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September  2019, 39(9): 5171-5183. doi: 10.3934/dcds.2019210

## Nonhydrostatic Pollard-like internal geophysical waves

 School of Mathematical Sciences, University College Cork, Western Road, Cork, Ireland

Received  August 2018 Revised  March 2019 Published  May 2019

We present a new exact and explicit Pollard-like solution describing internal water waves representing the oscillation of the thermocline in a nonhydrostatic model. The derived solution is a modification of Pollard's surface wave solution in order to describe internal water waves at general latitudes. The novelty of this model consists in the embodiment of transitional layers beneath the thermocline. We present a Lagrangian analysis of the nonlinear internal water waves and we show the existence of two modes of the wave motion.

Citation: Mateusz Kluczek. Nonhydrostatic Pollard-like internal geophysical waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5171-5183. doi: 10.3934/dcds.2019210
##### References:
 [1] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, UK, 2006. doi: 10.1017/CBO9780511734939. [2] J. P. Boyd, Dynamics of the Equatorial Ocean, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 2018. [3] A. Constantin, On the deep water wave motion, Journal of Physics A: Mathematical and General, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313. [4] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, volume 81., Society for Industrial and Applied Mathematics, Philadelphia, USA, 2011. CBMS-NSF Regional Conference Series in Applied Mathematics. doi: 10.1137/1.9781611971873. [5] A. Constantin, On the modelling of equatorial waves, Geophysical Research Letters, 39 (2012), 1-4. doi: 10.1029/2012GL051169. [6] A. Constantin, Some three-dimensional nonlinear equatorial flows, Journal of Physical Oceanography, 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1. [7] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, Journal of Physical Oceanography, 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1. [8] A. Constantin and P. Germain, Instability of some equatorially trapped waves, Journal of Geophysical Research: Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. [9] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophysical & Anstrophysical Fluid Dynamics, 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785. [10] 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, Physics Letters, Section A: General, Atomic and Solid State Physics, 380 (2016), 3007-3012. doi: 10.1016/j.physleta.2016.07.036. [11] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, Journal of Physical Oceanography, 46 (2016), 1935-1945. doi: 10.1175/JPO-D-15-0205.1. [12] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, Journal of Physical Oceanography, 46 (2016), 3585-3594. doi: 10.1175/JPO-D-16-0121.1. [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, Physics of Fluids, 29 (2017), 1-21. [14] A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, Journal of Fluid Mechanics, 820 (2017), 511-528. doi: 10.1017/jfm.2017.223. [15] B. Cushman-Roisin and J.-M. Becker, Introduction to Geophysical Fluid Dynamics. Physcial and Numerical Aspects, Academic Press, Waltham, USA, 2011. [16] L. Fan, H. Gao and Q. Xiao, An exact solution for geophysical trapped waves in the presence of an underlying current, Dynamics of Partial Differential Equations, 15 (2018), 201-214. doi: 10.4310/DPDE.2018.v15.n3.a3. [17] I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on Geophysical Flows, In S. Friedlander and D. Serre, editors, Handbook of Mathematical Fluid Dynamics, volume 4, pages 201–329. North-Holland, 2007. doi: 10.1016/S1874-5792(07)80009-7. [18] T. Garrison and R. Ellis, Oceanography: An Invitation to Marine Science, Cengage Learning, Boston, USA, 2014. [19] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, an Diego, USA, 1982. [20] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, European Journal of Mechanics, B/Fluids, 38 (2013), 18-21. doi: 10.1016/j.euromechflu.2012.10.001. [21] D. Henry, Internal equatorial water waves in the f-plane, Journal of Nonlinear Mathematical Physics, 22 (2015), 499-506. doi: 10.1080/14029251.2015.1113046. [22] D. Henry, Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces, Journal of Fluid Mechanics, 804 (2016), R1, 11 pp. doi: 10.1017/jfm.2016.544. [23] D. Henry, Exact equatorial water waves in the f-plane, Nonlinear Analysis: Real World Applications, 28 (2016), 284-289. doi: 10.1016/j.nonrwa.2015.10.003. [24] D. Henry, On three-dimensional Gerstner-like equatorial water waves, Phylosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170088, 16 pp. doi: 10.1098/rsta.2017.0088. [25] H.-C. Hsu, An exact solution for nonlinear internal equatorial waves in the f-plane approximation, Journal of Mathematical Fluid Mechanics, 16 (2014), 463-471. doi: 10.1007/s00021-014-0168-3. [26] H.-C. Hsu, An exact solution for equatorial waves, Monatshefte für Mathematik, 176 (2015), 143-152. doi: 10.1007/s00605-014-0618-2. [27] D. Ionescu-Kruse, On Pollard's wave solution at the equator, Journal of Nonlinear Mathematical Physics, 22 (2015), 523-530. doi: 10.1080/14029251.2015.1113050. [28] D. Ionescu-Kruse, Instability of Pollard's exact solution for geophysical ocean flows, Physics of Fluids, 28 (2016), 086601. doi: 10.1063/1.4959289. [29] D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phylosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 376 (2017), 20170090, 21 pp. doi: 10.1098/rsta.2017.0090. [30] D. Ionescu-Kruse, A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows, Journal of Differential Equations, 264 (2018), 4650-4668. doi: 10.1016/j.jde.2017.12.021. [31] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phylosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092. [32] W. S. Kessler and M. J. McPhaden, Oceanic equatorial waves and the 1991-93 El Niño, Journal of Climate, 8 (1995), 1757-1774. [33] M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, Journal of Mathematical Fluid Mechanics, 19 (2017), 305-314. doi: 10.1007/s00021-016-0281-6. [34] M. Kluczek, Equatorial water waves with underlying currents in the f-plane approximation, Applicable Analysis, 97 (2018), 1867-1880. doi: 10.1080/00036811.2017.1343466. [35] M. Kluczek, Exact Pollard-like internal water waves, Journal of Nonlinear Mathematical Physics, 26 (2019), 133-146. doi: 10.1080/14029251.2019.1544794. [36] J. N. Moum, J. D. Nash and W. D. Smyth, Narrowband oscillations in the upper equatorial ocean. Part Ⅰ: Interpretation as shear instabilities, Journal of Physical Oceanography, 41 (2011), 397-411. doi: 10.1175/2010JPO4450.1. [37] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag Berlin Heidelberg, Virginia, USA, 1987. [38] R. T. Pollard, Surface waves with rotation: An exact solution, Journal of Geophysical Research, 75 (1970), 5895-5898. doi: 10.1029/JC075i030p05895. [39] V. V. Prasolov, Polynomials, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 2004. doi: 10.1007/978-3-642-03980-5. [40] A. Rodríguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions, Annali di Matematica Pura ed Applicata (1923–), 197 (2018), 1787-1797. doi: 10.1007/s10231-018-0749-5. [41] A. Rodríguez-Sanjurjo, Internal equatorial water waves and wave-current interactions in the f-plane, Monatshefte für Mathematik, 186 (2018), 685-701. doi: 10.1007/s00605-017-1052-z. [42] R. Stuhlmeier, Internal Gerstner waves: Applications to dead water, Applicable Analysis, 93 (2014), 1451-1457. doi: 10.1080/00036811.2013.833609. [43] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press, New York, USA, 2006. [44] F. J. von Gerstner, Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Annalen der Physik, 32 (1809), 412-445.

show all references

##### References:
 [1] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, UK, 2006. doi: 10.1017/CBO9780511734939. [2] J. P. Boyd, Dynamics of the Equatorial Ocean, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 2018. [3] A. Constantin, On the deep water wave motion, Journal of Physics A: Mathematical and General, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313. [4] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, volume 81., Society for Industrial and Applied Mathematics, Philadelphia, USA, 2011. CBMS-NSF Regional Conference Series in Applied Mathematics. doi: 10.1137/1.9781611971873. [5] A. Constantin, On the modelling of equatorial waves, Geophysical Research Letters, 39 (2012), 1-4. doi: 10.1029/2012GL051169. [6] A. Constantin, Some three-dimensional nonlinear equatorial flows, Journal of Physical Oceanography, 43 (2013), 165-175. doi: 10.1175/JPO-D-12-062.1. [7] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, Journal of Physical Oceanography, 44 (2014), 781-789. doi: 10.1175/JPO-D-13-0174.1. [8] A. Constantin and P. Germain, Instability of some equatorially trapped waves, Journal of Geophysical Research: Oceans, 118 (2013), 2802-2810. doi: 10.1002/jgrc.20219. [9] A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophysical & Anstrophysical Fluid Dynamics, 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785. [10] 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, Physics Letters, Section A: General, Atomic and Solid State Physics, 380 (2016), 3007-3012. doi: 10.1016/j.physleta.2016.07.036. [11] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, Journal of Physical Oceanography, 46 (2016), 1935-1945. doi: 10.1175/JPO-D-15-0205.1. [12] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, Journal of Physical Oceanography, 46 (2016), 3585-3594. doi: 10.1175/JPO-D-16-0121.1. [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, Physics of Fluids, 29 (2017), 1-21. [14] A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, Journal of Fluid Mechanics, 820 (2017), 511-528. doi: 10.1017/jfm.2017.223. [15] B. Cushman-Roisin and J.-M. Becker, Introduction to Geophysical Fluid Dynamics. Physcial and Numerical Aspects, Academic Press, Waltham, USA, 2011. [16] L. Fan, H. Gao and Q. Xiao, An exact solution for geophysical trapped waves in the presence of an underlying current, Dynamics of Partial Differential Equations, 15 (2018), 201-214. doi: 10.4310/DPDE.2018.v15.n3.a3. [17] I. Gallagher and L. Saint-Raymond, On the influence of the Earth's rotation on Geophysical Flows, In S. Friedlander and D. Serre, editors, Handbook of Mathematical Fluid Dynamics, volume 4, pages 201–329. North-Holland, 2007. doi: 10.1016/S1874-5792(07)80009-7. [18] T. Garrison and R. Ellis, Oceanography: An Invitation to Marine Science, Cengage Learning, Boston, USA, 2014. [19] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, an Diego, USA, 1982. [20] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, European Journal of Mechanics, B/Fluids, 38 (2013), 18-21. doi: 10.1016/j.euromechflu.2012.10.001. [21] D. Henry, Internal equatorial water waves in the f-plane, Journal of Nonlinear Mathematical Physics, 22 (2015), 499-506. doi: 10.1080/14029251.2015.1113046. [22] D. Henry, Equatorially trapped nonlinear water waves in a β-plane approximation with centripetal forces, Journal of Fluid Mechanics, 804 (2016), R1, 11 pp. doi: 10.1017/jfm.2016.544. [23] D. Henry, Exact equatorial water waves in the f-plane, Nonlinear Analysis: Real World Applications, 28 (2016), 284-289. doi: 10.1016/j.nonrwa.2015.10.003. [24] D. Henry, On three-dimensional Gerstner-like equatorial water waves, Phylosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170088, 16 pp. doi: 10.1098/rsta.2017.0088. [25] H.-C. Hsu, An exact solution for nonlinear internal equatorial waves in the f-plane approximation, Journal of Mathematical Fluid Mechanics, 16 (2014), 463-471. doi: 10.1007/s00021-014-0168-3. [26] H.-C. Hsu, An exact solution for equatorial waves, Monatshefte für Mathematik, 176 (2015), 143-152. doi: 10.1007/s00605-014-0618-2. [27] D. Ionescu-Kruse, On Pollard's wave solution at the equator, Journal of Nonlinear Mathematical Physics, 22 (2015), 523-530. doi: 10.1080/14029251.2015.1113050. [28] D. Ionescu-Kruse, Instability of Pollard's exact solution for geophysical ocean flows, Physics of Fluids, 28 (2016), 086601. doi: 10.1063/1.4959289. [29] D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phylosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 376 (2017), 20170090, 21 pp. doi: 10.1098/rsta.2017.0090. [30] D. Ionescu-Kruse, A three-dimensional autonomous nonlinear dynamical system modelling equatorial ocean flows, Journal of Differential Equations, 264 (2018), 4650-4668. doi: 10.1016/j.jde.2017.12.021. [31] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phylosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092. [32] W. S. Kessler and M. J. McPhaden, Oceanic equatorial waves and the 1991-93 El Niño, Journal of Climate, 8 (1995), 1757-1774. [33] M. Kluczek, Exact and explicit internal equatorial water waves with underlying currents, Journal of Mathematical Fluid Mechanics, 19 (2017), 305-314. doi: 10.1007/s00021-016-0281-6. [34] M. Kluczek, Equatorial water waves with underlying currents in the f-plane approximation, Applicable Analysis, 97 (2018), 1867-1880. doi: 10.1080/00036811.2017.1343466. [35] M. Kluczek, Exact Pollard-like internal water waves, Journal of Nonlinear Mathematical Physics, 26 (2019), 133-146. doi: 10.1080/14029251.2019.1544794. [36] J. N. Moum, J. D. Nash and W. D. Smyth, Narrowband oscillations in the upper equatorial ocean. Part Ⅰ: Interpretation as shear instabilities, Journal of Physical Oceanography, 41 (2011), 397-411. doi: 10.1175/2010JPO4450.1. [37] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag Berlin Heidelberg, Virginia, USA, 1987. [38] R. T. Pollard, Surface waves with rotation: An exact solution, Journal of Geophysical Research, 75 (1970), 5895-5898. doi: 10.1029/JC075i030p05895. [39] V. V. Prasolov, Polynomials, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 2004. doi: 10.1007/978-3-642-03980-5. [40] A. Rodríguez-Sanjurjo, Global diffeomorphism of the Lagrangian flow-map for Pollard-like solutions, Annali di Matematica Pura ed Applicata (1923–), 197 (2018), 1787-1797. doi: 10.1007/s10231-018-0749-5. [41] A. Rodríguez-Sanjurjo, Internal equatorial water waves and wave-current interactions in the f-plane, Monatshefte für Mathematik, 186 (2018), 685-701. doi: 10.1007/s00605-017-1052-z. [42] R. Stuhlmeier, Internal Gerstner waves: Applications to dead water, Applicable Analysis, 93 (2014), 1451-1457. doi: 10.1080/00036811.2013.833609. [43] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press, New York, USA, 2006. [44] F. J. von Gerstner, Theorie der wellen samt einer daraus abgeleiteten theorie der deichprofile, Annalen der Physik, 32 (1809), 412-445.
The nonhydrostatic model and its flow regions for fixed $y$ and $\phi$. The thermocline separates two layers of ocean water with different but constant densities $\rho_+>\rho_0$ in a stable stratification [15,18,43]. The thermocline is described by a trochoid propagating with the phase speed $c$. The transitional layer $\cal{T}(t)$ provide a transition from the wave motion region to the motionless abyssal deep-water region of the ocean. The schematic is presented for fixed $\tilde{f} = f/\hat{f} = \tan\phi$
The angle of the inclination of particles' orbits is $\arctan(-d/a)$ [14,35] and is increasing with the latitude resulting in the three-dimensional profile of the internal water wave [35] (see figure 3). The upper and lower interface of the transitional layer becomes also inclined at the angle $\phi$ with respect to the local meridional axis. The inclination of the orbits and the interfaces is the result of the Earth's constant rotation
The schematic three-dimensional profile of the internal water waves. The cross-wave tilt is a result of Earth's rotation [14,35,38]. At the equator the wave profile is in the vertical plane [3]
The plot of the polynomial P(X) evaluated at 45$^\circ$ N for $(\rho_+-\rho_0)/\rho_0 = 4\times 10^{-3}$, $k = 6.28\times 10^{-2}$ $m^{-1}$. The upper plot shows two roots of order $O(1)$, whereas the lower plot presents two roots of order $O(\epsilon)$
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