  The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is
studied in the limit of strongly local interactions where the usual kinetic
equation, describing the time evolution of the spectral wave-action density,
can be approximated by a PDE. If the wave action is initially compactly
supported in frequency space, it is then redistributed by resonant interactions
producing the usual direct and inverse cascades, leading to the formation of
the K-Z spectra. The emphasis here is on the direct cascade. The evolution
proceeds by the formation of a self-similar front which propagates to the right
leaving a quasi-stationary state in its wake. This front is sharp in the sense
that the solution remains compactly supported until it reaches infinity. If the
energy spectrum has infinite capacity, the front takes infinite time to reach
infinite frequency and leaves the K-Z spectrum in its wake. On the other hand,
if the energy spectrum has finite capacity, the front reaches infinity within a
finite time, t*, and the wake is steeper than the K-Z spectrum. For this case,
the K-Z spectrum is set up from the right after the front reaches infinity. The
slope of the solution in the wake can be related to the speed of propagation of
the front. It is shown that the anomalous slope in the finite capacity case
corresponds to the unique front speed which ensures that the front tip contains
a finite amount of energy as the connection to infinity is made. We also
introduce, for the first time, the notion of entropy production in wave
turbulence and show how it evolves as the system approaches the stationary K-Z
spectrum.
