= 0, 
\end{array}
\right.$$ where $P$ denotes the pressure. As we are interested in the behavior of solutions to for high Reynolds number, without loss of generality we assume throughout the paper that $\nu \in (0,1]$.

The study of for small perturbations is an old problem in hydrodynamic stability, considered by both Rayleigh [@Rayleigh80] and Kelvin [@Kelvin87], as well as by many modern authors with new perspectives (see e.g. the classical texts [@DrazinReid81; @Yaglom12] and the references therein). Rayleigh and Kelvin both studied the linearization of , which is simply $$\begin{aligned}
 \label{def:2DNSE_Linear} 
\left\{
\begin{array}{l}
\partial_t \omega + y\partial_x \omega = \nu \Delta \omega, \\ 
\Delta \psi = \omega, \\ 
\omega(0) = \omega_{in}. 
\end{array} 
\right. \end{aligned}$$ Rayleigh first proved spectral stability (the absence of unstable eigenmodes) and Kelvin went further to explicitly solve . Indeed, if we denote by $\hat{\omega}(t,k,\eta)$ the Fourier transform of $\omega(t,x,y)$ (note $k \in {\mathbb Z}$, $\eta \in {\mathbb R}$), then it is straightforward to verify Kelvin’s solution

\[eq:KelvinSoln\] $$\begin{aligned}
\hat{\omega}(t,k,\eta) & = \hat{\omega}_{in}(k,\eta+kt)\exp\left[-\nu\int_0^t{\left\vertk\right\vert}^2 + {\left\vert\eta - k\tau + kt\right\vert}^2 d\tau \right], \\
\hat{\psi}(t,k,\eta) & = -\frac{\hat{\omega}_{in}(k,\eta+kt)}{k^2 + \eta^2}\exp\left[-\nu\int_0^t{\left\vertk\right\vert}^2 + {\left\vert\eta - k\tau + kt\right\vert}^2 d\tau \right]. \label{eq:psiKelvin}\end{aligned}$$

From we see directly that if $k\neq 0$ (that 