# American Institute of Mathematical Sciences

August 2019, 39(8): 4399-4414. doi: 10.3934/dcds.2019178

## Exact solution and instability for geophysical waves at arbitrary latitude

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2 Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, Bucharest 014700, Romania 3 Department of Mathematics, Hohai University, Nanjing 210098, China

* Corresponding author: Jifeng Chu

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  July 2017 Revised  January 2019 Published  May 2019

Fund Project: Jifeng Chu was supported by the National Natural Science Foundation of China (Grants No. 11671118 and No. 11871273). Yanjuan Yang was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2017B715X14)

We present an exact solution to the nonlinear governing equations in the $\beta$-plane approximation for geophysical waves propagating at arbitrary latitude on a zonal current. Such an exact solution is explicit in the Lagrangian framework and represents three-dimensional, nonlinear oceanic wave-current interactions. Based on the short-wavelength instability approach, we prove criteria for the hydrodynamical instability of such waves.

Citation: Jifeng Chu, Delia Ionescu-Kruse, Yanjuan Yang. Exact solution and instability for geophysical waves at arbitrary latitude. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4399-4414. doi: 10.3934/dcds.2019178
##### 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., 71–77, ASME, New York, 1987. [2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511734939. [3] A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313. [4] 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. [5] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879. [6] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169. [7] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1. [8] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1. [9] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. [10] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785. [11] 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. [12] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic circumpolar current, J. Phys. Oceanogr., 46 (2016), 358503594. doi: 10.1175/JPO-D-16-0121.1. [13] 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), 056604. [14] A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223. [15] B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011. [16] L. Fan and H. Gao, Instability of equatorial edge waves in the background flow, Proc. Amer. Math. Soc., 145 (2017), 765-778. doi: 10.1090/proc/13308. [17] L. Fan, H. Gao and Q. Xiao, An exact solution for geophysical trapped waves in the presence of an underlying current, Dyn. Partial Differ. Equ., 15 (2018), 201-214. doi: 10.4310/DPDE.2018.v15.n3.a3. [18] 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. [19] 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. [20] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. [21] A. E. Gill, Atmosphere-Ocean Dynamics, Academic, 1982. [22] D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7. [23] 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. [24] D. Henry, Exact equatorial water waves in the $f$-plane, Nonlinear Anal. Real World Appl., 28 (2016), 284-289. doi: 10.1016/j.nonrwa.2015.10.003. [25] D. Henry, Equatorially trapped nonlinear water waves in a $\beta$-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11 pp. doi: 10.1017/jfm.2016.544. [26] D. Henry, A modified equatorial $\beta$-plane approximation modelling nonlinear wave-current interactions, J. Differential Equations, 263 (2017), 2554-2566. doi: 10.1016/j.jde.2017.04.007. [27] D. Henry, On three-dimensional Gerstner-like equatorial water waves, Philos. Trans. Roy. Soc. A, 376 (2018), 20170088, 16 pp. doi: 10.1098/rsta.2017.0088. [28] D. Henry and H.-C. Hsu, Instability of Equatorial water waves in the $f$-plane, Discrete Contin. Dyn. Syst., 35 (2015), 909-916. doi: 10.3934/dcds.2015.35.909. [29] D. Henry and H.-C. Hsu, Instability of internal equatorial water waves, J. Differential Equations, 258 (2015), 1015-1024. doi: 10.1016/j.jde.2014.08.019. [30] H.-C. Hsu, An exact solution for equatorial waves, Monatsh. Math., 176 (2015), 143-152. doi: 10.1007/s00605-014-0618-2. [31] 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. [32] 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. [33] 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. [34] D. Ionescu-Kruse, Instability of Pollard's exact solution for geophysical ocean flows, Phys. Fluids, 28 (2016), 086601. doi: 10.1063/1.4959289. [35] D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2017), 20170090, 21pp. doi: 10.1098/rsta.2017.0090. [36] D. Ionescu-Kruse, A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows, J. Differential Equations, 264 (2018), 4650-4668. doi: 10.1016/j.jde.2017.12.021. [37] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Philos. Trans. R. Soc. A, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092. [38] M. Kluczek, Physical flow properties for Pollard-like internal water waves, J. Math. Phys., 59 (2018), 123102, 12pp. doi: 10.1063/1.5038657. [39] M. Kluczek, Exact Pollard-like internal eater waves, J. Nonlinear Math. Phys., 26 (2019), 133-146. doi: 10.1080/14029251.2019.1544794. [40] H. Lamb, Hydrodynamics, Reprint of the 1932 sixth edition. With a foreword by R. A. Caflisch [Russel E. Caflisch]. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. [41] S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444. [42] A. Lifschitz and E. Hameiri, Local stability conditions in fluid mechanics, Phys. Fluids, 3 (1991), 2644-2651. doi: 10.1063/1.858153. [43] A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 365501, 10pp. doi: 10.1088/1751-8113/45/36/365501. [44] A.-V. Matioc, Exact geophysical waves in stratified fluids, Appl. Anal., 92 (2013), 2254-2261. doi: 10.1080/00036811.2012.727987. [45] R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898. doi: 10.1029/JC075i030p05895. [46] A. Rodríguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for equatorially-trapped internal water waves, Nonlinear Anal., 149 (2017), 156-164. doi: 10.1016/j.na.2016.10.022. [47] A. Rodríguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions, Ann. Mat. Pura Appl., 197 (2018), 1787-1797. doi: 10.1007/s10231-018-0749-5. [48] S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves, Nonlinear Anal., 125 (2015), 725-731. doi: 10.1016/j.na.2015.06.017. [49] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.

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., 71–77, ASME, New York, 1987. [2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511734939. [3] A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313. [4] 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. [5] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res., 117 (2012), C05029. doi: 10.1029/2012JC007879. [6] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169. [7] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1. [8] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1. [9] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. [10] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785. [11] 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. [12] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic circumpolar current, J. Phys. Oceanogr., 46 (2016), 358503594. doi: 10.1175/JPO-D-16-0121.1. [13] 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), 056604. [14] A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223. [15] B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011. [16] L. Fan and H. Gao, Instability of equatorial edge waves in the background flow, Proc. Amer. Math. Soc., 145 (2017), 765-778. doi: 10.1090/proc/13308. [17] L. Fan, H. Gao and Q. Xiao, An exact solution for geophysical trapped waves in the presence of an underlying current, Dyn. Partial Differ. Equ., 15 (2018), 201-214. doi: 10.4310/DPDE.2018.v15.n3.a3. [18] 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. [19] 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. [20] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. [21] A. E. Gill, Atmosphere-Ocean Dynamics, Academic, 1982. [22] D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.S2.7. [23] 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. [24] D. Henry, Exact equatorial water waves in the $f$-plane, Nonlinear Anal. Real World Appl., 28 (2016), 284-289. doi: 10.1016/j.nonrwa.2015.10.003. [25] D. Henry, Equatorially trapped nonlinear water waves in a $\beta$-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11 pp. doi: 10.1017/jfm.2016.544. [26] D. Henry, A modified equatorial $\beta$-plane approximation modelling nonlinear wave-current interactions, J. Differential Equations, 263 (2017), 2554-2566. doi: 10.1016/j.jde.2017.04.007. [27] D. Henry, On three-dimensional Gerstner-like equatorial water waves, Philos. Trans. Roy. Soc. A, 376 (2018), 20170088, 16 pp. doi: 10.1098/rsta.2017.0088. [28] D. Henry and H.-C. Hsu, Instability of Equatorial water waves in the $f$-plane, Discrete Contin. Dyn. Syst., 35 (2015), 909-916. doi: 10.3934/dcds.2015.35.909. [29] D. Henry and H.-C. Hsu, Instability of internal equatorial water waves, J. Differential Equations, 258 (2015), 1015-1024. doi: 10.1016/j.jde.2014.08.019. [30] H.-C. Hsu, An exact solution for equatorial waves, Monatsh. Math., 176 (2015), 143-152. doi: 10.1007/s00605-014-0618-2. [31] 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. [32] 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. [33] 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. [34] D. Ionescu-Kruse, Instability of Pollard's exact solution for geophysical ocean flows, Phys. Fluids, 28 (2016), 086601. doi: 10.1063/1.4959289. [35] D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2017), 20170090, 21pp. doi: 10.1098/rsta.2017.0090. [36] D. Ionescu-Kruse, A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows, J. Differential Equations, 264 (2018), 4650-4668. doi: 10.1016/j.jde.2017.12.021. [37] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Philos. Trans. R. Soc. A, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092. [38] M. Kluczek, Physical flow properties for Pollard-like internal water waves, J. Math. Phys., 59 (2018), 123102, 12pp. doi: 10.1063/1.5038657. [39] M. Kluczek, Exact Pollard-like internal eater waves, J. Nonlinear Math. Phys., 26 (2019), 133-146. doi: 10.1080/14029251.2019.1544794. [40] H. Lamb, Hydrodynamics, Reprint of the 1932 sixth edition. With a foreword by R. A. Caflisch [Russel E. Caflisch]. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. [41] S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444. [42] A. Lifschitz and E. Hameiri, Local stability conditions in fluid mechanics, Phys. Fluids, 3 (1991), 2644-2651. doi: 10.1063/1.858153. [43] A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A, 45 (2012), 365501, 10pp. doi: 10.1088/1751-8113/45/36/365501. [44] A.-V. Matioc, Exact geophysical waves in stratified fluids, Appl. Anal., 92 (2013), 2254-2261. doi: 10.1080/00036811.2012.727987. [45] R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898. doi: 10.1029/JC075i030p05895. [46] A. Rodríguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for equatorially-trapped internal water waves, Nonlinear Anal., 149 (2017), 156-164. doi: 10.1016/j.na.2016.10.022. [47] A. Rodríguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions, Ann. Mat. Pura Appl., 197 (2018), 1787-1797. doi: 10.1007/s10231-018-0749-5. [48] S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves, Nonlinear Anal., 125 (2015), 725-731. doi: 10.1016/j.na.2015.06.017. [49] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.
The rotating framework with the origin at a point on the Earth's surface with latitude $\phi$, the $x$-axis chosen horizontally due east, the $y$-axis horizontally due north (in the tangent plane) and the z-axis vertically upward
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