# American Institute of Mathematical Sciences

July  2022, 21(7): 2447-2461. doi: 10.3934/cpaa.2022067

## 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

* Corresponding author

Received  October 2021 Revised  February 2022 Published  July 2022 Early access  March 2022

Fund Project: This work is partially supported by the National Natural Science Foundation of China (12161015), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA (grant Nos. 1/0358/20 and 2/0127/20)

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.

Citation: Fahe Miao, Michal Fečkan, Jinrong Wang. Exact solution and instability for geophysical edge waves. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2447-2461. doi: 10.3934/cpaa.2022067
##### 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.
The rotation frame of reference
The coordinate system for the sloping beach
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