Energy and Vorticity Spectra in Turbulent Superfluid {}^{4}He from T=0 to T_{\lambda}.

Energy and Vorticity Spectra in Turbulent Superfluid He from to .

Laurent Boué, Victor S. L’vov, Yotam Nagar, Sergey V. Nazarenko, Anna Pomyalov, and Itamar Procaccia Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
University of Warwick, Mathematics Institute Coventry, CV4 7AL, UK
Abstract

We discuss the energy and vorticity spectra of turbulent superfluid He in the entire temperature range from up to the phase transition “ point”, K. Contrary to classical developed turbulence in which there are only two typical scales, i.e. the energy injection and the dissipation scales , here the quantization of vorticity introduces two additional scales, the vortex core radius and the mean vortex spacing . We present these spectra for the super- and the normal-fluid components in the entire range of scales from to including the cross-over scale where the hydrodynamic eddy-cascade is replaced by the cascade of Kelvin waves on individual vortices. At this scale a bottleneck accumulation of the energy was found earlier at . We show that even very small mutual friction dramatically suppresses the bottleneck effect due to the dissipation of the Kelvin waves. Using our results for the spectra we estimate the Vinen “effective viscosity” in the entire temperature range and show agreement with numerous experimental observation for .

I Introduction

Superfluidity was discovered by Kapitza and by Allen and Misener in 1938 who demonstrated the existence of an inviscid fluid flow of He below K. In the same year, London linked the properties of the superfluid He to Bose-Einstein condensation.

Soon after, Landau and Tisza offered a “two fluid” model in which the dynamics of the superfluid He is described in terms of a viscous normal component and an inviscid superfluid component, each with its own density and and its own velocity field and . Already in 1955, Feynman realized Feynman () that the potential appearance of quantized vortex lines will result in a new type of turbulence, the turbulence of superfluids. The experimental verification of this prediction followed in the paper by Hall and Vinen a year later Vinen ().

An isolated vortex line is a stable topological defect in which the superfluid density drops to zero and the velocity . Here is the radial distance from the center that exceeds a core radius cm in He. The existence of quantized vortex lines1 (); 2 (); Vinen2008 () in superfluid tubulence introduces automatically additional length scales that do not exist in classical turbulence. In addition to , the density of vortex lines defines an “inter-vortex” average spacing denoted as .

The pioneering experimental observation of Maurer and Tabeling MT-98 () showed quite clearly that the large-scale energy spectrum of turbulent He above and below are indistinguishable. This and later experiments and simulations gave rise to the growing consensus that on scales much larger than the energy spectra of turbulent superfluids are very close to those of classical fluids if they are excited similarly BLR (). The understanding is that motions with scales correlate the vortex lines, organizing them into vortex bundles. At these large scales the quantization of the vortex lines becomes irrelevant and the large scale motions are similar to those of continuous hydrodynamic eddies. Obviously, since energy is cascaded by hydrodynamic eddies to smaller and smaller scales, we must reach a scale where the absence of viscous dissipation will require new physics.

Evidently, when the observation scales approach and below, the discreteness of the quantized vortex lines becomes crucial. Indeed, on such scales the dynamics of the vortex lines themselves become relevant including vortex reconnections and the excitation of Kelvin waves on the individual vortex lines. Kelvin waves exist also in classical hydrodynamics but here they become important in taking over the role of transferring the energy further down the scales. Their nonlinear interaction results in a so-called “weak wave turbulence” 92ZLF (); 11Nazar () supporting a mean energy flux towards shorter and shorter scales. Finally when the cascade reaches the core radius scale the energy is radiated away by quasiparticles (phonons in He ) VinenNiemela ().

Although the overall picture of superfluid turbulence described above seems quite reasonable, some important details are yet to be established. A particularly interesting issue is the physics on scales close to the crossover between the eddy-dominated and the Kelvin wave-dominated regimes of the spectrum. It was pointed out in LABEL:LNR-1 that the nonlinear transfer mechanism of weakly nonlinear Kelvin waves on sparse vortex lines is less efficient than the energy transfer mechanism due to strongly nonlinear eddy interactions in continuous fluids. This may cause an energy cascade stagnation at the crossover scale.

The present paper is motivated by some exciting new experimental and simulational developments Lathrop1 (); Lathrop2 (); Andreev (); Ladik (); Ladik1 (); Ladik2 (); Roche (); Manchester-exp (); Golov2 (); Karman (); rev4 (); exp (); Helsinki-exp (); Tsubota2008 (); araki (); stalp (); nimiela2005 (); exp1 (); Vin03-PRL (); cn (); exp2 (); exp3 (); exp4 (); RocheBarenghi () that call for a fresh analysis of the physics of superfluid turbulence in a range of temperatures and length scales. These developments include, among others, cryogenic flow visualization techniques by micron-sized solid particles and metastable helium molecules, that allow, e.g. direct observation of vortex reconnections, mean normal and superfluid velocity profiles in thermal counterflow  Lathrop1 (); Lathrop2 (); the observation of Andreev reflection by an array of vibrating wire sensors shedding light on the role of vortex dynamics in the formation of quantum turbulence, etc. Andreev (); the measurements of the vortex line density by the attenuation of second sound Ladik (); Ladik1 (); Ladik2 (); Roche () and by the attenuation of ion beams Manchester-exp (); Golov2 (). An important role in the recent developments is played by large-scale well-controlled apparata, like the Prague Ladik (); Ladik1 (); Ladik2 () and the Grenoble wind tunnelsRoche () , the Manchester spin-down Manchester-exp (); Golov2 (), the Grenoble Von Karman flows Karman (), the Helsinki rotating cryostat rev4 (); exp (); Helsinki-exp (), and some other experiments. Additional insight was provided by large-scale numerical simulations of quantum turbulence by the vortex-filament and other methods that gives direct access to detailed picture of vortex dynamics which is still unavailable in experiments, see also  Refs.Tsubota2008 (); araki (); Vin03-PRL (); cn (); RocheBarenghi (); exp3 ().

The stagnation of the energy cascade at the intervortex scale mentioned above is referred to as the bottleneck effect. This issue was studied in LABEL:LNR-1 in the approximation of a “sharp” crossover. LABEL:LNR-2 introduced a model of a gradual eddy-wave crossover in which both the eddy and the wave contributions to the energy spectrum of superfluid turbulence at zero temperature (see Eq. (10a)) were found as a continuous function of the wave vector . The main message of LABEL:LNR-2 is that the bottleneck phenomenon is robust and common to all the situations where the energy cascade experiences a continuous-to-discrete transition. The details of the particular mechanism of this transition are secondary. Indeed, most discrete physical processes are less efficient than their continuous counterparts 111It is interesting to make comparison with turbulence of weakly nonlinear waves where the main energy transfer mechanism is due to wavenumber and frequency resonances. In bounded volumes the set of wave modes is discrete and there are much less resonances between them than in the continuous case. Thus the energy cascades between scales are significantly suppressed.. On the other hand, particular mechanisms of the continuous-to-discrete transition can obviously lead to different strengths of the bottleneck effect.

The main goal of the present paper is to develop a theory of superfluid turbulence that analyzes the dynamics of turbulent superfluid He and computes its energy and vorticity spectra in the entire temperature range from up to the phase transition, “ point” K, and in the entire range of scales , from the outer (energy-containing, or energy-injection) scale down to the core radius . We put a particular focus on the crossover scales , where the bottleneck energy accumulation is expected LNR-1 (); LNR-2 ().

The main results of this paper are presented in Sec. II. Its introductory subsection,

II.1 Basic approximations and models,

overviews the basic physical mechanisms, which determine the behavior of superfluid turbulence and describes the set of main approximations and models, used for their description.
 
The rest of Sec. II is devoted to the following problems:

II.2. Temperature dependence of the energy spectra and the bottleneck effect in turbulent He;

II.3. Temperature dependence of the vorticity spectra;

II.4. Correlations between normal and superfluid motions and the energy exchange between components;

II.5. Temperature dependence of the effective superfluid viscosity in He.

Clearly, the basic physics of the large scale motions, differ from that of small scale motions. The same can be said about different regions of temperature: zero temperature limit, small, intermediate and large temperatures. It would be difficult to follow the full description of the physical picture of superfluid turbulence in all these regimes without clear understanding of the entire phenomenon as a whole. Therefore in Sec. II.1 we restricted ourselves to a panoramic overview of the main approximations and models, leaving detailed consideration of some important, but in some sense secondary issues, to the next two sections of the paper (Sec. III and Sec. IV). These include the analysis of the range of validity of the basic equations of motions, of the main approximations made in the derivation, and of the numerical procedures. These sections consist of the following subsections:

III.1. Coarse-grained, two-fluid, gradually-damped Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations;

III.2. Two-fluid Sabra shell-model of turbulent He;

IV.1. Differential approximations for the energy fluxes of the hydrodynamic and Kelvin wave motions;

IV.2. One-fluid differential model of the graduate eddy-wave crossover;

In the final Section  V we summarize our results on the temperature dependence of the energy spectra of the normal and superfluid components in the entire region of scales. We demonstrate in Fig. 5 that the computed temperature dependence of the effective viscosity agrees qualitatively with the experimental data in the entire temperature range. We consider this agreement as a strong evidence that our low-temperature, one fluid differential model and the high temperature coarse-grained gradually damped HVBK model capture the relevant basic physics of the turbulent behavior of He.

Ii Underlaying physics and the results

ii.1 Basic approximations and models

ii.1.1 Coarse-grained, two-fluid, gradually-damped HVBK equations

As we noticed in the Introduction, the large-scale motions of superfluid He (with characteristic scales ) are described using the two-fluid model as interpenetrating motions of a normal and a superfluid component with densities , and velocities , . Following LABEL:L199 we neglect variations of densities by considering them as functions of the temperature only, and . We also neglect both the bulk viscosity and the thermal conductivity. This results in the simplest form of the two incompressible-fluids model for superfluid He that have a form of the Euler equation for and the Navier-Stokes equation for , see e.g. Eqs. (2.2) and (2.3) in Donnely’s textbook 1 (). As motivated below, we add an effective superfluid viscosity term also in the superfluid equation, writing

(1a)
(1b)
Here , are the pressures of the normal and the superfluid components:
is the total density, is the kinematic viscosity of normal fluid.
The term describes the mutual friction between the superfluid and the normal components mediated by quantized vortices, which transfer momentum from the superfluid to the normal subsystem and vice versa. Following Ref. LNV (), we approximate it as follows:
(1c)
where is the characteristic superfluid vorticity.
The equations (1) are referred to as the Hall-Vinen-Bekarevich-Khalatnikov (or HVBK) coarse-grained model. The relevant parameters in these equations, are the densities and , the mutual friction parameters and the kinematic viscosity of the normal-fluid component normalized by .
The original HVBK model does not take into account the important process of vortex reconnection. In fact, vortex reconnections are responsible for the dissipation of the superfluid motion due to mutual friction. This extra dissipation can be modeled as an effective superfluid viscosity as suggested in LABEL:VinenNiemela:
(1d)

We have added a dissipative term proportional to to the standard HVBK model and the resulting Eqs. (1) [discussed in more details in Sec. III] will be referred to as the “gradually damped HVBK model”. We use this name to distinguish our model from the alternative “truncated HVBK model” suggested in LABEL:22 which was recently used for for numerical analysis of the effective viscosity in LABEL:Ladik2. We suspect that the sharp truncation introduced in the latter model creates an artificial bottleneck effect that is removed in the gradually damped model. The difference in predictions between the models will be further discussed in Sec. III.1.3.

ii.1.2 Two-fluid Sabra shell-model of turbulent He

The gradually damped HVBK Eqs. (1) provide an adequate basis for our studies of the large-scale statistics of superfluid turbulence. However their mathematical analysis is very difficult because of the same reasons that make the the Navier-Stokes equationsLP-Exact () difficult. The interaction term is much larger than the linear part of the equation (their ratio is the Reynolds number, Re), the nonlinear term is nonlocal both in the physical and in the wave-vector -space, the energy exchange between eddies of similar scales, that determines the statistics of turbulence, is masked by much larger kinematic effect of sweeping of small eddies by larger ones, etc.

Direct numerical simulations of the HVBK Eqs. (1) are even more difficult than the Navier-Stokes analog, being extremely demanding computationally, allowing therefore for a very short span of scales. A possible simplification is provided by shell models of turbulence 9 (); 10 (); 12 (); 13 (); 14 (); Sabra (); s2 (); s3 (); s4 (); s5 (); Bif (); s1 (); s2 (); s3 (); s4 (); s5 (). They significantly simplify the Navier-Stokes equations for space-homogeneous, isotropic turbulence of incompressible fluid. The idea is to consider the equations in wave vector -Fourier representation and to mimic the statistics of in the entire shell of wave numbers by only one complex shell velocity . The integer index is referred to as the shell index, and the shell wave numbers are chosen as a geometric progression , with being the shell-spacing parameter. This results in the ordinary differential equation

(2a)
Here the nonlinear term NL is linear in and quadratic in ( a functional of the set ), which usually involves shell velocities with . the kinetic energy is preserved by the nonlinear term. For example, in the popular Gledzer-Ohkitani-Yamada (GOY) shell model9 (); 10 ()
(2b)

where the asterisk stands for complex conjugation. In the limit and with , Eqs. (2) preserve the kinetic energy and has a second integral of motion . The traditional choice allows to associate with the helicity in the Navier-Stokes equations.

Note that the simultaneous rescaling , and with some factor results in a straightforward rescaling of the the time variable without any effect on the instantaneous stationary statistics of the model. Thus, the shell model (2) has only one fitting parameter , which has only little effect on the resulting statistics. The traditional choice allows to reasonably model the interactions in -space with an efficient energy exchange between modes of similar index .

We stress that with the above choice of parameters, , , and

(3)

the shell models reproduced well various scaling properties of space-homogeneous, isotropic turbulence of incompressible fluids, see Ref.Bif () and references therein. To mention just a few:

– the values of anomalous scaling exponents (see, e.g. Table I in Sabra ());

– the viscous corrections to the scaling exponents s2 ();

– the connection between extreme events (outliers) and multiscaling s3 ();

– the inverse cascade in two-dimensional turbulence s4 ();

– the strong universality in forced and decaying turbulence,s5 (), etc s1 (); s2 (); s3 (); s4 (); s5 ().

Therefore, we propose shell models are a possible alternative to the numerical solution of the HVBK Eqs. (1). This option was studied in Ref. 8 () which proposed a two-fluid GOY shell model for superfluid turbulence with an additional coupling by the mutual friction.

In our studies of superfluid turbulence Sabra1 (); Sabra2 (); BLPP-2013 () and below, we use the so-called Sabra-shell model Sabra (), with a different form of the nonlinear term:

(4)

The advantage of the Sabra model over the GOY model is that the resulting spectra do not suffer from the unphysical period-three oscillations, thanks to the strong statistical locality induced by the phase invariance Sabra (); Bif ().

We solved numerically the two-fluid Sabra-shell model form of the HVBK equations (2a) and (4) coupled by the mutual friction, for the shell velocities. Gathering enough statistics, we computed the pair- and cross-correlation functions of the normal- and the super-fluid shell velocities. This led to the energy spectra and together with the cross-correlation .

In the simulations we used shells. All the results are obtained by averaging over about 500 large eddy turnover times. The rest of details of the numerical implementation and simulations are given in Sec. III.2.

ii.1.3 Low temperature one-fluid eddy-wave model of superfluid turbulence

As we just explained, in the high-temperature region the fluid motions with scales are damped and motions with are faithfully described by the Sabra-shell model  (2a) and (4). In this approach we first solve the dynamical equation and then perform the statistical averaging numerically.

In the low temperature regime, , where the Kelvin wave motions of individual vortex lines are important this approach is no longer tenable. Instead, we adopt a different strategy, in which we first perform the statistical averaging analytically and then solve the resulting equations for the averaged quantities numerically.

To this end we begin with the dynamical Biot-Savart equation of motion for quantized vortex lines. Then we applied the Hamiltonian description to develop a “weak turbulence” formalism to the energy cascade by Kelvin waves 92ZLF (). This approach results in a closed form expression for the Kelvin wave energy spectra, derived in Ref. LN-09, :

(5a)
Here is the energy flux over small-scale region, , and . The value of the universal constant was estimated analytically in Ref. KW-2, . The dimensionless constant may be considered as the r.m.s. vortex line deflection angle at scale and is given by
(5b)

In the low-temperature region, , the density of the normal component is very small and due to very large kinematic viscosity it may be considered at rest. Therefore the large scale motions of He, , are governed by the first of HVBK Eq. (1a), which coincides with the Navier-Stokes equation in the limit . Therefore, in the hydrodynamic range of scales, , we can use the Kolmogorov-Obukhov –law Frisch () for the hydrodynamic energy spectrum:

(6)

Here is the energy flux over large scale range and is the Kolmogorov dimensionless constant.

Both spectra, (5) and (6) have the same -dependence, , but different powers of the energy flux. A way to match these spectra in the limit was suggested in LABEL:LNR-2. The idea was to adopt the differential approximations to the Kelvin-wave KW-T () and hydrodynamic-energy flux Leith67 (), based on their spectra (5a) and (6):

(7a)
(7b)
and to construct a differential approximation for the superfluid energy flux that is valid for all wave numbers (including the cross over scale):
(7c)

The additional cross-contributions and originate from the interaction of two types of motion, hydrodynamic and Kelvin waves.

For the total energy flux should be -independent, const. As explained in LABEL:LNR-2 this leads to an ordinary differential equation for the total superfluid energy . In this paper we generalize this approach to the full temperature interval with the help of the energy balance equation

(8)

The right-hand-side of this equation originates from the Vinen-Niemella viscosity in Eq. (1a) and accounts for the dissipation in the system.

Much more detailed description of this procedure can be found in Sec. IV.

ii.2 Temperature dependence of the energy spectra and the bottleneck effect in turbulent He

, K 0.43 0.55 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.16
0.003 0.007 0.014 0.026 0.045 0.0728 0.111 0.162 0.229 0.313 0.420 0.553 0.741 0.907
0.0025 0.0056 0.026 0.034 0.051 0.072 0.097 0.126 0.160 0.206 0.279 0.48 1.097
0.70 0.83 0.80 0.78 1.00 0.76 0.70 0.65 0.60 0.55 0.51 0.49 0.53 0.65 1.209
1.09 0.43 0.27 0.17 0.12 0.10 0.10 0.09 0.09 0.09 0.09 0.093 0.101 0.124 0.154
1179 148 38 11.1 4.62 2.34 1.32 0.84 0.56 0.39 0.29 0.22 0.182 0.167 0.170
0.0067 0.022 0.040 0.061 0.099 0.101 0.135 0.171 0.207 0.234 0.237 0.280 0.312 0.427 0.815
Table 1: The parameters of the superfluid He, taken from Refs. VinenNiemela (); DonnelyBarenghi98 (): the relative density of the normal component , the mutual friction parameter , the combination [which weakly depends temperature and is responsible for the mutual friction density in Eq. (1)], He-II kinematic viscosity ( is the dynamic viscosity) and the kinematic viscosity of the normal-fluid component ; the effective superfluid viscosity (inter- and extrapolation of Ref. VinenNiemela () results).

To discuss our results we define the energy spectrum of isotropic turbulence (in one-dimensional -space) such that

(9)

is the energy density of He per unit mass. Hereafter stands for the “proper averaging” which may be time averaging over long stationary dynamical trajectory or/and space averaging in the space-homogeneous case, or an ensemble averaging in the theoretical analysis. Assuming that the Navier-Stokes dynamics are ergodic, all these types of averaging are equivalent.

In the low temperature range, where He consists mainly of the superfluid component, we distinguish the spectrum of large scale hydrodynamic motions with , denoted as , from the spectrum of small scales Kelvin waves (with ), denoted as . The total superfluid energy spectrum is written as

(10a)
In the high temperature range, where the densities of the super-fluid and normal-fluid components are comparable, but Kelvin waves are fully damped, we will distinguish the spectrum of hydrodynamic motions of the superfluid component at large scales as , from that of the normal-fluid component, , omitting for brevity the subscript “”. In this temperature range, the total energy spectrum of superfluid He is written as
(10b)

The resulting energy spectra , and for a set of eleven temperatures from K to K are shown in Fig. 1.

(a) Low , one-fluid model (b) High , two fluid model
Figure 1: Color online. Log-Log plots of the energy spectra in superfluid He compensated by the inertial range scaling at different temperatures (shown as labels). Panel (a): Plots of the (compensated by and normalized by their values at ) full superfluid energy spectra (solid lines) and their hydrodynamic (large scale) parts (dashed lines) for the one-fluid model (Sec. IV.2). Panel (b): Plots of the (compensated by the anomalous scaling and normalized by their inertial range value) shell energies of the normal fluid component (solid lines) and of the superfluid component (dashed lines) for the two-fluid shell model (Sec. III.2).

ii.2.1 Low-temperature one-fluid energy spectra

First, we discuss the results for the eddy-wave model of superfluid turbulence [cf. Sec.  II.1.3], for the low temperature range K, which is shown in Fig. 1a. These spectra are compensated by a factor such that both the Kolmogorov-Obukhov-41 spectrum , Eq. (6) (for the hydrodynamic scales ) and the Lvov-Nazarenko spectrum , Eqs. (5a) show up as a plateau. These plateaus are clearly seen for the lowest shown temperature K. Moreover, the full energy spectrum (solid blue line) demonstrates the existence of an important bottleneck energy accumulation. We observe a large cross over region connecting the HD region , where with a much higher plateau for where .

In the cross over region the compensated energy spectrum is close to (cf. the black dashed line), meaning that depends on only weakly. In this region the energy spectrum is dominated by Kelvin waves, , while the energy flux in dominated by the HD eddy motions. Therefore we have here a flux-less regime of Kelvin waves. Without flux the situation resembles thermodynamic equilibrium, in which the Kelvin waves energy spectrum corresponds to energy equipartition between the degrees of freedom, i.e. const, as observed.

For the Kelvin waves energy spectrum at K is suppressed by the mutual friction, as explained in Sec. IV.1. In Fig.1a this part of the spectrum is not shown; however one sees progressive suppression of the energy spectra with temperature increasing from 0.44 K (around ) to K (around ). It is important to notice that for K the HD part of the spectrum is practically temperature independent; only the Kelvin waves energy spectra are suppressed by the temperature, cf. the coinciding dashed lines in Fig. 1a for and 0.49  K.

For K, the Kelvin wave contributions to the energy spectra are very small – the solid and the dashed lines for the same temperature are fairly close. Finally, the dashed and the solid lines for K practically coincide, i.e. the Kelvin waves are fully damped. This means that for K there is no need to account for the Kelvin wave motions on individual vortex lines, and the full description of the problem is captured by the coarse-grained HVBK.

(a) Low , one-fluid model (b) High , two fluid model
Figure 2: Color online. Color online. Linear-log plots of the vorticity spectral densities, , normalized by their values, at different temperatures. The total mean-square vorticity is proportional to the area under the plot. Panel (a): Superfluid vorticity spectra at low temperatures in the one-fluid model, Sec. IV.2. Panel (b): Normal and superfluid vorticity spectra (solid lines) and (dashed lines) at high temperatures in the two-fluid model, Sec. III.2.

ii.2.2 High-temperature two-fluid energy spectra

The energy spectra, obtained with the Sabra-shell model form of HVBK equations (2a) and (4), for temperatures K, are shown Fig. 1b for K (in blue), K (in magenta), K (in green) and K (in red). The lowest temperature in this two-fluid approach, K, was chosen for comparison with the highest temperature K in the one-fluid approach; see Fig. 1a. At K, which is a frequently used temperature in numerical simulations of superfluid turbulence, the normal fluid component is not negligible (), and the normal fluid kinematic viscosity is still much larger than that of the superfluid: . For , when , the kinematic viscosities are close to each other (see Tab. 1). At higher temperatures the normal fluid components play more and more important role until they dominate at K, when . At the highest temperature in this simulation, , close to , we have , and the effective superfluid kinematic viscosity is even larger than .

Shell-model simulations reproduce intermittency effects and therefore the scaling exponent of the energy spectra slightly differs from the KO-41 prediction, . For the chosen shell-model parametersBLPP-2013 (); Sabra () which is quite close to the experimental observations. For better comparison with the low-temperature one-fluid results of Fig. 1a, we show in Fig. 1b the normal (solid lines) and superfluid (dashed lines) energy spectra and , compensated by so that they exhibit a plateau in the inertial interval of scales.

As expected, for K, when , the superfluid and normal fluid spectra are very close, and similar to the spectra of classical fluids. In the inertial range they demonstrate the anomalous behavior ] with the scaling exponent . Moreover, due to the strong coupling between the normal and superfluid component (discussed below in Sec. II.4) the energy fluxes in both components are equal (see, e.g. Fig. 4), and therefore the energies are equal in the inertial interval as well, . Non-trivial behavior occurs only in the inertial-viscous crossover region; therefore the inertial interval is not shown in Fig. 1.

For , when , the viscous cutoff of the normal fluid’s spectrum, , occurs at much smaller than the cutoff of the superfluid spectrum, . To estimate the ratio , notice that in the KO-41 picture of turbulence may be found by balancing the eddy-turnover frequency,

(11)

with the viscous dissipation frequency . This gives the well known result

(12)

In our case . Therefore, neglecting the energy exchange between the super- and the normal-fluid components, we get an estimate:

(13)

For , when this gives —in a good agreement with the result in Fig. 1b. For K, the ratio of the viscosities is smaller (about , see Tab. 1). Therefore the difference in cutoffs is less pronounced. As expected, for K, when the situation is the opposite, and the superfluid component is damped at a smaller than the normal one.

Notice that there is no bottleneck energy accumulation in the spectra (see Figs. 1b) obtained using the shell model approximation of the gradually damped HVBK equations. This is qualitatively different from the results of the truncated HVBK model 22 (), which demonstrated a very pronounced bottleneck both in the normal and the superfluid components, e.g. at K. The latter would lead to a huge contribution to the mean square superfluid vorticity and, as a result, to a very small effective Vinen’s viscosity . This would definitely contradict the experimental observation shown in Fig. 5. We will discuss this issue in greater detail in Sec. II.5.

ii.3 Temperature dependence of the vorticity spectra in turbulent He

At this point we cannot compare our predictions for energy spectra with experimental observations, especially in the cross-over and in the small scale regions. This stems from the lack of small probes, see cf. the review BLR (). On the other hand, the attenuation of second sound or ion scattering may be used to measure the mean vortex line density in He or even its time and space dependence BLR (). In turn, the value can be expressed in terms of the mean-square superfluid vorticity via the quantum of circulation  Vin-2001 ():

(14)

Therefore, the information about the vorticity is very important from the viewpoint of comparison with available and future experiments.

By analogy with the energy spectra (9), let us define the power spectra of vorticity so that the mean-square vorticity is given by the integral:

(15)

In isotropic incompressible turbulence . Therefore we define

(16)

For brevity, we omit the subscript “” for the normal component; .

Plots of for different temperatures are shown in Figure 2. According to Eq. (15), the area under these plots is proportional to the total mean square vorticity, . Fig. 2a shows the results for the eddy-wave model (the corresponding energy spectra for the same temperatures are shown in Fig. 1a). One sees that the largest (and temperature independent) value of is reached for K: plots for and K practically coincide. Accordingly, the temperature range K may be considered as zero-temperature limit with the maximal value of (and correspondingly, the smallest value of , as we will discuss later). At temperatures above 0.5K the area under the plots decreases (and correspondingly, increases).

In Fig. 2b we show vorticity spectra of the normal-fluid (solid lines) and the superfluid components (dashed lines) for different temperatures obtained in the framework of the Sabra-shell model (the corresponding spectra are shown in Fig. 1b). Again, the area under the plots is proportional to the total mean square vorticity . One sees that for the lowest temperature K the normal fluid vorticity (blue solid line) is fully suppressed by the huge normal viscosity, while the superfluid vorticity is very large. At this temperature one can describe the superfluid He in the range of scales using a one-fluid approximation with zero normal-fluid velocity. This provides the main contribution to the vorticity. To some extent, this situation persists up to K, when the superfluid vorticity is still larger than the normal one, see Fig. 1b. As expected, for , when the normal and superfluid viscosities are compatible, the normal and superfluid vorticities are very close. For these and higher temperatures the analysis of our problems definitely calls for a two-fluid description.

ii.4 Correlations of normal and superfluid motions and energy exchange between components

High , two-fluid model
Figure 3: Color online. Cross-correlation coefficients , Eq. (20a) (solid lines) and , Eq. (20b) (dashed lines) for different temperatures. Color code is the same as in Fig. 1b and Fig. 2b: K – blue, K – cyan, K – green, and K – red.

ii.4.1 Correlations of the normal and superfluid velocities

It is often assumed (see e.g. Ref.VinenNiemela ()) that the normal and superfluid velocities are “locked” in the sense that

(17)

(at least in the inertial interval of scales). For quantitative understanding to which extent this assumption is statistically valid we consider the simplest possible case of stationary, isotropic and homogeneous turbulence. Here we introduce a cross-correlation function (in 1D -representation) of the normal and the superfluid velocities . This correlation function is defined using the simultaneous, one-point cross-velocity correlation similarly to Eq. (9):

(18)

If, for example, motions of the normal and the superfluid components at a given are completely correlated, then . If this is true for all scales, then Eq. (17) is valid.

It is natural to normalize by the normal and the superfluid energy densities, and . This can be reasonably done in one of two ways:

(19a)
(19b)

Both coefficients are equal to unity for fully locked superfluid and normal velocities, Eq. (17), and both vanish if the velocities are statistically independent. However, if , with then , but still . In any case .

In shell models, the coefficients and can be written as follows:

(20a)
(20b)

These objects are shown in Fig. 3. At first glance, it is surprising that the correlations (dashed lines in Fig. 3) for K persist for much larger wave vectors than , approaching . For example, for K (blue lines) vanishes at , while all the way up to . In this range of scales () , but , meaning that strongly damped normal velocity does not have its own dynamics and should be considered as “slaved” by the superfluid velocity. The damped velocity (normal or superfluid) at any temperature K would follow this ”slaved” dynamics.

A model expression of the cross-correlation in terms of the self-correlation functions and was found in LABEL:L199. In current notations it reads:

(21)

where the characteristic interaction frequencies (or turnover frequencies) of eddies in the normal and superfluid components, and , are given by Eq. (11) and is defined as:

(22)

The derivation of Eq. (21) in LABEL:L199 involves diagrammatic perturbation approach and is rather cumbersome. However the simplicity of the final result (21) motivated us to re-derive it in a simple and transparent way which is presented in the Appendix.

Let us analyze first Eq. (21) in the inertial interval of scales, where according to Fig. 1b, and the terms with the viscosities in the denominator may be neglected. In this case

where the viscous cutoff of the superfluid inertial interval is given by estimate (12). First of all we see that the correlation coefficient is governed by the dimensionless parameter which involves the mutual friction coefficient , as expected. What is less expected, is that this parameter, according to Tab. 1, depends on the temperature only weakly and is close to unity. Therefore, in the inertial interval we have:

and this expression is very close to unity. In the other words, in the inertial interval we expect the full locking of the normal and the superfluid velocities for all temperatures. This prediction fully agrees with the observations in Fig. 3.

Consider now case K, when according to the data in Fig. 1b and estimate (13). For we have:

(25)

Then Eq. (21) simplifies to the following form,

and it may be analyzed as follows:

Using (13), for we get

(26)

We see that the velocities decorrelate in the interval , as expected.

Estimating in the regime (25) is less simple, because it requires knowledge of the ratio in terms of and . Instead, we can directly use Eq. (61b), which in regime  (25) may be simplified (in the -representation) as follows:

(27)

i.e. is slaved by . Equation (27) immediately gives , but in full agreement with our results in Fig. 3. In particular, this means that our simple model of correlations between and , suggested in the Appendix, quantitatively correctly reflects the basic physics of this phenomenon.

High , two fluid model
Figure 4: Color online. Temperature dependence of the ratios: – horizontal black line; , Eq. (28a), – blue line with triangles, , Eq. (28a), – green line with diamonds, , Eq. (28b), – red line with circles.

ii.4.2 Energy dissipation and exchange due to mutual friction

Strong coupling of the normal and the superfluid velocities suppresses the energy dissipation and the energy exchange between the normal and the superfluid components caused by the mutual friction (which is proportional to , Eq. (1c)). Nevertheless, some dissipation due to the mutual friction is still there. Consider the ratio of the total injected energy to the total energy dissipated due to the viscosity in the normal and the superfluid components:

(28a)
a quantity plotted in Fig. 4 (red line with circles). Here and are the inertial range normal and superfluid energy fluxes. This ratio exceeds unity by about 10%, meaning that of the injected energy is dissipated by the mutual friction. As expected, this effect disappears at K, when the effective superfluid and normal fluid kinematic viscosities are matching (and therefore ).
The mutual friction has a significantly more important influence on the energy exchange between the normal and the superfluid components. The energy exchange can be quantified by a similar ratio defined for each fluid component,
(28b)

shown by a green line with diamonds and a blue line with triangles respectively in Fig. 4. At the lowest shown temperature K, we have meaning that only about 10% of the energy density (per unit mass) which is dissipated by the normal fluid component comes from the direct energy input. The rest of the energy density dissipated by viscosity (at large ) was transferred from the superfluid component by the mutual friction. This is because for K, we have and therefore the normal velocity becomes more damped at lower wavenumbers than the superfluid velocity, see Fig. 1b. Such an energy transfer by mutual friction from the superfluid component to the normal one increases (blue line with triangles) above unity. This effect is smaller than the one for because at low temperatures and the energy per unity volume is approximately conserved. As expected, there is no energy exchange between the components at K, when and ). At this temperature . Again, as expected for K, when (see Tab. 1) we have , meaning that the energy goes from the less damped normal component to the more damped superfluid one.

To understand why the energy exchange due to the mutual friction is larger than the energy dissipation by the mutual friction, notice that the energy exchange is proportional to the (small) velocity difference, , while the energy dissipation is proportional to the square of this parameter, .

Low , one-fluid model    High , two-fluid model
Figure 5: Color online. Comparison of the experimental, numerical and analytical results for the temperature dependence of the effective kinematic viscosities: Blue triangles – Manchester spin-down experiments Manchester-exp (); Green empty circles – Manchester ion-jet experiments Golov2 (); sea-green diamonds with error-bars – Prague counterflow experiments Ladik (); cian crosses with error-bars – Prague decay in grid co-flow experiments Ladik1 (); Magenta empty squares – Oregon towed grid experiments stalp ();Pink right triangles–Oregon towered grid experiments nimiela2005 (). Solid green line – experimental results DonnelyBarenghi98 () for the normal-fluid kinematic viscosity (normalized by the normal-fluid density); Dashed green line – He-II kinematic viscosity , (normalized by the total density) – see also Tab. 1. Thin black dash line – effective viscosity for the random vortex tangle , estimated by Eq. (31); Thick dot-dashed black line – the Vinen-Niemela estimate VinenNiemela () of the effective superfluid viscosity, , given by Eq. (32). Blue solid line – at K from numerical solution of Eqs. (51c) for the one-fluid differential model of gradual eddy-wave crossover; Red solid line – at K from numerical simulations in Sec. III.2 of gradually damped two-fluid HVBK Eqs. (1) in the Sabra shell-model approximation (33).

ii.5 Temperature dependence of the effective superfluid viscosity in He

The temperature-dependent effective (Vinen’s) viscosity is defined stalp () by the relation between the rate of energy-density (per unit mass) flux into turbulent superfluid, , and the vortex-line density, :

(29)

According to Eq. (15), is proportional to the area under the plots vs. , shown in Fig. 2 and discussed in Sec. II.3. These results allow us to determine the viscosity (analytically and numerically) in the entire temperature range from up to .

ii.5.1 Low temperature range

Consider first the temperature dependence of in the low temperature range K, shown in Fig. 5 by the solid blue line. This dependence is found in Sec. IV.2 in the framework of one-fluid model of gradual eddy-wave crossover Eqs. (51). As we mentioned, the largest (and temperature independent) value of (at fixed value of ) is reached for K. Accordingly, the temperature range K may be considered as a zero-temperature limit, at which reaches its smallest value. The results of the Manchester spin-down experiment Manchester-exp () are temperature independent as well (within the natural scatter of the data). The particular value of found in these experiments is probably accurate up to a numerical factor due to uncertainty in the determination of the outer scale of turbulence, taken in Ref. Manchester-exp () for simplicity as the size of the cube. Our low-temperature, one-fluid model (51) involves one fitting parameter, which determines the crossover scale in the blending function Eq.(48). This parameter affects the resulting value of and was chosen such as to meet its accepted experimental value .

At temperatures above 0.5 K, the area under the plots in Fig. 2a become smaller and smaller. This is caused by the suppression of the Kelvin wave spectra, which is more pronounced at larger temperature, as seen in Fig. 1a. The value of decreases with the temperature resulting in a progressive increase in as shown by the solid blue line in Fig. 5 together with the experimental (Manchester spin-downManchester-exp () and ion-jetGolov2 ()) values of . There is a reasonably good agreement between the temperature dependence of found in the framework of the one-fluid model of eddy-wave crossover at low temperatures and the experiments. Importantly, in the modeling we have used only one phenomenological parameter to fit the zero- limit of , while the temperature dependence of the latter follows from the model without any additional fitting.

ii.5.2 High temperature range

At K, Kelvin waves are already fully damped; see Fig. 1a. This means that for these temperatures we can use the coarse-grained HVBK Eqs. (1). Using the shell model approximation we find the temperature dependence of in the temperature range K as shown in Fig. 5 by the solid red line. In the intermediate temperature range KK this line overlaps with the blue solid line, showing the one-fluid results. The reason for this overlap is very simple: for KK, the Kelvin waves are already damped (see Fig. 1a), and the normal-fluid eddies at scales