David
McLaughlin
New York University-Courant Institute
dmac@CIMS.NYU.EDU
Dispersive wave turbulence is studied numerically for a class
of one-dimensional nonlinear wave equations. Both deterministic
and random (white noise in time) forcings are studied. Four
distinct stable spectra are observed -- the direct and inverse
cascades of weak turbulence (WT) theory, thermal equilibrium,
and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each
spectrum can describe long-time behavior, and each can be
only metastable (with quite diverse lifetimes) -- depending
on details of nonlinearity, forcing and dissipation. Cases
of a long-lived MMT transient state decaying to a state with
WT spectra, and vice-versa, are displayed. In the case of
freely decaying turbulence, without forcing, both cascades
of weak turbulence are observed. These WT states constitute
the clearest and most striking numerical observations of WT
spectra to date -- over four decades of energy, and three
decades of spatial, scales. Numerical experiments that study
details of the composition, coexistence, and transition between
spectra are then discussed, including: (i) for deterministic
forcing, sharp distinctions between focusing and defocusing
nonlinearities, including the role of long wave-length instabilities,
localized coherent structures, and chaotic behavior; (ii)
the role of energy growth in time to monitor the selection
of MMT or WT spectra; (iii) a second manifestation of the
MMT spectrum as it describes a self-similar evolution of the
wave, without temporal averaging; (iv) coherent structures
and the evolution of the direct and inverse cascades; and
(v) nonlocality (in k-space) in the transferral process.
