Random walk model for dual cascades in wave turbulence

Dual cascades in turbulent systems with two conserved quadratic quantities famously arise in both two-dimensional hydrodynamic turbulence and also in wave turbulence based on four-wave interactions. Examples for the latter include surface waves and nonlinear Schrödinger equations with cubic nonlinearity. However, numerical simulations in forced-dissipative equilibrium of two-dimensional turbulence and of a one-dimensional wave system reveal that the physical nature of their cascades is starkly different. This is demonstrated by comparing their spectra in a finite inertial range and by comparing the temporal fluctuations of their spectral fluxes. In particular, the flux fluctuations are much larger in the wave case and frequently lead to instantaneous flux values that have the opposite sign of the mean flux, a phenomenon that is completely absent in the hydrodynamic case. A simple random walk model for the dual cascade in wave turbulence is then formulated that is very successful in explaining these effects. In particular, the model is able to replicate the detailed shape of the observed turbulent spectrum in a finite inertial range, and it also offers a ready explanation for the large flux fluctuations. It is also shown that a nonlinear diffusion model for the wave system cannot explain the observed spectral shapes. Overall, this suggests that in wave turbulence the systematic spectral fluxes observed in a dual cascade do not require an irreversible dynamical mechanism, rather, they arise as the inevitable outcome of blind chance.