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Stability analysis of the boundary value problem modelling a two-layer ocean
Exact solution and instability for geophysical edge waves
1. | Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China |
2. | Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina 842 48, Bratislava, Slovakia |
3. | Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia |
We present an exact solution to the nonlinear governing equations in the $ \beta $-plane approximation for geophysical edge waves at an arbitrary latitude. Such an exact solution is derived in the Lagrange framework, which describes trapped waves propagating eastward or westward along a sloping beach with a shoreline parallel to the latitude line. Using the short-wavelength instability method, we establish a criterion for the instability of such waves.
References:
[1] |
A. Bennett, Lagrangian Fluid Dynamics, Cambrige University Press, Combridge, 2006.
doi: 10.1017/cbo9780511734939.![]() ![]() ![]() |
[2] |
B. J. Bayly, Three-dimensional instabilities in qusi-two-dimensional inviscid flows, Nonlinear Wave Inter. Fluids, (1987), 71–77. |
[3] |
J. Chu, D. Ionescu-Kruse and Y. Yang,
Exact solution and instability for geophysical trapped waves at arbitrary latitude, Discret. Contin. Dynam. Syst., 39 (2019), 4399-4414.
doi: 10.3934/dcds.2019178. |
[4] |
J. Chu, D. Ionescu-Kruse and Y. Yang, Exact solution and instability for geophysical waves with centripetal forces and at arbitrary latitude, J. Math. Fluid Mech., 21 (2019), 16pp.
doi: 10.1007/s00021-019-0423-8. |
[5] |
A. Constantin,
Edge waves along a sloping beach, J. Phys. A: Math. General, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[6] |
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. |
[7] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029, 8 pp.
doi: 10.1029/2012JC007879. |
[8] |
A. Constantin and P. Germain,
Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810.
doi: 10.1002/jgrc.20219. |
[9] |
A. Constantin,
Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[10] |
A. Constantin and R. S. Johnson,
An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945.
doi: 10.1175/JPO-D-15-0205.1. |
[11] |
A. Constantin and R. S. Johnson,
A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids, 29 (2017), 1935-1945.
doi: 10.1063/1.4984001. |
[12] |
A. Constantin and R. I. Ivanov,
Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.
doi: 10.1007/s00220-019-03483-8. |
[13] |
B. Cushman-Robisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, 2011. |
[14] |
U. T. Ehrenmark,
Oblique wave incidence on a plane beach: The classical problem recisired, J. Fluid Mech., 368 (1998), 291-319.
doi: 10.1017/S0022112098001888. |
[15] |
B. Elfrink and T. Baldock,
Hydrodynamics and sediment transport in the swash zone: A review and perspectives, Coast. Engineer., 45 (2002), 149-167.
doi: 10.1016/S0378-3839(02)00032-7. |
[16] |
L. Fan and H. Gao,
Instability of equatorial edge waves in the background flow, Proceed. Amer. Math. Soc., 145 (2017), 765-778.
doi: 10.1090/proc/13308. |
[17] |
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. |
[18] |
I. Gallagher and L. Saint-Raymond,
On the influence of the Earth's rotation on geophysical flows, Handbook Mathematical. Fluid Dynam., 4 (2017), 201-329.
doi: 10.1016/S1874-5792(07)80009-7. |
[19] |
F. Gerstner,
Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.
doi: 10.1002/andp.18090320808. |
[20] |
Y. Guan, M. Fečkan and J. Wang,
Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows, Discret. Contin. Dynam. Syst., 41 (2021), 1157-1176.
doi: 10.3934/dcds.2020313. |
[21] |
D. Henry and O. Mustafa,
Existence of solutions for a class of edge wave equations, Discret. Contin. Dynam. Syst. Series B, 6 (2006), 1113-1119.
doi: 10.3934/dcdsb.2006.6.1113. |
[22] |
D. Henry and H. C. Hsu,
Instability of internal equatorial water waves, J. Differ. Equ., 258 (2015), 1015-1024.
doi: 10.1016/j.jde.2014.08.019. |
[23] |
D. Henry,
A modified equatorial $\beta$-plane approximation modelling nonlinear wave-current interactions, J. Differ. Equ., 263 (2017), 2554-2566.
doi: 10.1016/j.jde.2017.04.007. |
[24] |
P. A. Hawd, A. J. Bowen and R. A. Holman,
Edge waves in the presence of strong longshore currents, J. Geophys. Res.: Oceans, 97 (1992), 11357-11371.
doi: 10.1029/92JC00858. |
[25] |
D. Ionescu-Kruse,
An exact solution for geophysical edge waves in the $f$-plane approximation, Nonlinear Anal.: Real World Appl., 24 (2015), 190-195.
doi: 10.1016/j.nonrwa.2015.02.002. |
[26] |
D. Ionescu-Kruse,
An exact solution for geophysical edge waves in the $\beta$-plane approximation, J. Math. Fluid Mech., 17 (2015), 699-706.
doi: 10.1007/s00021-015-0233-6. |
[27] |
D. Ionescu-Kruse,
Short-wavelength instabilities of edge waves in stratified water, Discret. Contin. Dynam. Syst., 35 (2015), 2053-2066.
doi: 10.3934/dcds.2015.35.2053. |
[28] |
D. Ionescu-Kruse,
Instability of equatorially trapped waves in stratified water, Ann. Mat. Pura Appl., 195 (2016), 585-599.
doi: 10.1007/s10231-015-0479-x. |
[29] |
R. S. Johnson,
Edge waves: Theories past and present, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 365 (2007), 2359-2376.
doi: 10.1098/rsta.2007.2013. |
[30] |
R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018), 19 pp.
doi: 10.1098/rsta.2017.0092. |
[31] |
P. D. Komar, Beach Processes and Sedimentation, Prentice Hall, New Jersey, 1998. |
[32] |
S. Leblanc,
Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254.
doi: 10.1017/S0022112004008444. |
[33] |
A. Lifschitz and E. Hameiri,
Local stability conditions in fluid dynamics, Phys. Fluids, 3 (1991), 2644-2651.
doi: 10.1063/1.858153. |
[34] |
A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A: Math. Theoret., 45 (2012), 365501, 10pp.
doi: 10.1088/1751-8113/45/36/365501. |
[35] |
F. Miao, M. Fečkan and J. Wang,
A new approach to study constant vorticity water flows in the $\beta$-plane approximation with centripetal forces, Dynam. Partial Differ. Equ., 18 (2021), 199-210.
doi: 10.4310/DPDE.2021.v18.n3.a2. |
[36] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. |
[37] |
A. Rodríguez-Sanjurjo, Instability of nonlinear wave-current interactions in a modified equatorial $\beta$-plane approximation, J. Math. Fluid Mech., 24 (2019), 12 pp.
doi: 10.1007/s00021-019-0427-4. |
[38] |
G. G. Stokes, Report on recent researches in hydrodynamics, Report of the British Association for the Advancement of Science, (1846), 1–20. |
[39] |
R. Stuhlmeter,
On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2013), 127-137.
doi: 10.1142/S1402925111001210. |
[40] |
J. Wang, M. Fečkan and W. Zhang, On the nonlocal boundary value problem of geophysical fluid flows, Zeit. für Math. Phys., 72 (2021), 18 pp.
doi: 10.1007/s00033-020-01452-z. |
[41] |
J. Wang, M. Fečkan and Y. Guan, Local and global analysis for discontinuous atmospheric Ekman equations, J. Dynam. Differ. Equ., (2021), 15 pp.
doi: 10.1007/s10884-021-10037-x. |
[42] |
G. B. Whitham,
Nonlinear effects in edge waves, J. Fluid Mech., 74 (1976), 353-368.
doi: 10.1017/S0022112076001833. |
[43] |
Y. Yang and X. Wang,
Exact and explicit internal water waves at arbitrary latitude with underlying currents, Dynam. Partial Differ. Equ., 17 (2020), 117-127.
doi: 10.4310/DPDE.2020.v17.n2.a2. |
[44] |
H. Yeh,
Nonlinear progressive edge waves: their instability and evolution, J. Fluid Mech., 152 (2006), 479-499.
doi: 10.1017/S0022112085000799. |
[45] |
C. Yih,
Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (2006), 765-767.
doi: 10.1017/S0022112066000983. |
[46] |
W. Zhang, M. Fečkan and J. Wang,
Positive solutions to integral boundary value problems from geophysical fluid flows, Monat. fur Math., 193 (2020), 901-925.
doi: 10.1007/s00605-020-01467-8. |
show all references
References:
[1] |
A. Bennett, Lagrangian Fluid Dynamics, Cambrige University Press, Combridge, 2006.
doi: 10.1017/cbo9780511734939.![]() ![]() ![]() |
[2] |
B. J. Bayly, Three-dimensional instabilities in qusi-two-dimensional inviscid flows, Nonlinear Wave Inter. Fluids, (1987), 71–77. |
[3] |
J. Chu, D. Ionescu-Kruse and Y. Yang,
Exact solution and instability for geophysical trapped waves at arbitrary latitude, Discret. Contin. Dynam. Syst., 39 (2019), 4399-4414.
doi: 10.3934/dcds.2019178. |
[4] |
J. Chu, D. Ionescu-Kruse and Y. Yang, Exact solution and instability for geophysical waves with centripetal forces and at arbitrary latitude, J. Math. Fluid Mech., 21 (2019), 16pp.
doi: 10.1007/s00021-019-0423-8. |
[5] |
A. Constantin,
Edge waves along a sloping beach, J. Phys. A: Math. General, 34 (2001), 9723-9731.
doi: 10.1088/0305-4470/34/45/311. |
[6] |
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. |
[7] |
A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029, 8 pp.
doi: 10.1029/2012JC007879. |
[8] |
A. Constantin and P. Germain,
Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810.
doi: 10.1002/jgrc.20219. |
[9] |
A. Constantin,
Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.
doi: 10.1175/JPO-D-12-062.1. |
[10] |
A. Constantin and R. S. Johnson,
An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945.
doi: 10.1175/JPO-D-15-0205.1. |
[11] |
A. Constantin and R. S. Johnson,
A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids, 29 (2017), 1935-1945.
doi: 10.1063/1.4984001. |
[12] |
A. Constantin and R. I. Ivanov,
Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.
doi: 10.1007/s00220-019-03483-8. |
[13] |
B. Cushman-Robisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, 2011. |
[14] |
U. T. Ehrenmark,
Oblique wave incidence on a plane beach: The classical problem recisired, J. Fluid Mech., 368 (1998), 291-319.
doi: 10.1017/S0022112098001888. |
[15] |
B. Elfrink and T. Baldock,
Hydrodynamics and sediment transport in the swash zone: A review and perspectives, Coast. Engineer., 45 (2002), 149-167.
doi: 10.1016/S0378-3839(02)00032-7. |
[16] |
L. Fan and H. Gao,
Instability of equatorial edge waves in the background flow, Proceed. Amer. Math. Soc., 145 (2017), 765-778.
doi: 10.1090/proc/13308. |
[17] |
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. |
[18] |
I. Gallagher and L. Saint-Raymond,
On the influence of the Earth's rotation on geophysical flows, Handbook Mathematical. Fluid Dynam., 4 (2017), 201-329.
doi: 10.1016/S1874-5792(07)80009-7. |
[19] |
F. Gerstner,
Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.
doi: 10.1002/andp.18090320808. |
[20] |
Y. Guan, M. Fečkan and J. Wang,
Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows, Discret. Contin. Dynam. Syst., 41 (2021), 1157-1176.
doi: 10.3934/dcds.2020313. |
[21] |
D. Henry and O. Mustafa,
Existence of solutions for a class of edge wave equations, Discret. Contin. Dynam. Syst. Series B, 6 (2006), 1113-1119.
doi: 10.3934/dcdsb.2006.6.1113. |
[22] |
D. Henry and H. C. Hsu,
Instability of internal equatorial water waves, J. Differ. Equ., 258 (2015), 1015-1024.
doi: 10.1016/j.jde.2014.08.019. |
[23] |
D. Henry,
A modified equatorial $\beta$-plane approximation modelling nonlinear wave-current interactions, J. Differ. Equ., 263 (2017), 2554-2566.
doi: 10.1016/j.jde.2017.04.007. |
[24] |
P. A. Hawd, A. J. Bowen and R. A. Holman,
Edge waves in the presence of strong longshore currents, J. Geophys. Res.: Oceans, 97 (1992), 11357-11371.
doi: 10.1029/92JC00858. |
[25] |
D. Ionescu-Kruse,
An exact solution for geophysical edge waves in the $f$-plane approximation, Nonlinear Anal.: Real World Appl., 24 (2015), 190-195.
doi: 10.1016/j.nonrwa.2015.02.002. |
[26] |
D. Ionescu-Kruse,
An exact solution for geophysical edge waves in the $\beta$-plane approximation, J. Math. Fluid Mech., 17 (2015), 699-706.
doi: 10.1007/s00021-015-0233-6. |
[27] |
D. Ionescu-Kruse,
Short-wavelength instabilities of edge waves in stratified water, Discret. Contin. Dynam. Syst., 35 (2015), 2053-2066.
doi: 10.3934/dcds.2015.35.2053. |
[28] |
D. Ionescu-Kruse,
Instability of equatorially trapped waves in stratified water, Ann. Mat. Pura Appl., 195 (2016), 585-599.
doi: 10.1007/s10231-015-0479-x. |
[29] |
R. S. Johnson,
Edge waves: Theories past and present, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 365 (2007), 2359-2376.
doi: 10.1098/rsta.2007.2013. |
[30] |
R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018), 19 pp.
doi: 10.1098/rsta.2017.0092. |
[31] |
P. D. Komar, Beach Processes and Sedimentation, Prentice Hall, New Jersey, 1998. |
[32] |
S. Leblanc,
Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254.
doi: 10.1017/S0022112004008444. |
[33] |
A. Lifschitz and E. Hameiri,
Local stability conditions in fluid dynamics, Phys. Fluids, 3 (1991), 2644-2651.
doi: 10.1063/1.858153. |
[34] |
A. V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A: Math. Theoret., 45 (2012), 365501, 10pp.
doi: 10.1088/1751-8113/45/36/365501. |
[35] |
F. Miao, M. Fečkan and J. Wang,
A new approach to study constant vorticity water flows in the $\beta$-plane approximation with centripetal forces, Dynam. Partial Differ. Equ., 18 (2021), 199-210.
doi: 10.4310/DPDE.2021.v18.n3.a2. |
[36] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1979. |
[37] |
A. Rodríguez-Sanjurjo, Instability of nonlinear wave-current interactions in a modified equatorial $\beta$-plane approximation, J. Math. Fluid Mech., 24 (2019), 12 pp.
doi: 10.1007/s00021-019-0427-4. |
[38] |
G. G. Stokes, Report on recent researches in hydrodynamics, Report of the British Association for the Advancement of Science, (1846), 1–20. |
[39] |
R. Stuhlmeter,
On edge waves in stratified water along a sloping beach, J. Nonlinear Math. Phys., 18 (2013), 127-137.
doi: 10.1142/S1402925111001210. |
[40] |
J. Wang, M. Fečkan and W. Zhang, On the nonlocal boundary value problem of geophysical fluid flows, Zeit. für Math. Phys., 72 (2021), 18 pp.
doi: 10.1007/s00033-020-01452-z. |
[41] |
J. Wang, M. Fečkan and Y. Guan, Local and global analysis for discontinuous atmospheric Ekman equations, J. Dynam. Differ. Equ., (2021), 15 pp.
doi: 10.1007/s10884-021-10037-x. |
[42] |
G. B. Whitham,
Nonlinear effects in edge waves, J. Fluid Mech., 74 (1976), 353-368.
doi: 10.1017/S0022112076001833. |
[43] |
Y. Yang and X. Wang,
Exact and explicit internal water waves at arbitrary latitude with underlying currents, Dynam. Partial Differ. Equ., 17 (2020), 117-127.
doi: 10.4310/DPDE.2020.v17.n2.a2. |
[44] |
H. Yeh,
Nonlinear progressive edge waves: their instability and evolution, J. Fluid Mech., 152 (2006), 479-499.
doi: 10.1017/S0022112085000799. |
[45] |
C. Yih,
Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (2006), 765-767.
doi: 10.1017/S0022112066000983. |
[46] |
W. Zhang, M. Fečkan and J. Wang,
Positive solutions to integral boundary value problems from geophysical fluid flows, Monat. fur Math., 193 (2020), 901-925.
doi: 10.1007/s00605-020-01467-8. |


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