  We study the dynamics of the two dimensional Navier-Stokes equations
linearized around a shear flow on a (non-square) torus which possesses exactly
two non-degenerate critical points. We obtain linear inviscid damping and
vorticity depletion estimates for the linearized flow that are uniform with
respect to the viscosity, and enhanced dissipation type decay estimates. The
main task is to understand the associated Rayleigh and Orr-Sommerfeld
equations, under the natural assumption that the linearized operator around the
shear flow in the inviscid case has no discrete eigenvalues. The key difficulty
is to understand the behavior of the solution to Orr-Sommerfeld equations in
three distinct regimes depending on the spectral parameter: the non-degenerate
case when the spectral parameter is away from the critical values, the
intermediate case when the spectral parameter is close to but still separated
from the critical values, and the most singular case when the spectral
parameter is inside the viscous layer.
