Renormalized flow theory of wave turbulence: Kolmogorov-Zakharov spectra as emergent asymptotic states

Abstract

We develop a continuous Wilsonian renormalized-flow theory of weak wave turbulence directly in spectral frequency space, for finite cascades in experimentally driven Newtonian fluids. The central quantity is a scale-dependent effective coupling that governs nonlinear transfer across logarithmic frequency shells and organizes the cascade as a finite renormalized branch. Within this formulation, the inertial interval is constructed dynamically as a plateau of the running flow, whose non-autonomous character is expressed through its explicit dependence on the logarithmic distance from the injection scale and thereby encodes the cumulative action of forcing and degradation along the cascade. The ultraviolet cutoff follows internally as the terminal scale at which the plateau branch ceases to exist, whereas the integrated spectral response is fixed by infrared matching to the injection scale. In this way, the finite inertial branch is determined by the renormalized dynamics itself, while Kolmogorov--Zakharov (KZ) spectra arise only as its asymptotic constant-flux scaling states. The theory applies to both capillary and gravity wave turbulence and admits a physically transparent realization in monochromatically driven discrete cascades, which fix the topology-dependent exponent structure of the renormalized flow.