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Energy transfer in resonant and near-resonant internal wave triads for weakly non-uniform stratifications. Part 1. Unbounded domain

Published online by Cambridge University Press:  15 July 2020

Saranraj Gururaj Open the ORCID record for Saranraj Gururaj [Opens in a new window]  and
Anirban Guha Open the ORCID record for Anirban Guha [Opens in a new window]
Saranraj Gururaj
Affiliation:
School of Science and Engineering, University of Dundee, DundeeDD1 4HN, UK
Anirban Guha*
Affiliation:
School of Science and Engineering, University of Dundee, DundeeDD1 4HN, UK
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, using multiple-scale analysis, we derive a generalized mathematical model for amplitude evolution, and for calculating the energy exchange in resonant and near-resonant global triads consisting of weakly nonlinear internal gravity wavepackets in weakly non-uniform density stratifications in an unbounded domain in the presence of viscous and rotational effects. Such triad interactions are one of the mechanisms by which high-wavenumber internal waves lead to ocean turbulence and mixing via parametric subharmonic instability. Non-uniform stratification introduces detuning – mismatch in the vertical wavenumber triad condition, which may strongly affect the energy transfer process. We investigate in detail how factors like wavepacket width, group speeds, nonlinear coupling coefficients, detuning and viscosity affect energy transfer in weakly varying stratification. We also investigate the effect of detuning on energy transfer in varying stratification for different daughter wave combinations of a fixed parent wave. We find limitations of the well-known ‘pump-wave approximation’ and derive a non-dimensional number, which can be evaluated from initial conditions, that can predict the maximum energy transferred from the parent wave during the later stages. Two additional non-dimensional numbers, based on various factors affecting energy transfer between near-resonant wavepackets, have also been defined. Moreover, we identify the optimal background stratification in a medium of varying stratification for the parent wave to form a triad with no detuning so that the energy transfer is maximum.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows Instability: Parametric instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A6
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Bourget, B., Dauxois, T., Joubaud, S. & Odier, P. 2013 Experimental study of parametric subharmonic instability for internal plane waves. J. Fluid Mech. 723, 1–20.CrossRefGoogle Scholar
Bourget, B., Scolan, H., Dauxois, T., Le Bars, M., Odier, P. & Joubaud, S. 2014 Finite-size effects in parametric subharmonic instability. J. Fluid Mech. 759, 739–750.CrossRefGoogle Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2 (2), 023068.CrossRefGoogle Scholar
Craik, A. D. D. & Adam, J. A. 1978 Evolution in space and time of resonant wave triads. I. The ‘pump-wave approximation’. Proc. R. Soc. Lond. A 363 (1713), 243–255.Google Scholar
Diamessis, P. J., Wunsch, S., Delwiche, I. & Richter, M. P. 2014 Nonlinear generation of harmonics through the interaction of an internal wave beam with a model oceanic pycnocline. Dyn. Atmos. Oceans 66, 110–137.CrossRefGoogle Scholar
Gayen, B. & Sarkar, S. 2013 Degradation of an internal wave beam by parametric subharmonic instability in an upper ocean pycnocline. J. Geophys. Res. Oceans 118 (9), 4689–4698.CrossRefGoogle Scholar
Grimshaw, R. 1988 Resonant wave interactions in a stratified shear flow. J. Fluid Mech. 190, 357–374.CrossRefGoogle Scholar
Grimshaw, R. 1994 Resonant wave interactions near a critical level in a stratified shear flow. J. Fluid Mech. 269, 1–22.CrossRefGoogle Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30 (4), 737–739.CrossRefGoogle Scholar
Holbrook, W. S., Fer, I. & Schmitt, R. W. 2009 Images of internal tides near the Norwegian continental slope. Geophys. Res. Lett. 36 (24), L00D10.CrossRefGoogle Scholar
Karimi, H. H. & Akylas, T. R. 2014 Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains. J. Fluid Mech. 757, 381–402.CrossRefGoogle Scholar
Karimi, H. H. & Akylas, T. R. 2017 Near-inertial parametric subharmonic instability of internal wave beams. Phys. Rev. Fluids 2, 074801.CrossRefGoogle Scholar
Lamb, K. G. 2007 Tidally generated near-resonant internal wave triads at a shelf break. Geophys. Res. Lett. 34 (18), L18607.CrossRefGoogle Scholar
Levine, M. D. 2002 A modification of the Garrett-Munk internal wave spectrum. J. Phys. Oceanogr. 32 (11), 3166–3181.2.0.CO;2>CrossRefGoogle Scholar
Lighthill, M. J. & Lighthill, J. 2001 Waves in Fluids. Cambridge University Press.Google Scholar
MacKinnon, J. A. & Winters, K. B. 2005 Subtropical catastrophe: significant loss of low-mode tidal energy at 28.9$^{\circ }$. Geophys. Res. Lett. 32 (15), L15605.CrossRefGoogle Scholar
Mathur, M. & Peacock, T. 2009 Internal wave beam propagation in non-uniform stratifications. J. Fluid Mech. 639, 133–152.CrossRefGoogle Scholar
Maurer, P., Joubaud, S. & Odier, P. 2016 Generation and stability of inertia–gravity waves. J. Fluid Mech. 808, 539–561.CrossRefGoogle Scholar
McEwan, A. D. & Plumb, R. A. 1977 Off-resonant amplification of finite internal wave packets. Dyn. Atmos. Oceans 2 (1), 83–105.CrossRefGoogle Scholar
St. Laurent, L. & Garrett, C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32 (10), 2882–2899.2.0.CO;2>CrossRefGoogle Scholar
St. Laurent, L. & Nash, J. D. 2004 An examination of the radiative and dissipative properties of deep ocean internal tides. Deep-Sea Res. II 51 (25), 3029–3042.CrossRefGoogle Scholar
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.CrossRefGoogle Scholar
Sutherland, B. R. 2016 Excitation of superharmonics by internal modes in non-uniformly stratified fluid. J. Fluid Mech. 793, 335–352.CrossRefGoogle Scholar
Varma, D. & Mathur, M. 2017 Internal wave resonant triads in finite-depth non-uniform stratifications. J. Fluid Mech. 824, 286–311.CrossRefGoogle Scholar
Vic, C., Garabato, A. C. N., Green, J. A. M., Waterhouse, A. F., Zhao, Z., Mélet, A., De Lavergne, C., Buijsman, M. C. & Stephenson, G. R. 2019 Deep-ocean mixing driven by small-scale internal tides. Nat. Commun. 10 (1), 1–9.CrossRefGoogle ScholarPubMed
Wunsch, S. 2017 Harmonic generation by nonlinear self-interaction of a single internal wave mode. J. Fluid Mech. 828, 630–647.CrossRefGoogle Scholar
Zhao, Z. & Alford, M. H. 2009 New altimetric estimates of mode-1 M2 internal tides in the central north Pacific ocean. J. Phys. Oceanogr. 39 (7), 1669–1684.CrossRefGoogle Scholar