Storms propagate over the ocean and create moving patches of strong winds that generate swell systems. Here, we describe the dynamics of wave generation under a moving storm by using a simple parametric model of wave development, forced by a temporally- and spatially-varying moving wind field. This framework reveals how surface winds under moving storms determine the origin and amplitude of swell events. Swell systems are expected to originate from locations different than the moving high-wind forcing regions. This is confirmed by a physically-informed optimization method that back-triangulates the common source locations of swell using their dispersion slopes, simultaneously measured at five wave-buoy locations. Hence, the parametric moving fetch model forced with reanalysis winds can predict the displacement between the highest winds and the observed swell source area when forced with reanalysis winds. The model further shows that the storm’s peak wind speed is the key factor determining swell energy since it determines surface wind gradients that lead to the spatial convergence of wave energy into a much smaller area than the wind fetch. This spatial wave energy convergence implies enhanced wave energy dissipation in this focusing area, slightly displaced from the maximum wind locations. This analysis provides an improved understanding of fetches for extra-tropical swell systems and may help to identify biases in swell forecast models, air-sea fluxes, and upper-ocean mixing estimations.

2D-parametric model is used to simulate waves under Tropical Cyclones (TCs). Set of equations describing either wind waves development and swell evolution, is solved using method of characteristics. Wave-rays patterns provide efficient visualization on how wave trains develop and travel through TC varying wind field and leave storm area as swell. The superposition of wave-trains rays exhibits coherent spatial patterns of significant wave height, peak wavelength and direction, depending on TC characteristics, - maximal wind speed (um), radius (Rm), and translation velocity (V). Group velocity resonance leads to appearance of waves with abnormal energy between the TC right and front sectors, further outrunning as swell through the TC front sector. Yet, when TC translation velocity exceeds a threshold value, waves cannot reach group velocity resonance, and travelling backwards, form a wake of swell systems trailing the forward moving TC. 2D-parametric model solutions are parameterized using 2D self-similar universal functions. Comparisons between self-similar solutions and measurements, demonstrate excellent agreement to warrant their use for scientific and practical applications. Self-similar solutions provide immediate estimates of azimuthal-radial distributions of wave parameters under TCs, solely characterized by arbitrary sets of um, Rm and V conditions. Self-similar solutions clearly divide TCs between slow TCs fulfilling conditions Rm/Lcr>1, and fast TCs corresponding to Rm/Lcr <1, where Lcr is a critical fetch. The region around Rm/Lc = 1 corresponds to the group velocity resonance conditions, leading to the largest possible waves generated by a TC.

A fully consistent 2D parametric model of waves development under spatially and temporally varying winds is suggested. The 2D model is based on first-principle conservation equations, consistently constrained by self-similar fetch-laws. Derived coupled equations written in the characteristic form provide practical means to rapidly assess how the energy, frequency and direction of dominant surface waves are distributed under varying wind forcing. For young waves, non-linear interactions are essential to drive the peak frequency downshift, and the wind energy input and wave breaking dissipation are the governing sources of the wave energy evolution. With a prescribed wind wave growth rate, proportional to ustar/c squared, wave breaking dissipation becomes a power-function of the dominant wave slope. Under uniform wind conditions, this growth rate imposes solutions for peak frequency and energy development to follow fetch-laws, with exponents q=-1/4 p=3/4 correspondingly. This set of exponents recovers the Toba’s laws, and imposes the wave breaking exponent equal to 3. A smooth transition from wind driven seas to swell is obtained. Varying wind direction is the only source to drive spectral peak direction changes. This can lead to occurrence of focusing/defocusing wave groups and formation of areas where wave-rays merge and cross. Solutions predict significant (but finite) local enhancements of the energy. Further propagating, wave rays diverge, leading to wave attenuation away from the storm area