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Wind generated equatorial Gerstner-type waves
National Research University Higher School of Economics, 25/12 Bol'shaya Pecherskaya str., Nizhny Novgorod, 603155, Russia |
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.
References:
[1] |
A. A. Abrashkin and E. I. Yakubovich,
Planar rotational flows of an ideal fluid, Sov. Phys. Dokl., 29 (1984), 370-374.
|
[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. |
[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. |
[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. |
[5] |
A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511734939.![]() ![]() ![]() |
[6] |
A. Constantin,
Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[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. |
[8] |
A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.
doi: 10.1029/2012GL051169. |
[9] |
A. Constantin, An exact solution for equatorially trapped waves, J. of Geophys. Res., 117 (2012), C05029.
doi: 10.1029/2012JC007879. |
[10] |
A. Constantin,
Some three dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[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. |
[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. |
[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.
|
[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. |
[15] |
W. Froude,
On the rolling of ships, Trans. Inst. Naval Arch., 3 (1862), 45-62.
|
[16] |
F. J. Gerstner,
Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 32 (1809), 412-445.
|
[17] |
O. A. Godin, Incompressible wave motion of compressible fluids, Phys. Rev. Lett., 108 (2012), 194501.
doi: 10.1103/PhysRevLett.108.194501. |
[18] |
O. A. Godin,
Finite-amplitude acoustic-gravity waves: Exact solutions, J. Fluid Mech., 767 (2015), 52-64.
doi: 10.1017/jfm.2015.40. |
[19] |
D. Henry,
On Gerstners water wave, J. Nonl. Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.S2.7. |
[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. |
[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. |
[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. |
[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. |
[24] |
G. Lamb, Hydrodynamics, 6th edition, Cambridge University Press, 1932. |
[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. |
[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. |
[27] |
E. Mollo-Christensen,
Allowable discontinuities in a Gerstner wave field, Phys. Fluids, 25 (1982), 586-587.
doi: 10.1063/1.863802. |
[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. |
[29] |
R. T. Pollard,
Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898.
doi: 10.1029/JC075i030p05895. |
[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.
|
[31] |
F. Reech,
Sur la théorie des ondes liquids périodiques, C. R. Acad. Sci. Paris, 68 (1869), 1099-1101.
|
[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. |
[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. |
[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. |
[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. |
[36] |
J. E. H. Weber, A note on trapped Gerstner waves, J. Geophys. Res., 117 (2011), C03048. |
[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. |
[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. |
[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. |
[40] |
V. E. Zakharov and E. A. Kuznetsov,
Hamiltonian formalism for nonlinear waves, Phys.-Usp., 40 (1997), 1087-1116.
|
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.
|
[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. |
[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. |
[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. |
[5] |
A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511734939.![]() ![]() ![]() |
[6] |
A. Constantin,
Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[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. |
[8] |
A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.
doi: 10.1029/2012GL051169. |
[9] |
A. Constantin, An exact solution for equatorially trapped waves, J. of Geophys. Res., 117 (2012), C05029.
doi: 10.1029/2012JC007879. |
[10] |
A. Constantin,
Some three dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[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. |
[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. |
[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.
|
[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. |
[15] |
W. Froude,
On the rolling of ships, Trans. Inst. Naval Arch., 3 (1862), 45-62.
|
[16] |
F. J. Gerstner,
Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 32 (1809), 412-445.
|
[17] |
O. A. Godin, Incompressible wave motion of compressible fluids, Phys. Rev. Lett., 108 (2012), 194501.
doi: 10.1103/PhysRevLett.108.194501. |
[18] |
O. A. Godin,
Finite-amplitude acoustic-gravity waves: Exact solutions, J. Fluid Mech., 767 (2015), 52-64.
doi: 10.1017/jfm.2015.40. |
[19] |
D. Henry,
On Gerstners water wave, J. Nonl. Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.S2.7. |
[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. |
[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. |
[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. |
[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. |
[24] |
G. Lamb, Hydrodynamics, 6th edition, Cambridge University Press, 1932. |
[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. |
[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. |
[27] |
E. Mollo-Christensen,
Allowable discontinuities in a Gerstner wave field, Phys. Fluids, 25 (1982), 586-587.
doi: 10.1063/1.863802. |
[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. |
[29] |
R. T. Pollard,
Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898.
doi: 10.1029/JC075i030p05895. |
[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.
|
[31] |
F. Reech,
Sur la théorie des ondes liquids périodiques, C. R. Acad. Sci. Paris, 68 (1869), 1099-1101.
|
[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. |
[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. |
[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. |
[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. |
[36] |
J. E. H. Weber, A note on trapped Gerstner waves, J. Geophys. Res., 117 (2011), C03048. |
[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. |
[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. |
[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. |
[40] |
V. E. Zakharov and E. A. Kuznetsov,
Hamiltonian formalism for nonlinear waves, Phys.-Usp., 40 (1997), 1087-1116.
|
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