  We consider the time-dependent 2D Ginzburg-Landau equation in the whole plane
with terms modeling impurities and applied currents. The Ginzburg-Landau
vortices are then subjected to three forces: their mutual repulsive
Coulomb-like interaction, the applied current pushing them in a fixed
direction, and the pinning force attracting them towards the impurities. The
competition between the three is expected to lead to complicated glassy
effects. We rigorously study the limit in which the number of vortices
$N_\epsilon$ blows up as the inverse Ginzburg-Landau parameter $\epsilon$ goes
to $0$, and we derive via a modulated energy method fluid-like mean-field
evolution equations. These results hold for parabolic, conservative, and
mixed-flow dynamics in appropriate regimes of $N_\epsilon\uparrow\infty$.
Finally, we briefly discuss some natural homogenization questions raised by
this study.
