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

Published online by Cambridge University Press:  14 May 2013

B. H. Burgess  and
T. G. Shepherd
B. H. Burgess*
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, Canada M5S 1A7
T. G. Shepherd
Affiliation:
Department of Physics, University of Toronto, Toronto, ON, Canada M5S 1A7 Department of Meteorology, University of Reading, Reading, Berkshire RG6 6BB, UK
*
†Email address for correspondence: [email protected]
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Abstract

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We study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in $\alpha $ turbulence, governed by $\partial \theta / \partial t+ J(\psi , \theta )= \nu {\nabla }^{2} \theta + f$, where $\theta = {(- \Delta )}^{\alpha / 2} \psi $ is generalized vorticity, and $\hat {\psi } (\boldsymbol{k})= {k}^{- \alpha } \hat {\theta } (\boldsymbol{k})$ 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 Fjørtoft analysis, which we extend to general $\alpha $, 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. Non-local triads contribute increasingly to the energy flux as $\alpha $ increases. More importantly, the energy cascade is downscale in the self-similar inertial range for $2. 5\lt \alpha \lt 10$. 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 $\alpha \lt 4$. However, downscale energy flux in the EDQNM self-similar inertial range for $\alpha \gt 2. 5$ leads us to predict that any inverse cascade for $\alpha \geq 2. 5$ will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for $\alpha \geq 2. 5$ is significantly steeper than the KLB prediction, while for $\alpha \lt 2. 5$ we obtain the KLB spectrum.

JFM classification

Turbulent Flows: Isotropic turbulence Turbulent Flows: Turbulent Flows Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 332 - 371
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence . The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
©2013 Cambridge University Press.

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