  The aim of this paper is to describe the long time behavior of solutions of
linearized Navier Stokes equations near a concave shear layer profile in the
long waves regime, namely for small horizontal Fourier variable $\alpha$, when
the viscosity $\nu$ vanishes. We show that the solutions converge exponentially
to $0$, except in some range of $\alpha$, namely for $\nu^{1/4} \lesssim
|\alpha| \lesssim \nu^{1/6}$, where there exists one unique unstable mode, with
an associated eigenvalue $\lambda$, such that $\Re \lambda$ is of order
$\nu^{1/4}$. In this regime we give a complete description of the solutions of
linearized Navier Stokes equations as the sum of the projection over the unique
exponentially growing mode and of an exponentially decaying term. The study of
this linear instability is a key point in the study of the nonlinear
instability of Prandtl bounday layers and of shear layer profiles.
