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Restricted equilibrium and the energy cascade in rotating and stratified flows

Published online by Cambridge University Press:  09 October 2014

Corentin Herbert ,
Annick Pouquet  and
Raffaele Marino
Corentin Herbert*
Affiliation:
National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA
Annick Pouquet
Affiliation:
National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309, USA
Raffaele Marino
Affiliation:
National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307, USA Institute for Chemical-Physical Processes - IPCF/CNR, Rende (CS), 87036, Italy
*
†Email address for correspondence: cherbert@ucar.edu
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Abstract

Most turbulent flows appearing in nature (e.g. geophysical and astrophysical flows) are subjected to strong rotation and stratification. These effects break the symmetries of classical, homogenous isotropic turbulence. In doing so, they introduce a natural decomposition of phase space in terms of wave modes and potential vorticity modes. The appearance of a new time scale, associated with the propagation of waves, hinders the understanding of energy transfers across scales. For instance, it is difficult to predict a priori whether the energy cascades downscale as in homogeneous isotropic turbulence or upscale as expected from balanced dynamics. In this paper, we suggest a theoretical approach based on equilibrium statistical mechanics for the ideal system, inspired by the restricted partition function formalism introduced in metastability studies. We focus on the qualitative features of the inviscid system, taking into account either all the modes or just the slow modes. Specifically, we show that at absolute equilibrium, i.e. when all the modes are considered, no negative temperature states exist, and the isotropic energy spectrum is close to equipartition. By contrast, when the statistics is restricted to the contributions of the slow modes, we find that in the presence of rotation, there exists a regime of negative temperature featuring an infrared divergence in both the isotropic and the axisymmetric average energy spectrum, characteristic of an inverse cascade regime. Such regimes are not allowed for purely stratified flows, even in the restricted ensemble, because the slow manifold then partitions into modes that carry potential vorticity on the one hand, and hydrostatically balanced but vorticity-free modes, the so-called vertical shear horizontal flows, on the other hand, which forbid the appearance of negative temperatures.

JFM classification

Turbulent Flows: Rotating turbulence Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 374 - 406
Copyright
© 2014 Cambridge University Press 

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