Numerical calculation of convection with reduced speed of sound technique

Abstract

Context. The anelastic approximation is often adopted in numerical calculations with low Mach numbers, such as those including stellar internal convection. This approximation requires so-called frequent global communication, because of an elliptic partial differential equation. Frequent global communication is, however, negative factor for the parallel computing performed with a large number of CPUs. Aims. We test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context of stellar internal convection. This reduction in the speed of sound leads to longer time steps despite the low Mach number, while the numerical scheme remains fully explicit and the mathematical system is hyperbolic, thus does not require frequent global communication. Methods. Two-and three-dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of solutions computed with different effective Mach numbers (owing to the reduction in the speed of sound) are compared to test the validity of our approach. Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number (based on the lower speed of sound) remains less than 0.7.

High-Resolution Calculations of the Solar Global Convection With the Reduced Speed of Sound Technique. I. The Structure of the Convection and the Magnetic Field Without the Rotation

“…When a moderate or zero reduction of the speed of sound is used in the near-surface layer while a large ξ around the bottom part of the convection zone is used, the properties of the thermal convection, even including the surface, can be properly investigated without undermining the physics. Hotta et al (2012b) confirmed that the inhomogeneous ξ is valid again when the Mach number is less than 0.7.…”

“…The reduced speed of sound technique (hereafter RSST, Rempel 2005Rempel , 2006Hotta et al 2012b) can overcome these drawbacks while avoiding the severe time step caused by the speed of sound. In the RSST the equation of continuity is replaced with…”

“…A simple algorithm and low cost of communication are significant factors that enable high resolution calculations. Hotta et al (2012b) investigated the validity of the RSST in a thermal convection problem. It was concluded that the RSST is valid when the Mach number defined with the RMS (root mean square) velocity and the reduced speed of soundĉ s is smaller than 0.7.…”

“…Hotta et al (2012b) suggest that the RSST is valid for the thermal convection under the criterion of v rms /ĉ s < 0.7, where v rms is the RMS (root mean square) convection velocity. Thus we can properly treat the convection with v rms < 1.3 km s −1 in this model.…”

Generation of twist on magnetic flux tubes at the base of the solar convection zone

“…We use the relation ρ = ρ 0 + ρ 1 , p = p 0 + p 1 , and s = s 0 + s 1 . The reduced speed of sound technique (hereafter RSST), which is introduced in Rempel (2005) and Hotta et al (2012b), is used in this study, and ξ denotes the ratio between the original and reduced speed of sound. The investigation of the validity of RSST for the flux rising is given in Hotta et al (in prep.).…”