  In this paper we investigate the inviscid limit $\nu \to 0$ for
time-quasi-periodic solutions of the incompressible Navier-Stokes equations on
the two-dimensional torus ${\mathbb T}^2$, with a small time-quasi-periodic
external force. More precisely, we construct solutions of the forced Navier
Stokes equation, bifurcating from a given time quasi-periodic solution of the
incompressible Euler equations and admitting vanishing viscosity limit to the
latter, uniformly for all times and independently of the size of the external
perturbation. Our proof is based on the construction of an approximate
solution, up to an error of order $O(\nu^2)$ and on a fixed point argument
starting with this new approximate solution. A fundamental step is to prove the
invertibility of the linearized Navier Stokes operator at a quasi-periodic
solution of the Euler equation, with smallness conditions and estimates which
are uniform with respect to the viscosity parameter. To the best of our
knowledge, this is the first positive result for the inviscid limit problem
that is global and uniform in time and it is the first KAM result in the
framework of the singular limit problems.
