On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrِdinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing. e sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform. nalysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.