# American Institute of Mathematical Sciences

August  2019, 39(8): 4533-4545. doi: 10.3934/dcds.2019186

## Shallow water models for stratified equatorial flows

 1 Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands 2 KTH Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25,100 44 Stockholm, Sweden

* Corresponding author: Ronald Quirchmayr

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

Fund Project: Both authors acknowledge the support of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) during the program "Mathematical Aspects of Physical Oceanography". R. Quirchmayr acknowledges the support of the European Research Council, Consolidator Grant No. 682537.

Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a detailed analysis of the effect of a given density distribution on the depth-structure of the corresponding traveling waves.

Citation: Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186
##### References:
 [1] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074. [2] J. L. Bona, P. E. Souganidis and W. Strauss, Stability and instability of solitary waves of korteweg-de vries type, Proc. Roy. Soc. London A, 411 (1987), 395-412.  doi: 10.1098/rspa.1987.0073. [3] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873. [4] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), Art. No. L05602. doi: 10.1029/2012GL051169. [5] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012), Art. No. C05029. doi: 10.1029/2012JC007879. [6] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr, 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1. [7] 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. [8] A. Constantin and R. I. Ivanov, A hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids., 27 (2015), Art. No. 086603. doi: 10.1063/1.4929457. [9] 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. [10] A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.  doi: 10.2991/jnmp.2008.15.s2.5. [11] 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. [12] 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. [13] A. Constantin and R. S. Johnson, Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography, Phys. Lett. A, 380 (2016), 3007-3012.  doi: 10.1016/j.physleta.2016.07.036. [14] 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. [15] B. Deconinck and T. Kapitula, The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A, 374 (2010), 4018-4022.  doi: 10.1016/j.physleta.2010.08.007. [16] M. W. Dingemans, Water Wave Propagation Over Uneven Bottoms, World Scientific, 1997. [17] P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, UK, 1989. doi: 10.1017/CBO9781139172059. [18] L. D. Faddeev and V. E. Zakharov, Kortweg-de Vries equation: A completely integrable Hamiltonian system, (Russian) Funkcional. Anal. i Prilovien., 5 (1971), 18–27. [19] A. V. Fedorov and J. N. Brown, Equatorial waves, Encyclopedia of Ocean Sciences, edited by J. Steele, 3679–3695, Academic Press: New York (2009). [20] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097. [21] A. Geyer and R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Phil. Trans. R. Soc. A, 376 (2018), 20170100, 12pp. doi: 10.1098/rsta.2017.0100. [22] 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. [23] R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal. Real World Appl., 34 (2017), 316-334.  doi: 10.1016/j.nonrwa.2016.09.010. [24] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.  doi: 10.1017/CBO9780511624056. [25] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224. [26] R. S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys., 22 (2015), 475-493.  doi: 10.1080/14029251.2015.1113042. [27] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19pp. doi: 10.1098/rsta.2017.0092. [28] T. Kappeler and J. Pöschel, KdV & KAM, Ergeb. der Math. und ihrer Grenzgeb., Springer, Berlin-Heidelberg-New York, 2003. doi: 10.1007/978-3-662-08054-2. [29] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739. [30] P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.  doi: 10.1002/cpa.3160210503. [31] P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978. [32] C. I. Martin, Dynamics of the thermocline in the equatorial region of the Pacific ocean, J. Nonlinear Math. Phys., 22 (2015), 516-522.  doi: 10.1080/14029251.2015.1113049. [33] S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.  doi: 10.1137/080721583.

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##### References:
 [1] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London A, 328 (1972), 153-183.  doi: 10.1098/rspa.1972.0074. [2] J. L. Bona, P. E. Souganidis and W. Strauss, Stability and instability of solitary waves of korteweg-de vries type, Proc. Roy. Soc. London A, 411 (1987), 395-412.  doi: 10.1098/rspa.1987.0073. [3] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873. [4] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012), Art. No. L05602. doi: 10.1029/2012GL051169. [5] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res. Oceans, 117 (2012), Art. No. C05029. doi: 10.1029/2012JC007879. [6] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr, 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1. [7] 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. [8] A. Constantin and R. I. Ivanov, A hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids., 27 (2015), Art. No. 086603. doi: 10.1063/1.4929457. [9] 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. [10] A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.  doi: 10.2991/jnmp.2008.15.s2.5. [11] 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. [12] 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. [13] A. Constantin and R. S. Johnson, Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography, Phys. Lett. A, 380 (2016), 3007-3012.  doi: 10.1016/j.physleta.2016.07.036. [14] 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. [15] B. Deconinck and T. Kapitula, The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A, 374 (2010), 4018-4022.  doi: 10.1016/j.physleta.2010.08.007. [16] M. W. Dingemans, Water Wave Propagation Over Uneven Bottoms, World Scientific, 1997. [17] P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, UK, 1989. doi: 10.1017/CBO9781139172059. [18] L. D. Faddeev and V. E. Zakharov, Kortweg-de Vries equation: A completely integrable Hamiltonian system, (Russian) Funkcional. Anal. i Prilovien., 5 (1971), 18–27. [19] A. V. Fedorov and J. N. Brown, Equatorial waves, Encyclopedia of Ocean Sciences, edited by J. Steele, 3679–3695, Academic Press: New York (2009). [20] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095-1097. [21] A. Geyer and R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Phil. Trans. R. Soc. A, 376 (2018), 20170100, 12pp. doi: 10.1098/rsta.2017.0100. [22] 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. [23] R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal. Real World Appl., 34 (2017), 316-334.  doi: 10.1016/j.nonrwa.2016.09.010. [24] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.  doi: 10.1017/CBO9780511624056. [25] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224. [26] R. S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys., 22 (2015), 475-493.  doi: 10.1080/14029251.2015.1113042. [27] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19pp. doi: 10.1098/rsta.2017.0092. [28] T. Kappeler and J. Pöschel, KdV & KAM, Ergeb. der Math. und ihrer Grenzgeb., Springer, Berlin-Heidelberg-New York, 2003. doi: 10.1007/978-3-662-08054-2. [29] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739. [30] P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21 (1968), 467-490.  doi: 10.1002/cpa.3160210503. [31] P. H. LeBlond and L. A. Mysak, Waves in the Ocean, Elsevier, Amsterdam, 1978. [32] C. I. Martin, Dynamics of the thermocline in the equatorial region of the Pacific ocean, J. Nonlinear Math. Phys., 22 (2015), 516-522.  doi: 10.1080/14029251.2015.1113049. [33] S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105.  doi: 10.1137/080721583.
Fig. 1a illustrates the fluid domain in the physical $({\bar x}, {\bar z})$-plane between the flat bed at ${\bar z}= 0$ and the free surface ${\bar z} = {\bar h}_0+{\bar \eta }(\cdot, {\bar t})$ at a certain instant of time ${\bar t}$. The average water level ${\bar h}_0$ is indicated by a dashed line, ${{\bar \lambda } }$ shows the distance between two consecutive crests and ${\bar a}$ is the vertical deviation of a typical crest from ${\bar h}_0$. Fig. 1b shows a prescribed depth dependent density distribution ${\bar \rho }({\bar z})$ with a significant density increase in the region between the two dotted horizontal lines close to the surface giving rise to a pycnocline
Solitary traveling wave solutions (26) of the surface equation (20) with linear density function $\rho (z) = 1+A(1-z)$. In Fig. 2a we see that for larger values of the parameter $A>0$ the profile becomes taller and also wider. In Fig. 2b we see how the amplitude of solutions decreases with depth $z\in [0, 1]$ like $(Az)^{-1}$ according to (27)
In Fig. 3a we see a plot of the density profile $\rho (z) = a_0-a_1\arctan\left(a_2(z-a_3)\right)$ defined in (29) for suitable choices of parameter values $a_i\in\mathbb{R}$ to model a density increase of $1\%$ from surface to bed. Fig. 3b shows a schematic representation of the fact that the amplitude of solitary wave solutions $\phi(\xi-c\tau, z)$ decays with depth inversely proportional to $\rho(z)$, cf. (27)
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