  We consider the two dimensional Navier-Stokes equations in vorticity form
with a stochastic forcing term given by a gaussian noise, white in time and
coloured in space. First, we prove existence and uniqueness of a weak (in the
Walsh sense) solution process $\xi$ and we show that, if the initial vorticity
$\xi_0$ is continuous in space, then there exists a space-time continuous
version of the solution. In addition we show that the solution $\xi(t,x)$
(evaluated at fixed points in time and space) is locally differentiable in the
Malliavin calculus sense and that its image law is absolutely continuous with
respect to the Lebesgue measure on $\mathbb{R}$.
