Thermal convection experiments in a liquid gallium layer subject to a uniform rotation and a uniform vertical magnetic field are carried out as a function of rotation rate and magnetic field strength. Our purpose is to measure heat transfer in a low-Prandtl-number (Pr = 0.023), electrically conducting fluid as a function of the applied temperature difference, rotation rate, applied magnetic field strength and fluid-layer aspect ratio. For Rayleigh-Bénard (non-rotating, non-magnetic) convection we obtain a Nusselt number-Rayleigh number law Nu = 0.129 Ra0.272±0.006 over the range 3.0 x 103 < Ra < 1.6 x 104. For non-rotating magnetoconvection, we find that the critical Rayleigh number RaC increases linearly with magnetic energy density, and a heat transfer law of the form Nu ∼ Ra1/2. Coherent thermal oscillations are detected in magnetoconvection at ∼ 1.4RaC. For rotating magnetoconvection, we find that the convective heat transfer is inhibited by rotation, in general agreement with theoretical predictions. At low rotation rates, the critical Rayleigh number increases linearly with magnetic field intensity. At moderate rotation rates, coherent thermal oscillations are detected near the onset of convection. The oscillation frequencies are close to the frequency of rotation, indicating inertially driven, oscillatory convection. In nearly all of our experiments, no well-defined, steady convective regime is found. Instead, we detect unsteady or turbulent convection just after onset.

Global scale magnetostrophic balance, in which Lorentz and Coriolis forces comprise the leading-order force balance, has long been thought to describe the natural state of planetary dynamo systems. This argument arises from consideration of the linear theory of rotating magnetoconvection. Here we test this long-held tenet by directly comparing linear predictions against dynamo modelling results. This comparison shows that dynamo modelling results are not typically in the global magnetostrophic state predicted by linear theory. Then, in order to estimate at what scale (if any) magnetostrophic balance will arise in nonlinear dynamo systems, we carry out a simple scaling analysis of the Elsasser number Λ, yielding an improved estimate of the ratio of Lorentz and Coriolis forces. From this, we deduce that there is a magnetostrophic cross-over length scale, [Formula: see text], where Λ_o is the linear (or traditional) Elsasser number, Rm_o is the system scale magnetic Reynolds number and D is the length scale of the system. On scales well above [Formula: see text], magnetostrophic convection dynamics should not be possible. Only on scales smaller than [Formula: see text] should it be possible for the convective behaviours to follow the predictions for the magnetostrophic branch of convection. Because [Formula: see text] is significantly smaller than the system scale in most dynamo models, their large-scale flows should be quasi-geostrophic, as is confirmed in many dynamo simulations. Estimating Λ_o ≃1 and Rm_o ≃10³ in Earth's core, the cross-over scale is approximately 1/1000 that of the system scale, suggesting that magnetostrophic convection dynamics exists in the core only on small scales below those that can be characterized by geomagnetic observations.

Global scale magnetostrophic balance, in which Lorentz and Coriolis forces comprise the leading-order force balance, has long been thought to describe the natural state of planetary dynamo systems. This argument arises from consideration of the linear theory of rotating magnetoconvection. Here we test this long-held tenet by directly comparing linear predictions against dynamo modelling results. This comparison shows that dynamo modelling results are not typically in the global magnetostrophic state predicted by linear theory. Then, in order to estimate at what scale (if any) magnetostrophic balance will arise in nonlinear dynamo systems, we carry out a simple scaling analysis of the Elsasser number Λ, yielding an improved estimate of the ratio of Lorentz and Coriolis forces. From this, we deduce that there is a magnetostrophic cross-over length scale, [Formula: see text], where Λ_o is the linear (or traditional) Elsasser number, Rm_o is the system scale magnetic Reynolds number and D is the length scale of the system. On scales well above [Formula: see text], magnetostrophic convection dynamics should not be possible. Only on scales smaller than [Formula: see text] should it be possible for the convective behaviours to follow the predictions for the magnetostrophic branch of convection. Because [Formula: see text] is significantly smaller than the system scale in most dynamo models, their large-scale flows should be quasi-geostrophic, as is confirmed in many dynamo simulations. Estimating Λ_o ≃1 and Rm_o ≃10³ in Earth's core, the cross-over scale is approximately 1/1000 that of the system scale, suggesting that magnetostrophic convection dynamics exists in the core only on small scales below those that can be characterized by geomagnetic observations.

Many dynamo studies extrapolate numerical model results to planetary conditions by empirically constructing scaling laws. The seminal work of Christensen and Aubert (2006) proposed a set of scaling laws that have been used throughout the geoscience community. These scalings make use of specially-constructed parameters that are independent of fluid diffusivities, anticipating that large-scale turbulent processes will dominate the physics in planetary dynamo settings. With these 'diffusion-free' parameterizations, the results of current numerical dynamo models extrapolate directly to fully-turbulent planetary core systems; the effects of realistic fluid properties merit no further investigation. In this study, we test the validity of diffusion-free heat transfer scaling arguments and their applicability to planetary conditions. We do so by constructing synthetic heat transfer datasets and examining their scaling properties alongside those proposed by Christensen and Aubert (2006). We find that the diffusion-free parameters compress and stretch the heat transfer data, eliminating information and creating an artificial alignment of the data. Most significantly, diffusion-free heat transfer scalings are found to be unrelated to bulk turbulence and are instead controlled by the onset of non-magnetic rotating convection, itself determined by the viscous diffusivity of the working fluid. Ultimately, our results, in conjunction with those of Stelzer and Jackson (2013) and King and Buffett (2013), show that diffusion-free scalings are not validated by current-day numerical dynamo datasets and cannot yet be extrapolated to planetary conditions.

A mechanism is proposed to explain the seismologically-inferred prograde rotation of the Earth's solid inner core in terms of the structure of convection in the fluid outer core. Numerical calculations of convection and dynamo action in the outer core exhibit excess temperatures inside the tangent cylinder surrounding the inner core. We show that this temperature difference generates a prograde thermal wind and a strong azimuthal magnetic field inside the tangent cylinder. Electromagnetic torques on the inner core derived from induced azimuthal magnetic fields and the ambient poloidal field equilibrate when the inner core angular velocity lags the nearby tangent cylinder fluid angular velocity by approximately 14%. The inferred prograde rotation of the inner core (1.1-3°/year relative to the mantle) can be produced by a very small (≃ 0.001 K) temperature anomaly within the tangent cylinder and indicates strong toroidal magnetic fields with peak intensities of 24-66 mT in that region of the core. Copyright 1996 by the American Geophysical Union.

The surface zonal winds observed in the giant planets form a complex jet pattern with alternating prograde and retrograde direction. While the main equatorial band is prograde on the gas giants, both ice giants have a pronounced retrograde equatorial jet.We use three-dimensional numerical models of compressible convection in rotating spherical shells to explore the properties of zonal flows in different regimes where either rotation or buoyancy dominates the force balance. We conduct a systematic parameter study to quantify the dependence of zonal flows on the background density stratification and the driving of convection.In our numerical models, we find that the direction of the equatorial zonal wind is controlled by the ratio of the global-scale buoyancy force and the Coriolis force. The prograde equatorial band maintained by Reynolds stresses is found in the rotation-dominated regime. In cases where buoyancy dominates Coriolis force, the angular momentum per unit mass is homogenized and the equatorial band is retrograde, reminiscent to those observed in the ice giants. In this regime, the amplitude of the zonal jets depends on the background density contrast with strongly stratified models producing stronger jets than comparable weakly stratified cases. Furthermore, our results can help to explain the transition between solar-like (i.e. prograde at the equator) and the "anti-solar" differential rotations (i.e. retrograde at the equator) found in anelastic models of stellar convection zones.In the strongly stratified cases, we find that the leading order force balance can significantly vary with depth. While the flow in the deep interior is dominated by rotation, buoyancy can indeed become larger than Coriolis force in a thin region close to the surface. This so-called "transitional regime" has a visible signature in the main equatorial jet which shows a pronounced dimple where flow amplitudes notably decay towards the equator. A similar dimple is observed on Jupiter, which suggests that convection in the planet interior could possibly operate in this regime. © 2013 Elsevier Inc.

The effect of spherical shell geometry on rapidly-rotating thermal convection is studied in a suite of high resolution three-dimensional numerical simulations. The geometry is characterized by the radius ratio, X = ri/ro, where ri is the inner shell radius, and ro is the outer shell radius. In this study, X is varied over the broad range 0.10 to 0.92 in calculations of Boussinesq rotating convection subject to isothermal, rigid boundary conditions. Simulations are performed at Prandtl number Pr = 1 and for Ekman numbers E = 10-3, 3 x 10-4 and 10-4. Near the onset of convection, the flow takes the form of rolls aligned parallel to the rotation axis and situated adjacent to the inner shell equator. The dimensionless azimuthal wavelength, λ c, of the rolls is found to be independent of the shell geometry, only varying with the Ekman number. The critical wave number, mc, of the columnar rolls increases in direct proportion to the inner boundary circumference. For our simulations the critical Rayleigh number Rac. at which convection first occurs varies in proportion to E-1.16 a result that is consistent with previous work on rotating convection. Furthermore, we find that Rac is a complex function of X. We obtain the relation Rac E1.16 = 0.21/X2+ 22.4√(1 - x)(1 + x), which adequately fits all our results. In supercritical convection calculations the flows form quasi-geostrophic sheet-like structures that are elongated in the radial direction, stretching from the inner boundary toward the outer boundary. © 2004 Taylor and Francis Ltd.

Turbulent, rapidly rotating convection has been of interest for decades, yet there exists no generally accepted scaling law for heat transfer behaviour in this system. Here, we develop an exact scaling law for heat transfer by geostrophic convection, Nu= (Ra/Rac)3 = 0. 0023 Ra 3 E4, by considering the stability of the thermal boundary layers, where Nu, Ra and E are the Nusselt, Rayleigh and Ekman numbers, respectively, and Rac is the critical Rayleigh number for the onset of convection. Furthermore, we use the scaling behaviour of the thermal and Ekman boundary layer thicknesses to quantify the necessary conditions for geostrophic convection: Ra ≈ E-3/2. Interestingly, the predictions of both heat flux and regime transition do not depend on the total height of the fluid layer. We test these scaling arguments with data from laboratory and numerical experiments. Adequate agreement is found between theory and experiment, although there is a paucity of convection data for low Ra E 3/2. © 2011 Cambridge University Press.