Spherically symmetric formation of localized vortex tangle around a heat source in superfluid 4 He
aa r X i v : . [ c ond - m a t . o t h e r] A p r Spherically symmetric formation of localized vortex tangle around a heat source insuperfluid He Sosuke Inui and Makoto Tsubota Department of Physics, Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan Department of Physics & Nambu Yoichiro Institute of Theoretical and Experimental Physics(NITEP) & The OCU Advanced Research Institute for Natural Science and Technology (OCARINA),Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan (Dated: May 1, 2020)We study the dynamical process of the vortex tangle development under a spherically symmetricthermal counterflow around a heat source submerged into a bulk superfluid He. We reveal a peculiarvortex dynamics that is unique to this geometry, which is greatly diverse from the vortex dynamics ina homogeneous counterflow. Two types of heater are considered here, namely, a spherical heater witha solid wall and a point-like heater. In both cases, a spherical vortex tangle is formed surroundingthe heater. The mechanism of vortex tangle development in the vicinity of a solid wall is stronglygoverned by Donnelly-Glaberson instability; while, far away from the heater or around a pointheater, the mechanism is governed by the dynamics of polarized vortex loops in radial counterflow.The decay process of such localized vortex tangles is also investigated and is compared with that ofhomogeneous vortex tangles.
I. INTRODUCTION
The superfluid He has an extremely high thermal con-ductivity and acts as an excellent coolant in experimentsconducted at extremely low temperatures . The highthermal conductivity reflects the two-fluid nature of thesuperfluid He. At 0 K, the macroscopic quantum effectgoverns the entire fluid and forms a ground state withoutany entropy. The type of flows allowed in such a systemis either a potential flow or a circulatory flow around afilamentary topological defect with a quantized circula-tion, i . e . a quantized vortex. On the other hand, at afinite temperature, the thermal excitations in the systemform a viscous fluid with nonzero entropy s , resulting ina so-called normal fluid .The highly effective thermal transport is achieved bythe normal fluid with velocity v n proportional to the heatflux q from a heat source per unit area, expressed as: q = ρ v n σT, (1)where ρ is the density of the fluid and σ ≡ s/ρV is thespecific entropy that is dependent on the temperature T . We consider a long closed pipe of cross-sectional area A with a heater of heating power W installed near oneend and filled up with superfluid He. The normal-fluidcomponent is driven from the hot end to the cold end,while the superfluid component is driven in the oppositedirection to conserve the total mass of the fluid in thepipe. The relative velocity v ns = | v n − v s | between thetwo fluids is thus given by, v ns = 1 ρ s σT WA , (2)where ρ s is the density of the superfluid component.While the heating power W is smaller than some crit-ical value W c , the relative velocity v ns increases linearlywith W . A series of milestone experiments pioneered by Vinen, in the late 1950s showed that the heat-ing power dependence is significantly modified above thecritical power W c . It was concluded that the modi-fication results from the formation of a quantized vortextangle, which is identified as a turbulent state of the su-perfluid component or quantum turbulence . Sincethe core of a quantized vortex of radius a ∼ ε ∼ ρ s κ / π with quantized cir-culation κ = h/m He ≈ − cm /s, where h is Planck’sconstant and m He is the mass of a He atom. This in-dicates that the energy transfer to the vortices results inthe increase in the total length or/and the length den-sity of a vortex tangle. With a fully developed vortextangle, where the vortex growth rate and the decay rateare balanced, the effective heat transport mediated bythe normal-fluid is disturbed, and its effective thermalconductivity drops significantly.The sudden drop in the thermal conductivity in thevicinity of a hot spot created in a superfluid He is aserious challenge in experiments because it may quenchthe entire system and prevent the fluid from workingas an efficient coolant. It is, therefore, important tounderstand how a vortex tangle evolves near a heaterunder a nonuniform thermal counterflow profile. Fur-thermore, understanding the dynamical process of vor-tex tangle propagation under such a counterflow wouldprovide insights on the “lost energy” in the micro big-bang experiment in Grenoble , as well as the “pecu-liar motions” of micron-sized particles trapped on thesuperfluid He surface . The majority of precedingexperimental and numerical studies mainly ad-dressed the properties of steady vortex tangles in a ther-mal counterflow in a narrow channel . However, the tan-gle formation processes in different thermal counterflowprofiles have rarely been studied until the recent numeri-cal works carried out by Varga and Sergeev et al . . Inthis paper, we present the results of our numerical sim-ulation on the investigation of localized inhomogeneousvortex tangles formed around a spherical heat source im-mersed in the superfluid. The decay process of such lo-calized vortex tangles is also compared with the homoge-neous ones in terms of the phenomenological parameter χ in the Vinen’s equation .We suppose a small number of remnant vortices in thesystem and simulate their time evolution based on thevortex filament model (VFM) under a spherically sym-metric steady thermal counterflow. In the simulations,vortices are allowed to reconnect to the spherical wall ofthe heater, so that a role of a solid boundary in the realexperiments can be investigated. Also, in this numericalstudy, we suppose a steady thermal counterflow, i . e . , afixed normal-fluid flow profile is prescribed, and distur-bances in its profile are neglected. The feedback on thenormal-fluid profile due to the vortices through the mu-tual friction is not discussed as it is beyond the scope ofthis paper.The paper is organized as follows: In section II, webriefly introduce our numerical scheme and the overviewof single vortex dynamics. In Sec. III, we present theresults of VFM simulations and discuss the dynamicalprocesses of the tangle development around a sphericallysymmetric thermal counterflow. The heat source is as-sumed to have a solid wall to which vortices can connect,and we also present the analysis of the structure of thevortex tangle. The heater discussed in Sec. IV is point-like, and we investigate the vortex evolution at some dis-tance apart from the heat source. Further, we discussthe characteristics of the decay process of the localizedvortex tangles. Finally, the summary is presented in Sec.V. II. DYNAMICS OF VORTICESA. Numerical Method: Vortex Filament Model
The motion of a quantized vortex follows the localsuperfluid flow, as described by Helmholtz’s theorems.Taking into account the temperature-dependent mutualfrictions, α and α ′ , the equation of motion of a vortexsegment at s ( ξ ) is d s ( ξ, t )d t = v s + α s ′ ( ξ ) × v ns − α ′ s ′ ( ξ ) × [ s ′ ( ξ ) × v ns ] , (3)with v s = v s, ind + v s, BS and v n = v n, ind . (4)Here, ξ is the arc-length parameterization of the vortexfilaments, and s ′ ( ξ ) is the unit normal vector along the vortex filament at ξ , and v s, ind and v n, ind refer to thevelocity fields thermally excited by the heater. The su-perfluid velocity v s, BS includes velocity contribution fromall the vortices, and is written in the form of Biot-Savartintegral; v s, BS = κ π Z L s ′ ( ξ ) × ( s ( ξ ) − s ( ξ )) | s ( ξ ) − s ( ξ ) | d ξ = v s, loc + v s, non-loc . (5)The divergence of the integral in the right-hand-sideof Eq. (5) as ξ → ξ can be avoided by separat-ing it into two terms, a localized induction velocity v s, loc ≈ β s ′ × s ′′ and a nonlocal contribution v s, non-loc ,where β = ( κ/ π ) ln( R/a ) and s ′′ is the second deriva-tive of s ( ξ ) with respect to ξ . The interaction betweenthe normal fluid and the vortices takes place through v ns = v n − v s in the mutual friction terms in Eq. (3).By discretizing the vortices in line segments of a suitableresolution ∆ ξ , we solve the integro-differential equation.Also, the time-integration is calculated with a suitabletime resolution ∆ t , adopting the fourth-order Runge-Kutta scheme.To simulate a realistic spherical heater onto which vor-tices can be trapped, calculation of vortex dynamics issubjected to a spherical solid boundary condition. Thecondition can be satisfied by finding a boundary inducedvelocity field v s,b such that( v s,BS + v s,b ) · ˆ n = 0 (6)holds at the surface of the sphere of radius r . Here, ˆ n isthe unit normal vector on the surface. Since the velocity v s,b satisfies the system of equations ∇ × v s,b = 0 ∇ · v s,b = 0 , (7)we can solve Eq.(7) in terms of the associated Legendrepolynomials . B. Motion of a Vortex Ring in Spherical ThermalCounterflow
For the sake of clarity in the later discussion, weshall consider a single vortex ring traveling toward andaway from a point heater immersed into a superfluid Hebath at T = 1 . a = 100 µ m, around which weprescribe the spherical thermal counterflow such thatthe relative velocity between normal fluid and super-fluid at some distance r from the center of the heateris v ns ( r ) = ρ s σT W πr ˆ r . This can be obtained by replac-ing the area A in Eq.(2) with 4 πr . We consider Eq.(3)for this case. By ignoring the nonlocal term in Eq.(5),the superfluid velocity v s in Eq.(4) becomes v s ≈ β s ′ × s ′′ − ρ n ρ s v L + R ˆ r , (8) { L { R = 1 / | s ′′ ( ξ ) | s ( ξ ) p = 1 R [ µ m ] L [ µ m] L [ µ m] [cm/s] [cm/s] (a) (b) (c) FIG. 1. (a) Schematics of a spherical heater and a vortexring of radius R separated by a distance L . In the absence ofthe heater, the vortex travels in the direction of s ′ × s ′′ . Theheater tends to “pull” the vortex toward it. (b) and (c) Thevector field representation of Eq.(9) at T = 1 . p = 0and p = 1, respectively (color online). The color indicates themagnitude of the flow p ˙ L + ˙ R at each location. where v ≡ W/ πρσT , and L is the distance between thecenters of the heater and the vortex ring. SubstitutingEq.(8) into Eq.(3), one obtains a system of differentialequations for the distance L and the vortex radius R ˙ L ≈ ( − p βR − ρ n ρ v L + R cos θ − ( − p αv L + R sin θ (9a)˙ R ≈ − ρ n ρ v L + R sin θ − α (cid:20) βR − ( − p v L + R cos θ (cid:21) , (9b)where θ = tan − ( R/L ) and the terms with α ′ are ne-glected. In Eq.(9), p is a parameter indicating the direc-tion of the ring propagation; p = 0 if the orientation ofthe vortex, i . e . s ′ × s ′′ , is radially outward, and p = 1 ifthat is radially inward. Figures 1(b) and (c) show Eq.(9)as vector fields ( ˙ L ( L, R ) , ˙ R ( L, R )) with p = 0 and p = 1,respectively, at T = 1 . L , R ) in the coordinate. In the case with p = 1, avortex of any radius R shrinks itself to vanish at somedistance L away or “collides with” the heater at L = 100 µ m with time. On the other hand, when p = 0, a vortexcould grow in size depending on its initial values, R and L .Though in a highly developed vortex tangle, this sim-ple single-loop analysis may not hold without modifica-tion, we can learn from this analysis that the orientationof each vortex ring that constitutes the tangle tends to orient itself in the outward direction. In preceding stud-ies with a simple channel flow geometry, the homogene-ity or/and isotropy of the vortex orientations are oftenassumed, which allows us to derive the Vinen’s equa-tion which predicts the time-evolution of a vortex linedensity . However, in the present study there is no guar-antee that the same equation holds because the vortexorientations are polarized. III. VORTEX TANGLE NEAR A SPHERICALHEATERA. Development of a Vortex Tangle
A sphere of a radius r = 0 . He. The length of each vortex segment∆ ξ is set to be within the range 1 ∼ × − mm. Now,the sphere is assumed to be heated homogeneously andsets up a steady thermal counterflow profile of the form v ns = v ns, r r ˆ r , (10)where v ns, is the magnitude of the relative counterflowvelocity at the surface r = r . In reality there shouldexist a “thermal boundary layer” with finite thicknesswhere the two fluids are accelerated and the profile inEq.(10) is not valid in the vicinity of the heater, as it isreported in Refs. . However, we ignore such effect inthis study, assuming the width of the layer is much lessthan the particle radius r .Figure 2 shows the time evolution of six seed vorticessymmetrically placed near a heater. In the simulation,we set the temperature of the system to be T = 1 .
3K and the counterflow velocity to be v ns, = 100 cm/s.Out of the six vortices, four are initially reconnected ontothe spherical surface. An end of the vortex line recon-nected to the heater surface is bent in order to meet theboundary condition of Eq.(6), i . e . , the vortex segment atthe surface needs to be perpendicular to the surface sothat the vertical velocity component vanishes where theymeet on the surface. As we discussed in Sec.II B, undera radial counterflow, the orientation of vortices are se-lected in a way that they tend to induce velocity awayfrom the heater, which seems to explain a spiral-shapestructure formed near the vortex-surface intersection (seeFig.3). The rapid growth of the tangle due to the spiral-shaped structure is also understood as the manifestationof the Donnelly-Glaberson (DG) instability . In thevicinity of the heater surface, the magnitude of the coun-terflow is strong, which makes the vortex lines unstableand excites Kelvin waves along them. Again, the Kelvinwaves with outward orientation are likely to be amplified;while the others tend to vanish because of the converg-ing counterflow. This growth mechanism driven by DGinstability would not hold for vortices that traveled farenough from the heater because the counterflow magni- (a) (d)(c)(b) FIG. 2. (a)-(d) Snapshots of development of a vortex tangle (blue filaments) around a heat source of radius 0 . t = 0 .
0, 2 .
0, 4 .
0, and 6 . (a) (d) (h)(g)(f) (c)(b)(e) FIG. 3. (a)-(h) Magnified snapshots of the development ofthe same vortex tangle. They are taken every after ∆ t = 0 . t = 4 . tude drops as 1 /r . We discuss the vortex tangle growthmechanism out of the DG instability regime in Sec. IV. B. Radial Vortex Line Density
The vortex line density of the tangle as a function ofthe radius is plotted in Fig.4. A single curve in the plotshows the radial vortex line density (RVLD) at time t in the entire tangle, and the change in color (blue → red) indicates the lapse of time (early → late). Althoughthe maximum RVLD has an upper bound due to thenumerical limitation ( L max ∼ ∆ ξ − ), the tangle seemsto grow unboundedly, as long as some vortices are keptreconnected onto the heater surface at r = r (= 0 . v s, ind . Atthe same time, a free vortex ring travels approximately r [ mm ] t [ m s ] L [ mm − ] FIG. 4. Radial vortex line density (RVLD) L (color online).A single curve plots RVLD of the tangle at some time t . r is the radial distance from the center of the heater of radius r = 0 . with a localized induction velocity v s, loc in Eq.(5), whichis inversely proportional to the local curvature radius.Therefore, a smaller radius of a vortex ring results ina stronger resistance to the counter flow that pulls ittoward the heater. In the vicinity of the surface, only afew vortices can survive in the strong counterflow, whileothers are ‘sucked’ into the heater. On the other hand,the counterflow becomes weaker as 1 /r , and more andmore vortices can overcome the pull as they go fartheraway from the heater. IV. VORTEX TANGLE AROUND A POINTHEATER
In Sec. III we have considered the case where the ini-tial seed vortices are placed relatively close to the heater,and some of them are reconnected onto it. In this sec-tion, however, we treat the heater as a point in order to (a) (d) (c)(b)(f) ( !(cid:0) ( (cid:1)(cid:2) ( (cid:3)(cid:4) d [cm] d [cm] d [cm] d [cm] R [ c m ] (e) (j) d [cm] D [ a r b . un i t ] d ≈ . FIG. 5. (a) - (e) Snapshots of vortices that grow to be a vortex tangle around the point heater. The parameters, v ns and T areset to be 50 cm/s and 1 . R (vertical axis) at some distance d (horizontal axis) with someorientation indicated by the loop polarization D (color from red to blue). See Eq.(11) and the text for the definition of D . investigate the tangle development process far away fromit, assuming no vortices are trapped on the heater. A. Development of a Tangle
Supposing a heater placed at the origin is point-like, weignore the boundary condition of Eq.(6), but the radiusof the heater is set to be r = 0 . ξ ∼ . × − mm for this simulation. Byplacing several seed vortices we investigated their devel-opment into a vortex tangle, and its dependence on thecounterflow magnitude v ns and the temperature T .A typical time-evolution process is summarized inFig.5. The panels (a)–(e) in Fig.5 show the snapshotsof the simulation. A single dot in the panels (f)–(j) ofFig.5 shows a vortex loop in the tangle corresponding tothe moments when the snapshots (a)–(e) are taken. Thehorizontal and vertical axes in the panels (f)–(j) repre-sent the radial distance d of a vortex loop, defined as theaverage distance between the origin and each vortex seg-ment within a single loop, and the size R of the loop isdefined as the average distance between each vortex seg-ment and the center of the loop. In terms of the lengthsdefined in Fig.1(a), the radius distance d is expressed as √ L + R . The color of the dot in the panels indicatesthe loop polarization D , which is defined as D = ˆ r · I a loop s ′ ( ξ ) × s ′′ ( ξ )d ξ/ N , (11)where ˆ r is the unit radial vector at the center of theloop, and N is a properly chosen numerical factor thatnormalizes the integral. The vortex loops with D ≈ D ≈ − d = R are drawn for reference in Fig.5(f)–(j). Apparently, themaximum loop size R max appears in the vicinity of theslopes at each panel, which indicates that the value of R max can be used as a good measure of the radius of thetangle.The existence of the large vortex loops of size compa-rable to the tangle size with polarization D ≈ v s, ind induced by the point heater.The relative counterflow profile v ns at the tangle front R f is given by Eq.(10), equating r = R f . Assuming that theconservation of mass is valid locally, the velocity v s, ind can be obtained from the following relation: ρ s v s, ind ( R f ) + ρ n v n, ind ( R f ) = 0 , (12)where v n, ind − v s, ind = v ns . Since a vortex ring of radius R travels with velocity v s ∼ β/R , the only vortex loopswith radius R c < β/v s, ind ( R f ) can travel against thecounterflow and gradually expand the “edge” of the tan-gle, while the other vortices are presumably well-confinedwithin the region of radius R f . B. Total Vortex Line Length (TVLL)and Decay Process
When it comes to the study of the homogeneous vortextangles, the vortex line density (VLD) L is the quantitythat is uniquely determined experimentally and compu-tationally and is frequently used to analyze the natureof the tangles. However, in the case of the localizedinhomogeneous vortex tangles, there remains some ar-bitrariness in the value of L depending on the schemeto estimate the volume V occupied by the vortex lines inthe tangle. The maximum value of spatially-independentVLD L may be achieved if the volume V is estimated bythe box-counting method, i . e . , dicing up the computa-tional space into cubes with suitable lengths, we count l [ c m ] t [s] t [ s ] T [K] v n s , [ c m / s ] (a) TVLL as function of time (b) TVLL peaks FIG. 6. (a) Total vortex line length (TVLL) l as functionof time t for T = 1 . v ns, = 30 cm/s. The solidcircle represents the maximum length l max . (b) TVLL peaks l max as function of T and v ns, (color online). The radii andthe depth of the circle’s face-color represent the maximumlength l max and the time it takes to reach the length at eachpoint in the parameter space ( T, v ns, ), respectively. How-ever, at ( T, v ns, ) = (2 . ,
20) and (2 . , l ( t ) tends to growunboundedly with time, so the radii do not reflect the actualvortex line length. the number of the cubes that contain the (segments of )vortices and calculate the total volume V BC . If we takethe lengths of the cubes to be the average vortex loop size,then L = l/V BC , where l is the total vortex line length(TVLL), gives the VLD averaged over the volume. Onthe other hand, since most of the vortices in the tangleare localized in a sphere of volume V loc = 4 πR /
3, theminimum value of L is achieved when we let the volume V be a constant such that V > V loc for all time. Thenthe TVLL l in this case is essentially the same as theVLD L , which is the quantity we shall mainly discuss inthis section.Figure 6(a) shows the profile of the TVLL l as a func-tion of time. Initially, l ( t ) grows rapidly within the regionof radius R f . As we have discussed in Sec. IV A, in theinitial growing stage, there are number of large vortexloops of the order of the tangle size. Since a large vortexloop cannot travel against the thermal counterflow, thosewith large radii or small local curvatures are essentiallyenclosed in the spherical region, inside of which vorticesgrow in length via mutual friction and in number becauseof the repeated reconnections. However, the growth ratedecreases, and the total length l ( t ) finds its maximumvalue l max . In the case of T = 1 . v ns, = 30cm/s, the tangle grows up to l max ≈
15 cm at time t ≈ l max on T and v ns, ,is summarized in Fig.6(b). The radii of the circles areproportional to the values of l max at ( T, v ns, ), and thethickness of the circle color indicates the time it takes toreach l max . Below T ≈ . l ( t ) hasa plateau for several seconds after a swift growth, dur-ing which the maximum value l max is attained. Above T > ∼ . α ′ becomes negative. If T > ∼ . v c , between v ns, = 20 cm/s and 30 cm/s, thatdetermines whether a tangle can develop or not. Abovethe critical velocity v c , the tangle may not be formed;while, below v c , the tangle tends to grow unboundedly.Aside from the “gradual decay” while the heater ison, we have also investigated the “free decay” turningoff the heater suddenly during the tangle developmentprocess. Figure 7(a) plots 1 /l as a function of time t .The broken curve represents the“gradual decay” wherethe heater is kept on. At time t free indicated by the solidcircles on the 1 /l plot, the point heater is turned off,and the time evolutions of the tangle for time t > t free are simulated. The solid branches departing from thebroken curve represent the results of the “free decay”simulations. After turning off the heater, the TVLL l tends to decay inversely proportional to time for severalseconds as it is indicated by the straight lines in Fig.7(a). What this indicates is that the form of the decayterm coincides with that of Vinen’s equation (VE) (cid:18) d L d t (cid:19) decay = − χ κ π L (13)that describes the time evolution of a homogeneous VLD L , where χ is a phenomenological parameter of orderunity. Substituting L = l/V BC into Eq.(13), we obtaind l d t − lV BC d V BC d t = − χ V BC κ π l . (14)Since the computational result shows that the secondterm in the left hand side of Eq.(14) is smaller than thefirst term by over an order of magnitude, we ignore thesecond term. Then, Eq.(14) has a simple solution of theform 1 l = 1 l + χ V BC κ π t, (15)where l is the initial TVLL at t = t free . Since L = l/V BC ,the value of χ can be found as a slope by rescaling thevertical axis 1 /l to be (2 π/κ )(1 / L ). Rescaling all the“gradual decay” and “free decay” profiles in Fig. 7(a), itturns out that all curves tends to collapse onto a singleline of the slope χ ≈ .
79, as we can see in Fig. 7(b).Considering the fact that the experimental value of χ in a homogeneous vortex tangle is of order unity, the t [s] / l [ c m − ] t [s] π κ L [ a r b . un i t ] χ ≈ . (a) (b) FIG. 7. (a) Comparison between “gradual decay” and “freedecay” (color online ). The system is kept at T = 1 . v ns, = 30cm/s. (b) Normalized 1 /l plots for both “gradual decay” and“free decay”. The TVLL l is normalized by the volume V BC . value we obtained here with the localized vortex tanglemay not be unreasonable, although the value of χ seemsto be sensitive to the volume V occupied by the vortexloops. Here, we have applied the box-counting method toestimate the volume V BC . However, the way of estimat-ing the occupied volume is not unique, and there remainssome uncertainty in the determination of the value of χ in the localized vortex tangles. Also, the approximationwhich allows us to obtain Eq.(15) only holds for suffi-ciently short time after turning off the heater, since theaverage distance among the vortices becomes larger thanthe average vortex loop size, which leads to the underesti-mation of the volume occupied by the vortices. The con-sequence of this effect can be observed in Fig. 7(b), i . e . ,the VLD L is overestimated, and the solid curves tendto drop below the broken curve whose slope is roughly χ ≈ .
79 after following it for at least a few second.
V. SUMMARY AND DISCUSSION
We performed simulations based on VFM in order toinvestigate the evolution of a vortex tangle around aspherically symmetric thermal counterflow in superfluid He. For the spherically symmetric heat source, we con-sidered two types of heaters. One is a sphere with solidboundary on which vortices can connect, and the otheris a point-like heater. In both the cases, only the steadythermal counterflow profile is prescribed and any modu-lations in the normal-fluid profile are not considered inthis work.Our simulations with a spherical heater reveal thata spherical vortex tangle is developed around it. Un-der a strong radial counterflow, the orientation of vortexloops tends to be polarized and they increase in size.When some vortices are connected to the solid wall ofthe heater, the tangle seems to grow unboundedly, form-ing a spiral-shape structure on vortex filaments. Thespiral structure may be identified as Kelvin waves thatare excited on the filaments due to the thermal coun-terflow (DG instability). The unbounded growth in vor-tex line would eventually lead our simulation to violateour assumption, namely, the disturbance in the normalfluid profile is no longer negligible. In the vicinity of theheater, our current numerical scheme is, thus, only validfor simulating the early stages of the vortex tangle de-velopment, where radial vortex line density (RVLD) is
L < ∼ cm − . If the normal fluid is greatly blockedby a thick wall of vortex tangle of L > ∼ cm − , then anon-negligible temperature gradient may be accumulatedwithin the tangle, as pointed out in the recent work bySergeev et al . .The simulations with a point-like heater resolve prob-lem of the unbounded growth. Since the radial counter-flow drops as 1 /r , the vortex creation via DG instabil-ity becomes less dominant as vortices get farther awayfrom the heat source. Again, a spherical vortex tangle isformed around the point heater, but in this case, a hollowregion is left inside the tangle, as the tangle front prop-agates radially outward. The tangle development seemsto have two phases; the production phase and the decayphase. During the production phase, vortices are moreor less confined in a small region of radius R f . Thus,the vortex loops in the tangle grow swiftly not only inlengths via mutual friction, but also in number becausethey repeatedly reconnect with each other within the re-gion. However, the growth is balanced by the decay dueto the mutual friction, and the TVLL finds its maximumvalue. During the decay period the number of small vor-tices in the tangle increases, and more vortices are able to“escape” the region of radius R f , overcoming the counter-flow. Since the extra dissipation of vortex loop reducesthe number of reconnections, the tangle starts to decay,which is what we call the “gradual decay”. If we turn off the point heater during the tangle development process,the tangle decays freely. We call such a decay processthe “free decay” in contrast to the the “gradual decay.”From the computational results we are able to extractthe information of the decay constant χ , although thevalue has an uncertainty since the volume that containsthe vortices can be chosen somewhat arbitrarily.In the present study, the normal-fluid profile is keptsteady, however, such an assumption may not hold inthe thermal boundary layer in the vicinity of the heatersurface, or in a highly dense vortex tangle as we havediscussed. We would like to address these issues in thefuture work by coupling the normal-fluid dynamics toVFM.M. T. acknowledges the support from JSPS KAKENHI(Grant No. JP17K05548, JP20H01855). S. V. Sciver,
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