Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T05:23:37.190Z Has data issue: false hasContentIssue false
  • English
  • Français

The nonlinear evolution of internal tides. Part 2. Lagrangian transport by periodic and modulated waves

Published online by Cambridge University Press:  08 September 2022

Bruce R. Sutherland Open the ORCID record for Bruce R. Sutherland [Opens in a new window]  and
Houssam Yassin Open the ORCID record for Houssam Yassin [Opens in a new window]
Bruce R. Sutherland*
Affiliation:
Department of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada Department of Earth & Atmospheric Sciences, University of Alberta, Edmonton, AB T6G 2E3, Canada
Houssam Yassin
Affiliation:
Department of Atmospheric and Oceanic Sciences, Princeton University, Princeton, NJ 08540-6654, USA
*
†Email address for correspondence: bruce.sutherland@ualberta.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine Lagrangian transport by a nonlinearly evolving vertical mode-1 internal tide in non-uniform stratification. In a companion paper (Sutherland & Dhaliwal, J. Fluid Mech., 2022, in press) it was shown that a parent internal tide can excite successive superharmonics that superimpose to form a solitary wave train. Despite this transformation, here we show that the collective forcing by the parent wave and superharmonics is effectively steady in time. Thus we derive relatively simple formulae for the Stokes drift and induced Eulerian flow associated with the waves under the assumption that the parent waves and superharmonics are long compared with the fluid depth. In all cases, the Stokes drift exhibits a mixed mode-1 and mode-2 vertical structure with the flow being in the waveward direction at the surface. If the background rotation is non-negligible, the vertical structure of the induced Eulerian flow is equal and opposite to that of the Stokes drift. This flow periodically increases and decreases at the inertial frequency with maximum magnitude twice that of the Stokes drift. When superimposed with the Stokes drift, the Lagrangian flow at the surface periodically changes from positive to negative over one inertial period. If the background rotation is zero, the induced Eulerian flow evolves non-negligibly in time and space for horizontally modulated waves: the depth below the surface of the positive Lagrangian flow becomes shallower ahead of the peak of the amplitude envelope and becomes deeper in the lee of the peak. These predictions are well-captured by fully nonlinear numerical simulations.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Waves in rotating fluids Geophysical and Geological Flows: Ocean processes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 948 , 10 October 2022 , A22
Check for updates
Copyright
© The Author(s), 2022. Published by 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

Alford, M.H., MacKinnon, J.A., Simmons, H.L. & Nash, J.D. 2016 Near-inertial internal gravity waves in the ocean. Annu. Rev. Mar. Sci. 8, 95–123.CrossRefGoogle ScholarPubMed
Baker, L. & Sutherland, B.R. 2020 The evolution of superharmonics excited by internal tides in non-uniform stratification. J. Fluid Mech. 891, R1.CrossRefGoogle Scholar
van den Bremer, T.S., Yassin, H. & Sutherland, B.R. 2019 Lagrangian transport by vertically confined internal gravity wavepackets. J. Fluid Mech. 864, 348–380.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M.E. 1998 On non-dissipative wave-mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.CrossRefGoogle Scholar
Dong, W., Bühler, O. & Smith, K.S. 2020 Mean flows induced by inertia-gravity waves in a vertically confined domain. J. Fluid Mech. 890, A6.CrossRefGoogle Scholar
Grimshaw, R.H.J. 1975 Nonlinear internal gravity waves and their interaction with the mean wind. J. Atmos. Sci. 32, 1779–1793.2.0.CO;2>CrossRefGoogle Scholar
Grimshaw, R.H.J. 1977 The modulation of an internal gravity-wave packet, and the resonance with the mean motion. Stud. Appl. Maths 56, 241–266.CrossRefGoogle Scholar
Grimshaw, R.H.J. & Helfrich, K.R. 2012 The effect of rotation on internal solitary waves. IMA J. Appl. Maths 77, 326–339.CrossRefGoogle Scholar
Hasselmann, K. 1970 Wave-driven inertial oscillations. Geophys. Fluid Dyn. 1, 463–502.CrossRefGoogle Scholar
Helfrich, K.R. & Grimshaw, R.H.J. 2008 Nonlinear disintegration of the internal tide. J. Phys. Oceanogr. 38, 686–701.CrossRefGoogle Scholar
Higgins, C., Vanneste, J. & van den Bremer, T.S. 2020 Unsteady Ekman–Stokes dynamics: implications for surface wave-induced drift of floating marine litter. Geophys. Res. Lett. 47, e2020GL089189.CrossRefGoogle Scholar
MacKinnon, J.A., et al. 2017 Climate process team on internal wave-driven ocean mixing. Bull. Am. Meteorol. Soc. 98, 2429–2454.CrossRefGoogle Scholar
McIntyre, M.E. 1973 Mean motions and impulse of a guided internal gravity wave packet. J. Fluid Mech. 60, 801–811.CrossRefGoogle Scholar
Onink, V., Wichmann, D., Delandmeter, P. & van Sebille, E. 2019 The role of Ekman currents, geostrophy, and Stokes drift in the accumulation of floating microplastic. J. Geophys. Res. 124, 1474–1490.CrossRefGoogle ScholarPubMed
Ostrovsky, L.A. & Stepanyants, Yu.A. 1989 Do internal solitons exist in the ocean? Rev. Geophys. 27, 293–310.CrossRefGoogle Scholar
Rainville, L. & Pinkel, R. 2006 Baroclinic energy flux at the Hawaiian Ridge: observations from the R/P FLIP. J. Phys. Oceanogr. 36, 1104–1122.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
Sutherland, B.R. & Dhaliwal, M.S. 2022 The nonlinear evolution of internal tides. Part 1: the superharmonic cascade. J. Fluid Mech. 948, A21.Google Scholar
Thorpe, S.A. 1968 On the shape of progressive internal waves. Phil. Trans. R. Soc. Lond. A 263, 563–614.Google Scholar
Wunsch, C. 1971 Note on some Reynolds stress effects of internal waves on slopes. Deep-Sea Res. 18, 583–591.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar
Wunsch, S. 2015 Nonlinear harmonic generation by diurnal tides. Dyn. Atmos. Oceans 71, 91–97.CrossRefGoogle Scholar