# American Institute of Mathematical Sciences

August  2017, 37(8): 4391-4398. doi: 10.3934/dcds.2017188

## Exact azimuthal internal waves with an underlying current

 Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received  January 2017 Revised  May 2017 Published  April 2017

In this paper, we present an explicit and exact solution of the nonlinear governing equations including Coriolis and centripetal terms for internal azimuthal waves with a uniform current in the $\beta$-plane approximation near the equator. This solution is described in the Lagrangian framework. The unidirectional azimuthal internal trapped are symmetric about the equator and propagate eastward above the thermocline and beneath the near-surface layer.

Citation: Hung-Chu Hsu. Exact azimuthal internal waves with an underlying current. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4391-4398. doi: 10.3934/dcds.2017188
##### References:
 [1] A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.  doi: 10.1088/0305-4470/34/45/311. [2] A. Constantin, The trajectories of particles in Stokes waves, Int. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [3] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299. [4] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.-Oceans, 117 (2012), C05029.  doi: 10.1029/2012JC007879. [5] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1. [6] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.-Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219. [7] A. Constantin, Some nonlinear, Equatorial trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. [8] 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. [9] 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. [10] A. V. Fedorov and W. K. Melville, Kelvin fronts on the equatorial thermocline, J. Phys. Oceanogr., 30 (2000), 1692-1705.  doi: 10.1175/1520-0485(2000)030<1692:KFOTET>2.0.CO;2. [11] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile (in German), Ann. Phys., 2 (1809), 412-445. [12] R. J. Greatbatch, Kelvin wave fronts, Rossby solitary waves and the nonlinear spin-up of the equatorial oceans, J. Geophys. Res., 90 (1985), 9097-9107.  doi: 10.1029/JC090iC05p09097. [13] D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. Art., 2006 (2006), ID23405, 13pp.  doi: 10.1155/IMRN/2006/23405. [14] 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. [15] D. Henry, Internal equatorial water waves in the f-plane, J. Nonlinear Mathematical Physics, 22 (2015), 499-506.  doi: 10.1080/14029251.2015.1113046. [16] D. Henry and H. C. Hsu, Instability of internal equatorial water waves, J. Differ. Equ., 258 (2015), 1015-1024.  doi: 10.1016/j.jde.2014.08.019. [17] D. Henry, Equatorially trapped nonlinear water waves in the β-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp.  doi: 10.1017/jfm.2016.544. [18] H. C. Hsu, Some nonlinear internal equatorial flow, Nonlinear Anal. Real World Appl., 18 (2014), 69-74.  doi: 10.1016/j.nonrwa.2013.12.011. [19] H. C. Hsu, An exact solution for nonlinear internal Equatorial waves in the f-plane approximation, J. Math. Fluid Mech., 16 (2014), 463-471.  doi: 10.1007/s00021-014-0168-3. [20] H. C. Hsu, Some nonlinear internal equatorial waves with a strong underlying current, Appl. Math. Lett., 34 (2014), 1-6.  doi: 10.1016/j.aml.2014.03.005. [21] H. C. Hsu, An exact solution for equatorial waves, Monatsh Math., 175 (2015), 143-152.  doi: 10.1007/s00605-014-0618-2. [22] H. C. Hsu and C. I. Martin, Free-surface capillary-gravity azimuthal equatorial flows, Nonlinear Anal., 144 (2016), 1-9.  doi: 10.1016/j.na.2016.05.019. [23] H. C. Hsu, Exact steady azimuthal equatorial internal waves in rotational stratified fluids, Preprint J. Math. Fluid Mech., (2017). [24] D. Ionescu-Kruse, An exact solution for geophysical edge waves in the f-plane approximation, Nonlinear Anal. Real World Appl., 24 (2015), 190-195.  doi: 10.1016/j.nonrwa.2015.02.002. [25] T. Izumo, The Equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the El Nino events in the tropical Pacific Ocean, Ocean Dyn., 55 (2005), 110-123. [26] J. N. Moum, J. D. Nash and W. D. Smyth, Narrowband oscillations in the upper equatorial ocean. Part Ⅰ: Interpretation as shear instability, J. Phys. Oceanogr., 41 (2011), 397-411.  doi: 10.1175/2010JPO4450.1.

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##### References:
 [1] A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731.  doi: 10.1088/0305-4470/34/45/311. [2] A. Constantin, The trajectories of particles in Stokes waves, Int. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5. [3] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299. [4] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.-Oceans, 117 (2012), C05029.  doi: 10.1029/2012JC007879. [5] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1. [6] A. Constantin and P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.-Oceans, 118 (2013), 2802-2810.  doi: 10.1002/jgrc.20219. [7] A. Constantin, Some nonlinear, Equatorial trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr., 44 (2014), 781-789. [8] 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. [9] 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. [10] A. V. Fedorov and W. K. Melville, Kelvin fronts on the equatorial thermocline, J. Phys. Oceanogr., 30 (2000), 1692-1705.  doi: 10.1175/1520-0485(2000)030<1692:KFOTET>2.0.CO;2. [11] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile (in German), Ann. Phys., 2 (1809), 412-445. [12] R. J. Greatbatch, Kelvin wave fronts, Rossby solitary waves and the nonlinear spin-up of the equatorial oceans, J. Geophys. Res., 90 (1985), 9097-9107.  doi: 10.1029/JC090iC05p09097. [13] D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. Art., 2006 (2006), ID23405, 13pp.  doi: 10.1155/IMRN/2006/23405. [14] 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. [15] D. Henry, Internal equatorial water waves in the f-plane, J. Nonlinear Mathematical Physics, 22 (2015), 499-506.  doi: 10.1080/14029251.2015.1113046. [16] D. Henry and H. C. Hsu, Instability of internal equatorial water waves, J. Differ. Equ., 258 (2015), 1015-1024.  doi: 10.1016/j.jde.2014.08.019. [17] D. Henry, Equatorially trapped nonlinear water waves in the β-plane approximation with centripetal forces, J. Fluid Mech., 804 (2016), R1, 11pp.  doi: 10.1017/jfm.2016.544. [18] H. C. Hsu, Some nonlinear internal equatorial flow, Nonlinear Anal. Real World Appl., 18 (2014), 69-74.  doi: 10.1016/j.nonrwa.2013.12.011. [19] H. C. Hsu, An exact solution for nonlinear internal Equatorial waves in the f-plane approximation, J. Math. Fluid Mech., 16 (2014), 463-471.  doi: 10.1007/s00021-014-0168-3. [20] H. C. Hsu, Some nonlinear internal equatorial waves with a strong underlying current, Appl. Math. Lett., 34 (2014), 1-6.  doi: 10.1016/j.aml.2014.03.005. [21] H. C. Hsu, An exact solution for equatorial waves, Monatsh Math., 175 (2015), 143-152.  doi: 10.1007/s00605-014-0618-2. [22] H. C. Hsu and C. I. Martin, Free-surface capillary-gravity azimuthal equatorial flows, Nonlinear Anal., 144 (2016), 1-9.  doi: 10.1016/j.na.2016.05.019. [23] H. C. Hsu, Exact steady azimuthal equatorial internal waves in rotational stratified fluids, Preprint J. Math. Fluid Mech., (2017). [24] D. Ionescu-Kruse, An exact solution for geophysical edge waves in the f-plane approximation, Nonlinear Anal. Real World Appl., 24 (2015), 190-195.  doi: 10.1016/j.nonrwa.2015.02.002. [25] T. Izumo, The Equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the El Nino events in the tropical Pacific Ocean, Ocean Dyn., 55 (2005), 110-123. [26] J. N. Moum, J. D. Nash and W. D. Smyth, Narrowband oscillations in the upper equatorial ocean. Part Ⅰ: Interpretation as shear instability, J. Phys. Oceanogr., 41 (2011), 397-411.  doi: 10.1175/2010JPO4450.1.
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