Hasselmann's kinetic equation for wind waves

Definition

Consider surface gravity waves on deep water, with wavevector $k \in \R^2$ and angular frequency $\omega(k) = \sqrt{g \abs{k}}$ given by the dispersion relation. Describe the sea by its wave action spectrum $N(k, x, t) \ge 0$, the energy spectrum in wavenumber space divided by $\omega(k)$. Write $N_j = N(k_j)$ and $\omega_j = \omega(k_j)$ for $j = 1, 2, 3$, and $N = N(k)$, $\omega = \omega(k)$. Hasselmann's kinetic equation is $$\frac{\partial N}{\partial t} + c_g \cdot \nabla_x N = S_{nl} + S_{in} + S_{ds}.$$ Here $c_g = \nabla_k \omega$ is the group velocity, and $S_{in}$ and $S_{ds}$ are the source terms for wind input and dissipation. The transfer by resonant four-wave interactions is $$S_{nl}(k) = \int T(k, k_1, k_2, k_3) \, \delta(k + k_1 - k_2 - k_3) \, \delta(\omega + \omega_1 - \omega_2 - \omega_3) \, \big[ N_1 N_2 N_3 + N N_2 N_3 - N N_1 N_3 - N N_1 N_2 \big] \, dk_1 \, dk_2 \, dk_3.$$ The kernel $T \ge 0$ is computed from the nonlinear equations of water waves.