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The steady-state form of large-amplitude internal solitary waves

Published online by Cambridge University Press:  10 November 2010

STUART E. KING ,
MAGDA CARR  and
DAVID G. DRITSCHEL
STUART E. KING*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, Fife KY16 9SS, UK
MAGDA CARR
Affiliation:
School of Mathematics and Statistics, University of St Andrews, Fife KY16 9SS, UK
DAVID G. DRITSCHEL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, Fife KY16 9SS, UK
*
†Email address for correspondence: stuart@mcs.st-and.ac.uk
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Abstract

A new numerical scheme for obtaining the steady-state form of an internal solitary wave of large amplitude is presented. A stratified inviscid two-dimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear time-dependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.

JFM classification

Geophysical and Geological Flows: Internal waves Waves/Free-surface Flows: Solitary waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 477 - 505
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Aigner, A., Broutman, D. & Grimshaw, R. 1999 Numerical simulations of internal solitary waves with vortex cores. Fluid Dyn. Res. 25, 315–333.CrossRefGoogle Scholar
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241–270.CrossRefGoogle Scholar
Benney, D. J. 1966 Long nonlinear waves in fluid flows. J. Math. Phys. 45, 52–63.CrossRefGoogle Scholar
Brown, D. J. & Christie, D. R. 1998 Fully nonlinear solitary waves in continuously stratified incompressible Boussinesq fluids. Phys. Fluids 10 (10), 2569–2586.CrossRefGoogle Scholar
Carr, M., Fructus, D., Grue, J., Jensen, A. & Davies, P. A. 2008 Convectively induced shear instability in large-amplitude internal solitary waves. Phys. Fluids 20, 12660.CrossRefGoogle Scholar
Cheung, T. K. & Little, C. G. 1990 Meterological tower, microbarograph array, and sodar observations of solitary-like waves in the nocturnal boundary layer. J. Atmos. Sci. 47, 2516–2536.2.0.CO;2>CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
Clarke, R. H., Smith, R. K. & Reid, D. G. 1981 The morning glory of the gulf of carpentaria: an atmospheric undular bore. Mon. Weath. Rev. 109, 1726–1750.2.0.CO;2>CrossRefGoogle Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29 (3), 593–607.CrossRefGoogle Scholar
Derzho, O. G. & Grimshaw, R. 1997 Solitary waves with a vortex core in a shallow layer of stratified fluid. Phys. Fluids 9, 3378–3385.CrossRefGoogle Scholar
Derzho, O. G. & Grimshaw, R. 2002 Solitary waves with recirculation zones in axisymmetric flows. J. Fluid Mech. 464, 217–250.CrossRefGoogle Scholar
Doviak, R. J., Chen, S. S. & Christie, D. R. 1991 A thunderstorm-generated solitary wave observation compared with theory for nonlinear waves in a sheared atmosphere. J. Atmos. Sci. 48, 87–111.2.0.CO;2>CrossRefGoogle Scholar
Doviak, R. J. & Christie, D. R. 1989 Thunderstorm-generated solitary waves: a wind shear hazard. J. Aircraft 26, 423.CrossRefGoogle Scholar
Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303.CrossRefGoogle Scholar
Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields. Q. J. R. Meteorol. Soc. 123, 1097–1130.Google Scholar
Dritschel, D. G. & Fontane, J. 2010 The combined Lagrangian advection method. J. Comput. Phys. 229, 5408–5417.CrossRefGoogle Scholar
Dubreil-Jacotin, L. 1932 Sur les ondes type permantentes dans les liquides hétérogènes. Atti dela Reale Academic Nationale dci Lincei 15 (6), 44.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.CrossRefGoogle Scholar
Fructus, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323–347.CrossRefGoogle Scholar
Funakoshi, M. & Oikawa, M. 1986 Long internal waves of large amplitude in a two-layer fluid. J. Phys. Soc. Jpn. 55 (1), 128–144.CrossRefGoogle Scholar
Grimshaw, R. 1969 On steady recirculating flow. J. Fluid Mech. 39, 695–703.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181–217.CrossRefGoogle Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
Helfrich, K. R. & White, B. L. 2010 A model for large-amplitude internal solitary waves with trapped cores. Nonlinear Process. Geophys. 17, 303–318.CrossRefGoogle Scholar
Lamb, K. G. 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109–144.CrossRefGoogle Scholar
Lamb, K. G. 2003 Shoaling solitary internal waves: on a criterion for the formation of waves with trapped cores. J. Fluid Mech. 478, 81–100.CrossRefGoogle Scholar
Lamb, K. G. 2008 On the calculation of the available potential energy of an isolated perturbation in a density-stratified fluid. J. Fluid Mech. 597, 415–427.CrossRefGoogle Scholar
Lamb, K. G. & Nguyen, V. T. 2009 Calculating energy flux in internal solitary waves with an application to reflectance. J. Phys. Oceanogr. 39, 559–580.CrossRefGoogle Scholar
Lamb, K. G. & Wan, B. 1998 Conjugate flows and flat solitary waves for a continuously stratified fluid. Phys. Fluids 10 (8), 2061–2079.CrossRefGoogle Scholar
Long, R. R. 1953 Some aspects of the flow of stratified fluids. Part I. A theoretical investigation. Tellus 5, 42–57.CrossRefGoogle Scholar
Manasseh, R., Ching, C.-Y. & Fernando, H. J. S. 1998 The transition from density-driven to wave dominated isolated flows. J. Fluid Mech. 361, 253–274.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Miles, J. W. 1980 Solitary waves. Annu. Rev. Fluid Mech. 12, 11–43.CrossRefGoogle Scholar
Moum, J. N., Farmer, D. M., Smyth, W. D., Armi, L. & Vagle, S. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33, 2093–2112.2.0.CO;2>CrossRefGoogle Scholar
Ostrovsky, L. A. & Stepanyants, Y. A. 1989 Do internal solitons exist in the ocean? Rev. Geophys. 27, 293–310.CrossRefGoogle Scholar
Rusås, P.-O. & Grue, J. 2002 Solitary waves and conjugate flows in a three-layer fluid. Eur. J. Mech. B/Fluids 21, 185–206.CrossRefGoogle Scholar
Scotti, A. & Pineda, J. 2004 Observation of very large and steep internal waves of elevation near the massachusetts coast. Geophys. Res. Lett. 31, L22307.CrossRefGoogle Scholar
Stanton, T. P. & Ostrovsky, L. A. 1998 Observations of highly nonlinear internal solitons over the continental shelf. Geophys. Res. Lett. 25 (14), 2695–2698.CrossRefGoogle Scholar
Stastna, M. & Lamb, K. G. 2002 Large fully nonlinear internal solitary waves: the effect of background current. Phys. Fluids 14 (9), 2987–2999.CrossRefGoogle Scholar
Turkington, B., Eydeland, A. & Wang, S. 1991 A computational method for solitary internal waves in a continuously stratified fluid. Stud. Appl. Maths 85, 93–127.CrossRefGoogle Scholar
Vlasenko, V., Brandt, P. & Rubino, A. 2000 Structure of large-amplitude internal solitary waves. J. Phys. Oceanogr. 30, 2172–2185.2.0.CO;2>CrossRefGoogle Scholar
White, B. L. & Helfrich, K. R. 2008 Gravity currents and internal waves in a stratified fluid. J. Fluid Mech. 616, 327–356.CrossRefGoogle Scholar
Yih, C.-S. 1960 Exact solutions for steady two-dimensional flow of a stratified fluid. J. Fluid Mech. 9, 161–174.CrossRefGoogle Scholar

King supplementary material

Movie 1 shows the nonlinear evolution of the vorticity field for the steady state pictured in figure 7(i) (and corresponds to figure 10). The region of the domain shown is [-0.5,0.5]x[0.5,1.0]. The zero vorticity region at the top of the domain remains stable and stagnant (in the moving frame). It can be seen that the diffusion of vorticity and the initially sharp gradients in vorticity and buoyancy lead to some weak fringing below the main area of the wave and within the stagnant region.

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Video 1 MB

King supplementary material

Movie 2 shows the nonlinear evolution of the vorticity field for the steady state pictured in figure 7(ii) (and corresponds to figure 11). Again the region of the domain shown is [-0.5,0.5]x[0.5,1.0]. From this movie it can be seen that the region at the top of the domain in which closed streamlines were found in the steady state solution is subject to an instability. This instability mixes the region at the top of the domain as time progresses.

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Video 1.3 MB

King supplementary material

Movie 3 shows the nonlinear evolution of the vorticity field for the steady state pictured in figure 9 (and corresponds to figure 12). Again the region of the domain shown is [-0.5,0.5]x[0.5,1.0]. The rotating core solution shown is subject to an instability at the rear stagnation point. This instability gives rise to a disturbance which is advected around the core of the wave. This modifies the wave sufficiently for it to slow.

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Video 2.7 MB