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

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.

💡 Research Summary

This paper investigates whether convection in a rotating spherical shell can exhibit diffusion‑free dynamics when driven by prescribed internal heating and cooling rather than by imposing heat fluxes at the inner and outer boundaries. Using the pseudo‑spectral code Dedalus, the authors solve the Boussinesq equations for a fluid with Prandtl number = 1, a radius ratio η = 0.8, and a thin heating/cooling layer of relative thickness δ̂ = 0.02. The heating function q(r) injects a constant flux F at the base of the convection zone and extracts the same amount at the top, thereby replacing the conductive boundary layers that dominate traditional boundary‑driven simulations.

The system is non‑dimensionalised with the shell depth d as length scale, a flux‑based temperature scale Λ = Fd/κ, and the free‑fall time τ = p d/Λ. The governing dimensionless parameters are the flux‑based Rayleigh number Ra_F = d⁴F/(ν κ²), the Taylor number Ta = 4Ω₀²d⁴/ν², and the flux‑based convective Rossby number Ro_cv,F = (Ra_F/Ta³)^{1/2}(dF)^{1/3}/(2Ω₀d). Importantly, Ro_cv,F is independent of the molecular diffusivities ν and κ, making it an ideal control parameter for probing diffusion‑free behaviour.

A suite of 38 simulations explores Ro_cv,F from 0.01 to 3.1, Ta from 10⁵ to 10⁹, and Ra_F from 3×10⁵ to 9.5×10⁹. The authors identify three dynamical regimes based on Ro_cv,F: (1) rotationally‑constrained (Ro_cv,F < 0.1) where all scales feel rotation, (2) rotationally‑influenced (0.1 ≤ Ro_cv,F < 1) where large scales are constrained but small scales can become inertial, and (3) rotationally‑unconstrained (Ro_cv,F ≥ 1) where rotation plays little role.

Key findings:

1. Diffusion‑free heat transport – The shell‑averaged temperature contrast ΔT and the Nusselt number collapse onto single curves when plotted against Ro_cv,F, regardless of the values of ν and κ. ΔT scales roughly as Ro_cv,F^{-0.6}, indicating that stronger rotation throttles the convective heat flux. This confirms that the bulk thermal response is governed solely by the Rossby number when internal heating/cooling is used.

2. Zonal flow transition – The direction of the equatorial zonal flow switches from prograde (Solar‑like) to retrograde (anti‑Solar) at Ro_cv,F ≈ 0.3–0.5. This transition is also independent of diffusivities, showing that the Rossby number alone determines the sign of the differential rotation in this setup.

3. Kinematic quantities – The Reynolds number Re and the small‑scale Rossby number Ro_ω exhibit scatter at fixed Ro_cv,F, reflecting their sensitivity to the viscous and thermal diffusion scales. Nevertheless, in the high‑supercriticality limit (Ra ≫ Ra_c) Re also approaches a diffusion‑free scaling, and the zonal flow structure becomes increasingly independent of ν and κ.

4. Regime dependence – In the rotationally‑constrained regime, both convective velocities and zonal‑flow amplitudes remain diffusivity‑dependent, whereas in the rotationally‑influenced and unconstrained regimes the bulk velocities become diffusion‑free.

The authors argue that the internal‑heating/cooling configuration eliminates the artificial conductive boundary layers that force traditional simulations to depend on molecular diffusivities. Consequently, the bulk thermal transport and the Rossby‑number‑controlled zonal‑flow reversal can be studied in a regime that is directly relevant to stellar and giant‑planet interiors, where the actual diffusivities are astronomically small.

Implications: This work provides a practical pathway to simulate astrophysical convection without the need to artificially inflate ν and κ. By focusing on the flux‑based Rossby number, one can obtain robust scaling laws for heat transport and differential rotation that are likely to hold in real stars and planets. Future extensions could incorporate compressibility, magnetic fields, and more realistic geometry, moving toward fully realistic models of stellar and planetary convection that retain diffusion‑free bulk dynamics.