Kinematic viscosity in solar convection simulations

¹ Institute of Physics, University of Graz, Universitätsplatz 5, 8010 Graz, Austria
² Now at: Institute for Theoretical Physics, Technical University at Brunswick, Mendelssohnstraße 3, 38106 Braunschweig, Germany
³ Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, 8042 Graz, Austria

^⋆ Corresponding author: johannes.tschernitz@tu-braunschweig.de

Received: 11 October 2024
Accepted: 6 July 2025

Abstract

Context. Numerical models are often used to improve our understanding of solar and stellar convection. To keep models numerically stable, direct numerical simulations (DNS) use various types of diffusivities. The kinematic viscosity, for example, is often chosen to be several orders of magnitude higher than realistic values. These high viscosities may distort the DNS results.

Aims. We test the effects of kinematic viscosity, hyperviscosity, and shock viscosity on the numerical stability for solar convection simulations with a finite-difference code. We investigated how their value ranges affect the size of the convection cells and the vertical plasma motions at grid distances of about 125 km.

Methods. We ran sets of convection simulations using the Pencil Code together with a density and temperature stratification that resembles the Sun. For simplicity and better understanding of the viscosity effects, we only used hydrodynamic simulations in a two-dimensional vertical plane with Cartesian coordinates, but allowed velocity vectors with three components (2.5D). Our physical domain included the upper 20 Mm of the convection zone and another 25 Mm of the solar atmosphere above the photosphere. To study each type of viscosity separately, we tested several parameters individually.

Results. We found differences in the length scale of the granules that depend on the kinematic viscosity ν. We also found that an asymptotic behavior develops for sufficiently low values of ν. An important numerically stabilizing factor is the shock viscosity, which acts in places where the kinematic viscosity is insufficient. Hyperviscosity has no significant effect on the numeric stability and length scales of the convection cells in our simulation runs.

Conclusions. We conclude that a kinematic viscosity of ν = 1.34 ⋅ 10⁸ m²/s or lower should be used for convection simulations with grid distances of about 125 km. The simulations became unstable when the kinematic viscosity was much lower than this. Shock viscosity provides additional numerical stability.