  In the setting of a compact Riemannian manifold of dimension $N\ge3$ we
provide a structural description of the limiting behaviour of the energy
measures of solutions to the parabolic Ginzburg-Landau equation. In particular,
we provide a decomposition of the limiting energy measure into a diffuse part,
which is absolutely continuous with respect to the volume measure, and a
concentrated part supported on a codimension $2$ rectifiable subset. We also
demonstrate that the time evolution of the diffuse part is determined by the
heat equation while the concentrated part evolves according to a Brakke flow.
This paper extends the work of Bethuel, Orlandi, and Smets.
