calculated a posteriori after each run

26

sionalized by the thermal diffusivity and the box height,the resulting Reynolds number is that defined in Eq. (8).

The turbulent Reynolds numbers tended to decreasewith Prandtl number for both Ra = 106 and 108

(Table 3, Fig. 33). The decrease in Reynolds numberwith Rayleigh number was stronger at Ra = 106 than atRa = 108. For Ra = 106:

Re Pr�1:4 ð10Þ

whereas for Ra = 108:

Re Pr�0:93 ð11Þ

There are two effects contributing to the decrease in theturbulent Reynolds number with Prandtl number. Thefirst is the direct effect of dividing the quantities v0 and l0by the Pr. The second effect is the that v0·l0 tended todecrease with Prandtl number. The smaller size of theeddies with increasing Prandtl number was also picked upvisually on the scalogram (e.g., Figs. 22 and 29).

Nusselt number

Several authors have proposed theories indicating thatthe Nusselt number should decrease with Prandtl num-ber according to either the power law NuPr–1/3 orNuPr–2/7 at constant Rayleigh number (Shraiman andSiggia 1990; Zaleski 2000).

We calculated the Nusselt numbers at the top of thebox as:

NuðzÞ ¼ vzTh iðzÞ � @z Th iðzÞ ð12Þ

where the velocity and temperature are averaged overthe horizontal layer. We have found a much weakerpower law dependence of the Nusselt number on Prandtlnumber at constant Rayleigh number in these experi-ments (Table 4, Fig. 34). For Ra = 106:

Nu Pr�0:13 ð13Þ

and for Ra = 108:

Nu Pr�0:05 ð14Þ

TheNusselt number appears to depend onlyweakly onthe Prandtl number, especially at higher Rayleigh num-ber. Thisweaker dependence of theNusselt number on thePrandtl number may be related to the fact that the simu-lations were two-dimensional. The 1/3 and 2/7 Nusseltnumber relationships are based on the wind of turbulencetheory for fluids with Prandtl numbers less than 7 (Zaleski2000). Grossmann and Lohse (2000) predicted that theserelationships would break down forPr >7as the wind ofturbulence decreased. We did see that the wind of turbu-lence decreased as Prandtl number increased (compareFig. 16 with Fig. 19). Therefore, it is not surprising thatwe found aNusselt–Prandtl number relationship differentfrom the 1/3 and 2/7 ones. This Nusselt number rela-tionship cannot be extrapolated up to the Prandtl numberof the Earth’s mantle (1025). The exponent in the Prandtl–Nusselt number relationship should decrease further asPrandtl number increases to around 105 or 106.

Three-dimensional convective effects

The systems discussed in this study are all two-dimen-sional. In 2-D systems, the vorticity vector is always

Fig. 33. Plot of log turbulent Re vs. log Pr

Table 4. Nusselt numbers

Rayleighnumber

Prandtlnumber

Aspectratio

Nusseltnumbera

106 7 3 2.4·105106 17 3 1.7·105106 40 3 1.6·105106 80 3 1.5·105106 100 3 1.7·105108 7 3 9.2·107108 17 3 8.2·107108 40 3 8.8·107108 60 3 6.4·107108 80 3 7.6·107108 100 3 4.6·107

aNusselt numbers were calculated a posteriori after each run

Fig. 34. Plot of log Nu vs. log Pr

27

perpendicular to the flowplane. Thismeans that no vortexstretching can occur in 2-D flows because the vorticitycannot be amplified by the velocity gradient. Vorticity canonly be weakened by viscous dissipation. A fundamentaldifference in the nature of turbulence in 2-D and 3-Dflowsis that energy in large vortices cannot be transported tosmaller scale vortices through vortex stretching; instead,in 2-D, energy at a given scale is transferred to larger scalesand dissipated at the system boundaries (Belmonte et al.2000). Because of the absence of vortex stretching, it ismore likely that fluctuations will be transported down-stream intact in 2-D than in 3-D flow (Belmonte et al.2000). Lateral mixing effects that can remove and intro-duce fluctuations would also beweaker in 2-D than in 3-Dflow (Belmonte et al. 2000).

Three-dimensional effects have important influenceson geophysical systems. The vertical velocity introducedby buoyancy in atmospheric flows induces vortexstretching that is responsible for the spiraling updraftsand downdrafts seen in thunderstorms and dust whirls(Cortese and Balachander 1993). For systems such asEuropa’s global ocean, rotation caused by Coriolis ef-fects may limit how far plumes can spread laterally(Thomson and Delaney 2001). One type of flow that ischaracterized by the formation of cigar-shaped vorticesthrough vortex stretching is known as ABC flow(Dombre et al. 1986; Aref and Brøns 1998). Thestretching ability of these flows leads to their ability toamplify magnetic fields and form dynamos (Gallowayand Frisch 1986). This type of flow has been suggestedas a mechanism for dynamo production in the Sun(Dorch 2000). Vortex stretching flows also are respon-sible for dynamos produced in the outer core of theEarth and the oceans of the Jovian moon Callisto andEuropa (Khurana et al. 1998). We will conduct futurestudies on 3-D finite Prandtl flows to further enhanceour understanding of these important flows in geo-physical systems.

Schmalzl et al. (2002) found that 3-D effects are mostimportant in flows with Prandtl numbers below 3 whenRa = 106 because of toroidal flow, which is neglected in2-D flow. They found two flow regimes. In the regime atPrandtl numbers below 3, rising currents are distortedby toroidal currents leading to distortion of currents andexchange of heat between plumes and the interior. Thisresults in diffusion being the main mechanism of heatflux. As the Prandtl number increased, the toroidal flowdecreased, and most of the heat was transported byplumes through the isothermal interior to the thermalboundary layers.

The simulations of Schmalzl et al. (2002) were done atRa = 106. Rescaling the convection equations by thefree-fall velocity rather than the thermal diffusivity givesa better understanding of the effect of increasing theRayleigh number on the toroidal flow. The free-fall ve-locity is given by:

U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiaDTgd

pð15Þ

where is a the coefficient of thermal expansion. Theconservation of momentum equation non-dimensional-ized by the free-fall velocity is:

@v

@t¼ v x þ T ez �rP þ

ffiffiffiffiffiffiPrRa

rr2v ð16Þ

where x is the vorticity. As the Rayleigh number in-creases, the diffusional term will drop out of the con-servation of momentum equation for Pr/Ra < 10–8. ForPr/Ra < 10–8, the toroidal velocity should not have asignificant effect at high enough Rayleigh number.

Conclusions

We have found that at a given Rayleigh number,increasing the Prandtl number can cause the convectionstyle to change from non-layered to layered. Othereffects of increasing the Prandtl number at a givenRayleigh number include an increase in the number ofplumes, a thinning of the plumes and boundary layers,and an increase in the horizontal shear. The overall sizeof the large eddies tend to decrease with Prandtl number.This is partially responsible for the decrease in Reynoldsnumber with Prandtl number.

As the Rayleigh number and Prandtl number wereincreased, the vorticity fields became more complicated.The wavelet transform helped us interpret these fields byallowing us to separate out the processes occurring atdifferent scales without losing the specific spatial loca-tion of the signals (van den Berg 1999). This allows us tosee the overall large-scale convection style even when itis masked by small-scale structures. The large scalescalogram slices were used to pick out large-scale cir-culation cells and layered structures. The small-scalescalogram slices allowed us to separate out smaller scaleprocesses, such as shear zones along circulation celledges and smaller vortices within larger circulation cells.

The E-max and k-max distributions were used tosynthesize the data in the scalogram (Bergeron et al.2000). The E-max gives the energy for the scale that hasthe maximum energy over a given range of scales. Itallows one to quickly find the most prominent feature inthat scale range. We found it useful for quickly pickingout features, such as shear zones at the smaller scaleswhere there was often signals from many sources. It wasespecially helpful for simplifying vorticity datasets insimulations at higher Rayleigh and Prandtl numberswhere a large number of vortices occurred over a widerange of scales.

The k-max maps out areas of sharp gradients. It wasused to draw a path of fluid movement through the box.The k-max could be used in simulations of crystalmushes to pick out areas of movement where anomalouscompositions might occur in the solidified fluid. Identi-fying areas of possible anomalous compositions is im-portant in the casting of metallic alloys and may also

28

prove useful in modeling flow in crystal mushes inmagma chambers, the inner and outer core boundary,and sea ice (Beckermann 2000).

We have found four major styles of convection in thelong-term solutions for simulations with an aspect-ratioof 3. For Ra = 106 and Pr = 7 to 80, there are fourconvection cells: two wide cells separated by two narrowcells. For Ra = 106 and Pr = 100 and Ra = 108 and Pr= 7 to 17, there are two convection cells with one cellthat is about 1.5 times wider than the other cell. AtPrandtl number 60 there are four narrow convectioncells. A major change in convection style is evident at Pr= 60. The increasing relative viscosity results in moreheat being transported by small-scale thermal plumesthan by large-scale plumes. At Ra = 108 and Pr = 100the four narrow cells divide into two layers of cells, andthe convection style becomes layered. This study indi-cates that the convection style is dependent on both theRayleigh and the Prandtl numbers. We will conductfuture studies in 3-D to help determine the effect of thethird dimension on these flows.

Because increasing the Prandtl number can greatlyincrease the shear in the fluid, the Prandtl number of thefluid may have significant effects on the type of defor-mation seen in a ‘‘lithospheric’’ layer above a convectingfinite Prandtl fluid. For example a large shear force cre-ated in the convecting brine-rich ocean of Europa (Carret al. 1998; Khurana et al. 1998) could explain some ofthe surface deformation seen on this Jovian moon.

We have also found that an increase in Prandtlnumber can have a large influence on both the thicknessof the boundary layers and the overall convection style,the scale at which plumes develop, the number of con-vection cells, and whether or not convection is layered.Wavelets have allowed us to analyze the convectionsimulations at different scales to better understand thesechanges in convection style.

We have found that the Reynolds number tends todecrease with the Prandtl number according to the lawsRePr–1.4 for Ra = 106 and RePr–0.93 for Ra = 108.This decrease in Reynolds number is expected as thePrandtl number increases because of the decreasing ve-locity and length scale along with the increasing vis-cosity.

Even with the significant effects of change in the con-vection style we found only a weak effect of the Prandtlnumber on the Nusselt number compared with the pro-posed 1/3 and2/7 laws (ShraimanandSiggia 1990;Zaleski2000). Grossmann and Lohse (2000) had predicted thebreakdown of these laws as Prandtl became greater than 7and the wind of turbulence, on which the laws are based,decreased. For Ra = 106, we found NuPr–0.13, and forRa = 108, we found NuPr–0.93. As with the Reynoldsnumber, the effect of the Prandtl number is weaker athigher Rayleigh numbers. The Prandtl–Nusselt numberrelationships found here should be valid for fluids at in-termediate Prandtl number. It is expected that the expo-nent in the relationshipwill decrease further as the Prandtlincreases towards 105 or 106.

The weakening of the Reynolds and Nusselt numberlaws matches the behavior seen in the temperature andvorticity fields. At Ra = 106, the lower Prandtl numberflows are dominated by large-scale convection cells withwell defined downwellings and upwellings that travelacross the whole box height. The increasing Prandtleffect significantly changes the convection by increasingthe importance of the small-scale features. Smaller scalethermal plumes that do not travel through the completebox height develop, and the large-scale downwellingsand upwellings are destroyed. At Ra = 108, the smallerscale features are already developed at lower Prandtlnumber, although they do tend to increase with Prandtlnumber.

There is a great need for future work to understandthe behavior of fluids with higher Prandtl numbers. Abetter understanding of heat flux in finite Prandtl num-ber convection will lead to a better understanding of theheat budget of planetary bodies with convecting globaloceans such as the Jovian moon Europa. It will alsoallow a better understanding of heat flux on the earlyEarth and moon during their magma ocean periods. Thestyle of convection in these magma oceans may also beimportant to the chemical differentiation of the earlyEarth and moon.

Acknowledgements We acknowledge Alain P. Vincent, DavidMunger, Ludek Vecsey, and Fabien Dubuffet for their help withthis study. We thank George Bergantz and Frank Spera for theirhelpful comments. Catherine Hier Majumder thanks the NSFFellowship program for providing support during this study. Thisresearch was also supported by the Complex Fluids Program ofDOE.

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