Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T13:57:09.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete Hamiltonian formalism for inertial waves in rotating fluids

Published online by Cambridge University Press:  13 October 2017

A. A. Gelash Open the ORCID record for A. A. Gelash [Opens in a new window] ,
V. S. L’vov  and
V. E. Zakharov
A. A. Gelash*
Affiliation:
Novosibirsk State University, Novosibirsk, 630090, Russia Institute of Thermophysics, SB RAS, Novosibirsk, 630090, Russia
V. S. L’vov
Affiliation:
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, 76100, Israel
V. E. Zakharov
Affiliation:
Novosibirsk State University, Novosibirsk, 630090, Russia Department of Mathematics, University of Arizona, AZ 857201 Tucson, USA Lebedev Physical Institute, RAS, Moscow, 119991, Russia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A complete Hamiltonian formalism is suggested for inertial waves in rotating incompressible fluids. Resonance three-wave interaction processes – decay instability and confluence of two waves – are shown to play a key role in the weakly nonlinear dynamics and statistics of inertial waves in the rapid rotation case. Future applications of the Hamiltonian approach to inertial wave theory are investigated and discussed.

JFM classification

Mathematical Foundations: Hamiltonian theory Instability: Nonlinear instability Turbulent Flows: Rotating turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 128 - 150
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 2160.CrossRefGoogle ScholarPubMed
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121.Google Scholar
Bordes, G., Moisy, F., Dauxois, T. & Cortet, P. P. 2012 Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid. Phys. Fluids 24 (1), 014105.Google Scholar
Brouzet, C., Sibgatullin, I. N., Scolan, H., Ermanyuk, E. V. & Dauxois, T. 2016 Internal wave attractors examined using laboratory experiments and 3D numerical simulations. J. Fluid Mech. 793, 109–131.Google Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.Google Scholar
Cambon, C., Rubinstein, R. & Godeferd, F. S. 2004 Advances in wave turbulence: rapidly rotating flows. New J. Phys. 6 (1), 73.Google Scholar
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier–Stokes equations. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 406, pp. 13–26. The Royal Society.Google Scholar
Dyachenko, A. I., Korotkevich, A. O. & Zakharov, V. E. 2003 Decay of the monochromatic capillary wave. JETP Lett. 77 (9), 477–481.Google Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68 (1), 015301.Google ScholarPubMed
Galtier, S. 2014 Theory for helical turbulence under fast rotation. Phys. Rev. E 89 (4), 041001.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Korotkevich, A. O., Dyachenko, A. I. & Zakharov, V. E. 2016 Numerical simulation of surface waves instability on a homogeneous grid. Physica D 321, 51–66.Google Scholar
Kuznetsov, E. A. 1972 Turbulence of ion sound in a plasma located in a magnetic field. Sov. Phys. JETP 35, 310.Google Scholar
Kuznetsov, E. A. & Mikhailov, A. V. 1980 On the topological meaning of canonical Clebsch variables. Phys. Lett. A 77 (1), 37–38.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Le Gal, P. 2013 Waves and instabilities in rotating and stratified flows. In Fluid Dynamics in Physics, Engineering and Environmental Applications, pp. 25–40. Springer.Google Scholar
di Leoni, P. C. & Mininni, P. D. 2016 Quantifying resonant and near-resonant interactions in rotating turbulence. J. Fluid Mech. 809, 821–842.Google Scholar
L’vov, V. S. 1994 Wave Turbulence under Parametric Excitation. Springer.Google Scholar
Lvov, Y. V., Polzin, K. L. & Tabak, E. G. 2004 Energy spectra of the ocean’s internal wave field: theory and observations. Phys. Rev. Lett. 92 (12), 128501.Google Scholar
Lvov, Y. V. & Tabak, E. G. 2001 Hamiltonian formalism and the Garrett–Munk spectrum of internal waves in the ocean. Phys. Rev. Lett. 87 (16), 168501.CrossRefGoogle ScholarPubMed
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F. P. A. 1997 Observation of an internal wave attractor in a confined, stably stratified fluid. Nature 388 (6642), 557–561.Google Scholar
Messio, L., Morize, C., Rabaud, M. & Moisy, F. 2008 Experimental observation using particle image velocimetry of inertial waves in a rotating fluid. Exp. Fluids 44 (4), 519–528.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467.Google Scholar
Nazarenko, S. 2011 Wave Turbulence. Springer.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Sagaut, P. & Cambon, C. 2008 Homogeneous Turbulence Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20 (1), 225–256.CrossRefGoogle Scholar
Scolan, H., Ermanyuk, E. & Dauxois, T. 2013 Nonlinear fate of internal wave attractors. Phys. Rev. Lett. 110 (23), 234501.Google Scholar
Sen, A., Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence. Phys. Rev. E 86 (3), 036319.Google ScholarPubMed
Waleffe, F. 1993 Inertial transfers in the helical decomposition. Phys. Fluids A 5 (3), 677–685.Google Scholar
Yakhot, V. & Zakharov, V. 1993 Hidden conservation laws in hydrodynamics; energy and dissipation rate fluctuation spectra in strong turbulence. Physica D 64 (4), 379–394.CrossRefGoogle Scholar
Zakharov, V. E. 1971 Hamiltonian formalism for hydrodynamic plasma models. Sov. Phys. JETP 33, 927–932.Google Scholar
Zakharov, V. E. & Kuznetsov, E. A. 1997 Hamiltonian formalism for nonlinear waves. Phys. Usp. 40 (11), 1087–1116.Google Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence 1. Wave Turbulence. Springer.CrossRefGoogle Scholar
Zakharov, V. E., L’vov, V. S. & Starobinets, S. S. 1971 Stationary nonlinear theory of parametric excitation of waves. Sov. Phys. JETP 32, 656.Google Scholar