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Thomas Gastine

Publications and source records attributed to Thomas Gastine.

At least 19 recordsLinked to original sources
arXiv2026

Motional induction in Ganymede's ocean

We investigate the magnetic signature of oceanic circulation in Ganymede's subsurface ocean using kinematic induction modeling. Our approach couples zonal jet flows from rotating thermal convection simulations with magnetic field models incorporating Ganymede's internal dynamo and external contributions from Jupiter. We solve the induction equation in spherical geometry for deep-ocean (493 km) and shallow-ocean (287 km) scenarios with varying magnetic Reynolds numbers. Ocean flows generate a predominantly toroidal magnetic field through the omega-effect, with a weaker poloidal component pervading beyond the conductive ocean layer. For some, but not all, induction configurations, analysis of the time-averaged Lowes-Mauersberger spectra reveals that ocean-induced signals dominate at spherical harmonic degrees $\ell \geq 4$. Deep ocean scenarios with magnetic Reynolds numbers above unity produce surface magnetic signals up to 9 nT. Our results demonstrate that Ganymede's intrinsic magnetic field creates favorable conditions for detecting subsurface ocean dynamics, thus emphasizing the need for low-altitude orbits for the Juice probe.

astro-ph.EP↗
arXiv2025

Online learning of subgrid-scale models for quasi-geostrophic turbulence in planetary interiors

Machine learning approaches to subgrid-scale (SGS) modelling are now well established in atmospheric and oceanic applications. Among these, online end-to-end learning, where the differentiable solver participates in the training, has shown particular promise. Yet, existing studies are largely restricted to idealised periodic domains, with no mechanical boundaries, precluding them from addressing the dynamics of bounded rotating flows relevant to planetary interiors. Here we consider two-dimensional quasi-geostrophic turbulence in a rapidly rotating annular bounded domain. We examine three configurations varying the geometry of the container and the rotation rate. The system exhibit key features such as zonal jets, Rossby waves, and, in the spherical shell geometry, a slow quasi-periodic inward drift of the jets. The spectral properties of the zonal and non-zonal flow can be understood in the framework of zonostrophic turbulence theory. We develop a differentiable solver for this system, which allows us to train SGS models online, over a time span of one turnover time, using coarse-grained data from direct numerical simulations. In all cases, a SGS model trained on a single turnover time accurately reproduce global integrated diagnostics, energy spectra as well as long-term dynamical behaviours ---such as jet migration--- occurring on timescales which exceed the training window by one order of magnitude. The online-trained model further outperforms classical hyperdiffusivity and Leith closure schemes, for which reducing the radial resolution is impractical. The resulting speed-up paves the way to further investigations of long-term geophysical fluid processes beyond the reach of direct numerical simulations.

physics.flu-dyn↗
arXiv2025

Core-surface kinematic control of polarity reversals in advanced geodynamo simulations

The geomagnetic field has undergone hundreds of polarity reversals over Earth's history, at a variable pace. In numerical models of Earth's core dynamics, reversals occur with increasing frequency when the convective forcing is increased past a critical level. This transition has previously been related to the influence of inertia in the force balance. Because this force is subdominant in Earth's core, concerns have been raised regarding the geophysical applicability of this paradigm. Reproducing the reversal rate of the past million years also requires forcing conditions that do not guarantee that the rest of the geomagnetic variation spectrum is reproduced. These issues motivate the search for alternative reversal mechanisms. Using a suite of numerical models where buoyancy is provided at the bottom of the core by inner-core freezing, we show that the magnetic dipole amplitude is controlled by the relative strength of subsurface upwellings and horizontal circulation at the core surface. A relative weakening of upwellings brings the system from a stable to a reversing dipole state. This mechanism is purely kinematic because it operates irrespectively of the interior force balance. It is therefore expected to apply at the physical conditions of Earth's core. Subsurface upwellings may be impeded by stable stratification in the outermost core. We show that with weak stratification levels corresponding to a nearly adiabatic core surface heat flow, a single model reproduces the observed geomagnetic variations ranging from decades to millions of years. \rev{In contrast with} the existing paradigm, reversals caused by this stable top core mechanism become more frequent when the level of stratification increases i.e. when the core heat flow decreases. This suggests that the link between mantle dynamics and magnetic reversal frequency needs to be reexamined.

physics.geo-ph↗
arXiv2023

Quasi-geostrophic convection-driven dynamos in a thick spherical shell

We present dynamos computed using a hybrid QG-3D numerical scheme in a thick spherical shell geometry. Our model is based on a quasi-geostrophic convection code extended with a 3D treatment of heat transport and magnetic induction. We find a collection of self-sustained, multipolar, weak field dynamos with magnetic energy one or two orders of magnitude lower than the kinetic energy. The poloidal magnetic energy is weak and, by construction, there is a lack of equatorially anti-symmetric components in the Buoyancy and Lorentz forces. This leads to configurations where the velocity field is only weakly impacted by the magnetic field, similar to dynamos found in 3D simulations where zonal flows and the $Ω$-effect dominate. The time-dependence of these dynamos is characterised by quasi-periodic oscillations that we attribute to dynamo waves. The QG-3D dynamos found so far are not Earth-like. The inability of our setup to produce strong, dipole-dominated, magnetic fields likely points to a missing ingredient in our QG flows, and a related lack of helicity and $α$-effect. The models presented here may be more relevant for studying stellar dynamos where zonal flows are known to dominate. This study was carried out at modest control parameters, however moving to lower Ekman numbers, when smaller values of both the magnetic and hydrodynamic Prandtl numbers can be of interest, our approach will be able to gain in efficiency by using relatively coarse grids for the 3D magnetic and temperature fields and a finer grid for the QG velocity field.

physics.geo-ph↗
arXiv2023

Comparison of Quasi-Geostrophic, Hybrid and 3D models of planetary core convection

We present investigations of rapidly-rotating convection in a thick spherical shell geometry relevant to planetary cores, comparing results from Quasi-Geostrophic, 3D and hybrid QG-3D models. The 170 reported calculations span Ekman numbers, $Ek$, between $10^{-4}$ and $10^{-10}$, Rayleigh numbers, $Ra$, between $2$ and $150$ times supercritical, and Prandtl numbers, $Pr$, between $10$ and $10^{-2}$. In general, we find convection is dominated by zonal jets at mid-depths in the shell, with thermal Rossby waves prominent close to the outer boundary when the driving is weaker. For the specific geometry studied here the hybrid method is best suited for studying convection at modest forcing, $Ra \leq 10 \, Ra_c$ when $Pr=1$, and departs from the 3D model results at higher $Ra$, displaying systematically lower heat transport. We find that the lack of equatorially anti-symmetric and $z$-correlations between temperature and velocity in the buoyancy force contributes to the weaker flows in the hybrid formulation. On the other hand, the QG models yield broadly similar results to the 3D models, for the specific range of parameters explored here. We cannot point to major disagreements between these two datasets at $Pr \geq 0.1$, although the QG model is effectively more strongly driven than the hybrid case. When $Pr$ is decreased, the range of agreement between the Hybrid and 3D models expands, indicating the hybrid method may be better suited to study convection in the regime $Pr \ll 1$. Previously proposed scaling laws for rapidly-rotating convection are retrieved: our simulations are overall well described by a triple balance between Coriolis, inertia and Archimedean forces with the length-scale of the convection following the diffusion-free Rhines-scaling.

physics.geo-ph↗
arXiv2021

Geomagnetic semblance and dipolar-multipolar transition in top-heavy double-diffusive geodynamo models

Convection in the liquid outer core of the Earth is driven by thermal and chemical perturbations. The main purpose of this study is to examine the impact of double-diffusive convection on magnetic field generation by means of 3D global geodynamo models, in the so-called "top-heavy" regime of double-diffusive convection, when both thermal and compositional background gradients are destabilizing. Using a linear eigensolver, we begin by confirming that, compared to the standard single-diffusive configuration, the onset of convection is facilitated by the addition of a second buoyancy source. We next carry out a systematic parameter survey by performing $79$ numerical dynamo simulations. We show that a good agreement between simulated magnetic fields and the geomagnetic field can be attained for any partitioning of the convective input power between its thermal and chemical components. On the contrary, the transition between dipole-dominated and multipolar dynamos is found to strongly depend on the nature of the buoyancy forcing. Classical parameters expected to govern this transition, such as the local Rossby number or the degree of equatorial symmetry of the flow, fail to capture the dipole breakdown. A scale-dependent analysis of the force balance instead reveals that the transition occurs when the ratio of inertial to Lorentz forces at the dominant length scale reaches $0.5$, regardless of the partitioning of the buoyancy power. The ratio of integrated kinetic to magnetic energy $E_k/E_m$ provides a reasonable proxy of this force ratio. Given that $E_k/E_m \ll 1$ in the Earth's core, the geodynamo is expected to operate far from the dipole-multipole transition. It hence appears that the occurrence of geomagnetic reversals is unlikely related to dramatic and punctual changes of the amplitude of inertial forces in the Earth's core, and that another mechanism must be sought.

physics.geo-ph↗
arXiv2020

Relating force balances and flow length scales in geodynamo simulations

In fluid dynamics, the scaling behaviour of flow length scales is commonly used to infer the governing force balance of a system. The key to a successful approach is to measure length scales that are representative of the energy contained in the solution (energetically relevant) and indicative of the established force balance (dynamically relevant). In numerical simulations of rotating convection and magneto-hydrodynamic dynamos in spherical shells, it has remained difficult to measure length scales that are both energetically and dynamically relevant, which has led to conflicting interpretations of the underlying force balance. By analysing an extensive set of magnetic and non-magnetic models, we focus on two length scales that achieve both energetic and dynamical relevance. The first one is the peak of the poloidal kinetic energy spectrum, which we successfully compare to crossover points on spectral representations of the force balance. In most dynamo models, this result confirms that the dominant length scale of the system is controlled by a quasi-geostrophic (QG-) MAC (Magneto-Archimedean-Coriolis) balance. In non-magnetic convection models, the analysis favours a QG-CIA (Coriolis-Inertia-Archimedean) balance. In dynamo models, we introduce a second energetically relevant length scale associated with the loss of axial invariance in the flow. We again relate this length scale to a crossover point in scale-dependent force balance diagrams, which marks the transition between large-scale geostrophy (the equilibrium of Coriolis and pressure forces) and small-scale magnetostrophy, where the Lorentz force overtakes the Coriolis force. Scaling analysis of these two energetically and dynamically relevant length scales suggests that the Earth's dynamo is controlled by a QG-MAC balance at a dominant scale of about 200 km, while magnetostrophic effects are deferred to scales smaller than 50 km.

physics.geo-ph↗
arXiv2020

Dynamo-based limit to the extent of a stable layer atop Earth's core

The existence of a stably stratified layer underneath the core-mantle boundary (CMB) has been recently revived by corroborating evidences coming from seismic studies, mineral physics and thermal evolution models. Such a layer could find its physical origination either in compositional stratification due to the accumulation of light elements at the top or the core or in thermal stratification due to the heat flux becoming locally sub-adiabatic. The exact properties of this stably-stratified layer, namely its size $\mathcal{H}_S$ and the degree of its stratification characterised by the Brunt-Väisälä frequency $N$, are however uncertain and highly debated. A stable layer underneath the CMB can have crucial dynamical impacts on the geodynamo. Because of the inhibition of the convective motions, a stable layer is expected to primarily act as a low-pass filter on the magnetic field, smoothing out the rapidly-varying and small-scale features by skin effect. To investigate this effect more systematically, we compute 70 global geodynamo models varying the size of the stably-stratified layer from 0 to 300~km and its amplitude from $N/Ω= 0$ to $N/Ω\simeq 50$, $Ω$ being the rotation rate. We show that the penetration of the convective flow in the stably-stratified layer is controlled by the typical size of the convective eddies and by the local variations of the ratio $N/Ω$. Using quantitative measures of the degree of morphological semblance between the magnetic field obtained in numerical models and the geomagnetic field at the CMB, we establish an upper bound for the stable layer thickness $\mathcal{H}_s < (N/Ω)^{-1} d_c$, $d_c$ being the horizontal size of the convective flow at the base of the stable layer. This defines a strong geomagnetic constraint on the properties of a stably-stratified layer beneath the CMB.

physics.geo-ph↗
arXiv2020

Magnetar formation through a convective dynamo in protoneutron stars

The release of spin-down energy by a magnetar is a promising scenario to power several classes of extreme explosive transients. However, it lacks a firm basis because magnetar formation still represents a theoretical challenge. Using the first three-dimensional simulations of a convective dynamo based on a protoneutron star interior model, we demonstrate that the required dipolar magnetic field can be consistently generated for sufficiently fast rotation rates. The dynamo instability saturates in the magnetostrophic regime with the magnetic energy exceeding the kinetic energy by a factor of up to 10. Our results are compatible with the observational constraints on galactic magnetar field strength and provide strong theoretical support for millisecond protomagnetar models of gamma-ray burst and superluminous supernova central engines.

astro-ph.HE↗
arXiv2019

Dynamo Action of Jupiter's Zonal Winds

The new data delivered by NASA's Juno spacecraft significantly increase our understanding of Jupiter's internal dynamics. The gravity data constrain the depth of the zonal flows observed at cloud level and suggest that they slow down considerably at a depth of about $0.96\,r_J$, where $r_J$ is the mean radius at the one bar level. Juno's magnetometer reveals the planet's internal magnetic field. We combine the new zonal flow and magnetic field models with an updated electrical conductivity profile to assess the zonal wind induced dynamo action, concentrating on the outer part of Jupiter's molecular hydrogen region where the conductivity increases very rapidly with depth. Dynamo action remains quasi-stationary and can thus reasonably be estimated where the magnetic Reynolds number remains smaller than one, which is roughly the region above $0.96\,r_J$. We calculate that the locally induced radial magnetic field reaches rms values of about $10^{-6}\,$T in this region and may just be detectable by the Juno mission. Very localized dynamo action and a distinct pattern that reflects the zonal wind system increases the chance to disentangle this locally induced field from the background field. The estimates of the locally induced currents also allow calculating the zonal flow related Ohmic heating and associated entropy production. The respective quantities remain below new revised predictions for the total dissipative heating and total entropy production in Jupiter for any of the explored model combinations. Thus neither Ohmic heating nor entropy production offer additional constraints on the depth of the zonal winds.

astro-ph.EP↗
arXiv2018

Dynamo Action in the Steeply Decaying Conductivity Region of Jupiter-like Dynamo Models

The Juno mission is delivering spectacular data of Jupiter's magnetic field, while the gravity measurements finally allow constraining the depth of the winds observed at cloud level. However, to which degree the zonal winds contribute to the planet's dynamo action remains an open question. Here we explore numerical dynamo simulations that include an Jupiter-like electrical conductivity profile and successfully model the planet's large scale field. We concentrate on analyzing the dynamo action in the Steeply Decaying Conductivity Region (SDCR) where the high conductivity in the metallic Hydrogen region drops to the much lower values caused by ionization effects in the very outer envelope of the planet. Our simulations show that the dynamo action in the SDCR is strongly ruled by diffusive effects and therefore quasi stationary. The locally induced magnetic field is dominated by the horizontal toroidal field, while the locally induced currents flow mainly in the latitudinal direction. The simple dynamics can be exploited to yield estimates of surprisingly high quality for both the induced field and the electric currents in the SDCR. These could be potentially be exploited to predict the dynamo action of the zonal winds in Jupiter's SDCR but also in other planets.

astro-ph.EP↗
arXiv2018

Spherical convective dynamos in the rapidly rotating asymptotic regime

Self-sustained convective dynamos in planetary systems operate in an asymptotic regime of rapid rotation, where a balance is thought to hold between the Coriolis, pressure, buoyancy and Lorentz forces (the MAC balance). Classical numerical solutions have previously been obtained in a regime of moderate rotation where viscous and inertial forces are still significant. We define a unidimensional path in parameter space between classical models and asymptotic conditions from the requirements to enforce a MAC balance and to preserve the ratio between the magnetic diffusion and convective overturn times (the magnetic Reynolds number). Direct numerical simulations performed along this path show that the spatial structure of the solution at scales larger than the magnetic dissipation length is largely invariant. This enables the definition of large-eddy simulations resting on the assumption that small-scale details of the hydrodynamic turbulence are irrelevant to the determination of the large-scale asymptotic state. These simulations are shown to be in good agreement with direct simulations in the range where both are feasible, and can be computed for control parameter values far beyond the current state of the art, such as an Ekman number $E=10^{-8}$. We obtain strong-field convective dynamos approaching the MAC balance and a Taylor state to an unprecedented degree of accuracy. The physical connection between classical models and asymptotic conditions is shown to be devoid of abrupt transitions, demonstrating the asymptotic relevance of classical numerical dynamo mechanisms. The fields of the system are confirmed to follow diffusivity-free, power-based scaling laws along the path.

physics.geo-ph↗
arXiv2017

Gravity darkening in late-type stars. The Coriolis effect

Recent interferometric data have been used to constrain the brightness distribution at the surface of nearby stars, in particular the so-called gravity darkening that makes fast rotating stars brighter at their poles than at their equator. However, good models of gravity darkening are missing for stars that posses a convective envelope. In order to better understand how rotation affects the heat transfer in stellar convective envelopes, we focus on the heat flux distribution in latitude at the outer surface of numerical models. We carry out a systematic parameter study of three-dimensional, direct numerical simulations of anelastic convection in rotating spherical shells. Restricting our investigations to hydrodynamical models with a thermal Prandtl number fixed to unity, we consider both thick and thin (solar-like) shells, and vary the stratification over three orders of magnitude. We measure the heat transfer efficiency in terms of the Nusselt number, defined as the output luminosity normalised by the conductive state luminosity. We report diverse Nusselt number profiles in latitude, ranging from brighter (usually at the onset of convection) to darker equator and uniform profiles. We find that the variations of the surface brightness are mainly controlled by the surface value of the local Rossby number: when the Coriolis force dominates the dynamics, the heat flux is weakened in the equatorial region by the zonal wind and enhanced at the poles by convective motions inside the tangent cylinder. In the presence of a strong background density stratification however, as expected in real stars, the increase of the local Rossby number in the outer layers leads to uniformisation of the surface heat flux distribution.

astro-ph.SR↗
arXiv2017

Physical conditions for Jupiter-like dynamo models

The Juno mission will measure Jupiter's magnetic field with unprecedented precision and provide a wealth of additional data that will allow to constrain the planet's interior structure and dynamics. Here we analyse 66 numerical simulations in order to explore the sensitivity of the dynamo-generated magnetic field to the planets interior properties. The degree l=4 field model VIP4 and up-to-date interior models based on ab initio simulations serve as benchmarks. Our results suggest that VIP4-like magnetic fields can be found for a number of different models. We find that whether we assume an ideal gas or use the more realistic interior model based on ab initio simulations makes no difference. However, two other factors are important. Low Rayleigh number leads to strong axial dipole contribution while the axial dipole dominance is lost when the convective driving is too strong. The required intermediate range that yields Jupiter-like magnetic fields depends on the other system properties. The second factor is the convective magnetic Reynolds number profile Rmc(r), a product of the non-axisymmetric flow velocity and electrical conductivity. We find that the depth where Rmc exceeds about 50 is a good proxy for the top of the dynamo region. When the dynamo region sits too deep, the axial dipole is once more too dominant due to geometric reasons. We extrapolated our results to Jupiter and the result suggests that the Jovian dynamo extends to 95% of the planetary radius. The zonal flows in our simulations are dominated by an equatorial jet largely confined to the molecular layer. Where the jet reaches down to higher electrical conductivities, it is gives rise to a secondary alpha-Omega dynamo that modifies the dipole-dominated field produced deeper in the planet. This secondary dynamo can lead to strong magnetic field patches at low latitudes that seem compatible with the VIP4 model.

astro-ph.EP↗
arXiv2016

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, while 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 employ 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 sub-dominant 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 inertia are smaller by a factor of at least 20 in the bulk of the fluid volume. Compared to non-magnetic convection at otherwise identical parameters, the dynamo flow is of larger scale, 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.

physics.geo-ph↗
arXiv2016

Reversal and amplification of zonal flows by boundary enforced thermal wind

Zonal flows in rapidly-rotating celestial objects such as the Sun, gas or ice giants form in a variety of surface patterns and amplitudes. Whereas the differential rotation on the Sun, Jupiter and Saturn features a super-rotating equatorial region, the ice giants, Neptune and Uranus harbour an equatorial jet slower than the planetary rotation. Global numerical models covering the optically thick, deep-reaching and rapidly rotating convective envelopes of gas giants reproduce successfully the prograde jet at the equator. In such models, convective columns shaped by the dominant Coriolis force typically exhibit a consistent prograde tilt. Hence angular momentum is pumped away from the rotation axis via Reynolds stresses. Those models are found to be strongly geostrophic, hence a modulation of the zonal flow structure along the axis of rotation, e.g. introduced by persistent latitudinal temperature gradients, seems of minor importance. Within our study we stimulate these thermal gradients and the resulting ageostrophic flows by applying an axisymmetric and equatorially symmetric outer boundary heat flux anomaly ($Y_{20}$) with variable amplitude and sign. Such a forcing pattern mimics the thermal effect of intense solar or stellar irradiation. Our results suggest that the ageostrophic flows are linearly amplified with the forcing amplitude $q^\star$ leading to a more pronounced dimple of the equatorial jet (alike Jupiter). The geostrophic flow contributions, however, are suppressed for weak $q^\star$, but inverted and re-amplified once $q^\star$ exceeds a critical value. The inverse geostrophic differential rotation is consistently maintained by now also inversely tilted columns and reminiscent of zonal flow profiles observed for the ice giants. Analysis of the main force balance and parameter studies further foster these results.

physics.geo-ph↗
arXiv2015

Helicity inversion in spherical convection as a means for equatorward dynamo wave propagation

We discuss here a purely hydrodynamical mechanism to invert the sign of the kinetic helicity, which plays a key role in determining the direction of propagation of cyclical magnetism in most models of dynamo action by rotating convection. Such propagation provides a prominent, and puzzling constraint on dynamo models. In the Sun, active regions emerge first at mid-latitudes, then appear nearer the equator over the course of a cycle, but most previous global-scale dynamo simulations have exhibited poleward propagation (if they were cyclical at all). Here, we highlight some simulations in which the direction of propagation of dynamo waves is altered primarily by an inversion of the kinetic helicity throughout much of the interior, rather than by changes in the differential rotation. This tends to occur in cases with a low Prandtl number and internal heating, in regions where the local density gradient is relatively small. We analyse how this inversion arises, and contrast it to the case of convection that is either highly columnar (i.e., rapidly rotating) or locally very stratified; in both of those situations, the typical profile of kinetic helicity (negative throughout most of the northern hemisphere) instead prevails.

astro-ph.SR↗
arXiv2015

Effect of shear and magnetic field on the heat-transfer efficiency of convection in rotating spherical shells

We study rotating thermal convection in spherical shells. We base our analysis on a set of about 450 direct numerical simulations of the (magneto)hydrodynamic equations under the Boussinesq approximation. The Ekman number ranges from $10^{-3}$ to $10^{-5}$. The supercriticality of the convection reaches about 1000 in some models. Four sets of simulations are considered: non-magnetic simulations and dynamo simulations with either free-slip or no-slip flow boundary conditions. The non-magnetic setup with free-slip boundaries generates the strongest zonal flows. Both non-magnetic simulations with no-slip flow boundary conditions and self-consistent dynamos with free-slip boundaries have drastically reduced zonal-flows. Suppression of shear leads to a substantial gain in heat-transfer efficiency, increasing by a factor of 3 in some cases. Such efficiency enhancement occurs as long as the convection is significantly influenced by rotation. At higher convective driving the heat-transfer efficiency tends towards that of the classical non-rotating Rayleigh-Bénard system. Analysis of the latitudinal distribution of heat flow at the outer boundary reveals that the shear is most effective at suppressing heat-transfer in the equatorial regions. Furthermore, we explore the influence of the magnetic field on the {\em non-zonal} flow components of the convection. For this we compare the heat-transfer efficiency of no-slip non-magnetic cases with that of the no-slip dynamo simulations. We find that at $E=10^{-5}$ magnetic field significantly affects the convection and a maximum gain of about 30\% (as compared to the non-magnetic case) in heat-transfer efficiency is obtained for an Elsasser number of about 3. Our analysis motivates us to speculate that convection in the polar regions in dynamos at $E=10^{-5}$ is probably in a `magnetostrophic' regime.

astro-ph.EP↗