Energy Conservation and Gravity Waves in Sound-proof Treatments of Stellar Interiors: Part I Anelastic Approximations

TL;DR

This paper examines how different sound-proof equations, especially anelastic approximations, handle gravity waves in stellar interiors, highlighting energy conservation issues and proposing better formulations for modeling subsonic stellar flows.

Contribution

It analyzes the energy conservation properties of various anelastic equations in stably-stratified atmospheres and offers recommendations for improving low-Mach number simulations in stellar physics.

Findings

Some anelastic treatments fail to conserve energy in stratified atmospheres.

One anelastic set of equations consistently conserves energy across atmospheres.

Numerical demonstrations of energy and pseudo-energy conservation issues.

Abstract

Typical flows in stellar interiors are much slower than the speed of sound. To follow the slow evolution of subsonic motions, various sound-proof equations are in wide use, particularly in stellar astrophysical fluid dynamics. These low-Mach number equations include the anelastic equations. Generally, these equations are valid in nearly adiabatically stratified regions like stellar convection zones, but may not be valid in the sub-adiabatic, stably stratified stellar radiative interiors. Understanding the coupling between the convection zone and the radiative interior is a problem of crucial interest and may have strong implications for solar and stellar dynamo theories as the interface between the two, called the tachocline in the Sun, plays a crucial role in many solar dynamo theories. Here we study the properties of gravity waves in stably-stratified atmospheres. In particular, we explore how gravity waves are handled in various sound-proof equations. We find that some anelastic treatments fail to conserve energy in stably-stratified atmospheres, instead conserving pseudo-energies that depend on the stratification, and we demonstrate this numerically. One anelastic equation set does conserve energy in all atmospheres and we provide recommendations for converting low-Mach number anelastic codes to this set of equations.