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Exploiting self-organized criticality in strongly stratified turbulence

Published online by Cambridge University Press:  23 December 2021

Gregory P. Chini*
Affiliation:
Program in Integrated Applied Mathematics, University of New Hampshire, Durham, NH03824, USA Department of Mechanical Engineering, University of New Hampshire, Durham, NH03824, USA
Guillaume Michel
Affiliation:
Institut Jean Le Rond d'Alembert, Sorbonne Université, CNRS, UMR 7190, ParisF-75005, France
Keith Julien
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO80309, USA
Cesar B. Rocha
Affiliation:
Department of Marine Sciences, University of Connecticut, Storrs, CT06269, USA
Colm-cille P. Caulfield
Affiliation:
BP Institute, University of Cambridge, CambridgeCB3 0EZ, UK Department of Applied Mathematics & Theoretical Physics, University of Cambridge, CambrdgeCB3 0WA, UK
*
Email address for correspondence: greg.chini@unh.edu

Abstract

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.

Type
JFM Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Chini et al. supplementary movie

MTQL simulation of strongly stratified Kolmogorov flow.

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Supplementary material: File

Chini et al. supplementary material

Python codes

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