A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

• Published in: Journal of mathematical fluid mechanics Vol. 21; no. 2; pp. 1 - 29
• Main Authors: Gui, Guilong, Liu, Yue, Sun, Junwei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2019; Springer Nature B.V
• Subjects: Approximation; Asymptotic methods; Classical and Continuum Physics; Computational fluid dynamics; Coriolis effect; Coriolis force; Earth rotation; Euler-Lagrange equation; Fluid mechanics; Fluid- and Aerodynamics; Free surfaces; Mathematical analysis; Mathematical Methods in Physics; Method of characteristics; Physical simulation; Physics; Physics and Astronomy; Rotation; Theoretical mathematics; Two dimensional models; Water waves; Wave breaking; Wave propagation; Rotation-Camassa–Holm equation; 35Q53; 35B30; shallow water; Coriolis effect; wave breaking; 35G25
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-019-0432-7

In the present study a mathematical model of the equatorial water waves propagating mainly in one direction with the effect of Earth’s rotation is derived by the formal asymptotic procedures in the equatorial zone. Such a model equation is analogous to the Camassa–Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.