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Using the vortex filament model with the full Biot-Savart law, we show that non-straight bundles of quantized vortex lines in HeII are structurally robust and can reconnect with each other maintaining their identity. We discuss vortex stretching in superfluid turbulence in many cases. We show that, during the bundle reconnection process, Kelvin waves of large amplitude are generated, in agreement with previous work and with the finding that helicity is produced by nearly singular vortex interactions in classical Euler flows. The reconnection events lead to changes in velocities, radius, number of points and total length. The existence of reconnections was confirmed by other authors using the model of nonlinear Schr?dinger equation (NLSE). Our results are agreed with the finding of other authors and extension to our numerical experiments.

It has been realised that the disordered motion of a tangle of quantized vortices play an important part in the behavior of superfluids. The flow is generally turbulent for high-velocity or high Reynolds number flows. Although turbulence has been studied intensely in many fields, it is still not yet well understood because it is a complicated dynamical phenomenon with strong nonlinearity [^{4}He exists in two phases: Helium I and Helium II [_{λ} = 2.1768 K [^{4}He is described by the two-fluid model, where the system consists of a viscous normal fluid has density and an inviscid superfluid has density with two independent velocity v_{n} and. The superfluid velocity field is given by, where is some constant determined by assuming that the action of a single helium atom in the fluid to be quantized in units ofPlanck’s constant, is the distance from the vortex axis and is the unit vector in the tangential direction [4,5]. The circulation of the vortices is quantized as

where is a path around the axis of the vortex and the constant is called the quantum of circulation

and is the mass of a ^{4}He atom. The constant is given by

From Equation (1), it follows that the superfluid velocity field around the vortex is

A quantived vortex is different from a vortex in a classical fluid where the circulation is quantized, which is contrary to a classical vortex that can have any value of circulation [

The dynamics of quantized vortices can be described by the Gross-Pitaevski (GP) model [

where is the order parameter filed, is an external potential, is a coupling constant, and is the mass of each particle. This description is most applicable in the limit of, where the normal fluid is absent. This model can also be used to explain phenomena related to vortex cores, such as nucleation and reconnection. However, the GP model is applicable to Bose-Einstein condensation of a dilute atomic Bose gas. It is, however, not applicable quantitatively to superfluid ^{4}He, which is not a weakly interacting Bose system. For more details, we refer to Refs. [8-10].

The second formulation for studying the dynamics of quantized vortices is called the vortex filament model. The aim of this work is to use this model under the full Biot-Savart law to make numrical simulations to examine the dynamics of inviscid vortex filaments, specifically what happens at vortex including reconnection events. This model was pioneered by Schwarz [11,12], where the vortex lines are numerically discretized by a large, variable number of points depends on the local radius of curvature. We will briefly describe this model in the next section.

As we mentioned in the Introduction, the quantized vortex has quantized circulation and the vortex core is very thin. These properties allow a quantized vortex to be considered as a vortex filament. In the vortex filament model a quantized vortex is represented as a filament passing through the fluid. In the case of superfluid ^{4}He, the vortex core radius is many orders of magnitude smaller than the average separation between vortices (typically, to cm in turbulence experiments), or any other scale of interest in the flow; it is therefore expedient to consider a superfluid vortex filament as a space curve of infinitesimal thickness in three dimensional space, where is the arc length and t is the time. The vortex lines are numerically discretized by a large variable number of points , which are called vortex points [13,14]. The governing equation of motion of the superfluid vortex lines is

where is the unit tangent, is the normal fluid velocity, and are temperature-dependent friction coefficients, and the total velocity of the vortex filament without dissipation is given by

where is the background superfluid velocity field, is a cutoff parameter corresponding to radius of the vortex filament, and are the lengths of the two adjacent line elements connected to the point and represents integration along the vortex line outside the region specified by and [5,15]. In the absence of normal fluid, vortex lines move with the local superfluid velocity where the friction coefficients, and, will be equal to zero. And so, the equation of vortex motion will be

In our numerical experiments we use the model of Schwarz which is explained in detail in Ref. [

So, for a vortex configuration modeled by vortex segments, we must solve a system of coupled first order differential equations.

Our simulations are performed in a cubic box of volume, where (typically, we chose or) with periodic boundary conditions. In this work, we use the vortex filament model in which vortex reconnections are performed by the numerical algorithm (rather than occurring as natural solution of the governing equation, as in the model of nonlinear Schrödinger equation (NLSE)) [18,19]. In our method, we firstly select these points which are candidates for a reconnection. This selection is based on the distance between those points, where the reconnection between two lines can takes place only if they have different directions. For more details, we refer the reader to Refs. [20-22].

By using the vortex filament method, we study the interaction of two non-straight vortex bundles (with sinusoidally) each one of them contains a given number M of (initially) non-straight parallel vortex strands, set (initially) at (vortex with antivortex bundles).

If two vortex strands become closer to each other than the local discretization along filaments, then, consistently with the orientation of the filaments, our numerical code reconnects the strands, provided that the total length is decreased [

In the case of bundles, the initial position of vortex strands within the same bundles is symmetric, and we found that the reconnection events are still possible in different values of M. When M = 3, we place three vortices at the corner of a equilateral triangle. The interaction between the bundles makes them to bend in the direction of each other, until, the first reconnection takes place. With the evolution, the second reconnection is found at, then they become free from each other at, and move away, see

A typical result for two bundles of five non-straight parallel vortex with sinusoidally strands each, where one of them against the other (vortex bundle with antivortex bundle), the radius of each bundle A = 0.0155804 cm, the distance between the closest point between the two non-straight bundles is shown in the upper-left panel of

than their size. Remarkably, vortex bundles survive reconnections with other bundles without disintegrating, but rather amplifying their vortex length, where the total length L increases by about as appear in the upperleft panel of

nection, see

These calculations are performed in a cubic periodic box, where.

This work deals with the numerical simulation of the motion of inviscid vortex filaments under full BiotSavart law. The calculations presented here were made when the periodic boundary conditions are applied in all axes. The very known fourth order Runge-Kutta method was used for the time evolution. The reconnection and vortex stretching in superfluid turbulence is concerned. The existence of reconnections was proved by Koplik and Levine [

The authors are grateful to Prof. Carlo F. Barenghi and Prof. Eed M. Darwish for useful discussions and a critical reading of the manuscript. This work is supported by the Deanship of Scientific Research at Taibah University (Project No. 1433/1790).