May  2015, 35(5): 2053-2066. doi: 10.3934/dcds.2015.35.2053

Short-wavelength instabilities of edge waves in stratified water

1. 

Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, RO-014700 Bucharest, Romania

Received  June 2014 Revised  July 2014 Published  December 2014

In this paper we make a detailed analysis of the short-wavelength instability method for barotropic incompressible fluids. We apply this method to edge waves in stratified water. These waves are unstable to short-wavelength perturbations if their steepness exceeds a specific threshold.
Citation: Delia Ionescu-Kruse. Short-wavelength instabilities of edge waves in stratified water. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2053-2066. doi: 10.3934/dcds.2015.35.2053
References:
[1]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows, in, Nonlinear Wave Interactions in Fluids, (1987), 71.   Google Scholar

[2]

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

[3]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[4]

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

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[6]

A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012).  doi: 10.1029/2012JC007879.  Google Scholar

[7]

A. Constantin and P. Germain, Instability of some equatorially trapped waves,, J. Geophys. Res.-Oceans, 118 (2013), 2802.  doi: 10.1002/jgrc.20219.  Google Scholar

[8]

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

[9]

M.-L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides heterogenes,, Atti Accad. Naz. Lincei, 15 (1932), 814.   Google Scholar

[10]

U. Ehrenmark, Oblique wave incidence on a plane beach: The classical problem revisited,, J. Fluid Mech., 368 (1998), 291.  doi: 10.1017/S0022112098001888.  Google Scholar

[11]

S. Friedlander and A. Lipton-Lifschitz, Localized instabilities in fluids,, in Handbook of Mathematical Fluid Dynamics, 2 (2003), 289.  doi: 10.1016/S1874-5792(03)80010-1.  Google Scholar

[12]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid,, Phys. Rev. Lett., 66 (1991), 2204.  doi: 10.1103/PhysRevLett.66.2204.  Google Scholar

[13]

S. Friedlander and M. M. Vishik, Instability criteria for steady flows of a perfect fluid,, Chaos, 2 (1992), 455.  doi: 10.1063/1.165888.  Google Scholar

[14]

S. Friedlander and V. Yudovich, Instabilities in fluid motion,, Not. Am. Math. Soc., 46 (1999), 1358.   Google Scholar

[15]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current,, J. Math. Fluid Mech., 16 (2014), 661.  doi: 10.1007/s00021-014-0175-4.  Google Scholar

[16]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.   Google Scholar

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006).  doi: 10.1155/IMRN/2006/23405.  Google Scholar

[18]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[19]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. B Fluids, 38 (2013), 18.  doi: 10.1016/j.euromechflu.2012.10.001.  Google Scholar

[20]

D. Henry and O. Mustafa, Existence of solutions for a class of edge wave equations,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1113.  doi: 10.3934/dcdsb.2006.6.1113.  Google Scholar

[21]

P.A. Howd, A.J. Bowen and R.A. Holman, Edge waves in the presence of strong longshore currents,, J. Geophys. Res., 97 (1992), 11357.  doi: 10.1029/92JC00858.  Google Scholar

[22]

D. Ionescu-Kruse, Instability of edge waves along a sloping beach,, J. Diff. Eqs., 256 (2014), 3999.  doi: 10.1016/j.jde.2014.03.009.  Google Scholar

[23]

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

[24]

R. S. Johnson, Edge waves: Theories past and present,, Phil. Trans. R. Soc. A, 365 (2007), 2359.  doi: 10.1098/rsta.2007.2013.  Google Scholar

[25]

D. D. Joseph, Stability of Fluid Motions I,, Springer Verlag, (1976).   Google Scholar

[26]

S. Leblanc, Local stability of Gerstner's waves,, J. Fluid Mech., 506 (2004), 245.  doi: 10.1017/S0022112004008444.  Google Scholar

[27]

N. R. Lebovitz and A. Lifschitz, Short-wavelength instabilities of Riemann ellipsoids,, Phil. Trans. R. Soc. Lond. A, 354 (1996), 927.  doi: 10.1098/rsta.1996.0037.  Google Scholar

[28]

A. Lifschitz, Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity,, Phys. Lett. A, 157 (1991), 481.  doi: 10.1016/0375-9601(91)91023-7.  Google Scholar

[29]

A. Lifschitz, On the instability of certain motions of an ideal incompressible fluid,, Advances Appl. Math., 15 (1994), 404.  doi: 10.1006/aama.1994.1017.  Google Scholar

[30]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics,, Phys. Fluids, 3 (1991), 2644.  doi: 10.1063/1.858153.  Google Scholar

[31]

A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A: Math. Theor., 45 (2012).  doi: 10.1088/1751-8113/45/36/365501.  Google Scholar

[32]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave,, Phys. Fluids, 25 (1982), 586.  doi: 10.1063/1.863802.  Google Scholar

[33]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach,, J. Nonlinear Math. Phys., 18 (2011), 127.  doi: 10.1142/S1402925111001210.  Google Scholar

[34]

G. B. Whitham, Lecture on Wave Propagation,, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, (1979).   Google Scholar

[35]

C. S. Yih, Note on edge waves in a stratified fluid,, J. Fluid Mech., 24 (1966), 765.  doi: 10.1017/S0022112066000983.  Google Scholar

[36]

C. S. Yih, Stratified flows,, Ann. Rev. Fluid Mech., 1 (1969), 73.  doi: 10.1146/annurev.fl.01.010169.000445.  Google Scholar

show all references

References:
[1]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows, in, Nonlinear Wave Interactions in Fluids, (1987), 71.   Google Scholar

[2]

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

[3]

A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723.  doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[4]

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

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[6]

A. Constantin, An exact solution for equatorially trapped waves,, J. Geophys. Res., 117 (2012).  doi: 10.1029/2012JC007879.  Google Scholar

[7]

A. Constantin and P. Germain, Instability of some equatorially trapped waves,, J. Geophys. Res.-Oceans, 118 (2013), 2802.  doi: 10.1002/jgrc.20219.  Google Scholar

[8]

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

[9]

M.-L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides heterogenes,, Atti Accad. Naz. Lincei, 15 (1932), 814.   Google Scholar

[10]

U. Ehrenmark, Oblique wave incidence on a plane beach: The classical problem revisited,, J. Fluid Mech., 368 (1998), 291.  doi: 10.1017/S0022112098001888.  Google Scholar

[11]

S. Friedlander and A. Lipton-Lifschitz, Localized instabilities in fluids,, in Handbook of Mathematical Fluid Dynamics, 2 (2003), 289.  doi: 10.1016/S1874-5792(03)80010-1.  Google Scholar

[12]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid,, Phys. Rev. Lett., 66 (1991), 2204.  doi: 10.1103/PhysRevLett.66.2204.  Google Scholar

[13]

S. Friedlander and M. M. Vishik, Instability criteria for steady flows of a perfect fluid,, Chaos, 2 (1992), 455.  doi: 10.1063/1.165888.  Google Scholar

[14]

S. Friedlander and V. Yudovich, Instabilities in fluid motion,, Not. Am. Math. Soc., 46 (1999), 1358.   Google Scholar

[15]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current,, J. Math. Fluid Mech., 16 (2014), 661.  doi: 10.1007/s00021-014-0175-4.  Google Scholar

[16]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.   Google Scholar

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006).  doi: 10.1155/IMRN/2006/23405.  Google Scholar

[18]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[19]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current,, Eur. J. Mech. B Fluids, 38 (2013), 18.  doi: 10.1016/j.euromechflu.2012.10.001.  Google Scholar

[20]

D. Henry and O. Mustafa, Existence of solutions for a class of edge wave equations,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1113.  doi: 10.3934/dcdsb.2006.6.1113.  Google Scholar

[21]

P.A. Howd, A.J. Bowen and R.A. Holman, Edge waves in the presence of strong longshore currents,, J. Geophys. Res., 97 (1992), 11357.  doi: 10.1029/92JC00858.  Google Scholar

[22]

D. Ionescu-Kruse, Instability of edge waves along a sloping beach,, J. Diff. Eqs., 256 (2014), 3999.  doi: 10.1016/j.jde.2014.03.009.  Google Scholar

[23]

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

[24]

R. S. Johnson, Edge waves: Theories past and present,, Phil. Trans. R. Soc. A, 365 (2007), 2359.  doi: 10.1098/rsta.2007.2013.  Google Scholar

[25]

D. D. Joseph, Stability of Fluid Motions I,, Springer Verlag, (1976).   Google Scholar

[26]

S. Leblanc, Local stability of Gerstner's waves,, J. Fluid Mech., 506 (2004), 245.  doi: 10.1017/S0022112004008444.  Google Scholar

[27]

N. R. Lebovitz and A. Lifschitz, Short-wavelength instabilities of Riemann ellipsoids,, Phil. Trans. R. Soc. Lond. A, 354 (1996), 927.  doi: 10.1098/rsta.1996.0037.  Google Scholar

[28]

A. Lifschitz, Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity,, Phys. Lett. A, 157 (1991), 481.  doi: 10.1016/0375-9601(91)91023-7.  Google Scholar

[29]

A. Lifschitz, On the instability of certain motions of an ideal incompressible fluid,, Advances Appl. Math., 15 (1994), 404.  doi: 10.1006/aama.1994.1017.  Google Scholar

[30]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics,, Phys. Fluids, 3 (1991), 2644.  doi: 10.1063/1.858153.  Google Scholar

[31]

A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach,, J. Phys. A: Math. Theor., 45 (2012).  doi: 10.1088/1751-8113/45/36/365501.  Google Scholar

[32]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave,, Phys. Fluids, 25 (1982), 586.  doi: 10.1063/1.863802.  Google Scholar

[33]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach,, J. Nonlinear Math. Phys., 18 (2011), 127.  doi: 10.1142/S1402925111001210.  Google Scholar

[34]

G. B. Whitham, Lecture on Wave Propagation,, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, (1979).   Google Scholar

[35]

C. S. Yih, Note on edge waves in a stratified fluid,, J. Fluid Mech., 24 (1966), 765.  doi: 10.1017/S0022112066000983.  Google Scholar

[36]

C. S. Yih, Stratified flows,, Ann. Rev. Fluid Mech., 1 (1969), 73.  doi: 10.1146/annurev.fl.01.010169.000445.  Google Scholar

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