A new nonlinear integral-differential equation describing Rossby waves and its related properties

Abstract

This paper focuses on algebraic Rossby solitary waves models in stratified fluids. Starting with the quasi-geostrophic potential vorticity equation, we present the Boussinesq-intermediate long-wave (Boussinesq-ILW) model for the first time using the multi-scale analysis and the perturbation expansion method. Compared with previous models describing algebraic Rossby solitary waves, the Boussinesq-ILW model is more general: the equation transforms into the Boussinesq-BO equation when h1→∞, and when [Ξ(B)]XXX changes to BXXXX, the model reduces to the Boussinesq equation. The conservation law is important for exploring the properties of the model, therefore we propose several common conservation laws: mass, momentum, and energy conservation. Finally, based on the trial function method, we obtain the solution of the Boussinesq-ILW equation and investigate the wave-wave interaction. The results show that the peak value of the Mach stem increases as the parameter γ increases, but the length becomes shorter and changes more rapidly.