Liquid sodium model of geophysical core convection

Physics of the Earth and Planetary Interiors 153 (2005) 136–149

Liquid sodium model of geophysical core convection Woodrow L. Shewa , Daniel P. Lathropb,∗ b

a Laboratoire de Physique, Ecole ´ Normale Sup´erieur, Lyon 69007, France IREAP, Department of Physics, Univerisity of Maryland, College Park, MD 20742, USA

Received 16 November 2004; received in revised form 28 February 2005; accepted 7 March 2005

Abstract Convective motions in Earth’s outer core are responsible for the generation of the geomagnetic field. We present liquid sodium convection experiments in a spherical vessel, designed to model the convective state of planetary cores such as the Earth’s. Heat transfer, azimuthal fluid velocities, and properties of temperature fluctuations were measured for different rotation rates and temperature drops across the convecting sodium. We observed small-scale convective motions with strong, large-scale azimuthal winds and developed turbulence despite the fact that convective heat transport was weak and the temperature profile was close to diffusive. In the context of Earth’s outer core, our observations suggest models which imply a thermal Rayleigh number Ra ≈ 6 × 1023 and a convective velocity near 2 × 10−4 m/s. Also, the energy spectrum of outer core may exhibit important structure down to length and time scales of 1 km and 30 days. Furthermore, we calculate an estimate of Ohmic dissipation, 0.1 TW, in the core based on the shape of experimentally observed power spectra. © 2005 Elsevier B.V. All rights reserved. Keywords: Rotating convection; Earth’s core; Sodium; Turbulent convection; Dynamo

1. Investigating Earth’s outer core Our knowledge of the state of Earth’s outer core has been built from observations, numerical and analytical models, and laboratory experiments. Each of these methods has advantages and shortcomings resulting in a complex, but far from complete answers to questions of dynamics in the outer core: how do fluid velocity, magnetic field, and thermal and compositional transport impact the processes of the geodynamo, mantle, and inner core? Much has been learned through seismology and magnetic field observations. Seismological models yield information about static properties of the core, while we primarily rely on magnetic field observations to reveal ∗ Corresponding author. Tel.: +1 301 405 1594; fax: +1 301 405 1670. E-mail address: [email protected] (D.P. Lathrop) URL: http://www.complex.umd.edu.

0031-9201/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2005.03.013

dynamics. Paleological, historical, and present day magnetic measurements reveal behavior from the enormous scales of pole reversals down to about 10 years and 1000 km. Unfortunately, the vast distance between the outer core and measurements at the surface obscures smaller, but also important, length and times scales. Modern computation has resulted in a new appreciation for the complexity of core dynamics, often closely mimicking the dominant force balances. Yet, realistic treatment of viscous effects remains out of reach. Experiments surpass the performance of numerics in this respect, achieving geophysically realistic turbulence, while falling far short of the full sampling of dynamics available in computations, often limited to a small number of point measurements. Over the past two decades, experiments have progressively pushed towards corelike force balances, producing a rich variety of insights into core dynamics, as well as the possibility of benchmarking increasingly realistic numerical simulations.

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The experiments presented here build upon a history of convection, rotating convection, and magnetoconvection experiments (Aubert et al., 2001; Aurnou and Olson, 2001; Brito et al., 2001; Cardin and Olson, 1992, 1994; Cioni et al., 1997; Carrigan and Busse, 1976, 1974, 1983; Chamberlain and Carrigan, 1986; Cordero and Busse, 1992; Gillet, 2004; Jaletzky, 1999; Manneville and Olson, 1996; Nataf, 2003; Sumita and Olson, 1999, 2002, 2000, 2003). Mimicking the Earth’s outer core, our convection vessel consists of a spherical outer wall and a concentric inner sphere one third the diameter. With a technique pioneered by Carrigan and Busse (1974), and since utilized by most of the above cited efforts, we heat the outer wall and cool the inner sphere while rapidly rotating the vessel giving rise to centrifugally driven thermal buoyancy forces. In addition to driving convection, the rotation induces Coriolis force effects, which are crucial for an accurate model of Earth’s outer core. Simulations have shown that centrifugally driven convection (cylindrically radial in direction) and that produced by gravity in Earth’s core (spherically radial in direction) are quite similar in character outside the polar regions, owing to the influence of Coriolis forces (Glatzmaier and Olson, 1993). Following this introduction, the paper begins with a description of the apparatus and experimental methods. Next we present our results along with explanatory arguments. Then, we summarize our measurements and discuss the implications for understanding the dynamics in Earth’s outer core. The last section contains conclusions and suggestions for extending the presented work with future experiments. 2. Experimental apparatus and methods The experimental system is centered around the convection vessel, which is illustrated in cross-section in Fig. 1. The vessel is spherical with a 60 cm diameter outer wall. The inner sphere is 20 cm diameter, concentric and rigidly attached to the outer wall by a 9 cm diameter shaft. About 110 liters of sodium occupies the annular volume between the inner and outer spheres. The rotation rate of the vessel ranged from 3 to 30 Hz (constant to within 0.75–0.05%). The centrifugal acceleration due to rotation, combined with an imposed temperature gradient between the inner and outer spheres drives thermal convection in the liquid sodium. The total heat transfer through the system ranged from 600 to 4700 W. The outer wall (2.54 cm thick titanium alloy: Ti–6Al–4V) was heated such that an approximate constant heat flux was maintained over the surface. The heat was provided by a stationary array of heat lamps

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Fig. 1. The experimental apparatus.

positioned outside of the vessel. Due to the nature of radiative heat transfer from the lamps to the outer wall, small departures from a constant heat flux boundary condition were due mostly to irregularities in the emissivity of the outer wall, which was painted matte black. The heat was extracted from the system by a cooling fluid which was circulated through the interior of the inner sphere (0.25 cm thick stainless steel wall) to produce an approximately constant temperature boundary at the inner sphere. The cooling fluid, kerosene, was pumped through the inner sphere at constant flow rate (around 20 l/min) and constant temperature (±0.1 ◦ C). For most of the presented results, the control parameters were rotation rate Ω and the temperature difference T between the inner and outer spheres. In terms of dimensionless numbers, the control parameters were the Ekman number E and Rayleigh number Ra, which are defined in Table 1. Also in Table 1, the properties of sodium are compared to estimates for the iron-rich fluid of Earth’s outer core. A typical experiment involved holding the rotation rate fixed while increasing T in steps. The maximum attainable T was limited by the available heating power; the cooling power was always able to match that of the heaters. Once statistical steady state was reached (typical transient of 20–40 min), data was acquired for about 20 min before changing the control parameters

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Table 1 Properties of sodium and experiment compared to conditions in Earth’s outer core Symbol ν κ α D Ω T E Ra Pr a b c d

Quantity (m2 /s)

Kinematic viscosity Thermal diffusivity (m2 /s) Thermal expansion coefficient (K−1 ) Shell gap (m) Rotation rate (rad/s) Temperature drop (K) Ekman (ν/2ΩD2 ) Rayleigh (αTΩ2 D4 /νκ) Prandtl (ν/κ)

Experiment

Earth’s outer core

7.1 × 10−7 (a)

10−6 (b) 5 × 10−6 (c) 10−5 (c) 2 × 106 (d) 7.3 × 10−5 700(b) 10−15 See Section 5.1 0.2

6.8 × 10−5 (a) 2.4 × 10−4 (a) 0.2 18.8–188.4 0.6–20.4 4.6 × 10−7 –5.5 × 10−8 4.2 × 106 –2.8 × 109 0.010

Values are taken from Ohse (1985). Values are taken from de Wijs et al. (1998). Values are taken from Roberts and Glatzmaier (2000). Values are taken from Dziewonski and Anderson (1981).

Table 2 Range of studied control parameters and associated dimensionless numbers Rotation rate (Hz)

T (◦ C)

Rayleigh

Ekman

Elsasser

3 5 10 15 20 25

2.1–19.8 2.9–20.4 1.7–18.9 0.7–18.1 1.1–16.0 0.6–14.7

4.2 × 106 –3.8 × 107 1.6 × 107 –1.1 × 108 3.8 × 107 –4.2 × 108 3.5 × 107 –9.1 × 108 9.8 × 107 –1.4 × 109 8.4 × 107 –2.8 × 109

4.6 × 10−7 2.8 × 10−7 1.4 × 10−7 9.2 × 10−8 6.9 × 10−8 5.5 × 10−8

– – 0–1.9 × 10−4 – – –

The upper limit for T was set by the power of the heater or the cooling capacity of the kerosene heat exchanger, depending on the rotation rate.

again. A few experiments were also conducted with applied magnetic fields, holding rotation rate and temperature drop fixed. In this case, a pair of Helmholtz configured electromagnetic coils produced magnetic fields up to 3 mT. The largest Elsasser number reached was Λ ≡ B2 /ρηΩµo = 1.9 × 10−4 , where B is the applied magnetic field strength, ρ is the fluid density, η is the magnetic diffusivity, and µo is the magnetic permeability. The ranges of investigated control parameters are delineated in Table 2. The observations and results presented here are derived from point measurements of temperature at various locations in the vessel using thermistors or thermocouples. Thermistor probes at two different depths within the outer wall of the vessel allowed measurement of heat flux. Two closely spaced temperature probes protruding slightly from the inner sphere surface near the equatorial plane were used to estimate average azimuthal fluid velocity. Another probe located about halfway between the inner sphere and outer sphere provided observations of temperature far from the vessel boundaries. 3. Experimental results Considered collectively, the velocity and temperature measurements of these experiments suggest a general

picture of the dynamics, which we describe here before delving into the supporting details. The velocity field seems to be well described by two parts: a large scale retrograde azimuthal wind and a smaller scale turbulent velocity responsible for the convective heat transfer. The scale of the azimuthal flow is characterized by Uφ ∼ ΩDαT , which suggests a balance between Coriolis and buoyancy forces (or possibly the presence of thermal winds). The smaller scale √ convective velocities seem to scale as Uc ∼ ΩD αT , reminiscent of a ballistic velocity estimate. The velocity field is most active near the inner sphere and seems to be characterized by a persistent length scale δ ∼ E1/3 D. A likely scenario is that buoyancy drives outward radial motion near the inner sphere where heat flux per unit area is greatest and these radial motions are deflected by Coriolis forces into an azimuthal flow. The flow is quite turbulent; the Reynolds number Re ≡ UD/ν based on the typical azimuthal flow is around 40,000. In spite of the turbulence, the heat transfer is still not more than twice that due to conduction. From temperature measurements at three different radii, we estimate the mean radial temperature profile is within 20% of the conductive profile throughout our measurements. The high thermal conductivity of liquid metals results in a state that is never far from diffusive. The turbulence of the velocity field cre-

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Fig. 2. Typical temperature time series are shown for rotations rates of 3 Hz (top), 15 Hz (middle), and 25 Hz (bottom), each with a low T (left) and a high T (right) example. These time series were acquired from a probe 3 cm above the equator and 0.25 cm away from the surface of the inner sphere.

ates fluctuations in temperature, which are swept around the vessel by the azimuthal flow. It is analysis of these temperature fluctuations that provide the majority of our results. 3.1. Temperature standard deviation Measurements of temperature in these experiments are the backbone of most of our results. The time series shown in Fig. 2 were measured 3 cm above the equator and 0.25 cm away from the surface of the inner sphere. As is evident in Fig. 2 the size of temperature fluctuations increase at larger T and decrease at higher Ω. The standard deviation of temperature fluctuations for many different rotation rates and temperature drops are shown in Fig. 3a. The trends suggested by the sample time series in Fig. 2 are clear. A simple argument based on advection of the temperature profile by convective velocity structures may be used to explain the observed behavior in standard deviation of temperature. Consider velocity structures in the flow of size δ ∼ E1/3 D. This scaling for convective vortex size is suggested by numerical and analytical predictions of flow structure near the onset of convection (Busse, 1970; Gillet, 2004; Roberts, 1968; Zhang and Schubert, 2000). If the velocity structures are

Fig. 3. (a) Standard deviation of temperature is plotted as a function of temperature drop for a range of rotation rates. Evidence for standard deviation scaling as E1/3 T is shown in (b) along with a linear fit with slope 4.0.

mixing a fraction δ/D of the full temperature drop T , as one might expect for a largely diffusive temperature profile, then the magnitude of temperature fluctuations would be σT ∼

δ T, D

σT ∼ E1/3 T.

(1) (2)

This scaling is borne out clearly in Fig. 3b. The dashed line in Fig. 3b shows the model σT = 4.0E1/3 T, with the coefficient determined by least squares.

(3)

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Fig. 4. The data in the probability density function are taken from many time series with different T , but all with rotation rate of 15 Hz. The source time series were all scaled by their standard deviation.

3.2. Temperature probability density functions A typical probability density function is shown in Fig. 4. The PDF was constructed from time series acquired close to the inner sphere equator. Although close to Gaussian, the PDF is slightly skewed towards cold events and slightly super-Gaussian in the positive tail. In general, it seems that low amplitude cooler-than-mean events are most likely, while extreme warm events are more likely than extreme cold events. Exponential tails have been observed in highly super-critical Rayleigh-B´enard convection in both stationary and rotating experiments (Hart et al., 2002; Liu and Ecke, 1997; Wu and Libchaber, 1992).

Fig. 5. (a) Unscaled azimuthal velocity for different rotation rates and temperature drops close to the inner sphere equator. (b) Rossby number based on azimuthal flow velocity is shown for increasing dimensionless temperature drop αTe , where Te ≡ T − Tc − Tadiabatic . A linear fit (dashed line) has a slope 3.5 and implies that Uφ ∼ ΩDαTe .

3.3. Azimuthal velocity Estimates of azimuthal velocities were obtained from temperature time series acquired from two probes separated azimuthally by 0.63 cm. The time series from the two probes appear very similar with one delayed in time with respect to the other on average, reflecting the underlying azimuthal velocity which is sweeping the temperature field past the probes. The separation distance between the probes is divided by the average delay time between the signals to obtain the magnitude of the azimuthal flow. The resulting velocities were always retrograde with respect to rotation, which is consistent with other experiments (Aubert et al., 2001; Gillet, 2004) and with the observed westward drift of magnetic field patterns originating in Earth’s outer core. Velocities for different rotation rates are plotted against T in Fig. 5a. Due to time resolution limitations,

the method of obtaining velocity measurements fails for large T and large rotation rates. These points are excluded from the presented data. Furthermore, data is only shown for values of T greater than the predicted onset of centrifugal convection (computed by Gillet, 2004). The velocity data may be presented in a dimensionless form using the following scalings. Velocity is normalized by DΩ (i.e. Rossby number) and the temperature drop T is replaced by αTe . The critical temperature drop Tc and the temperature drop due to compression effects Tadiabatic are subtracted from the full T to form Te ≡ T − Tc − Tadiabatic , which represents the temperature drop available to drive convection. The velocity, scaled in this way, is shown in Fig. 5b. The error bars shown in Fig. 5b come from uncertainty in time resolution (±0.01 s) and the combined uncertainty in temperature for T, Tc , and Tadiabatic (±0.2 ◦ C).

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There are several possible explanations for a velocity that scales as v ∼ DΩαT . The simplest idea stems from a balance between the buoyancy force and the Coriolis force in the Navier–Stokes equation, Ω2 rαT ∼ 2Ω × v,

(4)

where r is the perpendicular distance from the rotation axis. A qualitative justification for such a force balance comes from considering the motion of a convective plume under the influence of Coriolis forces. Buoyancy forces drive cold plumes away from the inner sphere. The initial radial trajectory of the plume is deflected by the Coriolis force into an azimuthal direction, which suggests azimuthal velocities experience the competition expressed in Eq. (4). An estimate of the magnitude of the values in Eq. (4) yields, Ω2 DαT ∼ ΩU.

(5)

One may then solve for the Rossby number to obtain a scaling that is consistent with that used in Fig. 5b, Ro =

U ∼ αT. ΩD

(6)

Thermal winds are another mechanism which may be responsible for the azimuthal flows we observe. Thermal winds are driven by the misalignment of the centrifugal acceleration with the temperature gradient. The approximate azimuthal flow due to thermal winds is vφ ≈ ΩDαT,

(7)

the same scaling as derived above. Aubert et al. (2001) have conducted tests in a similar geometry with a reversed temperature gradient (stable to convection) and report that the resulting flows are much smaller in magnitude than those observed in the presence of convection. Those results suggest that the azimuthal flows we observe are not generated by thermal winds, although our system was not capable of performing the same reverseT test. One should be cautious extending the our results to the Earth’s core. In the reasoning leading to Eqs. (6) and (7), two factors of Ω come from the centrifugally driven buoyancy force. For the Earth, the gravitationally driven buoyancy force is practically independent of rotation rate. In order to compare to Earth’s core, the scalings above must be put into a form using dimensionless numbers, whose definitions naturally account for this difference: κ vφ = 3.5 RaE. (8) D

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3.3.1. Comparison to previous azimuthal flow results Several previous research efforts have studied azimuthal flow in comparable systems to our own. In general their results have suggested velocities which increase less dramatically with T and Ω than the trends we presented above. Aubert et al. (2001) made extensive experimental measurements of convection and developed a scaling theory for typical vortex size, temperature fluctuations, as well as radial and azimuthal velocities. Extending work of Cardin and Olson (1994), they used the vertically averaged vorticity equation assuming Coriolis, buoyancy and inertial effects are in balance to derive vφ ∼ (RaPr −2 Nu)4/5 E9/10 . They found agreement between their experimental measurements and this scaling provided the approximation be made that the Nusselt number remained constant. To compare with our data where Ro ∼ αT (Fig. 5), we rewrite their scaling law in terms of Rossby number, T , and Ω with Nusselt dependence removed (i.e. constant Nu); their scaling law becomes Ro ∼ T 0.8 Ω−0.3 . As discussed in Section 3.4, we found that convective heat flux increases like Ω1/3 T , in other words, Nusselt number is not constant, rather Nu ≈ Ω1/3 + 1. If one uses this Nusselt number dependence Aubert’s scaling law becomes vφ ∼ T 0.8 Ω0.7 (Ω1/3 + 1)4/5 , which is a closer match to our observations, but still somewhat underpredicts the rate of increase of vφ with the control parameters. Christensen (2002) did numerical simulations of convection in spherical shells with free-slip boundaries and Pr = 1. He found that as Ekman number is decreased (E ≥ 10−5 ) the Rossby number based on zonal flow appeared to asymptotically approach the scaling Ro = 0.19(RaPr −1 E2 )2/5 . Again we may rewrite this in terms of our control parameters, Ro ∼ T 0.4 , and see the contrast with our results. Manneville and Olson (1996) measured zonal flows in a very similar experimental device using water for the convecting fluid. They observed azimuthal fluid velocities which increase with Rayleigh number, but reach a plateau for the highest values of Ra. The high T data in our results may indicate similar behavior, but higher Rayleigh numbers and a different measurement technique for velocities would be required to fully explore this hypothesis in our experiments. They also found complicated banded structures in the azimuthal flow which depended upon latitude and Rayleigh number. Aurnou and Olson (2001) did convection experiments in spherical shells, but focusing on the polar region within the tangent cylinder. They found significant zonal flows whose magnitude scales as vφ ∼ 2(B/2Ω)1/2 , where B = αgTNuκ/D, and g is the gravitational accelera-

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tion. This result is difficult to compare to ours since the centrifugal acceleration which drives our convection depends upon Ω and is perpendicular to the rotation axis, unlike g in the system studied by Aurnou et al. (2003). 3.4. Heat transfer Liquid sodium provides a lower Prandtl number than fluids used in previous convection experiments in similar systems (sodium: Pr ≈ 0.01, gallium: Pr ≈ 0.025, water: Pr ≈ 7, silicon oil: Pr ≈ 14). For two otherwise identical systems with low and high Prandtl numbers, the Nusselt number will always be larger for the high Pr system. In spite of the lower convective heat flux for low Pr systems, the flow is often quite turbulent even very close to onset due the relatively low viscosity. In our experiments as well as two different gallium convection experiments turbulence was observed very close to onset with a Nusselt number never exceeding 2 (Aubert et al., 2001; Aurnou et al., 2003). This contrasts sharply with water or silicon oil experiments which often exhibit very high Nusselt numbers. For example, Sumita and Olson (2003) have obtained Nusselt numbers as high as 186 using silicon oil. We measured heat transfer using two temperature probes embedded at different depths in the outer wall of the vessel at a position halfway between the equator and the top of the vessel. The data for multiple rotation rates and T values are shown in Fig. 6a. While some evidence for latitude dependent heat transfer is discussed in the upcoming section on measurements with applied magnetic field, detailed measurements have not yet been made to quantify this. We attempt to systematize our observations using a calculation inspired by a theory primarily developed by Grossmann, and Lohse for Rayleigh-B´enard convection (Grossmann and Lohse, 2000; Shraiman and Siggia, 1990). The basic idea is to relate global dissipation to an estimate of local dissipation in order to estimate Nu(Ra, E, Pr). By taking a volume average of the kinetic energy equation one may obtain the exact relation for the global viscous dissipation, global =

ν3 (Nu − 1)RaPr −2 , D4

(9)

(see Shew, 2004 for a detailed derivation). Consider also the component of dissipation associated with local convective velocity structures of spatial size δ ∼ E1/3 D and typical velocity U, local ≡ ν(∇v)2

(10)

Fig. 6. (a) The total heat flux is plotted for different rotation rates and a range of temperature drops. The dotted line represents the heat that would be conducted if the sodium were stationary. Note that the total flux is not much greater than the conductive flux – a feature of low Pr fluids. (b) The convective heat flux is plotted against TΩ1/3 .

local ∼ ν

U2 δ δ2 D

(11)

ν3 Re2 , (12) D4 E1/3 where the factor of δ/D is introduced to account for the fact that the region in which the dissipation occurs is a fraction of the whole volume. Another way to interpret the local estimate is that the dissipation is dominated by laminar viscous boundary layers of thickness E1/3 D. Aubert et al. (2001) have argued that dissipation should be dominated by boundary layers. Plaut and Busse (2002) have observed in numerical simulations the presence of thermally modified Stewartson layers on the vertical boundaries of cylindrical centrifugal convection with thickness that scales like E1/3 . Whether the dissipation is dominated by wall boundary layers or small scale local ∼

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free convective structures, we may equate the global and local dissipations using the assumption global = local + other , where local other .

(13)

Then we have (Nu − 1)RaPr −2 ∼ Re2 E−1/3 .

(14)

One√may estimate a convective velocity scale U ∼ ΩD αT by calculating the final velocity of a fluid parcel that is T colder than the neighboring fluid and subject to an acceleration Ω2 D over a distance D. Using this ballistic velocity estimate and putting the dimensionless numbers in terms of our control parameters Ω and T , we obtain, (Nu − 1)T ∼ TΩ1/3 .

(15)

The part of the heat transfer due only to convection is proportional to (Nu − 1)T . Our measured convective heat flux is shown in Fig. 6b for increasing TΩ1/3 . 3.5. Temperature power spectra Both the heat flux and standard deviation observations were explained with arguments using the predicted length scale for vortices at the onset of convection (δ ∼ E1/3 D). This begs the question: do we observe any periodicity associated with vortices close to onset? The time series in Fig. 2 suggests that this is not the case. We may be certain by examining power spectra of the temperature fluctuations Fig. 7. It is clear in the power spectra shown in Fig. 8 that the temperature signal is not periodic, showing broad band fluctuations with interesting structure down to the lowest rotation rates and T values. Several features are common to all the temperature spectra that we measured. Most noticeable is the knee marking a sharp change in slope towards high frequency. At frequencies above this knee, the spectra exhibit a behavior close to S(f ) ∼ f −5.7 . Significantly, we find that the knee frequency at which the steep slope begins obeys √ fc = 2.0Ω αT . (16) Although difficult to determine systematically and precisely, the knee was manually identified for each spectrum. Plotted in Fig. 8, these data are well described by Eq. (16). The estimated uncertainty in determining fc is indicated with one exemplary error bar in Fig. 8a. The frequencies are scaled by the rotation rate and compared to square root of T (dashed line) in Fig. 8b. Two-dimensional numerical simulations by Tran and Bowman suggest that the knee in the spectra marks the primary frequency at which energy is injected into the

Fig. 7. Temperature power spectra are shown for (a) three different T at one rotation rate and (b) three rotation rates at a constant T .

flow (Tran and Bowman, 2003, 2004). In other words, the knee frequency is associated with convective motions. √ The fact that the knee frequency behaves as f ∝ Ω t strengthens the arguments in the last section where we used the ballistic√estimate for the convective motions, Uconvective ∼ ΩD αT . At frequencies lower than the knee, we suggest that the temperature fluctuations are due to an inverse cascade of energy from the scale of convective motions up to the size of the container. The Coriolis forces in our experiments likely cause the flow to be nearly two-dimensional outside of viscous boundary layers. The inverse energy cascade is a commonly invoked view of dynamics in two-dimensional turbulence (Batchelor, 1969; Kraichnan, 1967; Tung and Orlando, 2003). Small scale vortices are continuously generated by buoyant forces and merge with one another to form larger and larger vortices. The energy which is not dissipated at the scale of injection, cascades to the largest scales where it is dissipated in viscous boundary layers.

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Fig. 9. The relative shift in the total heat flux, equatorial heat flux, and 45 ◦ latitude heat flux are shown for increasing applied magnetic field. The data is taken at a rotation rate of 10 Hz and T of 6.9 ◦ C.

Fig. 8. Temperature power spectra show a distinct transition to behavior with f −5.7 . The knee frequency fc , marking the transition was identified manually and plotted in (a). The knee frequency is shown scaled by rotation rate and compared to (T )1/2 (the √ dashed line) in (b); the observations are consistent with fc = 2.0Ω αT .

3.6. Magnetic field effects Due to time limitations and the unusual difficulty of these experiments, only a few measurements were obtained with applied magnetic fields. The resulting change in dynamics manifested in altered heat flux. The total heat flux required to maintain a given T was observed to drop with increased magnetic field. Furthermore, the spatial distribution of heat transfer changed. Heat flux increased near the equator and decreased at a polar angle of 45◦ . These results are summarized in Fig. 9 where relative shifts are shown for total heat flux, equatorial heat flux, and 45◦ latitude heat flux. We note that the experimental methods used to acquire equatorial and total heat flux measurements was affected slightly by rotation rate and coolant flow rate. For this reason, these measurements are absent in the discussions of heat transfer

above (Section 3.4) where a range of T and Ω are explored. In Fig. 9, we present relative shifts due to changes in magnetic field with Ω and coolant flow rate held constant. Magnetic fields can have a similar effect as rotation on fluid flow when the fluid is electrically conducting. The magnetic field suppresses gradients in the velocity field which are in the same direction as the magnetic field; the flow tends toward two-dimensional (Chandrasekhar, 1961, e.g.). Our observations of changes in heat flux with applied magnetic field are consistent with this idea. If fluid motion is confined to planes perpendicular to the rotation axis then heat flux will be concentrated near the equator. Furthermore, the sloping boundaries of the spherical vessel combined with the tendency towards 2D flow amounts to an overall suppression of flow, consistent with the decrease in global heat flux. One must exercise caution when comparing these results to the Earth or with numerical simulations. Unlike typical numerical models of linear magnetoconvection (Sakuraba, 2002; Jones et al., 2003, e.g.), the fluid motion in our experiments is highly turbulent. Unlike the Earth, on the other hand, our Elsasser number was about 10−4 ; the Elsasser number for Earth is likely order unity. In the case of Earth, contrary to our observations, magnetic fields may actually facilitate convection (Roberts and Glatzmaier, 2000, e.g.). While the Elsasser number in our experiments is small, the interaction parameter, N = B2 D/ρµo ηvφ , is of order unity. With N ∼ 1 and Λ  1, the Lorentz forces may in fact modify the flow, albeit less so than Coriolis forces. The interaction parameter is important in understanding magnetic field effects on turbulence at small scales, such as in boundary layers

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(Sisan et al., 2003, 2004; Davidson, 2004, e.g.). Our observations of changes in heat flux with applied magnetic field underline the dynamical importance of the interaction parameter. One should note that for Earth’s core, N may be greater than 100.

14 times critical, while the Nusselt number remained less than 2. With the ballistic estimate for convective velocities and convective length scales δ ∼ E1/3 D, an argument based on dissipation was used to derive a relationship for the heat transport due only to convection Φconv . We found

4. Summary of results

Φconv ∝ Ω1/3 T,

The convection experiments described in this paper were designed to model aspects of Earth’s outer core. We observed convective motions with strong azimuthal winds and developed turbulence despite the fact that convective heat transport was weak and the temperature profile was close to diffusive. In this section, we summarize the quantitative results and use them to make some simple predictions for the unobserved dynamics in the outer core. Our observations suggest a model for fluid velocities characterized by two scales: a large scale azimuthal flow and a smaller scale turbulent convective flow. The azimuthal flow was measured near the equator of the inner sphere and for all parameter values was retrograde with respect to rotation. Consistent with a balance between Coriolis and buoyancy forces, the magnitude of the azimuthal velocity was found to be,

with a dimensional proportionality coefficient of 61 W s1/3 ◦ C−1 . Power spectra of the temperature fluctuations exhibited a characteristic shape with a shallow slope at lower frequencies and a transition to a steeper slope at a well defined knee frequency. The knee frequency may be interpreted as the frequency at which convective motions inject energy into the flow. The knee frequency fc was found to be consistent with √ fc = 2.0Ω αT . (21)

Uφ = 3.5ΩDαT.

(17)

The Reynolds number based on this azimuthal flow ranged from 103 to 104 . The smaller scale convective velocities act to weakly mix the temperature gradient, resulting in a radial temperature profile which deviates from the conductive profile likely by less than 20%. Broad band power spectra and nearly Gaussian probability density functions of resulting temperature fluctuations indicate that the convective motions are turbulent. The temperature fluctuations are swept by the azimuthal flow past temperature probes, which are fixed in the rotating frame. Measurements of these fluctuations suggest that the typical length scale for the convective velocities varies as E1/3 D. This results in temperature signals with standard deviation, σT = 4.0E1/3 T.

(18)

Heat flux and the power spectra of temperature fluctuations suggest the small scale convective velocities are consistent with the ballistic estimate, √ Uconv ∼ ΩD αT . (19) Heat flux was found to rise with increasing rotation rate. At the highest rotation rates (E = 5.5 × 10−8 ) and largest temperature drops from the inner to outer sphere (Ra = 2.0 × 109 ), the Rayleigh number was about

(20)

At frequencies smaller than fc , power law slopes of the spectra range between 0 and −1.7. Above fc a slope of −5.7 ± 0.05 persists for all of the parameters we explored. We suppose that dissipation is dominated by the motions associated with the knee frequency and the remaining energy cascades to larger scales through the lower frequencies and to smaller scales through the higher frequencies. 5. Predictions for Earth’s outer core The experiments described here resemble Earth’s outer core in several ways. Both have turbulent velocity fields. This is revealed by the temperature fluctuations of the experiment and the broadband secular variation of Earth’s magnetic field. Both the experiment and Earth’s core have modest Nusselt numbers. Convective heat transfer in the experiment was never larger than the conductive heat transfer. Similarly, in Earth’s core, the heat conducted down the adiabat is thought to be responsible for most of the heat flux out of the core (Stacey and Loper, 1984, e.g.). Both systems appear to have large scale retrograde azimuthal flows, although different in magnitude. The Rossby number based on azimuthal flow measurements for the experiment was between 10−2 and 10−3 , while the Earth’s Rossby number based on the westward drift of the secular variation is of order 10−6 . (It is perhaps interesting to note that the Rossby number based on zonal flows on Jupiter are of order 10−2 .) The most important difference between the experiment and the Earth is likely the presence of a large magnetic field in Earth’s core, the geodynamo. The Elsasser number for Earth’s core is estimated near unity. During the few experiments with applied magnetic

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fields, our Elsasser number was only as large as 10−4 . A second, less important difference lies in the buoyancy force; Earth’s gravitation has spherical symmetry, while the centrifugal forces in the experiment have cylindrical symmetry. For the sake of better understanding, let us set aside the differences between the core and the experiment and explore the implications of our results for the Earth. 5.1. Azimuthal flow and core Rayleigh numbers Let us first consider the azimuthal flow. The expression for azimuthal velocity obtained in the experiment is κ Uφ = 3.5 RaE. (22) D Here and many of the arguments to follow, we rewrite the models from the experiment in terms of dimensionless numbers with hopes that differences between the experiment and the core will be naturally accounted for by the altered definitions of the dimensionless numbers. The westward drift of the secular variation is often associated with an azimuthal velocity of the liquid motion in the outer core; it is about 5 × 10−4 m/s (Roberts and Glatzmaier, 2000). If this azimuthal flow is caused by the same mechanisms responsible for that in our experiment, then this implies that the Rayleigh number in the Earth’s core is thermal Racore =

D (5 × 10−4 m/s)E−1 . 3.5κ

(23)

We use the estimates, E ∼ 10−15 (with ν ≈ 1 × 10−6 m2 /s (de Wijs et al., 1998), κ ≈ 5 × 10−6 m2 /s (Roberts and Glatzmaier, 2000) and the shell gap for the outer core, D ≈ 2 × 106 m to obtain thermal Racore = 6 × 1022 .

(24)

One might guess that compositional convection is governed by similar physics. If Earth’s azimuthal flow is driven by compositional convection, we may use a very similar argument to estimate the compositional Rayleigh number. In this case we replace κ with the compositional diffusion coefficient Dξ ≈ 10−9 m2 /s (Roberts and Glatzmaier, 2000) and arrive at, compositional Racore ∼ 1025 .

(25)

These Ra estimates are significantly lower than those made by Gubbins using an argument based on entropy (Gubbins, 2001). He estimated Rathermal ∼ 1029 and Racompositional ∼ 1038 .

5.2. Convective flow velocity We may also derive an estimate for the turbulent convective core velocities from our model of the convective heat flux. In the discussion of results above, we compared our observations to (Nu − 1)RaPr −2 ∼ Re2 E−1/3 .

(26)

where the Reynolds number is based on convective velocities. The Prandtl number is about 0.5, the Nusselt number is likely 1 < Nu < 2 (we take Nu = 2), and we use our estimate of Ra ≈ 6 × 1022 from above. Then Earth’s Reynolds number would be Re ≈ 4 × 108 . And since Re ≡ Uconv D/ν we can estimate the convective velocities in Earth’s outer core, Uconv ≈ 2 × 10−4 m/s,

(27)

which is the same magnitude as the azimuthal flow, estimated from the westward drift. One may also use this convective velocity to construct a magnetic Reynolds number Rm for the core. Uconv D (28) ≈ 2 × 102 , Rm ≡ η where the magnetic diffusivity η of the outer core is taken to be approximately η ≈ 2 m2 /s. 5.3. Time scales, length scales, and Ohmic dissipation Temperature power spectra from our experiments exhibit a dramatic drop in power law slope at a distinct time scale determined by the ballistic or convective frequency, √ fc = 2.0Ω αT . This is likely associated with the injection of energy into the flow by convective motions. There should exist a similar convective frequency in the outer core. The convective frequency in the Earth’s core can be estimated by
νκ fc ∼ 2.0 Ra. (29) D4 If we again use our previous estimate of Ra ≈ 6 × 1022 , we obtain a frequency of fc = 4 × 10−7 1/s. This yields a characteristic time for convective motions of τc ≈ 2.5 × 106 s ∼ 30 days.

(30)

If the frequency of fluctuations is due primarily to advection of a spatial structures in the temperature field, then one may estimate a length scale associated with the convective frequency using the westward drift velocity: δc ∼ τc Udrift ≈ 1 km.

(31)

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Further implications of the spectral behavior lie in the energy budget of the outer core. The minimum power required to drive the dynamo may be estimated by the Ohmic dissipation of the electric currents associated with the geomagnetic field. We assume, for the sake of this estimate, that the Earth’s magnetic field spectrum is similar to the temperature spectra we observe in our experiments. This implies significant power dissipation due to the magnetic field at length scales as small as 1 km. Ohmic dissipation in the core has been estimated by several different methods in the past few years. Anderson (2002) used arguments based on the necessary power to drive mantle convection to estimate 8 TW . Buffett (2002) used results from a numerical model to estimate 0.1–0.5 TW. Roberts et al. (2003) have estimated 1– 2 TW based on spectra from a different numerical model. Christensen and Tilgner (2004) developed a scaling theory from numerical and experimental data which predicts 0.2–0.5 TW. Let us attempt an estimate of our own with the speculative assumption that the shape of Earth’s magnetic field power spectrum is similar to the temperature power spectra which we observe in our experiments. The form of the magnetic power spectrum EB (k), where k is wavenumber, determines the Ohmic dissipation QΩ . One may estimate  ∞ η QΩ = V EB (k)k2 dk, (32) µo 0 where η is the magnetic diffusivity, µo is the vacuum magnetic permeability, and V is the volume of the core. A reasonable approximation of the spectra observed in our experiments may be expressed  for 0 ≤ k ≤ ko   A,    −B k EB (k) ≈ A (33) , for ko ≤ k ≤ kc  ko    0, for kc ≤ k. Here we have used Taylor’s hypothesis to estimate that the wavenumber and frequency spectra have a comparable form. In order to compare to Earth’s core, the flat part of the spectrum below ko may be matched with the observable part of the spectrum from the Earth’s magnetic field. Roberts et al. (2003) compute 6 MW of dissipation due to the observable part of Earth’s magnetic field alone. If we assume that the flat part of the spectrum accounts for this power, then A is of order unity. And B is the power law slope of the spectrum for the inverse-cascade range of wavenumbers between ko and the injection scale kc . Our measurements suggest 1 < B < 2. In Eq. (33),

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we assume that the contribution to Ohmic dissipation by structures with wavenumber above kc is negligible. Then the Ohmic dissipation is simply   η ko 2 η kc B 2−B QΩ = Ak dk + Ako k dk (34) µ 0 µ ko  3  k η kB k3−B ko3 − (35) QΩ = A o + o c µ 3 3−B 3−B If we now use the estimated length scale from above to approximate kc ∼ 1/δc = 10−3 m−1 , A ∼ 1, and assume ko ∼ 10−6 m−1 , then the Ohmic dissipation estimate for B = 1.5 is QΩ ≈ 0.1 TW.

(36)

One should note that this estimate is quite sensitive to the value of B, ranging from 10 GW (B = 2) to 10 TW (B = 1) and also neglects any contribution due to toroidal magnetic field in the Earth’s core. Although these estimates are not very restrictive in terms of the energy budget of the Earth’s core, it is perhaps significant that the shape of power spectra we observe experimentally and consequent Ohmic dissipation estimates are consistent with currently accepted energy budgets for the core. These estimates also highlight the importance of understanding both the spectral form and any relevant crossover length scales. In addition to dissipation estimates, our spectral measurements have implications for the history and evolution of the core. As discussed in detail by Labrosse (2003), the Ohmic dissipation of the geodynamo can be linked to the growth of the inner core. This link is established using a global entropy balance between the dissipation and all the energy sources in the outer core. If one is concerned with the evolution of inner core growth, one must also be concerned with how the magnetic field spectrum, and hence Ohmic dissipation, changes with Rayleigh number. For example, the shape of the spectrum may remain unchanged as Ra changes, with only it’s magnitude adjusting. In this case, Ohmic dissipation and therefore inner core growth, would scale conveniently with the intensity of the observable dipole (Labrosse and Macouin, 2003). Our experiments indicate a different scenario: the smallest important scales become larger as the forcing decreases. This behavior is characterized by the scaling √ of the convective frequency fc ∼ Ra, which implies kc ∼ Ra−1/2 . This type of spectral evolution with Ra is partially supported by ideas of Stevenson (1984), where it is argued that in order for the Elsasser number to remain close to unity, the magnitude of large scale magnetic field does not change. Instead, the decreases in total power dissipation as the core cools are compensated by

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increases in the smallest important length scales in the magnetic spectrum. 5.4. Magnetic field and heat flux Our experimental results indicate that at low Elsasser number the presence of magnetic fields decreases the total heat transfer, while increasing local heat transfer in some regions. This result suggests some simple implications for Earth’s outer core perhaps independent of Elsasser number. If magnetic field dynamics in the core cause local variations in heat transfer on large enough length scales, and long enough time scales, one might imagine that they would influence core-mantle thermal interactions and perhaps also persistent structures in the observable magnetic field (e.g. the Central Pacific low flux patch (Kelly and Gubbins, 1996; Johnson and Constable, 1998). Similar mechanisms may also affect inner core growth. If there is persistent structure in the magnetic field near the inner core, this will cause convection to occur more strongly in some regions than in others. The regions of the inner core surface where convection is most vigorous will experience more cooling and removal of light elements and therefor grow more rapidly. Although gravity ultimately forces it to be close to spherical, the inner core is likely formed by a process which is not spherically symmetric, owing to the non-spherical symmetry of the local magnetic field. The anisotropy of seismic models for the inner core may be linked to such heat flux considerations. Related ideas have been discussed by Bergman (1997), Buffett and Wenk (2001), Yoshida et al. (1996). We hope someday to make detailed observations of the spatial patterns of heat flux in our experiments at core-like Elsasser numbers. 6. Conclusions and acknowledgements In an effort to better understand the fluid dynamics of Earth’s outer core, we have conducted a series of rotating liquid sodium convection experiments in a spherical geometry. Using measurements of temperature dynamics and heat flux, we have developed models for the time and length scales of convection as well as the resulting fluid velocity and heat transfer for a range of Ekman and Rayleigh numbers (4.6 × 10−7 > E > 5.5 × 10−8 , 4.2 × 106 < Ra < 2.8 × 109 ). Extrapolating these models to Earth-like parameters, we have made estimates of Ra, convective fluid velocity, Ohmic dissipation, time and length scales, as well as some conjectures about the growth dynamics of the inner core.

Our results suggest further investigation of numerous topics. First, a broader and more thorough study with stronger applied magnetic fields may provide quantitative models for Elsasser number effects closer to Earthlike parameters. Second, it would be interesting to measure the dependence of heat flux on polar angle. Finally, the implementation of ultrasound for direct fluid velocity measurements, as has been done in several liquid metal experiments (Brito et al., 2001; Sisan et al., 2004, e.g.) would significantly contribute to our understanding of the dynamics. Because the design, construction, and instrumentation of these experiments was challenging, perhaps beyond the expectations of a reader who is not familiar with spinning 110 kg of sodium at 30 Hz, we owe a debt of gratitude for the work and advice of many people over the past 4 years. We thank Don Martin, Nicolas Gillet, Morgan Varner, Fred Cawthorne, John Rogers, J. Pyle, James Weldon, Chris Jones, and Gary Glatzmaier. This work was supported by the National Science Foundation of U.S.A. (Geophysics and Earth Sciences Instrumentation programs).

References Anderson, O.L., 2002. The power balance at the core-mantle boundary. Phys. Earth Planet. Int. 131, 1. Aubert, J., Brito, D., Nataf, H.-C., Cardin, P., Masson, J.-P., 2001. A systematic experimental study of rapidly rotating spherical convection in water and liquid gallium. Phys. Earth Planet. Int. 128, 51. Aurnou, J.M., Andreadis, S., Zhu, L., Olson, P.L., 2003. Experiments on convection in Earth’s core tangent cylinder. Earth Planet. Sci. Lett. 212, 119. Aurnou, J.M., Olson, P.L., 2001. Experiments on Rayleigh-B´enard convection, magnetoconvection and rotating magnetoconvection in liquid gallium. J. Fluid Mech. 430, 283. Batchelor, G.K., 1969. Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. II 12, 233. Bergman, M.I., 1997. Measurements of electric anisotropy due to solidification texturing and the implications for the Earth’s core. Nature 389, 60. Brito, D., Nataf, H.-C., Cardin, P., Aubert, J., Masson, J.-P., 2001. Ultrasonic doppler velocimetry in liquid gallium. Experiments in fluids 31, 653. Buffett, B.A., 2002. Estimates of heat flow in the deep mantle based on the power requirements of the geodynamo. Geophys. Res. Lett. 29. Buffett, B.A., Wenk, H.-R., 2001. Texturing of Earth’s inner core by Maxwell stresses. Nature 413, 60. Cardin, P., Olson, P.L., 1995. An experimental approach to thermochemical convection in the Earth’s core. Geophys. Res. Lett. 19. Cardin, P., Olson, P.L., 1994. Chaotic thermal convection in a rapidly rotating spherical shell: consequences for flow in the outer core. Phys. Earth Planet. Int. 82, 235.

W.L. Shew, D.P. Lathrop / Physics of the Earth and Planetary Interiors 153 (2005) 136–149 Cioni, S., Horanyi, S., Krebs, L., M¨uller, U., 1997. Temperature fluctuation properties in sodium convection. Phys. Rev. E 56, R3753. Busse, F.H., 1970. Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441. Carrigan, C.R., Busse, F.H., 1976. Laboratory simulation of thermal convection in rotating planets and stars. Science 191, 81. Carrigan, C.R., Busse, F.H., 1974. Convection induced by centrifugal buoyancy [Earth core]. J. Fluid Mech. 62, 579. Carrigan, C.R., Busse, F.H., 1983. An experimental and theoretical investigation of the onset of convection in rotating spherical shells. J. Fluid Mech 126, 287. Chamberlain, J.A., Carrigan, C.R., 1986. An experimental investigation of convection in a rotating sphere subject to time varying thermal boundary conditions. Geophys. Astrophys. Fluid Dynam. 8, 303. Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability. Dover Publications Inc., New York. Christensen, U.R., Tilgner, A., 2004. Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos. Nature 429, 169. Christensen, U.R., 2002. Zonal flow driven by strongly supercritical convection in rotating spherical shells. J. Fluid Mech. 470, 115. Cordero, S., Busse, F.H., 1992. Experiments on convection in rotating hemispherical shells: transition to a quasi-periodic state. Geophys. Res. Lett. 19, 733. Davidson, P.A., 2004. Turbulence: An Introduction for Scientists and Engineers. Oxford University Press, New York. de Wijs, G.A., Kresse, G., Vocadlo, L., Dobson, D., Alfe, D., Gillian, M.J., Price, G., 1998. The viscosity of liquid iron at the physical conditions of the Earth’s core. Nature 392, 805. Dziewonski, A.M., Anderson, D.L., 1981. Preliminary reference Earth model. Phys. Earth Planet. Int. 25, 297. Gillet, N., 2004. Personal comunication. Glatzmaier, G.A., Olson, P.L., 1993. Highly supercritical thermalconvection in a rotating spherical-shell: centrifugal vs radial gravity. Geophys. Astrophys. Fluid Dynam. 70, 113. Grossmann, S., Lohse, D., 2000. Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27. Gubbins, D., 2001. The Rayleigh number for convection in the Earth’s core. Phys. Earth Planet. Int. 128, 3. Hart, J.E., Kittelman, S., Ohlsen, D.R., 2002. Mean flow precession and temperature probability density functions in turbulent rotating convection. Phys. Fluids 14, 955. Jaletzky, M., 1999. Ph.D. Dissertation. University of Bayreuth. Johnson, C.L., Constable, C.G., 1998. Persistently anomalous Pacific geomagnetic fields. Geophys. Res. Lett. 25, 1011. Jones, C.A., Mussa, A.I., Worland, S.J., 2003. Magnetoconvection in a rapidly rotating sphere: the weak-field case. Proc. R. Soc. Lond., A 459, 773. Kelly, P., Gubbins, D., 1996. Persistent patterns in the geomagnetic field during the last 2.5 Myr. Surv. Geophys. 17, 173. Kraichnan, R.H., 1967. Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417. Labrosse, S., 2003. Thermal and magnetic evolution of the Earth’s core. Phys. Earth Planet. Int. 140, 127. Labrosse, S., Macouin, M., 2003. The inner core and the geodynamo. C.R. Geosci. 335, 37. Liu, Y.M., Ecke, R.E., 1997. Heat transport scaling in turbulent Rayleigh-B´enard conveciton: effects of rotation and Prandtl number. Phys. Rev. Lett. 79, 2257.

149

Manneville, J.-B., Olson, P.L., 1996. Banded convection in rotating fluid spheres and the circulation of the Jovian atmosphere. Icarus 122, 242. Nataf, H.-C., 2003. Dynamo and convection experiments. In: Jones, C., Soward, A., Zhang, K. (Eds.), Earth’s Core and Lower Mantle. Taylor and Francis, London. Ohse, R.W., 1985. Handbook of Thermodynamic and Transport Properties of Alkali Metals. Blackwell Scientific, Oxford. Plaut, E., Busse, F.H., 2002. Low-Prandtl number convection in a rotating cylindrical annulus. J. Fluid Mech. 464, 345. Roberts, P.H., 1968. On thermal instability of a rotating- fluid sphere containing heat sources. Phil. Trans. R. Soc. Lond., A 263, 93. Roberts, P.H., Jones, C.A., Calderwood, A.R., 2003. Energy fluxes and ohmic dissipation in the Earth’s core. In: Jones, C., Soward, A., Zhang, K. (Eds.), Earth’s Core and Lower Mantle. Taylor and Francis, London. Roberts, P.H., Glatzmaier, G.A., 2000. Geodynamo theory and simulations. Rev. Mod. Phys. 72, 1081. Shew, W.L., 2004. Ph.D. Dissertation. University of Maryland, College Park. Shraiman, B.I., Siggia, E.D., 1990. Heat transport in high-Rayleighnumber convection. Phys. Rev. A 42, 3650. Sisan, D.R., Mujica, N., Tillotson, W.A., Huang, Y.-M., Dorland, W., Hassam, A.B., Antonsen, T.M., Lathrop, D.P., 2004. Experimental observation and characterization of the magnetorotational instability. Phys. Rev. Lett. 93, 114502. Stacey, F.D., Loper, D.E., 1984. Thermal histories of the core and mantle. Phys. Earth Planet. Int. 36, 99. Sakuraba, A., 2002. Linear magnetoconvection in rotating fluid spheres permeated by a uniform axial magnetic field. Geophys. Astrophys. Fluid Dynam. 96, 291. Stevenson, D.J., 1984. The energy flux number and 3 types of planetary dynamo. Astronomische Nachrichten 305, 257. Sumita, I., Olson, P.L., 1999. A laboratory model for convection in Earth’s core driven by a thermally heterogeneous mantle. Science 286, 1547. Sumita, I., Olson, P.L., 2000. Laboratory experiments on high Rayleigh number thermal convection in a rapidly rotating hemispherical shell. Phys. Earth Planet. Int. 117, 153. Sumita, I. and Olson, P. L., 2002. Rotating thermal convection experiments in a hemispherical shell with heterogeneous boundary heat flux: implications for the Earth’s core. J. Geophys. Res. 107, ETG5–1-18. Sumita, I., Olson, P.L., 2003. Experiments on highly supercritical thermal convection in a rapidly rotating hemispherical shell. J. Fluid Mech. 492, 271. Tran, C.V., Bowman, J.C., 2003. On the dual cascade in twodimensional turbulence. Physica D 176, 242. Tran, C. V. and Bowman, J. C., 2004. Robustness of the inverse cascade in two-dimensional turbulence. Phys. Rev. E 69, Article No. 036303. Tung, K.K., Orlando, W.W., 2003. The k−3 and k−5/3 energy spectrum of atmospheric turbulence: Quasigeostropic two-level model simulation. J. Atmos. Sci. 60, 824. Wu, X.Z., Libchaber, A., 1992. Scaling relations in thermal turbulence. Phys. Rev. A 45, 842. Yoshida, S., Sumita, I., Kumazawa, M., 1996. Growth model of the inner core coupled with the outer core dynamics and the resulting elastic anisotropy. J. Geophys. Res. 101, 28085. Zhang, K., Schubert, G., 2000. Magnetohydrodynamics in rapidly rotating spherical systems. Annu. Rev. Fluid Mech. 32, 409.