  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.
