Hostname: page-component-669899f699-7xsfk Total loading time: 0 Render date: 2025-04-27T07:40:41.315Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear interfacial wave formation in three dimensions

Published online by Cambridge University Press:  23 February 2015

John Grue
John Grue*
Affiliation:
Mechanics Division, Department of Mathematics, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway
*
†Email address for correspondence: johng@math.uio.no
Get access Rights & Permissions [Opens in a new window]

Abstract

A three-dimensional two-layer, fully dispersive and strongly nonlinear interfacial wave model, including the interaction with a time-varying bottom topography, is developed. The method is based on a set of integral equations. The source and dipole terms are developed in series expansions in the vertical excursions of the interface and bottom topography, obtaining explicit inversion by Fourier transform. Calculations of strongly nonlinear interfacial waves with excursions comparable to the thinner layer depth show that the quadratic approximation of the method contains the essential dynamics, while the additional cubic terms always are small. Computations confirm the onset of wave train formation driven by topography, observed in experiments (Maxworthy, J. Geophys. Res., vol. 84(C1), 1979, pp. 338–346), depending on the Froude number and the topography height. Simulations of tidally driven three-dimensional internal wave formation show the formation of two wave trains per half tidal cycle for strong forcing and one wave train for weak forcing. Waves of both backward and forward curvature are calculated.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 735 - 762
Check for updates
Copyright
© 2015 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.)

Article purchase

Temporarily unavailable

References

Berntsen, J., Xing, J. & Davies, A. M. 2008 Numerical studies of internal waves at a sill: sensitivity to horizontal size and subgrid scale closure. Cont. Shelf Res. 28, 1376–1393.Google Scholar
Boegman, L. & Dorostkar, A.2011 Three-dimensional simulation of NLIW generation, propagation and breaking in Cayuga Lake. In 7th International Symposium Stratified Flows, Rome, Italy, pp. 22–26.Google Scholar
Clamond, D., Fructus, D. & Grue, J. 2007 A note on time integrators in water-wave simulations. J. Engng Maths 58, 149–156.Google Scholar
Clamond, D. & Grue, J. 2001 A fast method for fully nonlinear water wave computations. J. Fluid Mech. 447, 337–355.Google Scholar
Farmer, D. M. & Armi, L. 1988 The flow of Atlantic water through the Strait of Gibraltar. Prog. Oceanogr. 21 (1), 1–105.CrossRefGoogle Scholar
Fringer, O. B., Gerritsen, M. & Street, R. L. 2006 An unstructured-grid finite-volume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model. 14, 139–173.Google Scholar
Fructus, D., Carr, M., Grue, J., Jensen, A. & Davies, P. A. 2009 Shear induced breaking of large internal solitary waves. J. Fluid Mech. 620, 1–29.Google Scholar
Fructus, D., Clamond, D., Grue, J. & Kristiansen, Ø. 2005a An efficient model for three-dimensional surface wave simulations. Part I. Free space problems. J. Comput. Phys. 205, 665–685.Google Scholar
Fructus, D. & Grue, J. 2007 An explicit method for the nonlinear interaction between water waves and variable and moving bottom topography. J. Comput. Phys. 222, 720–739.Google Scholar
Fructus, D., Kharif, C., Francius, M., Kristiansen, Ø., Clamond, D. & Grue, J. 2005b Dynamics of crescent water wave patterns. J. Fluid Mech. 537, 155–186.Google Scholar
Grimshaw, R. & Helfrich, K. 2012 The effect of rotation on internal solitary waves. IMA J. Appl. Maths 1–14; doi:10.1093/imamat/hxs024.Google Scholar
Grue, J. 2002 On four highly nonlinear phenomena in wave theory and marine hydrodynamics. Appl. Ocean Res. 24, 261–274.Google Scholar
Grue, J. 2013a Havets indre bølger. Nature 137 (6), 210–226; (in Norwegian).Google Scholar
Grue, J. 2013b Interfacial wave motion of very large amplitude: formulation in three dimensions and numerical experiments. Procedia IUTAM 8, 129–143; ISSN 2210-9838.Google Scholar
Grue, J., Friis, H. A., Palm, E. & Rusås, P.-O. 1997 A method for computing unsteady fully nonlinear interfacial waves. J. Fluid Mech. 351, 223–252.Google Scholar
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.Google Scholar
Grue, J. & Palm, E. 1996 Wave drift damping of floating bodies in slow yaw motion. J. Fluid Mech. 319, 323–352.Google Scholar
Grue, J., Pelinovsky, E. N., Fructus, D., Kharif, C. & Talipova, T. 2008 Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami. J. Geophys. Res. 113, C05008.Google Scholar
Guo, C., Chen, X., Vlasenko, V. & Stashchuk, N. 2011 Numerical investigation of internal solitary waves from the Luzon Strait: generation process, mechanism and three-dimensional effects. Ocean Model. 38, 203–216.Google Scholar
Jackson, C. R., da Silva, J. C., Jeans, G., Alpers, W. & Caruso, M. J. 2013 Nonlinear internal waves in synthetic aperture radar imagery. Oceanography 26, 68–79.Google Scholar
Lai, Z., Chen, C., Cowles, G. & Beardsley, R. 2010 A nonhydrostatic version of FVCOM: 2. Mechanistic study of tidally generated nonlinear internal waves in Massachusetts Bay. J. Geophys. Res. 115, C12049.Google Scholar
Maxworthy, T. 1979 A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. J. Geophys. Res. 84 (C1), 338–346.Google Scholar
Melville, W. K. & Helfrich, K. R. 1987 Transcritical two-layer flow over topography. J. Fluid Mech. 178, 31–52.Google Scholar
Mikhlin, S. G. 1957 Integral Equations, p. 338. Pergamon.Google Scholar
Morozov, E. G., Parrilla-Barrerab, G., Velarde, M. G. & Scherbinina, A. D. 2003 The straits of Gibraltar and Kara Gates: a comparison of internal tides. Oceanol. Acta 26 (3), 231–241.Google Scholar
Nash, J. D. & Moum, J. N. 2005 River plumes as a source of large-amplitude internal waves in the coastal ocean. Nat. Lett. 437, 400–403.Google Scholar
Newman, J. N. 1977 Marine Hydrodynamics, p. 402. MIT Press.Google Scholar
Ostrovsky, L. 1978 Nonlinear internal waves in a rotating ocean. Oceanology 18 (2), 119–125.Google Scholar
Palmer, M. R., Inall, M. E. & Sharples, J. 2013 The physical oceanography of Jones Bank: a mixing hotspot in the Celtic Sea. Prog. Oceanogr. 117, 9–24.Google Scholar
Sveen, J. K., Guo, Y., Davies, P. A. & Grue, J. 2002 On the breaking of internal solitary waves at a rigde. J. Fluid Mech. 469, 161–188.Google Scholar
Vitousek, S. & Fringer, O. B. 2011 Physical vs. numerical dispersion in nonhydrostatic ocean modeling. Ocean Model. 40, 72–86.Google Scholar
Vlasenko, V., Sanchez Garrido, J. C., Stashchuk, N., Garcia Lafuente, J. & Losada, M. 2009 Three-dimensional evolution of large-amplitude internal waves in the Strait of Gibraltar. J. Phys. Oceanogr. 39, 2230–2246.Google Scholar
Vlasenko, V., Stashchuk, N., Inall, M. E. & Hopkins, J. 2014 Tidal energy conversion in a global hot spot: on the 3-D dynamics of baroclinic tides at the Celtic Sea shelf break. J. Geophys. Res. doi:10.1002/2013JC009708.Google Scholar
Vlasenko, V., Stashchuk, N., Palmer, M. R. & Inall, M. E. 2013 Generation of barolinic tides over an isolated underwater bank. J. Geophys. Res. 118, 1–14.Google Scholar
Zhang, Z., Fringer, O. B. & Ramp, S. R. 2011 Three-dimensional, nonhydrostatic numerical simulation of nonlinear internal wave generation and propagation in the South China Sea. J. Geophys. Res. 116, C05022.Google Scholar