Wilson-Fisher renormalization of discrete gravity-capillary wave turbulence in viscous fluids

TL;DR

This work experimentally realizes Wilson–Fisher renormalization in discrete gravity-capillary wave turbulence (DSWT) by introducing a running frequency scale $\omega$ and a dimensionless coupling $g(\omega)$, revealing two topology-defined universality classes: triadic capillary waves with $N=3$ and tetradic gravity waves with $N=4$. Viscosity acts as the relevant perturbation, producing a Kolmogorov cutoff $\omega_K$ and enabling Wilsonian crossover trajectories between the fixed points, as captured by WF beta functions and Wegner scaling. The authors define two measurable RG coordinates, $\bar{\Omega}_\nu$ and $\bar{\Sigma}$, and verify distinct renormalized scalings: $\bar{\Omega}_\nu \sim \bar{E}^{2/3}$ for CW and $\bar{\Omega}_\nu \sim \bar{E}^{1}$ for GW, while $\bar{\Sigma}_\omega \sim {\rm Re}^{2/(N-1)}$, giving $\bar{\Sigma} \sim {\rm Re}^{1}$ (CW) and ${\rm Re}^{2/3}$ (GW). Across fluids spanning six decades in Re, laser Doppler vibrometry confirms these predictions and the existence of two distinct cascades with different spectral exponents ($S(\omega)\sim \omega^{-17/6}$ for CW and $\omega^{-5/2}$ for GW) and dismutation features in GW. The study positions DSWT as a tunable laboratory for nonequilibrium Wilsonian criticality and as a bridge between discrete and continuous turbulence theories.

Abstract

We report an experimental realization of Wilson-Fisher renormalization in driven surface-wave turbulence across Newtonian fluids spanning nearly six decades in Raynolds number. Discrete capillary and gravity turbulence define two universality classes selected by interaction topology: triadic resonances for capillary waves and effectively tetradic scattering for gravity waves. Navier-Stokes viscosity is the relevant perturbation that renormalizes spectral transfer and terminates the cascade. The resulting framework predicts the Kolmogorov cutoff from the balance of nonlinear transfer and viscous damping, and Reynolds scaling of the integrated inertial spectral weight. Laser Doppler Vibrometry quantitatively confirms these renormalized scaling laws, establishing discrete gravity-capillary turbulence as a tunable laboratory for nonequilibrium crossoever criticality.

Figures (4)

• Figure 1: Frequency-domain signatures and inertial scaling of discrete surface-wave turbulence (DSWT) in viscous fluids. (a) Broadband surface-velocity power spectral density $S(\omega)$ showing the capillary-wave (CW; top) and gravity-wave (GW; bottom) cascades for water and high-viscosity oils (Oil 1: $10$, Oil 3: $10^{3}$ times the viscosity of water). Open arrows mark the excitation frequency $\omega_0$, filled ones the Kolmogorov cutoff $\omega_K$. Dashed guide lines indicate KZ scaling: $S(\omega)\sim\omega^{-17/6}$ (CW) and $S(\omega)\sim\omega^{-5/2}$ (GW). Inset: discrete inviscid CW modes persist across the inertial range, essentially independent of bulk viscosity. (b) CW spectrum illustrating three-wave ($N=3$) resonant mixing, with peaks at $\omega_1$, $\omega_2$, $2\omega_1$, $2\omega_2$, and $\omega_1+\omega_2$. (c) GW spectrum illustrating four-wave ($N=4$) transfer, with dominant peaks at $\omega_1$, $\omega_2$ and dismutation resonances.
• Figure 2: Kolmogorov cutoff as a Wilson--Fisher ultraviolet scale. (a) GW ($\omega_0=8\,\mathrm{Hz}$): $\omega_K$ vs forcing amplitude $A$. (b) CW ($\omega_0=27\,\mathrm{Hz}$): $\omega_K$ vs $A$. In both regimes, $\omega_K$ grows monotonically with forcing, defining the Kolmogorov exit scale. (c) Wegner scaling of the ultraviolet coordinate $\bar{\Omega}_\nu\equiv \omega_K/(\nu k_0^2)$ with $k_0=2\pi/\Lambda_0$ and $\Lambda_0$ the dispersion-selected forcing length, versus reduced injected energy $\bar{E}\equiv E/E_0$, with $E=\rho (A\omega_0)^2/2$ and $E_0=\rho(\nu/\Lambda_0)^2$ the viscous kinetic-energy scale at $\Lambda_0$ (see SM). Data collapse onto topology-selected branches: $\bar{\Omega}_\nu\sim \bar{E}^{1}$ (GW, $N=4$) and $\bar{\Omega}_\nu\sim \bar{E}^{2/3}$ (CW, $N=3$). Solid lines: Wilson--Fisher predictions from Eq. (\ref{['eq:omegamax']}).
• Figure 3: Reynolds-number renormalization of the cascade response. Reduced inertial spectral weight $\bar{\Sigma}\equiv\Sigma_{\mathrm{PSD}}/(\Lambda_0\omega_0)^2$ versus ${\rm Re}\equiv U_0\Lambda_0/\nu$ across all fluids, revealing two energetically separated universality classes: capillary-wave (CW, upper) and gravity-wave (GW, lower) turbulence. Data follow WF/Wegner scaling and collapse onto topology-controlled power laws $\bar{\Sigma}_\omega\sim{\rm Re}^{2/(N-1)}$: $\bar{\Sigma}\sim{\rm Re}^{1}$ for CW ($N=3$) and $\bar{\Sigma}_\omega\sim{\rm Re}^{2/3}$ for GW ($N=4$) (solid lines: WF predictions from Eq. (\ref{['eq:Sigma_Re']})), demonstrating interaction-topology renormalization of the global cascade response (SM; End Matter).
• Figure 4: Topology-dependent interaction vertices and WF saturation at fixed points. (a) Capillary-wave turbulence admits exact triads ($N=3$), yielding $\delta g\sim g^2$ and $\beta(g)=y_g g-B_3 g^2$. (b) Deep-water gravity-wave turbulence proceeds via effective tetrads ($N=4$) with finite resonance broadening (detuning insertion $\Delta\neq 0$), yielding $\delta g\sim g^3$ and $\beta(g)=y_g g-B_4 g^3$. Interaction topology fixes the nonlinear term in $\beta(g)$ and defines distinct RG universality classes.