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August  2019, 39(8): 4443-4453. doi: 10.3934/dcds.2019181

Wind generated equatorial Gerstner-type waves

Anatoly Abrashkin

National Research University Higher School of Economics, 25/12 Bol'shaya Pecherskaya str., Nizhny Novgorod, 603155, Russia

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  July 2018 Revised  December 2018 Published  May 2019

A class of non-stationary surface gravity waves propagating in the zonal direction in the equatorial region is described in the f-plane approximation. These waves are described by exact solutions of the equations of hydrodynamics in Lagrangian formulation and are generalizations of Gerstner waves. The wave shape and non-uniform pressure distribution on a free surface depend on two arbitrary functions. The trajectories of fluid particles are circumferences. The solutions admit a variable meridional current. The dynamics of a single breather on the background of a Gerstner wave is studied as an example.

Keywords: Lagrangian variables, f-plane approximation, exact solution, Gerstner wave, vorticity.
Mathematics Subject Classification: Primary: 76B15; Secondary: 86A05, 74G05, 70H03.
Citation: Anatoly Abrashkin. Wind generated equatorial Gerstner-type waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4443-4453. doi: 10.3934/dcds.2019181
References:
[1]

A. A. Abrashkin and E. I. Yakubovich, Planar rotational flows of an ideal fluid, Sov. Phys. Dokl., 29 (1984), 370-374.   Google Scholar

[2]

A. A. Abrashkin, D. A. Zen'kovich and E. I. Yakubovich, Matrix formulation of hydrodynamics and extension of Ptolemaic flows to three-dimensional motions, Radiophys. Quantum El., 39 (1996), 518-526.  doi: 10.1007/BF02122398.  Google Scholar

[3]

A. A. Abrashkin and A. Soloviev, Vortical freak waves in water under external pressure action, Phys. Rev. Lett., 110 (2013), 014501. doi: 10.1103/PhysRevLett.110.014501.  Google Scholar

[4]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Rational Mech. Anal., 204 (2012), 479-513.  doi: 10.1007/s00205-011-0483-2.  Google Scholar

[5] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511734939.  Google Scholar
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A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[7]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, 81(2011), SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.  Google Scholar

[8]

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

[9]

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

[10]

A. Constantin, Some three dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[11]

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

[12]

A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528.  doi: 10.1017/jfm.2017.223.  Google Scholar

[13]

M. L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides heterogenes, Atti. Accad. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., 6 (1932), 814-819.   Google Scholar

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U. Frisch and B. Villone, Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow, Eur. Phys. J. H, 39 (2014), 325-351.  doi: 10.1140/epjh/e2014-50016-6.  Google Scholar

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W. Froude, On the rolling of ships, Trans. Inst. Naval Arch., 3 (1862), 45-62.   Google Scholar

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D. Henry, On Gerstners water wave, J. Nonl. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[20]

D. Henry, An exact solution for 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

[21]

D. Henry, Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11 pp. doi: 10.1017/jfm.2016.544.  Google Scholar

[22]

D. Henry, Exact equatorial water waves in the f-plane, Nonl. Anal. Real World Appl., 28 (2016), 284-289.  doi: 10.1016/j.nonrwa.2015.10.003.  Google Scholar

[23]

M. Kluczek, Equatorial water waves with underlying currents in the f-plane approximation, Appl. Anal., 97 (2018), 1867-1880.  doi: 10.1080/00036811.2017.1343466.  Google Scholar

[24]

G. Lamb, Hydrodynamics, 6th edition, Cambridge University Press, 1932. Google Scholar

[25]

E. Mollo-Christensen, Gravitational and geostrophic billows: Some exact solutions, J. Atmosph. Sciences, 35 (1978), 1395–1398. doi: 10.1175/1520-0469(1978)035<1395:GAGBSE>2.0.CO;2.  Google Scholar

[26]

E. Mollo-Christensen, Edge waves in a rotating stratified fluid, an exact solution, J. Phys. Oceanogr., 9 (1979), 226-229.  doi: 10.1175/1520-0485(1979)009<0226:EWIARS>2.0.CO;2.  Google Scholar

[27]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave field, Phys. Fluids, 25 (1982), 586-587.  doi: 10.1063/1.863802.  Google Scholar

[28]

S. Monismith, H. Nepf, E. A. Cowen, L. Thais and J. Magnaudet, Laboratory observations of mean flows under surface gravity waves, J. Fluid Mech., 573 (2007), 131-147.  doi: 10.1017/S0022112006003594.  Google Scholar

[29]

R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898.  doi: 10.1029/JC075i030p05895.  Google Scholar

[30]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. Lond. A, 153 (1863), 127-138.   Google Scholar

[31]

F. Reech, Sur la théorie des ondes liquids périodiques, C. R. Acad. Sci. Paris, 68 (1869), 1099-1101.   Google Scholar

[32]

A. Rodrgues-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map defining equatorially-trapped internal water waves, Nonl. Anal., 149 (2017), 156-164.  doi: 10.1016/j.na.2016.10.022.  Google Scholar

[33]

A. Rodrgues-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions, Ann. Mat. Pura Appl., 197 (2018), 1787-1797.  doi: 10.1007/s10231-018-0749-5.  Google Scholar

[34]

S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining Equatorially trapped water waves, Nonl. Anal., 125 (2015), 725-731.  doi: 10.1016/j.na.2015.06.017.  Google Scholar

[35]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137.  doi: 10.1142/S1402925111001210.  Google Scholar

[36]

J. E. H. Weber, A note on trapped Gerstner waves, J. Geophys. Res., 117 (2011), C03048. Google Scholar

[37]

J. E. H. Weber, Do we observe Gerstner waves in wave tank experiments?, Wave motion, 48 (2011), 301-309.  doi: 10.1016/j.wavemoti.2010.11.005.  Google Scholar

[38]

J. E. H. Weber, An interfacial Gerstner-type trapped wave, Wave Motion, 77 (2018), 186-194.  doi: 10.1016/j.wavemoti.2017.12.002.  Google Scholar

[39]

C.-S. Yih, Note on edge waves in a stratified fluid, Advanced Series on Fluid Mechanics, (1991), 108–110. doi: 10.1142/9789812813084_0010.  Google Scholar

[40]

V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, Phys.-Usp., 40 (1997), 1087-1116.   Google Scholar

show all references

References:
[1]

A. A. Abrashkin and E. I. Yakubovich, Planar rotational flows of an ideal fluid, Sov. Phys. Dokl., 29 (1984), 370-374.   Google Scholar

[2]

A. A. Abrashkin, D. A. Zen'kovich and E. I. Yakubovich, Matrix formulation of hydrodynamics and extension of Ptolemaic flows to three-dimensional motions, Radiophys. Quantum El., 39 (1996), 518-526.  doi: 10.1007/BF02122398.  Google Scholar

[3]

A. A. Abrashkin and A. Soloviev, Vortical freak waves in water under external pressure action, Phys. Rev. Lett., 110 (2013), 014501. doi: 10.1103/PhysRevLett.110.014501.  Google Scholar

[4]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Rational Mech. Anal., 204 (2012), 479-513.  doi: 10.1007/s00205-011-0483-2.  Google Scholar

[5] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511734939.  Google Scholar
[6]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[7]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, 81(2011), SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.  Google Scholar

[8]

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

[9]

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

[10]

A. Constantin, Some three dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[11]

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

[12]

A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528.  doi: 10.1017/jfm.2017.223.  Google Scholar

[13]

M. L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides heterogenes, Atti. Accad. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., 6 (1932), 814-819.   Google Scholar

[14]

U. Frisch and B. Villone, Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow, Eur. Phys. J. H, 39 (2014), 325-351.  doi: 10.1140/epjh/e2014-50016-6.  Google Scholar

[15]

W. Froude, On the rolling of ships, Trans. Inst. Naval Arch., 3 (1862), 45-62.   Google Scholar

[16]

F. J. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 32 (1809), 412-445.   Google Scholar

[17]

O. A. Godin, Incompressible wave motion of compressible fluids, Phys. Rev. Lett., 108 (2012), 194501. doi: 10.1103/PhysRevLett.108.194501.  Google Scholar

[18]

O. A. Godin, Finite-amplitude acoustic-gravity waves: Exact solutions, J. Fluid Mech., 767 (2015), 52-64.  doi: 10.1017/jfm.2015.40.  Google Scholar

[19]

D. Henry, On Gerstners water wave, J. Nonl. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[20]

D. Henry, An exact solution for 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

[21]

D. Henry, Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11 pp. doi: 10.1017/jfm.2016.544.  Google Scholar

[22]

D. Henry, Exact equatorial water waves in the f-plane, Nonl. Anal. Real World Appl., 28 (2016), 284-289.  doi: 10.1016/j.nonrwa.2015.10.003.  Google Scholar

[23]

M. Kluczek, Equatorial water waves with underlying currents in the f-plane approximation, Appl. Anal., 97 (2018), 1867-1880.  doi: 10.1080/00036811.2017.1343466.  Google Scholar

[24]

G. Lamb, Hydrodynamics, 6th edition, Cambridge University Press, 1932. Google Scholar

[25]

E. Mollo-Christensen, Gravitational and geostrophic billows: Some exact solutions, J. Atmosph. Sciences, 35 (1978), 1395–1398. doi: 10.1175/1520-0469(1978)035<1395:GAGBSE>2.0.CO;2.  Google Scholar

[26]

E. Mollo-Christensen, Edge waves in a rotating stratified fluid, an exact solution, J. Phys. Oceanogr., 9 (1979), 226-229.  doi: 10.1175/1520-0485(1979)009<0226:EWIARS>2.0.CO;2.  Google Scholar

[27]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave field, Phys. Fluids, 25 (1982), 586-587.  doi: 10.1063/1.863802.  Google Scholar

[28]

S. Monismith, H. Nepf, E. A. Cowen, L. Thais and J. Magnaudet, Laboratory observations of mean flows under surface gravity waves, J. Fluid Mech., 573 (2007), 131-147.  doi: 10.1017/S0022112006003594.  Google Scholar

[29]

R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898.  doi: 10.1029/JC075i030p05895.  Google Scholar

[30]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. Lond. A, 153 (1863), 127-138.   Google Scholar

[31]

F. Reech, Sur la théorie des ondes liquids périodiques, C. R. Acad. Sci. Paris, 68 (1869), 1099-1101.   Google Scholar

[32]

A. Rodrgues-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map defining equatorially-trapped internal water waves, Nonl. Anal., 149 (2017), 156-164.  doi: 10.1016/j.na.2016.10.022.  Google Scholar

[33]

A. Rodrgues-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions, Ann. Mat. Pura Appl., 197 (2018), 1787-1797.  doi: 10.1007/s10231-018-0749-5.  Google Scholar

[34]

S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining Equatorially trapped water waves, Nonl. Anal., 125 (2015), 725-731.  doi: 10.1016/j.na.2015.06.017.  Google Scholar

[35]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137.  doi: 10.1142/S1402925111001210.  Google Scholar

[36]

J. E. H. Weber, A note on trapped Gerstner waves, J. Geophys. Res., 117 (2011), C03048. Google Scholar

[37]

J. E. H. Weber, Do we observe Gerstner waves in wave tank experiments?, Wave motion, 48 (2011), 301-309.  doi: 10.1016/j.wavemoti.2010.11.005.  Google Scholar

[38]

J. E. H. Weber, An interfacial Gerstner-type trapped wave, Wave Motion, 77 (2018), 186-194.  doi: 10.1016/j.wavemoti.2017.12.002.  Google Scholar

[39]

C.-S. Yih, Note on edge waves in a stratified fluid, Advanced Series on Fluid Mechanics, (1991), 108–110. doi: 10.1142/9789812813084_0010.  Google Scholar

[40]

V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, Phys.-Usp., 40 (1997), 1087-1116.   Google Scholar

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