arXiv subjects

T. Gastine

Publications and source records attributed to T. Gastine.

At least 19 recordsLinked to original sources

arXiv2024

Rotating convection with a melting boundary: an application to the icy moons

A better understanding of the ice-ocean couplings is required to better characterise the hydrosphere of the icy moons. Using global numerical simulations in spherical geometry, we have investigated here the interplay between rotating convection and a melting boundary. To do so, we have implemented and validated a phase field formulation in the open-source code \texttt{MagIC}. We have conducted a parameter study varying the influence of rotation, the vigour of the convective forcing and the melting temperature. We have evidenced different regimes akin to those already found in previous monophasic models in which the mean axisymmetric ice crust transits from pole-ward thinning to equator-ward thinning with the increase of the rotational constraint on the flow. The derivation of a perturbative model of heat conduction in the ice layer enabled us to relate those mean topographic changes to the underlying latitudinal heat flux variations at the top of the ocean. The phase change has also been found to yield the formation of sizeable non-axisymmetric topography at the solid-liquid interface with a typical size close to that of the convective columns. We have shown that the typical evolution timescale of the interface increases linearly with the crest-to-trough amplitude and quadratically with the mean melt radius. In the case of the largest topographic changes, the convective flows become quasi locked in the topography due to the constructive coupling between convection and ice melting. The tentative extrapolation to the planetary regimes yields $\mathcal{O}(10^2-10^3)$ meters for the amplitude of non-axisymmetric topography at the base of the ice layer of Enceladus and $\mathcal{O}(10^3-10^4)$ meters for Titan.

astro-ph.EP↗

arXiv2024

Fingering convection in a spherical shell

We use 123 three dimensional direct numerical simulations to study fingering convection in non-rotating spherical shells. We investigate the scaling behaviour of the flow lengthscale, the non-dimensional heat and compositional fluxes $Nu$ and $Sh$ and the mean convective velocity over the fingering convection instability domain defined by $1 \leq R_ρ< Le$, $R_ρ$ being the ratio of density perturbations of thermal and compositional origins and $Le$ the Lewis number. We show that the chemical boundary layers are marginally unstable and adhere to the laminar Prandtl-Blasius model, hence explaining the asymmetry between the inner and outer spherical shell boundary layers. We develop scaling laws for two asymptotic regimes close to the two edges of the instability domain, namely $R_ρ\lesssim Le$ and $R_ρ\gtrsim 1$. For the former, we develop novel power laws of a small parameter $ε$ measuring the distance to onset, which differ from theoretical laws published to date in Cartesian geometry. For the latter, we find that the Sherwood number $Sh$ gradually approaches a scaling $Sh\sim Ra_ξ^{1/3}$ when $Ra_ξ\gg 1$; and that the Péclet number accordingly follows $Pe \sim Ra_ξ^{2/3} |Ra_T|^{-1/4}$, $Ra_ξ$ being the chemical Rayleigh number. When the Reynolds number exceeds a few tens, we report on a secondary instability which takes the form of large-scale toroidal jets which span the entire spherical domain. Jets distort the fingers resulting in Reynolds stress correlations, which in turn feed the jet growth until saturation. This nonlinear phenomenon can yield relaxation oscillation cycles.

physics.flu-dyn↗

arXiv2024

Zonostrophic turbulence in the subsurface oceans of the Jovian and Saturnian moons

In order to characterize the global circulation of the subsurface ocean of Jovian and Saturnian moons, we analyze the properties of 21 three-dimensional simulations of Boussinesq thermal convection in a rapidly rotating spherical shell. Flow is driven by an adverse temperature contrast imposed across the domain, and is subjected to no-slip boundary conditions. We cover a region of parameter space previously unexplored by global simulations, both in terms of rapid rotation and vigor of convective forcing, closer to, yet still admittedly far from, the conditions appropriate for the subsurface ocean of Ganymede, Europa, Enceladus, and Titan. Our most extreme simulations exhibit a dynamic global circulation that combines powerful east-west zonal jets, planetary waves, and vortices. A spectral analysis of the kinetic energy distribution performed in cylindrical geometry reveals a high degree of anisotropy of the simulated flows. Specifically, the axisymmetric zonal energy spectra follow a steep $-5$ slope in wavenumber space, with the energy amplitude exclusively controlled by the rotation rate. In contrast, the non-axisymmetric residual spectra display a gentle $-5/3$ slope, with the energy amplitude controlled by the thermal buoyancy input power. This spectral behavior conforms with the theory of zonostrophic turbulence and allows us to propose tentative extrapolations of these findings to the more extreme conditions of icy satellites. By assuming that kinetic energy dissipates via Ekman friction we predict an upper bound for the zonal velocity ranging from a few centimeters per second for Enceladus to about one meter per second for Ganymede, with residual velocities smaller than the zonal velocity by an order of magnitude on each moon. These predictions yield typical jets size approaching the ocean depth of Titan, Ganymede and Europa and $10$ to $40\%$ of the ocean depth on Enceladus.

physics.flu-dyn↗

arXiv2022

Latitudinal regionalization of rotating spherical shell convection

Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynamics in polar regions can differ significantly from the lower latitude, equatorial dynamics. Yet most spherical shell convective scaling laws use globally-averaged quantities that erase latitudinal differences in the physics. Here we quantify those latitudinal differences by analyzing spherical shell simulations in terms of their regionalized convective heat transfer properties. This is done by measuring local Nusselt numbers in two specific, latitudinally separate, portions of the shell, the polar and the equatorial regions, $Nu_p$ and $Nu_e$, respectively. In rotating spherical shells, convection first sets in outside the tangent cylinder such that equatorial heat transfer dominates at small and moderate supercriticalities. We show that the buoyancy forcing, parameterized by the Rayleigh number $Ra$, must exceed the critical equatorial forcing by a factor of $\approx 20$ to trigger polar convection within the tangent cylinder. Once triggered, $Nu_p$ increases with $Ra$ much faster than does $Nu_e$. The equatorial and polar heat fluxes then tend to become comparable at sufficiently high $Ra$. Comparisons between the polar convection data and Cartesian numerical simulations reveal quantitative agreement between the two geometries in terms of heat transfer and averaged bulk temperature gradient. This agreement indicates that spherical shell rotating convection dynamics are accessible both through spherical simulations and via reduced investigatory pathways, be they theoretical, numerical or experimental.

physics.flu-dyn↗

arXiv2022

Transition from multipolar to dipolar dynamos in stratified systems

Observations of surface magnetic fields of cool stars reveal a large diversity of configurations. Although there is now a consensus that these fields are generated through dynamo processes occurring within the convective zone, the physical mechanism driving such a variety of field topologies is still debated. This paper discusses the possible origins of dipole and multipole-dominated morphologies using three-dimensional numerical simulations of stratified systems where the magnetic feedback on the fluid motion is significant. Our main result is that dipolar solutions are found at Rossby numbers up to 0.4 in strongly stratified simulations, where previous works suggested that only multipolar fields should exist. We argue that these simulations are reminiscent of the outlier stars observed at Rossby numbers larger than 0.1, whose large-scale magnetic field is dominated by their axisymmetric poloidal component. As suggested in previous Boussinesq calculations, the relative importance of inertial over Lorentz forces is again controlling the dipolar to multipolar transition. Alternatively, we find that the ratio of kinetic to magnetic energies can equally well capture the transition in the field morphology. We test the ability of this new proxy to predict the magnetic morphology of a few M-dwarf stars whose internal structure matches that of our simulations and for which homogeneous magnetic field characterization is available. Finally, the magnitude of the differential rotation obtained in our simulations is compared to actual measurements reported in the literature for M-dwarfs. In our simulations, we find a clear relationship between anti-solar differential rotation and the emergence of dipolar fields.

astro-ph.SR↗

arXiv2022

An assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus

We analyze the behaviour of an ensemble of time integrators applied to the semi-discrete problem resulting from the spectral discretization of the equations describing Boussinesq convection in a cylindrical annulus. The equations are cast in a vorticity-streamfunction formulation that yields a differential algebraic equation (DAE). The ensemble comprises 28 members: 4 implicit-explicit multistep schemes, 22 implicit-explicit Runge-Kutta (IMEX-RK) schemes, and 2 fully explicit schemes used for reference. The schemes are assessed for 11 different physical setups that cover laminar and turbulent regimes. Multistep and order 2 IMEX-RK methods exhibit their expected order of convergence under all circumstances. IMEX-RK methods of higher-order show occasional order reduction that impacts both algebraic and differential field variables. We ascribe the order reduction to the stiffness of the problem and, to a larger extent, the presence of the DAE. Using the popular Crank-Nicolson Adams-Bashforth of order 2 (CNAB2) integrator as reference, performance is defined by the ratio of maximum admissible time step to the cost of performing one iteration; the maximum admissible time step is determined by inspection of the time series of viscous dissipation within the system, which guarantees a physically acceptable solution. Relative performance is bounded between 0.5 and 1.5 across all studied configurations. Considering accuracy jointly with performance, we find that 6 schemes consistently outperform CNAB2, meaning that in addition to allowing for a more efficient calculation, the accuracy that they achieve at their operational limit of stability yields a lower error. In our most turbulent setup, where the behaviour of the methods is almost entirely dictated by their explicit component, 13 IMEX-RK integrators outperform CNAB2 in terms of accuracy and efficiency.

physics.flu-dyn↗

arXiv2021

Stable stratification promotes multiple zonal jets in a turbulent Jovian dynamo model

The ongoing NASA's Juno mission puts new constraints on the internal dynamics of Jupiter. Data gathered by its onboard magnetometer reveal a dipole-dominated surface magnetic field accompanied by strong localised magnetic flux patches. The gravity measurements indicate that the fierce surface zonal jets extend several thousands of kilometers below the cloud level before rapidly decaying below $0.94-0.96\,R_J$, $R_J$ being the mean Jovian radius at the one bar level. Several internal models suggest an intricate internal structure with a thin intermediate region in which helium would segregate from hydrogen, forming a compositionally-stratified layer. Here, we develop the first global Jovian dynamo which incorporates an intermediate stably-stratified layer between $0.82\,R_J$ and $0.86\,R_J$. Analysing the energy balance reveals that the magnetic energy is almost one order of magnitude larger than kinetic energy in the metallic region, while most of the kinetic energy is pumped into zonal motions in the molecular envelope. Those result from the different underlying force hierarchy with a triple balance between Lorentz, Archimedean and ageostrophic Coriolis forces in the metallic core and inertia, buoyancy and ageostrophic Coriolis forces controlling the external layers. The simulation presented here is the first to demonstrate that multiple zonal jets and dipole-dominated dynamo action can be consolidated in a global simulation. The inclusion of an stable layer is a necessary ingredient that allows zonal jets to develop in the outer envelope without contributing to the dynamo action in the deeper metallic region. Stable stratification however also smooths out the small-scale features of the magnetic field by skin effect. These constraints suggest that possible stable layers in Jupiter should be located much closer to the surface ($0.9-0.95\,R_J$).

astro-ph.EP↗

arXiv2019

Force balance in numerical geodynamo simulations: a systematic study

Dynamo action in the Earth's outer core is expected to be controlled by a balance between pressure, Coriolis, buoyancy and Lorentz forces, with marginal contributions from inertia and viscous forces. Current numerical simulations of the geodynamo, however, operate at much larger inertia and viscosity because of computational limitations. This casts some doubt on the physical relevance of these models. Our work aims at finding dynamo models in a moderate computational regime which reproduce the leading-order force balance of the Earth. By performing a systematic parameter space survey with Ekman numbers in the range $10^{-6} \leq E \leq 10^{-4}$, we study the variations of the force balance when changing the forcing (Rayleigh number, $Ra$) and the ratio between viscous and magnetic diffusivities (magnetic Prandtl number, $Pm$). For dipole-dominated dynamos, we observe that the force balance is structurally robust throughout the investigated parameter space, exhibiting a quasi-geostrophic (QG) balance (balance between Coriolis and pressure forces) at zeroth order, followed by a first-order MAC balance between the ageostrophic Coriolis, buoyancy and Lorentz forces. At second order this balance is disturbed by contributions from inertia and viscous forces. Dynamos with a different sequence of the forces, where inertia and/or viscosity replace the Lorentz force in the first-order force balance, can only be found close to the onset of dynamo action and in the multipolar regime. Our study illustrates that most classical numerical dynamos are controlled by a QG-MAC balance, while cases where viscosity and inertia play a dominant role are the exception rather than the norm.

physics.geo-ph↗

arXiv2019

pizza: An open-source pseudo-spectral code for spherical quasi-geostrophic convection

We present a new pseudo-spectral open-source code nicknamed pizza. It is dedicated to the study of rapidly-rotating Boussinesq convection under the 2-D spherical quasi-geostrophic approximation, a physical hypothesis that is appropriate to model the turbulent convection that develops in planetary interiors. The code uses a Fourier decomposition in the azimuthal direction and supports both a Chebyshev collocation method and a sparse Chebyshev integration formulation in the cylindrically-radial direction. It supports several temporal discretisation schemes encompassing multi-step time steppers as well as diagonally-implicit Runge-Kutta schemes. The code has been tested and validated by comparing weakly-nonlinear convection with the eigenmodes from a linear solver. The comparison of the two radial discretisation schemes has revealed the superiority of the Chebyshev integration method over the classical collocation approach both in terms of memory requirements and operation counts. The good parallelisation efficiency enables the computation of large problem sizes with $\mathcal{O}(10^4\times 10^4)$ grid points using several thousands of ranks. This allows the computation of numerical models in the turbulent regime of quasi-geostrophic convection characterised by large Reynolds $Re$ and yet small Rossby numbers $Ro$. A preliminary result obtained for a strongly supercritical numerical model with a small Ekman number of $10^{-9}$ and a Prandtl number of unity yields $Re\simeq 10^5$ and $Ro \simeq 10^{-4}$. pizza is hence an efficient tool to study spherical quasi-geostrophic convection in a parameter regime inaccessible to current global 3-D spherical shell models.

physics.flu-dyn↗

arXiv2016

Scaling regimes in spherical shell rotating convection

Rayleigh-Bénard convection in rotating spherical shells can be considered as a simplified analogue of many astrophysical and geophysical fluid flows. Here, we use three-dimensional direct numerical simulations to study this physical process. We construct a dataset of more than 200 numerical models that cover a broad parameter range with Ekman numbers spanning $3\times 10^{-7} \leq E \leq 10^{-1}$, Rayleigh numbers within the range $10^3 < Ra < 2\times 10^{10}$ and a Prandtl number unity. We investigate the scaling behaviours of both local (length scales, boundary layers) and global (Nusselt and Reynolds numbers) properties across various physical regimes from onset of rotating convection to weakly-rotating convection. Close to critical, the convective flow is dominated by a triple force balance between viscosity, Coriolis force and buoyancy. For larger supercriticalities, a subset of our numerical data approaches the asymptotic diffusivity-free scaling of rotating convection $Nu\sim Ra^{3/2}E^{2}$ in a narrow fraction of the parameter space delimited by $6\,Ra_c \leq Ra \leq 0.4\,E^{-8/5}$. Using a decomposition of the viscous dissipation rate into bulk and boundary layer contributions, we establish a theoretical scaling of the flow velocity that accurately describes the numerical data. In rapidly-rotating turbulent convection, the fluid bulk is controlled by a triple force balance between Coriolis, inertia and buoyancy, while the remaining fraction of the dissipation can be attributed to the viscous friction in the Ekman layers. Beyond $Ra \simeq E^{-8/5}$, the rotational constraint on the convective flow is gradually lost and the flow properties vary to match the regime changes between rotation-dominated and non-rotating convection. The quantity $Ra E^{12/7}$ provides an accurate transition parameter to separate rotating and non-rotating convection.

physics.flu-dyn↗

arXiv2015

Turbulent Rayleigh-Bénard convection in spherical shells

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a unity Prandtl number and Rayleigh numbers up to $10^9$. In this geometry, curvature and radial variations of the gravitationnal acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from $η=r_i/r_o=0.2$ to $η=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl-Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame, that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with $η=0.6$ and a gravity proportional to $1/r^2$. We show that our spherical models are consistent with the predictions of Grossmann \& Lohse's (2000) theory and that $Nu(Ra)$ and $Re(Ra)$ scalings are in good agreement with plane layer results.

physics.flu-dyn↗

arXiv2014

Explaining Jupiter's magnetic field and equatorial jet dynamics

Spacecraft data reveal a very Earth-like Jovian magnetic field. This is surprising since numerical simulations have shown that the vastly different interiors of terrestrial and gas planets can strongly affect the internal dynamo process. Here we present the first numerical dynamo that manages to match the structure and strength of the observed magnetic field by embracing the newest models for Jupiter's interior. Simulated dynamo action primarily occurs in the deep high electrical conductivity region while zonal flows are dynamically constrained to a strong equatorial jet in the outer envelope of low conductivity. Our model reproduces the structure and strength of the observed global magnetic field and predicts that secondary dynamo action associated to the equatorial jet produces banded magnetic features likely observable by the Juno mission. Secular variation in our model scales to about 2000 nT per year and should also be observable during the one year nominal mission duration.

astro-ph.EP↗

arXiv2014

Zonal flow scaling in rapidly-rotating compressible convection

The surface winds of Jupiter and Saturn are primarily zonal. Each planet exhibits strong prograde equatorial flow flanked by multiple alternating zonal winds at higher latitudes. The depth to which these flows penetrate has long been debated and is still an unsolved problem. Previous rotating convection models that obtained multiple high latitude zonal jets comparable to those on the giant planets assumed an incompressible (Boussinesq) fluid, which is unrealistic for gas giant planets. Later models of compressible rotating convection obtained only few high latitude jets which were not amenable to scaling analysis. Here we present 3-D numerical simulations of compressible convection in rapidly-rotating spherical shells. To explore the formation and scaling of high-latitude zonal jets, we consider models with a strong radial density variation and a range of Ekman numbers, while maintaining a zonal flow Rossby number characteristic of Saturn. All of our simulations show a strong prograde equatorial jet outside the tangent cylinder. At low Ekman numbers several alternating jets form in each hemisphere inside the tangent cylinder. To analyse jet scaling of our numerical models and of Jupiter and Saturn, we extend Rhines scaling based on a topographic $β$-parameter, which was previously applied to an incompressible fluid in a spherical shell, to compressible fluids. The jet-widths predicted by this modified Rhines length are found to be in relatively good agreement with our numerical model results and with cloud tracking observations of Jupiter and Saturn.

astro-ph.EP↗

arXiv2013

A Bcool magnetic snapshot survey of solar-type stars

Stellar magnetic field measurements obtained from spectropolarimetry offer key data for activity and dynamo studies, and we present the results of a major high-resolution spectropolarimetric Bcool project magnetic snapshot survey of 170 solar-type stars from observations with the Telescope Bernard Lyot and the Canada-France-Hawaii Telescope. For each target star a high signal-to-noise circularly polarised Stokes V profile has been obtained using Least-Squares Deconvolution, and used to detect surface magnetic fields and measure the corresponding mean surface longitudinal magnetic field ($B_{l}$). Chromospheric activity indicators were also measured. Surface magnetic fields were detected for 67 stars, with 21 of these stars classified as mature solar-type stars, a result that increases by a factor of four the number of mature solar-type stars on which magnetic fields have been observed. In addition, a magnetic field was detected for 3 out of 18 of the subgiant stars surveyed. For the population of K-dwarfs the mean value of $B_{l}$ ($|B_{l}|_{mean}$) was also found to be higher (5.7 G) than $|B_{l}|_{mean}$ measured for the G-dwarfs (3.2 G) and the F-dwarfs (3.3 G). For the sample as a whole $|B_{l}|_{mean}$ increases with rotation rate and decreases with age, and the upper envelope for $|B_{l}|$ correlates well with the observed chromospheric emission. Stars with a chromospheric S-index greater than about 0.2 show a high magnetic field detection rate and so offer optimal targets for future studies. This survey constitutes the most extensive spectropolarimetric survey of cool stars undertaken to date, and suggests that it is feasible to pursue magnetic mapping of a wide range of moderately active solar-type stars to improve understanding of their surface fields and dynamos.

astro-ph.SR↗

arXiv2013

From solar-like to anti-solar differential rotation in cool stars

Stellar differential rotation can be separated into two main regimes: solar-like when the equator rotates faster than the poles and anti-solar when the polar regions rotate faster than the equator. We investigate the transition between these two regimes with 3-D numerical simulations of rotating spherical shells. We conduct a systematic parameter study which also includes models from different research groups. We find that the direction of the differential rotation is governed by the contribution of the Coriolis force in the force balance, independently of the model setup (presence of a magnetic field, thickness of the convective layer, density stratification). Rapidly-rotating cases with a small Rossby number yield solar-like differential rotation, while weakly-rotating models sustain anti-solar differential rotation. Close to the transition, the two kinds of differential rotation are two possible bistable states. This study provides theoretical support for the existence of anti-solar differential rotation in cool stars with large Rossby numbers.

astro-ph.SR↗

arXiv2013

What controls the large-scale magnetic fields of M dwarfs?

Observations of active M dwarfs show a broad variety of large-scale magnetic fields encompassing dipole-dominated and multipolar geometries. We detail the analogy between some anelastic dynamo simulations and spectropolarimetric observations of 23 M stars. In numerical models, the relative contribution of inertia and Coriolis force in the global force balance -estimated by the so-called local Rossby number- is known to have a strong impact on the magnetic field geometry. We discuss the relevance of this parameter in setting the large-scale magnetic field of M dwarfs.

astro-ph.SR↗

arXiv2013

Zonal flow regimes in rotating anelastic spherical shells: an application to giant planets

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. We find that the direction of the equatorial zonal wind is controlled by the ratio of buoyancy and 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 homogenised 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 and "anti-solar" differential rotations 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 (rotation-dominated inside and buoyancy-dominated in a thin surface layer). 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.

astro-ph.EP↗

arXiv2012

What controls the magnetic geometry of M dwarfs?

Context: observations of rapidly rotating M dwarfs show a broad variety of large-scale magnetic fields encompassing dipole-dominated and multipolar geometries. In dynamo models, the relative importance of inertia in the force balance -- quantified by the local Rossby number -- is known to have a strong impact on the magnetic field geometry. Aims: we aim to assess the relevance of the local Rossby number in controlling the large-scale magnetic field geometry of M dwarfs. Methods: we explore the similarities between anelastic dynamo models in spherical shells and observations of active M-dwarfs, focusing on field geometries derived from spectropolarimetric studies. To do so, we construct observation-based quantities aimed to reflect the diagnostic parameters employed in numerical models. Results: the transition between dipole-dominated and multipolar large-scale fields in early to mid M dwarfs is tentatively attributed to a Rossby number threshold. We interpret late M dwarfs magnetism to result from a dynamo bistability occurring at low Rossby number. By analogy with numerical models, we expect different amplitudes of differential rotation on the two dynamo branches.

astro-ph.SR↗