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Critical transitions in thin layer turbulence

Published online by Cambridge University Press:  01 June 2017

Santiago Jose Benavides  and
Alexandros Alexakis Open the ORCID record for Alexandros Alexakis [Opens in a new window]
Santiago Jose Benavides
Affiliation:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139-4307, USA
Alexandros Alexakis*
Affiliation:
Laboratoire de Physique Statistique, École Normale Supérieure, CNRS, Université Pierre et Marie Curie, Université Paris Diderot, Paris 75005, France
*
†Email address for correspondence: alexakis@lps.ens.fr
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Abstract

We investigate a model of thin layer turbulence that follows the evolution of the two-dimensional motions $\boldsymbol{u}_{2D}(x,y)$ along the horizontal directions $(x,y)$ coupled to a single Fourier mode along the vertical direction ( $z$ ) of the form $\boldsymbol{u}_{q}(x,y,z)=[v_{x}(x,y)\sin (qz),v_{y}(x,y)\sin (qz),v_{z}(x,y)\cos (qz)]$ , reducing thus the system to two coupled, two-dimensional equations. The model, despite its simplicity and ad hoc construction, displays a rich behaviour. Its reduced dimensionality allows a thorough investigation of the transition from a forward to an inverse cascade of energy as the thickness of the layer $H=\unicode[STIX]{x03C0}/q$ is varied. Starting from a thick layer and reducing its thickness it is shown that two critical heights are met: (i) one for which the forward unidirectional cascade (similar to three-dimensional turbulence) transitions to a bidirectional cascade transferring energy to both small and large scales and (ii) one for which the bidirectional cascade transitions to a unidirectional inverse cascade when the layer becomes very thin (similar to two-dimensional turbulence). The two critical heights are shown to have different properties close to criticality that we are able to analyse with numerical simulations for a wide range of Reynolds numbers and aspect ratios.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Turbulent Flows: Turbulent Flows Turbulent Flows: Turbulent transition

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Papers
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Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 364 - 385
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© 2017 Cambridge University Press 

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