ICTS

We study instability of unidirectional flows for the linearized 2D alpha-Euler equations (which are a regularized version of the classical 2D Euler equations) on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a single vector  p in the integer lattice Z^2. A decomposition allows us to write the linearized operator L_B as a direct sum of one-dimensional difference operators L_{B,q} parametrized by some vectors q in Z^2. We consider those flows for which the set {q+np: n \in Z^2 } has exactly one point in the open disc of radius p and show that these steady states are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator L_{B,q}$ in terms of equations involving certain continued fractions.