Barotropic instability of shear flows

    We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and find the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. The addition of the Coriolis force is found to bring some fundamental changes to the stability of shear flows. Moreover, we study the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping near the shear flows. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian PDEs. The last one is to study the singular and non-resonant neutral modes by using hypergeometric functions and singular Sturm-Liouville theory. This is a joint work with Zhiwu Lin and Jincheng Yang.