  We study the properties of energy flux in wave turbulence via the
Majda-McLaughlin-Tabak (MMT) equation with a quadratic dispersion relation. One
of our purposes is to resolve the inter-scale energy flux $P$ in the stationary
state to elucidate its distribution and scaling with spectral level. More
importantly, we perform a quartet-level decomposition of $P=\sum_\Omega
P_\Omega$, with each component $P_\Omega$ representing the contribution from
quartet interactions with frequency mismatch $\Omega$, in order to explain the
properties of $P$ as well as study the wave-turbulence closure model. Our
results show that time series of $P$ closely follows a Gaussian distribution,
with its standard deviation several times its mean value $\overline{P}$. This
large standard deviation is shown to mainly result from the fluctuation (in
time) of the quasi-resonances, i.e., $P_{\Omega\neq 0}$. The scaling of
spectral level with $\overline{P}$ exhibits $\overline{P}^{1/3}$ and
$\overline{P}^{1/2}$ at high and low nonlinearity, consistent with the kinetic
and dynamic scalings respectively. The different scaling laws in the two
regimes are explained through the dominance of quasi-resonances ($P_{\Omega\neq
0}$) and exact resonances ($P_{\Omega= 0}$) in the former and latter regimes.
Finally, we investigate the wave-turbulence closure model, which connects
fourth-order correlators to products of pair correlators through a broadening
function $f(\Omega)$, sometimes argued to be a $sinc$ function in the theory.
Our numerical data show that consistent behavior of $f(\Omega)$ can only be
observed upon averaging over a large number of quartets, but with $f(\Omega)$
showing $f\sim 1/\Omega^\beta$ dependence with $\beta$ taking values between
$1.3$ and $1.6$.
