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On the appearance of internal wave attractors due to an initial or parametrically excited disturbance

Published online by Cambridge University Press:  02 January 2013

Janis Bajars ,
Jason Frank  and
Leo R. M. Maas
Janis Bajars
Affiliation:
Centrum Wiskunde & Informatica, PO Box 94079, 1090 GB Amsterdam, The Netherlands
Jason Frank*
Affiliation:
Centrum Wiskunde & Informatica, PO Box 94079, 1090 GB Amsterdam, The Netherlands
Leo R. M. Maas
Affiliation:
Royal Netherlands Institute for Sea Research, PO Box 59, 1790 AB Texel, The Netherlands
*
†Email address for correspondence: [email protected]
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Abstract

In this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler–Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 283 - 311
Copyright
©2013 Cambridge University Press

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References

Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics, 2nd edn. Springer.CrossRefGoogle Scholar
Bühler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271–293.CrossRefGoogle Scholar
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified spherical shell. J. Fluid Mech. 398, 271–297.CrossRefGoogle Scholar
Drijfhout, S. & Maas, L. R. M. 2007 Impact of channel geometry and rotation on the trapping of internal tides. J. Phys. Oceanogr. 37, 2740–2763.CrossRefGoogle Scholar
Echeverri, P., Yokossi, T., Balmforth, N. J. & Peacock, T. 2011 Tidally generated internal-wave attractors between double ridges. J. Fluid Mech. 669, 354–374.CrossRefGoogle Scholar
Fricker, P. & Nepf, H. 2000 Bathymetry, stratification, and internal seiche structure. J. Geophys. Res. 105, 14,237–14,251.CrossRefGoogle Scholar
Gerkema, T. & van Haren, H. 2012 Absence of internal tidal beams due to non-uniform stratification. J. Sea Res. doi:10.1016/j.seares.2012.03.008.CrossRefGoogle Scholar
Grisouard, N., Staquet, C. & Pairaud, I. 2008 Numerical simulation of a two-dimensional internal wave attractor. J. Fluid Mech. 614, 1–14.CrossRefGoogle Scholar
Hairer, E., Lubich, C. & Wanner, G. 2006 Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations. Springer.Google Scholar
Hazewinkel, J., van Breevoort, P., Dalziel, S. B. & Maas, L. R. M. 2008 Observations on the wavenumber spectrum and evolution of an internal wave attractor. J. Fluid Mech. 598, 373–382.CrossRefGoogle Scholar
Hazewinkel, J., Grisouard, N. G. & Dalziel, S. B. 2011 Comparison of laboratory and numerically observed scalar fields of an internal wave attractor. Eur. J. Mech. B 30 (1), 51–56.CrossRefGoogle Scholar
Hazewinkel, J., Tsimitri, C., Maas, L. R. M. & Dalziel, S. B. 2010 Observations on the robustness of internal wave attractors to perturbations. Phys. Fluids 22, 107102.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E. & Ratiu, T. S. 2002 The Euler–Poincaré equations in geophysical fluid dynamics. In Large-Scale Atmosphere-Ocean Dynamics II (ed. Roulstone, I. & Norbury, J.), pp. 251–300. Cambridge University Press.Google Scholar
John, F. 1941 The Dirichlet problem for a hyperbolic equation. Am. J. Math. 63, 141–154.CrossRefGoogle Scholar
Kopecz, S. 2006 Fractal internal wave patterns in a tilted square. Unpublished report, Kassel University.Google Scholar
Lam, F.-P. A. & Maas, L. R. M. 2008 Internal wave focusing revisited: a reanalysis and new theoretical links. Fluid Dyn. Res. 40 (2), 95–122.CrossRefGoogle Scholar
LaZerte, B. D. 1980 The dominating higher order vertical modes of the internal seiche in a small lake. Limnol. Oceanogr. 25 (S), 846–854.CrossRefGoogle Scholar
Leimkuhler, B. J. & Reich, S. 2004 Simulating Hamiltonian Dynamics. Cambridge University Press.Google Scholar
Lighthill, J. 1996 Internal waves and related initial-value problems. Dyn. Atmos. Oceans 23, 3–17.CrossRefGoogle Scholar
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15 (9), 2757–2782.CrossRefGoogle Scholar
Maas, L. R. M. 2009 Exact analytic self-similar solution of a wave attractor field. Physica D: Nonlinear Phenomena 238 (5), 502–505.CrossRefGoogle Scholar
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F. P. A. 1997 Observation of an internal wave attractor in a confined, stably stratified fluid. Nature 388, 557–561.CrossRefGoogle Scholar
Maas, L. R. M. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle Scholar
McEwan, A. D. & Robinson, R. M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67 (4), 667–687.CrossRefGoogle Scholar
McLachlan, R. I. 1995 Symplectic integration of Hamiltonian wave equations. Numer. Math. 66 (1), 465–492.CrossRefGoogle Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467–521.CrossRefGoogle Scholar
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 19–44.CrossRefGoogle Scholar
Olver, P. J. 1986 Applications of Lie Groups to Differential Equations. Springer.CrossRefGoogle Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Wave attractors in rotating fluids: a paradigm for ill-posed Cauchy problems. Phys. Rev. Lett. 85, 4277–4280.CrossRefGoogle ScholarPubMed
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
da Silva, J. C. B., Magalhães, J., Gerkema, T. & Maas, L. R. M. 2012 Internal solitary waves in the Red Sea: an unfolding mystery. Oceanography 25 (2), 96–107.CrossRefGoogle Scholar
Swart, A., Sleijpen, G. L. G., Maas, L. R. M. & Brandts, J. 2007 Numerical solution of the two-dimensional Poincaré equation. J. Comput. Appl. Math. 200 (1), 317–341.CrossRefGoogle Scholar
Tang, W. & Peacock, T. 2010 Lagrangian coherent structures and internal wave attractors. Chaos 20, 017508.CrossRefGoogle ScholarPubMed
Tilgner, A. 1999 Driven inertial oscillations in spherical shells. Phys. Rev. E 59, 1789–1794.CrossRefGoogle Scholar
Whitham, G. B. 1999 Linear and Nonlinear Waves, 2nd edn. Wiley.CrossRefGoogle Scholar