Kraichnan-Leith-Batchelor similarity theory and two-dimensional inverse cascades

We study the scaling properties and Kraichnan-Leith-Batchelor (KLB) theory of forced inverse cascades in generalized two-dimensional (2D) fluids ($\alpha$-turbulence models) simulated at resolution $8192^2$. We consider $\alpha=1$ (surface quasigeostrophic flow), $\alpha=2$ (2D vorticity dynamics) and $\alpha=3$. The forcing scale is well-resolved, a direct cascade is present and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both $\alpha=1$ and $\alpha=2$. The active scalar field for $\alpha=3$ contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction $-(7-\alpha)/3$ in all three systems. Since we stop the simulations well before the cascades have reached the domain scale, vortex formation and spectral steepening are not due to condensation effects; nor are they caused by large-scale dissipation, which is absent. One- and two-point pdfs, hyperflatness factors and structure functions indicate that the inverse cascades are intermittent and non-Gaussian over much of the inertial range for $\alpha=1$ and $\alpha=2$, while the $\alpha=3$ inverse cascade is much closer to Gaussian and non-intermittent. For $\alpha=3$ the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling $\mathcal{E}(k) \propto k^{-2}$ ($\alpha=1$) and $\mathcal{E}(k) \propto k^{-5/3}$ ($\alpha=2$) in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation ($\alpha=1$ and $\alpha=2$) and non-realizability ($\alpha=3$) cause 2D inverse cascades to deviate from the KLB predictions, but that the flow between the vortices exhibits KLB scaling and non-intermittent statistics for $\alpha=1$ and $\alpha=2$. The results will appear in \cite{BurgessEA2015}, which has been accepted to the \emph{Journal of Fluid Mechanics}