  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.
