Experiments on wave turbulence: the evolution and growth of second sound acoustic turbulence in superfluid 4He confirm self-similarity

We report our experiments on the formation of second sound acoustic turbulence in superfluid 4He. The initial growth in spectral amplitude follows power laws that steepen rapidly with increasing harmonic number n, corresponding to a propagating front in frequency space. The lower growth exponents agree well with analytic predictions and numerical modeling. The observed increase in the formation delay with n validates the concept of self-similarity in the growth of wave turbulence.


Introduction
Turbulence may be defined as a far-from-equilibrium state of a nonlinear physical system whose energy distribution extends over many degrees of freedom. It takes many different forms, of which vortex turbulence [1,2] is probably the best known. A common feature lies in energy transfer between the scale or frequency at which energy from an external source is pumped into the system and a different scale or frequency at which it can be transformed to heat via dissipative processes. Turbulence in systems of waves is widespread in nature and technology. It can arise e.g. among phonons in solids [3], in optical fibers and nonlinear optical media [4], on vibrating plates [5]- [7] and the surfaces of ferrofluids [8,9], in sound waves in oceanic waveguides [10], as magnetic turbulence in interstellar gases [11], in shock waves in the solar wind coupled to the Earth's magnetosphere [12], in surface waves on liquid hydrogen [13] and in second sound in superfluid helium [14,15]. Typically, there is a cascade-like transfer of turbulent energy towards the high frequency domain [14,16], which is where dissipation mainly occurs.
Of special interest is the application of the wave turbulence (WT) concept to the description of energy transfer in superfluid systems far from equilibrium, including disturbed superfluid 4 He [17], nonequilibrium exciton-polariton condensates in a cavity [18,19] and atomic Bose-Einstein condensates in a trap [20]. It has been shown theoretically and numerically [21,22] that the evolution of acoustic WT is of key importance in the late stages of condensate formation. Although the existing theory can in principle illuminate the nonequilibrium properties of superfluid systems, no experimental studies of turbulent evolution have previously been reported. The concept of self-similar formation of WT is widely accepted in theory [17,21,23,24] but, to our knowledge, this prediction has not previously been checked experimentally.
In what follows, we report the results of an experiment to study the evolutionary processes that occur during the build-up towards steady-state WT in superfluid 4 He, following the initial application of a periodic driving force to excite second sound in a resonant cavity. It was established earlier that second sound (temperature-entropy waves) provides an ideal model system for laboratory studies of steady-state wave turbulent phenomena, in particular because the nonlinear wave interactions can readily be adjusted and controlled. Under steady-state nonequilibrium conditions, with continuous energy injection and dissipation [14,15], acoustic turbulence is formed whose energy balance is highly nonlocal in K -space, and a direct flux of energy through the spectral scales is established. We now demonstrate that the formation of this turbulent state is self-similar and that, in its initial stages, it is described by power laws for 3 the amplitudes of the spectral harmonics. The evolution can be understood as a formation front propagating in frequency space towards the high frequency domain.
The velocity of a second sound traveling wave depends on its amplitude and, to a first approximation, can be written as where δT is the wave amplitude, u 20 is the velocity of a wave of infinitely small amplitude and α is the nonlinearity coefficient of second sound, which is determined by the relation [25] where C is the heat capacity per unit mass of liquid helium at constant pressure and T is the temperature.
The nonlinearity coefficient α may be either positive or negative, depending on the temperature and pressure [25]- [27]. Under saturated vapor pressure (SVP), in the region of roton second sound (i.e. at T > 0.9 K), the nonlinearity coefficient is positive (α > 0) at temperatures T < T α = 1.88 K (like the nonlinearity coefficient of conventional sound waves in ordinary media), but it is negative (α < 0) in the range T α < T < T λ . Here, T λ = 2.176 K is the temperature of the superfluid-to-normal (He II to He I) transition. At T = T α , the nonlinearity coefficient passes through zero. The ease with which α can be adjusted (by alteration of T ), and the large values that it can take, make second sound in He II ideal as a test bed for the study of nonlinear wave interactions and associated phenomena, including WT.
The frequency of second sound depends on its wave vector k as [28] ω = u 20 k 1 + λξ 2 k 2 + · · · , where ξ = ξ 0 (1 − T /T λ ) −2/3 , ξ 0 ∼ 2-3 A and λ ∼ 1. We emphasize that the dispersion of second sound is significant within the close vicinity of the superfluid transition (i.e. for T λ − T < 1 µK), but is very weak in the temperature range T < 2.1 K relevant to the present investigations. However, the presence of non-zero dispersion is of key importance in the formation of weak turbulence in He II. It is well established that if the dispersion is exactly equal to zero, ∂u 20 /∂k = 0, the main mechanism of energy transport through the frequency scales is quite different from turbulence and can be associated with shock wave creation [29]. When the dispersion is positive but small, ∂u 20 /∂k > 0 (e.g. in the case of second sound in He II, given by equation (2) where the leading nonlinear term is quadratic over the amplitude, see equation (1)), the wave turbulent state is formed in a narrow cone of k-vectors around the direction of propagation of the wave with k = k drive [16]. The relative phases of waves with k-vectors almost collinear to k drive (i.e. within the cone) are random, and a standard kinetic equation for waves can be used to describe the cascade-like, turbulent propagation of energy in the wave system [16,30]. However, the interaction of waves with non-collinear wave vectors is controlled by higher nonlinear terms and is relatively small. That is, wave turbulence in He II is nearly one-dimensional, provided that the dispersion given by equation (2) is weak, i.e. the temperature of the helium bath is not close to T λ . The formation of the wave turbulent regime in He II manifests itself in, for example, fluctuations in the wave field at high frequencies and in the establishment of a near-Gaussian probability distribution function for second sound wave amplitudes, as was observed earlier [31].

4
In section 2, we describe briefly the experimental arrangements. The results obtained are presented in section 3. In section 4, we discuss the results, present a numerical theory and consider the broader implications of this work.

Experimental details
The experiments used the second sound resonator described previously [14,32]: a thin film heater capped one end of a cylindrical quartz tube of length = 7 cm and diameter D = 15 mm. The other end was capped by a superconducting bolometer, whose transition temperature could be adjusted by changing an external magnetic field. The He II inside the cell was held under its saturated vapor pressure at a temperature of T = 2.08 K, where (T λ − T )/T λ = 0.041. The temperature signal from the bolometer was recorded by a digital oscilloscope and then transferred to a computer for analysis. The quality of the resonator reached Q ∼ 3000 at the higher resonance numbers (N > 10) but was less at lower resonances, behavior that is understandable in terms of losses due to normal fluid motion in the boundary layer next to the wall. Figure 1(a) plots a typical evolution envelope following the step-like application of a resonant periodic drive to the heater under conditions such that there is no inverse energy cascade [15]. The standing wave in δT evolves nonmonotonically and often (as here) exhibits asymmetry. At early times, after switching on the drive, only a response at the driving frequency is evident, as shown by the power spectral density plotted in figure 1(b), corresponding to arrow-b in figure 1(a). Some time later, the direct energy cascade has become established in the low frequency domain, whereas the wave amplitude at high frequency remains close to zero, e.g. harmonics are formed only up to ω/2π < 20 kHz in figure 1(c). At later times, the cascade is fully formed up to ∼ 80 kHz, as shown in figure 1(d), corresponding to arrow-d in figure 1(a).

Experimental results
We now examine how the harmonics of the driving force evolve with time by analysis of the heights of the Fourier spikes labeled 1-5 in figure 1(d). The set of results plotted in figure 2 corresponds to the time domain signal shown in figure 1(a). It is immediately evident that each of the harmonic amplitudes grows initially according to a power law, undergoes shallow oscillations and finally flattens out as the steady state is approached. It is also clear that the corresponding power-law exponents at early times increase with the harmonic number n and that the higher harmonics develop after short delays.

Discussion and theory
To try to account for these phenomena, we apply the numerical model used earlier [14] for the computation of steady-state spectra. Energy balance is governed by where b n (t) = (1/2)(B −1 n S n + iB n β n ) is the time-dependent canonical amplitude of second sound at the nth resonant frequency f n ; S n and β n are the space Fourier components ∼ cos(2π f n x/u 20 L) of the entropy and of the potential of the normal-superfluid relative velocity; B n = ( f n C/T ) 1/2 ; V n,n 1 ,n 2 ∝ α(nn 1 n 2 ) 1/2 describes the three-wave interaction; γ n = νn 2 models the viscous damping of second sound; and F d ∝ W is the amplitude of the force driving the nth resonant mode. In this representation, the wave spectrum can be calculated as A f ∝ B n (b n + b * n ). To capture the dependence of the formation process on the driving force amplitude, we compute the evolution of the second sound wave spectrum with time for the driving force F d = 1 (in numerical units). The steady-state spectrum established at large times, i.e. after all transient processes are finished at t form 5, is shown in figures 3(a) by circles. Figure 3(a) also demonstrates the evolution of the spectrum during the build-up process, similar to that observed in the experiments (figures 1(b)-(d)). Figure 3(b) shows the dependence on time of the amplitude of the wave at the driving frequency and of two higher harmonics computed for the evolution presented in figure 3(a). It  is evident that the dependences of the wave amplitudes on time during the formation process t < 2 are well described by a power-law function, where m = 2n − 1. We see also that the characteristic formation time is t form ∼ 5. We can account for the observed dependence (4) analytically within the framework of the model (3). It is evident from figure 3(b) that, at small times t t form , the wave amplitude is also small. During this era, it is the wave at the driving frequency that has the largest amplitude. We may therefore reduce the whole set of equation (3) to a simpler form dA 1 /dt = F d , dA n /dt = V n,1,n−1 A 1 A n−1 (n > 1). We neglect dissipation, which has a negligible effect early in 7 the growth process. The reduced equations imply that the wave at the driving frequency, n = 1, grows mainly due to interaction with the driving force, and the evolution of each n > 1 harmonic is defined by the interaction with its left-hand side neighbor, A n−1 , and with the fundamental wave, A 1 . From the reduced equations, one finds A 1 = F d × t for n = 1 and A n =Ṽ n F n d t (2n−1) , whereṼ n = (1/(2n − 1)!!) n k=2 V k,1,k−1 for n > 1 and !! signifies a double-factorial. The result obtained clearly agrees well with the numerical computations (figures 3(b) and equation (4)). The exponents m = 2n − 1 are universal and do not depend on the value of the nonlinearity coefficient (at V k,l,m ∝ α = 0) or on the driving amplitude F d . The model is valid for small t t form .
The results from the numerical model compare well with those from the experiments. In both cases, the initial growth in spectral amplitude takes the form of a power law whose exponent grows with increasing harmonic number n. For n = 1, 2, the measured exponents agree within experimental error with the values of 1 and 3 predicted by equation (4). For larger n, the experimental exponents still increase with n, but the agreement becomes only qualitative. We comment, however, that such discrepancies are to be expected, because at larger times when the low-frequency harmonics are not small with respect to their equilibrium values, nonlinear interactions other than those taken into account in the reduced equations start playing an essential role.
To analyze the details of the turbulence build-up process at larger t, we suppose that formation is self-similar with a finite formation time t * (i.e. of the explosion type for WT [16], corresponding to the case of a finite capacity spectrum [33]). In agreement with general recommendations [16,17,33], in this case the harmonic amplitude at frequency ω, time t, can be estimated as A(ω, t) = (t * − t) q f (ξ ), where ξ = ω(t * − t) p is the self-similar variable, f is a universal function that describes the evolution of the wave spectrum in the course of the build-up process, and q and p are positive exponents. The f function obeys the boundary conditions [16] f ∝ ξ −s at ξ 1 and f ξ s → 0 at ξ 1, where s ≈ 1 is close to the Kolmogorov index of the steady-state turbulent spectrum taken with opposite sign [14,33]. (The indices also satisfy the relation q = sp, to ensure the steady-state character of the spectrum formed for ξ 1; see below.) According to the boundary conditions, in the low frequency domain ξ 1, a spectrum A ω ∝ ω −s is formed, and in the high frequency domain ξ 1, the harmonic amplitudes are close to zero. At the characteristic transition frequency between the two domains, ω f , one has ξ ∼ 1 [17]. As a result, the transient process can be understood as the propagation of a formation front in frequency space from low to high frequencies, whose position at time t is This conclusion agrees with the experimental observations shown in figure 1 and in the numerical simulations of figure 3(a). For example, the position of the front at t = 0.66 s is marked by an arrow in figure 1(c). (Note that under experimental conditions the cascade is restricted at a high frequency ω b due to viscosity [14], hence the dependence (5) is only valid at To capture the details of the front propagation, we plot in figure 4 the characteristic delay time of harmonic formation as a function of n for the first 12 harmonics. The formation delay t n is defined as the interval between the application of the driving force at t on and the time t n when the nth harmonic rises from the common noise floor. The formation delay of the fundamental wave, n = 1, is negligible. The steady growth of t n with n in figure 4 corresponds to the finite propagation velocity of the front, dω f (t)/dt, in agreement with equation (5). It follows from (5) that the difference between the formation times of the nth and 1st harmonics is equal to t n − t 1 = const × (ω −1/ p 1 − ω −1/ p n ). Noting that the frequency of the nth harmonic is ω n = n × ω 1 , one obtains where C is a constant of the order of t form . The experimental points in figure 4 can be closely fitted by (6) (curves) for p = 5.0. Thus the experimental data are well described by the selfsimilar dependence (6).

Conclusions
We have reported the first experimental study of the processes involved in the creation of steadystate acoustic turbulence in a superfluid system. In the main, the results are well described by a numerical model, although there are points of detail that are not captured. These include the damped oscillations seen in the spectral amplitudes as the steady state is approached, and discrepancies in the measured and calculated power-law exponents describing the initial growth for n 3. Up to intermediate times, the build-up of turbulence is well described by the selfsimilar model (5). These same considerations should also be applicable to the other systems described in the Introduction for which, however, the experimental conditions and parameters are in many cases much harder to control.