Semi-convection in rotating spherical shells: flows, layers and dynamos

TL;DR

This paper investigates semi-convection in rotating spherical shells as a model for gas-giant interiors where an unstable thermal gradient coexists with a stabilising compositional gradient. It combines linear onset analysis using a SINGE eigensolver with non-linear 3D simulations (XSHELLS) to reveal a layering phase that merges into either a wide stably stratified layer over a central convective zone or a single convective region, depending on forcing and rotation. When magnetic fields are included, self-sustained dynamos arise, with field structure heavily influenced by flow regime: jet-dominated states tend to produce strongly dipolar, axisymmetric surface fields due to SSL filtering, while convection-dominated states yield more multipolar fields. The results identify a parameter range, notably around the rotationally driven balance captured by $R_B=\mathrm{Ek}^{3/2}\mathrm{Ra}_T/\Pr$, that can generate planetary-like magnetic fields, providing a mechanism for Saturn-like dynamos without imposed SSLs.

Abstract

Large regions of gaseous planets are thought to be stratified with an unstable thermal gradient, but a stabilising gradient of heavy element composition. Fluid in these regions is unstable to semi-convection, with motions driven by differences in the molecular diffusivity of temperature and composition, and could play a role in supporting planetary magnetic fields. Previous studies focus largely on local models in Cartesian boxes; here, we investigate semi-convection in rotating spherical shells. The onset of linear instability shows a transition between the two limits of rotating convection and non-rotating semi-convection. Non-linear simulations evolve into a system of concentric layers of relatively constant density, separated by narrow high-gradient regions. These layers gradually merge, resulting in a statistically steady state dominated by either a single convection region or a narrower convective zone beneath a stably stratified layer (SSL), depending on the strength of the thermal forcing compared to the rotation. When magnetic field generation is considered, our magnetohydrodynamic simulations exhibit self-sustained dynamo action. In cases where the turbulent convective region generates magnetic fields that are smoothed by zonal flows within the overlying SSL, the resulting field is strongly dipolar and axisymmetric, in encouraging agreement with Saturn's observed magnetic field. Within the regimes explored, both the Rossby number and the thickness of the SSL are well predicted by a single combination of control parameters. This enables the identification of a parameter range in which the generated magnetic fields resemble those of planetary dynamos.

Figures (15)

• Figure 1: Sketch of the physical setup for the problem.
• Figure 2: Linear onset of semi-convection computed with SINGE, for $\Delta R=0.5$, $\mathrm{Pr}=0.3$, $\mathrm{Sc}=3$, with three values of $\mathrm{Ek}$ and spherical harmonic order $1\leq m\leq 40$. Diagonal lines show (blue) $\mathrm{Ra}_T=\mathrm{Ra}_C$, (green) $\mathrm{Ra}_T=\mathrm{Ra}_C/L$, and (red) the limit of non-rotating semi-convection given by \ref{['eqn:limit_semiconvection']}. The pink lines show the position of transects for non-linear simulations, one with $R_\rho=1.2$ while $\mathrm{Ra}_C$ and $\mathrm{Ra}_T$ vary, the other for $\mathrm{Ra}_C=2.33\times10^8$, and $1<R_\rho<3$. Horizontal lines show the onset of thermal rotating convection, with values given in Table \ref{['tab:convectiononset']}.
• Figure 3: Results from an XSHELLS simulation with $\mathrm{Ek}=10^{-4}$, $\mathrm{Pr}=0.3$, $\mathrm{Sc}=3$, $\Delta R=0.5$, $\mathrm{Ra}_C=4.22\times 10^7$ and $R_\rho=1.2$. (a) Time series of the total kinetic energy $E_u$. Second and third row show snapshots of the flow at (b)--(e): $t=0.1$ and (f)--(i) $0.2t_\nu$, marked with vertical dashed lines in (a). (b),(f) Equatorial snapshots of the density perturbation $\rho=C-T$; (c),(g) Equatorial snapshots of the total density $\rho_0=C_0+C-T_0-T$; (d),(h) meridional snapshots of the radial velocity $u_r$, (e),(i) meridional snapshots of the azimuthal velocity $u_\phi$.
• Figure 4: Results from an XSHELLS simulation with $\mathrm{Ek}=10^{-4}$, $\mathrm{Pr}=0.3$, $\mathrm{Sc}=3$, $\Delta R=0.5$, $\mathrm{Ra}_C=1.33\times 10^9$ and $R_\rho=1.2$. (a) Time series of the total kinetic energy $E_u$. The second and third row show snapshots of the flow at (b)--(e): $t=0.02$ and (f)--(i) $0.04t_\nu$, marked with vertical dashed lines in (a). (b),(f) Equatorial snapshots of the density perturbation $\rho = C-T$; (c),(g) Equatorial snapshots of the total density $\rho_{tot}=C_0+C-T_0-T$; (d),(h) meridional snapshots of the radial velocity $u_r$, (e),(i) meridional snapshots of the azimuthal velocity $u_\phi$.
• Figure 5: (a) Flux ratio time-averaged over the linear phase of evolution $\overline{\gamma}$ as a function of $R_\rho$ for simulations along the transect $\mathrm{Ra}_C=2.33\times10^8$, $1\leq R_\rho\leq3$, with $\mathrm{Pr}=0.3$, $\mathrm{Sc}=3$, $\Delta R=0.5$, $\mathrm{Ek}=10^{-4}$. The values for $\overline{\gamma}$ are calculated (by \ref{['eqn:gamma']}) during the linear growth phase of each simulation. The vertical dashed line marks the limit of rotating semi-convection $R_{\rho,\text{onset}}=2.68$, calculated using SINGE. Points are coloured by the value of $\overline{\mathrm{Re}}$; to the right of the semi-convection limit. The dotted black line shows the limit of pure conduction $R_\rho/L$. (b),(c) Snapshots of the axisymmetric ($m=0$) component of the composition field $C$, for (b) $R_\rho=1.2$ and (c) $R_\rho=1.8$, labelled in red in (a). (d)--(e) radial profiles of the total axisymmetric composition field $C_{tot}=C_{axi}+C_0(r)$ along the white lines in (b)--(d), coloured by the value of $C_{axi}$. Dashed black lines show the conductive profile $C_0(r)$.