Weak Turbulent Kolmogorov Spectrum for Surface Gravity Waves

We study the long-time evolution of surface gravity waves on deep water excited by a stochastic external force concentrated in moderately small wave numbers. We numerically implemented the primitive Euler equations for the potential flow of an ideal fluid with free surface written in Hamiltonian canonical variables, using the expansion of the Hamiltonian in powers of nonlinearity of terms up to fourth order. We show that because of nonlinear interaction processes a stationary Fourier spectrum of a surface elevation close to $⟨|{\eta }_k{|}^2⟩\sim k^{-7/2}$ is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy.