Dynamics of turbulent plugs in a superfluid 4 He channel counterflow
DDynamics of turbulent plugs in a superfluid He channel counterflow
A. Pomyalov
Dept. of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel
Quantum turbulence in superfluid He-4 in narrow channels often takes the form of moving localizedvortex tangles. Such tangles, called turbulent plugs, also serve as building blocks of quantumturbulence in wider channels. We report on a numerical study of various aspects of the dynamicsand structure of turbulent plugs in a wide range of governing parameters. The unrestricted growthof the tangle in a long channel provides a unique view on a natural tangle structure includingsuperfluid motion at many scales. We argue that the edges of the plugs propagate as turbulentfronts, following the advection-diffusion-reaction dynamics. This analysis shows that the dynamicsof the two edges of the tangle have distinctly different nature. We provide an analytic solution ofthe equation of motion for the fronts that define their shape, velocities and effective diffusivity, andanalyze these parameters for various flow conditions.
I. INTRODUCTION
Quantum properties of liquid He become apparent when it is cooled below critical temperature T λ = 2 .
17 K.Quantized part of fluid vorticity, an inviscid superfluid,forms a quantum ground state. A gas of thermal excita-tions represents a viscous normal fluid with continuousvorticity. The vorticity quantization results in creationof thin quantum vortex lines of fixed circulation. Theselines form dense tangles that interacts with the normalfluid via mutual friction force.When placed in a channel with a temperature gradi-ent, two components of the superfluid He flow in oppo-site directions. The superfluid flows towards the heater,while the normal fluid moves away from it. Such a set-ting, called thermal counterflow, has been long used tostudy properties of superfluid He components andtheir interaction. Early experiments on the thermalcounterflow in He in narrow channels found a wide va-riety of scenarios of the vortex tangle dynamics .Propagating turbulent fronts and localized vortex tan-gles, or turbulent plugs, were observed in long thin glassand metal capillaries . Depending on conditions, theseplugs were either stationary, moving in one direction orexpanding both toward and away from the heater.The stationary, almost homogeneous tangles, fill-ing the whole channel, were found in relatively widechannels . In this case, the local variations of thevortex line density (VLD) buildup towards the stationaryregime were considered as transient effects and most ofthe attention turned to studies of the steady-state prop-erties with VLD as the main parameter of the system.Derivation of a set of closed equations for the descrip-tion of the quantum vortex tangle dynamics and statis-tics using only its macroscopic characteristics have beenan ultimate goal since early days of superfluid He stud-ies. The Vinen’s equation for the time evolution of thevortex line density L in a homogeneous tangle servedas a basis of most theoretical considerations for decades(see, for example, Refs. ). A microscopic theory bySchwarz introduced additional structural properties ofthe tangle, such as root-mean-square curvature and var- ious anisotropy parameters, as important ingredients ofthe theory . As was pointed out by Schwarz, thearguments leading to these equations for L apply onlyto the average time-dependent behavior near the steady-state, although are very often used in other situations.For a moving tangle, a number of theories pre-dicted that the plug’s motion is defined by drift asa whole with a constant velocity and a diffusion-likespreading. It was commonly assumed that the fully de-veloped homogeneous tangle is expanding into the lam-inar superfluid, having well-defined properties, the sameas for the stationary homogeneous tangle. No direct ex-perimental or numerical evidence, supporting these as-sumptions, was found so far. The only numerical studyof such a moving turbulent plug by Schwarz was car-ried out within an approximation that ignores non-localinteractions between vortex lines and was fully focusedon the conditions that allow sustaining the quantum tur-bulence.Recent advances in the experimental techniques, in-cluding flow visualization , as well as increasing com-puting power, renewed the interest to the spatial inho-mogeneity due to presence of channel walls andspatially-resolved investigations of the transient behaviorin the thermal counterflow . The latter work showedthat the vortex tangle that eventually fills the wholechannel, grows starting from a number of remnant vortexrings. These rings first form separate localized turbulentplugs, that later merge. Remarkably, the structural prop-erties of the large-scale tangle became homogeneous soonafter the merger, while the vortex line density distribu-tion remained inhomogeneous much longer, as reflectedby very different VLD build-up patterns at different lo-cations in the channel.In this paper, we study the dynamical and structuralproperties of localized turbulent plugs in the wide rangeof flow conditions. Unlike previous simulations of thethermal counterflow in the channel, here the vortex tan-gle development in the streamwise direction is undis-turbed by artificial self-interactions, caused by periodicboundary conditions. Such conditions are routinely usedto ensure the quick creation of a dense tangle that is ho-mogeneous in the streamwise direction. Although conve- a r X i v : . [ c ond - m a t . o t h e r] F e b nient, this approach does not allow to study the naturalstructure of the tangle and the local influence of the meansuperfluid velocity on the vortex lines motion.The paper is organized as follows. In Sec. II we con-sider the vortex tangle motion as a whole and the distri-bution of the vortex line density in the developing tangle.We start by introducing in Sec. II A important notionsand parameters of the thermal counterflow in superfluid He. Then we describe the numerical setup (Sec. II B)and the chosen ways for the characterization of the de-veloping vortex tangle in the channel (Sec. II C). Next,we consider the spatio-temporal evolution of the tanglevortex line density (Sec. II D), while peculiarities of thetransient processes are discussed in Sec. II E. The large-scale superfluid motion, created inside the vortex tan-gle, is described in Sec. II F. In Sec. II G we focus on thestructural properties of the developed tangle. Section IIIis devoted to the study of the VLD front dynamics andstructure. First, we overview important information fromthe turbulent front propagation studies, relevant for thecurrent work (Sec. III A). Next, we derive an equation ofmotion for VLD, that describes the evolution of the tan-gle edges, or fronts (Sec. III B), show that the two tanglefronts have different nonlinearity types (Sec. III C), con-sider the closure for the nonlinear term (Sec. III D) andsolve the equation of front motion analytically for thefront shape (Sec. III E). Then we discuss the parameters,that describe the front propagation: the front velocities(Sec. III F) and the effective diffusivity (Sec. III G). InSec. IV we summarize our findings.
II. DYNAMICS OF A TURBULENT PLUGSA. The counterflow turbulence in the channel
As already mentioned, at temperatures below T λ =2 . He become a superfluid. In this state, thesuperfluid He of the density ρ is often described inthe framework of the two-fluid model as an interpene-trating mixture of a normal fluid with the density ρ n and a superfluid component of the density ρ s , such that ρ s + ρ n = ρ and the components’ contributions ρ s , ρ n arestrongly temperature dependent .The normal-fluid component has low viscosity and con-tinuous vorticity, while the superfluid is inviscid and itsvorticity is constrained to vortex-line singularities of coreradius a ≈ − cm with fixed circulation κ = h/M ≈ − cm /s, where h is Planck’s constant and M is themass of the He atom. The two components are coupledby the mutual friction force. Under the influence of thetemperature gradient applied along the channel, the nor-mal fluid is moving away from the heater with a meanvelocity V n . At the same time, the superfluid is movingtowards the heater with the mean velocity V s , creating arelative, or a counterflow, velocity V ns = V n − V s , propor-tional to the applied heat flux. The chaotic tangle of vor-tex lines is then generated from pre-existing remnant vor- 𝐻𝐻𝑦𝑦 𝑥𝑥𝑧𝑧
FIG. 1: Numerical setup. The simulations are set up in along planar channel of a width H . The normal-fluid velocityis oriented towards positive x -direction. tex loops due to the coupling by temperature-dependentmutual friction force. The governing parameters thatdefine the dynamics and the structure of the tangle are,therefore, the counterflow velocity and the temperature,while the geometric constraints, such as channel dimen-sions, influence the inhomogeneity of the vortex tangle. B. Numerical setup
The simulations were set up in a long planar channel ofa width H (see Fig. 1). To describe dynamics of the vor-tex lines we use the vortex filament method for thechannel flow . The vortex lines are parameterizedby curves s ( ξ, t ) and discretized in a set of points withthe resolution ∆ ξ = 0 .
001 cm. The equation of motionfor such a line point is d s ( ξ, t ) dt = V drift ( s , t ) = V s ( s , t ) + V mf ( s , t ) , (1) V mf ( s ) , t = ( α − α (cid:48) s (cid:48) × (cid:1) s (cid:48) × V ns ( s , t ) , where s (cid:48) is the unit vector along the vortex linesand α, ˜ α are the temperature-dependent mutual frictionparameters . Here the right-hand-side of Eq. (1) repre-sents the drift velocity of the tangle V drift . The superfluidvelocity V s = V BS + V , (2) V BS ( s , t ) = κ π (cid:90) Ω s − s | s − s | × ds = V loc + V nl (3)accounts for the tangle contribution V BS ( s , t ) and themean superfluid velocity V that is defined by the coun-terflow condition of zero mass-flux. In its turn, V BS may be further divided into the local part, produced bythe scales up to local radius of curvature R = 1 / | s (cid:48)(cid:48) | , V loc = β ( s (cid:48) × s (cid:48)(cid:48) ) , β = ( κ/ π ) ln( R/a ) and the nonlocalvelocity V nl which is produced by the rest of the tan-gle Ω. The mutual friction part V mf ( s , t ) describes theinteraction with the normal fluid via counterflow veloc-ity V ns ( s , t ) = V n − V s ( s , t ). The material parametersof He, used in the simulations, are listed in Table I.The time resolution for the vortex line point is set by theforth-order Runge-Kutta stability criterion.To generate the counterflow, we use two time-independent prescribed wall-normal profiles of thestream-wise projection of the normal-fluid velocity -0.5 0 0.500.20.40.60.811.2
ParabolicFlattened
FIG. 2: Normal-fluid velocity profiles normalized by the meanvalue ˜ V n = V n / (cid:104) V n (cid:105) . The shape of the flattened profile is de-fined by a combination of six Legendre polynomials, such thatit has the same (cid:104) V n (cid:105) as the corresponding parabolic profile. V x n ( y ), shown in Fig. 2. The parabolic profile corre-sponds to the laminar normal-fluid velocity. It was ob-served experimentally at low heat fluxes. At larger heatfluxes, when the normal fluid loses its stability but notyet become fully turbulent, its profile flattens . Sim-ilar flattening of the normal-fluid velocity profile wasfound in simulations with a two-way coupling of the fluidcomponents . Although such a fully coupled dynam-ics gives the most reliable description of the superfluid He, it is still computationally prohibitive for sufficientlylarge systems and long propagation times. Therefore weignore the back-influence of the superfluid component onthe normal fluid and model the expected normal-fluid ve-locity flattening by imposing thecorresponding time-independent profile (dashed line inFig. 2).The mean superfluid velocity V is dynamically de-fined by the zero-mass-flux condition ρ n (cid:104) V n (cid:105) v + ρ s (cid:104) V (cid:105) v = 0 , (4)where (cid:104) ... (cid:105) v denotes global volume averaging and V in-clude a contribution of the superfluid velocity induced bythe vortex tangle, calculated on a dense grid. This contri-bution, although small, is not negligible and grows withthe development of the spatially-inhomogeneous tangle.To mimic solid walls in the wall-normal direction, theboundary conditions on the wall are s (cid:48) ( ± H/
2) = (0 , ± V y s ( ± H/
2) = 0. In the spanwise direction, periodicconditions were used. To ensure free evolution of the de-veloping tangle, we use open conditions in the streamwisedirection. In this way, the properties of the tangle edges,moving as fronts, as well as the natural structure of thetangle bulk, can be studied.The vortex tangles at all conditions were initiated us-ing the same set of 8 vortex loops of similar sizes R (cid:28) H and different orientations. The loops were placed at aparticular streamwise location, 4 circular loops in thebulk and 4 half-circular loops at the walls. The difference TABLE I: Material properties of He used in the simula-tions. T , K 1.3 1.65 1.9 ρ n /ρ s α α (cid:48) in the dynamics of these tangles, therefore, originatesfrom the flow conditions only, allowing comparison.The tangle dynamics was studied at three tempera-tures T = 1 . , .
65 and 1 . V n and a narrow channelwidth H = 0 . (cid:104) V n (cid:105) as for the corresponding parabolic profile). The simula-tion parameters are listed in Table II, columns − z -directionwas always equal to H . Despite the periodic boundaryconditions in the spanwise direction, the interaction be-tween the vortex lines and their images is an importantfactor in the current setting. The study of the influence ofthe slit aspect ratio on the tangle dynamics is beyond thescope of this paper. The tangle evolution was followeduntil a well-developed bulk tangle was formed, such thatthe final length of the tangle is about 4 − H . The actualfinal time of evolution t f varies for different conditions. C. Characterization of the tangle
To characterize the developing tangle, we calculate thetime-dependent two-dimensional (2D) ( x, y )-spatial dis-tribution of tangle properties, averaged over spanwise z direction, at equispaced time moments. To this end, wedefine a fixed grid with the a resolution ∆ x = 0 . y = 0 . over parts of thetangle Ω (cid:48) that fall into a grid cell V (cid:48) = ∆ x × ∆ y × H .In such a way we obtain the vortex line density L ,the curvature of the vortex lines κ ≡ | s (cid:48)(cid:48) | , the meansquare curvature (cid:104) κ (cid:105) , the ratio c = (cid:104) κ (cid:105) / L , the lo-cal binormal I (cid:96) = (cid:104) s (cid:48) × s (cid:48)(cid:48) (cid:105) and its anisotropy index I † (cid:96) = (cid:104) s (cid:48) × s (cid:48)(cid:48) (cid:105) / (cid:104)| s (cid:48)(cid:48) |(cid:105) , the contributions to the tangle driftvelocity, as defined by right-hand-side of (1) and variousterms of the balance equation, defined by Eq. (16). Inthe above definitions, the arguments ( x, y, t ) were omit-ted for clarity. To compare the results for different flowconditions, we use dimensionless quantities, normalizedusing the mean counterflow velocity calculated from thezero-mass-flux condition V = (cid:104) V n (cid:105) y (1 + ρ n /ρ s ) and thecirculation quantum κ . The procedures for calculation ofvarious profiles are described in Appendix A.To measure the velocity of front propagation, it is cus-tomary to choose a threshold value of propagating quan- FIG. 3: VLD evolution. (a)-(b) T = 1 . U c = 3 cm/c. (c)-(d)- T = 1 . U c = 1 cm/c. Panels (a) and (c) show L ( x, y )distribution at t = 0 . t f / , t f / t f with the top snapshot corresponding to the early stages of the dynamics and thebottom snapshot corresponding to t f . Panels (b) and (d) show the time evolution of VLD averaged over y direction L ( x, t ).Both cases correspond to the parabolic profile of V n and the channel width H = 0 . L are color-coded asshown by colorbars in panels (b) and (d).TABLE II: Parameters of simulations by columns: ( V n profile: P denoteparabolic profile, F denote for flattened profile; ( U c ; ( (cid:104) V n (cid:105) . For the parabolic profile, (cid:104) V n (cid:105) = − / U c ; ( V = (cid:104) V n (cid:105) y (1 + ρ n /ρ s ); ( L core0 ; ( L wall0 . The error-bars for L j account for the standarddeviation from the mean values. .1 2 3 4 5 6 7 8 9Run T , K Type H U c (cid:104) V n (cid:105) V L core0 × − L wall0 × − − cm − . ± .
03 0 . ± .
12 P 0.1 3 2 2.11 0 . ± .
05 1 . ± .
43 P 0.1 4 2.66 2.84 2 . ± .
06 3 . ± .
84 1.3 F 0.1 − . ± . . ± .
25 P 0.15 3 2 2.12 1 . ± . . ± .
46 P 0.2 3 2 2.11 7 . ± . . ± .
27 P 0.1 1.5 1 1.22 0 . ± .
02 1 . ± .
38 1.65 P 0.1 2 1.66 1.63 1 . ± .
03 2 . ± .
59 F 0.1 − . ± .
02 1 . ± .
210 P 0.1 1 0.66 1.19 1 . ± .
02 2 . ± .
311 P 0.1 1.2 0.8 1.42 2 . ± .
03 3 . ± .
312 1.9 P 0.1 1.5 1 1.36 4 . ± . . ± .
613 F 0.1 − . ± .
05 2 . ± .
614 P 0.15 1 0.66 1.18 1 . ± .
04 2 . ± .
215 P 0.2 1 0.66 1.17 1 . ± .
06 1 . ± . tity and to follow the change of its position. To avoidinevitable freedom in the choice of the threshold value L , we use here a different approach. Instead of followinga single threshold value, we find the velocity that allowsto collapse whole edge of the tangle to a single shape.It turned out that such an approach gives a very robustmeasurement of the velocity, allowing simultaneously tostudy the front shape. The speeds of both VLD frontswere measured over the time interval when the tanglebulk is formed and the fronts do not change their shape during propagation. The details on the procedure aredescribed in Appendix B. The values of bulk VLD inthe channel core and near the walls are listed in Table II,columns −
9. The error-bars here and in Figs. 15, 18,correspond to the standard deviation around the meanvalues.
D. Evolution of VLD
The examples of the evolution of the vortex line den-sity at low and high temperatures are shown in Fig. 3.These examples illustrate the main difference in the flowconditions that crucially affect the tangle dynamics. Thevortex tangle is advected by the superfluid velocity field.At low T , the mean superfluid velocity V is weak dueto small the fraction of the normal fluid [cf. Eq. (4)].The tangle dynamics is governed mostly by the tangle-induced velocity and a net tangle displacement is neg-ligible, as is illustrated in Fig. 3b. On the other hand,at high temperature, V and V n are comparable and thetangle is flushed along the channel by the mean superfluidvelocity, see Fig. 3d. Under all conditions, the vortex tan-gle develops as a moving turbulent plug. At T = 1 . .Two edges of the tangle are different: a narrow and sharpedge is formed in the direction of V n and a wide and lesssteep edge in the direction of V s . As we show later, theseedges move with constant velocities and without chang-ing their shape. We, therefore, label them as a hot front(moving the direction of normal-fluid velocity away fromthe heater) and a cold front (moving in the direction ofmean superfluid velocity toward the heater). In furtheranalysis we distinguish a near-wall and a core regionsin the wall-normal direction and a bulk and the frontsregions in the streamwise direction, see Fig. 22.To characterize the distribution of VLD along andacross the tangle, we plot its streamwise profiles in Fig. 4and wall-normal profiles in Fig. 5. The dynamics of VLD,obtained with the parabolic normal-fluid velocity at vari-ous values of U c , differ mostly by the duration of transientbehavior in the tangle core and the mean value of VLDin the bulk of the tangle.In Fig. 4 we compare the streamwise VLD profiles forthe parabolic and for the flattened normal-fluid profilesat similar t f . At low T = 1 . V n ( y ), shown inFig. 4a, is still not fully developed (see Sec. II E for de-tails). The length of the tangle in both cases is about3.5 H . The edges of the tangles reached similar stream-wise positions indicating similar fronts velocities. At high T = 1 . L ( x ) for both V n profiles is almosthomogeneous in the tangle bulk. Here, however, the hotedge moved faster for the flattened V n ( y ), leading to ashorter plug. The mean VLD in the bulk, in this case, isabout 20% higher than for the parabolic profile.To compare the wall-normal VLD profiles for various flow conditions, we plot in Fig. 5 a dimensionless VLD L † ( y ) = L ( y ) κ / ( V ) . Here we compare L † ( y ) forthe parabolic and flattened V n ( y ) for the narrow chan-nel H = 0 . U c , in panels (b) and (d).First of all, we note that the wall-normal profiles L ( y )are consistent with the profiles obtained in the steady-state tangles with the VLD peaking near the wallat about the intervortex distance. The flattened profiles[thin red lines in panels (a) and (c)] have larger VLD val-ues in the tangle core. The L † ( y ) near the walls is higherfor the parabolic V n profile at T = 1 . T = 1 . V n ( y ) is homogeneous not only over thecore region but also over a large part of the near-wallregion, especially at T = 1 . V n ( y ), indicating that the increase of VLDnear the walls is indeed related to the boundary effect.These profiles are compared in Fig. 5b,d. At both tem-peratures, the tangles for widest channels H = 0 . T the VLD just did not reach the expected values,while at high T the tangle is formed by merging of twoindependent vortex plugs [similar to shown in Fig. 3d].The resulting streamwise inhomogeneity does not allowto properly resolve the near-wall region in the profilescalculated over narrow tangle bulk. However, it can beclearly seen that, as the channel become wider, the rangeof nearly-flat VLD distribution extends from the core tothe near-wall region, in a way similar to the flow, gener-ated by the flattened V n ( y ) profile. Comparing the VLDprofiles for H = 0 . . T the near-wall VLD is similar for bothchannel width. The normalized positions of the peaksdo not change with the channel width, meaning that thepeaks appear further from the wall for wider channels.To rationalize these observations we plot in Fig. 6 theprofiles of various velocities, normalized by the counter-flow velocity V † = V /V . We start with the streamwisecomponent of the superfluid velocity V x s ( y ) = V + V x BS .Near the walls V s < V . The main difference between the superfluid ve-locity behavior at low T [panels (a) and (b)] and at high T [panels (c) and (d)] is in the channel core, where at T = 1 . V s >
0, while at T = 1 . V s <
0. As aresult, at low T the value of V ns is smaller than U c , whileat high T , V ns is larger than V n everywhere in the channeland homogeneous across the core even for the parabolic V n profile. Furthermore, as is shown in Fig. 7a, the shapeof V ns ( y ) remain almost unchanged with increasing H atlow temperature while becoming flat over an increasinglylarger part of the channel as the channel become widerat high T , Fig. 7b. Since the tangle dynamics is definedby V ns according to Eq. (1), such behavior may explainthe tendency for a more homogeneous VLD distribution
10 15 2000.511.52 10
10 15 2000.511.52 10 T=1.3 K T=1.3 K (c)
T=1.9 KP FT=1.9 KFP (b)(a) (d)
FIG. 4: The stream-wise VLD profiles L ( x ) for T = 1 . T = 1 . V n ( “P”) was used in panel (a) [ U c = 3 cm/s] and panel (c) [ U c = 1 cm/s]. In the panels (b) and (d), the flattened V n profiles (“F”) were used, with the same (cid:104) V n (cid:105) as in the panels (a) and (c), respectively. The profiles for the walls regionare shown by thick blue lines and for the core region by thin red lines. Vertical dot-dashed lines denote the edges of the bulkregion. The channel width is H = 0 . t = t f . -0.5 0 0.502468 10 -3 -0.5 0 0.502468 10 -3 -0.5 0 0.5051015202530 -0.5 0 0.5051015202530(a) P H=0.15H=0.1 H=0.2F (b) FT=1.3 K T=1.3 KH=0.15H=0.1H=0.2 P T=1.9 K (d)
T=1.9 K (c)
FIG. 5: The wall-normal profiles of normalized VLD profiles L † ( y ) for T = 1 . T = 1 . L † ( y ) profiles for the parabolic (P) and the flattened (F) V n profiles at H = 0 . L † ( y ) profiles for the parabolic V n and different channel widths labeled in the figures by their values.All profiles at T = 1 . U c = 3 cm/s, the profiles at T = 1 . U c = 1 cm/s. Here, and inFigs. 6-8, dashed vertical lines denote edges of the channel core. Thin solid black lines are placed at the intervortex distancefrom the walls. In each panel only one intervortex distance is shown to avoid clutter. in wider channels at high T than at low temperature.The superfluid velocity plays an additional role in thedynamics. It is usually assumed that the overall tanglemotion is defined by the superfluid velocity V . How-ever, as is shown in Fig. 3b,d, the fronts of the tanglemay both move in the direction of V , or only cold frontmoves with V , while the hot front moves in the oppo-site direction. It is natural to associate the direction ofthe cold front motion with the direction of the super-fluid velocity V s near the walls, while the direction of the hot front motion with the direction of V s in the core ofthe channel. Such an assumption is further supportedin Fig. 8, where we plot the V s and V ns velocities for theintermediate T = 1 .
65 K, and Fig. 9, where we plot theevolution of the corresponding L ( x, t ). Here the super-fluid velocity in the channel core is close to zero and thebehavior of the hot front is very sensitive to the flowconditions. The superfluid velocity for the flattened V n profile, shown in Fig. 8 as the blue dashed line, is negligi-ble at the center of the channel and the corresponding hot -0.5 0 0.500.511.5 VsVnsVn -0.5 0 0.500.511.5
VsVnsVn -0.5 0 0.5-0.500.511.5
VsVnsVn -0.5 0 0.5-0.500.511.5
VsVnsVn
PT=1.3 K FT=1.3 K FT=1.9 K P T=1.9 K (c) (d)(b)(a)
FIG. 6: The wall-normal profiles of normalized velocities V x † s , V † n and V † ns . Panels (a) and (b) compare the profiles for theparabolic (“P”) and the flattened V n (“F”) profiles for T = 1 . U c = 3 cm/c, panels (c) and (d) compare the velocity profilesfor T = 1 . U c = 1 cm/c. The channel width is H = 0 . V n ( y ) profiles, bluedot-dashed lines denote the full streamwise superfluid velocity V x s ( y ) = V + V x BS ( y ), solid red lines denote the profiles of thecounterflow velocity V ns ( y ) = V n ( y ) − V x s ( y ). -0.4 -0.2 0 0.2 0.400.20.40.60.811.21.41.61.8 H=0.1 PH=0.1 FH=0.15H=0.2 -0.5 0 0.500.511.5
H=0.1 PH=0.1 FH=0.15H=0.2 (b)
T=1.3 K T=1.9 K (a)
FIG. 7: The wall-normal profiles of normalized V † ns profiles forthe parabolic at various channel widths and for the flattened V n (dashed lines) profiles for (a) T = 1 . T = 1 . H = 0 . H = 0 .
15 cm, dot-dashed lines correspond to H = 0 . VLD front [Fig. 9c] is stationary. The hot VLD front inthe flow generated by the parabolic V n ( y ) with U c = 1 . V s (0) (cid:46)
0, has hardly settled [Fig. 9a],despite relatively long propagation time. On the otherhand, at larger U c = 2 cm/s, we clearly see in Fig. 8b thehot front moving opposite to the direction of V . Thecold fronts under all conditions move with V , althoughthe front speeds differ. Here, the cold front speed for the -0.5 0 0.5-0.500.511.5 T=1.65 K
FIG. 8: The wall-normal profiles of normalized V s and V ns profiles for T = 1 .
65 K and the channel width H = 0 . V n with U c = 1 . U c = 2 cm/s and dashed line to the flat-tened profile of V n . flattened V n profile is smaller than for the correspondingparabolic V n ( y ), consistently with a smaller value of V s at the walls. The fronts speeds are not equal to V s at thewall or in the core, although they are clearly related. E. Transient Dynamics
In this Section, we consider the transient dynamics ofthe growing turbulent plugs for different conditions. Herewe compare the changes in the shape of the tangle, plot-
FIG. 9: VLD dynamics for T = 1 .
65 K and various V n profiles: (a) parabolic profile wih U c = 1 . U c = 2 cm/s and (c) flattened V n profile, corresponding to (a).FIG. 10: Rescaled normalized VLD profiles L † ( x ) for the parabolic V n at various time moments for (a) T = 1 . U c = 3cm/sand (b) T = 1 . U c = 1cm/s in the narrow channel H = 0 . L ( x, y ) for T = 1 . U c = 3 cm/s, H = 0 .
15 cm. ting in Figs. 10 and Fig. 11 the dimensionless VLD L † ( x )for the core and for the near-walls regions, rescaled tothe tangle width at each of the presented three timemoments. In this way, the scaled coordinate X = 0corresponds to the cold edge of the tangle and X = 1corresponds to the hot edge. The earliest time momentcorresponds to the time when the three-dimension (3D)tangle was formed and the latest to the time when thebulk region and two fronts are fully developed.The tangle dynamics for the parabolic V n profile isshown in Fig. 10. The main feature of these profiles isthe asymmetry with respect to the center of the tangle.The wall profiles, shown by dashed lines, rise along alltangle length and the asymmetry is relatively mild. Thecore profiles, shown by solid lines, on the other hand, arevery asymmetric, with the hot side growing faster thanthe cold side. We can see that at T = 1 . T = 1 . L † in the wall and in the core region,in accordance with Fig. 5c. Notably, also here the coreregion first develops closer to the hot front (i.e in thedirection of V n ), despite the fact that V s in the core isoriented in this case in the opposite direction.The main reason for this asymmetry is the spatial dis-tribution of the driving velocity. As is shown in AppendixC, due to enhanced VLD production in the channel corein the hot front region, and the transverse VLD flux thatmoves the vortex lines toward the walls, the parabolicwall-normal profile of the normal-fluid is translated intoa transient VLD distribution that reminds a horseshoeshape: L is higher near the walls and near the hot edgein the core of the channel, as is shown in Fig. 10c. Theclearly visible hump in the earliest core VLD profile (e.g.blue solid line, labeled “ t = 3” in Fig. 10a corresponds tothe central part of the horseshoe. With the developmentof the tangle, the hump is redistributed to the rest of thecore region and becomes less prominent, although it doesnot disappear completely even when the bulk value of L is established over a large part of the core. This horse-shoe shape of the most dense part of the growing tanglelasts longer for larger U c and wider channels. Such anasymmetry of the tangle, that appears from the very be- -3 -3 t=3 t=1.5 t=1 FIG. 11: Rescaled normalized VLD profiles L † ( x ) for the flattened V n at various time moments for (a) T = 1 . U c = 3cm/sand (b) T = 1 . U c = 1cm/s. Solid lines denote the core VLD profile, dashed lines denote the wall profiles. ginning of the tangle development leads to very differentinitial conditions for the formation of plug fronts (see alsoAppendix C).One may argue that such a scenario may not berealized in the real counterflow due to flattening ofthe normal-fluid profile and therefore more even initialVLD distribution. However, as we show in Fig. 11 andFig. 26(c,f), the streamwise tangle asymmetry is initiallypresent even if the flattened V n ( y ) is imposed, with VLDgrowing faster at the hot edge of the plug. This transientbehavior does not last long in this case, however, the hotfront remains stepper than the cold front, similar to thetangles formed under the parabolic V n . F. Large-scale superfluid motion
In simulations of homogeneous tangles under triply-periodic boundary conditions, the presence of meannormal-fluid and superfluid velocities is accounted for bya constant and space-homogeneous counterflow velocity,while the tangle-induced velocity is artificially random-ized by interactions with image vortex lines. In simu-lations of superfluid turbulence in the channel with pe-riodic streamwise conditions, the translation invarianceis broken in the wall-normal direction, creating super-fluid motion from the center of the channel towards thewalls. Still, in the streamwise direction, the variation ofthe vortex lines velocity is not taken into account.In our simulations, the tangle has finite streamwiselength and the superfluid velocity varies along the tangleas well as across it. The drift velocity of vortex lines V drift that include the mean velocity as well as all contributionsof the tangle-induced velocity represent the superfluidmotion at all scales that are formed in our system. InFig. 12 we plot the tangle drift velocity for T = 1 . x , while in thecore its motion is oriented toward larger x values, ed-dies of various sides are formed. For parabolic V n profile,at weak driving velocity U c = 2 cm/s, Fig. 12a, many circular eddies with sizes that are much larger than theintervortex distance (cid:96) but smaller than the channel size H , are formed. At strong driving velocity U c = 4 cm/s,Fig. 12b, two dominant vorticies of the system size H/ V n profile, Fig. 12c, we can see both the system size motionand smaller eddies. Please note, that near the walls thetangle velocity contributions are oriented perpendicularto the channel walls due to no-slip boundary conditions.It is the mean superfluid velocity that moves the vortexlines near the walls and helps to create the large-scale ed-dies. At higher temperatures, the dominant V sweepsthe tangle along the channel and masks the presence ofsmaller superfluid motions, similar to the sweeping ve-locity in classical fluids. However, analysis of the relativedrift velocity V drft − V clearly shows the presence ofthese motions also in the vortex tangles at higher T . G. Structural parameters c and I † (cid:96) In the microscopic description of the tangle dynam-ics very important role is played by two structural pa-rameters: the local binormal I (cid:96) = (cid:104) s (cid:48) × s (cid:48)(cid:48) (cid:105) and the ratiobetween the vortex line density and the mean-square cur-vature c . These parameters contribute to the terms ofthe equation of motion for L , responsible for the produc-tion and annihilation of the vortex line length, respec-tively [cf. Eqs. (10)-(14)]. In the homogeneous tangles,these parameters are constants, while in the channel flowthey depend on the position in the channel. The behav-ior of these parameters at the edges of the tangle was notstudied so far.The profiles of the coefficient c are shown in Fig. 13for T = 1 . T = 1 . c exhibit fluctuations along thetangle with a relatively large amplitude, especially when0
14 16 18-0.500.5(a) 13 14 15 16 17 18-0.500.5 (b)
13 14 15 16 17 18-0.500.5 (c)
FIG. 12: Tangle drift velocity V drfit for T = 1 . U c = 2 cm/s, (b) U c = 4cm/s and (c) flattened V n profile. Thearrows direction shows the local orientation of the velocity, the size of the arrows is proportional to its magnitude.FIG. 13: The coefficient c at various conditions. The profiles for T = 1 . V n with U c = 3 cm/s, labeled “P”, (b) the flattened profile, labeled “F”, (c) the corresponding wall-normalprofiles. The profiles for T = 1 . V n with U c = 1 cm/s and (e) the flattenedprofile; (f) the corresponding wall-normal profiles. Dot-dashed and dashed black lines mark the edges of the bulk and the coreregions for the streamwise and for the wall-normal profiles, respectively. Thin solid lines in panels (c) and (f) are placed at theintervortex distance from the corresponding walls. the flow is driven by the parabolic V n . The fluctuationsare less pronounced in the wall-normal profiles [panels(c)and (f)]. These profiles have a somewhat different aver-aging scope, however, we incline to attribute these fluc-tuations to the streamwise inhomogeneity of both VLDand the curvature, that do not match exactly. Neverthe-less, the values of c are fairly constant along the tangle,with the same values observed also the hot front region.The behavior of c in the cold front regions is different,with the tendency of becoming larger at low T . Similarbehavior is observed for other values of U c (not shown).When the flow is driven by the flattened V n profile, thevalues of c may be considered almost constant across thechannel. Its behavior changes only within the intervor-tex distance from the wall, where the values of VLD drop very quickly, while the square curvature keeps its valuesalmost until the wall. Conversely, when the driving veloc-ity has parabolic profile, c has the largest values in thecenter of the channel and decreases linearly towards thewalls until the intervortex distance is reached. Then itincreases, in a similar way as for the flattened V n profile,even reaching similar values at the wall. These largervalues in the core of the channel, as compared to thenear-walls region, are observed along all the tangle bulkand in the hot front. On the other hand, the values of c in the flows driven by flattened V n may be consideredalmost space-homogeneous, except for the cold front andvery near the walls, and similar to the values of c in thechannel core, observed for the parabolic normal flow.The dominant contribution to the production of vortex1 FIG. 14: The streamwise component of the index I † (cid:96),x at various conditions. The profiles for T = 1 . V n with U c = 3 cm/s, labeled “P”, (b) the flattened profile, labeled “F”, (c)the corresponding wall-normal profiles. The profiles for T = 1 . V n with U c = 1 cm/s and (e) the flattened profile; (f) the corresponding wall-normal profiles. Vertical dot-dashed lines mark the edgesof the bulk of the streamwise profiles, vertical dashed lines mark the edges of the core for the wall-normal profiles. Thin solidlines in panels (c) and (f) are placed at the intervortex distance from the corresponding walls. -3 -3 FIG. 15: The mean values of (a) c and (b) I † (cid:96),x in the core of the channel. The labels (A-C) correspond to current simulations(A: T = 1 . T = 1 .
65 K, green symbols, C: T = 1 . ◦ denote front velocities for the parabolic V n and various U c , (cid:5) corresponds to the flattened V n profile, (cid:46) and (cid:47) denote channel width H = 0 .
15 cm and H = 0 . T = 1 . , . . c and I † (cid:96) for the homogeneous tangle from Ref. with the GEC reconnection criterion, the same as used in this paper. In panel(a), thick dot-dashed lines, labeled G ( T = 1 .
65 K) and H ( T = 1 . c from Ref. inthe range of intervortex distances [4 × − − × − ]cm. Filled squares with error-bars, labeled ”I”, denote the results ofsimulations in the channel with parallel solid plates from Ref. in the range of temperatures T = 1 . − . lines has the term that depend of the streamwise pro-jection of the local binormal I (cid:96),x . In Fig. 14 we plot itsvalues normalized by the mean curvature I † (cid:96),x = I (cid:96),x / κ .The general behavior of I † (cid:96),x is similar to that of c . Wetherefore point out main differences. Looking at thewall-normal profiles, Fig. 14c,f, we notice that I † (cid:96),x is al- most homogeneous over the channel core for parabolicflows at both temperatures, crossing over to a linear de-crease toward the walls beyond the core region. It doesnot increase significantly very near the wall, although at T = 1 . T = 1 . I † (cid:96),x ex-tend further toward to walls, especially at high T . Nev-ertheless, the difference between the mean values of thechannel core and the near-walls region persists along thetangle bulk, even in this case. The values of I † (cid:96),x in bothfronts regions differ from the bulk, even if we take into ac-count strong fluctuations in its streamwise distribution.Interestingly, the shape of wall-normal profiles of c and I (cid:96),x in the flows, generated by the flattened V n pro-files, does not depend on V ns and the channel widthat both high and low T , although for different reasons.At low T , the curvature is only weakly dependent onthe distance from the wall, while VLD strongly peaksnear the walls. At high T , wall-normal profiles of L aremore homogeneous, but the curvature, in this case, de-creases toward the walls more strongly. The resulting y -distributions of c , Fig. 13c,f, are very similar. Thewall-normal distribution of I † (cid:96),x is fully defined by thestreamwise component of the binormal that is large inthe center of the channel and quickly decreases towardthe walls. Its shape is only slightly altered by similardistribution of (cid:104)| s (cid:48)(cid:48) |(cid:105) , Fig. 14c,f.To get an idea of how these results are related to otherknown measurements, we compare in Fig. 13a the valuesof c for the channel core with the results of simulationsof the homogeneous tangles , in the planar channel and with the experimental results for the range of in-tervortex distances, typical for our simulations. We havechosen to compare the values for the core of the channelbecause the experiments were carried out in wide chan-nels, where the core behavior is expected to dominate.These values are also expected to be more comparablewith c in the homogeneous tangle.As is clearly seen, the calculated values of c do notdepend on the intervortex distance within the range usedin our simulations. The temperature dependence agreeswith previous results, i.e. larger c at lower temperatures.Our current results agree well with the values obtained inthe homogeneous vortex tangles , shown by thin dashedlines. The values of c obtained in numerical simulationsof the vortex tangle in the flow between parallel plates for temperatures between 1 . − . c were calculated as averages overthe whole channel and are expected to be lower than thevalues in the channel core. With this in mind, they agreereasonably well with our results for T = 1 . for T = 1 . (cid:96) = 4 × − − × − cm, the fit becomes un-reliable and the experimental points tend to scatter, seeFig.11 in Ref. . The experimental values for T = 1 . T = 1 . I † (cid:96),x in the channel core areshown Fig. 13b. Note that this parameter measures align-ment of the local velocity V loc with the direction of the counterflow velocity. Similar to c , the index I † (cid:96),x is fairlyconstant for a given temperature, being larger for higher T . These values are somewhat higher than those ob-tained in the homogeneous tangles , including the valuesobtained in the flow driven by the flattened normal-fluidprofile (marked by diamond symbols). The temperaturemismatch ( T = 1 .
65 K in our simulation vs T = 1 . ) may account for the difference at the intermedi-ate temperature, however, the trend is systematic acrossthe temperatures. III. FRONT DYNAMICS AND ANALYSIS OFTHE VLD BALANCE EQUATIONA. Background Overview
Interface motion and front propagation in fluids aresubject of intensive studies in various fields of knowl-edge. Perhaps most well known are chemical reactionfronts in liquids , population dynamics of ecologicalcommunities , and combustion . The mathematicaldescription of those phenomena is based on partial differ-ential equations (PDE) for the evolution of the concen-tration of the reacting species and the evolution of thevelocity field. The two PDEs for the reactants and the ve-locity field are usually coupled, often in a nontrivial way.A mathematical simplification can be obtained by ne-glecting the back-reaction of the reactant on the velocityfield, which evolves independently. Such simplification isusually justified for the laminar velocity field. Even insuch a limit, the front dynamics is still nontrivial andit is described by a so-called advection-reaction-diffusion (ARD) equation ∂θ/∂t + u ( r , t ) · ∇ θ = D ∇ θ + F ( θ ) , (5)where θ ( r , t ) ∈ [0 ,
1] is the reactant concentration, D is the diffusivity and F ( θ ) is the reaction term. Thefront interface is in general two-dimensional, although inmany cases it is sufficient to consider its motion only inone direction. In a typical model situation the localizedinitial conditions are used, i.e. θ ( r , → r → −∞ and θ → r → ∞ . In this case, the reaction front will move towardspositive r . Here θ = 0 is an unstable state and θ = 1 astable one, therefore F ( θ ) satisfies the condition F (0) = F (1) = 0 , F ( θ ) > , if 0 < θ < . (6)It was shown that if there is no advection, the frontspeed converges to a limiting velocity v , defined by amarginal stability condition. In a moving fluid, it is nat-ural to expect that the front will propagate with anaverage (turbulent) speed v f > v . The turbulent frontspeed v f is defined by relative importance of the flow char-acteristics, such as the relevant system size Λ, advectingvelocity u , the diffusivity D , and the typical time scale τ r of the reaction term F ( θ ) = f ( θ ) /τ r . The shape of F ( θ ),3or more specifically the value θ at which it has largestslope, also plays a very important role. Two types ofits functional dependence are of particular importance:(1) a Fischer-Kolmogorov-Petrovskii-Peskunov (FKPP)nonlinearity F ( θ ) = θ (1 − θ ), or in general, any con-vex function F (cid:48)(cid:48) ( θ ) <
0; (2) an Arrhenius (or ignition)nonlinearity F ( θ ) = exp − θ c /θ (1 − θ ). Here the param-eter θ c is an activation concentration, below which thereis almost no production.In case of FKPP nonlinearity, the maximum slope F ( θ )occurs at θ = 0. Such fronts are called pulled fronts andtheir dynamics is fully determined by the region θ ≈
0, asif pulled by the leading edge. When the maximum slopeof F ( θ ) occurs at θ >
0, the front is pushed by the non-linear interior. The allowed velocity of the pulled frontshas to satisfy the condition (cid:112) DF (cid:48) (0) ≤ v min < (cid:115) D sup θ F ( θ ) θ , (7)where F ( θ ) /θ is the measure of the growth rate. ForFKPP dynamics F ( θ ) /θ = F (cid:48) (0) and for localized initialconditions v min = v = 2 (cid:112) DF (cid:48) (0).For pushed fronts, the minimal front velocity v min isalways larger than v . In both cases, depending on thesteepness of the initial conditions the asymptotic frontspeed may relax to the minimal v min or remain larger.There exists a vast literature on the front propagationin various flows. We concentrate on the laminar shearflow of ADR type and summarize several important re-sults. For details see Refs. and references therein. • The front velocity is bounded by K u < v f
Now we return to the channel counterflow of the super-fluid He and relate the properties of the model system,described in the previous section, to the dynamics of theturbulent vortex tangle.Here the role of the dimensionless variable θ inadvection-reaction-diffusion equation (5) is played by thenormalized VLD L = L / L , where L is the equilibrium vortex line density in the bulk of the tangle. The equa-tion of motion for L ( r, t ) in the channel may be writtenas ∂ t L ( r , t )+ ∇ [ V drift ( r ) L ( r , t )] = ˜ D ∇ L ( r , t )+ F [ L ( r , t )] , (8)where ˜ D is the effective diffusivity of VLD and V drift isthe tangle drift velocity, see Eq. (1). We follow Schwarz’smicroscopic approach and recall that the rate of elon-gation of the vortex line segment δξ is1 δξ dδξdt = α ( V ns ( s , t ) · ( s (cid:48) × s (cid:48)(cid:48) ) − | s (cid:48) × s (cid:48)(cid:48) | ) (9)+ s (cid:48) · V nl (cid:48) − α (cid:48) s (cid:48)(cid:48) · V ns . Integration of Eq. (9) over the vortex tangle gives for theright-hand-side (RHS) term F ( L ) F = P + P + P − D , (10) P = α L V (cid:48) (cid:90) Ω (cid:48) ( V − V nl ) · ( s (cid:48) × s (cid:48)(cid:48) ) dξ , (11) P = 1 L V (cid:48) (cid:90) Ω (cid:48) s (cid:48) · V (cid:48) nl dξ , (12) P = − α (cid:48) L V (cid:48) (cid:90) Ω (cid:48) s (cid:48)(cid:48) · V ns dξ , (13) D = α L V (cid:48) (cid:90) Ω (cid:48) V loc · ( s (cid:48) × s (cid:48)(cid:48) ) dξ . (14)Here P is usually named the production term since it isresponsible for most of the vortex line elongation. Thelast term D is traditionally termed the decay term sinceit represents the annihilation of vortex-line length duringvortex dynamics and reconnections. Two other terms P and P also represent the production of the vortex-linelength. In the homogeneous, tangle P vanish by symme-try. The term P is usually omitted due to smallness. Weinclude all terms since P and P become non-negligibleat low T near the walls (see Appendix C). Each term isproportional to L due to integration over dξ and divisionby L . At this stage, we retain the integral representationof F ( L ).Using the same approach, the VLD flux is defined as J = 1 L V (cid:48) (cid:90) Ω (cid:48) V drift dξ = V L + 1 L V (cid:48) (cid:90) Ω (cid:48) ( V BS + V mf ) dξ . (15)As was shown in Sec. II D, the bulk VLD and othertangle properties in the core of the channel and near thewalls are different but well defined. Therefore, insteadof taking into account full 3D structure of the tangle, aswell as 2D front interface, we consider the dynamics of thecore and the wall regions separately as one-dimensional(1D).However, to get 1D equation for L ( x ), it is not suffi-cient to only account for the streamwise component ofEq. (8). Although the transverse diffusion is negligible,the transverse VLD flux J y is an important factor in the4 FIG. 16: The profiles of ˜ F j ( L ) vs L [panels (a) and (b) for the cold and hot fronts, respectively] and ˜ F j ( L ) /L vs L [panels(c) and (d) for the cold and hot fronts, respectively] for T = 1 . V n are shown by solid lines,the profiles for the flattened V n are shown by dashed lines and denoted as “P” and “F”, respectively. The lines for the channelcore and for the walls region are labeled in the figure. -0.15 -0.1 -0.05010203040 T=1.3 Khot,wall hot,corecold,wallscold,core(a) -0.3 -0.25 -0.20102030405060 T=1.65 Khot,corehot,wall cold,wallscold,core(b) -0.7 -0.6 -0.5 -0.4 -0.301020304050 T=1.9 Khot,wallhot,core cold,wallscold,core(c) FIG. 17: Maximum growth rate sup L [ ˜ F j ( L ) /L ] at various flow conditions. In all panels, ◦ denote front velocities for parabolic V n and various U c , (cid:5) corresponds to the flattened V n profile, (cid:46) and (cid:47) denote channel widths H = 0 .
15 cm and H = 0 . V is shown by dashed lines, which serve to guide the eye only. Different data sets aremarked in the figure by labels of the same color that point to the corresponding symbols. inhomogeneous tangle dynamics , moving VLD fromthe channel core towards the walls. We move it to RHSof Eq. (8), such that after averaging of the core and wallsregions, it will serve as an additional decay term in thechannel core and as an additional production term nearthe walls. In such a way we get the ARD-type equationfor the normalized VLD L ( x, t ) for the core (labeled as’“c”) and for the walls (labeled as “w”) regions: ∂L j ( x, t ) ∂t + ∂ J j ( x, t ) ∂x = ˜ D j ∂ L j ( x, t ) ∂x + ˜ F j [ L ( x, t )] , J j ( x, t ) = V L j ( x, t ) + ˜ J jx ( x, t ) (16)˜ F j [ L ( x, t )] = F j [ L ( x, t )] − ∂ J jy ( x, t ) ∂y , j ∈ { c , w } . (17)The longitudinal tangle-induced flux ˜ J x = J x − V L helps to redistribute the vortex line density along thetangle. We account for it by replacing V → V x s . Herewe neglected the streamwise component of the mutualfriction contribution to the drift velocity V x mf as it con- tributes only about 1% to the value of V x drift . The modi-fied “reaction” term ˜ F j [ L ( x, t )] includes the contributionfrom the transverse flux. Since, in this formulation, theeffective diffusivity is a parameter that depends on theflow conditions, we allow for different values of ˜ D j forthe channel core and for near-walls regions. Moreover,the values may differ in the tangle bulk and in the frontsregions.Using this framework, we analyze in the rest of thisSection various aspects of the propagation of the fronts,including the type of the fronts, their speeds, shapes, andthe effective diffusivity. C. Properties of ˜ F ( L ) To identify the type of nonlinearity in the Eq. (16), wecalculate the front profile for ˜ F ( L ), as described in Ap-pendix B. The dependencies of ˜ F ( L ) and ˜ F ( L ) /L on L in the front regions , calculated for T = 1 .
13 14 15 16 17 18012 10 -3 FIG. 18: The coefficient B † for various conditions. Streamwise profiles for (a) T = 1 . U c = 3 cm/s and (b) T = 1 . U c = 1 cm/s. Thin red lines correspond to the core profiles, thick blue lines denote near-wall profiles. Vertical dot-dashedlines mark the edges of the tangle bulk. In calculation of B for the core and for the walls regions we used the correspondinginstantaneous values of β, c and L and then averaged over time. (c) B † averaged over tangle bulk. Symbols, denoting variousflow conditions, are the same as in Fig. 15.
13 14 15 16 17 180123 13 14 15 16 17 180123 2 3 4 5 6 7 80123 2 3 4 5 6 7 80123PT=1.3 K FT=1.3 K(a) (c)(d)(b) PFT=1.9 KT=1.9 K
FIG. 19: The ratio C for various conditions. The profiles for the parabolic V n with (a) T = 1 . U C = 3 cm/s and (c) T = 1 . U c = 1 cm/s. The profiles of C for the corresponding flattened profiles are shown in (b) for T = 1 . T = 1 . C = 1. in Fig. 16. The results for the parabolic profile as shownby solid lines, for the flattened profile– by dashed lines.As is clearly seen, the L dependence of ˜ F ( L ) is differentfor the hot fronts [panel (a)] and for the cold fronts [panel(b)]. The hot fronts are of the FKPP type, i.e the largestrate of growth sup L [ ˜ F ( L ) /L ] is at L →
0, as is shown inFig. 16(d), while for the cold fronts [Fig. 16(c)], it is foundcloser to the center part of the front. This property is ro-bust and observed all flow conditions, and for both typesof the V n profiles, although at T = 1 . L = 0than at low T . Despite complicated shapes of ˜ F ( L ) forvarious flow conditions, the values of the largest rate ofgrowth sup L [ ˜ F ( L ) /L ], shown in Fig. 17, depend linearlyon V . The dependencies for the walls and the core re-gions differ even for the same front region, with hot frontsbeing stronger dependent on the advecting velocity than the cold fronts. In particular, sup L [ ˜ F ( L ) /L ] for the coldfront in the channel core is almost V independent.Most of attempts to find equation of motion for thevortex line density so far dealt with steady-state tanglesand represented P and D in Eq. (10) as functions of L and V ns only for the homogeneous tangles, adding thecurvature, the binormal, and their derivatives in theinhomogeneous case. In the current situation of the in-homogeneous and growing tangle we can not expect aunique closure. Aiming at the analysis of front dynam-ics, we make use of the fact that at least the hot frontsare of FKPP type. We then seek to represent Eq. (17) ina general form ˜ F ( L ) = A L − B L , (18)where coefficients A and B have dimensions [1/s] andmay depend on the position and time.6 D. Closure for ˜ F ( L ) The main idea behind all the proposed closures isto take slowly-varying fields out of the average along thevortex lines. The resulting closure form is a product ofslowly-varying macroscopic properties of the flow [suchas V ns ( x, y, t )] and of the tangle [such as c ( x, y, t ) and I (cid:96),x ( x, y, t )]. In Appendix C we discuss various contribu-tions to the ˜ F ( L ). For our current analysis, however, wedo not need all of them.We start with the last term in Eq. (18). It is read-ily associated with the term D , Eq. (14), as the relation D ∝ L was shown experimentally and rationalizedtheoretically for the steady-state homogeneous vortextangles. We use here the following form of this depen-dence for the dimensionless VLD L : D ≈ αβ (cid:104) κ (cid:105) L = B L ; B = αβc L , (19)where the relation (cid:104) κ (cid:105) = c L was used. In the homoge-neous tangle, where c is a constant, the coefficient B isalso a constant for a given temperature. In the inhomoge-neous developing tangle, the mean-square curvature (cid:104) κ (cid:105) in the bulk of the tangle is almost homogeneous acrossthe channel, while the vortex line density is not. There-fore, the coefficient c has more complicated behavior, asis shown in Fig. 13. Nevertheless, when c is averagedover the core and near-walls regions separately, the clo-sure Eq. (19) works quite well, especially at low T , as isshown in Appendix C, Fig. 24.It turned out that the values of B are very weakly de-pendent on the position in the channel. The differencebetween the values of c in the channel core and near thewalls is compensated by the corresponding difference inthe values of L , such that B is almost constant every-where in the channel, with the exception of the immedi-ate vicinity of the tangle edge, where the measurementsof c become unreliable. To compare B for various flowconditions, we plot in Fig. 18 the dimensionless B † = B κ/ ( V ) ≈ α c Γ , (20)where in the right-most relation we took into account that ln( R/a ) / (4 π ) ≈ κ L / ( V ) is a di-mensionless coefficient relating the steady-state homo-geneous VLD and the counterflow velocity. The coeffi-cient B † is expected to be a rising function of the tem-perature, but to have only weak dependence on other flowconditions. The streamwise profiles of B † are illustratedfor T = 1 . U c = 3 cm/s (Fig. 18a) and T = 1 . U c = 1 cm/s (Fig. 18b). The values of B † averaged overtangle bulk are summarized in Fig. 18c. As expected, thecoefficients B † grow with the temperature, but otherwise,despite some scatter, are almost independent of the flowconditions. Note that, in accordance with the behaviorof c , B † is larger in the flow generated by the flattened V n profile (diamonds), that in the flow, generated by thecorresponding parabolic profile (circles), for similar V . Using almost constancy of B over entire tangle, we canassociate τ dec ≡ ( B ) − with some characteristic time, inthis case of the tangle decay, and further rewrite˜ F ( L ) = Lτ dec ( C − L ) , C = AB . (21)In the steady-state homogeneous tangle, C = 1. Again,there is no a-priori reason to expect that this relationwill hold in the current situation. However, as is shownin Fig. 19, up to natural fluctuations, C ≈ T = 1 . T theyare less dissipative. The closeness of the ratio C to unityindicate that we correctly account for all the relevantcontributions to the ˜ F ( L ) in Eq. (17). E. Solution of VLD equation of motion
Having defined the functional form for ˜ F ( L ) and tak-ing, for now, C = 1, we can return to Eq. (16) and rewriteit as ∂ t L j ( x, t ) + V x s ∂ x L j ( x, t ) = (22)˜ D j ∂ x,x L j ( x, t ) + 1 /τ dec L j ( x, t )[1 − L j ( x, t )] . We now switch to dimensionless variables (omitting forshortness the index j ) τ = t/τ dec , z = x/σ, σ = (cid:113) ˜ Dτ dec , (23) w = V x s /V diff , V diff = σ/τ dec , (24)to rewrite Eq. (22) as ∂ τ L j + w j ∂ z L j = ∂ z,z L j + L j − ( L j ) . (25)Comparing with Eq. (5), we see that Eq. (25) is the ARDequation of FKPP type for the vortex line density, whichfor front velocities v f > (cid:113) ˜ D/τ d admits a traveling wavesolution ζ = c ( z − V f τ ) with the dimensionless front speed V f = v f /V diff . Substituting this solution to (25), we getan equation that defines the velocity and the shape ofthe front:[ c j v j ∂ ζ + c j ∂ ζ,ζ ] L j + L j − ( L j ) = 0 , (26) v j = V j f − w j . A similar equation was obtained by Nemirovskii for1D front propagation, using the original Vinen’s form for F ( L ) = α V i L / − β V i L , and solved numerically for thefront speed, with the parameters estimated for the ho-mogeneous steady-state vortex tangle by Schwarz andthe diffusion constant D ≈ . κ .7 -0.2 -0.15 -0.1 -0.05-0.1-0.0500.050.10.150.2 -0.3 -0.25 -0.2 -0.15 -0.1-0.2-0.100.1 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3-0.6-0.5-0.4-0.3-0.2-0.10Hot front T=1.3 KCold front Hot frontCold front (c) T=1.65 K T=1.9 KHot frontCold front (a) (b)
FIG. 20: Front velocities as a function of mean superfluid velocity for (a) T = 1 . T = 1 .
65 K and (c) T = 1 . ◦ denote front velocities for the parabolic V n and various U c , (cid:5) corresponds to the flattened V n profile, (cid:46) and (cid:47) denote channel width H = 0 .
15 cm and H = 0 . FIG. 21: The cold and hot fronts shapes for various conditions: (a,b) V n with T = 1 . U C = 3 cm/s ; (c,d) T = 1 .
3K andflattened V n profile; (e,f) T = 1 . U c = 1 cm/s; (g,h) T = 1 .
9K and flattened V n profile. The cold front shapes are shown inpanels (a,c,e) and (g), the hot front shapes ares shown in panels (b,d,f) and (h). Thin red lines correspond to the core fronts,thick blue lines denote near-wall fronts. Cyan dashed lines denote fits for the near-wall fronts, red dashed lines denote fits forthe fronts in the channel core. The equation (26) may be solved analytically usingTahn method . A general form of these solutions,symmetric with respect to the direction of propagation,reads: L ( ζ ) = 14 (cid:2) ± tanh ζ (cid:3) , c = 12 √ , v = ∓ √ , (27) or, relaxing the requirement that C = 1, L ( ζ ) = C (cid:2) ± tanh ζ (cid:3) , c = √C √ , v = ∓ √C√ , (28)Returning to the original dimensional variables L ( x ) = 14 (cid:2) ± tanh( 1 λ [ x − v f t ]) (cid:3) , (29) λ = 2 σ (cid:112) / C , v f = ± V diff (cid:112) C / V x s , (30)8where λ is the front width. As we can see, the effectivediffusion constant and the characteristic decay time de-fine both the front width and the front velocity via thediffusion spread σ and its speed V diff .The similar (symmetric) solution was postulated inRef. without derivation, assuming F ( L ) based on Vi-nen’s form of F ( L ) for the case of thermal counterflow in the presence of a wall.However, as we know now, the hot and cold fronts areof different types. Strictly speaking, only hot fronts areof FKPP type (pulled) and fulfill the underlying assump-tions for the solution. Nevertheless, we may hope that,at least at high temperatures, the solution will describereasonably well also the cold fronts.Recalling that ˜ D depends on the flow conditions andtherefore may be different for the channel core and nearthe walls, we get solutions for four fronts: L j, c ( x ) = 14 (cid:2) λ j, c [ x − v cf t ]) (cid:3) , (31) L j, h ( x ) = 14 (cid:2) − tanh( 1 λ j, h [ x − v hf t ]) (cid:3) , (32)where λ j, c , λ j, h are the widths of the corresponding frontsand v c f , v h f are the corresponding front velocities. Here aword of caution is in order. The solutions Eq. (31)-(32) donot describe any transient behavior, such as VLD humpin the channel core near the hot front, strong VLD fluc-tuations at the fronts at low T , or effects of the differenttype of the nonlinearity for the cold fronts. Since frontsof the studied tangles most probably did not reach theexpected limiting shapes, all parameters are consideredas effective and corresponding to the chosen time t f .The mean front shapes were calculated using the pro-cedure described in Appendix B and fitted with the so-lutions Eq. (31)-(32) to obtain the front velocities v hf , v cf and front widths λ j, c , λ j, h . F. Front velocities and shapes
The front speeds are shown in Fig. 20 as a function ofthe mean superfluid velocity V . It is clearly seen that v f depends linearly on the advecting velocity, with hot andcold front speeds having opposite trends, independent ofthe actual orientation of the hot front velocity. All datafor a given temperature are well fit by the same lineardependence, shown as black dashed lines. Note that thefront velocities are the same for the channel core and thenear-walls regions. This point requires additional atten-tion. As we mentioned earlier, the hot front is lead bythe channel core, while the cold front is defined by thenear-wall region. Moreover, the superfluid velocity inthe bulk of the channel, as we showed in Fig. 8, is closeto the corresponding v f . This raises a natural question,how the hot front velocity near the channel walls becomeequal to that in the core and similarly, the cold front ve-locity in the core becomes equal to v cf near the channelwalls? The answer lies in the action of the transverse flux TABLE III: Onset front velocities v ∗ f and corresponding meansuperfluid velocities V , ∗ s . The error-bars reflect the sensitiv-ity of the linear fitting procedure. T , K 1.3 1.65 1.9 v ∗ f , cm/s − . ± . − . ± . − . ± . V ∗ s , cm/s − . ± . − . ± . − . ± . ∂ J jy ( x, t ) /∂y that changes very strongly in the fronts re-gions, but is almost constant along the tangle, see Fig. 26and Appendix C. In this way, the hot front near the wallis formed by VLD brought by the flux from the channelcore and its velocity matches the velocity at the core onlyvery close to the tangle edge. Similarly, the superfluid ve-locity in the core of the channel is quickly changed to v cf by the transverse flux which in this region brings VLDfrom the walls toward the channel core. Here, the flux ismuch weaker than in the hot front region and the devel-opment of the cold front in the channel core is a result ofa complicated interplay of various mechanisms, leadingto long-lasting transient behavior.At low T , | v hf | > | v cf | for the same advecting velocity V , while at high T the relation is opposite. This obser-vation is in agreement with the early experiments is thincapillaries . Moreover, the front velocities, observed inRef. for T = 1 .
34 K at low heat fluxes, are similar to v f measured in our simulations at T = 1 . V ∗ s atwhich the front speeds are expected to be the same and,therefore, only one front can propagate. It is natural toassociate the corresponding v ∗ f with the onset of the frontsolution in the counterflow. Since without advecting flow(or more specifically, the counterflow V ns ), the counter-flow turbulence does not exist, the onset front velocity v ∗ f > v . Note that at all temperatures, the values of V ∗ s are larger than the critical V , below which the vortextangle is not formed. The values of V were estimatedfrom the √L = γ ( V ns − v c ) dependence and the coun-terflow condition. The onset fronts velocities v ∗ f and V ∗ s are listed in the Table III.The representative front shapes, together with theirfits with the solution Eq. (31)-(32), are shown in Fig. 21.The x -axis shows the distance from the front edge ( X =0) for the core and the walls regions separately. First ofall, we note the presence of the narrow VLD hump, lo-calized between the tangle bulk and the hot front in thetangles driven by the parabolic V n profiles, Fig. 21a,c.This hump is not formed when the normal-fluid veloc-ity profile is flattened, Fig. 21b,d. In all cases, the hotfronts are 2 − L in the cold front shapes, well seen for thenear-wall front shapes at both temperatures, is a sign ofnon-FKPP non-linearity and is not accounted for by thesolution. However, the solution Eq. (31) describes rea-9sonably well the overall cold front shapes, especially athigh temperatures, at which the fronts are well-formedand developed. G. Effective diffusivity
The importance of the diffusion mechanism for the de-cay of inhomogeneous tangle was studied theoretically and numerically for the decaying tangles at T = 0 Kwith most recent estimates of the effective diffusion con-stant in the range (0 . − κ . The presence of dissipativewalls reduces the values of the effective diffusion con-stant, while in the 3D unbounded vortex tangle thevalue of the effective diffusion constant was found to beclose to 0 . κ .Using the relation between the front width and theeffective diffusion constant, Eqs. (23) and (30), we canestimate ˜ D for various conditions. For that, we rewriteEq. (30) as ˜ D j, c = ( λ j, c ) τ d , ˜ D j, h = ( λ j, h ) τ d , (33)where we retain C = 1 and τ d = Const . for given condi-tions. To get an idea of what behavior to expect from ˜ D we rewrite (33) (omitting indices j , c and h for clarity)as ˜ D = λ B
24 = λ B † ( V ) κ . (34)The temperature dependence of ˜ D is therefore mostlydefined by B † , the dependence on the driving velocityby ( V ) and the influence of other flow conditions, in-cluding the spatial dependence – by the front width λ .There is no systematic dependence of the front widthon the driving velocity. Recall that the cold fronts arewider than the hot fronts, such that for the given T and V , ˜ D j, c > ˜ D j, h with the difference reaching up to anorder of magnitude. The typical front width range de-creases with temperature, such that λ c ∼ (0 . − .
1) cmat T = 1 . T = 1 . λ c ∼ (0 . − .
05) cm.The hot fronts are more narrow: λ h ∼ (0 . − .
05) cmat T = 1 . λ h ∼ (0 . − .
03) cm at T = 1 . B † grows with T . As a result, forthe studied range of flow conditions, at the cold front,the typical values are ˜ D c ∼ (0 . − . κ , while at thehot fronts ˜ D h ∼ (0 . − . κ and are larger for highertemperatures. This T -dependence is more prominent forthe flows driven by the parabolic normal-fluid velocity.The representative values of ˜ D , calculated according toEq. (33), are listed in Table IV. It is important to re-member that the effective diffusivity ˜ D is not a materialproperty of superfluid He, but a dynamical property ofpropagating fronts in the particular flow conditions, in-cluding different nonlinear processes in the front regions.In addition, the values listed in the table correspond to the reached stage of the tangle development and are sen-sitive to the presence of the transient processes in thetangle core. Nevertheless, since the order of magnitudeof ˜ D is the same for the flows driven by the parabolic andby flattened V n profiles at all studied temperatures, thesevalues may be considered as a robust dynamical propertyof the propagating fronts in the channel counterflow.The values of ˜ D at the hot front are remarkably close tothe values of the effective diffusion constant found numer-ically in the bounded and unbounded bulk tanglesat zero temperature. We do not have a reliable measureof the diffusion in the bulk of the tangle. However, sincethe values of many of the tangle properties in the bulkare similar to those in the hot front region, we suggestthat also the values of ˜ D in the tangle bulk would besimilar to those in the hot front region at least in theorder of the magnitude. IV. DISCUSSION
Our simulations of the quantum vortex tangles thatdevelop freely in the channel from localized initial con-ditions under the influence of the counterflow velocity,give a unique insight into their natural dynamics andstructure. Despite a wide variety of the flow conditionsexperienced by the vortex lines that influence the localdynamics, there are many common features.In particular, the tangles may be divided into regionsaccording to their dynamics. The regions near the tan-gle edges exhibit front dynamics. The dynamics of tanglebulk is more similar to that of the steady-state stationarytangles. In the bulk, the parts of the tangle that developnear the channel wall, are first to reach equilibrium VLDand grow almost symmetrically with respect of the direc-tion of the counterflow velocity. On the other hand, thetransient tangle dynamics in the channel core is slower,with notable asymmetry and preferential growth of VLDtoward the hot front, resulting in the long-lasting stream-wise inhomogeneity. This behavior is similar at high andlow temperatures, despite the different direction of thehot front propagation. This asymmetry is originatedfrom the production of the vortex line length, stronglypeaked in the channel core within the hot front region.The only difference between the dynamics at differentvelocities of the driving normal fluid and even its wall-normal profile is the duration of the transient behaviorand degree of the inhomogeneity of resulting vortex tan-gle. Conversely, the structural properties of the vortextangle, such as the ratio between the curvature and thevortex line density and preferential orientation of the lo-cal velocity, reach their steady-state distributions as soonas the tangle become three-dimensional, with core valuessimilar to those obtained in the simulations of the steady-state vortex tangles and the experimental estimates.The VLD is higher near the walls than in the chan-nel core, peaking at about the intervortex distance, inagreement with the results of simulations of steady-state0 T = 1 . T = 1 .
65 K T = 1 . V n ( y ) P F P F P F˜ D core,c /κ . ± . . ± . . ± . . ± . . ± . . ± . D wall,c /κ . ± . . ± . . ± . . ± . . ± . . ± . D core,h /κ . ± .
005 0 . ± .
01 0 . ± .
02 0 . ± .
02 0 . ± .
01 0 . ± . D wall,h /κ . ± .
01 0 . ± .
04 0 . ± .
02 0 . ± .
01 0 . ± .
05 0 . ± . V n profile, respectively. The error-bars account for C = 1 ± . λ and B . The flow conditions are the same as in Fig. 13 and Fig. 14. tangles in the channel. This difference between the chan-nel core and the near-wall regions is less prominent whenthe flow is driven by the normal-fluid velocity with theflattened profile. A similar trend of relatively flat VLDdistribution in the channel core, that extends towards thewalls, was observed in simulations with wider channels atall temperatures.An explicit account for the advecting mean superfluidvelocity allowed us to detect a superfluid motion of var-ious scales within the vortex tangle. The largest scalesof this motion reach the channel size at strong drivingvelocity. When normal-fluid velocity profile is flattened,as is expected in the turbulent flow, superfluid motionsexist at many scales. The presence of this large-scalesuperfluid motion is reflected in the streamwise inhomo-geneity of various tangle properties. The typical periodof the fluctuations is of the order H/
2, corresponding tothe largest eddies formed in the tangle.The analysis of the dynamics of the fronts in the frame-work of the advection-diffusion-reaction equation givesunexpected results. The two fronts are driven by differ-ent parts of the flow and have a different type of non-linearity of the generalized production term. The hotfronts are “pulled”, i.e. driven by the flow in the channelcore and the leading edge dominate in defining their highsteepness and the propagation speed. The cold fronts, onthe other hand, are lead by the near-walls tangle and are“pushed” by the nonlinear interior. A low density ”foot”moves before the tangle, and only at about quarter of thefront width the VLD start to rise fast. These fronts arewide and the shape difference between the channel coreand near the walls is larger. In accordance with ADR dy-namics, the front velocities are linearly proportional tothe advecting mean superfluid velocity, with common de-pendence for all conditions at a given temperature. Theanalytic solution of the equation of motion Eq. (16) fitswell the overall front shapes for all conditions, while itdoes not describe the transient effects near the hot frontsand the effects of the non-FKPP nonlinearity at the coldfronts. These solutions allow extracting the effective dif-fusivity which is flow-dependent and different at the hotand at the cold fronts. The values of the effective diffu-sivity measured the hot fronts agree in the order of mag-nitude with recent estimates from simulations at T = 0K. Appendix A: Calculation of various profiles.
The wall-normal and the streamwise profiles of variousquantities are calculated according to the scheme shownin Fig. 22. The division into different zones is somewhatarbitrary, however, we have checked that the values ofthe tangle properties are robust with respect to the vari-ation of the zones boundaries within 2 mesh-sizes. Forillustration we use the vortex line density L . The wall-normal profiles were obtained by averaging the 2D mapsover the bulk region of the tangle defined at each timemoment and further averaged over last t av = 0 . t f − t av ( dashed lines) and t f (solid lines). The streamwise profiles were calculated for t f by averaging over the core and near-walls regions sep-arately. In cases where the behavior at two near-wallsregions was similar, they were averaged together. Thestreamwise profiles of structural properties, such as c ,and various terms of the balance equation, in addition toaveraging over core and near-walls regions, were averagedover last 0 . L , the points near the edge of the tangle, where L ( x ) ≈
0, were omitted in calculation of the time averageand not shown.The intervortex distance (cid:96) = L − / , shown in the wall-normal y -profiles as vertical thin black lines, is calculatedhere at time t f by averaging L over bulk in x -directionand over near-walls region in y -direction. Appendix B: Front shape.
The fronts of the tangles propagate without shapechange. To show this, we shift the x -positions of thestreamwise VLD profiles L ( x ), corresponding to the timeperiod when the bulk and the fronts are fully developed,to the left and to the right, such that the correspondingtangle edges overlap. This procedure is used to measurethe front speeds v c f and v h f that allow such an overlap.The original profiles L ( x ) are shown in Fig. 23b. The re-sult of the cold front collapse is plotted in panel(a) andof the hot front collapse– in panel(c). Clearly, the frontshape does not change during this time period. To ob-1 FIG. 22: Schematic representation of various averaging zones. (a) 2D map of L ( x, y ) (cm − ) in which various averaging zonesare marked. (b) streamwise profiles (cid:104)L ( x ) (cid:105) y averaged over two near-wall zones and over the core in the y direction. (c) Thewall-normal profile (cid:104)L ( y ) (cid:105) x averaged over the tangle bulk in the x -direction. (d) the streamwise profiles (cid:104)L ( x ) (cid:105) y , in which twonear-wall zones are averaged together. In panels (c) and (d) the shaded area shows variation of VLD between t f − t av (thindashed lines) and t f (think solid lines). tain the front shape we calculate the dimensionless VLD L = L / L , where L is the mean VLD in the bulk of thetangle. Since the values of VLD differ in the core of thechannel and near the walls, we treat these regions sep-arately. We further average these profiles over the timeperiod of 0.2 s. In such a way we obtain four shapes,for the cold front for the hot front in the core and in thenear-walls regions, shown in Fig. 21.The same procedure was used to obtain the frontshapes of other quantities of interest. Appendix C: Terms of balance equation.
In this section, we provide a detailed description ofvarious contributions to F ( L ) used in the analysis of thefront dynamics. As was shown in Sec. III D, the spa-tial distribution of the decay term D ≈ αβc L L es-sentially follows L . This representation faithfully de-scribes the integral form (14) not only on average inthe steady-state tangle but also locally and instanta-neously, including the transient stage of the dynamics,as is shown in Fig. 24. To allow comparison, the dimen-sionless values D † = D κ/ ( V ) are plotted. The modelslightly overestimates the decay term at high T , but oth-erwise should be considered very adequate everywherein the tangle. Note that the strong streamwise inho-mogeneity, amplified compared to VLD, is well repro-duced by the model. The situation is different with theproduction term. Directly interpreting the model formas a product of average slowly-varying fields, we get for P = α (cid:104) V x ns,nl (cid:105)(cid:104) s (cid:48) × s (cid:48)(cid:48) (cid:105) x L ≈ αV I (cid:96),x L . So far, the prob- lem of the closure for P amounted to the question how todescribe I (cid:96),x in terms of L and V ns . As it followsfrom the discussion in Sec. II D and II G, in the inhomo-geneous flows, there is no simple answer to this question.Additional complication arises at low T , at which thecontributions of P = (cid:104) s (cid:48) · V (cid:48) nl (cid:105) L and P = − α (cid:48) V ns (cid:104) κ (cid:105) L near the walls are not negligible. We do not attempt hereto find the best model representation, but rather pointout additional difficulties brought up by the presence oflarge-scale superfluid motion.The wall-normal profiles of the dimensionless P † = P κ/ ( V ) contributions to the production term areshown in Fig. 25. The main contribution P , shown bypurple dotted lines, is peaking in the channel core, whereit is almost constant, then quickly decreasing toward thewalls. This behavior is very similar to I (cid:96),x ( y ) at all stud-ied temperatures, with differences in the near-wall be-havior. For the parabolic V n profiles, Fig. 25a-c, at hightemperature, P remains non-zero even very close to thewalls, at intermediate T = 1 .
65 K P drops to zero atabout intervortex distance from the wall, while at low T it becomes negligible already at about 2 (cid:96) from the nearestwall. Two other contributions, P and P are negligiblecompared to P in the channel core, gradually increasingtoward the walls and attaining the largest values at thedistance (cid:96) from them. Here we see the largest differencebetween the high and low T behavior. At T = 1 . P and P may be safely neglected ev-erywhere in the channel. At T = 1 .
65 K the contributionof P becomes important, while at T = 1 . P and P are dominant in near the walls, such that over-all production in this region is about half of that in thechannel core. As a results, the total production y -profile2 (a) (b) (c) FIG. 23: The hot and cold front shapes. A series of streamwise VLD profiles corresponds to last 1 s of the evolution of thewalls region, T = 1 . V n . (a) The profiles are collapsed using the cold front speed, (b) the original profiles, (c) theprofiles are collapsed using the hot front speed.FIG. 24: The decay term Eq. (14) (thick lines) and its model form Eq. (19) with 95% confidence interval (shaded area) atdifferent conditions. (a,c) The streamwise profiles for the channel core for (a) T = 1 . V n with U c = 3 cm/s and(c) T = 1 . V n with U c = 1 cm/s. (b,d) The wall-normal profiles for the conditions of (a) and (c), respectively,and matching flows with flattened V n profile. Dot-dashed black lines mark the edges of the bulk and the core regions for thestreamwise and for the wall-normal profiles, respectively. Thin solid lines in panels (b),(d) are placed at the intervortex distancefrom the corresponding walls. The profiles are calculated as described in Appendix A. For normalization in Eq. (14), L core0 wasused for the streamwise profiles in (a) and (c) and L wall0 for the wall-normal profiles in (b) and (d). becomes similar to that for the flattened V n profile atthis temperature, Fig. 25d, although in the latter case P has the dominant contribution (about 90%) every-where in the channel. For this type of the V n profile, thecontributions of P and P may be neglected at all tem-peratures, especially at high T . The difference betweenthe production in the channel core and in the near-wallregions is much smaller than for the parabolic V n pro-files. These features are even more pronounced at highertemperatures.To see how the VLD production is distributed alongthe tangle, we plot in Fig. 26(a-c) the streamwise profilesof P † and of the total production P † + P † + P † for thesame conditions as in Fig. 25. We do not show the profilesfor T = 1 .
65 K, as they represent an intermediate caseand do not bring more information.First of all, we can clearly distinguish the bulk, thehot and the cold front regions. In the tangle bulk, theproduction is almost constant, up to fluctuations thatare stronger in the channel core than in the near-wallsregion. In accordance with profiles shown in Fig. 25, the contribution of P (thin lines) is dominant at high T ,Fig. 26b, both in the core and near walls, as well as forthe flows generated by the flattened V n profiles, Fig. 26c.At low T , Fig. 26a, P constitutes about half of the totalproduction in the near-walls region.In the hot front region, the production in the corehas a pronounced peak in the channel core, very closeto the tangle edge, which is dominated by P . TheVLD produced in this region is then taken to the wallsby the transverse flux, as is well seen in Fig. 26(d-f)where we plot ∂ J † y ( x, t ) /∂y, for the dimensionless J † y = J y κ/ ( V ) . Although this peak is not as pronounced inthe flows generated by the flattened V n profiles, the pro-duction is still stronger in the channel core than near thewalls. The horseshoe shape of the VLD distribution, asin Fig. 10c, is the result of this dominant production inthe channel core and the outward flux in the hot frontregion.The situation is completely different in the cold frontregion, where the production and the fluxes are stronglysuppressed. Here, the production, the decay, and the3fluxes balance each other in a manner that strongly de- pend on the flow conditions. R. J. Donnelly,
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FIG. 25: The production terms: P (Eq. (11), purple dotted line), P (Eq. (12), green dashed line), P (Eq. (13), browndot-dashed line) and their sum (blue solid line) at different conditions. (a) T = 1 . V n with U c = 3 cm/s, (b) T = 1 .
65 K, parabolic V n with U c = 1 . T = 1 . V n with U c = 1 cm/s, (d) T = 1 . V n profile. Dashed black lines mark the edges of the core region. Thin solid lines are placed at the intervortex distance from thecorresponding walls.
13 14 15 16 17 180123 10 -3 -3 (a) (b) (c)corewalls core corewalls wallsT=1.3 K P T=1.9 K P T=1.3 K F13 14 15 16 17 18-1-0.500.511.5 10 -3 -3
12 14 16 18-1-0.500.511.5 10 -3 T=1.3 K T=1.9 K T=1.3 K FP corewalls corewallsP(d) (e) (f)corewalls
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