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

Abstract

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...

Statistical Properties of Directional Ocean Waves: The Role of the Modulational Instability in the Formation of Extreme Events

“…This is particularly true if the ratio between the wave steepness and the spectral bandwidth, known as the Benjamin-Feir Index (BFI), is large [4]. We mention that rogue waves have also been recently observed in optical systems [9] and in acoustic turbulence in He II [10] where giant waves are observed during an inverse cascade process.We emphasize that in many different fields of physics (plasmas [11,12], nonlinear optics [13,14], chargedparticle beam dynamics [15,16]) the modulational instability plays an important role; under suitable physical conditions a nonlinear Schrödinger equation can be derived and the modulational instability can be analyzed directly with this equation [2]. A major question which has to be addressed (and is the subject of the present Letter) concerns the role of the modulational instability in two dimensional propagation.…”

“…This is particularly true if the ratio between the wave steepness and the spectral bandwidth, known as the Benjamin-Feir Index (BFI), is large [4]. We mention that rogue waves have also been recently observed in optical systems [9] and in acoustic turbulence in He II [10] where giant waves are observed during an inverse cascade process.…”

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“…Such waves may also appear in other environments such as oceanic rogue waves [32,33], stock market crashes [34], optical rogue waves [35,36], superfluid helium [37], Bose-Einstein condensates [38], atmospheric physics [39] and opposing current flow [40]. The rogue waves in plasmas may be generated due to wavewave interactions [41], and their evolution can also be described by the NLS equation for modulated wave packets.…”

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