  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 obtain the sharp stability boundary in the whole parameter space,
which corrects previous results in the fluid literature. Our new results are
confirmed by more accurate numerical computation. The addition of the Coriolis
force is found to bring fundamental changes to the stability of shear flows.
Moreover, we study dynamical behaviors near the shear flows, including the
bifurcation of nontrivial traveling wave solutions and the linear inviscid
damping. 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 using Sturm-Liouville theory and hypergeometric functions.
