Direct numerical simulations of the -mechanism

II. Nonlinear saturation and the Hertzsprung progression

Laboratoire d'Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS, 14 avenue Edouard Belin, 31400 Toulouse, France e-mail: thomas.gastine@ast.obs-mip.fr

Received: 2 April 2008
Accepted: 24 July 2008

Abstract

Context. We study the κ-mechanism that excites radial oscillations in Cepheid variables.

Aims. We address the mode couplings that manage the nonlinear saturation of the instability in direct numerical simulations (DNS).

Methods. We project the DNS fields onto an acoustic subspace built from the regular and adjoint eigenvectors that are solutions to the linear oscillation equations.

Results. We determine the time evolution of both the amplitude and kinetic energy of each mode that propagates in the DNS. More than 98% of the total kinetic energy is contained in two modes that correspond to the linearly-unstable fundamental mode and the linearly-stable second overtone. Because the eigenperiod ratio is close to , we discover that the nonlinear saturation is due to a 2:1 resonance between these two modes. An interesting application of this result concerns the reproduction of Hertzsprung's progression observed in Bump Cepheids.