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Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current
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Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces
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 |
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.
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] | |
[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] | |
[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.
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