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

Abstract

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 $\mathbf{q}\in {\mathbb{Z}}^2$ 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 $\Vert \mathbf{p}\Vert$. 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.

1 figure