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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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August
2019, 39(8): 4471-4486.
doi: 10.3934/dcds.2019183
Geophysical internal equatorial waves of extreme form
Department of Computing & Mathematics, Waterford Institute of Technology, Waterford, Ireland |
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.
Citation:
Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete and Continuous Dynamical Systems,
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. |
| [2] |
A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.
doi: 10.1017/CBO9780511734939.
|
| [3] |
B. Buffoni and J. Toland, Analytic 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. 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. |
| [42] |
S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988. |
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. |
| [2] |
A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.
doi: 10.1017/CBO9780511734939.
|
| [3] |
B. Buffoni and J. Toland, Analytic 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. 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. |
| [42] |
S. H. C. Wacogne, Dynamics of the Equatorial Undercurrent and Its Termination, PhD thesis, Massachusetts Institute of Technology, 1988. |
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
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