Remark on the Stability of Energy Maximizers for the 2D Euler equation on ${\mathbb{T}}^2$

It is well-known that the first energy shell, $${\mathcal{S}}_1^{c_0}:=\{\alpha \cos (x+\mu )+\beta \cos (y+\lambda ):{\alpha }^2+{\beta }^2=c_0\mathrm{\&}(\mu ,\lambda )\in {\mathbb{R}}^2\}$$ of solutions to the 2d Euler equation is Lyapunov stable on ${\mathbb{T}}^2$. This is simply a consequence of the conservation of energy and enstrophy. Using the idea of Wirosoetisno and Shepherd \cite{WS}, which is to take advantage of conservation of a properly chosen Casimir, we give a simple and quantitative proof of the $L^2$ stability of single modes up to translation. In other words, each $${\mathcal{S}}_1^{\alpha ,\beta }:=\{\alpha \cos (x+\mu )+\beta \cos (y+\lambda ):(\mu ,\lambda )\in {\mathbb{R}}^2\}$$ is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases $\alpha =0,$ $\beta =0$, and $\alpha =\pm \beta$ may be markedly less stable than the others.