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July  2012, 11(4): 1549-1561. doi: 10.3934/cpaa.2012.11.1549

Effects of shear flow on KdV balance - applications to tsunami

Raphael Stuhlmeier 1,

1. 

Fakultät für Mathematik, University of Vienna, Nordbergstr. 15, 1090 Vienna, Austria

Received  March 2011 Revised  June 2011 Published  January 2012

Building upon recent work in the applicability of soliton theory to tsunami propagation, we discuss the effects of shear flow on the KdV balance. This leads in the shallow-water limit to the Burns condition, and we see that for shear which does not yield critical layer solutions, the speeds determined by the Burns condition arise again in the KdV balance. In the event of waves propagating counter to the shear, KdV dynamics arise earlier, while their appearance is delayed in the case of waves propagating with the shear, the magnitude of this effect depending on the surface shear velocity.
Keywords: tsunami., solitons, Korteweg--de Vries equation, Burns condition, shear flow.
Mathematics Subject Classification: 76B15, 35C20, 35Q31, 35Q5.
Citation: Raphael Stuhlmeier. Effects of shear flow on KdV balance - applications to tsunami. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1549-1561. doi: 10.3934/cpaa.2012.11.1549
References:
[1]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.

[2]

T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), 97-116. doi: 10.1017/S0022112062000063.

[3]

D. J. Benney and R. F. Bergeron, A new class of nonlinear waves in parallel flows (Nonlinear waves in parallel shear flows, discussing laminar flow breakdown due to free stream disturbances), Stud. Appl. Math., 48 (1969), 181-204.

[4]

N. Bowditch, "The American Practical Navigator," National Imagery and Mapping Agency, Bethesda, MD, 1995.

[5]

J. C. Burns, Long waves in running water, Proc. Camb. Phil. Soc., 4 (1953), 695-706. doi: 10.1017/S0305004100028899.

[6]

A. Constantin, On the deep water wave motion, J. Phys. A, Math. Gen., 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.

[7]

A. Constantin, On the propagation of tsunami waves, with emphasis on the tsunami of 2004, Discrete Cont. Dyn.-B, 12 (2009), 525-537. doi: 10.3934/dcdsb.2009.12.525.

[8]

A. Constantin, On the relevance of soliton theory to tsunami modelling, Wave Motion, 46 (2009), 420-426. doi: 10.1016/j.wavemoti.2009.05.002.

[9]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.

[10]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[11]

A. Constantin and D. Henry, Solitons and Tsunamis, Z. Naturforsch, 64a (2009), 65-68.

[12]

A. Constantin and R. S. Johnson, Modelling tsunamis, J. Phys. A, 39 (2006), L215-L217. doi: 10.1088/0305-4470/39/14/L01.

[13]

A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73. doi: 10.2991/jnmp.2008.15.s2.5.

[14]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dyn. Res., 40 (2008), 175-211. doi: 10.1088/0169-5983/42/3/038901.

[15]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[16]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[17]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409. doi: 10.1017/S0022112070001349.

[18]

F. Gerstner, Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile, Abh. böhm. Ges. Wiss., 1 (1804).

[19]

J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2: Comparison with experiments, J. Fluid Mech., 65 (1974), 289-314. doi: 10.1017/S002211207400139X.

[20]

J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 3: Oscillatory waves, J. Fluid Mech., 84 (1978), 337-358. doi: 10.1017/S0022112078000208.

[21]

D. Henry, On Gerstner's Water Wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.s2.7.

[22]

R. S. Johnson, On the nonlinear critical layer below a nonlinear unsteady surface wave, J. Fluid Mech., 167 (1986), 327-351. doi: 10.1017/S0022112086002847.

[23]

R. S. Johnson, On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid. Dyn., 57 (1991), 115-133. doi: 10.1080/03091929108225231.

[24]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'' Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.

[25]

M. Lakshmanan, Integrable Nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 31-49. doi: 10.1007/978-3-540-71256-5_2.

[26]

J. Lighthill, "Waves in fluids,'' Cambridge University Press, Cambridge, 1978.

[27]

F. Omori, On tsunamis around Japan, Rep. Imp. Earthquake Comm., 34 (1902), 5-79.

[28]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. Roy. Soc. London Ser. A, 153 (1863), 127-138.

[29]

H. Segur, The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1, J. Fluid Mech., 59 (1973), 721-736. doi: 10.1017/S0022112073001813.

[30]

H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 3-29. doi: 10.1007/978-3-540-71256-5_1.

[31]

H. Segur, Integrable models of waves in shallow water, in "Probability, Geometry, and Integrable Systems for Henry McKean's Seventy-Fifth Birthday'' (eds. M. Pinsky and B. Birnir), MSRI Publ. (2008), 345-371.

[32]

R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960, Discrete Cont. Dyn.-B, 12 (2009), 623-632. doi: 10.3934/dcdsb.2009.12.623.

[33]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.

[34]

F. Ursell, The long-wave paradox in the theory of gravity waves, Math. Proc. Cambridge Philos. Soc., 49 (1953), 685-694. doi: 10.1017/S0305004100028887.

[35]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[36]

C. S. Yih, Surface waves in flowing water, J. Fluid Mech., 51 (1972), 209-220.

show all references

References:
[1]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.

[2]

T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), 97-116. doi: 10.1017/S0022112062000063.

[3]

D. J. Benney and R. F. Bergeron, A new class of nonlinear waves in parallel flows (Nonlinear waves in parallel shear flows, discussing laminar flow breakdown due to free stream disturbances), Stud. Appl. Math., 48 (1969), 181-204.

[4]

N. Bowditch, "The American Practical Navigator," National Imagery and Mapping Agency, Bethesda, MD, 1995.

[5]

J. C. Burns, Long waves in running water, Proc. Camb. Phil. Soc., 4 (1953), 695-706. doi: 10.1017/S0305004100028899.

[6]

A. Constantin, On the deep water wave motion, J. Phys. A, Math. Gen., 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.

[7]

A. Constantin, On the propagation of tsunami waves, with emphasis on the tsunami of 2004, Discrete Cont. Dyn.-B, 12 (2009), 525-537. doi: 10.3934/dcdsb.2009.12.525.

[8]

A. Constantin, On the relevance of soliton theory to tsunami modelling, Wave Motion, 46 (2009), 420-426. doi: 10.1016/j.wavemoti.2009.05.002.

[9]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.

[10]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[11]

A. Constantin and D. Henry, Solitons and Tsunamis, Z. Naturforsch, 64a (2009), 65-68.

[12]

A. Constantin and R. S. Johnson, Modelling tsunamis, J. Phys. A, 39 (2006), L215-L217. doi: 10.1088/0305-4470/39/14/L01.

[13]

A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73. doi: 10.2991/jnmp.2008.15.s2.5.

[14]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dyn. Res., 40 (2008), 175-211. doi: 10.1088/0169-5983/42/3/038901.

[15]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[16]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x.

[17]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409. doi: 10.1017/S0022112070001349.

[18]

F. Gerstner, Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile, Abh. böhm. Ges. Wiss., 1 (1804).

[19]

J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2: Comparison with experiments, J. Fluid Mech., 65 (1974), 289-314. doi: 10.1017/S002211207400139X.

[20]

J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 3: Oscillatory waves, J. Fluid Mech., 84 (1978), 337-358. doi: 10.1017/S0022112078000208.

[21]

D. Henry, On Gerstner's Water Wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.s2.7.

[22]

R. S. Johnson, On the nonlinear critical layer below a nonlinear unsteady surface wave, J. Fluid Mech., 167 (1986), 327-351. doi: 10.1017/S0022112086002847.

[23]

R. S. Johnson, On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid. Dyn., 57 (1991), 115-133. doi: 10.1080/03091929108225231.

[24]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'' Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.

[25]

M. Lakshmanan, Integrable Nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 31-49. doi: 10.1007/978-3-540-71256-5_2.

[26]

J. Lighthill, "Waves in fluids,'' Cambridge University Press, Cambridge, 1978.

[27]

F. Omori, On tsunamis around Japan, Rep. Imp. Earthquake Comm., 34 (1902), 5-79.

[28]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. Roy. Soc. London Ser. A, 153 (1863), 127-138.

[29]

H. Segur, The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1, J. Fluid Mech., 59 (1973), 721-736. doi: 10.1017/S0022112073001813.

[30]

H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 3-29. doi: 10.1007/978-3-540-71256-5_1.

[31]

H. Segur, Integrable models of waves in shallow water, in "Probability, Geometry, and Integrable Systems for Henry McKean's Seventy-Fifth Birthday'' (eds. M. Pinsky and B. Birnir), MSRI Publ. (2008), 345-371.

[32]

R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960, Discrete Cont. Dyn.-B, 12 (2009), 623-632. doi: 10.3934/dcdsb.2009.12.623.

[33]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.

[34]

F. Ursell, The long-wave paradox in the theory of gravity waves, Math. Proc. Cambridge Philos. Soc., 49 (1953), 685-694. doi: 10.1017/S0305004100028887.

[35]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[36]

C. S. Yih, Surface waves in flowing water, J. Fluid Mech., 51 (1972), 209-220.

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