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

Short-wavelength instabilities of edge waves in stratified water

Delia Ionescu-Kruse 1,

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.
Keywords: localized instability analysis, stratification, Short-wavelength method, edge waves, explicit solutions..
Mathematics Subject Classification: Primary: 76E99, 76B15, 76B70; Secondary: 35Q35, 35B36, 34E2.
Citation: Delia Ionescu-Kruse. Short-wavelength instabilities of edge waves in stratified water. Discrete and Continuous Dynamical Systems, 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, edited by R. W. Miksad et al., ASME, New York, (1987), 71-77.

[2]

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

[3]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.

[4]

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

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[6]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879.

[7]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.-Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219.

[8]

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

[9]

M.-L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides heterogenes, Atti Accad. Naz. Lincei, 15 (1932), 814-819.

[10]

U. Ehrenmark, Oblique wave incidence on a plane beach: The classical problem revisited, J. Fluid Mech., 368 (1998), 291-319. doi: 10.1017/S0022112098001888.

[11]

S. Friedlander and A. Lipton-Lifschitz, Localized instabilities in fluids, in Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), North-Holland, 2 (2003), 289-354. doi: 10.1016/S1874-5792(03)80010-1.

[12]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66 (1991), 2204-2206. doi: 10.1103/PhysRevLett.66.2204.

[13]

S. Friedlander and M. M. Vishik, Instability criteria for steady flows of a perfect fluid, Chaos, 2 (1992), 455-460. doi: 10.1063/1.165888.

[14]

S. Friedlander and V. Yudovich, Instabilities in fluid motion, Not. Am. Math. Soc., 46 (1999), 1358-1367.

[15]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667. doi: 10.1007/s00021-014-0175-4.

[16]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., (2006), Art. ID 23405, 13 pp. doi: 10.1155/IMRN/2006/23405.

[18]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7.

[19]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21. doi: 10.1016/j.euromechflu.2012.10.001.

[20]

D. Henry and O. Mustafa, Existence of solutions for a class of edge wave equations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1113-1119. doi: 10.3934/dcdsb.2006.6.1113.

[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), 11357-11371. doi: 10.1029/92JC00858.

[22]

D. Ionescu-Kruse, Instability of edge waves along a sloping beach, J. Diff. Eqs., 256 (2014), 3999-4012. doi: 10.1016/j.jde.2014.03.009.

[23]

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

[24]

R. S. Johnson, Edge waves: Theories past and present, Phil. Trans. R. Soc. A, 365 (2007), 2359-2376. doi: 10.1098/rsta.2007.2013.

[25]

D. D. Joseph, Stability of Fluid Motions I, Springer Verlag, New York, 1976.

[26]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444.

[27]

N. R. Lebovitz and A. Lifschitz, Short-wavelength instabilities of Riemann ellipsoids, Phil. Trans. R. Soc. Lond. A, 354 (1996), 927-950. doi: 10.1098/rsta.1996.0037.

[28]

A. Lifschitz, Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity, Phys. Lett. A, 157 (1991), 481-487. doi: 10.1016/0375-9601(91)91023-7.

[29]

A. Lifschitz, On the instability of certain motions of an ideal incompressible fluid, Advances Appl. Math., 15 (1994), 404-436. doi: 10.1006/aama.1994.1017.

[30]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics, Phys. Fluids, 3 (1991), 2644-2651. doi: 10.1063/1.858153.

[31]

A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A: Math. Theor., 45 (2012), 365501, 10 pp. doi: 10.1088/1751-8113/45/36/365501.

[32]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave, Phys. Fluids, 25 (1982), 586-587. doi: 10.1063/1.863802.

[33]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137. doi: 10.1142/S1402925111001210.

[34]

G. B. Whitham, Lecture on Wave Propagation, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 61. Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1979.

[35]

C. S. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), 765-767. doi: 10.1017/S0022112066000983.

[36]

C. S. Yih, Stratified flows, Ann. Rev. Fluid Mech., 1 (1969), 73-110. doi: 10.1146/annurev.fl.01.010169.000445.

show all references

References:
[1]

B. J. Bayly, Three-dimensional instabilities in quasi-two dimensional inviscid flows, in Nonlinear Wave Interactions in Fluids, edited by R. W. Miksad et al., ASME, New York, (1987), 71-77.

[2]

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

[3]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.

[4]

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

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[6]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879.

[7]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.-Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219.

[8]

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

[9]

M.-L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides heterogenes, Atti Accad. Naz. Lincei, 15 (1932), 814-819.

[10]

U. Ehrenmark, Oblique wave incidence on a plane beach: The classical problem revisited, J. Fluid Mech., 368 (1998), 291-319. doi: 10.1017/S0022112098001888.

[11]

S. Friedlander and A. Lipton-Lifschitz, Localized instabilities in fluids, in Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), North-Holland, 2 (2003), 289-354. doi: 10.1016/S1874-5792(03)80010-1.

[12]

S. Friedlander and M. M. Vishik, Instability criteria for the flow of an inviscid incompressible fluid, Phys. Rev. Lett., 66 (1991), 2204-2206. doi: 10.1103/PhysRevLett.66.2204.

[13]

S. Friedlander and M. M. Vishik, Instability criteria for steady flows of a perfect fluid, Chaos, 2 (1992), 455-460. doi: 10.1063/1.165888.

[14]

S. Friedlander and V. Yudovich, Instabilities in fluid motion, Not. Am. Math. Soc., 46 (1999), 1358-1367.

[15]

F. Genoud and D. Henry, Instability of equatorial water waves with an underlying current, J. Math. Fluid Mech., 16 (2014), 661-667. doi: 10.1007/s00021-014-0175-4.

[16]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., (2006), Art. ID 23405, 13 pp. doi: 10.1155/IMRN/2006/23405.

[18]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7.

[19]

D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids, 38 (2013), 18-21. doi: 10.1016/j.euromechflu.2012.10.001.

[20]

D. Henry and O. Mustafa, Existence of solutions for a class of edge wave equations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1113-1119. doi: 10.3934/dcdsb.2006.6.1113.

[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), 11357-11371. doi: 10.1029/92JC00858.

[22]

D. Ionescu-Kruse, Instability of edge waves along a sloping beach, J. Diff. Eqs., 256 (2014), 3999-4012. doi: 10.1016/j.jde.2014.03.009.

[23]

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

[24]

R. S. Johnson, Edge waves: Theories past and present, Phil. Trans. R. Soc. A, 365 (2007), 2359-2376. doi: 10.1098/rsta.2007.2013.

[25]

D. D. Joseph, Stability of Fluid Motions I, Springer Verlag, New York, 1976.

[26]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444.

[27]

N. R. Lebovitz and A. Lifschitz, Short-wavelength instabilities of Riemann ellipsoids, Phil. Trans. R. Soc. Lond. A, 354 (1996), 927-950. doi: 10.1098/rsta.1996.0037.

[28]

A. Lifschitz, Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity, Phys. Lett. A, 157 (1991), 481-487. doi: 10.1016/0375-9601(91)91023-7.

[29]

A. Lifschitz, On the instability of certain motions of an ideal incompressible fluid, Advances Appl. Math., 15 (1994), 404-436. doi: 10.1006/aama.1994.1017.

[30]

A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics, Phys. Fluids, 3 (1991), 2644-2651. doi: 10.1063/1.858153.

[31]

A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A: Math. Theor., 45 (2012), 365501, 10 pp. doi: 10.1088/1751-8113/45/36/365501.

[32]

E. Mollo-Christensen, Allowable discontinuities in a Gerstner wave, Phys. Fluids, 25 (1982), 586-587. doi: 10.1063/1.863802.

[33]

R. Stuhlmeier, On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2011), 127-137. doi: 10.1142/S1402925111001210.

[34]

G. B. Whitham, Lecture on Wave Propagation, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 61. Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin-New York, 1979.

[35]

C. S. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), 765-767. doi: 10.1017/S0022112066000983.

[36]

C. S. Yih, Stratified flows, Ann. Rev. Fluid Mech., 1 (1969), 73-110. doi: 10.1146/annurev.fl.01.010169.000445.

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