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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 & 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. doi: 10.1007/s00222-007-0088-4. Google Scholar

[2]

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

[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. Google Scholar

[4]

N. Bowditch, "The American Practical Navigator,", National Imagery and Mapping Agency, (1995). Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1016/j.euromechflu.2010.09.008. Google Scholar

[10]

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

[11]

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

[12]

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

[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. doi: 10.2991/jnmp.2008.15.s2.5. Google Scholar

[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. doi: 10.1088/0169-5983/42/3/038901. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

F. Gerstner, Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile,, Abh. b{\, 1 (1804). Google Scholar

[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. doi: 10.1017/S002211207400139X. Google Scholar

[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. doi: 10.1017/S0022112078000208. Google Scholar

[21]

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

[22]

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

[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. doi: 10.1080/03091929108225231. Google Scholar

[24]

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

[25]

M. Lakshmanan, Integrable Nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. doi: 10.1007/978-3-540-71256-5_2. Google Scholar

[26]

J. Lighthill, "Waves in fluids,'', Cambridge University Press, (1978). Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[30]

H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004,, in, (2007), 3. doi: 10.1007/978-3-540-71256-5_1. Google Scholar

[31]

H. Segur, Integrable models of waves in shallow water,, in, (2008), 345. Google Scholar

[32]

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

[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. doi: 10.1017/S0022112088002423. Google Scholar

[34]

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

[35]

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

[36]

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

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. doi: 10.1007/s00222-007-0088-4. Google Scholar

[2]

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

[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. Google Scholar

[4]

N. Bowditch, "The American Practical Navigator,", National Imagery and Mapping Agency, (1995). Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1016/j.euromechflu.2010.09.008. Google Scholar

[10]

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

[11]

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

[12]

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

[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. doi: 10.2991/jnmp.2008.15.s2.5. Google Scholar

[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. doi: 10.1088/0169-5983/42/3/038901. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

F. Gerstner, Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile,, Abh. b{\, 1 (1804). Google Scholar

[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. doi: 10.1017/S002211207400139X. Google Scholar

[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. doi: 10.1017/S0022112078000208. Google Scholar

[21]

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

[22]

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

[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. doi: 10.1080/03091929108225231. Google Scholar

[24]

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

[25]

M. Lakshmanan, Integrable Nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. doi: 10.1007/978-3-540-71256-5_2. Google Scholar

[26]

J. Lighthill, "Waves in fluids,'', Cambridge University Press, (1978). Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[30]

H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004,, in, (2007), 3. doi: 10.1007/978-3-540-71256-5_1. Google Scholar

[31]

H. Segur, Integrable models of waves in shallow water,, in, (2008), 345. Google Scholar

[32]

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

[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. doi: 10.1017/S0022112088002423. Google Scholar

[34]

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

[35]

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

[36]

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

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