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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

A. Constantin, The trajectories of particles in Stokes waves, Int. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[3]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.  Google Scholar

[4]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.-Oceans, 117 (2012), C05029.  doi: 10.1029/2012JC007879.  Google Scholar

[5]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile (in German), Ann. Phys., 2 (1809), 412-445.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

H. C. Hsu, An exact solution for equatorial waves, Monatsh Math., 175 (2015), 143-152.  doi: 10.1007/s00605-014-0618-2.  Google Scholar

[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.  Google Scholar

[23]

H. C. Hsu, Exact steady azimuthal equatorial internal waves in rotational stratified fluids, Preprint J. Math. Fluid Mech., (2017).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

J. N. MoumJ. 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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. Constantin, The trajectories of particles in Stokes waves, Int. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[3]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.  Google Scholar

[4]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.-Oceans, 117 (2012), C05029.  doi: 10.1029/2012JC007879.  Google Scholar

[5]

A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr., 43 (2013), 165-175.  doi: 10.1175/JPO-D-12-062.1.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile (in German), Ann. Phys., 2 (1809), 412-445.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

H. C. Hsu, An exact solution for equatorial waves, Monatsh Math., 175 (2015), 143-152.  doi: 10.1007/s00605-014-0618-2.  Google Scholar

[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.  Google Scholar

[23]

H. C. Hsu, Exact steady azimuthal equatorial internal waves in rotational stratified fluids, Preprint J. Math. Fluid Mech., (2017).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

J. N. MoumJ. 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.  Google Scholar

[1]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147

[2]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[3]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[4]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159

[5]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[6]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[7]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[8]

Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094

[9]

Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020055

[10]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

[11]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[12]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106

[13]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[14]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[15]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[16]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[17]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047

[18]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[19]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364

[20]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (50)
  • HTML views (56)
  • Cited by (0)

Other articles
by authors

[Back to Top]