-
Previous Article
Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry
- DCDS Home
- This Issue
-
Next Article
Livšic theorem for banach rings
Exact azimuthal internal waves with an underlying current
Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan |
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.
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. 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. |
[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. 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. |
[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). 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. |
[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. 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. |
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. |
[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. 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. |
[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. 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. |
[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). 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. |
[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. 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. |
[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
Tools
Metrics
Other articles
by authors
[Back to Top]