Diffusion-Free Dynamics in Rotating Spherical Shell Convection Driven By Internal Heating and Cooling

TL;DR

The paper addresses whether interior convection in rotating spherical shells driven by internal heating and cooling can exhibit diffusion-free dynamics, avoiding the boundary-layer diffusivity constraints that plague boundary-driven models. Using 38 Boussinesq, rotating spherical-shell simulations with an internally prescribed heating/cooling profile implemented in Dedalus, the authors map the parameter space with $Ra_{\text{F}}$, $Ta$, $Pr$, and the diffusion-free control $Ro_{\text{cv,F}} = \left( Ra_{\text{F}} / Ta^{3/2} Pr^{2} \right)^{1/3}$, demonstrating that several bulk properties (notably the radial temperature contrast $\Delta T$ and convective heat transport $Nu$) become independent of $\nu$ and $\kappa$ in the rotationally-uninfluenced and rotationally-influenced regimes. They identify three dynamical regimes—rotationally-constrained ($Ro_{\text{cv,F}}<0.1$), rotationally-influenced ($0.1\leq Ro_{\text{cv,F}}<1$), and rotationally-uninfluenced ($Ro_{\text{cv,F}}\geq1$)—and show corresponding scaling relations: Nu follows diffusion-free MLT in the uninfluenced regime, diffusion-free RMLT in the influenced regime, and an offset RMLT in the constrained regime, with high supercriticality sometimes restoring Nu ∝ $Ra^{1/2}$. The Reynolds number shows a diffusion-free ultimate scaling in the non-rotating/uninfluenced cases, but follows diffusivity-dependent VAC or CIA balances in the constrained cases. These results imply that diffusion-free dynamics can be realized in interior convection models using internal heating/cooling, providing a path to more faithful extrapolations to stars and giant planets. Future work should incorporate compressibility and magnetism to assess robustness in more realistic settings.

Abstract

The bulk properties of convection in stellar and giant planet interiors are often assumed to be independent of the molecular diffusivities, which are very small. By contrast, simulations of this process in rotating, spherical shells, which are typically driven by conductive boundary heat fluxes, generally yield results that depend on the diffusivity. This makes it challenging to extrapolate these simulation results to real objects. However, laboratory models and Cartesian-box simulations suggest that diffusion-free dynamics are more readily obtained if convection is driven using prescribed internal heating and cooling instead of boundary fluxes. Here, we apply this methodology to simulations of Boussinesq, hydrodynamic rotating spherical shell convection. We find that this set-up unambiguously yields diffusion-free behaviour for some bulk properties of the convection, such as the radial temperature contrast and the convective heat transport. Moreover, the transition from prograde to retrograde equatorial zonal flow is diffusion-free and only depends on the convective Rossby number. The diffusivity dependence of other bulk properties is regime-dependent. In simulations that are rotationally constrained, the convective velocities, and the strength and structure of the zonal flow, are diffusion-dependent, although the zonal flow appears to approach a diffusion-free state for sufficiently high supercriticality. In simulations that are uninfluenced by rotation, or are only influenced by rotation at large scales, diffusion-free convective velocities and zonal flows are obtained. The result that many aspects of our idealised simulations are diffusion-free has promising implications for the development of realistic stellar and giant planet convection models that can access diffusion-free regimes.

Figures (6)

• Figure 1: Summary of simulations presented in this study. Panel a) shows the input parameters $Ra_{\text{F}}$ and $Ta$ used for each experiment. Panel b) shows the combination $Ra Ta^{-\frac{2}{3}}$ as a proxy for the supercriticality (computed using the 'output' Rayleigh number given by Equation \ref{['eq:Ra']}). Panel c) shows $Re$ (defined by Equation \ref{['eq:Re']}), computed using the r.m.s. fluctuating velocity, as a measure of the degree of turbulence obtained in each simulation. Finally, panel d) shows the output Rossby numbers $Ro_{\text{bulk}}$ and $Ro_{\omega}$ (defined in Equation \ref{['eq:Ro_alt']}) as a measure of the degree of rotational constraint at the largest spatial scale, and smaller scales, respectively, plotted against $Ro_{\text{cv,F}}$. The marker colour denotes the dynamical regime occupied by a simulation, and the marker shape denotes the configuration of the differential rotation. In panels a)--c), simulations that share the same $Ro_{\text{cv,F}}$ are connected by lines. Three lines are emphasised in bold, corresponding to $Ro_{\text{cv,F}}=3.11\times10^{-2}, 6.70\times10^{-2},\ \text{and}\ 3.11\times10^{-1}$ (from bottom to top in each panel). In panel d), the grey dashed lines are proportional to $Ro_{\text{cv,F}}$ and included as an eye guide.
• Figure 2: Pseudodimensional output data from each experiment. Panel a) shows the shell-averaged temperature difference across the CZ. Panel b shows the equatorial zonal velocity, averaged between $\pm5^{\circ}$ latitude and over the depth of the CZ. Panels c) and d) show the kinetic energy, averaged over the volume of the CZ. KE$_{\text{fluc}}$ is computed using the fluctuating component of the velocity, and DRKE is computed using the axially-symmetric component of the azimuthal velocity. In panels a) and c), lines corresponding to scaling predictions from RMLT and due to CIA balance (see text), respectively, are shown as eye guides. See footnote 4 for details of the re-dimensionalisation procedure. Instances for which only one experiment exists for a given $Ro_{\text{cv,F}}$ are marked with the $\dagger$ symbol.
• Figure 3: Snapshots of the perturbation temperature for six experiments. The experiments shown in the top row are in the rotationally-uninfluenced regime, those in the middle row are in the rotationally-influenced regime, and those in the bottom row are in the rotationally-constrained regime. In each panel, the inner surface corresponds to $r=4.2$ and the outer surface corresponds to $r=4.8$.
• Figure 4: Snapshots of the perturbation temperature for two further experiments. The upper panel shows a simulations with $Ro_{\text{cv,F}}=1.44\times10^{-1}$ that is classified as rotationally-influenced. In this experiment, the influence on the dynamics is more apparent visually than is the case for the two rotationally-influenced simulations with $Ro_{\text{cv,F}}=3.11\times10^{-1}$ shown in Figure \ref{['fig:snaps']}. The lower panel shows a rotationally constrained case with $Ro_{\text{cv,F}}=6.70\times10^{-2}$ (as in Figure \ref{['fig:snaps']}, bottom row). This simulation is more dissipative than those with $Ro_{\text{cv,F}}=6.70\times10^{-2}$ shown in Figure \ref{['fig:snaps']}. In the upper panel, the colourbar maximum has been rescaled by $0.7$ to enhance the visibility of features at lower latitudes.
• Figure 5: Azimuthally-averaged angular velocity $\Omega$ normalised by the rotation rate $\Omega_{0}$ (see Equation \ref{['eq:Om']}) for two series of simulations with fixed $Ro_{\text{cv,F}}$. The top row shows $Ro_{\text{cv,F}}=3.11\times10^{-1}$ and the bottom row shows $Ro_{\text{cv,F}}=6.70\times10^{-2}$. For a given $Ro_{\text{cv,F}}$, different combinations of $Ra_{\text{F}}$ and $Ta$ correspond to different diffusivities. In the slices above, the diffusivity decreases from left to right. The dashed lines show the edges of the CZ at $r_{\text{i}}=4$ and $r_{\text{o}}=5$.