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Perron-Frobenius theory and frequency convergence for reducible substitutions
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 |
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.
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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