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Low-order Boussinesq models based on $\unicode[STIX]{x1D70E}$ coordinate series expansions

Published online by Cambridge University Press:  01 June 2020

James T. Kirby Open the ORCID record for James T. Kirby [Opens in a new window]
James T. Kirby*
Affiliation:
Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of Delaware, NewarkDE 19716, USA
*
†Email address for correspondence: [email protected]
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Abstract

We derive weakly dispersive Boussinesq equations using a $\unicode[STIX]{x1D70E}$ coordinate for the vertical direction, employing a series expansion in powers of $\unicode[STIX]{x1D70E}$. We restrict attention initially to the case of constant still-water depth $h$ in order to simplify subsequent analysis, and consider equations based on expansions about the bottom elevation $\unicode[STIX]{x1D70E}=0$, and then about a reference elevation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}}$ in order to improve linear dispersion properties. We use a perturbation analysis, suggested recently by Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38), to show that the resulting models are not subject to the trough instability studied there. A similar analysis is performed to develop a model for interfacial waves in a two-layer fluid, with comparable results. We argue, by extension, that a necessary condition for eliminating trough instabilities is that the model’s nonlinear dispersive terms should not contain still-water depth $h$ and surface displacement $\unicode[STIX]{x1D702}$ separately.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Internal waves Instability: Nonlinear instability
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , R3
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Barthelemy, X., Banner, M. L., Peirson, W. L., Fedele, F., Allis, M. & Dias, F. 2018 On a unified breaking onset threshold for gravity waves in deep and intermediate depth water. J. Fluid Mech. 841, 463–488.CrossRefGoogle Scholar
Bellotti, G. & Brocchini, M. 2002 On using Boussinesq-type equations near the shoreline: a note of caution. Ocean Engng 29, 1569–1575.CrossRefGoogle Scholar
Brocchini, M. 2013 A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics. Proc. R. Soc. Lond. A 469, 20130496.CrossRefGoogle ScholarPubMed
Chen, Q. 2006 Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. J. Engng Mech. 132 (2), 220–230.Google Scholar
Choi, W., Barros, R. & Jo, T.-C. 2009 A regularized model for strongly nonlinear internal solitary waves. J. Fluid Mech. 629, 73–85.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
Derakhti, M., Kirby, J. T., Banner, M. L., Grilli, S. T. & Thomson, J. 2020 A unified breaking-onset criterion for surface gravity water waves in arbitrary depth. J. Geophys. Res.: Oceans (in press).CrossRefGoogle Scholar
Durán, A., Dutykh, D. & Mitsotakis, D. 2018 Peregrine’s system revisited. In Nonlinear Waves and Pattern Dynamics (ed. Abcha, N., Pelinovsky, E. & Mutabazi, I.), pp. 3–43. Springer.CrossRefGoogle Scholar
Kennedy, A. B., Kirby, J. T., Chen, Q. & Dalrymple, R. A. 2001 Boussinesq-type equations with improved nonlinear performance. Wave Motion 33, 225–243.CrossRefGoogle Scholar
Kirby, J. T. 2016 Boussinesq models and their application to coastal processes across a wide range of scales. ASCE J. Waterway Port Coast. Ocean Engng 142 (6), 03116005.Google Scholar
Kirby, J. T., Shi, F., Tehranirad, B., Harris, J. C. & Grilli, S. T. 2013 Dispersive tsunami waves in the ocean: Model equations and sensitivity to dispersion and coriolis effects. Ocean Model. 62, 39–55.CrossRefGoogle Scholar
Liska, R., Margolin, L. & Wendroff, B. 1995 Nonhydrostatic two-layer models of incompressible flow. Comput. Maths Applics. 29, 25–37.CrossRefGoogle Scholar
Madsen, P. A. & Fuhrman, D. R. 2020 Trough instabilities in Boussinesq formulations for water waves. J. Fluid Mech. 889, A38.CrossRefGoogle Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. ASCE J. Waterway Port Coast. Ocean Engng 119 (6), 618–638.CrossRefGoogle Scholar
Peregrine, D. H. 1966 Calculations of the development of an undular bore. J. Fluid Mech. 25 (2), 321–330.CrossRefGoogle Scholar
Serre, F.1953 Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche (Juin-Juillet), 374–388.Google Scholar
Shi, F., Kirby, J. T., Harris, J. C., Geiman, J. D. & Grilli, S. T. 2012 A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Model. 43–44, 36–51.CrossRefGoogle Scholar
Su, C. H. & Gardner, C. S. 1969 Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10 (3), 536–539.CrossRefGoogle Scholar
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Bousinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar