In this paper, we perform direct statistical simulations of a model of two-dimensional flow that exhibits a transition from jets to vortices. The model employs two-scale Kolmogorov forcing, with energy injected directly into the zonal mean of the flow. We compare these results with those from direct numerical simulations. For square domains, the solution takes the form of jets, but as the aspect ratio is increased, a transition to isolated coherent vortices is found. We find that a truncation at second order in the equal-time but nonlocal cumulants that employ zonal averaging (zonal CE2) is capable of capturing the form of the jets for a range of Reynolds numbers as well as the transition to the vortex state but, unsurprisingly, is unable to reproduce the correlations found for the fully nonlinear (non-zonally symmetric) vortex state. This result continues the program of promising advances in statistical theories of turbulence championed by Kraichnan.
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