The quantization of vortex lines in superfluids requires the introduction of their density L(r, t) in the description of quantum turbulence. The space homogeneous balance equation for L(t), proposed by Vinen on the basis of dimensional and physical considerations, allows a number of competing forms for the production term P. Attempts to choose the correct one on the basis of time-dependent homogeneous experiments ended inconclusively. To overcome this difficulty we announce here an approach that employs an inhomogeneous channel flow which is excellently suitable to distinguish the implications of the various possible forms of the desired equation. We demonstrate that the originally selected form which was extensively used in the literature is in strong contradiction with our data. We therefore present a new inhomogeneous equation for L(r, t) that is in agreement with our data and propose that it should be considered for further studies of superfluid turbulence.

The main goal of this paper is to present a comprehensive characterization of well developed vortex tangles in a turbulent counterflow in quantum fluids (with a laminar normal fluid component). We analyze extensive numerical simulations using the vortex filament method, solving the full Biot-Savart equations for the vortex dynamics in a wide range of temperatures and counter-flow velocities. In addition to a detailed analysis of traditional characteristics such as vortex line density, anisotropic and curvature parameters of the vortex tangle, we stress other dynamical and statistical characteristics which are either much less studied or even unstudied. The latter include reconnection rates, mean mutual friction forces, drift velocities and the probability distribution functions of various tangle parameters: the loop length, the line curvature, the mean curvature of loops with a given length, etc. During these studies we compare the three main reconnection procedures which are widely used in the literature, and identify which properties are strongly affected by the choice of the reconnection criteria and which of them are practically insensitive to the reconnection procedure. The conclusion is that the vortex filament method in the framework of the Biot-Savart equation sufficiently robust and well suited for the description of the steady state vortex tangle in a quantum counterflow. The Local-Induction Approximation to this equation may be successfully used to analytically establish relationships between mean characteristics of the stochastic vortex tangle.

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.

We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid 4 He at low and ultra-low temperatures, 0 T 1.1 K. Using high resolution vortex filament simulations with full Biot-Savart vortex dynamics, we show that for ultra-low temperatures T 0.5 K, when the mutual friction parameters α ≃ α ′ < 10 −5 , the vortex reconnections excite Kelvin waves with wave lengths λ of the order of the inter-vortex distance ℓ. These excitations cascade down to the resolution scale ∆ξ which in our simulations is of the order ∆ξ ∼ ℓ/100. At this scale the Kelvin waves are numerically damped by a line-smoothing procedure, that is supposed to mimic the dissipation of Kelvin waves by phonon and roton emission at the scale of the vortex core. We show that the Kelvin waves cascade is statistically important: the shortest available Kelvin waves at the end of the cascade determine the mean vortex line curvature S, giving S 30/ℓ and play major role in the tangle decay at ultra-low temperatures below 0.6 K. The found dependence of ℓS on the resolution scale ∆ξ agrees with the L'vov-Nazarenko energy spectrum of weakly-interacting Kelvin waves, ELN ∝ k −5/3 rather than the spectrum ELN ∝ k −1 , suggested by Vinen for turbulence of Kelvin waves with large amplitudes. We also show that already at T = 0.8 K, when α and α ′ are still very low, α ≃ α ′ < 10 −3 , the Kelvin wave cascade is fully damped, vortex lines are very smooth, S ≃ 2/ℓ and the tangle decay is predominantly caused by the mutual friction.