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On energy exchanges between eddies and the mean flow in quasigeostrophic turbulence

Published online by Cambridge University Press:  17 December 2019

William Barham  and
Ian Grooms Open the ORCID record for Ian Grooms [Opens in a new window]
William Barham
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Ian Grooms*
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
*
†Email address for correspondence: [email protected]
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Abstract

We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.

JFM classification

Geophysical and Geological Flows: Geostrophic turbulence Geophysical and Geological Flows: Quasi-geostrophic flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A3
Copyright
© 2019 Cambridge University Press

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