Conduction in low Mach number flows: Part I Linear & weakly nonlinear regimes

TL;DR

This paper compares temperature and entropy diffusion models in low Mach number flows, finding similar results in linear and weakly nonlinear regimes but potential differences in turbulent convection.

Contribution

It provides a detailed comparison of temperature and entropy diffusion in linear and weakly nonlinear regimes using simulations of compressible and reduced equations.

Findings

Temperature and entropy diffusion yield similar results in linear regimes.

Pseudo-incompressible equations show larger errors with temperature diffusion.

Differences may become significant in strongly turbulent convection.

Abstract

Thermal conduction is an important energy transfer and damping mechanism in astrophysical flows. Fourier's law - the heat flux is proportional to the negative temperature gradient, leading to temperature diffusion - is a well-known empirical model of thermal conduction. However, entropy diffusion has emerged as an alternative thermal conduction model, despite not ensuring the monotonicity of entropy. This paper investigates the differences between temperature and entropy diffusion for both linear internal gravity waves and weakly nonlinear convection. In addition to simulating the two thermal conduction models with the fully compressible Navier-Stokes equations, we also study their effects in the reduced, "sound-proof" anelastic and pseudo-incompressible equations. We find that in the linear and weakly nonlinear regimes, temperature and entropy diffusion give quantitatively similar results, although there are some larger errors in the pseudo-incompressible equations with temperature diffusion due to inaccuracies in the equation of state. Extrapolating our weakly nonlinear results, we speculate that differences between temperature and entropy diffusion might become more important for strongly turbulent convection.