Observation of an Inverse Energy Cascade in Developed Acoustic Turbulence in Superfluid Helium

We report observation of an inverse energy cascade in second sound acoustic turbulence in He II. Its onset occurs above a critical driving energy and it is accompanied by giant waves that constitute an acoustic analogue of the rogue waves that occasionally appear on the surface of the ocean. The theory of the phenomenon is developed and shown to be in good agreement with the experiments.

Figure 1

Evolution of the observed wave shape in the resonator (left column) and of the power spectrum of second sound standing waves (right) with increasing drive frequency ${\omega }_d$ near the 96th resonance: ${\omega }_d/2\pi =9530.8\mathrm{Hz}$ (a),(b) and 9535.2 Hz (c),(d). The ac heat flux density was $W=42\mathrm{mW}/{\mathrm{cm}}^2$. The temperature $T=2.08\mathrm{K}$ corresponded to negative nonlinearity. The fundamental and first harmonic in (b),(d) are indicated by vertical (green and blue) arrows at $\sim 10$ and 20 kHz; the low-frequency domain where the subharmonics appear is indicated by the horizontal range in (d) under the leftmost vertical (red) arrows. The horizontal arrows in (a),(c) indicate the fundamental period of a wave at the driving frequency.

Figure 2

The energy $E_{\mathrm{LF}}$ contained in the low-frequency part of the spectrum as a function of the ac heat flux density $W$, while driving near to the 96th resonance for $T\simeq 2.08\mathrm{K}$. The threshold value of $W$, marked by the (green) arrow, was $W_c=10.4\mathrm{mW}/{\mathrm{cm}}^2$. The points are from experiment; dashed lines are guides to the eye. Inset: The dependence of $E_{\mathrm{LF}}$ on ${\omega }_d$, measured for $W=55.6\mathrm{mW}/{\mathrm{cm}}^2$; the (red) arrow labels the maximum value of $E_{\mathrm{LF}}$, which is taken to the main figure.

Figure 3

Transient evolution of the 2nd sound wave amplitude $\delta T$ after a steplike shift of the driving frequency to the 96th resonance at time $t=0.397\mathrm{s}$. Signals in frames 1 and 3 are similar to those obtained in steady-state measurements, Figs. 1a, 1c, respectively. Formation of isolated “rogue” waves is clearly evident. Inset: Example of a rogue wave, enlarged from frame 2.

Figure 4

Instantaneous spectra in frames 1 and 3 of Fig. 3. The lower (blue) spectrum, for frame 1, shows the direct cascade only; the upper (orange) spectrum, for frame 3, shows both the direct and inverse cascades. The (green) arrow indicates the fundamental peak at the driving frequency. Inset: Evolution of the wave energy in the low-frequency and high-frequency domains is shown by the (orange) squares and (blue) triangles respectively; (black) arrows mark the positions of frames 1 and 3.

Figure 5

Dependence of the ac heat flux density $W$ at which the instability develops on the dimensionless frequency detuning $\Delta =({\omega }_d-{\omega }_n)/{\omega }_n$ of the driving force frequency ${\omega }_d$ from a cavity resonance ${\omega }_n$. Numerical calculations (line) are compared with measurements (points) for driving at the 96th resonance. Horizontal bars mark the widths of the hysteretic region where second sound exists in a metastable state. Inset: Bifurcation diagram showing regions of stability (unshaded) and regions of instability (yellow shaded) against the generation of subharmonics. The soft instability occurs over the (orange) line between the (green) critical points at $\pm {\Delta }^{*}$; outside them lies the hard instability; $W^{*}$ is the threshold value of the instability.