Approaching the asymptotic regime of rapidly rotating convection: boundary layers versus interior dynamics

Critical Prandtl Number for Heat Transfer Enhancement in Rotating Convection

Rotation, which stabilizes flow, can enhance the heat transfer in Rayleigh-Bénard convection (RBC) through Ekman pumping. In this Letter, we present the results of our direct numerical simulations of rotating RBC, providing a comprehensive analysis of this heat transfer enhancement relative to nonrotating RBC in the parameter space of Rayleigh number (Ra), Prandtl number (Pr), and Taylor number (Ta). We show that for a given Ra, there exists a critical Prandtl number (Pr_{cr}) below which no significant heat transfer enhancement occurs at any rotation rate, and an optimal Prandtl number (Pr_{opt}) at which maximum heat transfer enhancement occurs at an optimal rotation rate (Ta_{opt}). Notably, Pr_{cr}, Pr_{opt}, Ta_{opt}, and the maximum heat transfer enhancement all increase with increasing Ra. We also demonstrate a significant heat transfer enhancement up to Ra=2×10^{10} and predict that the enhancement would become even more pronounced at higher Ra, provided Pr is also increased commensurately.

Reynolds number scaling and energy spectra in geostrophic convection

We report flow measurements in rotating Rayleigh-Bénard convection in the rotationally-constrained geostrophic regime. We apply stereoscopic particle image velocimetry to measure the three components of velocity in a horizontal cross-section of a water-filled cylindrical convection vessel. At a constant, small Ekman number Ek = 5 × 10^-8 we vary the Rayleigh number Ra between 10¹¹ and 4 × 10¹² to cover various subregimes observed in geostrophic convection. We also include one nonrotating experiment. The scaling of the velocity fluctuations (expressed as the Reynolds number Re) is compared to theoretical relations expressing balances of viscous-Archimedean-Coriolis (VAC) and Coriolis-inertial-Archimedean (CIA) forces. Based on our results we cannot decide which balance is most applicable here; both scaling relations match equally well. A comparison of the current data with several other literature datasets indicates a convergence towards diffusion-free scaling of velocity as Ek decreases. However, the use of confined domains leads at lower Ra to prominent convection in the wall mode near the sidewall. Kinetic energy spectra point at an overall flow organisation into a quadrupolar vortex filling the cross-section. This quadrupolar vortex is a quasi-two-dimensional feature; it only manifests in energy spectra based on the horizontal velocity components. At larger Ra the spectra reveal the development of a scaling range with exponent close to -5/3, the classical exponent for inertial-range scaling in three-dimensional turbulence. The steeper Re(Ra) scaling at low Ek and development of a scaling range in the energy spectra are distinct indicators that a fully developed, diffusion-free turbulent bulk flow state is approached, sketching clear perspectives for further investigation.

Magnetic damping of jet flows in quasi-two-dimensional Rayleigh-Bénard convection

The mechanism responsible for the damping of the large-scale, azimuthally directed jets observed at Jupiter's surface is not well known, but electromagnetic forces are suspected to play a role as the planet's electrical conductivity increases radially with depth. To isolate the jet damping process, we carry out a suite of direct numerical simulations of quasi-two-dimensional, horizontally periodic Rayleigh-Bénard convection with stress-free boundary conditions in the presence of an external, vertical magnetic field. Jets, punctuated by intermittent convective bursts, develop at Rayleigh numbers (Ra, ratio of buoyancy to diffusion) beyond 10^{5} when the magnetic field is relatively weak. Five primary flow regimes are found by varying 10^{3}≤Ra≤10^{10} and the Chandrasekhar number (Ch, ratio of Lorentz to viscosity) 0≤Ch≤10^{6}: (i) steady convection rolls, (ii) steady magneto-columns, (iii) unsteady to turbulent magneto-plumes, (iv) horizontally drifting magneto-plumes, and (v) jets with intermittent turbulent convective bursts. We parse the parameter space using transitions derived from the interaction parameter (N, ratio of Lorentz to inertia). The transition to the regime dominated by jets has the most immediate applications to the magnetic damping of Jovian jet flows, where the separation between jets and a magnetically constrained system occurs at a jet-based interaction parameter value of N_{J}≈1. We approximate the value of the Jovian interaction parameter as a function of depth, and find that the jets may brake at ≈6000 km below the surface, which is deeper than recent estimates from NASA's Juno mission. This suggests that mechanisms in addition to electromagnetic forces are likely required to fully truncate the jets.

Force balance in rapidly rotating Rayleigh-Bénard convection

The force balance of rotating Rayleigh-Bénard convection regimes is investigated using direct numerical simulation on a laterally periodic domain, vertically bounded by no-slip walls. We provide a comprehensive view of the interplay between governing forces both in the bulk and near the walls. We observe, as in other prior studies, regimes of cells, convective Taylor columns, plumes, large-scale vortices (LSVs) and rotation-affected convection. Regimes of rapidly rotating convection are dominated by geostrophy, the balance between Coriolis and pressure-gradient forces. The higher-order interplay between inertial, viscous and buoyancy forces defines a subdominant balance that distinguishes the geostrophic states. It consists of viscous and buoyancy forces for cells and columns, inertial, viscous and buoyancy forces for plumes, and inertial forces for LSVs. In rotation-affected convection, inertial and pressure-gradient forces constitute the dominant balance; Coriolis, viscous and buoyancy forces form the subdominant balance. Near the walls, in geostrophic regimes, force magnitudes are larger than in the bulk; buoyancy contributes little to the subdominant balance of cells, columns and plumes. Increased force magnitudes denote increased ageostrophy near the walls. Nonetheless, the flow is geostrophic as the bulk. Inertia becomes increasingly more important compared to the bulk, and enters the subdominant balance of columns. As the bulk, the near-wall flow loses rotational constraint in rotation-affected convection. Consequently, kinetic boundary layers deviate from the expected behaviour from linear Ekman boundary layer theory. Our findings elucidate the dynamical balances of rotating thermal convection under realistic top/bottom boundary conditions, relevant to laboratory settings and large-scale natural flows.

Experimental observation of the geostrophic turbulence regime of rapidly rotating convection

The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the "geostrophic turbulence" regime of rapidly rotating convection. However, the latter regime has remained elusive in the laboratory, despite a worldwide effort to design ever-taller rotating convection cells over the last decade. Building on a recent experimental approach where convection is driven radiatively, we report heat transport measurements in quantitative agreement with this scaling regime, the experimental scaling law being validated against direct numerical simulations (DNS) of the idealized setup. The scaling exponent from both experiments and DNS agrees well with the geostrophic turbulence prediction. The prefactor of the scaling law is greater than the one diagnosed in previous idealized numerical studies, pointing to an unexpected sensitivity of the heat transport efficiency to the precise distribution of heat sources and sinks, which greatly varies from planets to stars.

Diffusion-Free Scaling in Rotating Spherical Rayleigh-Bénard Convection

Direct numerical simulations are employed to reveal three distinctly different flow regions in rotating spherical Rayleigh-Bénard convection. In the high-latitude region I vertical (parallel to the axis of rotation) convective columns are generated between the hot inner and the cold outer sphere. The mid-latitude region I I is dominated by vertically aligned convective columns formed between the Northern and Southern hemispheres of the outer sphere. The diffusion-free scaling, which indicates bulk-dominated convection, originates from this mid-latitude region. In the equator region I I I , the vortices are affected by the outer spherical boundary and are much shorter than in region I I .

Rotation suppresses giant-scale solar convection

The entire Sun completes a full rotation in roughly 28 d. Within the outer 30% of the solar interior, turbulent thermal convection powers fluid outward. Rotation deflects the fluid and determines the morphology of eddies and large-scale shear. Such flows are the ultimate agents of astrophysical and planetary magnetic field generation, one of the most important open problems in all of science. Our results make theoretical predictions regarding the Sun’s internal flow structure and rotational constraint. We predict tall and slender vortices persisting throughout much of the convection zone under the sway of strong rotation. We also clarify previous observational discrepancies and explain why such structures have been hard to reproduce in numerical simulations.

The observational absence of giant convection cells near the Sun’s outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller dominant length scales. We specifically predict a characteristic convection length scale of roughly 30-Mm throughout much of the convection zone, implying weak flow amplitudes at 100- to 200-Mm giant cells scales, representative of the total envelope depth. Our reasoning is such that Coriolis forces primarily balance pressure gradients (geostrophy). Background vortex stretching balances baroclinic torques. Both together balance nonlinear advection. Turbulent fluxes convey the excess part of the solar luminosity that radiative diffusion cannot. We show that these four relations determine estimates for the dominant length scales and dynamical amplitudes strictly in terms of known physical quantities. We predict that the dynamical Rossby number for convection is less than unity below the near-surface shear layer, indicating rotational constraint.

Competition between Ekman Plumes and Vortex Condensates in Rapidly Rotating Thermal Convection

We perform direct numerical simulations of rotating Rayleigh-Bénard convection (RRBC) of fluids with low (Pr=0.1) and high (Pr≈5) Prandtl numbers in a horizontally periodic layer with no-slip bottom and top boundaries. No-slip boundaries are known to actively promote the formation of plumelike vertical disturbances, through so-called Ekman pumping, that control the ambient flow at sufficiently high rotation rates. At both Prandtl numbers, we demonstrate the presence of competing large-scale vortices (LSVs) in the bulk. Strong buoyant forcing and rotation foster the quasi-two-dimensional turbulent state of the flow that leads to the upscale transfer of kinetic energy that forms the domain-filling LSV condensate. The Ekman plumes from the boundary layers are sheared apart by the large-scale flow, yet we find that their energy feeds the upscale transfer. Our results of RRBC simulations substantiate the emergence of large-scale flows in nature regardless of the specific details of the boundary conditions.

Numerical study of heat transfer in Rayleigh-Bénard convection under rarefied gas conditions

The focus of this research is to delineate the thermal behavior of a rarefied monatomic gas confined between horizontal hot and cold walls, physically known as rarefied Rayleigh-Bénard (RB) convection. Convection in a rarefied gas appears only for high temperature differences between the horizontal boundaries, where nonlinear distributions of temperature and density make it different from the classical RB problem. Numerical simulations adopting the direct simulation Monte Carlo approach are performed to study the rarefied RB problem for a cold to hot wall temperature ratio equal to r=0.1 and different rarefaction conditions. Rarefaction is quantified by the Knudsen number, Kn. To investigate the long-time thermal behavior of the system two ways are followed to measure the heat transfer: (i) measurements of macroscopic hydrodynamic variables in the bulk of the flow and (ii) measurements at the microscopic scale based on the molecular evaluation of the energy exchange between the isothermal wall and the fluid. The measurements based on the bulk and molecular scales agreed well. Hence, both approaches are considered in evaluations of the heat transfer in terms of the Nusselt number, Nu. To characterize the flow properly, a modified Rayleigh number (Ra_{m}) is defined to take into account the nonlinear temperature and density distributions at the pure conduction state. Then the limits of instability, indicating the transition of the conduction state into a convection state, at the low and large Froude asymptotes are determined based on Ra_{m}. At the large Froude asymptote, simulations following the onset of convection showed a relatively small range for the critical Rayleigh (Ra_{m}=1770±15) that flow instability occurs at each investigated rarefaction degree. Moreover, we measured the maximum Nusselt values Nu_{max} at each investigated Kn. It was observed that for Kn≥0.02, Nu_{max} decreases linearly until the transition to conduction at Kn≈0.03, known as the rarefaction limit for r=0.1, occurs. At the low Froude (parametric) asymptote, the emergence of a highly stratified flow is the prime suspect of the transition to conduction. The critical Ra_{m} in which this transition occurs is then determined at each Kn. The comparison of this critical Rayleigh versus Kn also shows a linear decrease from Ra_{m}≈7400 at Kn=0.02 to Ra_{m}≈1770 at Kn≈0.03.

Boundary Zonal Flow in Rotating Turbulent Rayleigh-Bénard Convection

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one, whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like Ra^{1/4}Ek^{2/3} where the Ekman number Ek decreases with increasing rotation rate.

Velocity and acceleration statistics in rapidly rotating Rayleigh-Bénard convection

Background rotation causes different flow structures and heat transfer efficiencies in Rayleigh-Bénard convection (RBC). Three main regimes are known: rotation-unaffected, rotation-affected and rotation-dominated. It has been shown that the transition between rotation-unaffected and rotation-affected regimes is driven by the boundary layers. However, the physics behind the transition between rotation-affected and rotation-dominated regimes are still unresolved. In this study, we employ the experimentally obtained Lagrangian velocity and acceleration statistics of neutrally buoyant immersed particles to study the rotation-affected and rotation-dominated regimes and the transition between them. We have found that the transition to the rotation-dominated regime coincides with three phenomena; suppressed vertical motions, strong penetration of vortical plumes deep into the bulk and reduced interaction of vortical plumes with their surroundings. The first two phenomena are used as confirmations for the available hypotheses on the transition to the rotation-dominated regime while the last phenomenon is a new argument to describe the regime transition. These findings allow us to better understand the rotation-dominated regime and the transition to this regime.

Geomagnetic polar minima do not arise from steady meridional circulation

Observations of the Earth's magnetic field have revealed locally pronounced field minima near each pole at the core-mantle boundary (CMB). The existence of the polar magnetic minima has long been attributed to the supposed large-scale overturning circulation of molten metal in the outer core: Fluid upwells within the inner core tangent cylinder toward the poles and then diverges toward lower latitudes when it reaches the CMB, where Coriolis effects sweep the fluid into anticyclonic vortical flows. The diverging near-surface meridional circulation is believed to advectively draw magnetic flux away from the poles, resulting in the low intensity or even reversed polar magnetic fields. However, the interconnections between polar magnetic minima and meridional circulations have not to date been ascertained quantitatively. Here, we quantify the magnetic effects of steady, axisymmetric meridional circulation via numerically solving the axisymmetric magnetohydrodynamic equations for Earth's outer core under the magnetostrophic approximation. Extrapolated to core conditions, our results show that the change in polar magnetic field resulting from steady, large-scale meridional circulations in Earth's outer core is less than [Formula: see text] of the background field, significantly smaller than the [Formula: see text] polar magnetic minima observed at the CMB. This suggests that the geomagnetic polar minima cannot be produced solely by axisymmetric, steady meridional circulations and must depend upon additional tangent cylinder dynamics, likely including nonaxisymmetric, time-varying processes.

Turbulent superstructures in Rayleigh-Bénard convection

Turbulent Rayleigh-Bénard convection displays a large-scale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the onset of convection. Here we report numerical simulations of turbulent convection in fluids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to 10⁷. We identify characteristic scales and times that separate the fast, small-scale turbulent fluctuations from the gradually changing large-scale superstructures. The characteristic scales of the large-scale patterns, which change with Prandtl and Rayleigh number, are also correlated with the boundary layer dynamics, and in particular the clustering of thermal plumes at the top and bottom plates. Our analysis suggests a scale separation and thus the existence of a simplified description of the turbulent superstructures in geo- and astrophysical settings.

Regimes of Coriolis-Centrifugal Convection

Centrifugal buoyancy affects all rotating turbulent convection phenomena, but is conventionally ignored in rotating convection studies. Here, we include centrifugal buoyancy to investigate what we call Coriolis-centrifugal convection (C^{3}), characterizing two so far unexplored regimes, one where the flow is in quasicyclostrophic balance (QC regime) and another where the flow is in a triple balance between pressure gradient, Coriolis and centrifugal buoyancy forces (CC regime). The transition to centrifugally dominated dynamics occurs when the Froude number Fr equals the radius-to-height aspect ratio γ. Hence, turbulent convection experiments with small γ may encounter centrifugal effects at lower Fr than traditionally expected. Further, we show analytically that the direct effect of centrifugal buoyancy yields a reduction of the Nusselt number Nu. However, indirectly, it can cause a simultaneous increase of the viscous dissipation and thereby Nu through a change of the flow morphology. These direct and indirect effects yield a net Nu suppression in the CC regime and a net Nu enhancement in the QC regime. In addition, we demonstrate that C^{3} may provide a simplified, yet self-consistent, model system for tornadoes, hurricanes, and typhoons.

Magnetism, dynamo action and the solar-stellar connection

The Sun and other stars are magnetic: magnetism pervades their interiors and affects their evolution in a variety of ways. In the Sun, both the fields themselves and their influence on other phenomena can be uncovered in exquisite detail, but these observations sample only a moment in a single star's life. By turning to observations of other stars, and to theory and simulation, we may infer other aspects of the magnetism-e.g., its dependence on stellar age, mass, or rotation rate-that would be invisible from close study of the Sun alone. Here, we review observations and theory of magnetism in the Sun and other stars, with a partial focus on the "Solar-stellar connection": i.e., ways in which studies of other stars have influenced our understanding of the Sun and vice versa. We briefly review techniques by which magnetic fields can be measured (or their presence otherwise inferred) in stars, and then highlight some key observational findings uncovered by such measurements, focusing (in many cases) on those that offer particularly direct constraints on theories of how the fields are built and maintained. We turn then to a discussion of how the fields arise in different objects: first, we summarize some essential elements of convection and dynamo theory, including a very brief discussion of mean-field theory and related concepts. Next we turn to simulations of convection and magnetism in stellar interiors, highlighting both some peculiarities of field generation in different types of stars and some unifying physical processes that likely influence dynamo action in general. We conclude with a brief summary of what we have learned, and a sampling of issues that remain uncertain or unsolved.

Inertial Wave Turbulence Driven by Elliptical Instability

The combination of elliptical deformation of streamlines and vorticity can lead to the destabilization of any rotating flow via the elliptical instability. Such a mechanism has been invoked as a possible source of turbulence in planetary cores subject to tidal deformations. The saturation of the elliptical instability has been shown to generate turbulence composed of nonlinearly interacting waves and strong columnar vortices with varying respective amplitudes, depending on the control parameters and geometry. In this Letter, we present a suite of numerical simulations to investigate the saturation and the transition from vortex-dominated to wave-dominated regimes. This is achieved by simulating the growth and saturation of the elliptical instability in an idealized triply periodic domain, adding a frictional damping to the geostrophic component only, to mimic its interaction with boundaries. We reproduce several experimental observations within one idealized local model and complement them by reaching more extreme flow parameters. In particular, a wave-dominated regime that exhibits many signatures of inertial wave turbulence is characterized for the first time. This regime is expected in planetary interiors.

Approaching a realistic force balance in geodynamo simulations

Earth sustains its magnetic field by a dynamo process driven by convection in the liquid outer core. Geodynamo simulations have been successful in reproducing many observed properties of the geomagnetic field. However, although theoretical considerations suggest that flow in the core is governed by a balance between Lorentz force, rotational force, and buoyancy (called MAC balance for Magnetic, Archimedean, Coriolis) with only minute roles for viscous and inertial forces, dynamo simulations must use viscosity values that are many orders of magnitude larger than in the core, due to computational constraints. In typical geodynamo models, viscous and inertial forces are not much smaller than the Coriolis force, and the Lorentz force plays a subdominant role; this has led to conclusions that these simulations are viscously controlled and do not represent the physics of the geodynamo. Here we show, by a direct analysis of the relevant forces, that a MAC balance can be achieved when the viscosity is reduced to values close to the current practical limit. Lorentz force, buoyancy, and the uncompensated (by pressure) part of the Coriolis force are of very similar strength, whereas viscous and inertial forces are smaller by a factor of at least 20 in the bulk of the fluid volume. Compared with nonmagnetic convection at otherwise identical parameters, the dynamo flow is of larger scale and is less invariant parallel to the rotation axis (less geostrophic), and convection transports twice as much heat, all of which is expected when the Lorentz force strongly influences the convection properties.

Transitions in turbulent rotating convection: A Lagrangian perspective

Using measurements of Lagrangian acceleration in turbulent rotating convection and accompanying direct numerical simulations, we show that the transition between turbulent states reported earlier [e.g., S. Weiss et al., Phys. Rev. Lett. 105, 224501 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.224501] is a boundary-layer transition between the Prandtl-Blasius type (typical of nonrotating convection) and Ekman type.

Heat-transport enhancement in rotating turbulent Rayleigh-Bénard convection

We present new Nusselt-number (Nu) measurements for slowly rotating turbulent thermal convection in cylindrical samples with aspect ratio Γ=1.00 and provide a comprehensive correlation of all available data for that Γ. In the experiment compressed gasses (nitrogen and sulfur hexafluride) as well as the fluorocarbon C_{6}F_{14} (3M Fluorinert FC72) and isopropanol were used as the convecting fluids. The data span the Prandtl-number (Pr) range 0.74<Pr<35.5 and are for Rayleigh numbers (Ra) from 3×10^{8} to 4×10^{11}. The relative heat transport Nu_{r}(1/Ro)≡Nu(1/Ro)/Nu(0) as a function of the dimensionless inverse Rossby number 1/Ro at constant Ra is reported. For Pr≈0.74 and the smallest Ra=3.6×10^{8} the maximum enhancement Nu_{r,max}-1 due to rotation is about 0.02. With increasing Ra, Nu_{r,max}-1 decreased further, and for Ra≳2×10^{9} heat-transport enhancement was no longer observed. For larger Pr the dependence of Nu_{r} on 1/Ro is qualitatively similar for all Pr. As noted before, there is a very small increase of Nu_{r} for small 1/Ro, followed by a decrease by a percent or so, before, at a critical value 1/Ro_{c}, a sharp transition to enhancement by Ekman pumping takes place. While the data revealed no dependence of 1/Ro_{c} on Ra, 1/Ro_{c} decreased with increasing Pr. This dependence could be described by a power law with an exponent α≃-0.41. Power-law dependencies on Pr and Ra could be used to describe the slope S_{Ro}^{+}=∂Nu_{r}/∂(1/Ro) just above 1/Ro_{c}. The Pr and Ra exponents were β_{1}=-0.16±0.08 and β_{2}=-0.04±0.06, respectively. Further increase of 1/Ro led to further increase of Nu_{r} until it reached a maximum value Nu_{r,max}. Beyond the maximum, the Taylor-Proudman (TP) effect, which is expected to lead to reduced vertical fluid transport in the bulk region, lowered Nu_{r}. Nu_{r,max} was largest for the largest Pr. For Pr=28.9, for example, we measured an increase of the heat transport by up to 40% (Nu_{r}-1=0.40) for the smallest Ra=2.2×10^{9}, even though we were unable to reach Nu_{r,max} over the accessible 1/Ro range. Both Nu_{r,max}(Pr,Ra) and its location 1/Ro_{max}(Pr,Ra) along the 1/Ro axis increased with Pr and decreased with Ra. Although both could be given by power-law representations, the uncertainties of the exponents are relatively large.

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions

The onset of dynamo action is investigated within the context of a newly developed low Rossby, low magnetic Prandtl number, convection-driven dynamo model. This multiscale model represents an asymptotically exact form of an α^{2} mean field dynamo model in which the small-scale convection is represented explicitly by finite amplitude, single mode solutions. Both steady and oscillatory convection are considered for a variety of horizontal planforms. The kinetic helicity is observed to be a monotonically increasing function of the Rayleigh number. As a result, very small magnetic Prandtl number dynamos can be found for sufficiently large Rayleigh numbers. All dynamos are found to be oscillatory with an oscillation frequency that increases as the strength of the convection is increased and the magnetic Prandtl number is reduced. Kinematic dynamo action is strongly controlled by the profile of the helicity; single mode solutions which exhibit boundary layer behavior in the helicity show a decrease in the efficiency of dynamo action due to the enhancement of magnetic diffusion in the boundary layer regions. For a given value of the Rayleigh number, lower magnetic Prandtl number dynamos are excited for the case of oscillatory convection in comparison to steady convection. With regard to planetary dynamos, these results suggest that the low magnetic Prandtl number dynamos typical of liquid metals are more easily driven by thermal convection than by compositional convection.

A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell

We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth’s core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E^1/2, where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1 m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E^−1/2 or E⁻¹). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth’s core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤ 10⁻⁵.