We study the linear stability problem of the stationary solution $\psi^* = −\cos y$ for the Euler equation on a 2-dimensional flat torus of sides $2\pi L$ and $2\pi$. We show that $\psi^*$ is stable if $L \in (0, 1)$ 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.
Keywords:
Euler equation, shear flows, linear stability
Citation:
Butta P., Negrini P., On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus, Regular and Chaotic Dynamics,
2010, Volume 15, Number 6,
pp. 637-645