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, $$\mathcal{S}_1^{c_0}:=\{\alpha \cos(x+\mu)+\beta\cos(y+\lambda): \alpha^2+\beta^2=c_0\,\, \&\,\, (\mu,\lambda)\in\mathbb{R}^2\}$$ 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.