Letter to the Editor

Surface effects and turbulent pressure

Assessing the Gas-Γ₁ and Reduced-Γ₁ empirical models

¹ LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
e-mail: kevin.belkacem@obspm.fr
² Univ. of Appl. Sciences Technikum Wien, Dept. Applied Mathematics and Physics, Höchstädtplatz 6, 1200 Wien, Austria
³ Wolfgang-Pauli-Institute c/o Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
⁴ Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig Weg 3, 37077 Göttingen, Germany

Received: 29 December 2020
Accepted: 14 January 2021

Abstract

The application of the full potential of stellar seismology is made difficult by the improper modelling of the upper-most layers of solar-like stars and their influence on the modelled frequencies. Our knowledge of these so-called ‘surface effects’ has improved thanks to the use of 3D hydrodynamical simulations, however, the calculation of eigenfrequencies relies on empirical models for the description of the Lagrangian perturbation of turbulent pressure, namely: the reduced-Γ₁ model (RGM) and the gas-Γ₁ model (GGM). Starting from the fully compressible turbulence equations, we derived both the GGM and RGM models by using a closure to model the flux of turbulent kinetic energy. We find that both models originate from two terms: the source of turbulent pressure due to compression produced by the oscillations and the divergence of the flux of turbulent pressure. We also demonstrate that they are both compatible with the adiabatic approximation and, additionally, that they imply a number of questionable assumptions, mainly with respect to mode physics. Among other hypotheses, it is necessary to neglect the Lagrangian perturbation of the dissipation of turbulent kinetic energy into heat and the Lagrangian perturbation of buoyancy work.