# American Institute of Mathematical Sciences

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

## Internal Gerstner waves on a sloping bed

 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.
Citation: Raphael Stuhlmeier. Internal Gerstner waves on a sloping bed. Discrete and Continuous Dynamical Systems, 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), 479-513. doi: 10.1007/s00205-011-0483-2. [2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, 2006. doi: 10.1017/CBO9780511734939. [3] B. Bolzano, Leben franz joseph ritters von gerstner, Abhandlungen der kön. böhmischen Gesellschaft der Wissenschaften, (1837). [4] A. Constantin, Edge waves along a sloping beach, J. Phys. A: Mathematical General, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311. [5] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [6] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879. [7] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), 1-4. doi: 10.1029/2012GL051169. [8] A. Constantin, Some three-dimensional non-linear equatorial flows, J. Phys. Oceanography, 43 (2013), 165-175. [9] A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344. doi: 10.1016/j.nonrwa.2007.03.003. [10] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.-Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. [11] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [12] A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2006), 1-18. doi: 10.1007/s00021-005-0214-2. [13] D. Farmer and J. Smith, Nonlinear internal waves in a fjord, in Elsevier Oceanography Series (editor, J. C. Nihoul), 23 (1978), 465-493. doi: 10.1016/S0422-9894(08)71294-7. [14] W. Froude, On the rolling of ships, Transactions of the Institution of Naval Architects, 11 (1861), 180-227. doi: 10.1080/03071847309433595. [15] C. Garrett and W. Munk, Internal waves in the ocean, Annual Review of Fluid Mechanics, (1979), 339-369. doi: 10.1146/annurev.fl.11.010179.002011. [16] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten theorie der deichprofile, Abh. der kön. böhmischen Gesellschaft der Wissenschaften, (1804). [17] F. Gerstner, Theorie der Wellen, Ann. Phys., 32 (1809), 412-445. doi: 10.1002/andp.18090320808. [18] K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Ann. Rev. Fluid Mech., 38 (2006), 395-425. doi: 10.1146/annurev.fluid.38.050304.092129. [19] D. Henry, On the deep-water stokes wave flow, Int. Math. Res. Not., 2008 (2008), 7 pp. doi: 10.1093/imrn/rnn071. [20] 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. [21] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056. [22] R. S. Johnson, Edge waves: Theories past and present, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2359-2376. doi: 10.1098/rsta.2007.2013. [23] H. Kalisch, Periodic traveling water waves with isobaric streamlines, J. Non-linear Math. Phys., 11 (2004), 461-471. doi: 10.2991/jnmp.2004.11.4.3. [24] P. D. Komar, Beach processes and sedimentation, Prentice-Hall, 1976. [25] A.-V. Matioco, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 10 pp. doi: 10.1142/S1402925112400098. [26] A.-V. Matioc and B.-V. Matioc, On periodic water waves with Coriolis effects and isobaric streamlines, J. Nonlinear Math. Phys., 19 (2012), 15 PP. doi: 10.1142/S1402925112400098. [27] E. G. Morozov and A. V. Marchenko, Short-period internal waves in an arctic Fjord (Spitsbergen), Izv. Atmos. Ocean. Phy., 48 (2012), 401-408. doi: 10.1134/S0001433812040123. [28] L. Mysak, Topographically trapped waves, Ann. Rev. Fluid Mech., 12 (1980), 45-76. [29] F. Nansen, The Norwegian North Polar Expedition 1893-1896, Scientific Results, Volume 5, The Fridtjof Nansen Fund for the Advancement of Science, Christiania, 1906. [30] J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1982. doi: 10.1115/1.3157711. [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), 127-138. doi: 10.1098/rstl.1863.0006. [32] F. Reech, Sur la theorie des ondes liquides periodiques, Comptes Rendus Acad. Sci. Paris, 68 (1869), 1099-1101. [33] G. G. Stokes, Report on recent researches in hydrodynamics, in Brit. Assoc. Rep., 1846. doi: 10.1017/CBO9780511702242.011. [34] G. G. Stokes, On the theory of oscillatory waves, Trans. Camb. Phil. Soc., 8 (1847), 441-455. doi: 10.1017/CBO9780511702242.013. [35] R. Stuhlmeier, Internal Gerstner waves: applications to dead water, Appl. Anal., to appear. doi: 10.1080/00036811.2013.833609. [36] C. Truesdell, The Kinematics of Vorticity, Indiana University Press, Bloomington, 1954. [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), 1687-1702. doi: 10.1098/rsta.2011.0450. [38] C. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), 765-767. doi: 10.1017/S0022112066000983.

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##### References:
 [1] A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Rat. Mech. Anal., 204 (2012), 479-513. doi: 10.1007/s00205-011-0483-2. [2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, 2006. doi: 10.1017/CBO9780511734939. [3] B. Bolzano, Leben franz joseph ritters von gerstner, Abhandlungen der kön. böhmischen Gesellschaft der Wissenschaften, (1837). [4] A. Constantin, Edge waves along a sloping beach, J. Phys. A: Mathematical General, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311. [5] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [6] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879. [7] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), 1-4. doi: 10.1029/2012GL051169. [8] A. Constantin, Some three-dimensional non-linear equatorial flows, J. Phys. Oceanography, 43 (2013), 165-175. [9] A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344. doi: 10.1016/j.nonrwa.2007.03.003. [10] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.-Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. [11] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [12] A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2006), 1-18. doi: 10.1007/s00021-005-0214-2. [13] D. Farmer and J. Smith, Nonlinear internal waves in a fjord, in Elsevier Oceanography Series (editor, J. C. Nihoul), 23 (1978), 465-493. doi: 10.1016/S0422-9894(08)71294-7. [14] W. Froude, On the rolling of ships, Transactions of the Institution of Naval Architects, 11 (1861), 180-227. doi: 10.1080/03071847309433595. [15] C. Garrett and W. Munk, Internal waves in the ocean, Annual Review of Fluid Mechanics, (1979), 339-369. doi: 10.1146/annurev.fl.11.010179.002011. [16] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten theorie der deichprofile, Abh. der kön. böhmischen Gesellschaft der Wissenschaften, (1804). [17] F. Gerstner, Theorie der Wellen, Ann. Phys., 32 (1809), 412-445. doi: 10.1002/andp.18090320808. [18] K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Ann. Rev. Fluid Mech., 38 (2006), 395-425. doi: 10.1146/annurev.fluid.38.050304.092129. [19] D. Henry, On the deep-water stokes wave flow, Int. Math. Res. Not., 2008 (2008), 7 pp. doi: 10.1093/imrn/rnn071. [20] 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. [21] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056. [22] R. S. Johnson, Edge waves: Theories past and present, Philos. Trans. Roy. Soc. London Ser. A, 365 (2007), 2359-2376. doi: 10.1098/rsta.2007.2013. [23] H. Kalisch, Periodic traveling water waves with isobaric streamlines, J. Non-linear Math. Phys., 11 (2004), 461-471. doi: 10.2991/jnmp.2004.11.4.3. [24] P. D. Komar, Beach processes and sedimentation, Prentice-Hall, 1976. [25] A.-V. Matioco, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 10 pp. doi: 10.1142/S1402925112400098. [26] A.-V. Matioc and B.-V. Matioc, On periodic water waves with Coriolis effects and isobaric streamlines, J. Nonlinear Math. Phys., 19 (2012), 15 PP. doi: 10.1142/S1402925112400098. [27] E. G. Morozov and A. V. Marchenko, Short-period internal waves in an arctic Fjord (Spitsbergen), Izv. Atmos. Ocean. Phy., 48 (2012), 401-408. doi: 10.1134/S0001433812040123. [28] L. Mysak, Topographically trapped waves, Ann. Rev. Fluid Mech., 12 (1980), 45-76. [29] F. Nansen, The Norwegian North Polar Expedition 1893-1896, Scientific Results, Volume 5, The Fridtjof Nansen Fund for the Advancement of Science, Christiania, 1906. [30] J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1982. doi: 10.1115/1.3157711. [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), 127-138. doi: 10.1098/rstl.1863.0006. [32] F. Reech, Sur la theorie des ondes liquides periodiques, Comptes Rendus Acad. Sci. Paris, 68 (1869), 1099-1101. [33] G. G. Stokes, Report on recent researches in hydrodynamics, in Brit. Assoc. Rep., 1846. doi: 10.1017/CBO9780511702242.011. [34] G. G. Stokes, On the theory of oscillatory waves, Trans. Camb. Phil. Soc., 8 (1847), 441-455. doi: 10.1017/CBO9780511702242.013. [35] R. Stuhlmeier, Internal Gerstner waves: applications to dead water, Appl. Anal., to appear. doi: 10.1080/00036811.2013.833609. [36] C. Truesdell, The Kinematics of Vorticity, Indiana University Press, Bloomington, 1954. [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), 1687-1702. doi: 10.1098/rsta.2011.0450. [38] C. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), 765-767. doi: 10.1017/S0022112066000983.
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