<p>The study is based on a simple parametric model, which is an extension of the self-similarity theory for surface waves generated by a wind field. According to the original similarity concept, the development of wind waves can be fully described using the scale of the fetch length (or time) and wind velocity. The aim of the work is to develop a parametric model to describe the wave generation in arbitrary spatio-temporal wind field. We assume that in this case similarity laws are also fulfilled, i.e., the rate of spectral the peak frequency and wave energy change is completely determined by the wave age. The source function is written in a form providing the stationary solution that corresponds to the well-known fetch law, confirmed in numerous experiments.</p><p>In order to extend the equations to the two-dimensional case, when the wind change occurs in both directions, it is assumed that the relations stay valid if the wind speed is replaced by its component in spectral peak direction. In this case, the system of equations should be supplemented by an expression for the evolution of spectral peak direction, describing its adaptating to the direction of non-uniform wind.</p><p>The algorithm for solving the complete system of equations describing the evolution of wave height, spectral peak frequency, its propagation direction and focusing/defocusing of wave energy, is based on the method of characteristics. To simulate the evolution of waves in a hurricane, we use the calculation in a non-stationary reference system associated with the hurricane. Coordinates, wave peak frequency, energy and direction are calculated along ray trajectory at every discrete time moment. To increase the stability of the numerical scheme, an implicit 4th-order Runge-Kutta method is used.</p><p>Test calculations were carried out for the case of the wave development from the coast with a uniform wind and then for an inhomogeneous cyclonic wind field for different hurricane speeds. The calculations reproduce the anisotropy of the energy distribution inside the hurricane and the effect of wave trapping by a moving cyclone. A comparison of the results with available field measurements of wave parameters in tropical cyclones showed their good agreement. The proposed algorithm can be used in wave forecast models and can serve for deeper understanding the wave field formation in extreme conditions.</p><p>The work was supported by Russian Science Foundation via grant 17-77-30019 and the Ministry of Education and Science of the Russian Federation under the State Assignment No. 0827-2018-0003.</p>
