American Institute of Mathematical Sciences

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August  2014, 34(8): 3183-3192. doi: 10.3934/dcds.2014.34.3183

Internal Gerstner waves on a sloping bed

Raphael Stuhlmeier 1,

1. 

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

Received  September 2013 Revised  October 2013 Published  January 2014

We provide an explicit solution to the full, nonlinear governing equations for gravity water waves describing internal edge waves along a sloping bed. This solution is based on the Gerstner edge wave. We discuss the relation of this internal, trochoidal edge wave to the analogous wave found in the linear theory, compare it with the classical Gerstner wave, as well as discuss the inclusion of Coriolis forces in the f-plane approximation.
Keywords: Internal waves, Gerstner waves., edge waves.
Mathematics Subject Classification: Primary: 76B15; Secondary: 76B5.
Citation: Raphael Stuhlmeier. Internal Gerstner waves on a sloping bed. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3183-3192. doi: 10.3934/dcds.2014.34.3183
References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows,, Arch. Rat. Mech. Anal., 204 (2012), 479. doi: 10.1007/s00205-011-0483-2. Google Scholar

[2]

A. Bennett, Lagrangian Fluid Dynamics,, Cambridge University Press, (2006). doi: 10.1017/CBO9780511734939. Google Scholar

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B. Bolzano, Leben franz joseph ritters von gerstner,, Abhandlungen der kön. böhmischen Gesellschaft der Wissenschaften, (1837). Google Scholar

[4]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A: Mathematical General, 34 (2001), 9723. doi: 10.1088/0305-4470/34/45/311. Google Scholar

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A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar

[6]

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

[7]

A. Constantin, On the modelling of equatorial waves,, Geophys. Res. Lett., 39 (2012), 1. doi: 10.1029/2012GL051169. Google Scholar

[8]

A. Constantin, Some three-dimensional non-linear equatorial flows,, J. Phys. Oceanography, 43 (2013), 165. Google Scholar

[9]

A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves,, Nonlinear Anal. Real World Appl., 9 (2008), 1336. doi: 10.1016/j.nonrwa.2007.03.003. Google Scholar

[10]

A. Constantin and P. Germain, Instability of some equatorially trapped waves,, J. Geophys. Res.-Oceans, 118 (2013), 2802. doi: 10.1002/jgrc.20219. Google Scholar

[11]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Commun. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20299. Google Scholar

[12]

A. Constantin and G. Villari, Particle trajectories in linear water waves,, J. Math. Fluid Mech., 10 (2006), 1. doi: 10.1007/s00021-005-0214-2. Google Scholar

[13]

D. Farmer and J. Smith, Nonlinear internal waves in a fjord,, in Elsevier Oceanography Series (editor, 23 (1978), 465. doi: 10.1016/S0422-9894(08)71294-7. Google Scholar

[14]

W. Froude, On the rolling of ships,, Transactions of the Institution of Naval Architects, 11 (1861), 180. doi: 10.1080/03071847309433595. Google Scholar

[15]

C. Garrett and W. Munk, Internal waves in the ocean,, Annual Review of Fluid Mechanics, (1979), 339. doi: 10.1146/annurev.fl.11.010179.002011. Google Scholar

[16]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten theorie der deichprofile,, Abh. der kön. böhmischen Gesellschaft der Wissenschaften, (1804). Google Scholar

[17]

F. Gerstner, Theorie der Wellen,, Ann. Phys., 32 (1809), 412. doi: 10.1002/andp.18090320808. Google Scholar

[18]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves,, Ann. Rev. Fluid Mech., 38 (2006), 395. doi: 10.1146/annurev.fluid.38.050304.092129. Google Scholar

[19]

D. Henry, On the deep-water stokes wave flow,, Int. Math. Res. Not., 2008 (2008). doi: 10.1093/imrn/rnn071. Google Scholar

[20]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. - B/Fluids, 38 (2013), 18. doi: 10.1016/j.euromechflu.2012.10.001. Google Scholar

[21]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997). doi: 10.1017/CBO9780511624056. Google Scholar

[22]

R. S. Johnson, Edge waves: Theories past and present,, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2359. doi: 10.1098/rsta.2007.2013. Google Scholar

[23]

H. Kalisch, Periodic traveling water waves with isobaric streamlines,, J. Non-linear Math. Phys., 11 (2004), 461. doi: 10.2991/jnmp.2004.11.4.3. Google Scholar

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P. D. Komar, Beach processes and sedimentation,, Prentice-Hall, (1976). Google Scholar

[25]

A.-V. Matioco, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A, 45 (2012). doi: 10.1142/S1402925112400098. Google Scholar

[26]

A.-V. Matioc and B.-V. Matioc, On periodic water waves with Coriolis effects and isobaric streamlines,, J. Nonlinear Math. Phys., 19 (2012). doi: 10.1142/S1402925112400098. Google Scholar

[27]

E. G. Morozov and A. V. Marchenko, Short-period internal waves in an arctic Fjord (Spitsbergen),, Izv. Atmos. Ocean. Phy., 48 (2012), 401. doi: 10.1134/S0001433812040123. Google Scholar

[28]

L. Mysak, Topographically trapped waves,, Ann. Rev. Fluid Mech., 12 (1980), 45. Google Scholar

[29]

F. Nansen, The Norwegian North Polar Expedition 1893-1896, Scientific Results, Volume 5,, The Fridtjof Nansen Fund for the Advancement of Science, (1906). Google Scholar

[30]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer, (1982). doi: 10.1115/1.3157711. Google Scholar

[31]

W. J. M. Rankine, On the exact form of waves near the surface of deep water,, Philos. Trans. Roy. Soc. London Ser. A, 153 (1863), 127. doi: 10.1098/rstl.1863.0006. Google Scholar

[32]

F. Reech, Sur la theorie des ondes liquides periodiques,, Comptes Rendus Acad. Sci. Paris, 68 (1869), 1099. Google Scholar

[33]

G. G. Stokes, Report on recent researches in hydrodynamics,, in Brit. Assoc. Rep., (1846). doi: 10.1017/CBO9780511702242.011. Google Scholar

[34]

G. G. Stokes, On the theory of oscillatory waves,, Trans. Camb. Phil. Soc., 8 (1847), 441. doi: 10.1017/CBO9780511702242.013. Google Scholar

[35]

R. Stuhlmeier, Internal Gerstner waves: applications to dead water,, Appl. Anal., (). doi: 10.1080/00036811.2013.833609. Google Scholar

[36]

C. Truesdell, The Kinematics of Vorticity,, Indiana University Press, (1954). Google Scholar

[37]

M. Umeyama, Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry,, Philos. Trans. Roy. Soc. London Ser. A, 370 (2012), 1687. doi: 10.1098/rsta.2011.0450. Google Scholar

[38]

C. Yih, Note on edge waves in a stratified fluid,, J. Fluid Mech., 24 (1966), 765. doi: 10.1017/S0022112066000983. Google Scholar

show all references

References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows,, Arch. Rat. Mech. Anal., 204 (2012), 479. doi: 10.1007/s00205-011-0483-2. Google Scholar

[2]

A. Bennett, Lagrangian Fluid Dynamics,, Cambridge University Press, (2006). doi: 10.1017/CBO9780511734939. Google Scholar

[3]

B. Bolzano, Leben franz joseph ritters von gerstner,, Abhandlungen der kön. böhmischen Gesellschaft der Wissenschaften, (1837). Google Scholar

[4]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A: Mathematical General, 34 (2001), 9723. doi: 10.1088/0305-4470/34/45/311. Google Scholar

[5]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar

[6]

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

[7]

A. Constantin, On the modelling of equatorial waves,, Geophys. Res. Lett., 39 (2012), 1. doi: 10.1029/2012GL051169. Google Scholar

[8]

A. Constantin, Some three-dimensional non-linear equatorial flows,, J. Phys. Oceanography, 43 (2013), 165. Google Scholar

[9]

A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves,, Nonlinear Anal. Real World Appl., 9 (2008), 1336. doi: 10.1016/j.nonrwa.2007.03.003. Google Scholar

[10]

A. Constantin and P. Germain, Instability of some equatorially trapped waves,, J. Geophys. Res.-Oceans, 118 (2013), 2802. doi: 10.1002/jgrc.20219. Google Scholar

[11]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Commun. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20299. Google Scholar

[12]

A. Constantin and G. Villari, Particle trajectories in linear water waves,, J. Math. Fluid Mech., 10 (2006), 1. doi: 10.1007/s00021-005-0214-2. Google Scholar

[13]

D. Farmer and J. Smith, Nonlinear internal waves in a fjord,, in Elsevier Oceanography Series (editor, 23 (1978), 465. doi: 10.1016/S0422-9894(08)71294-7. Google Scholar

[14]

W. Froude, On the rolling of ships,, Transactions of the Institution of Naval Architects, 11 (1861), 180. doi: 10.1080/03071847309433595. Google Scholar

[15]

C. Garrett and W. Munk, Internal waves in the ocean,, Annual Review of Fluid Mechanics, (1979), 339. doi: 10.1146/annurev.fl.11.010179.002011. Google Scholar

[16]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten theorie der deichprofile,, Abh. der kön. böhmischen Gesellschaft der Wissenschaften, (1804). Google Scholar

[17]

F. Gerstner, Theorie der Wellen,, Ann. Phys., 32 (1809), 412. doi: 10.1002/andp.18090320808. Google Scholar

[18]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves,, Ann. Rev. Fluid Mech., 38 (2006), 395. doi: 10.1146/annurev.fluid.38.050304.092129. Google Scholar

[19]

D. Henry, On the deep-water stokes wave flow,, Int. Math. Res. Not., 2008 (2008). doi: 10.1093/imrn/rnn071. Google Scholar

[20]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. - B/Fluids, 38 (2013), 18. doi: 10.1016/j.euromechflu.2012.10.001. Google Scholar

[21]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997). doi: 10.1017/CBO9780511624056. Google Scholar

[22]

R. S. Johnson, Edge waves: Theories past and present,, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2359. doi: 10.1098/rsta.2007.2013. Google Scholar

[23]

H. Kalisch, Periodic traveling water waves with isobaric streamlines,, J. Non-linear Math. Phys., 11 (2004), 461. doi: 10.2991/jnmp.2004.11.4.3. Google Scholar

[24]

P. D. Komar, Beach processes and sedimentation,, Prentice-Hall, (1976). Google Scholar

[25]

A.-V. Matioco, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A, 45 (2012). doi: 10.1142/S1402925112400098. Google Scholar

[26]

A.-V. Matioc and B.-V. Matioc, On periodic water waves with Coriolis effects and isobaric streamlines,, J. Nonlinear Math. Phys., 19 (2012). doi: 10.1142/S1402925112400098. Google Scholar

[27]

E. G. Morozov and A. V. Marchenko, Short-period internal waves in an arctic Fjord (Spitsbergen),, Izv. Atmos. Ocean. Phy., 48 (2012), 401. doi: 10.1134/S0001433812040123. Google Scholar

[28]

L. Mysak, Topographically trapped waves,, Ann. Rev. Fluid Mech., 12 (1980), 45. Google Scholar

[29]

F. Nansen, The Norwegian North Polar Expedition 1893-1896, Scientific Results, Volume 5,, The Fridtjof Nansen Fund for the Advancement of Science, (1906). Google Scholar

[30]

J. Pedlosky, Geophysical Fluid Dynamics,, Springer, (1982). doi: 10.1115/1.3157711. Google Scholar

[31]

W. J. M. Rankine, On the exact form of waves near the surface of deep water,, Philos. Trans. Roy. Soc. London Ser. A, 153 (1863), 127. doi: 10.1098/rstl.1863.0006. Google Scholar

[32]

F. Reech, Sur la theorie des ondes liquides periodiques,, Comptes Rendus Acad. Sci. Paris, 68 (1869), 1099. Google Scholar

[33]

G. G. Stokes, Report on recent researches in hydrodynamics,, in Brit. Assoc. Rep., (1846). doi: 10.1017/CBO9780511702242.011. Google Scholar

[34]

G. G. Stokes, On the theory of oscillatory waves,, Trans. Camb. Phil. Soc., 8 (1847), 441. doi: 10.1017/CBO9780511702242.013. Google Scholar

[35]

R. Stuhlmeier, Internal Gerstner waves: applications to dead water,, Appl. Anal., (). doi: 10.1080/00036811.2013.833609. Google Scholar

[36]

C. Truesdell, The Kinematics of Vorticity,, Indiana University Press, (1954). Google Scholar

[37]

M. Umeyama, Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry,, Philos. Trans. Roy. Soc. London Ser. A, 370 (2012), 1687. doi: 10.1098/rsta.2011.0450. Google Scholar

[38]

C. Yih, Note on edge waves in a stratified fluid,, J. Fluid Mech., 24 (1966), 765. doi: 10.1017/S0022112066000983. Google Scholar

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