Brad Hindman, Department of Applied Mathematics, University of Colorado Boulder

The Tyranny of Sound: Sound-proofing the fluid equations

Numerical simulations of convection in low-mass stars, the Earth’s atmosphere, giant planets, and many other astrophysical objects all must face the tyranny of sound. Generally, sound waves propagate quickly and have high frequencies; thus, the typical timescale associated with acoustics is far shorter than those arising from convection and large-scale circulations. In a numerical simulation, this short timescale ensures through the CFL condition that sound waves control the size of the timestep that can be taken while still maintaining numerical stability. The difference can be dramatic. For example, at the base of the Sun’s convection zone, the speed of sound is roughly 200 km/s while the convective flow speed is on the order of 20 m/s . A numerical simulation that is forced to track
sound waves for stability will need to take 10,000 times as many time steps to evolve the solution for the same duration as a simulation that could ignore the acoustic wave field. This inflation of the necessary computational work is particular onerous since the immense timescale difference between the deep convection and the sound waves indicates that the two phenomena are essentially decoupled.  I present three different qpproximations that remove sound waves as a solution of the fluid equations: Boussinesq, anelastic, and pseudo-incompressible.  I will discuss when each is relevent and when each fails. In particular, I will explore the differences between the anelastic and the pseudo-incompressible approximation and demonstrate that the latter is almost always a better approximation to make.