The energy spectrum of the superfluid turbulence without the normal fluid is studied numerically under the vortex filament model. Time evolution of the Taylor-Green vortex is calculated under the full nonlocal Biot-Savart law. It is shown that for k < 2π/l, the energy spectrum is very similar to the Kolmogorov's -5/3 law which is the most important statistical property of the conventional turbulence, where k is the wave number of the Fourier component of the velocity field and l the average intervortex spacing. The vortex length distribution becomes to obey a scaling property reflecting the self-similarity of the tangle. 67.40.Vs, 67.40.Bz Particular attention has been focused recently on the similarity between superfluid turbulence and conventional turbulence [1][2][3]. Early work of the superfluid turbulence has been concerned with the counterflow where the normal fluid and superfluid flow oppositely [4], having no classical analog. However Stalp et al. studied recently the superfluid turbulence produced by the towed grid, thus finding the similarity between the superfluid turbulence and the conventional turbulence above 1.4 K [2]. They observed indirectly the Kolmogorov law which is one of the most important statistical properties of the conventional turbulence. This is understood by the idea that the superfluid and the normal fluid are likely to be coupled together by the mutual friction between them, and to behave like a conventional fluid [1,5]. Since the normal fluid is negligible at mK temperatures, an important question arises: even free from the normal fluid, is the superfluid turbulence still similar to the conventional turbulence or not?As the physical model to describe the vortex dynamics in superfluid He at very low temperatures, two types of models are well-known: the Gross-Pitaevskii (GP) equation which describes the motion of a weakly interacting Bose condensate, and the vortex filament model governed by the incompressible Euler dynamics. The former reduces to the Euler vortex filament model when variations of the wave function over scales of the order of the superfluid healing length are neglected. The GP equation includes such complicated compressible effects as the radiation of sound from the vortex lines [6,7], the vortexsound interactions, etc. In order to consider the intrinsic property of superfluid turbulence in a simpler situation, we study the energy spectrum of the 3D velocity field induced by the vortex tangle in the absence of the normal fluid under the vortex filament model.The energy spectrum of the vortices in superfluid was numerically calculated by other authors. Nore et al. studied the energy spectrum of the decaying superfluid turbulence by using the GP equation, and finding the transient spectrum for small k has the Kolmogorov law [3]. However at late stage some complicated compressible effects become dominant. On the other hand, an advantage of the vortex filament model compared with the GP equation is the followings. First this model enables us to calculate the energy s...

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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

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The theory of inhomogeneous superfluid turbulence is developed on the basis of kinetics of merging and splitting vortex loops. Vortex loops composing the vortex tangle can move as a whole with some drift velocity depending on their structure and length. The flux of length, energy, momentum, etc., executed by the moving vortex loops takes place. The situation here is exactly the same as in usual classical kinetic theory, with the difference being that the "carriers" of various physical quantities are not the point particles but extended objects ͑vortex loops͒, which possess an infinite number of degrees of freedom, with highly involved dynamics. We suggest to complete our investigation, based on the supposition that vortex loops have a Brownian structure, with the only degree of freedom being, lengths of loops l. This concept allows us to study the dynamics of the vortex tangle on the basis of the kinetic equation for the distribution function n͑l , t͒-the density of a loop in the space of their lengths. Imposing the coordinate dependence on the distribution function n͑l , r , t͒ and modifying the "kinetic" equation with regard to an inhomogeneous situation, we are able to investigate various problems on the transport processes in superfluid turbulence. In this paper, we evaluate the flux of the vortex line density L͑x , t͒ due to the gradient of this quantity. The corresponding evolution of quantity L͑x , t͒ obeys the diffusion type equation, as it can be expected from dimensional analysis. The diffusion coefficient is arrived at from calculation of the ͑size-dependent͒ free path and drift velocity of the vortex loops, and takes the value 2.2, which exceeds approximately 20-fold the value obtained in early numerical simulation. We discuss the probable reason for this large discrepancy. We use the diffusion equation to describe the decay of the vortex tangle at a very low temperature. Comparison with recent experiments on decay of the superfluid turbulence is presented.

A description of the chaotic vortex tangle in superfluid turbulent He II is developed. Unlike current phenomenological theory dealing with only the macroscopic variable, the vortex line density L v (t) and describing thereby only the macroscopic hydrodynamic phenomena, our approach allows us to describe effects due to the arrangement of the vortex tangle and the interaction of lines. To develop this approach we introduce a trial distribution function in the space of vortex loop configurations which absorbs all properties of superfluid turbulence known both from experiment and from numerical simulations. This trial distribution function is built in terms of the path integral. A number of allowed configurations is obtained evaluating the path integral with constraints connected with the established properties of the vortex tangle. Using the trial distribution function we also build the characteristic ͑generating͒ functional which allows us to evaluate any average over the vortex loop configuration. On the basis of the developed approach we briefly discuss some simple statistical characteristics of the vortex tangle. A more extended example of the developed approach studying superfluid mass current induced by vortex tangle is reported in a subsequent paper.

We submit the results of the numerical experiment on the decay of the quantum turbulence in the absence of the normal component of the superfluid helium. Computations were fulfilled inside a fixed domain with the use of the vortex filament method. The purpose of this study was to ascertain the role of the various factors arising in the numerical procedure, such as change in length in the reconnection processes, the procedures regulating the amount of points on the lines, eliminations of very small loops below the space resolution as well as the evaporation of the loops from the volume. We would like to stress that the widely accepted mechanism-a cascadelike transfer of the energy by nonlinear Kelvin waves (and radiation of sound)-was not considered. One of the reasons is that the space resolution along the lines did not allow to detect generation of high harmonics, moreover, particularly to get harmonics, which really radiate sound. In addition, the use of the method assumes that the fluid is incompressible. Numerical simulations have been performed for the cubic domain with transparent walls embedded in an unbounded space, and for a cube with solid smooth walls. Calculations showed that in the case of unlimited space the decay of quantum turbulence caused by the evaporation of vortex loops, which is implemented in a diffusion-like manner. The rate of the attenuation of the vortex line density agrees with the one, predicted by the theory of diffusion of nonuniform vortex tangles. In the case of a cube with solid walls, the main decay is also due to the diffusion of the vortex loops to boundaries. The vortex loops, whose ends glide on a smooth wall, execute the sophisticated motion (especially when they jump from the one face to the other) with many subsequent reconnections. As a result, there appear smaller and smaller loops with a size of few spatial resolutions, which were removed from the calculation. Indirect comparison with the experiments shows that the time of decay agrees with the measured data.

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