Jing Yu, Hui Liu, Jian Song*
College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, Inner Mongolia, China.
*Corresponding author:Jian Song
The National Natural Science Foundation of China (Grant No. 42275052); The Inner Mongolia Autonomous Region “Talents Revitalizing Inner Mongolia” Project Team Initiative (Grant No. 2025TEL06); The Natural Science Foundation of Inner Mongolia (Grant No. 2024MS04015); The Autonomous Region Colleges and Universities Basic Research Operating Expenses Project (Grant No. ZTY2024081).
Abstract
Based on the classical Euler equations for incompressible fluids, geometric corrections are added to consider the deviation of the free surface from a spherical Earth under gravity and the effect of the full Coriolis force. This derivation leads to the governing equations for equatorial flow under the modified β-plane approximation. The equation contains nonlinear terms, which pose significant challenges in solving it. By using the exact solution of the governing equations in Lagrangian coordinates and systematically analyzing the matrices, it is shown that this flow is positively pressured. Incorporating the dynamical constraints at the free surface, the exact solution is substituted into the governing equations to obtain the partial derivatives of pressure Px, Py, Pz, which are then converted into Lagrangian coordinates Pa, Pb, based on Clairaut’s theorem (stating that continuous differentiable functions have equal second-order mixed partial derivatives, Pab = Pba), an explicit dispersion relation consistent with the exact solution is derived via this equality of pressure derivatives. The resulting findings provide an analytical foundation for the instability analysis of equatorial geophysical waves.
References
[1] Constantin A, Johnson RS. The dynamics of waves interacting with the Equatorial Undercurrent. Geophys Astrophys Fluid Dyn. 2015;109(4):311-58. doi:10.1080/03091929.2015.1066785.
[2] Constantin A. Some three-dimensional nonlinear equatorial flows. J Phys Oceanogr. 2013;43:165-75.
[3] Constantin A, Johnson RS. On the modelling of large-scale atmospheric flow. J Differ Equ. 2021;285:751-98.
doi:10.1016/j.jde.2021.02.051.
[4] Cushman-Roisin B, Beckers JM. Chapter 21—Equatorial dynamics. In: International Geophysics. Vol 101. ; 2011:709-15. doi:10.1016/B978-0-08-093065-8.00021-6.
[5] Henry D. A modified equatorial beta-plane approximation modelling nonlinear wave–current interactions. J Differ Equ. 2017;263(5):2554-66. doi:10.1016/j.jde.2017.04.007.
[6] Benzoni-Gavage S, Aubert S, Coulombel JF. Boundary conditions for Euler equations. AIAA J. 2003;41:56-63. doi:10.2514/2.1923.
[7] Constantin A. An exact solution for equatorially trapped waves. J Geophys Res Oceans. 2012;117:C05029. doi:10.1029/2012JC007879.
[8] Chu J, Ionescu-Kruse D, Yang Y. Exact solution and instability for geophysical waves with centripetal forces and at arbitrary latitude. J Math Fluid Mech. 2019;21:19. doi:10.1007/s00021-019-0436-4.
[9] Henry D. Exact solutions modelling nonlinear atmospheric gravity waves. J Math Fluid Mech. 2024;26:6.
doi:10.1007/s00021-023-00795-8.
[10] Mai XL, Li W. Exact solutions for the Schrödinger equation with higher-order dispersive terms and cubic-quintic nonlinear terms. J Beijing Univ Chem Technol. 2021;48:114-20.
[11] Yao F, Liu KJ, Yu XD, et al. Kinetic dispersion relation of fast magnetosonic waves and its impact on the scattering of radiation belt electrons. Geophys Res Lett. 2023;50:103292. doi:10.1029/2023GL103292.
[12] Smith CJ, Kramer RJ, Myhre G, et al. Effective radiative forcing and adjustments in CMIP6 models. Atmos Chem Phys. 2020;20(16):9591-618. doi:10.5194/acp-20-9591-2020.
[13] Henry D, Martin CI. Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates. Dyn Partial Differ Equ. 2018;15(4):337-49. doi:10.4310/DPDE.2018.v15.n4.a3.
How to cite this paper
Exact Nonlinear Equatorially Trapped Waves in a Modified β-plane Approximation for Atmospheric Flows
How to cite this paper: Jing Yu, Hui Liu, Jian Song. (2025) Exact Nonlinear Equatorially Trapped Waves in a Modified β-plane Approximation for Atmospheric Flows. Journal of Applied Mathematics and Computation, 9(4), 224-232.
DOI: http://dx.doi.org/10.26855/jamc.2025.12.001