Skip to main content Accessibility help
×
×
Home

Non-canonical Hamiltonian structure and Poisson bracket for two-dimensional hydrodynamics with free surface

  • A. I. Dyachenko (a1), P. M. Lushnikov (a1) (a2) and V. E. Zakharov (a1) (a3)

Abstract

We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable $w$ into the fluid’s area, with the real line of $w$ mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov (J. Appl. Mech. Tech. Phys., vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to the Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identify powerful reductions that allow one to find general classes of particular solutions.

Copyright

Corresponding author

Email address for correspondence: plushnik@math.unm.edu

References

Hide All
Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics. Springer.
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58, 977999.10.1103/RevModPhys.58.977
Chalikov, D. & Sheinin, D. 1998 Direct modeling of one-dimensional nonlinear potential waves. Adv. Fluid Mech. 17, 207258.
Chalikov, D. & Sheinin, D. 2005 Modeling of extreme waves based on equation of potential flow with a free surface. J. Comput. Phys. 210, 247273.10.1016/j.jcp.2005.04.008
Chalikov, D. V. 2016 Numerical Modeling of Sea Waves. Springer.
Cole, M. W. & Cohen, M. H. 1969 Image-potential-induced surface bands in insulators. Phys. Rev. Lett. 23, 1238.10.1103/PhysRevLett.23.1238
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 7383.
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532540.
Crowdy, D. G. 1999 Circulation-induced shape deformations of drops and bubbles: exact two-dimensional models. Phys. Fluids 11, 28362845.
Crowdy, D. G. 2000a A new approach to free surface Euler flows with capillarity. Stud. Appl. Maths 105, 3558.
Crowdy, D. G. 2000b Hele-Shaw flows and water waves. J. Fluid Mech. 409, 223242.10.1017/S0022112099007685
Dyachenko, A. I. 2001 On the dynamics of an ideal fluid with a free surface. Dokl. Math. 63 (1), 115117.
Dyachenko, A. I., Kachulin, D. I. & Zakharov, V. E. 2013 On the nonintegrability of the free surface hydrodynamics. JETP Lett. 98, 4347.
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. & Zakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 7379.
Dyachenko, A. I. & Zakharov, V. E. 1994 Is free surface hydrodynamics an integrable system? Phys. Lett. A 190 (2), 144148.
Dyachenko, S. A., Lushnikov, P. M. & Korotkevich, A. O. 2016 Branch cuts of Stokes wave on deep water. Part I. Numerical solution and Padé approximation. Stud. Appl. Maths 137, 419472.
Edelman, V. S. 1980 Levitated electrons. Sov. Phys. Usp. 23, 227244.
Fefferman, C. 1971 Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77, 587588.
Fefferman, C. & Stein, E. M. 1972 H p spaces of several variables. Acta Math. 71, 137193.10.1007/BF02392215
Flierl, G. R., Morrison, P. J. & Swaminathan, R. V.2018 Jovian vortices and jets.arXiv:1809.08671.
Gakhov, F. D. 1966 Boundary Value Problems. Pergamon.
Galin, L. A. 1945 Unsteady filtration with free surface. Dokl. Akad. Nauk SSSR 47, 246249.
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095.10.1103/PhysRevLett.19.1095
Grant, M. A. 1973 Standing Stokes waves of maximum height. J. Fluid Mech. 60 (3), 593604.10.1017/S0022112073000364
Hilbert, D. 1905 Üeber eine Anwendung der Integralgleichungen auf ein Problem der Funktionentheorie. In Verhandlungen des dritten internationalen Mathematiker Kongresses in Heidelberg 1904 (ed. Krazer, A.), pp. 233240. Teubner.
Howison, S. D. 1986 Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Maths 46, 2026.
Kharif, C. & Pelinovsky, E. 2003 Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. (B/Fluids) 22, 603634.
Kuznetsov, E. A. & Lushnikov, P. M. 1995 Nonlinear theory of the excitation of waves by a wind due to the Kelvin–Helmholtz instability. J. Expl Theor. Phys. 81, 332340.
Landau, L. D. & Lifshitz, E. M. 1989 Fluid Mechanics, 3rd edn., vol. 6. Pergamon.
Lushnikov, P. M. & Zubarev, N. M. 2018 Exact solutions for nonlinear development of a Kelvin–Helmholtz instability for the counterflow of superfluid and normal components of helium II. Phys. Rev. Lett. 120, 204504.10.1103/PhysRevLett.120.204504
Lushnikov, P. M. 2016 Structure and location of branch point singularities for Stokes waves on deep water. J. Fluid Mech. 800, 557594.10.1017/jfm.2016.405
Lushnikov, P. M. & Zakharov, V. E. 2005 On optimal canonical variables in the theory of ideal fluid with free surface. Physica D 203, 929.
Meison, D., Orzag, S. & Izraely, M. 1981 Applications of numerical conformal mapping. J. Comput. Phys. 40, 345360.
Mineev-Weinstein, M., Wiegmann, P. B. & Zabrodin, A. 2000 Integrable structure of interface dynamics. Phys. Rev. Lett. 84 (22), 51065109.
Mineev-Weinstein, M. B. & Dawson, S. P. 1994 Class of nonsingular exact solutions for Laplacian pattern formation. Phys. Rev. E 50, R24R27.
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467521.
Morrison, P. J. 2005 Hamiltonian and action principle formulations of plasma physics. Phys. Plasmas 12, 058102.
Novikov, S., Manakov, S. V., Pitaevskii, L. P. & Zakharov, V. E. 1984 Theory of Solitons: The Inverse Scattering Method. Springer.
Ovsyannikov, L. V. 1973 Dynamics of a fluid. M.A. Lavrent’ev Institute of Hydrodynamics Sib. Branch USSR Ac. Sci. 15, 104125.
Pandey, J. N. 1996 The Hilbert Transform of Schwartz Distributions and Applications. Wiley.
Penney, W. G. & Price, A. T. 1952 Part II. Finite periodic stationary gravity waves in a perfect liquid. Phil. Trans. R. Soc. Lond. A 244, 254284.
Polubarinova-Kochina, P. Y. 1945 On motion of the contour of an oil layer. Dokl. Akad. Nauk SSSR 47, 254257.
Polyanin, A. D. & Manzhirov, A. V. 2008 Handbook of Integral Equations, 2nd edn. Chapman and Hall/CRC.
Ruban, V. P. 2005 Quasiplanar steep water waves. Phys. Rev. E 71, 055303R.
Rudin, W. 1986 Real and Complex Analysis, 3rd edn. McGraw-Hill.
Shikin, V. B. 1970 Motion of helium ions near a vapor–liquid surface. Sov. Phys. JETP 31, 936.
Shraiman, B. I. & Bensimon, D. 1984 Singularities in nonlocal interface dynamics. Phys. Rev. A 30, 28402844.
Stoker, J. J. 1957 Water Waves. Interscience.
Stokes, G. G. 1880 On the theory of oscillatory waves. Math. Phys. Papers 1, 197229.
Tanveer, S. 1991 Singularities in water waves and Rayleigh–Taylor instability. Proc. R. Soc. Lond. A 435, 137158.
Tanveer, S. 1993 Singularities in the classical Rayleigh–Taylor flow: formation and subsequent motion. Proc. R. Soc. Lond. A 441, 501525.
Tanveer, S. 1996 Some analytical properties of solutions to a two-dimensional steadily translating inviscid bubble. Proc. R. Soc. Lond. A 452, 13971410.
Titchmarsh, E. C. 1948 Introduction to the Theory of Fourier Integrals, 2nd edn. Clarendon.
Weinstein, A. 1983 The local structure of Poisson manifolds. J. Differ. Geom. 18, 523557.
Wilkening, J. 2011 Breakdown of self-similarity at the crests of large-amplitude standing water waves. Phys. Rev. Lett. 107, 184501.10.1103/PhysRevLett.107.184501
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on a surface. J. Appl. Mech. Tech. Phys. 9 (2), 190194.
Zakharov, V. E. & Dyachenko, A. I.2012 Free-surface hydrodynamics in the conformal variables. arXiv:1206.2046.
Zakharov, V. E., Dyachenko, A. I. & Prokofiev, A. O. 2006 Freak waves as nonlinear stage of Stokes wave modulation instability. Eur. J. Mech. (B/Fluids) 25, 677692.10.1016/j.euromechflu.2006.03.004
Zakharov, V. E., Dyachenko, A. I. & Vasiliev, O. A. 2002 New method for numerical simulation of nonstationary potential flow of incompressible fluid with a free surface. Eur. J. Mech. (B/Fluids) 21, 283291.10.1016/S0997-7546(02)01189-5
Zakharov, V. E. & Faddeev, L. D. 1971 Korteweg–de Vries equation: a completely integrable Hamiltonian system. Funct. Anal. Applics. 5, 280287.
Zakharov, V. E. & Kuznetsov, E. A. 1997 Hamiltonian formalism for nonlinear waves. Usp. Fiz. Nauk 167, 11371167.
Zakharov, V. E., Lvov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I. Springer.
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62.
Zubarev, N. M. 2000 Charged-surface instability development in liquid helium: an exact solution. JETP Lett. 71, 367369.
Zubarev, N. M. 2002 Exact solutions of the equations of motion of liquid helium with a charged free surface. J. Expl Theor. Phys. 94, 534544.
Zubarev, N. M. 2008 Formation of singularities on the charged surface of a liquid-helium layer with a finite depth. J. Expl Theor. Phys. 107, 668678.10.1134/S1063776108100154
Zubarev, N. M. & Kochurin, E. A. 2014 Interaction of strongly nonlinear waves on the free surface of a dielectric liquid in a horizontal electric field. JETP Lett. 99, 627631.10.1134/S0021364014110125
Zubarev, N. M. & Zubareva, O. V. 2006 Nondispersive propagation of waves with finite amplitudes on the surface of a dielectric liquid in a tangential electric field. Tech. Phys. Lett. 32, 886888.
Zubarev, N. M. & Zubareva, O. V. 2008 Stability of nonlinear waves on the ideal liquid surface in a tangential electric field. Tech. Phys. Lett. 34, 535537.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed