On the particle motion in geophysical deep water waves traveling over uniform currents
Author:
Anca-Voichita Matioc
Journal:
Quart. Appl. Math. 72 (2014), 455-469
MSC (2010):
Primary 76B15; Secondary 74G05, 37N10
DOI:
https://doi.org/10.1090/S0033-569X-2014-01337-5
Published electronically:
April 23, 2014
MathSciNet review:
3237559
Full-text PDF Free Access
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References |
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Additional Information
Abstract: We describe a family of exact Gerstner-type solutions for the geophysical equatorial deep water wave problem in the $f$-plane approximation. These Gerstner-type waves are two-dimensional and travel with constant speed over a uniform horizontal current. The particle paths in the presence and absence of the Coriolis force are also analyzed in dependence of the current strength.
References
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- Anca-Voichita Matioc, On particle trajectories in linear water waves, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 4275–4284. MR 2683875, DOI https://doi.org/10.1016/j.nonrwa.2010.05.014
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- Anca-Voichita Matioc, On particle trajectories in linear deep-water waves, Commun. Pure Appl. Anal. 11 (2012), no. 4, 1537–1547. MR 2900801, DOI https://doi.org/10.3934/cpaa.2012.11.1537
- Bogdan-Vasile Matioc, Regularity results for deep-water waves with Hölder continuous vorticity, Appl. Anal. 92 (2013), no. 10, 2144–2151. MR 3169153, DOI https://doi.org/10.1080/00036811.2012.718335
- Bogdan-Vasile Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math. 70 (2012), no. 2, 393–405. MR 2953110, DOI https://doi.org/10.1090/S0033-569X-2012-01261-1
- Anca-Voichita Matioc, An explicit solution for deep water waves with Coriolis effects, J. Nonlinear Math. Phys. 19 (2012), no. suppl. 1, 1240005, 8. MR 2999399, DOI https://doi.org/10.1142/S1402925112400050
- Erik Mollo-Christiansen. Gravitational and geostrophic billows: Some exact solutions. J. Atmos. Sci, 35:1395–1398, 1978.
- Hisashi Okamoto and Mayumi Sh\B{o}ji, The mathematical theory of permanent progressive water-waves, Advanced Series in Nonlinear Dynamics, vol. 20, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1869386
- Joseph Pedlosky. Geophysical fluid dynamics. Springer, New York, 1979.
- W. J. M. Rankine. On the exact form of waves near the surface of fluids. Philos. Trans. R. Soc. Lond. A, 153:127–138, 1863.
- J. J. Stoker, Water waves, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1992. The mathematical theory with applications; Reprint of the 1957 original; A Wiley-Interscience Publication. MR 1153414
- Raphael Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys. 18 (2011), no. 1, 127–137. MR 2786939, DOI https://doi.org/10.1142/S1402925111001210
- J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1–48. MR 1422004, DOI https://doi.org/10.12775/TMNA.1996.001
References
- Adrian Constantin, Edge waves along a sloping beach, J. Phys. A 34 (2001), no. 45, 9723–9731. MR 1876166 (2002j:76015), DOI https://doi.org/10.1088/0305-4470/34/45/311
- Adrian Constantin, On the deep water wave motion, J. Phys. A 34 (2001), no. 7, 1405–1417. MR 1819940 (2002b:76010), DOI https://doi.org/10.1088/0305-4470/34/7/313
- Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390 (2007j:35240), DOI https://doi.org/10.1007/s00222-006-0002-5
- Adrian Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. MR 2867413
- Adrian Constantin. On the modelling of Equatorial waves. Geophys. Res. Lett., 39:L05602, 2012.
- Adrian Constantin, Mats Ehrnström, and Gabriele Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl. 9 (2008), no. 4, 1336–1344. MR 2422547 (2009f:35265), DOI https://doi.org/10.1016/j.nonrwa.2007.03.003
- Adrian Constantin, Mats Ehrnström, and Erik Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), no. 3, 591–603. MR 2362244 (2009c:35359), DOI https://doi.org/10.1215/S0012-7094-07-14034-1
- Adrian Constantin and Joachim Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math. 15 (2004), no. 6, 755–768. MR 2144685 (2006b:76013), DOI https://doi.org/10.1017/S0956792504005777
- Adrian Constantin and Joachim Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech. 498 (2004), 171–181. MR 2256915 (2007f:76023), DOI https://doi.org/10.1017/S0022112003006773
- Adrian Constantin and Joachim Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2) 173 (2011), no. 1, 559–568. MR 2753609 (2012a:35367), DOI https://doi.org/10.4007/annals.2011.173.1.12
- Adrian Constantin and Walter Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), no. 4, 533–557. MR 2604871 (2011b:76017), DOI https://doi.org/10.1002/cpa.20299
- Adrian Constantin and Gabriele Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech. 10 (2008), no. 1, 1–18. MR 2383410 (2009a:76017), DOI https://doi.org/10.1007/s00021-005-0214-2
- Isabelle Gallagher and Laure Saint-Raymond. On the influence of the earth’s rotation on geophysical flows. Handbook of Mathematical Fluid Dynamics, 4:201–329, 2007.
- Franz Gerstner. Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys., 2:412–445, 1809.
- David Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. , posted on (2006), Art. ID 23405, 13. MR 2272104 (2007k:76017), DOI https://doi.org/10.1155/IMRN/2006/23405
- David Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys. 14 (2007), no. 1, 1–7. MR 2287829 (2007k:76018), DOI https://doi.org/10.2991/jnmp.2007.14.1.1
- David Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 87–95. MR 2434727 (2009k:76020), DOI https://doi.org/10.2991/jnmp.2008.15.s2.7
- Delia Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 13–27. MR 2434722 (2009g:76015), DOI https://doi.org/10.2991/jnmp.2008.15.s2.2
- Robin Stanley Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555 (99m:76017)
- Henrik Kalisch, Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys. 11 (2004), no. 4, 461–471. MR 2097657 (2005m:35237), DOI https://doi.org/10.2991/jnmp.2004.11.4.3
- James Lighthill, Waves in fluids, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2001. Reprint of the 1978 original. MR 1872073 (2002i:76001)
- Anca-Voichita Matioc, On particle trajectories in linear water waves, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 4275–4284. MR 2683875 (2012b:76019), DOI https://doi.org/10.1016/j.nonrwa.2010.05.014
- Anca-Voichita Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A 45 (2012), no. 36, 365501, 10. MR 2967917, DOI https://doi.org/10.1088/1751-8113/45/36/365501
- Anca-Voichita Matioc, An explicit solution for deep water waves with Coriolis effects, J. Nonlinear Math. Phys. 19 (2012), no. suppl. 1, 1240005, 8. MR 2999399, DOI https://doi.org/10.1142/S1402925112400050
- Anca-Voichita Matioc, On particle trajectories in linear deep-water waves, Commun. Pure Appl. Anal. 11 (2012), no. 4, 1537–1547. MR 2900801, DOI https://doi.org/10.3934/cpaa.2012.11.1537
- Bogdan-Vasile Matioc. Regularity results for deep-water waves with Hölder continuous vorticity. Appl. Anal. 92 (10) (2013), 2144–2151. DOI 10.1080/00036811.2012.718335. MR 3169153
- Bogdan-Vasile Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math. 70 (2012), no. 2, 393–405. MR 2953110, DOI https://doi.org/10.1090/S0033-569X-2012-01261-1
- Anca-Voichita Matioc, An explicit solution for deep water waves with Coriolis effects, J. Nonlinear Math. Phys. 19 (2012), no. suppl. 1, 1240005, 8. MR 2999399, DOI https://doi.org/10.1142/S1402925112400050
- Erik Mollo-Christiansen. Gravitational and geostrophic billows: Some exact solutions. J. Atmos. Sci, 35:1395–1398, 1978.
- Hisashi Okamoto and Mayumi Shōji, The mathematical theory of permanent progressive water-waves, Advanced Series in Nonlinear Dynamics, vol. 20, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR 1869386 (2003h:76016)
- Joseph Pedlosky. Geophysical fluid dynamics. Springer, New York, 1979.
- W. J. M. Rankine. On the exact form of waves near the surface of fluids. Philos. Trans. R. Soc. Lond. A, 153:127–138, 1863.
- James J. Stoker, Water waves, Wiley Classics Library, John Wiley & Sons Inc., New York, 1992. The mathematical theory with applications; Reprint of the 1957 original; A Wiley-Interscience Publication. MR 1153414 (92m:76029)
- Raphael Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys. 18 (2011), no. 1, 127–137. MR 2786939 (2012g:76033), DOI https://doi.org/10.1142/S1402925111001210
- John Francis Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1–48. MR 1422004 (97j:35130)
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Additional Information
Anca-Voichita Matioc
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria
Email:
anca.matioc@univie.ac.at
Keywords:
Gravity deep-water waves,
Gerstner’s wave,
Coriolis effects,
Lagrangian coordinates
Received by editor(s):
June 22, 2012
Published electronically:
April 23, 2014
Article copyright:
© Copyright 2014
Brown University