A ONE-DIMENSIONAL MODEL FOR DISPERSIVE WAVE TURBULENCE

A family of one-dimensional nonlinear dispersive wave equations is int roduced as a model for assessing the validity of weak turbulence theor y for random waves in an unambiguous and transparent fashion. These mo dels have an explicitly solvable weak turbulence theory which is devel oped here, with Kolmogorov-type wave number spectra exhibiting interes ting dependence on parameters in the equations. These predictions of w eak turbulence theory are compared with numerical solutions with dampi ng and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number. It is established that the quas i-Gaussian random phase hypothesis of weak turbulence theory is an exc ellent approximation in the numerical statistical steady state. Nevert heless, the predictions of weak turbulence theory fail and yield a muc h flatter (\k\(-1/3)) spectrum compared with the steeper (\k\(-3/4)) s pectrum observed in the numerical statistical steady state. The reason s for the failure of weak turbulence theory in this context are elucid ated here. Finally, an inertial range closure and scaling theory is de veloped which successfully predicts the inertial range exponents obser ved in the numerical statistical steady states.