  A semi-discretization in time, according to a full implicit Euler scheme, for
a 2D dissipative quasi geostrophic equation, is studied. We prove existence,
uniqueness and regularity results of the solution to the predicted
discretization, in the subcritical case for any initial data in $\dot{L}^2$.
Hence, we define an infinite semi-discrete dynamical system, then we prove the
existence and the regularity of the corresponding global attractor, for a
source term $f$ in $\dot{L}^{p_{\alpha}}$, for a fixed $p_{\alpha} =
\frac{2}{1-\alpha} $.
