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

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

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.

Keywords: Hamiltonian functional, capillary-gravity effects, piecewise constant vorticity, depth-dependent currents, stratification.
Mathematics Subject Classification: Primary:35Q31, 35Q35, 76B15, 76D45;Secondary:76D3.
Citation: Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete and Continuous Dynamical Systems, 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.

[2]

A. Constantin, On the modelling of equatorial waves Geophys. Res. Lett. 39 (2012), L05602. doi: 10.1029/2012GL051169.

[3]

A. Constantin, An exact solution for equatorially trapped waves J. Geophys. Res. : Oceans 117 (2012), C05029. doi: 10.1029/2012JC007879.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

M. Giaquinta and S. Hildebrandt. Calculus of Variations I, Springer-Verlag, Berlin, 1996.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Amer. Math. Soc. , Providence, RI, 2013. doi: 10.1090/surv/188.

[23]

P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978.

[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.

[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.

[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.

[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.

[28]

S.-A. Maslowe, Critical layers in shear flows, Ann. Rev. Fluid Mech., 18 (1986), 405-432. 

[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.

[30]

D. P. Nicholls, Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007), 44-74.  doi: 10.1002/gamm.200790009.

[31]

R. Quirchmayr, On the existence of benthic storms, J. Nonl. Math. Phys., 22 (2015), 540-544.  doi: 10.1080/14029251.2015.1113053.

[32]

G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. 

[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. 

[34]

G. Thomas, Wave-current interactions: an experimental and numerical study, J. Fluid Mech., 216 (1990), 303-315. 

[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.

[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.

[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.

[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.

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.

[2]

A. Constantin, On the modelling of equatorial waves Geophys. Res. Lett. 39 (2012), L05602. doi: 10.1029/2012GL051169.

[3]

A. Constantin, An exact solution for equatorially trapped waves J. Geophys. Res. : Oceans 117 (2012), C05029. doi: 10.1029/2012JC007879.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

M. Giaquinta and S. Hildebrandt. Calculus of Variations I, Springer-Verlag, Berlin, 1996.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

D. Lannes, The Water Waves Problem. Mathematical Analysis and Asymptotics, Amer. Math. Soc. , Providence, RI, 2013. doi: 10.1090/surv/188.

[23]

P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978.

[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.

[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.

[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.

[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.

[28]

S.-A. Maslowe, Critical layers in shear flows, Ann. Rev. Fluid Mech., 18 (1986), 405-432. 

[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.

[30]

D. P. Nicholls, Boundary perturbation methods for water waves, GAMM-Mitt., 30 (2007), 44-74.  doi: 10.1002/gamm.200790009.

[31]

R. Quirchmayr, On the existence of benthic storms, J. Nonl. Math. Phys., 22 (2015), 540-544.  doi: 10.1080/14029251.2015.1113053.

[32]

G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. 

[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. 

[34]

G. Thomas, Wave-current interactions: an experimental and numerical study, J. Fluid Mech., 216 (1990), 303-315. 

[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.

[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.

[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.

[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.

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