Planetary core dynamics and convective heat transfer scalingy

Transcription

1 Geophysical and Astrophysical Fluid Dynamics, Vol. 101, Nos. 5 6, October December 2007, Planetary core dynamics and convective heat transfer scalingy J. M. AURNOU* Department of Earth and Space Sciences, University of California, Los Angeles, CA (Received 30 November 2006; in final form 8 April 2007) In this article, analysis of a compilation of recent core dynamics models focuses on the properties of non-magnetic rotating convection. Numerical simulations and laboratory experiments are shown to agree qualitatively in cases with similar control parameter values. Small-scale convection columns interact with the spherical shell to generate larger-scale zonal flows. Comparing quantitative results, heat transfer data is compiled from eight recent studies of core convection. The data are plotted in terms of Nusselt and Rayleigh numbers, Nu and Ra, and also in terms of diffusivity-free, modified Nusselt and flux Rayleigh numbers, Nu and Ra Q. Based on the compiled results, it appears that an asymptotic scaling law for heat transfer in planetary core convection models has yet to be determined. Numerical modeling results support a Nu ðra Q Þ0:55 scaling law (equivalently, Nu Ra 1:2 ), which is likely to be affected by the Nu Ra Q onset scaling. In contrast, laboratory experiments, which reach more extreme parameter values, support a Nu ðra Q Þ0:3 scaling law (Nu Ra 0:4 ). Given this significant disagreement, further laboratory and numerical modeling is needed in the fully turbulent, rapidly-rotating regime. Investigation of this regime, especially using realistic low Prandtl number fluids, will require considerable effort. Numerical models will require efficient parallelization and massive computational resources in order to resolve the range of structures that exist in strongly turbulent, rotating flows. Laboratory models will have to minimize the effects of finite thermal conductivity boundaries, which can alter the scaling behavior at high heat transfer rates. Keywords: Core dynamics; Convection; Heat transfer; Planetary dynamos 1. Introduction Planetary dynamos convert the kinetic energy of electrically-conducting core fluid motions into magnetic energy. Because core flows are thought to be driven by thermochemical buoyancy forces, researchers have sought to understand the dynamics * aurnou@ucla.edu. Tel.: Fax: ya contribution in memory of Stephen Zatman. Geophysical and Astrophysical Fluid Dynamics ISSN print/issn online ß 2007 Taylor & Francis DOI: /

2 328 J. M. Aurnou of buoyancy-driven flows and how such convective motions can generate dynamo action. In addition to buoyancy forces, core flows are also strongly affected by Coriolis forces and Lorentz forces. In cases where the Coriolis forces dominate over the Lorentz forces, fluid motions tend to become two-dimensional along the direction of the rotation axis (see figure 1). It is estimated that Coriolis and Lorentz forces tend to be comparable in planetary core settings based on theoretical arguments (e.g. Fearn 1998) and from extrapolations of planetary magnetic field observations (e.g. Stevenson 2003). However, numerical simulations have shown that nearly two-dimensional, axially-aligned motions persist even for strong magnetic fields, with Lorentz forces up to 10 times stronger than Coriolis forces (Olson and Glatzmaier 1995, Busse 2002). Thus, it is reasonable to assume that non-magnetic rotating convection experiments provide insight into the dynamics of planetary core processes. The non-dimensional parameters that describe thermal convection in rapidly rotating spherical shells are defined in table 1. Typical parameter values in present-day models are far from the estimated values for planetary cores. However, it maybe be possible to simulate the convection dynamics without having to replicate the planetary core parameter values. Many physical systems have well-defined, limiting asymptotic behaviors at either large or small values of the control parameters. If models show that a system has such a limiting scaling behavior, then extrapolation to planetary parameter values may be justified. For example, the schematic in figure 2 shows a typical function f that varies with control parameter x. For intermediate values of x, the function f(x) varies strongly with x. Hexagonal symbols mark out a set of hypothetical experimental results for Figure 1. Schematic showing the two basic convective flow structures thought to exist in planetary cores: large-scale axisymmetric zonal flows and small-scale convection columns.

3 this intermediate regime. If a scaling law is fit to these data, it will not correctly model the behavior of the system in the planetary parameter regime. However, if experiments are carried out in the asymptotic regime (square symbols), the scaling law should extrapolate well to the more extreme values that describe planetary conditions. In searching for asymptotic scaling laws for core dynamics, recent work has developed in two main areas: laboratory models and numerical models. These Table 1. Non-dimensional parameters and their estimated values in models and in Earth s core. The upper five rows are input parameters; the lower three rows are output parameters. Kinematic viscosity is ; is angular velocity; D ¼ r o r i is fluid layer thickness; T is thermal expansivity; g o is outer boundary gravity; T is imposed temperature difference; is thermal diffusivity; is magnetic diffusivity; r i and r o are inner and outer sherical shell boundaries, respectively; Q is characteristic heat flux; k is thermal conductivity; and u is characteristic velocity. Parameter Ratio Definition Models Earth s core Ekman Rayleigh Prandtl Magnetic Prandtl Radius ratio Nusselt Reynolds Rossby Viscosity Coriolis Buoyancy Dissipation Viscous diffusion Thermal diffusion Viscous diffusion Magnetic diffusion Inner shell radius Outer shell radius Total heat transfer Conduction Inertia Viscosity Inertia Coriolis Scaling core heat transfer 329 E ¼ =D Ra ¼ T g o TD 3 = Pr ¼ = 0:1 10 0:01 0:1 Pm ¼ = 0: ¼ r i =r o 0:33 0: Nu ¼ QD=kT Re ¼ ud= Ro ¼ u=d Figure 2. Schematic showing behavior of function f(x). For large values of control parameter x, f(x) follows a single asymptotic scaling law, e.g. f x (solid black line). The short-dashed and dot-dashed lines represent extrapolations of two experimental studies into the asymptotic regime. For experiments carried out using smaller control parameter values, the scaling behavior of f varies with x (hexagons). To accurately extrapolate results to the extreme parameter values that befit planetary conditions, experiments must be carried out in the asymptotic regime (squares).

4 330 J. M. Aurnou approaches provide complementary ways of simulating core convection processes. In section 2 of this article, a qualitative comparison is made of flow fields produced in recent core dynamics models. This is followed in section 3 by a quantitative comparison of heat transfer in a number of thermal convection models. This comparison indicates that the scaling law derived from numerical modeling results is not the same as the scaling laws derived from laboratory experiments. Reasons for this discrepancy and comments relevant to future modeling efforts are discussed in section 4. Lastly, section 5 summarizes this study s findings. 2. Comparison of planetary core dynamics models In rapidly rotating flows, the velocity field tends not to vary along the direction of the rotation u! =@z 0. This is called the Taylor Proudman theorem (Tritton 1987) and the resulting flows are called quasigeostrophic. The rigidification of the flow in the axial ^z-direction greatly increases the buoyancy forcing required to drive convection (Chandrasekhar 1961, Dormy et al. 2004). Furthermore, rapidly-rotating convection onsets via small-scale axial columns, shown schematically in figure 1. The azimuthal wavenumber, m, at the onset of convection scales as m E 1=3 and this scaling can persist well into the non-linear regime (Stellmach and Hansen 2004, Gillet and Jones 2006). Figure 3 shows sideview images from two models of planetary core convection. Figure 3(a) shows color contours of axial vorticity, ^z ð r! u! Þ, from a numerical dynamo calculation carried out by M.H. Heimpel, which uses isothermal, mechanically rigid boundaries and radial gravity. The Rayleigh number in this model is ten times the critical value at which convection onsets, Ra 10 Ra C. Figure 3(b) shows shear structures in a rapidly rotating spherical shell of convecting water (Cardin and Olson 1994). In this laboratory experiment, Ra 50 Ra C, the approximately isothermal boundaries are mechanically rigid and centrifugal forces provides the effective gravitational acceleration. Both images verify that narrow, axially-aligned columns are robust features that exist beyond the onset of convection. In addition to small-scale convection columns, large-scale zonal (m ¼ 0) flows are also predicted to develop in planetary core convection. Convection columns in rapidly rotating flows tend to conserve a quantity called potential vorticity (PV). Potential vorticity is comprised here of two main components, local PV perturbations and background PV. Gradients in background PV arise due to variation of axial column height in the spherical shell (e.g. Heimpel and Aurnou 2007). Because the height of axial columns decreases with cylindrical radius at low latitudes, potential vorticity conservation causes convection columns extending outward in cylindrical radius to become tilted in the prograde azimuthal direction. The prograde tilt of the small-scale convection columns generates Reynolds stresses that drive large-scale axially-invariant zonal flows (Zhang 1992). Axially-varying zonal flows called thermal winds can also develop in rapidly-rotating convection. These may dominate flow in the Earth s core at high latitudes inside the axial tangent cylinder that circumscribes

5 Scaling core heat transfer 331 Figure 3. Sideview images of convective structures in core convection models. Rotation axes are oriented towards the top of the page in both images. (a) Color contours of axial vorticity from an unpublished numerical dynamo model from M. Heimpel with E ¼ 10 4, Ra=Ra C ¼ 10, Pr ¼ 1, Pm ¼ 5 and isothermal, rigid boundaries. (b) Shear structures in rapidly rotating convection, imaged using Kalliroscope flakes in water, from the laboratory experiments of Cardin and Olson (1994) having E ¼ 2:5 10 6, Ra=Ra C 50 and Pr ¼ 7. Note that the convective structures in panels (a) and (b) are all closely aligned with the rotation axis. the inner core equator (Olson and Aurnou 1999, Aurnou et al. 2003, Sreenivasan and Jones 2005, 2006). Figure 4 shows equatorial flow patterns from three different models of core flow in the region outside the tangent cylinder. Figure 4(a) shows convective flow structures from a thermal convection experiment carried out in a rotating hemispherical shell of water subject to a parabolic effective gravitational acceleration (Sumita and Olson 2000). Convection columns form near the inner boundary and are then buoyantly-driven cylindrically outwards towards the outer boundary, forming thin, axially-aligned, spiral sheets. The prograde spiralling of the convective sheets occurs because motions in the hemispherical shell conserve potential vorticity. This spiralling produces azimuthal Reynolds stresses in the fluid that can drive large-scale zonal flows. Figure 4(b) shows temperature contours from the quasigeostrophic thermal convection model of Aubert et al. (2003). In this numerical model, it is assumed that fluid motions remains nearly perfectly aligned along the axial direction. Making this quasigeostrophic (QG) approximation, Aubert et al. (2003) solve for the two-dimensional (2D) convective motions in the equatorial plane, while simultaneously requiring the motions to conserve PV in a spherical cavity. The equatorial boundaries are isothermal and mechanically-rigid. Furthermore, the large-scale zonal flows are acted upon by internal viscous stresses as well as parameterized boundary stresses. The case shown here is qualitatively similar to the laboratory experimental results of Sumita and Olson (2000): spiraling sheets of fluid extend from the inner boundary towards the outer boundary.

6 332 J. M. Aurnou Figure 4(c) shows vorticity contours from a QG numerical model of mechanicallydriven shear flow in a spherical cavity (Schaeffer and Cardin 2005). In this 2D model, the outer boundary has a step-wise change in rotation rate at a cylindrical radius of 0:35 r o. This discontinuity in rotation rate produces a shear layer and, equivalently, a step in axial vorticity. For sufficiently large steps in vorticity, this shear layer becomes unstable due to Rossby wave instabilities. Qualitatively similar to thermal Rossby waves that develop in rapidly rotating convection (Busse 2002; Aubert et al. 2003), the spiralling vorticity contours compare well with the thermallydriven rapidly rotating cases shown in the other two panels of this figure. The various experimental and numerical images in figures 3 and 4 show the robust features of rapidly-rotating flows in planetary core geometries: azimuthally-narrow columnar structures co-existing with large-scale zonal flows. Both these types of motions are essential to the generation processes of planetary magnetic fields (Gubbins and Roberts 1987). However, the models we have shown have been carried out at relatively moderate degrees of supercriticality (Ra950 Ra C ). Although the flows in planetary cores are likely to be quasigeostrophic (low E and Ro values), they are far more turbulent than the flows in these models. Furthermore, only a limited number of rotating convection models have been made using realistic, low Pr fluid properties, even though fundamentally different modes can arise in such cases (Zhang 1994, Simitev and Busse 2005). Figure 4. Equatorial flows in core dynamics models, viewed from above. The sense of the rotation is right-handed and out of the plane of the page. (a) Convective flow structures in a hemispherical shell of water from the laboratory experiments of Sumita and Olson (2000) with E ¼ 4:7 10 6, Ra=Ra C ¼ 5:9, Pr ¼ 7. (b) Isotherm contours of convection from the quasigeostrophic numerical model of Aubert et al. (2003), with E ¼ 9:7 10 6, Ra=Ra C 5, Pr ¼ 7. (c) Vorticity contours at the onset of instability in the quasigeostrophic numerical model of mechanically driven, E ¼ 10 6 flow in the split-sphere configuration of Schaeffer and Cardin (2005).

7 3. Comparison of heat transfer in core convection models In the preceding section, the basic flow structures of rotating spherical shell convection have been discussed in terms of recent experimental and numerical models of core flow. These models show that small-scale columns and large-scale zonal flows are both essential features of rapidly rotating convection. However, this only establishes qualitative agreement between the different models. In this section, a quantitative comparison of core convection models focusses on the scaling of total heat transfer. Plots of convective heat transfer (figures 5, 8, 9 and 11) use data taken from eight core convection studies, which seeks to adequately represent recent numerical and laboratory experiments. Data from laboratory experiments are denoted by horizontally-pointed solid triangles. Data from numerical studies are represented by the following symbols: stress-free boundary cases have symbols with yellow centers; cases with rigid boundaries have white centers; rotating convection cases are denoted by circular symbols and dynamo cases are denoted by upward-pointed triangles. All the studies are carried out using spherical or hemispherical shell geometries with Earth-like radius ratios near 0.35, except for the studies of Aurnou and Olson (2001) and Stellmach and Hansen (2004), which use plane-layer configurations. Further information on the studies is given in table Classical Nusselt-Rayleigh number scaling Scaling core heat transfer 333 Thermally-driven convective heat transfer is parameterized by the relationship between the Nusselt and Rayleigh numbers. The Nusselt number is the ratio of the total heat transfer normalized by the conductive heat transfer across the fluid layer. The Rayleigh number is the non-dimensional strength of buoyancy forces in the fluid. At small Ra values, dissipative effects dominate buoyancy effects and, therefore, convective motions can not occur. In this regime Nu ¼ 1 because the heat transfer is due to thermal conduction only. In non-rotating, plane layer convection subject to Figure 5. (a) Nusselt-Rayleigh rotating convection results from the numerical studies Tilgner and Busse (1997), Christensen (2002) and Aubert (2005), and from the experimental studies of Sumita and Olson (2000, 2003). The black line ( ¼ 1:14 0:02) is the best fit to the E ¼ results from Christensen (2002). The red line ( ¼ 0:41 0:02) is the best fit to the Nu 20 results from Sumita and Olson (2003). (b) The same data plotted in terms of ðra Ra C Þ=Ra C. The black line ( ¼ 0:78 0:05) is the best fit to the compiled Nu 2 results from the three numerical studies. The Nu < 1 data from Sumita and Olson (2000) likely arise from measurement error at small temperature differences across the fluid layer.

8 334 J. M. Aurnou isothermal, mechanically rigid boundary conditions, the critical value of the Rayleigh number (where convection onsets) has a fixed value of Ra C ¼ At Ra > Ra C, convective motions contribute to heat transfer across the fluid layer and Nu > 1. In general, convection causes Nu to increase monotonically with Ra and their functional relationship can be expressed as Nu Ra where the value of may vary in different convective regimes (e.g. Spiegel 1971, Kek and Muller 1993). In rapidly-rotating convection, the critical Rayleigh number increases with the rate of rotation. This occurs because increasingly larger buoyancy forces are required to break the Taylor Proudman constraint. In the asymptotic limit of rapid rotation (E! 0), the critical Rayleigh number varies with Ekman number as Ra C E 4=3 (Chandrasekhar 1961, Jones et al. 2000, Dormy et al. 2004, Zhang and Liao 2004). Planetary core convection is also affected by magnetic fields, which may act to lower the critical Rayleigh number in the regime where Lorentz and Coriolis forces are comparable (Chandrasekhar 1961, Eltayeb and Roberts 1970, Fearn 1998). However, we focus here specifically on the effects of rotation. The Rayleigh number in the Earth s liquid outer core is estimated to be greater than (Gubbins 2001, Aurnou et al. 2003) and would plot far off the right edge of figure 5(a). It is not clear in what regime core convection is occurring, in part, because the degree of supercriticality, ðra Ra C Þ=Ra C, is unknown. To better understand this system, it is necessary to have good estimates of heat flow from the Earth s core in conjunction with accurate heat transfer scaling laws. Figure 5(a) shows Nu Ra data from five studies of rotating convection in spherical shell geometries. (The parameter ranges covered in these studies are given in table 2.) The data follow individual Nu Ra curves that each correspond to a particular Ekman number value. The lower the Ekman number, the greater the value of the critical Rayleigh number, Ra C, where the convective motions onset and the Nusselt number departs from unity. For the Nu02 numerical simulation results (circles), ð1þ Table 2. Information on the studies used in figures 5 through 11. The upper four rows are the numerical studies; the lower four rows are the experimental studies. Abreviations are the following. Methods: Num ¼ numerical experiments; Lab ¼ Laboratory experiments. Geometries: SpShl ¼ spherical shell, HemiShl ¼ hemispherical shell. Gravity: Vrtl ¼ vertical, Pblc ¼ parabolic; Cfgl ¼ centrifugal. Colons (:) denote parameter value ranges. The magnetic Prandtl numbers from Aurnou and Olson (2001) and Shew and Lathrop (2005) are shown in parantheses to denote that only non-magnetic, rotating convection data are used from their studies. Study Method Geometry Gravity Fluid E Pr Pm Ra Max(Nu) Tilgner & Busse (1997) Num SpShl Radial 2e-3: 6e : 10 4e3: 8e5 4.1 Christensen (2002) Num SpShl Radial 3e-4: 1e-5 1 3e5: 1e Stellmach & Hansen (2004) Num Planar Vrtl 2e-4: 1e-6 1: 30 1: 30 2e6: 2e Aubert (2005) Num SpShl Radial 1e-4: 1e-5 0.1: 1 0: 10 9e5: 4e Sumita & Olson (2000) Lab HemiShl Pblc H 2 O 5e e7: 8e8 3.8 Aurnou & Olson (2001) Lab Planar Vrtl Ga 4e-2: 7e (2e-6) 3e3: 3e4 1.5 Sumita & Olson (2003) Lab HemiShl Pblc Si-oil 5e e9: 1e Shew & Lathrop (2005) Lab SpShl Cfgl Na 3e-7: 1e (8e-6) 1e8: 2e9 2.1

9 the Nu Ra curves follow similar trends and have scaling exponents which range from 1:08 1:18. On average, 1:14 for these curves. In contrast, the laboratory data (triangles) from Sumita and Olson (2003) with Nu020 increases more slowly with Ra and has a scaling exponent of ¼ 0:41 0:03. This is far from the estimates from the numerical studies and raises the possibility of a different heat transfer relationship at parameter values that have yet to be modeled numerically. It should also be noted that Sumita and Olson s (2003) experiments, carried out using Pr ¼ 14 silicone oil, produce a scaling exponent close to 5/12 as proposed by Ierley et al. (2006) for infinite Pr fluid in the asymptotic high Ra limit. In figure 5(b) the same Nu Ra data is re-plotted using ðra Ra C Þ=Ra C as the abscissa. This change in x-axis scaling compresses the data towards a single curve. The Nu Ra scaling exponent for the compiled Nu 2 numerical data is ¼ 0:78 0:05, which differs from the 1:14 fits for the individual Nu Ra curves. This difference most likely occurs because the data in figure 5(b) does not collapse precisely onto a single curve, due to difficulties in estimating Ra C in the individual studies (Al-Shamali et al. 2004, Dormy et al. 2004) Heat transfer scaling using diffusivity-free parameters Scaling core heat transfer 335 Turbulent flows may not be dependent on the fluid s microscopic diffusivities (e.g. Brito et al. 2004). For instance, convective heat transfer may only weakly depend on the fluid s thermal properties in the regime where convective flow structures first fill the fluid volume but thin thermal boundary layers have yet to form. Another argument, given by Kraichnan (1962), posits that extremely turbulent Rayleigh Be nard convection will destroy the thermal boundary layers and the heat flow will not depend on the local thermal properties of the fluid (see also Spiegel 1971). In this so-called ultimate convection regime, it is predicted that the convective heat transport will scale as ¼ 1=2 (corresponding below to ¼ 1=3). Bounding estimates for fully-developed turbulent Rayleigh Be nard convection predict that ¼ 1=2 is the maximum possible value of the Nu Ra scaling exponent (e.g. Howard 1963, Vitanov 2005). Recent work by Aubert et al. (2001), Christensen (2002) and Christensen and Aubert (2006) have sought to construct scaling laws based on diffusivity-free versions of the control parameters. In this parameterization, the Rayleigh number can be replaced by the modified flux Rayleigh number, Ra Q, and the Nusselt number is replaced by the modified Nusselt number, Nu, given in table 3. The modified flux Rayleigh number is defined such that rotational forces dominate buoyancy forces in the Ra Q 1 regime and buoyancy effects dominate over rotation in the Ra Q 1 regime Table 3. Diffusivity-free non-dimensional parameters and their estimated values in models and in Earth s core. Fluid density is and c p is specific heat capacity. Parameter Definition Models Earth s core Modified flux Rayleigh Modified Nusselt Ra Q ¼ RaNuE3 Pr 2 ¼ T g o Q=c p 3 D Nu ¼ NuE Pr ¼ Q=c pd

10 336 J. M. Aurnou (Christensen and Aubert 2006). The modified flux Rayleigh number does not depend on the fluid s kinematic viscosity or its thermal diffusivity. The modified Nusselt number does not depend on the thermal conductivity of the fluid. In this parameterization, it is implicitly assumed that strongly turbulent, rapidly-rotating convection is not controlled by the boundary layers in which the microscopic diffusivities dominate the physics. Instead, the dynamics are controlled by the turbulent transports in the fluid bulk. Should such arguments prove correct, studies of core convection will be greatly simplified. For example, since the fluid s thermal properties would not affect the asymptotic scaling behavior, it would be possible to implement realistic core convection experiments using high Pr silicone oils instead of more difficult to handle, low Pr liquid metals. To test this parameterization, heat transfer data from recent core convection models is plotted in terms of Nu and Ra Q, yielding best-fit scaling relationships of the form: Nu ðra Q Þ : It can be shown by manipulating the definitions of Nu and Ra Q that and relate to one another as ¼ 1 ð2þ or, equivalently, ¼ 1 þ, ð3þ as plotted in figure 6. The critical values of Nu and Ra Q differ in their behaviors from the critical values of Nu and Ra. Whereas the critical value of the Nusselt number is fixed at Nu ¼ 1, the critical value of the modified Nusselt number varies as Nu ¼ EPr 1. Since the critical Rayleigh number in rapidly rotating convection varies as Ra C E 4=3, the critical value of the modified flux Rayleigh number is found to vary as Ra Q E 5=3 Pr 2. Thus, for fixed Pr, the critical values of modified Nusselt and flux Rayleigh scale as Nu ðra Q Þ0:6, as shown in figure 7. For fixed E, the critical values of modified Nusselt and flux Rayleigh scale as Nu ðra Q Þ0:5. Because the critical values follow similar trends for varying E or Pr, data near the onset of convection will always trend roughly along the solid line shown in figure 7. Thus, it is important not to solve for using subcritical or weakly supercritical Nu Ra Q Figure 6. Plot of equation (3), which relates the Nu Ra scaling exponent and the Nu Ra Q scaling exponent. Short dashed line: ( ¼ 0:41, ¼ 0:29). Long dashed line: ( ¼ 1:22, ¼ 0:55).

11 Scaling core heat transfer 337 data as it will tend to scale with between 0.5 and 0.6, irrespective of the actual scaling behavior of the system in the asymptotic, strongly supercritical regime. Note that in the Earth s core rough estimates of Nu and Ra Q are and 10 13, respectively. Some present models of rotating convection can already reach extremely low Ra Q values that are of order However, most of these models are carried out using parameter values that are all far from planetary values, with E values of order 10 5, Pr values of order 10 and Ra / Ra C values that are typically less than 10. In fitting scaling laws to the experimental and numerical modeling results, only non-transitional, supercritical data are employed. Transitional data correspond to cases made in the complex Ra Q regime where buoyancy and Coriolis effects are comparable (e.g. Aurnou et al. 2007). Instead, data points are only considered in the rapidly-rotating Ra Q 10 2 range and in the buoyancy-dominated Ra Q 102 range. In addition, supercritical data correspond to cases with significant convective heat transfer. This is taken to correspond to Nu 2 in the majority of cases. However, in laboratory experiments using liquid metals, where convective heat transfer can remain small even with vigorous convection, it is assumed that Nu 1:1 cases are supercritical. Figure 8 shows Nu Ra Q results taken from numerical modeling studies. Figure 8(a), which illustrates the basic behavior of this data, shows the E ¼ and results from Christensen (2002). As expected, the lower E data have lower Nu and Ra Q values. The rapidly-rotating, supercritical convection results Figure 7. Critical values of the modified Nusselt and flux Rayleigh numbers: Nu ¼ EPr 1 and Ra Q ¼ c E 5=3 Pr 2 with c taken to be unity. Solid line with open circles: E varying from 10 2 to 10 15, Pr fixed at 0.1. Dashed line with solid squares: E fixed at 10 5, Pr varying from 10 3 to For varying E, onset ¼ 0:6. For varying Pr, onset ¼ 0:5. Thus, near the onset of convection, varying both E and Pr should typically yield onset 0:55 scaling.

12 338 J. M. Aurnou (defined above) line up well and scale as ¼ 0:55 0:01, in agreement with Christensen s (2002) analysis. Overall this ¼ 0:55 scaling fits the supercritical data well. However, the highest Ra Q data points at each E value are trending away from this fit towards lower values. This implies that their may be a different heat transfer relationship for more strongly supercritical cases. Figure 8(b) shows Nu Ra Q results from numerical cases that employ stress-free boundary conditions. The supercritical cases are best fit by a ¼ 0:55 0:01 scaling law. The highest Ra Q cases from Tilgner and Busse (1987) depart from this trend because their Ra Q values are close to 1. Therefore, these cases are unlikely to be in the asymptotic regime of rapidly rotating convection. Figure 8. Modified Nusselt vs. modified flux Rayleigh results from numerical studies. (a) Christensen (2002): E ¼ ð3 10 4, ), ¼ 0.35, Pr ¼ 1 and stress-free boundaries. (b) Stress-free boundary results from rotating convection cases for various E and Pr values. Tilgner and Busse (1997): ¼ 0:40; Stellmach and Hansen (2004): unit-aspect ratio, Cartesian, planar geometry stress-free, isothermal top and bottom boundaries and periodic sidewall conditions. (c) Dynamo cases from Stellmach and Hansen (2004) and the ¼ 0.35, rigid boundary models of Aubert (2005). (d) Compiled results from numerical studies. The solid black line denotes the Nu ¼ 0:09ðRa QÞ 0:550:01 power law fit to the compiled rapidly-rotating, supercritical data (i.e., Nu 2; Ra Q 10 2 ).

13 Scaling core heat transfer 339 Figure 8(c) shows Nu Ra Q results from dynamo cases of Stellmach and Hansen (2004) and Aubert (2005). The best fit scaling law for these cases has ¼ 0:54 0:01. Note again that trends towards lower values with increasing Ra Q for each individual E-value curve. Figure 8(d) is a compilation of all the numerical cases. This compiled numerical data set is best fit by a ¼ 0:55 0:01 scaling law (which corresponds to 1:2) and is in good agreement with the rotating convection study of Christensen (2002) and the dynamo study of Christensen and Aubert (2006). However, the change in trend for the most strongly supercritical cases may indicate that the ¼ 0:55 scaling law has a limited range of applicability. Figure 9 is a Nu Ra Q compilation of the heat transfer data shown in figures 5 and 8. In addition, this plot includes non-magnetic rotating convection data from the liquid metals experiments of Aurnou and Olson (2001) and Shew and Lathrop (2005). As in figure 8, the best fit to the Nu 2, Ra Q <10 2 numerical data yields ¼ 0:55 0:01. However, different -values are found for the laboratory Figure 9. Compilation of Nu Ra Q results from the numerical models shown in Figure 8 and the laboratory models of Sumita and Olson (2000), Aurnou and Olson (2001), Sumita and Olson (2003) and Shew and Lathrop (2005). The solid black line is the ¼ 0:55 0:01 best fit to the Nu 2, Ra Q 10 2 numerical data. The short dashed line is the ¼ 0:29 0:02 best fit to the Ra Q > rapidly-rotating, strongly supercritical results of Sumita and Olson (2003). The long dashed line is the ¼ 0:34 0:02 power law fit to the Ra Q 100 buoyancy-dominated results of Aurnou and Olson (2001).

14 340 J. M. Aurnou experimental data. The slowly rotating Ra Q >1 cases of Aurnou and Olson (2001) are in the buoyancy dominated regime. Their Ra Q >102 cases have a scaling exponent of ¼ 0:34 0:02. The rapidly-rotating, strongly supercritical cases of Sumita and Olson (2003) follow a roughly similar trend and are best fit by ¼ 0:29 0:01. The decreased value of their scaling exponent is closer to the results of Julien et al. (1996) and Liu and Ecke (1997), which both found that convective heat transfer in the supercritical, rotating regime may scale similarly to weakly rotating or non-rotating convection such that 0:29 (corresponding to 0.22). Lastly, note the fundamentally different convective heat transfer scaling in the experiments of Shew and Lathrop (2005), which were carried out using high thermal conductivity, low Pr liquid sodium as the working fluid. Their results highlight the need for further experiments in liquid metals. 4. Discussion The numerical results of figure 8 appear to broadly support the hypothesis that a ¼ 0:55 scaling regime exists in which heat transfer depends predominantly on the quasigeostrophic convective motions, and only weakly on the microscopic diffusivities of the fluid. However, many of the highest Ra data points appear to depart from this trend. If the supercritical data continues to depart from this trend, then the compiled data will become segmented, similarly to figure 5(a). To test this, figure 10 re-plots the results from figure 9 with reduced y-axis Nu ðra Q Þ 0:55. Here, the supercritical Nu results that follow the proposed (Ra Q Þ0:55 scaling will line up horizontally. This occurs in the range Ra Q Considering that Ra Q E3, the region of horizontal alignment represents a relatively limited range of parameter space. Furthermore, in all the numerical data presented here, ðra Ra C Þ=Ra C 9130 and the Reynolds number is for the convective flow field. Such cases may not be in the low Rossby number, high Reynolds number regime that is relevant to planetary core dynamics (Glatzmaier 2002). Figure 10. Reduced Nu* vs. Ra Q results. Data that follows Nu ðra QÞ 0:55 scaling behavior should plot along horizontal lines. Compiled supercritical numerical results appear to follow such a trend in the range Ra Q910 4.

15 Scaling core heat transfer 341 The results of figures 9 and 10 show that further numerical and laboratory core convection experiments must be carried out at ðra Ra C Þ=Ra C values approaching or exceeding 10 3 in order to determine whether the 0:55 scaling of Christensen (2002) or the 0:3 scaling of Sumita and Olson (2003) correctly describe the fullyturbulent, rapidly rotating regime. These extreme cases will be expensive to simulate numerically. The agreement in figure 8(b) of the scaling exponent for the spherical shell calculations and the plane layer calculations of Stellmach and Hansen (2004) suggest that massively parallelized, planar calculations may be an efficient approach. However, the Nusselt numbers are all less than 2 in these calculations, so it is possible that the apparent agreement between plane layer and spherical calculations is due to the -scaling behavior at the onset of convection (figure 7), instead of the scaling in the supercritical regime. Two-dimensional QG calculations are also less expensive than 3D spherical shell calculations. However, QG models are limited to solving the heat equation in the equatorial plane, which may lead to asymptotic scaling behavior that differs from 3D models (Aubert et al. 2003). Laboratory experiments carried out at extreme Ra and Nu values must account for the finite thermal conductivity of the bounding experimental materials. In particular, the resistance to heat transfer through the boundaries must not exceed the resistance to heat transfer across the fluid layer. When the heat transfer resistance of the boundaries exceeds that of the convecting fluid, Nu Ra data trends towards smaller -values. (See Chilla et al. (2004) and Verzicco (2004) for detailed treatments of this problem.) This decrease in is not caused by lowered convective efficiency; rather, it is due to the thermal choke imposed by the resistance of the experimental device. In analogy with Ohm s law, the thermal resistance R is defined via the relationship T ¼ QR, where T is the temperature drop and Q is the heat flux. Using Fourier s law of conduction leads to the relationship R b ¼ D b =k b in solid materials, where D b is the thickness of the boundary material and k b is its thermal conductivity. In a convecting fluid R f ¼ D f =ðk f NuÞ. Thus, the Biot number, Bi, which is the ratio of the boundary material s to the fluid s heat transfer resistance, is Bi ¼ R b ¼ Nu k f R f k b Db D f : ð4þ Ideally, Bi should be less than in order to maintain isothermal boundary conditions (Özis ik 1980, Verzicco 2004). Equation (4) shows that it is difficult to maintain ideal thermal boundary conditions in experiments that use high thermal conductivity liquid metals as their working fluids and in experiments that attain very high Nu values. This may explain the limited Nu variation observed in the liquid sodium experiments of Shew and Lathrop (2005), which employed extremely high thermal conductivity fluid and a mechanically strong but relatively low conductivity titanium alloy outer spherical shell. To keep Bi 1, the thermal conductivity of the boundary material should exceed that of the working fluid and the thickness of the boundary should be kept small. However, Bi will always approach unity when Nu becomes sufficiently large and this will then cause the Nu Ra scaling exponent to decrease in value. For example, rough estimates give that Bi 1 in the Nu 0 20 cases of Sumita and Olson (2003) due to inefficient thermal coupling to the laboratory

16 342 J. M. Aurnou Figure 11. Nusselt Rayleigh data from Sumita and Olson (2000, 2003). The ¼ 0:36 0:09 power law is a fit to all the Nu > 2 data from Sumita and Olson (2000). The ¼ 0:41 0:02 power law is a fit to the Ra > data from Sumita and Olson (2003). Thermistor measurements showed that the interior temperature becomes isothermalized for Ra (vertical dashed line). Here we estimate that the Biot number, Bi, is of order unity for Nu in their device (horizontal dashed line). air that sets the device s outer boundary temperature condition (see figure 11). Although Sumita and Olson s (2003) 0:41 scaling exponent agrees well with the asymptotic bounding estimate of Ierley et al. (2006), our rough estimate of Bi suggests that their highest ðra Ra C Þ=Ra C data may be affected by non-ideal thermal boundary conditions. 5. Summary Numerical and experimental core dynamics simulations have been shown to qualitatively agree in cases with similar control parameter values. To compare core dynamics models quantitatively, heat transfer data have been compiled from recent models of rotating convection. Analysis of this data indicates that numerical experiments and laboratory experiments produce markedly different scaling laws. Furthermore, it is not clear that these data are representative of the asymptotic regime of rapidly-rotating, fully turbulent convection in planetary cores. The compiled heat transfer data set has also been used to compare Nu Ra and Nu Ra Q heat transfer scaling formats. The Nu Ra Q scaling acts to spread the data over many orders of magnitude following a 0:5 to 0.6 onset scaling trend. Although this is visually appealing, in some cases details may be hidden that are more obvious in the Nu Ra format. However, used together, it is possible that these two scaling approaches may be complementary. The Nu Ra format allows one to determine the supercriticality of the convection; the Nu Ra Q format allows one to determine the importance of rotational effects.

17 The results of numerical simulations suggests that a diffusivity-free scaling regime ( 1:22; 0:55) may exist in the regime where convection occurs throughout the fluid volume, but well-defined thermal boundary layers have yet to form. However, once convection becomes strong enough to isothermalize the fluid interior, it is likely that convective heat transfer will become controlled by the physical properties of the thermal boundary layers. Such a behavior is suggested by the results of Sumita and Olson (2003), whose laboratory measurements in the ðra Ra C Þ=Ra C 300, Ra Q 10 7 range follow a heat transfer scaling behavior ( 0:41; 0:29) similar to the buoyancy dominated experiments of Aurnou and Olson (2001). To test the robustness of these scaling exponents in a planetary core-like setting, numerical and laboratory simulations of rapidly rotating convection, rotating magnetoconvection and dynamo action must be carried out in the strongly turbulent regime using realistic, low Prandtl number fluids. Acknowledgements Scaling core heat transfer 343 I wish to thank the NSF Geophysics Programs for research support; Andy Jackson and Jon Mound for constructive reviews; Eric King, Jerome Noir, Peter Olson, Glenn Simpson and Krista Soderlund for fruitful discussions; and Julien Aubert, Woody Shew and Ikuro Sumita for kindly providing me with their data. References Al-Shamali, F.M., Heimpel, M.H. and Aurnou, J.M., Varying the spherical shell geometry in rotating thermal convection, Geophys. Astrophys. Fluid Dyn., 2004, 98, Aubert, J., Steady zonal flows in spherical shell fluid dynamos. J. Fluid Mech., 2005, 542, Aubert, J., Gillet, N. and Cardin, P., Quasigeostrophic models of convection in rotating spherical shells. Geophys. Geochem. Geosys., 2003, 4, doi: /2002GC Aurnou, J.M. and Olson, P.L., Experiments on Rayleigh Benard convection, magnetoconvection and rotating magnetoconvection in liquid gallium, J. Fluid Mech., 2001, 430, Aurnou, J.M., Andreadis, S., Zhu, L. and Olson, P.L., Experiments on convection in Earth s core tangent cylinder. Earth Planet. Sci. Lett., 2003, 212, Aurnou, J.M., Heimpel, M.H. and Wicht, J., The effects of vigorous mixing in a convective model of zonal flow on the Ice Giants. Icarus, 2007, doi: /j.icarus Busse, F.H., Convective flows in rapidly rotating spheres and their dynamo action. Phys. Fluids, 2002, 14, Brito, D., Aurnou, J.M. and Cardin, P., Turbulent viscosity measurements relevant to planetary core-mantle dynamics. Phys. Earth Planet. Int., 2004, 141, 3 8. Cardin, P. and Olson, P.L., Chaotic thermal convection in a rapidly rotating spherical shell: consequences for flow in the outer core. Phys. Earth Planet. Int., 1994, 82, Chandrasekhar, S., Hydrodynamic and Magnetohydronamic Stability, pp , 1961 (Oxford University Press: Oxford). Chilla, F., Rastello, M., Chaumat, S. and Castaing, B., Ultimate regime in Rayleigh Be nard convection: the role of plates. Phys. Fluids, 2004, 16, Christensen, U.R., Zonal flow driven by strongly supercritical convection in rotating spherical shells. J. Fluid Mech., 2002, 470, Christensen, U.R. and Aubert, J., Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields. Geophys. J. Int., 2006, doi: / j x x.

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