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Wave turbulence in a rotating channel

Published online by Cambridge University Press:  13 February 2014

Julian F. Scott
Julian F. Scott*
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, Université de Lyon, Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

This paper describes wave-turbulence closure and its consequences for rapidly rotating (i.e. small Rossby number) turbulence confined by two infinite, parallel walls perpendicular to the rotation axis. Expressing the flow as a combination of inertial waveguide modes leads to a spectral matrix, whose diagonal elements express the distribution of energy over modes and whose off-diagonal elements represent correlations between modes of different orders. In preparation for wave-turbulence closure, the flow is decomposed into two-dimensional and wave components. The former is found to evolve as if it were a classical, two-dimensional, non-rotating flow, but with wall friction due to Ekman pumping by the boundary layers. Evolution equations for the wave-component elements of the spectral matrix are derived using a wave-turbulence approach. Detailed analysis of these equations shows that, surprisingly, the two-dimensional component has no effect on wave-component energetics. As expected for wave turbulence, energy transfer between wave modes is via resonant triads and takes place at times $O(\varepsilon ^{-2})$ multiples of the rotational period, where $\varepsilon $ is the Rossby number. Despite playing no role in wave-mode energetics, the two-dimensional component produces decay of the off-diagonal elements of the spectral matrix on a time scale that is small compared with $O(\varepsilon ^{-2})$ rotation periods. There are thus three asymptotically distinct stages in the evolution of the turbulence in the limit of small Rossby number: the two-dimensional flow begins to evolve at the usual large-eddy turnover time scale ($O(\varepsilon ^{-1})$ multiples of the rotation period) and continues to develop thereafter. This is followed by decorrelation of different wave orders and finally evolution of the wave energy spectra due to resonant interactions.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Rotating turbulence Turbulent Flows: Wave-turbulence interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 741 , 25 February 2014 , pp. 316 - 349
Copyright
© 2014 Cambridge University Press 

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References

Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.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
Boffetta, G. & Ecke, R. E. 2012 Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451.Google Scholar
Bourouiba, L. 2008 Discreteness and resolution effects in rapidly rotating turbulence. Phys. Rev. E 78, 056309.CrossRefGoogle ScholarPubMed
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, 302–332.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68, 015301.Google Scholar
Kartashova, E. A. 1994 Weakly nonlinear theory of finite-size effects in resonators. Phys. Rev. Lett. 72, 2013–2016.Google Scholar
Kraichnan, R. H. & Montgomery, D. 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619.Google Scholar
Lvov, Y. V., Nazarenko, S. & Pokorni, B. 2006 Discreteness and its effects on water-wave turbulence. Physica D 218, 24–35.Google Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer.Google Scholar
Newell, A. C. & Rumpf, B. 2011 Wave turbulence. Annu. Rev. Fluid Mech. 43, 59–78.Google Scholar
Pushkarev, A. N. 1999 On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves. Eur. J. Mech. (B/Fluids) 18, 345–351.Google Scholar
Racz, J.-P. & Scott, J. F. 2008 Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. Part 1. Linear theory. J. Fluid Mech. 595, 265–290.CrossRefGoogle Scholar
Smith, L. M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large-scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11, 1608–1622.Google Scholar
Teitelbaum, T. & Mininni, P. D. 2012 Decay of Batchelor and Saffman rotating turbulence. Phys. Rev. E 86, 066320.CrossRefGoogle ScholarPubMed
Turner, L. 1999 Macroscopic structures of inhomogeneous, Navier–Stokes turbulence. Phys. Fluids 11, 2367–2380.Google Scholar
Waleffe, F. 1993 Inertial transfers in the helical decomposition. Phys. Fluids A5, 677–685.Google Scholar
Yang, Y.-T., Su, W.-D. & Wu, J.-Z. 2010 Helical-wave decomposition and applications to channel turbulence with streamwise rotation. J. Fluid Mech. 662, 91–122.CrossRefGoogle Scholar
Zakharov, V. E., Korotkevich, A. O., Pushkarev, A. N. & Dyachenko, A. I. 2005 Mesoscopic Wave Turbulence. J. Expl Theor. Phys. Lett. 82 (8), 487–491.Google Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I. Springer.Google Scholar