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Decay of vortex tangle in superfluid 4He at very low temperatures
Physica B 284}288 (2000) 79}80
Decay of vortex tangle in super#uid He at very low temperatures M. Tsubota *, T. Araki , S.K. Nemirovskii Faculty of Science, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan Institute of Thermophysics, Academy of Science, Novosibirsk 630090, Russia
Abstract Recently, the Lancaster group observed the free decay of vortex tangle in super#uid He at mK temperatures. Since the system at such low temperatures is free from the normal #uid and the usual mutual friction, the mechanism of the free decay is unknown. In order to understand these phenomena, this work studies numerically the vortex dynamics without the mutual friction. The absence of mutual friction prevents the vortex from smoothing. The resulting kinked structure promotes vortex reconnection, thus making lots of vortex loops with small size. Such cascade process as breakup of vortices to smaller ones and the nonlocal interaction can decay the length of vortices. 2000 Elsevier Science B.V. All rights reserved. Keywords: Cascade process; Vortex dynamics; Vortex tangle
1. Introduction In connection with the experiments by McClintock et al. [1], we study numerically the vortex dynamics in the absence of mutual friction. Our numerical method is very similar to that of Schwarz [2,3] and is described in our previous paper [4]. A quantized vortex is represented by a vortex "lament that moves subject to the super#ow velocity "eld given by the Biot}Savart expression. The velocity of the "lament comes from the local-induced "eld due to a curved line element acting on itself and the nonlocal "eld made by the other line elements. When two vortices approach within a critical distance, they are assumed to reconnect [2,3].
atures [2,3]. Fig. 1 (a) shows VT made under an applied #ow and the mutual friction. Switching o! the applied #ow and the mutual friction changes this VT to that of Fig. 1(b) in a short time. The absence of mutual friction makes VT very kinked, so that lots of small vortex loops appear because of reconnection. Since such cascade process as breakup of vortices to smaller ones is assumed to continue down to microscopic scale, we eliminate the smallest loops which meet the reconnection condition. The line}length density (LLD) of this VT decays because of this cascade process. The decay rate is little a!ected by the space resolution.
3. Full nonlocal calculation 2. LIA calculation First, we made a calculation with the localized induction approximation (LIA) which has been successful in the dynamics of the vortex tangle (VT) at "nite temper* Corresponding author. E-mail address: [email protected] (M. Tsubota)
LIA restricts the vortex to a purely binormal motion which conserves the length, thus dropping the nonuniform stretching action due to the nonlocal terms. Since the full nonlocal calculation for VT requires much computing time, we run that calculation for a more dilute con"guration (Figs. 2 and 3). The "rst four vortex rings of the same radius 8.75;10\ cm are placed symmetrically inside a cube of the side 5;10\ cm. The periodic boundary conditions are adopted for all three directions.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 0 4 4 - X
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M. Tsubota et al. / Physica B 284}288 (2000) 79}80
Fig. 1. Vortex tangle at (a) ¹"1.6 and (b) 0 K. Fig. 3. Decay of LLD for the vortices of Fig. 2.
vortices collide sometimes. LLD ¸ depends on the competition between the stretch due to the nonlocal term and the disappearance of the smallest loops. First ¸ increases just a little because of the just superior stretch. After the vortices are divided to many small ones, however, the cascade process decreases ¸. The similar nonlocal calculation was made for the #uid con"ned in the solid boundary; LLD decays more rapidly. We also made the LIA calculation for the same initial condition. The decay rate of LLD is lower than that of the nonlocal calculation. The nonlocal interaction makes more reconnections, thus promoting the cascade process. Another important mechanism for the tangle decay is the dissipation by radiation of sound [5]. This e!ect will be taken into account in the future.
Fig. 2. Decay of four vortices, (a) 0, (b) 0.1, (c) 0.5, and (d) 3 s.
Four rings move toward the center of the cubic; the four vortices resulting after reconnection moves outside oppositely. While propagating, the vortices become kinked and then small kinked parts are cut o! by reconnection. The periodic boundary condition makes the
References [1] P.V.E. McClintock et al., in these Proceedings (LT-22), Physica B (2000). [2] K.W. Schwarz, Phys. Rev. B 31 (1985) 5782. [3] K.W. Schwarz, Phys. Rev. B 38 (1988) 2398. [4] M. Tsubota, S. Maekawa, Phys. Rev. B 47 (1993) 12 040. [5] D.C. Samuels, C.F. Barenghi, Phys. Rev. Lett. 81 (1998) 4381.
Decay of vortex tangle in super#uid He at very low temperatures M. Tsubota *, T. Araki , S.K. Nemirovskii Faculty of Science, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan Institute of Thermophysics, Academy of Science, Novosibirsk 630090, Russia
Abstract Recently, the Lancaster group observed the free decay of vortex tangle in super#uid He at mK temperatures. Since the system at such low temperatures is free from the normal #uid and the usual mutual friction, the mechanism of the free decay is unknown. In order to understand these phenomena, this work studies numerically the vortex dynamics without the mutual friction. The absence of mutual friction prevents the vortex from smoothing. The resulting kinked structure promotes vortex reconnection, thus making lots of vortex loops with small size. Such cascade process as breakup of vortices to smaller ones and the nonlocal interaction can decay the length of vortices. 2000 Elsevier Science B.V. All rights reserved. Keywords: Cascade process; Vortex dynamics; Vortex tangle
1. Introduction In connection with the experiments by McClintock et al. [1], we study numerically the vortex dynamics in the absence of mutual friction. Our numerical method is very similar to that of Schwarz [2,3] and is described in our previous paper [4]. A quantized vortex is represented by a vortex "lament that moves subject to the super#ow velocity "eld given by the Biot}Savart expression. The velocity of the "lament comes from the local-induced "eld due to a curved line element acting on itself and the nonlocal "eld made by the other line elements. When two vortices approach within a critical distance, they are assumed to reconnect [2,3].
atures [2,3]. Fig. 1 (a) shows VT made under an applied #ow and the mutual friction. Switching o! the applied #ow and the mutual friction changes this VT to that of Fig. 1(b) in a short time. The absence of mutual friction makes VT very kinked, so that lots of small vortex loops appear because of reconnection. Since such cascade process as breakup of vortices to smaller ones is assumed to continue down to microscopic scale, we eliminate the smallest loops which meet the reconnection condition. The line}length density (LLD) of this VT decays because of this cascade process. The decay rate is little a!ected by the space resolution.
3. Full nonlocal calculation 2. LIA calculation First, we made a calculation with the localized induction approximation (LIA) which has been successful in the dynamics of the vortex tangle (VT) at "nite temper* Corresponding author. E-mail address: [email protected] (M. Tsubota)
LIA restricts the vortex to a purely binormal motion which conserves the length, thus dropping the nonuniform stretching action due to the nonlocal terms. Since the full nonlocal calculation for VT requires much computing time, we run that calculation for a more dilute con"guration (Figs. 2 and 3). The "rst four vortex rings of the same radius 8.75;10\ cm are placed symmetrically inside a cube of the side 5;10\ cm. The periodic boundary conditions are adopted for all three directions.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 0 4 4 - X
80
M. Tsubota et al. / Physica B 284}288 (2000) 79}80
Fig. 1. Vortex tangle at (a) ¹"1.6 and (b) 0 K. Fig. 3. Decay of LLD for the vortices of Fig. 2.
vortices collide sometimes. LLD ¸ depends on the competition between the stretch due to the nonlocal term and the disappearance of the smallest loops. First ¸ increases just a little because of the just superior stretch. After the vortices are divided to many small ones, however, the cascade process decreases ¸. The similar nonlocal calculation was made for the #uid con"ned in the solid boundary; LLD decays more rapidly. We also made the LIA calculation for the same initial condition. The decay rate of LLD is lower than that of the nonlocal calculation. The nonlocal interaction makes more reconnections, thus promoting the cascade process. Another important mechanism for the tangle decay is the dissipation by radiation of sound [5]. This e!ect will be taken into account in the future.
Fig. 2. Decay of four vortices, (a) 0, (b) 0.1, (c) 0.5, and (d) 3 s.
Four rings move toward the center of the cubic; the four vortices resulting after reconnection moves outside oppositely. While propagating, the vortices become kinked and then small kinked parts are cut o! by reconnection. The periodic boundary condition makes the
References [1] P.V.E. McClintock et al., in these Proceedings (LT-22), Physica B (2000). [2] K.W. Schwarz, Phys. Rev. B 31 (1985) 5782. [3] K.W. Schwarz, Phys. Rev. B 38 (1988) 2398. [4] M. Tsubota, S. Maekawa, Phys. Rev. B 47 (1993) 12 040. [5] D.C. Samuels, C.F. Barenghi, Phys. Rev. Lett. 81 (1998) 4381.