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 pseudo-incompressible 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.

Keywords:

• Rotating compressible convection
• anelastic approximation
• pseudo-incompressible
• sound-proof

Acknowledgements

We would like to thank two anonymous reviewers for comments that greatly improved the manuscript. We gratefully acknowledge that Prof. Pascale Garaud from UCSC provided her NRK routines, which were the basis of our linear convection code.

Notes

No potential conflict of interest was reported by the authors.

1 Christensen-Dalsgaard et al. (1996) for example specify values for $\mathit{ϵ}/D$ of $\mathrm{O}({10}^{-4})$ in the bulk of the solar convection zone.

2 Note firstly that the prefactor in the perturbational pressure term equals the squared Mach number (see appendix B) and secondly that $-{\mathit{\rho }}_0/c_p{\mathit{\partial }}_t{s}_1$, which is the dimensional form of the $-\mathit{ϵ}{\widehat{\mathit{\rho }}}_0{\widehat{\mathit{\partial }}}_t{\widehat{s}}_1$ term in the continuity equation (18a(18a) $$\begin{array}{cc}\hfill \frac{\mathit{ϵ}D}{\mathit{\gamma }-1}\frac{1}{{\widehat{T}}_0}{\widehat{\mathit{\partial }}}_t{\widehat{p}}_1-\mathit{ϵ}{\widehat{\mathit{\rho }}}_0{\widehat{\mathit{\partial }}}_t{\widehat{s}}_1=& -\widehat{\mathbf{\nabla }}\mathbf{·}\left({\widehat{\mathit{\rho }}}_0\widehat{\mathit{v}}\right),\hfill \end{array}$$(18a) ), equals the temporal derivative of the pseudo-incompressible density ${\mathit{\partial }}_t{\mathit{\rho }}^{\ast }$ as first defined in Durran (1989).

3 Compare figures (1(a)–(d)) to Calkins et al. (2015b) figures (3(a)–(d)), respectively. Note that they plot slightly different values than we do. While they plot ${\widehat{k}}_{c}E{k}^{1/3}$, we plot ${\widehat{k}}_c$, which effectively makes our values 100 times larger for the $Ek={10}^{-6}$ cases shown. Moreover they scale their critical frequency ${\widehat{\mathit{\omega }}}_c^{\mathrm{C}alkins}$ with a timescale inferred from the free-fall velocity based on values at the top boundary, see their equation (2.6). We, however, use reference values at the bottom boundary. The relation between our critical frequency and theirs is given by ${\widehat{\mathit{\omega }}}_c=exp[{(N_{\mathit{\rho }})}^{1/(2n)}]{\widehat{\mathit{\omega }}}_c^{\mathrm{C}alkins}$. See also their supplementary material for critical Rayleigh numbers.

Additional information

Funding

Support was provided for J.V. by the National Aeronautics and Space Administration [grant number OPR NNX13AK94G], [grant number PGG NNX14AN70G]. The simulations were run at UCSC on the Hyades supercomputer obtained with an MRI grant from the National Science Foundation.