Evolution of Wind-Generated Shallow-Water Waves in the Framework of a Modified Kadomtsev–Petviashvili Equation

In a recent paper, denoted by MG24 in this text, we used a modified Korteweg–de Vries (KdV) equation to describe the evolution of wind-driven water wave packets in shallow water. The modifications were several forcing/friction terms describing wave growth due to critical-level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave growth due to turbulent wave stress in the air near the interface, and wave decay due to a turbulent bottom boundary layer. The outcome was a KdV–Burgers type of equation that can be a stable or unstable model depending on the forcing/friction parameters. In most cases that we examined, many solitary waves are generated, suggesting the formation of a soliton gas. In this paper, we extend that model in the horizontal direction transverse to the wind forcing to produce a similarly modified Kadomtsev–Petviashvili equation (KPII for water waves in the absence of surface tension). A modulation theory is described for the cnoidal and solitary wave solutions of the unforced KP equation, focusing on the forcing/friction terms and the transverse dependence. Then, using similar initial conditions to those used in MG24, that is a sinusoidal wave with a slowly varying envelope, but supplemented here with a transverse sinusoidal term, we find through numerical simulations that the radiation field upstream is enhanced, but that a soliton gas still emerges downstream as in MG24.