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On the stability and bifurcation of the non-rotating Boussinesq equation with the Kolmogorov forcing at a low Péclet number. (English) Zbl 1444.76058
Summary: 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 \(a\in[\sqrt{3}/4,\sqrt{3}/2)\), 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.
76E20 Stability and instability of geophysical and astrophysical flows
76E15 Absolute and convective instability and stability in hydrodynamic stability
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35B32 Bifurcations in context of PDEs
Full Text: DOI
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