Aspects of Two-Mode Probability Density Function in Weak Wave Turbulence

Abstract

We derive the time evolution of the two-mode amplitude probability density function. Using this equation, we derive conditions for the existence of a zero flux steady-state solution. We also derive the equation for a vortex solution and show that the product of two one-mode steady-state solutions can be a two-mode steady-state solution only when an extra condition is satisfied. With this extra condition assumed, we plot the flux of probability vector on two mode's plane. It is shown that this flux lines circulate around the center (n(a), n(b)), which are the mean values of the two mode's amplitude square.