  In this paper, we numerically study the wave turbulence of surface gravity
waves in the framework of Euler equations of the free surface. The purpose is
to understand the variation of the scaling of the spectra with wavenumber $k$
and energy flux $P$ at different nonlinearity levels under different
forcing/free-decay conditions. For all conditions (free decay, narrow- and
broadband forcing) we consider, we find that the spectral forms approach wave
turbulence theory (WTT) solution $S_\eta\sim k^{-5/2}$ and $S_\eta\sim P^{1/3}$
at high nonlinearity levels. With the decrease of nonlinearity level, the
spectra for all cases become steeper, with the narrow-band forcing case
exhibiting the most rapid deviation from WTT. To interpret these spectral
variations, we further investigate two hypothetical and disputable mechanisms
about bound waves and finite-size effect. Through a tri-coherence analysis, we
find that the finite-size effect is present in all cases, which is responsible
for the overall steepening of the spectra and reduced capacity of energy flux
at lower nonlinearity levels. The fraction of bound waves in the domain
generally decreases with the decrease of nonlinearity level, except for the
narrow-band case, which exhibits a transition at some critical nonlinearity
level below which a rapid increase is observed. This increase serves as the
main reason for the fastest deviation from WTT with the decrease of
nonlinearity in the narrow-band forcing case.
