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The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio

Published online by Cambridge University Press:  17 November 2017

T. S. van den Bremer Open the ORCID record for T. S. van den Bremer [Opens in a new window]  and
B. R. Sutherland Open the ORCID record for B. R. Sutherland [Opens in a new window]
T. S. van den Bremer*
Affiliation:
School of Engineering, University of Edinburgh, The King’s Buildings, Robert Stevenson Road, Edinburgh EH9 3FB, UK
B. R. Sutherland
Affiliation:
Departments of Physics, and Earth and Atmospheric Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2E3
*
†Email address for correspondence: [email protected]
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Abstract

We examine the wave-induced flow of small-amplitude, quasi-monochromatic, three-dimensional, Boussinesq internal gravity wavepackets in a uniformly stratified ambient. It has been known since Bretherton (J. Fluid Mech., vol. 36 (4), 1969, pp. 785–803) that one-, two- and three-dimensional wavepackets induce qualitatively different flows. Whereas the wave-induced mean flow for compact three-dimensional wavepackets consists of a purely horizontal localized circulation that translates with and around the wavepacket, known as the Bretherton flow, such a flow is prohibited for a two-dimensional wavepacket of infinite spanwise extent, which instead induces a non-local internal wave response that is long compared with the streamwise extent of the wavepacket. One-dimensional (horizontally periodic) wavepackets induce a horizontal, non-divergent unidirectional flow. Through perturbation theory for quasi-monochromatic wavepackets of arbitrary aspect ratio, we predict for which aspect ratios which type of induced mean flow dominates. We compose a regime diagram that delineates whether the induced flow is comparable to that of one-, two- or compact three-dimensional wavepackets. The predictions agree well with the results of fully nonlinear three-dimensional numerical simulations.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 834 , 10 January 2018 , pp. 385 - 408
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Copyright
© 2017 Cambridge University Press 

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References

Acheson, D. J. 1976 On over-reflexion. J. Fluid Mech. 77, 433–472.Google Scholar
Akylas, T. R. & Tabaei, A. 2005 Resonant self-acceleration and instability of nonlinear internal gravity wavetrains. In Frontiers of Nonlinear Physics (ed. Litvak, A.), pp. 129–135. Institute of Applied Physics.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave action and its relatives. J. Fluid Mech. 89, 647–664.Google Scholar
van den Bremer, T. S. & Sutherland, B. R. 2014 The mean flow and long waves induced by two-dimensional internal gravity wavepackets. Phys. Fluids 26, 106601.Google Scholar
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow. Q. J. R. Meteorol. Soc. 92, 466–480.Google Scholar
Bretherton, F. P. 1969 On the mean motion induced by gravity waves. J. Fluid Mech. 36 (4), 785–803.Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302, 529–554.Google Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.Google Scholar
Bühler, O. 2014 Waves and Mean Flows, 2nd edn. Cambridge University Press.Google Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.Google Scholar
Bühler, O. & McIntyre, M. E. 2003 Remote recoil: a new wave–mean interaction effect. J. Fluid Mech. 492, 207–230.Google Scholar
Dosser, H. V. & Sutherland, B. R. 2011a Anelastic internal wavepacket evolution and stability. J. Atmos. Sci. 68, 2844–2859.Google Scholar
Dosser, H. V. & Sutherland, B. R. 2011b Weakly nonlinear non-Boussinesq internal gravity wavepackets. Physica D 240, 346–356.Google Scholar
Drazin, P. G. 1977 On the instability of an internal gravity wave. Proc. R. Soc. Lond. A 356, 411–432.Google Scholar
Grimshaw, R. H. J. 1972 Nonlinear internal gravity waves in a slowly varying medium. J. Fluid Mech. 54, 193–207.Google Scholar
Kataoka, T. & Akylas, T. R. 2013 Stability of internal gravity wave beams to three-dimensional modulations. J. Fluid Mech. 736, 67–90.Google Scholar
Kataoka, T. & Akylas, T. R. 2015 On three-dimensional internal gravity wave beams and induced large-scale mean flows. J. Fluid Mech. 769, 621–634.Google Scholar
Klostermeyer, J. 1991 Two-dimensional and three-dimensional parametric instabilities in finite amplitude internal gravity waves. Geophys. Astrophys. Fluid Dyn. 61, 1–25.CrossRefGoogle Scholar
Mied, R. R. 1976 The occurrence of parametric instabilities in finite-amplitude internal gravity waves. J. Fluid Mech. 78, 763–784.Google Scholar
Scinocca, J. F. & Shepherd, T. G. 1992 Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations. J. Atmos. Sci. 49, 5–27.Google Scholar
Shrira, V. I. 1981 On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves. Intl J. Non-Linear Mech. 16, 129–138.Google Scholar
Sutherland, B. R. 2006a Internal wave instability: wave–wave versus wave-induced mean flow interactions. Phys. Fluids. 18, 074107.Google Scholar
Sutherland, B. R. 2006b Weakly nonlinear internal wavepackets. J. Fluid Mech. 569, 249–258.Google Scholar
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.Google Scholar
Tabaei, A. & Akylas, T. R. 2007 Resonant long–short wave interactions in an unbounded rotating stratified fluid. Stud. Appl. Maths 119, 271–296; TA07.Google Scholar
Voisin, B. 1991 Internal wave generation in uniformly stratified fluids. Part 1. Green’s function and point sources. J. Fluid Mech. 231, 439–480.Google Scholar
Wagner, G. L. & Young, W. R. 2015 Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech. 785, 401–424.CrossRefGoogle Scholar
Xie, J.-H. & Vanneste, J. 2015 A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143–169.Google Scholar

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