# American Institute of Mathematical Sciences

January  2017, 37(1): 387-404. doi: 10.3934/dcds.2017016

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

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

* Corresponding author: Calin Iulian Martin

Received  March 2016 Revised  May 2016 Published  November 2016

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.

Citation: Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016
##### References:
 [1] 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. Google Scholar [2] A. Constantin, On the modelling of equatorial waves Geophys. Res. Lett. 39 (2012), L05602. doi: 10.1029/2012GL051169. Google Scholar [3] A. Constantin, An exact solution for equatorially trapped waves J. Geophys. Res. : Oceans 117 (2012), C05029. doi: 10.1029/2012JC007879. Google Scholar [4] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] 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. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] 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.Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] M. Giaquinta and S. Hildebrandt. Calculus of Variations I, Springer-Verlag, Berlin, 1996. Google Scholar [16] 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. Google Scholar [17] D. Henry, Internal equatorial water waves in the $f$-plane, J. Nonl. Math. Phys., 22 (2015), 499-506. doi: 10.1080/14029251.2015.1113046. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [22] D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Amer. Math. Soc. , Providence, RI, 2013. doi: 10.1090/surv/188. Google Scholar [23] P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978.Google Scholar [24] 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. Google Scholar [25] C. I. Martin, Surface tension effects in the equatorial ocean dynamics, Monatshefte für Mathematik, (2015), 1-8. doi: 10.1007/s00605-015-0858-9. Google Scholar [26] 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. Google Scholar [27] 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. Google Scholar [28] S.-A. Maslowe, Critical layers in shear flows, Ann. Rev. Fluid Mech., 18 (1986), 405-432. Google Scholar [29] 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. Google Scholar [30] D. P. Nicholls, Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007), 44-74. doi: 10.1002/gamm.200790009. Google Scholar [31] R. Quirchmayr, On the existence of benthic storms, J. Nonl. Math. Phys., 22 (2015), 540-544. doi: 10.1080/14029251.2015.1113053. Google Scholar [32] G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. Google Scholar [33] 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. Google Scholar [34] G. Thomas, Wave-current interactions: an experimental and numerical study, J. Fluid Mech., 216 (1990), 303-315. Google Scholar [35] 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. Google Scholar [36] E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005. Google Scholar [37] 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. Google Scholar [38] 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. Google Scholar

show all references

##### References:
 [1] 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. Google Scholar [2] A. Constantin, On the modelling of equatorial waves Geophys. Res. Lett. 39 (2012), L05602. doi: 10.1029/2012GL051169. Google Scholar [3] A. Constantin, An exact solution for equatorially trapped waves J. Geophys. Res. : Oceans 117 (2012), C05029. doi: 10.1029/2012JC007879. Google Scholar [4] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] 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. Google Scholar [9] 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. Google Scholar [10] 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. Google Scholar [11] 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.Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] M. Giaquinta and S. Hildebrandt. Calculus of Variations I, Springer-Verlag, Berlin, 1996. Google Scholar [16] 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. Google Scholar [17] D. Henry, Internal equatorial water waves in the $f$-plane, J. Nonl. Math. Phys., 22 (2015), 499-506. doi: 10.1080/14029251.2015.1113046. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar [20] 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. Google Scholar [21] 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. Google Scholar [22] D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Amer. Math. Soc. , Providence, RI, 2013. doi: 10.1090/surv/188. Google Scholar [23] P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978.Google Scholar [24] 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. Google Scholar [25] C. I. Martin, Surface tension effects in the equatorial ocean dynamics, Monatshefte für Mathematik, (2015), 1-8. doi: 10.1007/s00605-015-0858-9. Google Scholar [26] 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. Google Scholar [27] 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. Google Scholar [28] S.-A. Maslowe, Critical layers in shear flows, Ann. Rev. Fluid Mech., 18 (1986), 405-432. Google Scholar [29] 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. Google Scholar [30] D. P. Nicholls, Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007), 44-74. doi: 10.1002/gamm.200790009. Google Scholar [31] R. Quirchmayr, On the existence of benthic storms, J. Nonl. Math. Phys., 22 (2015), 540-544. doi: 10.1080/14029251.2015.1113053. Google Scholar [32] G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. Google Scholar [33] 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. Google Scholar [34] G. Thomas, Wave-current interactions: an experimental and numerical study, J. Fluid Mech., 216 (1990), 303-315. Google Scholar [35] 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. Google Scholar [36] E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005. Google Scholar [37] 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. Google Scholar [38] 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. Google Scholar
 [1] Shu-Ming Sun. Existence theory of capillary-gravity waves on water of finite depth. Mathematical Control & Related Fields, 2014, 4 (3) : 315-363. doi: 10.3934/mcrf.2014.4.315 [2] Mark Jones. The bifurcation of interfacial capillary-gravity waves under O(2) symmetry. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1183-1204. doi: 10.3934/cpaa.2011.10.1183 [3] Frédéric Rousset, Nikolay Tzvetkov. On the transverse instability of one dimensional capillary-gravity waves. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 859-872. doi: 10.3934/dcdsb.2010.13.859 [4] Kenta Ohi, Tatsuo Iguchi. A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1205-1240. doi: 10.3934/dcds.2009.23.1205 [5] Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929 [6] Didier Bresch, Jacques Simon. Western boundary currents versus vanishing depth. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 469-477. doi: 10.3934/dcdsb.2003.3.469 [7] Kristoffer Varholm. Solitary gravity-capillary water waves with point vortices. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3927-3959. doi: 10.3934/dcds.2016.36.3927 [8] Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219 [9] Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163 [10] D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure & Applied Analysis, 2004, 3 (2) : 197-216. doi: 10.3934/cpaa.2004.3.197 [11] Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114 [12] Yu-Xia Wang, Wan-Tong Li. Combined effects of the spatial heterogeneity and the functional response. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 19-39. doi: 10.3934/dcds.2019002 [13] Anass Belcaid, Mohammed Douimi, Abdelkader Fassi Fihri. Recursive reconstruction of piecewise constant signals by minimization of an energy function. Inverse Problems & Imaging, 2018, 12 (4) : 903-920. doi: 10.3934/ipi.2018038 [14] Nicolay M. Tanushev, Luminita Vese. A piecewise-constant binary model for electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 423-435. doi: 10.3934/ipi.2007.1.423 [15] Xiaoping Fang, Youjun Deng. Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement. Inverse Problems & Imaging, 2018, 12 (3) : 733-743. doi: 10.3934/ipi.2018031 [16] Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691 [17] Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457 [18] Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475 [19] Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045 [20] G. A. Athanassoulis, K. A. Belibassakis. New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth. Conference Publications, 2007, 2007 (Special) : 75-84. doi: 10.3934/proc.2007.2007.75

2018 Impact Factor: 1.143