A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows

Calin Iulian Martin

doi:10.3934/dcds.2017016

Article Contents

2017, Volume 37, Issue 1: 387-404. Doi: 10.3934/dcds.2017016

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A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows

Calin Iulian Martin^,

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

* Corresponding author: Calin Iulian Martin
* Corresponding author: Calin Iulian Martin

Received: March 2016

Revised: May 2016

Published: September 2017

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Abstract

Under consideration here are two-dimensional rotational stratified water flows driven by gravity and surface tension, bounded below by a rigid flat bed and above by a free surface. The distribution of vorticity and of density is piecewise constant-with a jump across the interface separating the fluid of bigger density from the lighter fluid adjacent to the free surface. The main result is that the governing equations for the two-layered rotational stratified flows, as described above, admit a Hamiltonian formulation.

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Mathematics Subject Classification: Primary:35Q31, 35Q35, 76B15, 76D45;Secondary:76D33.

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References

	T. B. Benjamin and P. J. Olver , Hamiltonian structures, symmetries and conservation laws for water waves, J. Fluid Mech., 125 (1982) , 137-185. doi: 10.1017/S0022112082003292.
	A. Constantin, On the modelling of equatorial waves Geophys. Res. Lett. 39 (2012), L05602. doi: 10.1029/2012GL051169.
	A. Constantin, An exact solution for equatorially trapped waves J. Geophys. Res. : Oceans 117 (2012), C05029. doi: 10.1029/2012JC007879.
	A. Constantin and P. Germain , Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013) , 2802-2810. doi: 10.1002/jgrc.20219.
	A. Constantin , Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014) , 781-789. doi: 10.1175/JPO-D-13-0174.1.
	A. Constantin and R. S. Johnson , The dynamics of waves interacting with the Equatorial Undercurrent, Geophysical and Astrophysical Fluid Dynamics, 109 (2015) , 311-358. doi: 10.1080/03091929.2015.1066785.
	A. Constantin and R. S. Johnson , An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016) , 1935-1945. doi: 10.1175/JPO-D-15-0205.1.
	A. Constantin , R. Ivanov and E. Prodanov , Nearly-Hamiltonian Structure for Water Waves with Constant Vorticity, J. Math. Fluid Mech., 10 (2008) , 224-237. doi: 10.1007/s00021-006-0230-x.
	A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis volume 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.
	A. Constantin and E. Varvaruca , Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011) , 33-67. doi: 10.1007/s00205-010-0314-x.
	A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers, to appear in Acta Mathematica arxiv: 1407.0092.
	A. Constantin and R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids Physics of Fluids27 (2015), 086603. doi: 10.1063/1.4929457.
	A. Constantin , R. Ivanov and C. I. Martin , Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016) , 1417-1447. doi: 10.1007/s00205-016-0990-2.
	W. Craig , P. Guyenne and H. Kalisch , Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005) , 1587-1641. doi: 10.1002/cpa.20098.
	M. Giaquinta and S. Hildebrandt. Calculus of Variations I, Springer-Verlag, Berlin, 1996.
	D. Henry , An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013) , 18-21. doi: 10.1016/j.euromechflu.2012.10.001.
	D. Henry , Internal equatorial water waves in the $f$-plane, J. Nonl. Math. Phys., 22 (2015) , 499-506. doi: 10.1080/14029251.2015.1113046.
	D. Henry and H.-C. Hsu , Instability of internal equatorial waves, J. Differential Equations, 258 (2015) , 1015-1024. doi: 10.1016/j.jde.2014.08.019.
	D. Henry , Exact equatorial water waves in the $f$ -plane, Nonlinear Anal. Real World Appl., 28 (2016) , 284-289. doi: 10.1016/j.nonrwa.2015.10.003.
	D. Ionescu-Kruse , Instability of equatorially trapped waves in stratified water, Ann. Mat. Pura Appl., 195 (2016) , 585-599. doi: 10.1007/s10231-015-0479-x.
	V. Kozlov and N. Kuznetsov , Dispersion relation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014) , 971-1018. doi: 10.1007/s00205-014-0787-0.
	D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Amer. Math. Soc. , Providence, RI, 2013. doi: 10.1090/surv/188.
	P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978.
	C. I. Martin , Dynamics of the thermocline in the equatorial region of the Pacific Ocean, J. Nonl. Math. Phys., 22 (2015) , 516-522. doi: 10.1080/14029251.2015.1113049.
	C. I. Martin , Surface tension effects in the equatorial ocean dynamics, Monatshefte für Mathematik, (2015) , 1-8. doi: 10.1007/s00605-015-0858-9.
	C. I. Martin , Hamiltonian structure for rotational capillary waves in stratified flows, J. Differential Equations, 261 (2016) , 373-395. doi: 10.1016/j.jde.2016.03.013.
	C. I. Martin and B.-V. Matioc , Existence of Wilton ripples for water waves with constant vorticity and capillary effects, SIAM J. Appl. Math., 73 (2013) , 1582-1595. doi: 10.1137/120900290.
	S.-A. Maslowe , Critical layers in shear flows, Ann. Rev. Fluid Mech., 18 (1986) , 405-432.
	A. -V. Matioc, Steady internal water waves with a critical layer bounded by the wave surface, J. Nonl. Math. Phys. , 19 (2012), 1250008, 21 pp. doi: 10.1142/S1402925112500088.
	D. P. Nicholls , Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007) , 44-74. doi: 10.1002/gamm.200790009.
	R. Quirchmayr , On the existence of benthic storms, J. Nonl. Math. Phys., 22 (2015) , 540-544. doi: 10.1080/14029251.2015.1113053.
	G. Stokes , On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847) , 441-455.
	C. Swan , I. P. Cummins and R. L. James , An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001) , 273-304.
	G. Thomas , Wave-current interactions: an experimental and numerical study, J. Fluid Mech., 216 (1990) , 303-315.
	E. Wahlén , A Hamiltonian formulation of water waves with constant vorticity, Lett. Math. Phys., 79 (2007) , 303-315. doi: 10.1007/s11005-007-0143-5.
	E. Wahlén , Steady water waves with a critical layer, J. Differential Equations, 246 (2009) , 2468-2483. doi: 10.1016/j.jde.2008.10.005.
	J. Wilkening and V. Vasan, Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem, in Nonlinear wave equations: analytic and computational techniques, 175–210, Contemp. Math. , 635, Amer. Math. Soc. , Providence, RI, 2015. doi: 10.1090/conm/635/12713.
	V. E. Zakharov , Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968) , 190-194. doi: 10.1007/BF00913182.