# American Institute of Mathematical Sciences

August  2019, 39(8): 4783-4796. doi: 10.3934/dcds.2019195

## On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density

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

* Corresponding author: Biswajit Basu

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

Received  October 2018 Revised  January 2019 Published  May 2019

The aim of the paper is to develop an exact solution relating to a system of model equations representing ocean flows with Equatorial Undercurrent and thermocline in the presence of linear variation of density with depth. The system of equations is generated from the Euler equations represented in a suitable rotating frame by following a careful asymptotic approach.The study in this paper is motivated by the recently developed Constantin-Johnson model [13] for Pacific flows with undercurrent and the exact results provided therein. The model formulated is two-layered, three-dimensional and nonlinear with a symmetric structure about the equator. The equations contain Coriolis effect and is consistent with $\beta$ - plane approximation. Exact results of the asymptotic system of equations have been derived in a region close to the equator.

Citation: Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195
##### References:
 [1] B. Basu, Some numerical investigations into a nonlinear three-dimensional model of Pacific equatorial ocean dynamics, Deep-Sea Res. II, 160 (2019), 7-15. [2] B. Basu, One a three-dimensional nonlinear model of Pacific equatorial ocean dynamics: Velocities and flow paths, Oceanography, 31(3) (2018), 51-58. [3] 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. [4] M. A. Cane, The response of an equatorial ocean to simple wind stress patterns: Ⅱ. Numerical results, J. Mar. Res., 6 (1979), 335-398. [5] J. R. Charney, Non-linear theory of a wind-driven homogeneous layer near the equator, Deep Sea Res., 6 (1959/60), 303-310.  doi: 10.1016/0146-6313(59)90089-9. [6] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.  doi: 10.1029/2012JC007879. [7] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.  doi: 10.1029/2012GL051169. [8] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219. [9] 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. [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, 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. [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, Phy. of Fluids, 29 (2017), 056604. [14] T. Cromwell, Circulation in a meridional plane in the central equatorial Pacific, J. Mar. Res., 12 (1953), 196-213. [15] H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino, Springer Science & Business Media, 2005. [16] 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. [17] N. P. Fofonoff and R. B. Montgomery, The equatorial undercurrent in the light of the vorticity equation, Tellus, 7 (1955), 518-521. [18] A. E. Gill, Atmosphere-ocean dynamics, Academic Press, New York, 2016. [19] A. E. Gill, The equatorial current in a homogeneous ocean, Deep Sea Res., 81 (1971), 421-431. [20] A. E. Gill, Models of equatorial currents, Proc. Numerical Models of Ocean Circulation, Nat. Acad. Sc., (1975), 181-203. [21] 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. [22] 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. [23] D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474.  doi: 10.1016/j.jde.2017.01.001. [24] G. C. Johnson, M. J. McPhaden and E. Firing, Equatorial Pacific Ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr., 31 (2001), 839-849.  doi: 10.1175/1520-0485(2001)031<0839:EPOHVD>2.0.CO;2. [25] 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.  doi: 10.1016/S0079-6611(02)00021-6. [26] W. S. Kessler, The circulation of the eastern tropical Pacific: A review, Progr. Oceanogr., 69 (2006), 181-217.  doi: 10.1016/j.pocean.2006.03.009. [27] 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. [28] 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. [29] J. P. McCreary Jr, Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech., 17 (1985), 359-409. [30] 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. [31] W. D. McKee, The wind-driven equatorial circulation in a homogeneous ocean, Deep Sea Res., 20 (1973), 889-899.  doi: 10.1016/0011-7471(73)90107-1. [32] J. Pedlosky, Thermocline theories, in General Circulation of the Ocean, Springer, (1987), 55–101. doi: 10.1007/978-1-4612-4636-7_2. [33] A. R. Robinson, An investigation into the wind as the cause of the equatorial undercurrent, J. Mar. Res., 24 (1966), 179-204. [34] H. Stommel, Wind-drift near the equator, Deep Sea Res., 6 (1960), 298-302.  doi: 10.1016/0146-6313(59)90088-7. [35] L. D. Talley, G. L. Pickard, W. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, Elsevier, London, 2011. [36] G. Veronis, An approximate theoretical analysis of the equatorial undercurrent, Deep Sea Res., 6 (1959/60), 318-327.  doi: 10.1016/0146-6313(59)90091-7.

show all references

##### References:
 [1] B. Basu, Some numerical investigations into a nonlinear three-dimensional model of Pacific equatorial ocean dynamics, Deep-Sea Res. II, 160 (2019), 7-15. [2] B. Basu, One a three-dimensional nonlinear model of Pacific equatorial ocean dynamics: Velocities and flow paths, Oceanography, 31(3) (2018), 51-58. [3] 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. [4] M. A. Cane, The response of an equatorial ocean to simple wind stress patterns: Ⅱ. Numerical results, J. Mar. Res., 6 (1979), 335-398. [5] J. R. Charney, Non-linear theory of a wind-driven homogeneous layer near the equator, Deep Sea Res., 6 (1959/60), 303-310.  doi: 10.1016/0146-6313(59)90089-9. [6] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), C05029.  doi: 10.1029/2012JC007879. [7] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), L05602.  doi: 10.1029/2012GL051169. [8] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219. [9] 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. [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, 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. [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, Phy. of Fluids, 29 (2017), 056604. [14] T. Cromwell, Circulation in a meridional plane in the central equatorial Pacific, J. Mar. Res., 12 (1953), 196-213. [15] H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino, Springer Science & Business Media, 2005. [16] 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. [17] N. P. Fofonoff and R. B. Montgomery, The equatorial undercurrent in the light of the vorticity equation, Tellus, 7 (1955), 518-521. [18] A. E. Gill, Atmosphere-ocean dynamics, Academic Press, New York, 2016. [19] A. E. Gill, The equatorial current in a homogeneous ocean, Deep Sea Res., 81 (1971), 421-431. [20] A. E. Gill, Models of equatorial currents, Proc. Numerical Models of Ocean Circulation, Nat. Acad. Sc., (1975), 181-203. [21] 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. [22] 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. [23] D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474.  doi: 10.1016/j.jde.2017.01.001. [24] G. C. Johnson, M. J. McPhaden and E. Firing, Equatorial Pacific Ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr., 31 (2001), 839-849.  doi: 10.1175/1520-0485(2001)031<0839:EPOHVD>2.0.CO;2. [25] 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.  doi: 10.1016/S0079-6611(02)00021-6. [26] W. S. Kessler, The circulation of the eastern tropical Pacific: A review, Progr. Oceanogr., 69 (2006), 181-217.  doi: 10.1016/j.pocean.2006.03.009. [27] 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. [28] 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. [29] J. P. McCreary Jr, Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech., 17 (1985), 359-409. [30] 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. [31] W. D. McKee, The wind-driven equatorial circulation in a homogeneous ocean, Deep Sea Res., 20 (1973), 889-899.  doi: 10.1016/0011-7471(73)90107-1. [32] J. Pedlosky, Thermocline theories, in General Circulation of the Ocean, Springer, (1987), 55–101. doi: 10.1007/978-1-4612-4636-7_2. [33] A. R. Robinson, An investigation into the wind as the cause of the equatorial undercurrent, J. Mar. Res., 24 (1966), 179-204. [34] H. Stommel, Wind-drift near the equator, Deep Sea Res., 6 (1960), 298-302.  doi: 10.1016/0146-6313(59)90088-7. [35] L. D. Talley, G. L. Pickard, W. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, Elsevier, London, 2011. [36] G. Veronis, An approximate theoretical analysis of the equatorial undercurrent, Deep Sea Res., 6 (1959/60), 318-327.  doi: 10.1016/0146-6313(59)90091-7.
A schematic of the structure of the equatorial ocean flow
The rotating frame of reference based on tangent plane, with the $\overline{x}$ axis chosen horizontally due east, the $\overline{y}$ axis horizontally due north and the $\overline{\overline z}$ axis vertically upward
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