Spectral non-locality, absolute equilibria and Kraichnan-Leith-Batchelor phenomenology in two-dimensional turbulent energy cascades

We study the degree to which Kraichnan-Leith-Batchelor (KLB) phenomenology describes two-dimensional energy cascades in alpha turbulence, governed by delta theta/delta t + J(psi), theta) = upsilon del(2)theta + f, where theta = (-Delta)(alpha/2) psi is generalized vorticity, and (psi) over bar (k)= k-alpha(theta) over bar (k) in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow (alpha = 1), regular two-dimensional flow (alpha = 2) and rotating shallow flow (alpha = 3), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjortoft analysis, which we extend to general ff, and point out their limitations. Using an alpha-dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Nonlocal triads contribute increasingly to the energy flux as alpha increases. More importantly, the energy cascade is downscale in the self-similar inertial range for 2.5 < alpha < 10. At alpha = 2.5 and alpha = 10, the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for alpha < 4. However, downscale energy flux in the EDQNM self-similar inertial range for alpha > 2.5 leads us to predict that any inverse cascade for alpha > 2.5 will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for alpha > 2.5 is significantly steeper than the KLB prediction, while for alpha < 2.5 we obtain the KLB spectrum.