Validity of sound-proof approaches in rapidly-rotating compressible convection: marginal stability versus turbulence

Abstract

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudoincompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases close to the parameters corresponding to the failure of the anelastic approach, which is shown to occur when the sound-crossing time of the domain exceeds a rotation time scale, i.e. for rotational Mach numbers greater than one. Concerning the supercritical case, which is naturally characterised by smaller rotational Mach numbers, we find that the anelastic approximation does not show unphysical behaviour. Growth rates computed with the linearised anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in a highly supercritical, rotating, compressible, low Prandtl number regime show good agreement. However, this nonlinear test example is for only a moderately low convective Rossby number of 0.14. ARTICLE HISTORY

Radius ratio dependency of the instability of fully compressible convection in rapidly rotating spherical shells

“…However, for small Prandtl numbers and rapidly rotating cases Calkins, Julien & Marti (2015a) found that the time derivative of the density reaches a non-negligible level, indicating that the anelastic approximation is not suitable. Verhoeven & Glatzmaier (2018) investigated linear compressible convection within a Newtonian ideal gas in a rotating plane layer geometry, and found that the anelastic approach breaks down when the sound-crossing time of the computational domain exceeds the rotation time scale. Liu et al (2019) studied the onset of fully compressible convection in a rapidly rotating spherical shell by linear stability analysis.…”

Acoustic and inertial modes in planetary-like rotating ellipsoids

“…For simplicity, we could use sound-proof formulations of the compressible equations (e.g. the anelastic approximation), but their applicability is still debated [19][20][21]. Thus, an unambiguous fully compressible theory of the inertial modes in the presence of planetary-like density (and pressure) variations is desirable.…”