Instability of unidirectional flows for the 2D $alpha$-Euler equations

We study stability of unidirectional flows for the linearized 2D $alpha$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $mathbf p in mathbb Z^{2}$. We linearize the $alpha$-Euler equation and write the linearized operator $L_{B} $ in $ell^{2}(mathbb Z^{2})$ as a direct sum of one-dimensional difference operators $L_{B,mathbf q}$ in $ell^{2}(mathbb Z)$ parametrized by some vectors $mathbf qinmathbb Z^2$ such that the set ${mathbf q +n mathbf p:n in mathbb Z}$ covers the entire grid $mathbb Z^{2}$. The set ${mathbf q +n mathbf p:n in mathbb Z}$ can have zero, one, or two points inside the disk of radius $|mathbf p|$. We consider the case where the set ${mathbf q +n mathbf p:n in mathbb Z}$ has exactly one point in the open disc of radius $mathbf p$. We show that unidirectional flows that satisfy this condition 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 $L_{B,mathbf q}$ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.