Collapse of statistical equilibrium in large-scale hydroelastic turbulent waves

TL;DR

This work experimentally investigates the free decay of large-scale hydroelastic turbulent waves that initially reside in statistical equilibrium (SE). By combining space-time profiling and spectral analyses, it shows that large-scale tensional waves follow a Rayleigh-Jeans–like SE and, upon stopping forcing, decay via a two-stage process where the final viscous stage yields a total-energy law $\mathcal{E}(t) \propto (t-t_\nu)^{-8/7}$, derived from the SE spectrum and linear damping. The dissipation time scale is shown to obey $\tau^{-1}(\omega) \propto \omega^{7/6}$, matching experimental measurements across frequencies and tensions. The results validate the theoretical energy-budget framework in Fourier space and suggest broad applicability to other decaying turbulence systems that commence from SE at large scales.

Abstract

At scales larger than the forcing scale, some out-of-equilibrium turbulent systems (such as hydrodynamic turbulence, wave turbulence, and nonlinear optics) exhibit a state of statistical equilibrium where energy is equipartitioned among large-scale modes, in line with the Rayleigh-Jeans spectrum. Key open questions now pertain to either the emergence, decay, collapse, or other nonstationary evolutions from this state. Here, we experimentally investigate the free decay of large-scale hydroelastic turbulent waves, initially in a regime of statistical equilibrium. Using space- and time-resolved measurements, we show that the total energy of these large-scale tensional waves decays as a power law in time. We derive an energy decay law from the theoretical initial equilibrium spectrum and the linear viscous damping, as no net energy flux is carried. Our prediction then shows a good agreement with experimental data over nearly two decades in time, for various initial effective temperatures of the statistical equilibrium state. We further identify the dissipation mechanism and confirm it experimentally. Our approach could be applied to other decaying turbulence systems, with the large scales initially in statistical equilibrium.

Figures (6)

• Figure 1: Top: Scheme of the experimental setup to study hydroelastic waves (lateral view). Left: top view. Bottom: Typical hydroelastic wave field, $\eta(x,y)$, obtained from Fourier transform profilometry.
• Figure 2: Experimental dispersion relation $f$ versus $[k/(2\pi)]^{3/2}$ (black circle). Small-scale random forcing $f_p\in[50, 100]$ Hz. Red line: polynomial fit of Eq. (\ref{['eq:LDR']}) with only one fitting parameter, the tension $T$, which yields $T=6.8$ N m$^{-1}$. Inset: Spatiotemporal spectrum of the wave height, $S_\eta(k,\omega)$, versus $f\equiv \omega/(2\pi)$ and $k/(2\pi)$. Solid white line: linear dispersion relation of Eq. (\ref{['eq:LDR']}). Black line: lower bound [$\min(f_p)$] of the random forcing range.
• Figure 3: (a) Spectrogram $S_\xi(f,t^*)$ of the wave amplitude versus time $t^*$ (log-scale) and frequency $f$. Log-scale colorbar. For $t<t_0$: small-scale forcing ($f_p\in[50,100]$ Hz) and SE regime at large scales [$f< \min(f_p)$]. Forcing is stopped at time $t_0$ (vertical white solid line). For $t_0\leq t < t_\nu$: initial decaying regime. Final decay starts at time $t_\nu$ (vertical white dot-dashed line). Applied tension $T=6.8$ N m$^{-1}$. Inset: temporal decay of wave amplitude $\xi(t)$ before ($t<t_0$) and after ($t> t_0$) forcing stops. (b) Normalized energy spectrum versus the normalized time $(t-t_\nu)/\tau^{\mathrm{exp}}(\omega^*)$ for various Fourier modes $\omega^*/(2\pi)\in[2,70]$ Hz (symbols). The black dashed line has a $-2$ slope corresponding to the exponential decay of Eq. \ref{['eq:Eomega']}. Applied tension $T=5$ N m$^{-1}$. Inset: Normalized energy spectrum versus unrescaled time $t-t_\nu$ for Fourier modes $\omega^*/(2\pi)=$ 3.2($\circ$), 6.4 ($\lhd$), and 20 ($\Box$) Hz.
• Figure 4: Experimental dissipation time scale $\tau^{\mathrm{exp}}(\omega)$ of hydroelastic tensional waves versus frequency $\omega/(2\pi)$. The red dashed line corresponds to Eq. (\ref{['eq:tau']}) with no fitting parameter. Horizontal dotted line: temporal cutoff due to the spectrogram computation (short time interval $\delta t=0.5$ s). Cutoff frequency: $1/\delta t = 2$ Hz. Applied tension $T=5$ N m$^{-1}$.
• Figure 5: Energy spectrum $E_\xi(\omega,t)$ at different times, before ($t<t_\nu$ - top blue curve) and after ($t>t_\nu$ - other curves) the final decay starts. Obtained from vibrometer measurements at times $t-t_\nu \in[-0.4, 0.04, 0.7, 1.3, 2.9, 4, 6.2, 8.4]$ s (from top to bottom). The arrow indicates the direction of time. Black dashed lines correspond to Eq. \ref{['eq:Eomega']} for the same times, except the blue dashed line, for $t-t_\nu=-0.4$ s (i.e., $t-t_0=0.1$ s), corresponding to Eq. \ref{['eq:SE_omega']}. Black dotted line: cutoff frequency, $1/\delta t$, due to the spectrogram computation (short time interval of $\delta t = 0.22$ s). Grey region: initial forcing range, $f_p\in[50,100]$ Hz. Applied tension $T=6.8$ N m$^{-1}$.