  In wave turbulence, it has been believed that statistical properties are well
described by the weak turbulence theory, in which nonlinear interactions among
wavenumbers are assumed to be small. In the weak turbulence theory, separation
of linear and nonlinear time scales derived from the weak nonlinearity is also
assumed. However, the separation of the time scales is often violated even in
weak turbulent systems where the nonlinear interactions are actually weak. To
get rid of this inconsistency, closed equations are derived without assuming
the separation of the time scales in accordance with Direct-Interaction
Approximation (DIA), which has been successfully applied to Navier--Stokes
turbulence. The kinetic equation of the weak turbulence theory is recovered
from the DIA equations if the weak nonlinearity is assumed as an additional
assumption. It suggests that the DIA equations is a natural extension of the
conventional kinetic equation to not-necessarily-weak wave turbulence.
