A nonlocal shallow-water model arising from the full water waves with the Coriolis effect

TL;DR

This paper derives a nonlocal shallow-water model incorporating Earth's rotation effects, analyzing wave-breaking phenomena and the influence of Coriolis force using advanced mathematical techniques.

Contribution

It introduces a novel bi-Hamiltonian shallow-water model with nonlocal nonlinearities that accounts for Earth's rotation, extending previous models with longer-term solution accuracy.

Findings

The model accurately predicts wave behavior over longer timescales.

Coriolis force influences wave-breaking and blow-up criteria.

The solution's deviation relates to horizontal velocity at a specific depth.

Abstract

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. 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.