Interaction Renormalization and Validity of Kinetic Equations for Turbulent States

We consider turbulence of waves interacting weakly via four-wave scattering (sea waves, plasma waves, spin waves, etc.). In the first order in the interaction, a closed kinetic equation has stationary solutions describing turbulent cascades. We show that the higher-order terms generally diverge both at small (IR) and large (UV) wave numbers for direct cascades. The analysis up to the third order identifies the most UV-divergent terms. To gain qualitative analytic control, we sum a subset of the most UV divergent terms, to all orders, giving a perturbation theory free from UV divergence, showing that turbulence becomes independent of the dissipation scale when it goes to zero. On the contrary, the IR divergence (present in the majority of cases) makes the effective coupling parametrically larger than the naive estimate and growing with the pumping scale L (similar to anomalous scaling in fluid turbulence). In such cases, the kinetic equation does not describe wave turbulence even of arbitrarily small level at a given k if kL is large enough that is the cascade is sufficiently long. We show that the character of strong turbulence is determined by whether the effective four-wave interaction is enhanced or suppressed by collective effects. The enhancement possibly signals that strong turbulence is dominated by multiwave bound states (solitons, shocks, cusps), similar to confinement in quantum chromodynamics.