Renormalized quasi-resonant transport in surface-wave turbulence

A renormalized coarse-grained transport formulation is developed for finite surface-wave turbulence in logarithmic frequency space. Shell elimination generates a running nonlinear activity on a restricted quasi-resonant channel whose frequency width is set by the nonlinear linewidth, while physical stationarity is imposed on the shell energy current rather than on the coupling itself. Microscopic Hamiltonian homogeneity together with constant flux fixes the underlying weak-turbulence scaling class and reproduces the classical capillary and gravity Kolmogorov–Zakharov spectra independently of the renormalization group (RG), providing its asymptotic consistency benchmark. The Wilsonian flow instead supplies the finite coarse-grained kinetics: it determines whether the corresponding stationary-current trajectory can be dynamically supported, how its effective coupling and linewidth run, and where the stationary weak branch loses validity. In squared-steepness variables, this exposes two distinct realizations: capillary transport follows a genuinely running trajectory, whereas deep-water gravity is the marginal scale-independent case, coinciding with fixed coupling in the ideal RG-supported interior.