On the stability problem of stationary solutions for the euler equation on a 2-dimensional torus

We study the linear stability problem of the stationary solution ψ * = −cos y for the Euler equation on a 2-dimensional flat torus of sides 2 πL and 2 π . We show that ψ * is stable if L ∈ (0, 1) and that exponentially unstable modes occur in a right neighborhood of L = n for any integer n . As a corollary, we gain exponentially instability for any L large enough and an unbounded growth of the number of unstable modes as L diverges.