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A non-local formulation of rotational water waves

Published online by Cambridge University Press:  08 November 2011

A. C. L. Ashton  and
A. S. Fokas
A. C. L. Ashton*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 129 - 148
Copyright
Copyright © Cambridge University Press 2011

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References

1. Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562 (1), 313–343.CrossRefGoogle Scholar
2. Ablowitz, M. J. & Haut, T. S. 2010 Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions. J. Phys. A: Math. Theor. 43, 434005.CrossRefGoogle Scholar
3. Adams, R. A. 1975 Sobolev Spaces, Pure and Applied Mathematics , vol. 65. Academic.Google Scholar
4. Amick, C. J., Fraenkel, L. E. & Toland, J. F. 1982 On the stokes conjecture for the wave of extreme form. Acta Mathematica 148 (1), 193–214.CrossRefGoogle Scholar
5. Benjamin, T. B. 1962 The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12 (01), 96–116.CrossRefGoogle Scholar
6. Chow, K.W. 1989 A second-order solution for the solitary wave in a rotational flow. Phys. Fluids A 1 (7), 1235–1239.CrossRefGoogle Scholar
7. Constantin, A. & Strauss, W. 2002 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Maths 57 (4), 481–527.CrossRefGoogle Scholar
8. Constantin, A., Ivanov, R. & Prodanov, E. 2008 Nearly-Hamiltonian structure for water waves with constant vorticity. J. Math. Fluid Mech. 10 (2), 224–237.CrossRefGoogle Scholar
9. Constantin, A. & Kartashova, E. 2009 Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. EPL 86, 29001.CrossRefGoogle Scholar
10. Craig, W. & Sternberg, P. 1988 Symmetry of solitary waves. Commun. Part. Diff. Equ. 13 (5), 603–633.CrossRefGoogle Scholar
11. Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108 (1), 73–83.CrossRefGoogle Scholar
12. Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675 (1), 141–167.CrossRefGoogle Scholar
13. Fokas, A.S. 1997 A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. A 453 (1962), 1411–1443.CrossRefGoogle Scholar
14. Fokas, A.S. 2000 On the integrability of linear and nonlinear partial differential equations. J. Math. Phys. 41 (6), 4188–4237.CrossRefGoogle Scholar
15. Haut, T.S. & Ablowitz, M.J. 2009 A reformulation and applications of interfacial fluids with a free surface. J. Fluid Mech. 631 (1), 375–396.CrossRefGoogle Scholar
16. Olver, P. 1984 Hamiltonian perturbation theory and water waves. Contemp. Maths 28, 231–249.CrossRefGoogle Scholar
17. Sun, S.M. 1999 Non-existence of truly solitary waves in water with small surface tension. Proc. R. Soc. A 455 (1986), 2191–2228.CrossRefGoogle Scholar
18. Da Silva, A.F.T. & Peregrine, D.H. 1988 Steep, steady surfaces waves on water of finite depth with constant vorticity. J. Fluid Mech. 195 (1), 281–302.CrossRefGoogle Scholar
19. Wu, S. 1997 Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130 (1), 39–72.CrossRefGoogle Scholar
20. Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.CrossRefGoogle Scholar