  As presented in Annenkov & Shrira (2009), when a surface gravity wave field
is subjected to an abrupt perturbation of external forcing, its spectrum
evolves on a ``fast'' dynamic time scale of $O(\varepsilon^{-2})$, with
$\varepsilon$ a measure of wave steepness. This observation poses a challenge
to wave turbulence theory that predicts an evolution with a kinetic time scale
of $O(\varepsilon^{-4})$. We revisit this unresolved problem by studying the
same situation in the context of a one-dimensional Majda-McLaughlin-Tabak (MMT)
equation with gravity wave dispersion relation. Our results show that the
kinetic and dynamic time scales can both be realised, with the former and
latter occurring for weaker and stronger forcing perturbations, respectively.
The transition between the two regimes corresponds to a critical forcing
perturbation, with which the spectral evolution time scale drops to the same
order as the linear wave period (of some representative mode). Such fast
spectral evolution is mainly induced by a far-from-stationary state after a
sufficiently strong forcing perturbation is applied. We further develop a
set-based interaction analysis to show that the inertial-range modal evolution
in the studied cases is dominated by their (mostly non-local) interactions with
the low-wavenumber ``condensate'' induced by the forcing perturbation. The
results obtained in this work should be considered to provide significant
insight into the original gravity wave problem.
