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Evolution of weak, homogeneous turbulence with rotation and stratification

Published online by Cambridge University Press:  11 January 2024

J.F. Scott Open the ORCID record for J.F. Scott [Opens in a new window]  and
C. Cambon
J.F. Scott*
Affiliation:
Ecole Centrale de Lyon, CNRS, Universite Claude Bernard Lyon 1, INSA Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique (UMR5509), 69134 Ecully CEDEX, France
C. Cambon
Affiliation:
Ecole Centrale de Lyon, CNRS, Universite Claude Bernard Lyon 1, INSA Lyon, Laboratoire de Mécanique des Fluides et d'Acoustique (UMR5509), 69134 Ecully CEDEX, France
*
† Email address for correspondence: julian.scott@ec-lyon.fr
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Abstract

This article concerns the time-evolution, spectral structure and scaling of weak turbulence subject to rotation and stable stratification. The flow is expressed as a combination of particular solutions, referred to as modes, of the linearised governing equations without viscosity or diffusion. Modes are of two types: oscillatory ones which represent inertial-gravity waves and time-independent ones that express a non-propagating (NP) component of the flow. The presence of the NP component, which plays an active role in the dynamics apart from in the case of pure rotation, renders wave-turbulence analysis problematic because the NP mode is non-dispersive. Equations are derived for the time evolution of the modal amplitudes, evolution which is due to nonlinearity and visco-diffusion. Subsequent analysis assumes that one or other (or both) of the Rossby and Froude numbers is small (weak turbulence). Given this assumption, the NP component is found to evolve independently of the wave one and a numerical scheme, similar to, though significantly different from classical direct numerical simulation, is used to determine its time evolution. The treatment of the wave component assumes its amplitude is large compared with the NP one, otherwise there are seemingly intractable difficulties of closure in the analysis. Given this further assumption, the wave component decouples from the NP one. Evolution equations for the wave spectra are derived using wave-turbulence analysis and are integrated numerically. As might be expected, these equations indicate that nonlinear coupling of wave modes is dominated by resonances. Results are given for both the NP and wave components.

JFM classification

Turbulent Flows: Wave-turbulence interactions Turbulent Flows: Stratified turbulence Turbulent Flows: Rotating turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 979 , 25 January 2024 , A17
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52, 4410–4428.2.0.CO;2>CrossRefGoogle Scholar
Bellet, F. 2003 Etude asymptotique de la turbulence d'ondes en rotation. PhD thesis, Ecole Centrale de Lyon.Google Scholar
Bellet, F., Godeferd, F.S., Scott, J.F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121.CrossRefGoogle Scholar
Benney, D.J. & Newell, A.C. 1969 Random wave closures. Stud. Appl. Maths 48, 29–53.CrossRefGoogle Scholar
Benney, D.J. & Saffman, P.G. 1966 Nonlinear interactions of random waves in a dispersive medium. Proc. R. Soc. A 289, 301–320.Google Scholar
Cambon, C. 2001 Turbulence and vortex structures in rotating and stratified flows. Eur. J. Mech. (B/Fluids) 20, 489–510.CrossRefGoogle Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.CrossRefGoogle Scholar
Cambon, C., Mansour, N.N. & Godeferd, F.S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303–332.CrossRefGoogle Scholar
Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. 1988 Spectral Methods in Fluid Dynamics. Springer-Verlag.CrossRefGoogle Scholar
Charney, J.G. 1948 On the scale of atmospheric motions. Geofys. Publ. Oslo 17 (2), 1–17.Google Scholar
Charney, J.G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Coleman, G.N., Ferziger, J.H. & Spalart, P.R. 1992 Direct simulation of the stably stratified turbulent Ekman layer. J. Fluid Mech. 244, 677–712 (Corrigendum: J. Fluid Mech. 252, 721).CrossRefGoogle Scholar
Deng, Y. & Hani, Z. 2021 On the derivation of the wave kinetic equation for NLS. Forum Maths Pi 9, e6 1–37.Google Scholar
Embid, P.F. & Madja, A.J. 1998 Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers. Geophys. Astrophys. Fluid Dyn. 87, 1–50.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68, 015301.CrossRefGoogle ScholarPubMed
Godeferd, F.S. & Cambon, C. 1994 Detailed investigation of energy transfers in homogeneous stratified turbulence. Phys. Fluids 6, 2084–2100.CrossRefGoogle Scholar
Godeferd, F.S. & Staquet, C. 2003 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large-scale and small-scale anisotropy. J. Fluid Mech. 486, 115–159.CrossRefGoogle Scholar
Haugen, N.E.L. & Brandenburg, A. 2004 Inertial range scaling in numerical turbulence with hyperviscosity. Phys. Rev. E 70, 026405.CrossRefGoogle ScholarPubMed
Herbert, C., Pouquet, A. & Marino, R. 2014 Restricted equilibrium and the energy cascade in rotating and stratified flows. J. Fluid Mech. 758, 374–406.CrossRefGoogle Scholar
Hossain, M. 1994 Reduction in the dimensionality of turbulence due to a strong rotation. Phys. Fluids 6, 1077–1080.CrossRefGoogle Scholar
Liechtenstein, L., Godeferd, F.S. & Cambon, C. 2005 Nonlinear formation of structures in rotating stratified turbulence. J. Turbul. 6, 1–18.CrossRefGoogle Scholar
Marino, R., Mininni, P.D., Rosenberg, D.L. & Pouquet, A. 2013 Inverse cascades in rotating stratified turbulence: fast growth of large scales. Europhys. Lett. 102, 44006.CrossRefGoogle Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer.CrossRefGoogle Scholar
Newell, A.C. & Rumpf, B. 2011 Wave turbulence. Annu. Rev. Fluid Mech. 43, 59–78.CrossRefGoogle Scholar
Orszag, S.A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.CrossRefGoogle Scholar
Orszag, S.A. & Patterson, G.S. 1972 Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76–79.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics. Springer-Verlag.CrossRefGoogle Scholar
Sagaut, P. & Cambon, C. 2018 Homogeneous Turbulence Dynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Smith, L.M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.CrossRefGoogle Scholar
Waleffe, F. 1991 Non-linear interactions in homogeneous turbulence with and without background rotation. In Annual Research Briefs, Center for Turbulence Research.Google Scholar
Zakharov, V.E., Lvov, V.S. & Falkovich, G.E. 1992 Kolmogorov Spectra of Turbulence I – Wave Turbulence. Springer-Verlag.CrossRefGoogle Scholar
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