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Numerical Simulation of a Weakly Nonlinear Model for Internal Waves

Published online by Cambridge University Press:  20 August 2015

Robyn Canning Gregory  and
David P. Nicholls
Robyn Canning Gregory*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
David P. Nicholls*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA
*
Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

Internal waves arise in a wide array of oceanographic problems of both theoretical and engineering interest. In this contribution we present a new model, valid in the weakly nonlinear regime, for the propagation of disturbances along the interface between two ideal fluid layers of infinite extent and different densities. Additionally, we present a novel high-order/spectral algorithm for its accurate and stable simulation. Numerical validation results and simulations of wave-packet evolution are provided.

Keywords

76B0776B1576B5565M70Internal wavesweakly nonlinear waveshigh-order spectral methodsDirichlet Neumann operatorssurface formulation
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1461 - 1481
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Benjamin, T. B. and Bridges, T. J., Reappraisal of the Kelvin-Helmholtz problem I- Hamiltonian structure, J. Fluid Mech., 333 (1997), 301–325.CrossRefGoogle Scholar
[2]Benjamin, T. B. and Bridges, T. J., Reappraisal of the Kelvin-Helmholtz problem II-interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333 (1997), 327–373.Google Scholar
[3]Burden, R.and Faires, J. D., Numerical Analysis, Brooks/Cole Publishing Co., Pacific Grove, CA, sixth edition, 1997.Google Scholar
[4]Bona, J. L., Lannes, D. and Saut, J.-C., Asymptotic models for internal waves, J. Math. Pures Appl., 89(6) (2008), 538–566.Google Scholar
[5]Choi, W. and Camassa, R., Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83–103.Google Scholar
[6]Craig, W. and Groves, M. D., Normal forms for wave motion in fluid interfaces, Wave Motion, 31(1) (2000), 21–41.CrossRefGoogle Scholar
[7]Craig, W., Guyenne, P. and Kalisch, H., A new model for large amplitude long internal waves, Comptes Rendus Mechanique, 332(7) (2004), 525–530.Google Scholar
[8]Craig, W., Guyenne, P. and Kalisch, H., Hamiltonian long-wave expansions for free surfaces and interfaces, Commun. Pure Appl. Math., 58(12) (2005), 1587–1641.Google Scholar
[9]Coifman, R. and Meyer, Y., Nonlinear harmonic analysis and analytic dependence, in Pseu-dodifferential Operators and Applications (Notre Dame, Ind., 1984), pages 71–78, Amer. Math. Soc., 1985.Google Scholar
[10]Craig, W. and Sulem, C., Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73–83.Google Scholar
[11]Guyenne, P., Lannes, D. and Saut, J.-C., Well-posedness of the Cauchy problem for models of large amplitude internal waves, Nonlinearity, 23(2) (2010), 237–275.Google Scholar
[12]Gottlieb, D. and Orszag, S. A., Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26.Google Scholar
[13]Haut, T. S. and Ablowitz, M. J., A reformulation and applications of interfacial fluids with a free surface, J. Fluid Mech., 631 (2009), 375–396.CrossRefGoogle Scholar
[14]Helfrich, K. and Melville, W., Long nonlinear internal waves, Ann. Rev. Fluid Mech., 38 (2006), 395–425.Google Scholar
[15]Jackson, C. R., The internal wave altas, http://internalwavealtas.com.Google Scholar
[16]Koop, C. G. and Butler, G., An investigation of internal solitary waves in a two-fluid system, J. Fluid Mech., 112 (1981), 225–251.CrossRefGoogle Scholar
[17]Kakleas, M. and Nicholls, D. P., Numerical simulation of a weakly nonlinear model for water waves with viscosity, J. Sci. Comput., 42(2) (2010), 274–290.CrossRefGoogle Scholar
[18]Lamb, H., Hydrodynamics, Cambridge University Press, Cambridge, sixth edition, 1993.Google Scholar
[19]Mei, C. C., Numerical methods in water-wave diffraction and radiation, Ann. Rev. Fluid Mech., 10 (1978), 393–416.Google Scholar
[20]Milder, D. M., An improved formalism for rough-surface scattering of acoustic and electromagnetic waves, in Proceedings of SPIE-The International Society for Optical Engineering (San Diego, 1991), volume 1558, pages 213–221, Int. Soc. for Optical Engineering, Bellingham, WA, 1991.Google Scholar
[21]Milder, D. M., An improved formalism for wave scattering from rough surfaces, J. Acoust. Soc. Am., 89(2) (1991), 529–541.CrossRefGoogle Scholar
[22]Milder, D. M. and Sharp, H. T., Efficient computation of rough surface scattering, in Mathematical and Numerical Aspects of Wave Propagation Phenomena (Strasbourg, 1991), pages 314–322, SIAM, Philadelphia, PA, 1991.Google Scholar
[23]Milder, D. M. and Sharp, H. T., An improved formalism for rough surface scattering II: numerical trials in three dimensions, J. Acoust. Soc. Am., 91(5) (1992), 2620–2626.CrossRefGoogle Scholar
[24]Mercier, M. J., Vasseur, R. and Dauxois, T., Resurrecting dead-water phenomenon, Nonlinear Pro. Geophys., 18 (2011), 193–208.Google Scholar
[25]Nguyen, H. Y. and Dias, F., A Boussinesq system for two-way propagation of interfacial waves, Phys. D, 237(18) (2008), 2365–2389.Google Scholar
[26]Nicholls, D. P., Boundary perturbation methods for water waves, GAMM-Mitteilungen, 30(1) (2007), 44–74.Google Scholar
[27]Nicholls, D. P. and Reitich, F., A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sec. A, 131(6) (2001), 1411–1433.CrossRefGoogle Scholar
[28]Nicholls, D. P. and Reitich, F., On analyticity of traveling water waves, Proc. Roy. Soc. Lond. A, 461(2057) (2005), 1283–1309.Google Scholar
[29]Nicholls, D. P. and Reitich, F., Rapid, stable, high-order computation of traveling water waves in three dimensions, Euro. J. Mech. B Fluids, 25(4) (2006), 406–424.CrossRefGoogle Scholar
[30]Scardovelli, R. and Zaleski, S., Direct numerical simulation of free-surface and interfacial flow, Ann. Rev. Fluid Mech., 31 (1999), 567–603.Google Scholar
[31]Tsai, W.-T. and Yue, D. K. P., Computation of nonlinear free-surface flows, in Ann. Rev. Fluid Mech., 28, 249–278, Annual Reviews, Palo Alto, CA, 1996.Google Scholar
[32]Yeung, R. W., Numerical methods in free-surface flows, Ann. Rev. Fluid Mech., 14 (1982), 395–442.Google Scholar
[33]Zakharov, V., Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190–194.Google Scholar