Local and non-local energy spectra of superfluid $^3$He turbulence
L. Biferale, D. Khomenko, V. L'vov, A. Pomyalov, I. Procaccia, G. Sahoo
aa r X i v : . [ c ond - m a t . o t h e r] J a n Local and non-local energy spectra of superfluid He turbulence
L. Biferale, D. Khomenko, V. L’vov, A. Pomyalov, I. Procaccia and G. Sahoo
Below the phase transition temperature T c ≃ − K He-B has a mixture of normal and super-fluid components. Turbulence in this material is carried predominantly by the superfluid component.We explore the statistical properties of this quantum turbulence, stressing the differences from thebetter known classical counterpart. To this aim we study the time-honored Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid turbulence. We combine pseudo-spectral directnumerical simulations with analytic considerations based on an integral closure for the energy flux.We avoid the assumption of locality of the energy transfer which was used previously in both analyticand numerical studies of the superfluid He-B turbulence. For
T < . T c , with relatively weakmutual friction, we confirm the previously found “subcritical” energy spectrum E ( k ), given by asuperposition of two power laws that can be approximated as E ( k ) ∝ k − x with an apparent scalingexponent < x ( k ) <
3. For
T > . T c and with strong mutual friction, we observed numericallyand confirmed analytically the scale-invariant spectrum E ( k ) ∝ k − x with a ( k -independent) expo-nent x > x ∼ T ≈ . T c .In the near-critical regimes we discover a strong enhancement of intermittency which exceeds by anorder of magnitude the corresponding level in classical hydrodynamic turbulence. Introduction
Helium below the phase transition temperatures T λ ≃ . He and T c ≃ − K in He can be described asconsisting of two coupled, interpenetrating fluids. Onefluid is inviscid with quantized vorticity, and the secondis viscous with a continuous vorticity. Consequently, su-perfluid turbulence is even more complex than turbulencein classical fluids. Moreover, the present knowledge ofmany aspects of superfluid turbulence is still not fullydeveloped despite the many decades since the discoveryof superfluidity, see, e.g. Refs. . The subject offersmany opportunities for new approaches and new discov-eries.From the experimental point of view the study of thestatistical properties of superfluid turbulence is still dif-ficult, even with the use of state-of-the-art technologies.The very low values of T λ and T c limit severely any vi-sual access, and in addition pose problems for adequatesensors . Nevertheless new experiments are emerging,requiring parallel theoretical efforts. Theoretical progressrequires developing direct numerical simulations (DNS)which presently are the only way to reach a completedescription of the evolution of the normal and superfluidvelocity components. Such data offer access to the statis-tical properties of superfluid turbulence. In the presentpaper we study the physics of superfluid He turbulence,using the fact that it is simpler problem than turbulencein He, due to very high viscosity of the normal compo-nent, which may be considered laminar.The energy spectra E ( k ) in space-homogeneous, steadyand isotropic turbulence in superfluid He were studiedanalytically within the algebraic approximation for theenergy flux in Ref. (see also Eq. (2b) below). Numeri-cally the issue was studied using the Sabra-shell model inRef. . The two papers considered the large-scale ve-locity fluctuations with k < π/ℓ , where ℓ is the mean dis-tance between quantized vortex lines. It was shown thatthe mutual friction between normal and superfluid com- ponents suppresses E ( k ) with respect of the Kolmogorov-1941 (K41) prediction : E K41 ( k ) = C K ε / k − / . (1)Here ε is the energy flux over scales, equal in this caseto the rate of energy input into the system at k = k : ε = ε ( k ); C K ∼ He turbulence was analyzed byLvov, Nazarenko and Volovik (LNV) in Ref. : d ε ( k ) dk + Ω E ( k ) = 0 , Ω = α ( T ) Ω T . (2a)Here α ( T ) is the temperature dependent dimensionlessmutual friction parameter and Ω T is the root mean square(rms) turbulent vorticity. The wavenumber-dependentenergy flux over scales, ε ( k ), was approximated in Ref. using K41-type dimensional reasoning, similar to Eq. (1): ε ( k ) = (cid:2) E ( k ) (cid:14) C K (cid:3) / k / = 83 (cid:2) E ( k ) (cid:3) / k / , (2b)as suggested by Kovasznay .The ordinary differential Eq. (2) has an analytical so-lution : E ( k ) = E K41 ( k ) h − Ω † + Ω † (cid:16) k k (cid:17) / i , where (3a)Ω † = ΩΩ cr , Ω cr = 54 q k E , E ≡ E ( k ) . (3b)For Ω † < k × = k (cid:2) Ω † (cid:14) (1 − Ω † ) (cid:3) / , (3c)that breaks the scaling invariance, predicting for Ω < Ω cr a superposition of two scaling laws:– For small k ≪ k × , the LNV spectrum (3a) takes a“critical” form E cr ( k ) = E (cid:0) k /k (cid:1) . (3d)– For large enough k ≫ k × , the K41 spectrum (1) isrecovered, but with the energy flux ε ∞ < ε . The dif-ference ε − ε ∞ is dissipated by the mutual friction. For k ∼ k × , the energy spectrum can be roughly approxi-mated as E ( k ) ∝ k − x with an apparent scaling exponent < x ( k ) < k × increases with α ( T ) andfor some critical value of α cr ∼ k < k < π/ℓ .For α ( T ) > α cr , the spectrum (3a) becomes “super-critical” and terminates at some final k ∗ that depends on α ( T ): E s ( k ) ∝ k − (cid:2) k / ∗ − k / (cid:3) . (3e)All types of the LNV spectra (subcritical, critical and su-percritical) where observed in Sabra-shell model simula-tions (see Refs. and for a general review on shell mod-els). However, the analytical LNV model is based onan uncontrolled algebraic approximation for the energyflux (2b); the shell-model of turbulence, used in Ref. , isalso an uncontrolled simplification of the basic equationsof motion for the superfluid velocity field. Therefore, theproblem of turbulent energy spectra in superfluid Herequires further investigation.In this paper we report results of a first (to the best ofour knowledge) DNS study of the statistical propertiesof a space-homogeneous, steady and isotropic turbulencein superfluid He. We provide results on the turbulentenergy spectra, the velocity and vorticity structure func-tions at different temperatures 0 < T < . T cr , the en-ergy balance and intermittency effects. To these aims weuse the gradually-damped version of the Hall-Vinen -Bekarevich-Khalatnikov (HVBK) coarse-grained two-fluid model Eq. (4) as suggested in Ref. . We expectthis model to describe properly the turbulent velocityfluctuations in superfluid He and He as long as thetheir scales exceed the mean intervortex distance ℓ .The paper is organized as follows: • Section I is devoted to an analytical descriptionof the statistical properties of the steady, homogeneous,isotropic, incompressible turbulence of superfluid He.This should serve as a basis for further studies ofsuperfluid turbulence in more complicated or/andrealistic cases: anisotropic turbulence, transient regimes,two-fluid turbulence of counterflowing, thermally driven,superfluid He turbulence, etc.In Sec. I A we present the gradually damped HVBKEqs. (4);In Sec. I B we introduce the required statistical objects. In Sec. I C we adapt the integral closure to obtain theenergy spectrum when the energy transfer overscales is not local.In Sec. I D we analyze the relations between the struc-ture functions of the velocity and vorticity fieldswith the sub- and super-critical energy spectra E ( k ). These are required for the analysis of theDNS data. • Section II presents the DNS results for the statisticsof superfluid turbulence in He, together with a compar-ison with the theoretical expectations.In Sec. II A we shortly describe the details of the numer-ical procedure;In Sec. II B we present the DNS results for the energyspectra obtained for different values of mutual fric-tion frequency Ω in the subcritical, critical and su-percritical regimes. We demonstrate their quanti-tative agreement with the corresponding theoreti-cal predictions, given by Eqs. (3a), (3d) and (16);In Sec. II C we report a significant enhancement of inter-mittency in near-critical regimes of superfluid Heturbulence, revealed by analysing the second- andfourth-order structure functions of the velocity andvorticity differences;In Sec. II D we analyze the energy balance in the en-tire region of k , shedding light on the origin of thesubcritical, critical and supercritical regimes of theenergy spectra;In Sec. II E we present and analyze the DNS results forthe energy and enstropy time evolution, showinghow the large and small scale turbulent fluctua-tions are correlated (or uncorrelated) in differentregimes;Section II F clarifies the relation between the mutualfriction frequency Ω and the temperature T in pos-sible experiments. • Section III summarizes our findings. For the conve-nience of the reader we present here the main results:The numerical subcritical energy spectra for different
T < . T cr (see Tab. I and Fig. 1a), are in goodagreement with the LNV prediction (3a) with a sin-gle fitting parameter b ≈ . in Eq. (3b).At T ≈ . T cr (corresponding to Ω = 0 . E cr ∝ /k .The numerically observed supercritical energy spectraat T > . T cr exhibit a scale-invariant behavior E ( k ) ∝ k − x , Eq. (16a) with the scaling exponent x > x ∼ T ≈ . T cr .Relaxing the assumption of locality by using integralclosure for the energy flux (11), we confirmed ana-lytically the scale-invariant spectrum E ( k ) ∝ k − x ,Eq. (16a) with the variable scaling exponent x thatdepends on the temperature in a qualitative agree-ment with the DNS observation.In the near-critical regimes we observed significant in-crease in turbulent fluctuations of superfluid veloc-ity and vorticity at small scales, typical for inter-mittency. I. ANALYTIC DISCUSSION OF THESTATISTICS OF HE TURBULENCEA. Gradually damped HVBK-equations forsuperfluid He-B turbulence
Large scale turbulence in superfluid He can be de-scribed by the Landau-Tisza two-fluid model in which theinterpenetrating normal and superfluid components havedensities ρ n , ρ s and velocity fields u n ( r , t ), u s ( r , t ), re-spectively. The gradually damped version of the coarse-grained HVBK equations for incompressible motions ofsuperfluids with constant densities has the form of twoNavier-Stokes equations supplemented by mutual fric-tion: ∂ u s ∂t + ( u s · ∇ ) u s − ρ s ∇ p s = ν s ∆ u s + f ns , (4a) ∂ u n ∂t + ( u n · ∇ ) u n − ρ n ∇ p n = ν n ∆ u n − ρ s ρ n f ns , (4b) p n = ρ n ρ [ p + ρ s | u s − u n | ] , p s = ρ s ρ [ p − ρ n | u s − u n | ] , f ns ≃ α ( T ) Ω T ( u n − u s ) . (4c)Here p n , p s are the pressures of the normal and the super-fluid components. ρ ≡ ρ s + ρ n is the total density, ν n isthe kinematic viscosity of normal fluid component. Thedissipative term with the Vinen’s effective superfluid vis-cosity ν s was added in Ref. to account for the energydissipation at the intervortex scale ℓ due to vortex re-connections and similar effects. A qualitative estimate ofthe effective viscosity ν s ≃ ακρ s /ρ follows from a modelof a random vortex tangle moving in a quiescent normalcomponent .The approximate Eq. (4c) for the mutual friction force f ns was suggested in Ref. . It involves the tempera-ture dependent dimensionless mutual friction parame-ters α ( T ) and rms superfluid turbulent vorticity Ω T . Inisotropic turbulenceΩ T ≡ (cid:10) | ω | (cid:11) ≈ Z k E s ( k ) dk , (5)where E s ( k ) is the one-dimensional (1D) energy spec-trum, normalized such that the total energy density perunit mass E s = R E s ( k ) dk . Note that in Eq. (4) we did not account for the reactivepart of the mutual friction , proportional to anothertemperature dependent parameter α ′ . As was shown inRef. , this force leads to a renormalization of the non-linear terms in Eq. (4a) by a factor (1 − α ′ ). DividingEq. (4a) by this factor, we see that (besides the renor-malization of time) we get also the renormalization of α ⇒ e α = α/ (1 − α ′ ) in Eq. (4c), which now reads: f ns ≃ Ω ( u n − u s ) , Ω = e α ( T ) Ω T , e α = α/ (1 − α ′ ) . (6)Ideally, the turbulent vorticity Ω T should be calcu-lated self-consistently, at each time step. However we usea simplified version, by first solving Eqs. (4) with somevalue of Ω, then calculating Ω T by Eq. (5) with the ob-served E s ( k ) and finally finding α DNS = Ω / Ω T . Afterthat we identify the temperature to which the particularsimulation corresponds by comparing with known exper-imental values α ( T ) = α DNS . We have verified that in thepresent range of parameters, simulations with a constantvalue of Ω and self-consistent simulations give similar re-sults.
B. Statistical description of space-homogeneous,isotropic turbulence of superfluid He
1. Definition of 1-D energy spectra and cross-correlations
Traditionally one describes the energy distributionover scales in a space-homogeneous, isotropic case usingthe one-dimensional (1D) energy spectrum E ( k ), definedby Eq. (9). To clarify this definition we need to recallsome well known relationships.Fourier transforms are defined with the following nor-malization: u n,s ( r , t ) ≡ Z d k (2 π ) e u n,s ( k , t ) exp( i k · r ) , (7a) e u n,s ( k , t ) = Z d r u n,s ( r , t ) exp( − i k · r ) . (7b)Next we define the simultaneous correlations and cross-correlations in k -representation, [proportional to δ ( k + q )and δ ( k + q + p ) due to the space homogeneity]: h e u n ( k , t ) · e u n ( q , t ) i = (2 π ) F nn ( k ) δ ( k + q ) , (8a) h e u s ( k , t ) · e u s ( q , t ) i = (2 π ) F ss ( k ) δ ( k + q ) , (8b) h e u n ( k , t ) · e u s ( q , t ) i = (2 π ) F ns ( k ) δ ( k + q ) , (8c) (cid:10)e u ξ s ( k , t ) e u β s ( q , t ) e u γ s ( p , t ) (cid:11) = (2 π ) F ξβγ sss ( k , q , p ) δ ( k + q + p ) . (8d)In the isotropic case the correlations F nn , F ss and F ns become independent of the direction of k , being functionsof the wavenumber k only. This allows us to introducethe one-dimensional energy spectra E s , E n and the cross-correlation E ns as follows: E n ( k ) = k π F nn ( k ) , E s ( k ) = k π F ss ( k ) ,E ns ( k ) ≡ k π F ns ( k ) . (9)
2. Energy balance equation
To derive the energy balance equation for E s ( k, t ) wefirst need to Fourier transform Eq. (4a) to get the equa-tion for e u s ( k , t ). Next, using Eq. (8b) and Eq. (9), wearrive to the required balance equation: ∂E s ( k ) ∂t + Tr( k ) + D ν ( k ) + D α ( k ) = 0 , (10a)D ν = 2 ν s k E s ( k ) , D α = 2 Ω (cid:2) E s ( k ) − E ns ( k ) (cid:3) . (10b)Here D ν describes the energy dissipation, caused by theeffective viscosity. The term D α is responsible for theenergy dissipation by the mutual friction with the char-acteristic frequency Ω given by Eqs. (4c) and (5).The energy transfer term Tr( k ) in Eq. (10a) originatesfrom the nonlinear terms in the HVBK Eqs. (4a) andhas the same form as in classical turbulence (see, e.g.Refs. ):Tr( k ) = 2 Re n Z V ξβγ ( k , q , p ) F ξβγ ( k , q , p ) × δ ( k + q + p ) d q d p (2 π ) o , (10c) V ξβγ ( k , q , p ) = i (cid:16) δ ξξ ′ − k ξ k ξ ′ k (cid:17) × (cid:16) k β δ ξ ′ γ + k γ δ ξ ′ β (cid:17) . (10d)Importantly, Tr( k ) preserves the total turbulent kineticenergy: Z k Tr( k ′ ) dk ′ = 0 and therefore can be writtenin the divergent form:Tr( k ) = ∂ ε ( k ) dk , (10e)where ε ( k ) is the energy flux over scales. C. Supercritical energy spectra
1. LNR integral closure
To relax the assumption of the local energy trans-fer in deriving the supercritical superfluid energy spec-trum, we use the integral closure, introduced by L’vov,Nazarenko and Rudenko (LNR). The main approxima-tion in this closure is the presentation of the third order velocity correlation function F ξβγ sss in Eq. (10c) as a prod-uct of the vertex V , Eq. (10d), two second order correla-tions F ss ( k j ), Eq. (8b), and response (Green’s) functions.This closure is widely used in analytic theories of classi-cal turbulence, for example in the Eddy-damped quasi-normal Markovian closure (EDQNM) (see, e.g. booksRef. ). Keeping in mind the uncontrolled characterof this approximation, LNR further simplified the result-ing approximation for isotropic turbulence by replacing d q d p δ ( k + q + p ) in Eq. (10c) with 3-dimensional vec-tors k , q , and p by q dq p dp δ ( k + q + p ) / ( k + q + p )with one-dimensional vectors k , q , and p varying inthe interval ( −∞ , + ∞ ). The next simplification is thereplacement of the interaction amplitude V ξβγ ( k , q , p ),Eq. (10d) by its scalar version ( ik ). The resulting LNRclosure can be written as follows:Tr( k ) = A k π Z ∞−∞ q dq p dp δ ( k + q + p )2 π ( k + q + p ) (11) × k F ss ( | q | ) F ss ( | p | ) + q F ss ( | k | ) F ss ( | p | ) + p F ss ( | q | ) F ss ( | k | )Γ( | k | ) + Γ( | q | ) + Γ( | p | ) . Here A is a dimensionless parameter of the order of unityand Γ( k ) is the typical relaxation frequencies on the scale k . The LNR model (11) satisfies all the general closurerequirements: it conserves energy, R Tr( k ) dk = 0 forany F k ; Tr( k ) = 0 for the thermodynamic equilibriumspectrum F k =const and for the cascade K41 spectrum F ( k ) ∝ | k | − / . Importantly, the integrand in Eq. (11)has the correct asymptotic behavior at the limits ofsmall and large q/k , as required by the sweeping-freeBelinicher-L’vov representation, see Ref. . This meansthat the model (11) adequately reflects contributions ofthe extended interaction triads and thus can be used forthe analysis of the supercritical spectra.
2. Supercritical spectra with non-local energy transfer
As was shown in Ref. , the eddy life time in He tur-bulence is restricted by the mutual friction, which dom-inates the dissipation due to the effective viscosity ν s k and the turbulent viscosity, caused by the eddy inter-actions. Therefore we can safely approximate Γ( k ) inEq. (11) by Ω. Omitting further the (uncontrolled) pref-actors of the order of unity and using Eq. (9), we rewriteTr( k ) in Eq. (10a) as followsTr( k ) ≃ − A k Ω ∞ Z −∞ dq dp δ ( k + q + p ) k + q + p (12a) × (cid:2) k E s ( | q | ) E s ( | p | ) + q E s ( | k | ) E s ( | p | ) + p E s ( | q | ) E s ( | k | ) (cid:3) . Here A is uncontrolled dimensionless parameter, pre-sumably of the order of unity. Recall, that in He tur-bulence E n ≪ E s and E ns ≪ E s . This allows us to sim-plify the mutual friction dissipation term D α to the formD α ( k ) ≈ E s ( k ). Hereafter we consider only superfluidcomponent and omit the superscript ”s“ in notations. Weshow below that in the supercritical regime the viscousdissipation term D ν ( k ) is vanishingly small with respectto the mutual friction term D α ( k ) and therefore can beneglected in the balance Eq. (10a). Thus, in the station-ary case Eq. (10a) can be presented in a simple form:Tr( k ) + 2 Ω E ( k ) = 0 . (12b)The integral (12a) diverges in the regions q ≪ k or p ≪ k .For these wavenumbers it can be approximated as:Tr( k ) ≃ − A k Ω ∞ Z −∞ E ( | q | )Ψ( k, q ) dq , (13)Ψ( k, q ) = k E ( | k + q | ) − ( k + q ) E ( | k | ) k + q + ( k + q ) . One sees that for q = 0 Ψ( k,
0) = 0 and the term whichis linear in q in the expansion does not contribute to theintegral (13). Therefore the main contribution to thisintegral in the region q ≪ k originates from the secondterm of the expansion:Ψ( k, q ) ≃ q ∂ Ψ ∂q (cid:12)(cid:12)(cid:12) q =0 = q h k E ′′ ( k ) − E ′ ( k ) i . (14)Here ′ indicates the derivative with respect to k . Nowthe energy balance Eqs. (12) can be simplified as follows: A Ω T k h k E ′′ ( k ) − E ′ ( k ) i = 4 Ω E ( k ) , (15)where Ω T is given by Eq. (5). Equation (15) has the scaleinvariant solutions E ( k ) ∝ k − x , (16a)with A Ω T x ( x −
1) = 8 Ω . (16b)The whole approach is valid if the main contribution tothe integral (5) comes from the region q ≪ k max , i.e forsupercritical cases with x >
3. With logarithmic accu-racy we can also include the critical case with x = 3.This allows us to estimate the new critical value of Ω forsupercritical regimes (with x > e Ω cr = Ω T √ A/ . (16c)Now we can rewrite Eq. (16b) as: x ( x −
1) = 6 (Ω ‡ ) , Ω ‡ ≡ Ω / e Ω cr , x > . (16d)We thus conclude that for the integral closure (12a) thattakes into account the long-distance energy transfer in k -space, the supercritical spectra do not terminate at somefinal value of k [as with the algebraic closure (2b)], butbehave like E ( k ) ∝ k − x with a scaling exponent x > ‡ . D. Relations between structure functions andenergy spectra a. Velocity structure function S ( r ) vs E ( k ) . Con-sider full 2 nd -order velocity structure function S ( r ) ≡ (cid:10) | v ( r + R ) − v ( R ) | (cid:11) , (17a)which is related to the 3D energy spectrum F ( k ) asfollows: S ( r ) = Z d k (2 π ) | − exp( i k · r ) | F ( k ) (17b)= 2 Z d k (2 π ) (cid:2) − cos( k · r ) (cid:3) F ( k ) . In spherical coordinates: S ( r ) = 2 Z E ( k ) h − sin( kr ) kr i dk . (18)Let us analyze convergence of this integral for scale-invariant spectra E ( k ) ∝ k − x . In the ultraviolet (UV)region (for k r ≫
1) the oscillating term ( ∝ sin( k r )) canbe neglected and the integral (18) converges if x >
1. Inthe infrared (IR) region (for small k ≪ − sin( k r ) / ( k r )] ≃ ( k r ) / x <
3. We con-clude that for the integral (18) the window of convergence(more often is referred to as the locality window ) is:1 < x < , Locality window for S integral. (20a)In this window, the leading contribution to the inte-gral (18) comes from the region k r ∼ S ( r ) ∝ r y , y = x − . (20b)This is a well know relationship. For example, for theK41 spectrum with x = 5 / y = 2 / k interval can be approximated as E ( k ) ∝ k − x with ≤ x ≤
3) are local and we can use for the estimateof the S the scaling relation (20b). We also see thatwhen exponent x approaches the critical value x = 3,the S scaling approaches the viscous limit with y = 2.For x = 3, S ( r ) ∝ r with logarithmic corrections, notdetectable with our resolution.In the supercritical region ( x > S -integral(18)formally IR-diverges and the integration region has tobe restricted from below by some k , similarly to theintegral (5). Together with Eq. (19), this gives the viscousbehavior for any x > S ( r ) ≃ ( r Ω T ) / . (21) TABLE I: Parameters used in the simulations by columns: ( ν s : the effective viscosity of the superfluid component; ( u srms : the rms velocity of the superfluid component; ( Re s λ = u srms λ/ν s : the Taylor-microscale Reynolds number, where λ = 2 πL s h u ih ω i is the Taylor microscale; ( ε s ν : the meanenergy dissipation rate for the superfluid component due to viscosity; ( ε stot : total mean energy dissipation rate for thesuperfluid component; ( η s = √ ν s /u srms ; ( T s0 = L/u srms : large-eddy-turnover time. The temperature dependence of e a is taken from Ref. (see Fig. 7). In all simulations: the number of collocation points along each axis is N = 1024; the sizeof the periodic box is L = 2 π ; the kinematic viscosity of the normal component is ν n = 10; the range of forced wavenumbers k ϕ = [0 . , . cr ≈ Ω = 0 . ν s u s rms Re sλ ε sν ε s tot η s T s Ω cr e Ω cr ≈ Ω † = Ω ‡ = Ω T / e α ( T ) T /T c Eq. (6) × × Eq. (25) 0 .
18 Ω T Ω / Ω cr Ω / e Ω cr Eq. (5) Eq. (6)1 0 5 1 .
14 0 4 . .
95 4 .
95 1 .
14 17.7 0 − ∞
02 0 .
25 5 0 .
89 0 .
28 3 . .
85 3 .
57 0 .
89 7.4 0 . −
41 164 0.193 0 . .
95 0 .
53 3 . .
34 5 . .
95 10.4 0 . −
21 42 0.274 0 . .
81 0 .
86 2 . .
015 4 .
38 0 .
81 2.2 0 . − . .
79 1 .
13 2 . . . .
79 0.9 − . .
75 1 .
46 2 . .
001 5 . .
75 0.55 − . .
57 4 .
42 1 . . .
53 0 .
57 0.3 − . . . . . . −
24 1.4 0.3 0.72 b. Vorticity structure function T ( r ) vs E ( k ) . Con-sider now 2 nd -order vorticity structure function T ( r ) ≡ (cid:10) | ω ( r + R ) − ω ( R ) | (cid:11) , (22a)which is related to the 3D energy spectrum F ( k ) as fol-lows: T ( r ) = Z d k (2 π ) | − exp( i k · r ) | k F ( k ) (22b)= 2 Z k E ( k ) h − sin( kr ) kr i dk . By analogy, we can immediately find the locality win-dow of this integral3 < x < , Locality window of T integral. (23a)Within this window T ( r ) ∝ r z , z = x − . (23b)It is also clear that for x > T ( r ) takesthe form T ( r ) ≃ r Z q E ( q ) dq ∼ r Ω T k . (23c) II. STATISTICS OF HE TURBULENCE:DNS RESULTS AND THEIR ANALYSISA. Numerical procedure
We carried out a series of DNSs of Eqs. (4a) and (4b)using a fully de-aliased pseudospectral code up to 1024 collocation points in a triply periodic domain of size L =2 π . In the numerical evolution, to get to a stationarystate we further stir the velocity field of the normal andsuperfluid components with a random Gaussian forcing: h ϕ u ( k , t ) · ϕ ∗ u ( q , t ′ ) i = Φ( k ) δ ( k − q ) δ ( t − t ′ ) b P ( k ) , (24)where b P ( k ) is a projector assuring incompressibility andΦ( k ) = Φ k − ; the forcing amplitude Φ is nonzeroonly in a given band of Fourier modes: k ϕ ∈ [0 . , .
5] .Time integration is performed with a 2nd order Adams-Bashforth scheme with viscous term exactly integrated.The parameters of the Eulerian dynamics for all runs arereported in Table I.
B. Energy spectra
1. Critical spectrum
The numerical energy spectra are shown in Fig. 1. Aswas predicted in Ref. , at some particular “critical” valueof the mutual friction (value of Ω = Ω cr in our currentnotations) there exists the self-similar balance betweenthe energy flux and the mutual-friction energy dissipa-tion, that leads to the scale-invariant critical spectrum E s ( k ) ∝ k − , Eq. (3d). As one sees in Figs. 1, the com-pensated spectrum for Ω = 0 . ≈ . < Ω cr we see the subcritical spectra, lying abovethe critical one. In this case, the energy at small k is dis-sipated by the mutual friction and approximately E ( k ) ∼ a b −1 k k E s ( k ) k − Ω = 0 . Ω = 0 . Ω = 0 . Ω = 0 . Ω = 0 −10 −5 k k E s ( k ) k − k − . k − . k − . Ω = 0 . Ω = 1 . Ω = 2 . Ω = 5 FIG. 1: The normalized energy spectra E s ( k ) = E ( k ) /E compensated by k : subcritical [Panel (a)] and supercritical [Panel(b)] (solid lines) for different values of Ω. The critical spectrum (with Ω = 0 .
9) is shown in both panels. The dashed linesin Panel (a) are the LNV-prediction (3a) for the subcritical spectra with one fitting parameter in Eq.(25) ( b = 0 .
5) for allΩ < .
9. The horizontal dashed lines in both panels show the critical spectrum. Other dashed lines in Panel (b) represent thescale-invariant spectra (16a) with an Ω-dependent exponent x .(a) (b) −6 −4 −2 ˜ r S ( ˜ r ) ˜ r ˜ r ˜ r ˜ r . ˜ r / Ω = 0 Ω = 0 . Ω = 0 . Ω = 0 . Ω = 0 . Ω = 1 . Ω = 2 . Ω = 5 −5 ˜ r T ( ˜ r ) ˜ r . ˜ r . ˜ r ˜ r ˜ r Ω = 0 Ω = 0 . Ω = 0 . Ω = 0 . Ω = 1 . Ω = 2 . Ω = 5 (c) (d) −5 ˜ r S ( ˜ r ) ˜ r / ˜ r ˜ r . ˜ r . ˜ r Ω = 0 Ω = 0 . Ω = 0 . Ω = 0 . Ω = 0 . Ω = 1 . Ω = 2 . Ω = 5 −5 ˜ r T ( ˜ r ) ˜ r ˜ r ˜ r Ω = 0 Ω = 0 . Ω = 0 . Ω = 0 . Ω = 1 . Ω = 2 . Ω = 5 FIG. 2: Color online. The second and forth-order velocity S (˜ r ) and S (˜ r ) [Panels (a),(c)] and vorticity T (˜ r ) and T (˜ r ) [Panels(b),(d)] structure functions for different Ω. The straight dashed lines with the estimates of the apparent scaling exponents serveto guide the eye only. k − . For larger k , the k -independent mutual friction dis-sipation can be neglected compared to the energy flux(with the inverse interaction time γ ( k ) ∼ k p kE ( k )) and E ( k ) can have K41 tail with the energy flux ε ∞ < ε input ,that for even larger k is dissipated by viscosity.
2. Subcritical LNV spectra
The analytical LNV-model of the subcritical spec-tra, based on the local in k -space algebraical closure (2b),was shortly presented in the Introduction. It results inEqs. (3) for E cr ( k, Ω) formally without explicit fitting pa-rameter. Nevertheless, having in mind simplification (4c)for the mutual friction, valid up to dimensionless factorof the order of unity and the uncontrolled character ofEq. (2b) for the energy flux, we replace in Eq. (3b) thenumerical factor by a fitting parameter b ≈ .
5. NowΩ cr = b q k E . (25)Fig. 1a compares the numerical results with the ana-lytical LNV-spectra (3a) with Ω cr given by Eq. (25).A good agreement between DNS and analytical spec-tra (3a) (with b ≈ .
5) allows us to conclude that thealgebraic LNV-model with the build-in locality of theenergy transfer adequately describes the basic physicalphenomena of the subcritical regime in superfluid Heturbulence.
3. Supercritical spectra
According to LNV model , for Ω > ˜Ω cr we expect su-percritical spectra, i.e. the energy is mainly dissipatedby the mutual friction and E s ( k ) falls below the criti-cal spectrum k − . As we pointed out, the energy trans-fer in this regime is not local anymore and a simple al-gebraic closure (2b) fails. Instead, we adopted an inte-gral closure (11) and predicted the scale-invariant spectra E s ( k ) ∝ k − x , Eq. (16a), with the exponent x , estimatedby Eq. (16b). As we see in Fig. 1b, the supercritical en-ergy spectra are indeed scale-invariant over more thana decade of k (decaying by 13 decades for Ω = 5). Thescaling exponent x increases with Ω ‡ = Ω / e Ω cr as qualita-tively predicted by Eq. (16b), although much slower. Forexample, Ω ‡ ≈ . .
1, see line ( x model ≃ . x num ≃ .
7. This disagreement increases with Ω ‡ .Here we should note that the particular form (11) of theintegral closure was chosen just for simplicity. We canuse much more sophisticated kind of a two-point inte-gral closure, like EDQNM, or Kraichnan’s Lagrangian-history direct interaction approximation , etc. Howeverthe result will be qualitatively similar: a scale-invariantsolution with the exponent x that increases with Ω ‡ .We again conclude that the suggested model (now withthe integral closure) describes qualitatively the physics of the supercritical regime of the superfluid He turbulencewith the balance between the energy flux from k ∼ k di-rectly to a given k ≫ k , [the left hand side of Eq. (15)],where it is dissipated by the mutual friction [the righthand side of Eq. (15)]. This balance equation results inthe power-like law E sp ∝ k − x , in agreement with theDNS results. The actual value of the exponent x de-pends on the details of the uncontrolled integral closure.A detailed analysis of the closure problem, including con-tribution of next order terms in perturbation approach,and comprehensive numerical simulations would be re-quired to achieve better understanding of the statisticsof the supercritical regimes of superfluid He turbulence
C. Enhancement of intermittency in critical andsubcritical regimes of superfluid He turbulence
Current Sec. II C is devoted to the discussion of the nu-merically found velocity and vorticity structure functions S ( r ), S ( r ) and T ( r ), T ( r ) and to comparison theirscaling with the corresponding theoretical predictions.The most important physical observation is a significantamplification of the velocity and vorticity fluctuations inthe critical and subcritical regimes (for 0 . ≤ Ω ≤ . enhancement of intermittency in superfluid He turbulence .
1. 2 nd -order structure functions of the velocity andvorticity S ( r ) and T ( r ) Consider scaling behavior of the velocity 2 nd -orderstructure function S (˜ r ) for different Ω, shown in Fig. 2aas a function of a dimensionless distance ˜ r = r/η . Forthe classical hydrodynamic turbulence (Ω = 0, blackline), S (˜ r ) demonstrates the expected behavior: a vis-cous regime, with S (˜ r ) ∝ ˜ r for small r followed bythe K41 regime, with S (˜ r ) ∝ ˜ r ζ , with ζ = 2 / ζ ( ζ ≈ .
70 instead of ζ = 2 / ≈ .
67) is not visibleon the scale of Fig. 2a and will be discussed below. Thespectrum for Ω = 0 .
25 (brown line) behaves similarly tothe classical case Ω = 0, just with larger cross-over valueof ˜ r . For larger subcritical values of Ω = 0 . . S (˜ r ) ∝ ˜ r behav-ior for small r is now followed by an apparent scalingbehavior S (˜ r ) ∝ ˜ r ζ with < ζ < . ζ ≈ .
0, while for Ω = 0 .
7, theapparent exponent ζ ≈ .
4, and become close to ζ ≈ .
9. Note thatfor much larger Reynolds numbers, these apparent expo-nents are expected to appear only around r × ∼ /k × . (a) (b) ˜ r F v ( ˜ r ) ˜ r − . ˜ r − . Ω = 1 . Ω = 2 . Ω = 5 Ω = 0 Ω = 0 . Ω = 0 . Ω = 0 . Ω = 0 . ˜ r F ω ( ˜ r ) Ω = 0 Ω = 0 . Ω = 0 . Ω = 1 . Ω = 0 . Ω = 0 . Ω = 2 . Ω = 5 (c) (d) Ω F v ( Ω ) ˜ r = 2000˜ r = 5000˜ r = 1000˜ r = 500˜ r = 250 Ω F ω ( Ω ) ˜ r = 5000˜ r = 1000˜ r = 500˜ r = 250˜ r = 2000 FIG. 3: Color online. The velocity F v (˜ r ) = S (˜ r ) /S (˜ r ) and vorticity F ω (˜ r ) = T (˜ r ) /T (˜ r ) flatness vs ˜ r for different Ω [Panels(a) and (b)] and vs Ω for different ˜ r [Panels (c) and (d)]. The straight dashed lines with the estimates of the apparent scalingexponents serve to guide the eye only. For r ≪ r × the apparent exponent should approach theclassical value ζ = 2 / r ≫ r × – the criticalvalue ζ = 1.As explained in Sec. I D, in the supercritical regime,when E s ( k ) ∝ k − x with x >
3, the integral (18) losses itslocality and is dominated by small r , where the velocityfield can be considered as smooth. In this regime theviscous behavior S (˜ r ) ∝ ˜ r is expected for all Ω > . S (˜ r ) ∝ ˜ r is disconnected fromthe energy scaling E ∝ k − x . The vorticity structure func-tion T (˜ r ) is more informative for this regime, because,as shown in Sec. I D, the vorticity field is not smooth for x < T (˜ r ) for different Ω.Consider first the test case Ω = 0, shown by a blackline. For very small ˜ r , when 1 / ˜ r exceeds viscous cutoffof the energy spectrum, we see the viscous behavior ∝ ˜ r , followed by the saturation region T (˜ r ) ≃ const. Asexplained in Sec. I D, this is because the energy spectrumexponent x = 5 / below the lower edge of the vorticity locality window (23a). For x <
3, the integral (22b) isdominated by large k in the interval πr < k < k max and T (˜ r ) becomes r -independent, as observed.In Figs. 2 we present two cases with x within the lo-cality window for vorticity (23a), 3 < x <
5: Ω = 0 . x ≈ . x ≈ .
66. According to ourasymptotical (for infinitely large scaling interval) predic-tion (23b), we expect for these cases z ≈ z ≈ . z ≈ . z ≈ .
8. Having relatively short scal-ing interval, we consider this agreement as acceptable.For even stronger mutual friction Ω = 2 . x ≈ . x ≈ . above the upper edge of the vorticity locality win-dow (23a). In this case integral (22b) diverges at lowerlimit, giving T (˜ r ) ≃
43 ˜ r ∞ Z k min k E ( k ) dk ∝ ˜ r . (26)as is indeed observed in Fig. 2b.0 (a) (b) (c) −4 −2 k D ( k ) D ν ( k ) d ǫ k dk Ω = 0 −4 −2 k D ( k ) D ν D α d ǫ k dk Ω = 0 . −4 −2 k D ( k ) D α D ν d ǫ k dk Ω = 0 . (d) (e) (f) k ǫ k D tot ν D tot α ǫ k Ω = 0 k ǫ k D tot α D tot ν ǫ k Ω = 0 . k ǫ k D tot ν D tot α ǫ k Ω = 0 . FIG. 4: Color online. The differential [Panels (a),(b),(c)] and the integral [Panels (d),(e),(f)] energy balances in the subcriticalregimes with Ω = 0 [Panels (a),(d)], Ω = 0 . .
2. 4 nd -order structure functions, flatnesses andenhancement of intermittency Consider now 4 th -order structure functions of the ve-locity and vorticity S (˜ r ) and T (˜ r ), shown in Fig. 2c andFig. 2d, for the subcritical and supercritical regimes. Asis well known, for the Gaussian statistics or, in a moregeneral case, for the “mono-scaling” statistics, the fourth-order structure functions are proportional to the squareof the second one: S (˜ r ) ∝ S (˜ r ) and T (˜ r ) ∝ T (˜ r ).We find such a behavior for very small ˜ r . For the clas-sical case Ω = 0 (Fig. 2c) we see again scaling exponent ζ close to the standard K41 value 4 / ≈ .
33 with in-termittency corrections, hardy visible on this scale. Forlarger Ω, the subcritical LNV spectrum (3a) becomes asuperposition of two scaling laws and, as we mentionedin the Introduction, in the vicinity of a crossover wavenumber k × may be approximated as k − x with an appar-ent scaling exponent < x ( k ) <
3. Indeed, we see inFig. 2c that the apparent value of ζ definitely deviatefrom 4/3, approaching, for example, ζ ≈ . . ζ ≈ . .
7. Such a steepening of the struc-ture functions spectra is caused by the energy dissipationby mutual friction (see Fig. 1).More importantly, upon increase in Ω the apparentscaling of the velocity field progressively deviates fromthe self-similar behavior type with S (˜ r ) ∝ S (˜ r ) and ζ = 2 ζ . For example, for Ω = 0 . ζ ≈ . < ζ ≈ . ξ = 2 ζ − ζ ≈ .
3) and for Ω = 0 . ξ ≈ . F v (˜ r ) and F ω (˜ r ), defined as: F v (˜ r ) = S (˜ r ) /S (˜ r ) , F ω (˜ r ) = T (˜ r ) /T (˜ r ) . (27)For the Gaussian and mono-scaling statistics, F v (˜ r ) and F ω (˜ r ) must be ˜ r -independent. In particular, for theGaussian statistics F v (˜ r ) = F ω (˜ r )=3. As is evident inFig. 3a and Fig. 3b, the intermittency corrections, hardlyvisible for structure functions for Ω = 0, are clearly ex-posed by the flatness. The velocity flatness F v (˜ r ) for thiscase (black solid line in Fig. 3a) approximately follow theintermittent exponent for turbulence in classical fluids ξ cl ≈ .
15, which is close to the experimental values forboth the longitudinal and transversal structure functions(for previous experimental and numerical works on in-termittency in the classical space-homogeneous isotropicturbulence see Refs. ) . As the mutual friction be-come stronger, the apparent exponent ξ increases, reach-ing its maximum ξ max ≈ . ≈ ξ cl at Ω = 0 .
7. The vor-ticity flatness F ω (˜ r )[Fig. 3b] too reaches its maximum forsmall ˜ r at slightly larger value of Ω ≈ .
9. This is a clearevidence of significant enhancement of intermittency inthe near-critical regimes of superfluid He turbulence.Additional important information can be found inFigs. 3c and 3d, where Ω-dependence of the velocity andvorticity flatnesses is shown for different ˜ r . The sharppeak appears for Ω . .
9. In the small ˜ r range, the ve-locity flatness F v (˜ r ) for Ω = 0 . > . D. Energy balance
The direct information about the relative importanceof the energy dissipation by the effective viscosity andby the mutual friction can be obtained from an analysisof the energy balance, shown in Figs. 4. The energy bal-ance for the classical turbulence (Ω = 0) is presented inFig. 4a. As expected, the energy input at a shell with agiven wave number k , Tr( k ) = dε ( k ) /dk (green line) iscompensated by the viscous dissipation D ν = 2 ν s E s ( k )(red line). The discrepancy in the region of very small k is caused by the energy pumping, which is not accountedin the balance Eq. (10a). Sometimes it is more convenientto discuss a “global” energy balance, analyzing insteadof the “local” in k balance Eq. (10a) its integral from k =to a given k . In the stationary case this gives: ε ( k ) = ε − D tot ν ( k ) − D tot α ( k ) , (28a)D tot ν ( k ) = Z k D ν ( q ) dq, D tot α ( k ) = Z k D α ( q ) dq . (28b)As we see in Fig. 4d (for Ω = 0), the energy flux overscales ε ( k ) is almost constant up to k ≃
20 and thendecreases due to the viscous dissipation. Accordingly, E s ( k, ∝ k − / . Minor upward deviation from thisbehavior may be a numerical artifact.The energy balance in the subcritical regime of thesuperfluid He turbulence, shown in Figs. 4b and 4e forΩ = 0 . . k ) in a given k (shown by green lines) is balanced bythe mutual friction dissipation D α ( k ) (shown by the bluelines). Only for large k &
75, the viscous dissipation be-gin to dominate. Nevertheless, as seen in Figs. 4e and 4f,the total contribution to the energy dissipation is dom-inated by the mutual friction everywhere. As expected,for larger and larger Ω the crossover wave number k × ,at which the local dissipation by viscosity and by mu-tual friction are equal, increases (compare Fig. 4b withΩ = 0 . .
7) and reaches k max forthe critical regime with Ω = 0 . k ≃ k max .In the supercritical regime, shown in Figs. 5e and 5f forΩ = 1 . k ) = dε ( k ) /dk (green lines) is fully compensated by the mutual friction dissi-pation(blue lines).The global energy balance, shown in Figs. 5, confirmsthis physical picture. E. Energy and enstropy time evolution
We consider here evolution of the total superfluid en-ergy E s ( t ) E s ( t ) = Z E s ( k, t ) dk (29a)and enstrophy 1 / T ( t ). As expected, in the subcriti-cal regime, when E ( k, t ) has apparent slope ∝ k − x with < x ( k ) <
3, the integral (29a) for total energy E ( t )is dominated by the small k ∼ k min , while the inte-gral (5) for total enstropy Ω T ( t ) is dominated by the large k ∼ k max . Therefore, for the large ratio k max /k min (inour case k max /k min ∼ ), one expects an uncorrelatedbehavior of E ( t ) and Ω T ( t ) in case of a well developed tur-bulent cascade. This behavior is confirmed in Figs. 6a,6b and 6c. In the supercritical regime, with the slope x >
3, both E ( t ) and Ω T ( t ) are dominated by the small k ∼ k min and have to be well correlated, as is indeed seenin Figs. 6e and 6f. However, in the critical regime (Fig. 6dwith Ω = 0 . E ( t ) and Ω T ( t ) are still uncorrelated be-cause E ( t ) is dominated by k ∼ k min , while Ω T ( t ) hasequal contributions from all k . F. Relation between Ω and temperature T ofpossible experiments Up to now we have considered Ω as a free parame-ter that determines the mutual friction by Eq. (4c), inwhich Ω T is given by Eq. (5). After the simulation witha prescribed Ω was completed, we numerically computedΩ T , using found energy spectra and Eq. (5), see Tab. I.Now, using Eq. (4c) we can find e α = Ω / Ω T for a givenΩ in the simulations. The parameter e α in He stronglydepend on temperature, as reported in and shown inFig. 7. Using these data, we can find T corresponding tothe simulations with any prescribed Ω. III. SUMMARY
This paper examined the basic statistical properties ofthe large-scale, homogeneous, steady, isotropic quantumturbulence in superfluid He, developing further someprevious results . Direct numerical simulations of thegradually damped version of the HVBK coarse-grainedtwo-fluid model of the superfluid He, Eqs. (4) wereperformed using pseudo-spectral methods in a fully pe-riodic box with a grid resolution of N = 1024 . Theanalytic study was based on the LNR integral closure for2 (a) (b) (c) −4 −2 k D ( k ) D ν ( k ) D α ( k ) d ǫ k dk Ω = 0 . −4 −2 k D ( k ) D α ( k ) D ν ( k ) d ǫ k dk Ω = 1 . −4 −2 k D ( k ) D α ( k ) D ν ( k ) d ǫ k dk Ω = 5 (d) (e) (f) k ǫ k D tot α D tot ν ǫ k Ω = 0 . −10123456 k ǫ k D tot ν D tot α ǫ k Ω = 1 . k ǫ k D tot α ǫ k D tot ν Ω = 5 FIG. 5: Color online. The differential [Panels (a),(b),(c)] and the integral [Panels (d),(e),(f)] the energy balances in the criticaland the supercritical regimes with Ω = 0 . . . t/T E ( t ) , Ω T ( t ) Ω T ( t ) / h Ω T ( t ) i E s ( t ) / h E s ( t ) i Ω = 0 t/T E ( t ) , Ω T ( t ) E s ( t ) / h E s ( t ) i Ω T ( t ) / h Ω T ( t ) i Ω = 0 . t/T E ( t ) , Ω T ( t ) E s ( t ) / h E s ( t ) i Ω T ( t ) / h Ω T ( t ) i Ω = 0 . (d) (e) (f) t/T E ( t ) , Ω T ( t ) E s ( t ) / h E s ( t ) i Ω T ( t ) / h Ω T ( t ) i Ω = 0 . t/T E ( t ) , Ω T ( t ) E s ( t ) / h E s ( t ) i Ω = 1 . Ω T ( t ) / h Ω T ( t ) i t/T E ( t ) , Ω T ( t ) Ω = 5 E s ( t ) / h E s ( t ) i Ω T ( t ) / h Ω T ( t ) i FIG. 6: Color online. The energy (red lines) and enstrophy (blue lines) time evolutions in the subcritical regime normalizedby mean-in-time values [Panels (a),(b),(c) with Ω = 0 , . , . .
9] and the supercritical regime[Panels (e),(f) with Ω = 1 . , . the energy flux , Eq. (11), adapted for He turbulence inEq. (12a). Both the DNS and the analytic approaches donot use the assumption of locality of the energy transferbetween scales. The main findings are:1. The direct numerical simulations confirmed the pre- viously found subcritical (3a) and critical (3d) energyspectra and showed that for
T < . T c (see Tabl. I)the analytic prediction are in a good quantitative agree-ment with the DNS results, using a single fitting param-eter b for all temperatures. The reason for this agree-3 Fig. 1. Mutual friction parameter = (1 /α as a function of temperature.In superfluid dynamics this parameter, composed of the dissipative mutual friction) and the reactive mutual friction ) corresponds to the Reynolds number Re of viscous hydrodynamics. Typically, when Re >
1, turbulence becomes possible inthe bulk volume between interacting evolving vortices. This transition to turbulenceas a function of temperature can readily be observed in He-B (at 0 59 ), while inHe II it would be within 01 K from the lambda temperature and has not beenidentified yet. a vortex front propagates along a rotating cylinder of circular cross section(Sec. 3), and the decay of a nearly homogeneous isotropic vortex tangle in su-perfluid He (Sec. 4), created by suddenly stopping the rotation of a containerwith square cross section.Turbulent flow in superfluid He-B and He is generally described by thesame two-fluid hydrodynamics of an inviscid superfluid component with singly-quantized vortex lines and a viscous normal component. The two componentsinteract via mutual friction. There are generic properties of turbulence thatare expected to be common for both superfluids. However, there are also inter-esting differences which extend the range of the different dynamic phenomenawhich can be studied in the He superfluids:In typical experiments with He-B, unlike with He, the mutual frictionparameter can be both greater and smaller than unity (Fig. 1) – thisallows the study of the critical limit for the onset of turbulence at(Sec. 2);The viscosity of the normal component in He-B is four orders of magni-tude higher than in He, hence the normal component in He-B is rarelyturbulent, which amounts to a major simplification at finite temperatures(but not in the = 0 limit with a vanishing normal component);While the critical velocity for vortex nucleation is much smaller in He-B,pinning on wall roughness is also weaker; this makes it possible to createvortex-free samples, which are instrumental in the transitional processesstudied in Secs. 2 and 3; on the other hand, the ever-present remanent vor-tices in superfluid He are expected to ease the production of new vortices,which becomes important in such experiments as spin-up from rest;
FIG. 7: Temperature dependence of the mutual friction pa-rameter e α ( T ) = α/ (1 − α ′ ), taken from Ref. ment is that in the subcritical regime the energy transferover scales is indeed local, in accordance with the ba-sic assumptions in Refs. . In the critical regime with E ( k ) ∝ k − , the exact locality of the energy transfer fails:all the scales contribute equally to the transfer of energyto the turbulent fluctuations with a given k . This leadsto a logarithmic corrections to the spectrum E ( k ) ∝ k − that cannot be detected with our DNS resolution.2. For T > . T c , when the mutual friction exceedssome critical value, we observed in DNS and confirmedanalytically the scale-invariant spectrum E ( k ) ∝ k − x with a ( k -independent) exponent x >
3. The exponent x increases gradually with the temperature, reaching in oursimulation the value x ≈ T ≈ . T c . The reason for this behavior of the supercritical spectra with x > k from the energy containing region at small k .3. We analyzed the 2 nd -order structure functions of thevelocity and vorticity S ( r ) and T ( r ) and demonstratedthat although their r -dependence can be rigorously foundfrom the energy spectrum E ( k ), their r -dependence ismuch less informative that the k -dependence of E ( k ).4. The 4 nd -order structure functions of the velocityand vorticity S ( r ) and T ( r ) provide important addi-tional [with respect to E ( k )] information about the statis-tics of quantum turbulence in the superfluid He. Wediscover a strong enhancement of intermittency in thenear-critical regimes with the level of turbulent fluctu-ations exceeding the corresponding level in the classicalturbulence by about an order of magnitude.5. The analysis of the energy balance and of the energyand enstrophy time evolution in various (subcritical, crit-ical and supercritical) regimes, confirms the discoveredphysical picture of the quantum He turbulence with thelocal and non-local energy transfer, in which the relativeimportance of the energy dissipation by the effective vis-cosity and by the mutual friction depends in a predictedway on the temperature and the wavenumber.We propose that these analytic and numerical findingsin the description of the statistical properties of steady,homogeneous, isotropic and incompressible turbulence ofsuperfluid He should serve as a basis for further stud-ies of superfluid turbulence in more complicated or/andrealistic cases: anisotropic turbulence, transient regimes,two-fluid turbulence of thermally driven counterflows insuperfluid He turbulence, etc. C. F. Barenghi, L. Skrbek, and K. R. Sreenivasan, Proc.Nat. Acad. Sci. USA , 4647-4652(2014). C. F. Barenghi, V. S. Lvovb, and P.-E. Roche, Proc. Nat.Acad. Sci. USA , 4683-4690(2014). V. Eltsov, R. Hanninen, M. Krusius, Proc. Nat. Acad. Sci.USA ,4711(2014). S.N. Fisher, M.J. Jackson, Y.A. Sergeev, V. Tsepelin, Natl.Acad. Sci. USA , 4659(2014). V. S. L’vov, S. V. Nazarenko and G. E. Volovik, JETPLetters, , iss.7 pp. 535-539(2004). L. Bou´e, V.S. L’vov, A. Pomyalov, and I. Procaccia, Phys.Rev. B, V. S. L’vov, S. V. Nazarenko, O. Rudenko, Phys. Rev. B , 024520(2007). U. Frisch,
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