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. Google Scholar

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511734939.  Google Scholar
[3]

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

[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.  Google Scholar

[5]

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

[6]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169.  Google Scholar

[7]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.  Google Scholar

[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.  Google Scholar

[17]

L. FanH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

A. E. Gill, Atmosphere-Ocean Dynamics, Academic, 1982. Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H.-C. Hsu, An exact solution for equatorial waves, Monatsh. Math., 176 (2015), 143-152.  doi: 10.1007/s00605-014-0618-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

D. Ionescu-Kruse, Instability of Pollard's exact solution for geophysical ocean flows, Phys. Fluids, 28 (2016), 086601. doi: 10.1063/1.4959289.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. Kluczek, Physical flow properties for Pollard-like internal water waves, J. Math. Phys., 59 (2018), 123102, 12pp. doi: 10.1063/1.5038657.  Google Scholar

[39]

M. Kluczek, Exact Pollard-like internal eater waves, J. Nonlinear Math. Phys., 26 (2019), 133-146.  doi: 10.1080/14029251.2019.1544794.  Google Scholar

[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.  Google Scholar

[41]

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

[42]

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

[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.  Google Scholar

[44]

A.-V. Matioc, Exact geophysical waves in stratified fluids, Appl. Anal., 92 (2013), 2254-2261.  doi: 10.1080/00036811.2012.727987.  Google Scholar

[45]

R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898.  doi: 10.1029/JC075i030p05895.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.   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, edited by R. W. Miksad et al., 71–77, ASME, New York, 1987. Google Scholar

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511734939.  Google Scholar
[3]

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

[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.  Google Scholar

[5]

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

[6]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602. doi: 10.1029/2012GL051169.  Google Scholar

[7]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[15]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.  Google Scholar

[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.  Google Scholar

[17]

L. FanH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

A. E. Gill, Atmosphere-Ocean Dynamics, Academic, 1982. Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

H.-C. Hsu, An exact solution for equatorial waves, Monatsh. Math., 176 (2015), 143-152.  doi: 10.1007/s00605-014-0618-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

D. Ionescu-Kruse, Instability of Pollard's exact solution for geophysical ocean flows, Phys. Fluids, 28 (2016), 086601. doi: 10.1063/1.4959289.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

M. Kluczek, Physical flow properties for Pollard-like internal water waves, J. Math. Phys., 59 (2018), 123102, 12pp. doi: 10.1063/1.5038657.  Google Scholar

[39]

M. Kluczek, Exact Pollard-like internal eater waves, J. Nonlinear Math. Phys., 26 (2019), 133-146.  doi: 10.1080/14029251.2019.1544794.  Google Scholar

[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.  Google Scholar

[41]

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

[42]

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

[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.  Google Scholar

[44]

A.-V. Matioc, Exact geophysical waves in stratified fluids, Appl. Anal., 92 (2013), 2254-2261.  doi: 10.1080/00036811.2012.727987.  Google Scholar

[45]

R. T. Pollard, Surface waves with rotation: An exact solution, J. Geophys. Res., 75 (1970), 5895-5898.  doi: 10.1029/JC075i030p05895.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.   Google Scholar
Figure 1.  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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