Wind generated equatorial Gerstner-type waves

Previous Article
Existence of solutions for nonlinear operator equations
DCDS Home
This Issue
Next Article
Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

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

Wind generated equatorial Gerstner-type waves

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
[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, 6^th 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, 6^th 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

[1]	David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909
[2]	Igor E. Pritsker and Richard S. Varga. Weighted polynomial approximation in the complex plane. Electronic Research Announcements, 1997, 3: 38-44.
[3]	Vishal Vasan, Katie Oliveras. Pressure beneath a traveling wave with constant vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3219-3239. doi: 10.3934/dcds.2014.34.3219
[4]	Keisuke Hakuta, Hisayoshi Sato, Tsuyoshi Takagi. On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$. Advances in Mathematics of Communications, 2016, 10 (2) : 221-228. doi: 10.3934/amc.2016002
[5]	Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007
[6]	Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219
[7]	Jifeng Chu, Delia Ionescu-Kruse, Yanjuan Yang. Exact solution and instability for geophysical waves at arbitrary latitude. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4399-4414. doi: 10.3934/dcds.2019178
[8]	R. Bartolo, Anna Maria Candela, J.L. Flores, Addolorata Salvatore. Periodic trajectories in plane wave type spacetimes. Conference Publications, 2005, 2005 (Special) : 77-83. doi: 10.3934/proc.2005.2005.77
[9]	M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1221-1235. doi: 10.3934/cpaa.2013.12.1221
[10]	Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[11]	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
[12]	Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
[13]	Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[14]	Jerry Bona, Jiahong Wu. Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1141-1168. doi: 10.3934/dcds.2009.23.1141
[15]	Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007
[16]	Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[17]	Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901
[18]	Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893
[19]	Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367
[20]	Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

American Institute of Mathematical Sciences