American Institute of Mathematical Sciences

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August  2019, 39(8): 4471-4486. doi: 10.3934/dcds.2019183

Geophysical internal equatorial waves of extreme form

Tony Lyons

Department of Computing & Mathematics, Waterford Institute of Technology, Waterford, Ireland

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

Received  August 2018 Revised  October 2018 Published  May 2019

The existence of internal geophysical waves of extreme form is confirmed and an explicit solution presented. The flow is confined to a layer lying above an eastward current while the mean horizontal flow of the solutions is westward, thus incorporating flow reversal in the fluid.

Keywords: Equatorial flows, geophysical internal waves, extreme waves.
Mathematics Subject Classification: Primary: 76B55, 76B15; Secondary: 35Q31.
Citation: Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183
References:
[1]

C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214. doi: 10.1007/BF02392728. Google Scholar

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006. doi: 10.1017/CBO9780511734939. Google Scholar
[3] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton University Press, 2003. doi: 10.1515/9781400884339. Google Scholar
[4]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the equatorial undercurrent, J. Nonlin. Math. Phys., 22 (2015), 531-539. doi: 10.1080/14029251.2015.1113052. Google Scholar

[5]

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

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873. Google Scholar

[7]

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

[8]

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

[9]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar

[10]

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

[11]

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

[12]

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

[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]

R. Courant, Differential and Integral Calculus, John Wiley & Sons, Inc., New York, 1988. Google Scholar

[16]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011. Google Scholar

[17]

M. L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides, Atti. Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., 15 (1932), 814-819. Google Scholar

[18]

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

[19]

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

[20]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, Pearson International Edition, Addison Wesley, 2002.Google Scholar

[21]

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

[22]

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

[23]

D. Henry, On three-dimensional Gerstner-like equatorial water waves, Phil. Trans. R. Soc. A, 376 (2018), 20170088, 16pp. doi: 10.1098/rsta.2017.0088. Google Scholar

[24]

D. Henry and H. C. Hsu, Instability of internal equatorial water waves, J. Diff. Eqn., 258 (2015), 1015-1024. doi: 10.1016/j.jde.2014.08.019. Google Scholar

[25]

D. Henry and S. Sastre-Gomez, Mean flow velocities and mass transport for equatorially-trapped water waves with an underlying current, J. Math. Fluid Mech., 18 (2016), 795-804. doi: 10.1007/s00021-016-0262-9. Google Scholar

[26]

D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2018), 20170090, 21pp. doi: 10.1098/rsta.2017.0090. Google Scholar

[27]

M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19 (2017), 305-314. doi: 10.1007/s00021-016-0281-6. Google Scholar

[28]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Disc. Contin. Dyn. Sys. Ser. A, 34 (2014), 3095-3107. doi: 10.3934/dcds.2014.34.3095. Google Scholar

[29]

T. Lyons, The pressure distribution in extreme Stokes waves, Nonlin. Anal. Real World Appl., 31 (2016), 77-87. doi: 10.1016/j.nonrwa.2016.01.008. Google Scholar

[30]

T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209–218. doi: 10.1007/s00021-016-0249-6. Google Scholar

[31]

T. Lyons, The dynamic pressure in deep-water extreme Stokes waves, Phil. Trans. R. Soc. A, 376 (2018), 20170095, 13pp. doi: 10.1098/rsta.2017.0095. Google Scholar

[32]

C. I. Martin, Dynamics of the thermocline in the equatorial region of the pacific ocean, J. Nonlin. Math. Phys., 22 (2015), 516-522. doi: 10.1080/14029251.2015.1113049. Google Scholar

[33]

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

[34]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc. London A, 153 (1863), 127-138. Google Scholar

[35]

A. Rodriguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Equatorially-trapped internal water waves, Nonlin. Anal.: Theor., Meth. Appl., 149 (2017), 156-164. doi: 10.1016/j.na.2016.10.022. Google Scholar

[36]

A. Rodriguez-Sanjurjo and M. Kluczek, Mean flow properties for equatorially trapped internal water wave–current interactions, Appl. Anal., 96 (2017), 2333-2345. doi: 10.1080/00036811.2016.1221943. Google Scholar

[37]

K. W. S and M. M. J, Oceaninc equatorial waves and the 1991–1993 El Niño, J. Climate, 8 (1995), 1757-1774. Google Scholar

[38]

S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves, Nonlin. Anal., 125 (2015), 725-731. doi: 10.1016/j.na.2015.06.017. Google Scholar

[39]

G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Mathematical and Physical Papers, 1 (1880), 225-228. Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Meth. Nonlin. Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[41]

G. K. Vallis, D. J. Parker and S. M. Tobias, A simple system for moist convection: The rainy-Benard model, J. Fluid Mech., 862 (2019), 162-199. doi: 10.1017/jfm.2018.954. Google Scholar

[42]

S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988.Google Scholar

show all references

References:
[1]

C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math., 148 (1982), 193-214. doi: 10.1007/BF02392728. Google Scholar

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006. doi: 10.1017/CBO9780511734939. Google Scholar
[3] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton University Press, 2003. doi: 10.1515/9781400884339. Google Scholar
[4]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the equatorial undercurrent, J. Nonlin. Math. Phys., 22 (2015), 531-539. doi: 10.1080/14029251.2015.1113052. Google Scholar

[5]

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

[6]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873. Google Scholar

[7]

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

[8]

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

[9]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar

[10]

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

[11]

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

[12]

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

[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]

R. Courant, Differential and Integral Calculus, John Wiley & Sons, Inc., New York, 1988. Google Scholar

[16]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011. Google Scholar

[17]

M. L. Dubreil-Jacotin, Sur les ondes de type permanent dans les liquides, Atti. Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., 15 (1932), 814-819. Google Scholar

[18]

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

[19]

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

[20]

H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, Pearson International Edition, Addison Wesley, 2002.Google Scholar

[21]

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

[22]

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

[23]

D. Henry, On three-dimensional Gerstner-like equatorial water waves, Phil. Trans. R. Soc. A, 376 (2018), 20170088, 16pp. doi: 10.1098/rsta.2017.0088. Google Scholar

[24]

D. Henry and H. C. Hsu, Instability of internal equatorial water waves, J. Diff. Eqn., 258 (2015), 1015-1024. doi: 10.1016/j.jde.2014.08.019. Google Scholar

[25]

D. Henry and S. Sastre-Gomez, Mean flow velocities and mass transport for equatorially-trapped water waves with an underlying current, J. Math. Fluid Mech., 18 (2016), 795-804. doi: 10.1007/s00021-016-0262-9. Google Scholar

[26]

D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2018), 20170090, 21pp. doi: 10.1098/rsta.2017.0090. Google Scholar

[27]

M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, J. Math. Fluid Mech., 19 (2017), 305-314. doi: 10.1007/s00021-016-0281-6. Google Scholar

[28]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Disc. Contin. Dyn. Sys. Ser. A, 34 (2014), 3095-3107. doi: 10.3934/dcds.2014.34.3095. Google Scholar

[29]

T. Lyons, The pressure distribution in extreme Stokes waves, Nonlin. Anal. Real World Appl., 31 (2016), 77-87. doi: 10.1016/j.nonrwa.2016.01.008. Google Scholar

[30]

T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209–218. doi: 10.1007/s00021-016-0249-6. Google Scholar

[31]

T. Lyons, The dynamic pressure in deep-water extreme Stokes waves, Phil. Trans. R. Soc. A, 376 (2018), 20170095, 13pp. doi: 10.1098/rsta.2017.0095. Google Scholar

[32]

C. I. Martin, Dynamics of the thermocline in the equatorial region of the pacific ocean, J. Nonlin. Math. Phys., 22 (2015), 516-522. doi: 10.1080/14029251.2015.1113049. Google Scholar

[33]

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

[34]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. R. Soc. London A, 153 (1863), 127-138. Google Scholar

[35]

A. Rodriguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Equatorially-trapped internal water waves, Nonlin. Anal.: Theor., Meth. Appl., 149 (2017), 156-164. doi: 10.1016/j.na.2016.10.022. Google Scholar

[36]

A. Rodriguez-Sanjurjo and M. Kluczek, Mean flow properties for equatorially trapped internal water wave–current interactions, Appl. Anal., 96 (2017), 2333-2345. doi: 10.1080/00036811.2016.1221943. Google Scholar

[37]

K. W. S and M. M. J, Oceaninc equatorial waves and the 1991–1993 El Niño, J. Climate, 8 (1995), 1757-1774. Google Scholar

[38]

S. Sastre-Gomez, Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves, Nonlin. Anal., 125 (2015), 725-731. doi: 10.1016/j.na.2015.06.017. Google Scholar

[39]

G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Mathematical and Physical Papers, 1 (1880), 225-228. Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Meth. Nonlin. Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001. Google Scholar

[41]

G. K. Vallis, D. J. Parker and S. M. Tobias, A simple system for moist convection: The rainy-Benard model, J. Fluid Mech., 862 (2019), 162-199. doi: 10.1017/jfm.2018.954. Google Scholar

[42]

S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988.Google Scholar

Figure 1.  The rotating $ \left(x,y,z\right) $-coordinate system fixed to the surface of the Earth. The $ x $-axis points due-east, the $ y $-axis points due-north and the $ z $-axis points vertically upwards from the surface
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Figure 2.  A cross section of the flow in the equatorial plane $ y = 0 $ with a flow wavelength of $ L = 200\,\mathrm{m} $. The average depth of the near surface layer is $ 60\,\mathrm{m} $, while the thermocline lies at an average depth of $ 120\,\mathrm{m} $. The transitional layer begins at $ 160\, \mathrm{m} $ approximately, while the motionless layer begins at $ 200\,\mathrm{m} $ beneath the free surface
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Figure 3.  The circular paths in the equatorial plane $ y = 0 $ traced by particles in the layer $ \mathcal{M}(t) $ are circles with centre $ (q,r-d_0) $ whose radius increases with depth. The wavelength of the solution shown is $ L = 200\ \mathrm{m} $. The particle trajectories are counterclockwise and the particle positions shown in the figure are at time $ t = 0 $
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Figure 4.  A pictorial outline for the proof of Theorem 3.2, confirming the existence of a solution of the implicit relation $ G(s,r) = G(0,0) $ of extreme form
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Figure 5.  The left-hand panel shows the graph $ (s,r_0(s)) $ when $ \kappa = 0.5 $ and $ L = 200 $ meters. The parameter $ d_0 = 120 $ ensures a mean equatorial depth of $ 120\, $m. In the right-hand panel the corresponding thermocline profile in the fluid domain, at the equatorial locations indicated in the figure
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Figure 6.  The thermocline surface when $ L = 200\, $m, $ r_0^* = 0\, $m and $ d_0 = 120\, $m, with $ \kappa = 0.5 $. The surface is viewed from beneath and the value of $ \kappa $ has been chosen to emphasise the extreme behaviour of the wave along the equator at the wave-troughs
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