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Geophysical internal equatorial waves of extreme form

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

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  • 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.

    Mathematics Subject Classification: Primary: 76B55, 76B15; Secondary: 35Q31.


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  • 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

    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

    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 $

    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

    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

    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

  • [1] C. J. AmickL. 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.
    [2] A. BennettLagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511734939.
    [3] B. Buffoni and  J. TolandAnalytic Theory of Global Bifurcation, Princeton University Press, 2003.  doi: 10.1515/9781400884339.
    [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.
    [5] A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.  doi: 10.1088/0305-4470/34/7/313.
    [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.
    [7] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012). doi: 10.1029/2012JC007879.
    [8] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012). doi: 10.1029/2012GL051169.
    [9] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.
    [10] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanog., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.
    [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.
    [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.
    [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] R. Courant, Differential and Integral Calculus, John Wiley & Sons, Inc., New York, 1988.
    [16] B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, vol. 101, Academic Press, 2011.
    [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. 
    [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.
    [19] F. Gerstner, Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Ann. Phys., 2 (1809), 412-445. 
    [20] H. Goldstein, C. P. Poole and J. L. Safko, Classical Mechanics, Pearson International Edition, Addison Wesley, 2002.
    [21] D. Henry, On Gerstner's water wave, J. Nonlin. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.S2.7.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [33] A. V. Matioc, Exact geophysical waves in stratified fluids, Appl, Anal., 92 (2013), 2254-2261.  doi: 10.1080/00036811.2012.727987.
    [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. 
    [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.
    [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.
    [37] K. W. S and M. M. J, Oceaninc equatorial waves and the 1991–1993 El Niño, J. Climate, 8 (1995), 1757-1774. 
    [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.
    [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. 
    [40] J. F. Toland, Stokes waves, Topol. Meth. Nonlin. Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.
    [41] G. K. VallisD. 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.
    [42] S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988.
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