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

Abstract

We study the degree to which Kraichnan-Leith-Batchelor (KLB) phenomenology describes two-dimensional energy cascades in α turbulence, governed by ∑θ/ ∑t+ J(Ψ, θ)= ν∇2 θ+ f, where θ= (-Δ)α/2 Ψ is generalized vorticity, and Ψ(k)= k-αθ (k) in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow ( α= 1), regular two-dimensional flow ( α= 2) and rotating shallow flow ( α= 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 α, and point out their limitations. Using an α-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. Non-local triads contribute increasingly to the energy flux as α increases. More importantly, the energy cascade is downscale in the self-similar inertial range for 2. 5 <4. However, downscale energy flux in the EDQNM self-similar inertial range for α>2. 5 leads us to predict that any inverse cascade for α≥2. 5 will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for α≥2. 5 is significantly steeper than the KLB prediction, while for α<2. 5 we obtain the KLB spectrum. © Cambridge University Press 2013.