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July  2022, 21(7): 2271-2290. doi: 10.3934/cpaa.2022085

On the nonlinear three-dimensional models in equatorial ocean flows

Biswajit Basu

School of Engineering, Trinity College Dublin, Dublin 2, Ireland

Received  December 2021 Revised  March 2022 Published  July 2022 Early access  May 2022

The paper focusses on some of the recent breakthroughs in the development of models for nonlinear, three-dimensional Equatorial oceanic flows by Constantin and Johnson. The unique character of the formulations is in the systematic approach followed, while making approximations as required, and consequently assessing the implications. These Constantin-Johnson type of models are general enough, as effects such as that of Earth's rotation, Coriolis term, stratification, thermocline, pycnocline, density variations and vertical velocities can be accounted for. Exact solutions based on the use of singular perturbation theory have been obtained for several different cases and situations. The novelty in the models lies in the introduction of a quasi-stream-function which facilitates the derivation of the solutions. Analytical results are supplemented with some numerical illustrations to provide a flavour of the complex flow structures involved. Insights are provided into the velocity field and flow paths, indicating the presence of cellular structures, upwelling/downwelling and subsurface ocean 'bridges'.

Keywords: Equatorial undercurrent, rotating frame, density variation, Euler equation, asymptotic analysis.
Mathematics Subject Classification: Primary: 35Q31, 35Q35; Secondary: 76B70.
Citation: Biswajit Basu. On the nonlinear three-dimensional models in equatorial ocean flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2271-2290. doi: 10.3934/cpaa.2022085
References:
[1]

B. Basu, One a three-dimensional nonlinear model of Pacific equatorial ocean dynamics: Velocities and flow paths, Oceanography, 31 (2018), 51-58. 

[2]

B. Basu, Some numerical investigations into a nonlinear three-dimensional model of Pacific equatorial ocean dynamics, Deep-Sea Res. II, 160 (2019), 7-15. 

[3]

B. Basu, On an exact solution of a nonlinear three-dimensional model in coean flows with equatorial undercurrent and linear variation in density, Discr. Cont. Dyn. Sys., 39 (2019), 4783-4796.  doi: 10.3934/dcds.2019195.

[4]

M. A. Cane, The response of an equatorial ocean to simple wind stress patterns: I. Model formulation and analytical results, J. Mar. Res., 37 (1979), 232-252. 

[5]

M. A. Cane, The response of an equatorial ocean to simple wind stress patterns: II. Numerical results, J. Mar. Res., 6 (1979), 335-398. 

[6]

J. R. Charney, Non-linear theory of a wind-driven homogeneous layer near the equator, Deep Sea Res., 6 (1959/60), 303-310. 

[7]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.

[8]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.

[9]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. 

[10]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810. 

[11]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layers, Phys. Fluids, 27 (2015), 086603. 

[12]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[13]

A. Constantin, R. I. Ivanov and C. I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447.  doi: 10.1007/s00205-016-0990-2.

[14]

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.

[15]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945. 

[16]

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), 21 pp. doi: 10.1063/1.4984001.

[17]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of Equatorial ocean flows, J. Phys. Oceanogr., 49 (2019), 2029-2042. 

[18]

A. Constantin and R. S. Johnson, Ekman-type solutions for shallow-water flows on a rotating sphere: A new perspective on a classical problem, Phys. Fluids., 31 (2019), 021401. 

[19]

T. Cromwell, Circulation in a meridional plane in the central equatorial Pacific, J. Mar. Res., 12 (1953), 196-213. 

[20]

H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino, Springer Science & Business Media, 2005.

[21]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopeida of Ocean Sciences edited by Steele, J, Academic Press, San Diego, (2009), 3679–3695.

[22]

N. P. Fofonoff and R. B. Montgomery, The equatorial undercurrent in the light of the vorticity equation, Tellus, 7 (1955), 518-521. 

[23]

A. E. Gill, The equatorial current in a homogeneous ocean, Deep Sea Res., 81 (1971), 421-431. 

[24]

A. E. Gill, Models of equatorial currents, Proc. Numerical Models of Ocean Circulation, Nat. Acad. Sc., (1975), 181–203.

[25] A. E. Gill, Atmosphere-ocean dynamics, Academic Press, New York, 2016. 
[26]

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.

[27]

D. Henry, Equatorially trapped nonlinear water waves in a $\beta$ -plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp. doi: 10.1017/jfm.2016.544.

[28]

D. Henry and C. I. Martin, Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification, J. Differ. Equ., 266 (2019), 6788-6808.  doi: 10.1016/j.jde.2018.11.017.

[29]

D. Henry and C. I. Martin, Azimuthal equatorial flows with variable density in spherical coordinate, Arch. Ration. Mech. Anal., 233 (2019), 497-512.  doi: 10.1007/s00205-019-01362-z.

[30]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889-3904.  doi: 10.1088/1361-6544/ab801f.

[31]

D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differ. Equ., 262 (2017), 4451-4474.  doi: 10.1016/j.jde.2017.01.001.

[32]

R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal.: RWA, 34 (2017), 316-334.  doi: 10.1016/j.nonrwa.2016.09.010.

[33]

G. C. Johnson, M. J. McPhaden and E. Firing, Equatorial Pacific Ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr., 31 (2001), 839-849. 

[34]

G. C. Johnson, B. M. Sloyan, W. S. Kessler and K. E. McTaggart, Direct measurements of upper ocean currents and water properties across the tropical Pacific during the 1990s, Progr. Oceanogr., 52 (2002), 31-61. 

[35]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.

[36]

K. B. Karanauskas, J. Jakoboski, T. M. S. Johnston, W. B. Owens, D. L. Rudnick and R. E. Todd, The Pacific Equatorial Undercurrent in three generationsof global climate models and glider observations, J. Geophys. Res.: Oceans, 125 (2020), 12pp.

[37]

W. S. Kessler, The circulation of the eastern tropical Pacific: A review, Progr. Oceanogr., 69 (2006), 181-217. 

[38]

C. I. Martin, Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation, Ann. Mat. Pura Appl., 196 (2017), 2253-2260.  doi: 10.1007/s10231-017-0663-2.

[39]

C. I. Martin, Azimuthal equatorial flows in spherical coordinates with discontinuous stratification, Phys. Fluids, 33 (2021), 026602. 

[40]

K. Marynets, The modeling of the equatorial undercurrent using the Navier-Stokes equations in rotating spherical coordinates, Appl. Anal., 100 (2019), 2069-2077.  doi: 10.1080/00036811.2019.1673375.

[41]

K. Marynets, A hyperbolic-type azimuthal velocity model for equatorial currents, Appl. Anal., 101 (2022), 1147-1155.  doi: 10.1080/00036811.2020.1774054.

[42]

J. P. McCreary, A linear stratified ocean model of the equatorial undercurrent, Phil. Trans. Roy. Soc. London A, 298 (1981), 603-635.  doi: 10.1098/rsta.1981.0002.

[43]

J. P. McCreary Jr, Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech., 17 (1985), 359-409. 

[44]

J. P. McCreary Jr and P. Lu, Interaction between the subtropical and equatorial ocean circulations: the subtropical cell, J. Phys. Oceanogr., 24 (1994), 466-497. 

[45]

W. D. McKee, The wind-driven equatorial circulation in a homogeneous ocean, Deep Sea Res., 20 (1973), 889-899. 

[46]

J. Pedlosky, Thermocline theories, in General Circulation of the Ocean, Springer, (1987), 55–101.

[47]

A. R. Robinson, An investigation into the wind as the cause of the equatorial undercurrent, J. Mar. Res., 24 (1966), 179-204. 

[48]

H. Stommel, Wind-drift near the equator, Deep Sea Res., 6 (1960), 298-302. 

[49]

L. D. Talley, G. L. Pickard, W. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, Elsevier, London, 2011.

[50]

G. Veronis, An approximate theoretical analysis of the equatorial undercurrent, Deep Sea Res., 6 (1959/60), 318-327. 

[51]

R. H. Zhang, L. M. Rothstein and A. J. Busalaccchi, Origin of upper-ocean warming and El Niño change on decadal time scales in the tropical Pacific Ocean, Nature, 391 (1998), 879-883. 

show all references

References:
[1]

B. Basu, One a three-dimensional nonlinear model of Pacific equatorial ocean dynamics: Velocities and flow paths, Oceanography, 31 (2018), 51-58. 

[2]

B. Basu, Some numerical investigations into a nonlinear three-dimensional model of Pacific equatorial ocean dynamics, Deep-Sea Res. II, 160 (2019), 7-15. 

[3]

B. Basu, On an exact solution of a nonlinear three-dimensional model in coean flows with equatorial undercurrent and linear variation in density, Discr. Cont. Dyn. Sys., 39 (2019), 4783-4796.  doi: 10.3934/dcds.2019195.

[4]

M. A. Cane, The response of an equatorial ocean to simple wind stress patterns: I. Model formulation and analytical results, J. Mar. Res., 37 (1979), 232-252. 

[5]

M. A. Cane, The response of an equatorial ocean to simple wind stress patterns: II. Numerical results, J. Mar. Res., 6 (1979), 335-398. 

[6]

J. R. Charney, Non-linear theory of a wind-driven homogeneous layer near the equator, Deep Sea Res., 6 (1959/60), 303-310. 

[7]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.

[8]

A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.

[9]

A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. 

[10]

A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810. 

[11]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layers, Phys. Fluids, 27 (2015), 086603. 

[12]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[13]

A. Constantin, R. I. Ivanov and C. I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Ration. Mech. Anal., 221 (2016), 1417-1447.  doi: 10.1007/s00205-016-0990-2.

[14]

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.

[15]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945. 

[16]

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), 21 pp. doi: 10.1063/1.4984001.

[17]

A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of Equatorial ocean flows, J. Phys. Oceanogr., 49 (2019), 2029-2042. 

[18]

A. Constantin and R. S. Johnson, Ekman-type solutions for shallow-water flows on a rotating sphere: A new perspective on a classical problem, Phys. Fluids., 31 (2019), 021401. 

[19]

T. Cromwell, Circulation in a meridional plane in the central equatorial Pacific, J. Mar. Res., 12 (1953), 196-213. 

[20]

H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino, Springer Science & Business Media, 2005.

[21]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopeida of Ocean Sciences edited by Steele, J, Academic Press, San Diego, (2009), 3679–3695.

[22]

N. P. Fofonoff and R. B. Montgomery, The equatorial undercurrent in the light of the vorticity equation, Tellus, 7 (1955), 518-521. 

[23]

A. E. Gill, The equatorial current in a homogeneous ocean, Deep Sea Res., 81 (1971), 421-431. 

[24]

A. E. Gill, Models of equatorial currents, Proc. Numerical Models of Ocean Circulation, Nat. Acad. Sc., (1975), 181–203.

[25] A. E. Gill, Atmosphere-ocean dynamics, Academic Press, New York, 2016. 
[26]

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.

[27]

D. Henry, Equatorially trapped nonlinear water waves in a $\beta$ -plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp. doi: 10.1017/jfm.2016.544.

[28]

D. Henry and C. I. Martin, Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification, J. Differ. Equ., 266 (2019), 6788-6808.  doi: 10.1016/j.jde.2018.11.017.

[29]

D. Henry and C. I. Martin, Azimuthal equatorial flows with variable density in spherical coordinate, Arch. Ration. Mech. Anal., 233 (2019), 497-512.  doi: 10.1007/s00205-019-01362-z.

[30]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889-3904.  doi: 10.1088/1361-6544/ab801f.

[31]

D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differ. Equ., 262 (2017), 4451-4474.  doi: 10.1016/j.jde.2017.01.001.

[32]

R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal.: RWA, 34 (2017), 316-334.  doi: 10.1016/j.nonrwa.2016.09.010.

[33]

G. C. Johnson, M. J. McPhaden and E. Firing, Equatorial Pacific Ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr., 31 (2001), 839-849. 

[34]

G. C. Johnson, B. M. Sloyan, W. S. Kessler and K. E. McTaggart, Direct measurements of upper ocean currents and water properties across the tropical Pacific during the 1990s, Progr. Oceanogr., 52 (2002), 31-61. 

[35]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056.

[36]

K. B. Karanauskas, J. Jakoboski, T. M. S. Johnston, W. B. Owens, D. L. Rudnick and R. E. Todd, The Pacific Equatorial Undercurrent in three generationsof global climate models and glider observations, J. Geophys. Res.: Oceans, 125 (2020), 12pp.

[37]

W. S. Kessler, The circulation of the eastern tropical Pacific: A review, Progr. Oceanogr., 69 (2006), 181-217. 

[38]

C. I. Martin, Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation, Ann. Mat. Pura Appl., 196 (2017), 2253-2260.  doi: 10.1007/s10231-017-0663-2.

[39]

C. I. Martin, Azimuthal equatorial flows in spherical coordinates with discontinuous stratification, Phys. Fluids, 33 (2021), 026602. 

[40]

K. Marynets, The modeling of the equatorial undercurrent using the Navier-Stokes equations in rotating spherical coordinates, Appl. Anal., 100 (2019), 2069-2077.  doi: 10.1080/00036811.2019.1673375.

[41]

K. Marynets, A hyperbolic-type azimuthal velocity model for equatorial currents, Appl. Anal., 101 (2022), 1147-1155.  doi: 10.1080/00036811.2020.1774054.

[42]

J. P. McCreary, A linear stratified ocean model of the equatorial undercurrent, Phil. Trans. Roy. Soc. London A, 298 (1981), 603-635.  doi: 10.1098/rsta.1981.0002.

[43]

J. P. McCreary Jr, Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech., 17 (1985), 359-409. 

[44]

J. P. McCreary Jr and P. Lu, Interaction between the subtropical and equatorial ocean circulations: the subtropical cell, J. Phys. Oceanogr., 24 (1994), 466-497. 

[45]

W. D. McKee, The wind-driven equatorial circulation in a homogeneous ocean, Deep Sea Res., 20 (1973), 889-899. 

[46]

J. Pedlosky, Thermocline theories, in General Circulation of the Ocean, Springer, (1987), 55–101.

[47]

A. R. Robinson, An investigation into the wind as the cause of the equatorial undercurrent, J. Mar. Res., 24 (1966), 179-204. 

[48]

H. Stommel, Wind-drift near the equator, Deep Sea Res., 6 (1960), 298-302. 

[49]

L. D. Talley, G. L. Pickard, W. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, Elsevier, London, 2011.

[50]

G. Veronis, An approximate theoretical analysis of the equatorial undercurrent, Deep Sea Res., 6 (1959/60), 318-327. 

[51]

R. H. Zhang, L. M. Rothstein and A. J. Busalaccchi, Origin of upper-ocean warming and El Niño change on decadal time scales in the tropical Pacific Ocean, Nature, 391 (1998), 879-883. 

Figure 1.  (A) The rotating frame of reference for the Earth with the North pole, South pole and the Equator indicated (as well as the rotation direction around the polar axis). (B) The spherical coordinate system
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Figure 2.  (A) Horizontal velocity, $ v $. (B) Vertical velocity, $ w $ (see [2])
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Figure 3.  Streamlines in the $ y $-$ \zeta $ plane at $ x = 0.5 $ for $ \omega = 0.3 $ (see [2])
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Figure 4.  Evolution of in-plane velocity amplitude maps in the $ y $-$ \zeta $ plane along the azimuthal $ x $ axis for $ \omega = 0.3 $ (see [2])
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Figure 5.  3D flow paths for quadratic-quartic EUC profile with paths initiating from points (A) near surface (B) at intermediate depth (see [1])
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Figure 6.  (A) Contour plot of the meridional (nondimensional) velocity component (B) Plots of two examples of upwelling (red - on the left) and downwelling (green - on the right) (see [17])
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