Vortex Tangle Dynamics Without Mutual Friction in Superfluid 4He

Abstract

A recent experiment has shown that a tangle of quantized vortices in superfluid 4 He decayed even at mK temperatures where the normal fluid was negligible and no mutual friction worked. Motivated by this experiment, this work studies numerically the dynamics of the vortex tangle without the mutual friction, thus showing that a self-similar cascade process, whereby large vortex loops break up to smaller ones, proceeds in the vortex tangle and is closely related with its free decay. This cascade process which may be covered with the mutual friction at higher temperatures is just the one at zero temperature Feynman proposed long ago. The full Biot-Savart calculation is made for dilute vortices, while the localized induction approximation is used for a dense tangle. The former finds the elementary scenario: the reconnection of the vortices excites vortex waves along them and makes them kinked, which could be suppressed if the mutual friction worked. The kinked parts reconnect with the vortex they belong to, dividing into small loops. The latter simulation under the localized induction approximation shows that such cascade process actually proceeds self-similarly in a dense tangle and continues to make small vortices. Considering that the vortices of the interatomic size no longer keep the picture of vortex, the cascade process leads to the decay of the vortex line density. The presence of the cascade process is supported also by investigating the classification of the reconnection type and the size distribution of vortices. The decay of the vortex line density is consistent with the solution of the Vinen's equation which was originally derived on the basis of the idea of homogeneous turbulence with the cascade process. The cascade process revealed by this work is an intrinsic process in the superfluid system free from the normal fluid. The obtained result is compared with the recent Vinen's theory which discusses the Kelvin wave cascade with sound radiation. 67.40.Vs, 67.40.Bz

Vortex simulations on a 3-sphere

“…Our previous calculations on T 3 give I = 0.76 [18]. Other simulations [7,9,13,19] find similar values for I at α = 0.1, with a decrease towards 2/3 as α decreases. As shown in Figure 9, the vortex orientation within the 3-sphere tangles is dramatically different.…”

On the derivation of the HOMFLYPT polynomial invariant for fluid knots

“…(i) As experiments and direct numerical simulations of superfluid turbulence show (Tsubota, Araki & Nemirovskii 2000;Leadbeater et al 2003;Kondaurova et al 2014), vortex tangles evolve through continuous production and interaction of small(er)-scale unlinked, unknotted vortex loops. Evidence shows that these loops are planar ring-like structures, with almost no writhe and no twist (superfluid vortices are like empty tubes with no internal structure); thus, their internal helicity is either zero or negligible.…”

Quantum turbulence—from superfluid helium to atomic Bose–Einstein condensates

“…Except for the thin core region, the superflow velocity field has a classically well-defined meaning and can be described by ideal fluid dynamics [11,31]. The velocity at point r due to a filament is given by the Biot-Savart expression,…”

“…At finite temperatures, mutual friction occurs between the vortex core and the normal flow. A numerical simulation method on this model has been described in detail elsewhere [11,12,31]. A vortex filament is represented by a single string of points separated by distance ∆ξ.…”