On the stability and bifurcation of the non-rotating Boussinesq equation with the Kolmogorov forcing at a low Péclet number

Abstract

This study examines the stability and potential bifurcations of a stratified shear flow governed by the non-rotating incompressible Boussinesq equation at a low Péclet number. For the ratio of the vertical scale to the horizontal scale of a stratified flow a∈[3/4,3/2), it is shown that there exists a threshold for the Reynold number Re above which the steady stratified shear flow driven by the Kolmogorov forcing becomes linearly unstable. As a result, the Boussinesq equation exhibits a Hopf bifurcation. To further determine the type of the Hopf bifurcation, the model is reduced to a low-order system whose numerical analyses reveal that the Hopf bifurcation is supercritical. That is, a stable periodic solution emerges, which describes an oscillating thermal convection in a highly stratified shear flow arising in the atmosphere or interior of many stellar systems with low Péclet numbers.