Quasiadiabatic Decay of Capillary Turbulence on the Charged Surface of Liquid Hydrogen

We study the free decay of capillary turbulence on the charged surface of liquid hydrogen. We find that decay begins from the high frequency end of the spectral range, while most of the energy remains localized at low frequencies. The apparent discrepancy with the self-similar theory of nonstationary wave turbulent processes is accounted for in terms of a quasiadiabatic decay wherein fast nonlinear wave interactions redistribute energy between frequency scales in the presence of finite damping at all frequencies. Numerical calculations based on this idea agree well with experimental data.

Figure 1

(a) The measured signal $P(t)$. The periodic driving force was switched off at time $t=1.8\mathrm{s}$. (b) Evolution of the turbulent power spectrum during the decay, calculated over $P(t)$. Grey shading indicates frequency components in the power spectrum whose $P_{\omega }^2$ exceeds the threshold ${10}^4$ (a.u.) in the graph below. (c) Instantaneous power spectra calculated at times indicated by the arrows in (b): curve 1 in the inset corresponds to time $t=2\mathrm{s}$; curve 2 corresponds to $t=2.5\mathrm{s}$. The spectra shown represent an ensemble average over ten identical measurements. The dashed line in (c) corresponds to the powerlike dependence $P_{\omega }^2\sim {\omega }^{-7/2}$ predicted by theory [6, 7].

Figure 2

Decay of the turbulent spectrum. The occupation number $n_{\omega }$ as a function of renormalized frequency calculated using (1) at three renormalized times: $t=0$, the initial steady-state spectrum (full curve); $t=3$ (dashes); $t=10$ (dots). The straight line corresponds to $n_{\omega }\sim {\omega }^{-7/2}$ as predicted by the theory of capillary turbulence [6, 7].