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. DOI: 10.1103/PhysRevLett.101.065303 PACS numbers: 67.25.dk, 47.20.Ky, 47.27.ÿi, 67.25.dt A highly excited state of a system with numerous degrees of freedom, characterized by a directional energy flux through frequency scales, is referred to as turbulent [1,2]. Like the familiar manifestations of vortex turbulence in fluids, turbulence can also occur in systems of waves, e.g., turbulence of sound waves in oceanic waveguides [3], magnetic turbulence in interstellar gases [4], shock waves in the solar wind and their coupling with Earth's magnetosphere [5], and phonon turbulence in solids [6]. Following the ideas of Kolmogorov, the universally accepted picture says that nonlinear wave interactions give rise to a cascade of wave energy towards shorter and shorter wavelengths until, eventually, it becomes possible for viscosity to dissipate the energy as heat. Experiments and calculations show that, most of the time, the Kolmogorov picture is correct [2,7,8].We demonstrate below that this picture is incomplete. Our experiments with second sound (temperature-entropy) waves in He II show that, contrary to the conventional wisdom, acoustic wave energy can sometimes flow in the opposite direction too. We note that inverse energy cascades are known in 2-dimensional incompressible liquids and Bose gases [9], and have been considered for quantized vortices [10].We find that energy backflow in our acoustic system is attributable to a decay instability (cf. the kinetic instability in turbulent systems [11]), controlled mainly by nonlinear decay of the wave into two waves of lower frequency governed by the energy (frequency) conservation law [2] ! 1 ! 2 ! 3 . Here ! i u 20 k i is the frequency of a linear wave of wave vector k i and u 20 is the second sound velocity at negligibly small amplitude. The instability manifests itself in the generation of subharmonics. A quite similar parametric process, due to 4-wave scattering (modulation instability), is thought to be responsible for the generation of large wind-driven ocean waves [12]. Decay instabilities (especially threshold and nearthreshold behavior) have been studied for, e.g., spin waves [13], magnetohydrodynamic waves in plasma [14], and interacting first and second sound waves in superfluid helium near the superfluid transition [15].We now discuss what happens to a system of acoustic waves far beyond the decay threshold. Modeling the resultant nonlinear wave transformations in the laboratory is a potentially fruitful approach that has already yielded important results for, e.g., the turbulent decay of capillary waves on the surface of liquid H 2 [16]. Here, we exploit the special pro...

Based on measurements of nonlinear second-sound resonances in a high-quality resonator, we have observed a steady-state wave energy cascade in He II involving a flux of energy through the spectral range towards high frequencies. We show that the energy balance in the wave system is nonlocal in K space and that the frequency scales of energy pumping and dissipation are widely separated. The wave amplitude distribution follows a power law over a wide range of frequencies. Numerical computations yield results in agreement with the experimental observations. We suggest that second-sound cascades of this kind may be useful for model studies of acoustic turbulence.

Rogue waves have been observed in superfluid helium. The experimental system consists of high intensity second sound (temperature-entropy) waves within a resonant cavity. Under steady state conditions, with a constant oscillatory driving force at the resonant frequency, the waves are turbulent and there are fluxes of energy towards both high and low frequencies. Rogue waves appear under the nonequilibrium conditions that prevail shortly after the drive has been switched on, prior to establishment of the steady state. The experiment is described briefly, relevant results are presented and discussed theoretically in terms of nonlinear wave interactions, and possible connections to rogue waves on the ocean are considered.

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.

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