Weakly nonlinear stability analysis of salt-finger convection in a longitudinally infinite cavity

Abstract

This paper is a two-dimensional linear and weakly nonlinear stability analyses of the three-dimensional problem of Chang et al. [“Three-dimensional stability analysis for a salt-finger convecting layer,” J. Fluid Mech. 841, 636–653 (2018)] concerning salt-finger convection, which is seen when there is sideways heating and salting along the vertical walls along with a linear variation of temperature and concentration on the horizontal walls. A two-dimensional linear stability analysis is first carried out in the problem with the knowledge that the result could be different from those of a three-dimensional study. A two-dimensional weakly nonlinear stability analysis, that is, then performed points to the possibility of the occurrence of sub-critical motions. Stability curves are drawn to depict various instability regions. With the help of a detailed stability analysis, the stationary mode is shown to be the preferred one compared to oscillatory. Local nonlinear stability analysis of the system is done in a neighborhood of the critical Rayleigh number to predict a sub-critical instability region. The existence of a stable solution at the onset of a weakly nonlinear convective regime is indicated, allowing one to perform a bifurcation study in the problem. Heat and mass transports are discussed by analyzing the Nusselt number, Nu, and Sherwood number, Sh, respectively. A simple relationship is obtained between the Nusselt number and the Sherwood number exclusively in terms of the Lewis number, Le.