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 ( �-turbulence models) simulated at resolution 8192 2 . We consider � = 1 (surface quasigeostrophic flow), � = 2 (2D vorticity dynamics) and � = 3. The forcing scale is well-resolved, a direct cascade is pre sent and there is no large-scale dissipation. Coherent vortices spanning a range of sizes, most larger than the forcing scale, are present for both � = 1 and � = 2. The active scalar field for � = 3 contains comparatively few and small vortices. The energy spectral slopes in the inverse cascade are steeper than the KLB prediction (7 �)/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 � = 1 and � = 2, while the � = 3 inverse cascade is much closer to Gaussian and non-intermittent. For � = 3 the steep spectrum is close to that associated with enstrophy equipartition. Continuous wavelet analysis shows approximate KLB scaling E(k) / k 2 (� = 1) and E(k) / k 5/3 (� = 2) in the interstitial regions between the coherent vortices. Our results demonstrate that coherent vortex formation (� = 1 and � = 2) and non-realizability (� = 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 � = 1 and � = 2. The results will appear in Burgess et al. (2015), which has been accepted to the Journal of Fluid Mechanics.