Vortex annihilation and inverse cascades in two dimensional superfluid turbulence

We study two dimensional superfluid turbulence by employing an effective description valid in the limit where the density of superfluid vortices is parametrically small. At sufficiently low temperatures the effective description yields an inverse cascade with Kolmogorov energy spectrum $E(k) \sim k^{-5/3}$\mathrm{.}Denotingthenumberofvorticesasafunctionoftimeby$N(t)$,wefindthatthevortexannihilationratescaleslike$\dot N \sim N^{5/3}$instateswithaninversecascadeand$\dot N \sim N^2$ for laminar flow.