THE QUASI-STREAMFUNCTION FORMALISM

Miao Tian, William Cottrell, Alex Sheremet, Jane Smith

Abstract


This paper provides a review of our recent developments in reformulating the quasi streamfunction (Ψ) formalism proposed by Kim et al. (2001) to relax the common constraint of kinematic bottom boundary condition. A restricted form of the Hamilton’s principle for irrotational flows is formulated only on surface variables. This transforms the problem to dynamical equations on the surface and a constraint equation related to the interior water column. The interior solution can be applied to express Ψ in terms of the natural canonically conjugate variable. The modified Ψ-formalism promises to provide a natural framework for the study of wave over arbitrary bathymetry and in the presence of strong shear flow if Clebsch variables are included. We demonstrate the formalism for horizontally homogeneous flows over mild topography, where asymptotic formulations for the Hamiltonian and Lagrangian are derived. The Hamiltonian shows consistency with Zakharov’s results up to the cubic order and the Lagrangian is written in terms of measurable variables.

Keywords


stream function; variational principle; hamiltonian; lagrangian

References


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Agnon, Y. and A. Sheremet, 1997. Stochastic nonlinear shoaling of directional spectra, Journal of Fluid Mechanics, 345, 79-99.http://dx.doi.org/10.1017/S0022112097006137

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Hasselmann, K., 1963. On the non-linear energy transfer in a gravity wave spectrum, Part II: Conservation theorems - wave-particle analogy - irreversibility, Journal of Fluid Methanics, 15/2, 273-281.http://dx.doi.org/10.1017/S0022112063000239

Hasselmann, K., 1963. On the non-linear energy transfer in a gravity-wave spectrum, Part III: Evaluation of the energy flux and swell-sea interaction for a Neumann spectrum, Journal of Fluid Methanics, 15/3, 385-398.http://dx.doi.org/10.1017/S002211206300032X

Kim, J., J.W. and K.J. Bai, R.C. Ertekin, W.C. Webster, 2001. A derivation of the Green-Naghdi equations for irrotational flows, Journal of Engineering Mathematics, 40, 17-42.http://dx.doi.org/10.1023/A:1017541206391

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Kim, J.W. and K.J. Bai, 2004. A new complementary mild slope equation, Journal of Fluid Methanics, 511, 25-40.http://dx.doi.org/10.1017/S0022112004007840

Luke, J.C., 1967. A variational principle for a fluid with a free surface, Journal of Fluid Mechanics, 27, 395.http://dx.doi.org/10.1017/S0022112067000412

Mei C.C., Mei M. Stiassnie, and D.K.-P. Yue, 2005. Theory and applications of ocean surface waves, Adv. Series on Ocean Eng. 23 (Expanded Edition), World Scientific.

Miles, J.W., 1977. On Hamilton's principle for surface waves. Journal of Fluid Mechanics, 27, 395.

Seliger R.L. and G.B. Whitham, 1968. Variational principles in continuum mechanics. Proceedings of the Royal Society, London, A 305, 1-25.http://dx.doi.org/10.1098/rspa.1968.0103

Toledo, Y. and Y. Agnon, 2009. Nonlinear refraction-diffraction of water waves: the complementary mildslope equations,Journal of Fluid Methanics, 641, 509-52.http://dx.doi.org/10.1017/S0022112009992369

Zakharov, V., 1968. Stability of periodic waves of finite amplitude on a surface of deep fluid, Journal of Applied Mechanical Technology Physics, 2, 190-198.

Zakharov, V., 1999. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid, European Journal of Mechanics, B - Fluids, 18 (3), 327-344.http://dx.doi.org/10.1016/S0997-7546(99)80031-4

Agnon, Y., A. Sheremet, J. Gonsalves, and M. Stiassnie, 1993. A unidirectional model for shoaling gravity waves, Coastal Engineering, 20, 29-58.http://dx.doi.org/10.1016/0378-3839(93)90054-C Agnon, Y. and A. Sheremet, 1997. Stochastic nonlinear shoaling of directional spectra, Journal of Fluid Mechanics, 345, 79-99.http://dx.doi.org/10.1017/S0022112097006137 Boussinesq, J., 1871. Theorie de l'Intumescence liquide appele onde solitaire ou de translation. Comptes Rendus Acad. Sci. Paris 72, 755-759. Bretherton, F.P., 1970. A note on Hamilton's principle for perfect fluids, Journal of Fluid Mechanics, 44, 19-31.http://dx.doi.org/10.1017/S0022112070001660 Clebsch, A., 1857. Uber eine allgemeine Transformation der hydrodynamischen Gleichungen,J. Reine Angew. Math. 54, 293-313.http://dx.doi.org/10.1515/crll.1857.54.293 Hasselmann, K., 1961. On the non-linear energy transfer in a gravity-wave spectrum, Part I: General theory, Journal of Fluid Mechanics, 12, 481-500.http://dx.doi.org/10.1017/S0022112062000373 Hasselmann, K., 1963. On the non-linear energy transfer in a gravity wave spectrum, Part II: Conservation theorems - wave-particle analogy - irreversibility, Journal of Fluid Methanics, 15/2, 273-281.http://dx.doi.org/10.1017/S0022112063000239 Hasselmann, K., 1963. On the non-linear energy transfer in a gravity-wave spectrum, Part III: Evaluation of the energy flux and swell-sea interaction for a Neumann spectrum, Journal of Fluid Methanics, 15/3, 385-398.http://dx.doi.org/10.1017/S002211206300032X Kim, J., J.W. and K.J. Bai, R.C. Ertekin, W.C. Webster, 2001. A derivation of the Green-Naghdi equations for irrotational flows, Journal of Engineering Mathematics, 40, 17-42.http://dx.doi.org/10.1023/A:1017541206391 Kim, J., J.W. and K.J. Bai, R.C. Ertekin, W.C. Webster, 2003. A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth: The Irrotational Green-Naghdi Model, Journal of Offshore Mechanics Arctic Engineering, 125, 25-32.http://dx.doi.org/10.1115/1.1537722 Kim, J.W. and K.J. Bai, 2004. A new complementary mild slope equation, Journal of Fluid Methanics, 511, 25-40.http://dx.doi.org/10.1017/S0022112004007840 Luke, J.C., 1967. A variational principle for a fluid with a free surface, Journal of Fluid Mechanics, 27, 395.http://dx.doi.org/10.1017/S0022112067000412 Mei C.C., Mei M. Stiassnie, and D.K.-P. Yue, 2005. Theory and applications of ocean surface waves, Adv. Series on Ocean Eng. 23 (Expanded Edition), World Scientific. Miles, J.W., 1977. On Hamilton's principle for surface waves. Journal of Fluid Mechanics, 27, 395. Seliger R.L. and G.B. Whitham, 1968. Variational principles in continuum mechanics. Proceedings of the Royal Society, London, A 305, 1-25.http://dx.doi.org/10.1098/rspa.1968.0103 Toledo, Y. and Y. Agnon, 2009. Nonlinear refraction-diffraction of water waves: the complementary mildslope equations,Journal of Fluid Methanics, 641, 509-52.http://dx.doi.org/10.1017/S0022112009992369 Zakharov, V., 1968. Stability of periodic waves of finite amplitude on a surface of deep fluid, Journal of Applied Mechanical Technology Physics, 2, 190-198. Zakharov, V., 1999. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid, European Journal of Mechanics, B - Fluids, 18 (3), 327-344.http://dx.doi.org/10.1016/S0997-7546(99)80031-4

Agnon, Y., A. Sheremet, J. Gonsalves, and M. Stiassnie, 1993. A unidirectional model for shoaling gravity waves, Coastal Engineering, 20, 29-58


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