August  2014, 34(8): 3135-3153. doi: 10.3934/dcds.2014.34.3135

A boundary integral formulation for particle trajectories in Stokes waves

1. 

Instituto Nacional de Matemática Pura e Aplicada/IMPA, Est. D. Castorina, 110, J. Botânico, Rio de Janeiro, RJ 22460-320, Brazil

Received  July 2013 Revised  September 2013 Published  January 2014

Recently important theorems have been established presenting qualitative results for particle trajectories below a Stokes wave. A diversity of orbit patterns were described, including the case of a closed orbit when a Stokes wave propagates in the presence of an adverse current. In this work these results are revisited in a quantitative fashion through a boundary integral formulation which leads to very accurate numerical simulations of particle trajectories. The boundary integral formulation allows the accurate evaluation of the vector field of the (particle's) dynamical system, without resorting to a series expansion and a small parameter. Accurate trajectories are benchmarked against well known expansions for weakly nonlinear waves. Simulations are then performed beyond this regime. Closed orbits are found in the presence of an adverse current, as well as non-smooth trajectories that have not been reported. These occur for both adverse and favorable currents.
Citation: André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135
References:
[1]

D. P. Bertsekas, Nonlinear Programming, $2^{nd}$ edition, Athena Scientific, Belmont, Massachusetts, 1999. doi: 10.1038/sj.jors.2600425.

[2]

H.-K. Chang, Y.-Y. Chen and J.-C. Liou, Particle trajectories of nonlinear gravity waves in deep water, Ocean Engineering, 36 (2009), 324-329. doi: 10.1016/j.oceaneng.2008.12.007.

[3]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[4]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, 2011. doi: 10.1137/1.9781611971873.

[5]

A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6.

[6]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[7]

A. Constantin, M. Ehrnström and E. Wahlén, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1-18. doi: 10.1016/j.nonrwa.2007.03.003.

[8]

A. Constantin and W. A. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.

[9]

A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fulid Mech., 10 (2008), 1336-1344. doi: 10.1007/s00021-005-0214-2.

[10]

M. W. Dingemans, Water Waves Propagation Over Uneven Bottoms, World Scientific, Singapore, 1997. doi: 10.1142/1241-part1.

[11]

J. D. Fenton, A fifth-order Stokes theory for steady waves, Journal of Waterway, Port, Coastal and Ocean Engineering, 111 (1985), 216-234. doi: 10.1061/(ASCE)0733-950X(1985)111:2(216).

[12]

J. D. Fenton, Nonlinear wave theories, The Sea: Ocean Engineering Science, 9 (1990), 1-18.

[13]

P. Guidotti, A new first-kind boundary integral formulation for the Dirichlet-to-Neumman map in 2D, J. Comput. Phy., 190 (2008), 325-345. doi: 10.1016/S0021-9991(03)00277-8.

[14]

D. Henry, On the deep-water Stokes wave flow, IMRN, 2008 (2008), 1-7. doi: 10.1093/imrn/rnn071.

[15]

M. Isobe, H. Nishimura and K. Horikawa, Expressions of Pertubation Solutions for Conservative Waves by Using Wave Height, Proceedings of 33rd annual conference of JSCE 1978, 760-761.

[16]

F. John, Partial Differential Equations, $4^{th}$ edition, Springer-Verlag, New York, 1982.

[17]

I. G. Jonsson and L. Arneborg, Energy properties and shoaling of higher-order stokes waves on current, Ocean Engineering, 22 (1995), 819-857. doi: 10.1016/0029-8018(95)00008-9.

[18]

H. Lamb, Hydrodynamics, Cambridge, Univ. Press, 1895.

[19]

M. S. Longuet-Higgins, Eulerian and Lagrangian aspects of surface waves, J. Fluid Mech., 173 (1986), 683-707. doi: 10.1017/S0022112086001325.

[20]

, Available at: http://w3.impa.br/~nachbin/AndreNachbin/Stokes_Waves.html.

[21]

H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves, World Scientific, River Edge, NJ, 2001.

[22]

H. Okamoto and M. Shōji, Trajectories of fluid particles in a periodic water wave, Phil. Trans. R. Soc. A, 370 (2012), 1661-1676. doi: 10.1098/rsta.2011.0447.

[23]

F. Ruellan and A. Wallet, Trajectoires internes das un clapotis partiel, La Houille Blanche, 5 (1950), 483-489.

[24]

L. Skjelbreia and J. Hendrinck, Fifth Order Gravity Wave Theory, JProceedings of 7th conference on coastal engineering, ASCE 1960, 184-196.

[25]

J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Intersciene Publ. Inc., New York, 1957.

[26]

G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. doi: 10.1017/CBO9780511702242.013.

[27]

L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719598.

[28]

M. Umeyama, Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry, Phil. Trans. R. Soc. A, 370 (2012), 1687-1702. doi: 10.1098/rsta.2011.0450.

[29]

F. Ursell, Mass transport in gravity waves, Proc. Cambridge Phil. Soc., 40 (1953), 145-150. doi: 10.1017/S0305004100028140.

[30]

G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974.

show all references

References:
[1]

D. P. Bertsekas, Nonlinear Programming, $2^{nd}$ edition, Athena Scientific, Belmont, Massachusetts, 1999. doi: 10.1038/sj.jors.2600425.

[2]

H.-K. Chang, Y.-Y. Chen and J.-C. Liou, Particle trajectories of nonlinear gravity waves in deep water, Ocean Engineering, 36 (2009), 324-329. doi: 10.1016/j.oceaneng.2008.12.007.

[3]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[4]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, 2011. doi: 10.1137/1.9781611971873.

[5]

A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6.

[6]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.

[7]

A. Constantin, M. Ehrnström and E. Wahlén, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1-18. doi: 10.1016/j.nonrwa.2007.03.003.

[8]

A. Constantin and W. A. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.

[9]

A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fulid Mech., 10 (2008), 1336-1344. doi: 10.1007/s00021-005-0214-2.

[10]

M. W. Dingemans, Water Waves Propagation Over Uneven Bottoms, World Scientific, Singapore, 1997. doi: 10.1142/1241-part1.

[11]

J. D. Fenton, A fifth-order Stokes theory for steady waves, Journal of Waterway, Port, Coastal and Ocean Engineering, 111 (1985), 216-234. doi: 10.1061/(ASCE)0733-950X(1985)111:2(216).

[12]

J. D. Fenton, Nonlinear wave theories, The Sea: Ocean Engineering Science, 9 (1990), 1-18.

[13]

P. Guidotti, A new first-kind boundary integral formulation for the Dirichlet-to-Neumman map in 2D, J. Comput. Phy., 190 (2008), 325-345. doi: 10.1016/S0021-9991(03)00277-8.

[14]

D. Henry, On the deep-water Stokes wave flow, IMRN, 2008 (2008), 1-7. doi: 10.1093/imrn/rnn071.

[15]

M. Isobe, H. Nishimura and K. Horikawa, Expressions of Pertubation Solutions for Conservative Waves by Using Wave Height, Proceedings of 33rd annual conference of JSCE 1978, 760-761.

[16]

F. John, Partial Differential Equations, $4^{th}$ edition, Springer-Verlag, New York, 1982.

[17]

I. G. Jonsson and L. Arneborg, Energy properties and shoaling of higher-order stokes waves on current, Ocean Engineering, 22 (1995), 819-857. doi: 10.1016/0029-8018(95)00008-9.

[18]

H. Lamb, Hydrodynamics, Cambridge, Univ. Press, 1895.

[19]

M. S. Longuet-Higgins, Eulerian and Lagrangian aspects of surface waves, J. Fluid Mech., 173 (1986), 683-707. doi: 10.1017/S0022112086001325.

[20]

, Available at: http://w3.impa.br/~nachbin/AndreNachbin/Stokes_Waves.html.

[21]

H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves, World Scientific, River Edge, NJ, 2001.

[22]

H. Okamoto and M. Shōji, Trajectories of fluid particles in a periodic water wave, Phil. Trans. R. Soc. A, 370 (2012), 1661-1676. doi: 10.1098/rsta.2011.0447.

[23]

F. Ruellan and A. Wallet, Trajectoires internes das un clapotis partiel, La Houille Blanche, 5 (1950), 483-489.

[24]

L. Skjelbreia and J. Hendrinck, Fifth Order Gravity Wave Theory, JProceedings of 7th conference on coastal engineering, ASCE 1960, 184-196.

[25]

J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Intersciene Publ. Inc., New York, 1957.

[26]

G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. doi: 10.1017/CBO9780511702242.013.

[27]

L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719598.

[28]

M. Umeyama, Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry, Phil. Trans. R. Soc. A, 370 (2012), 1687-1702. doi: 10.1098/rsta.2011.0450.

[29]

F. Ursell, Mass transport in gravity waves, Proc. Cambridge Phil. Soc., 40 (1953), 145-150. doi: 10.1017/S0305004100028140.

[30]

G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974.

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