We present a self‐consistent analytic theory of the wind‐driven sea—the weak‐turbulent theory. The base statement of the theory is that the four‐wave resonant interactions play the leading role in the energy balance of the wind‐driven sea. We study the exact solution of the stationary wave kinetic equation and the self‐similar solutions of wave kinetic equation both in fetch‐ and duration‐limited cases. The theory makes possible to explain the nature of universal power‐like energy spectra as well as the power‐like dependence of the total wave energy and peak frequency from the fetch and the duration.

We determine asymptotic behavior of the coupling coefficient for four-wave interactions of deep water gravity waves in the limiting case when two wave vectors of interacting waves are small with respect to the other two (“long-short interactions”). It makes possible to find numerically dimensionless Kolmogorov constants for the powerlike Kolmogorov–Zakharov spectra. The results obtained are crusially important for comparison of the weak turbulent theory with the experiments and the natural observations.

The Hasselmann equation for the nonlinear interactions of deep-water gravity waves differs from other four-wave kinetic equations by the interaction coefficient. The explicit formula for this coefficient (e.g. Krasitskii, J. Fluid. Mech., vol. 272, 1994, pp. 1–20; Zakharov, Eur. J. Mech. B/Fluids, vol. 18. issue 3, 1999, pp. 327–344) is of great complexity and leaves its properties obscured. We provide analytical results for the behaviour of the coefficient in different domains. The Phillips curve and discrete interaction approximation-like quadruplets are studied in detail. The coupling coefficient for the long–short wave interactions is calculated and found to be surprisingly small. This smallness greatly reduces the non-locality of the interactions.

The Kolmogorov spectrum of waves on water is a result of cascading energy via four-wave interactions. For this spectrum in the isotropic case, we introduce basic small perturbations and calculate the decrement of their damping depending on the frequency. We confirm the stability of the Kolmogorov spectrum. The calculation results are applicable for analyzing the stability of numerical methods for solving the kinetic equation. We show that using the discrete-interaction approximation strongly reduces the damping in certain cases.

A new algorithm is used for detailed numerical study of the evolution of isotropic swell in a homogeneous ocean. It is shown that the Zakharov-Filonenko spectrum occurs in an explosive manner in a short time. The Kolmogorov constant of the solution is estimated numerically.