Energy Spectrum of Superfluid Turbulence with No Normal-Fluid Component

Abstract

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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