Energy spectra of developed superfluid turbulence

Abstract

Turbulence spectra in superfluids are modified by the nonlinear energy dissipation caused by the mutual friction between quantized vortices and the normal component of the liquid. We have found a new state of fully developed turbulence which occurs in some range of two Reynolds parameters characterizing the superfluid flow. This state displays both the Kolmogorov-Obukhov -scaling law E k ∝ k −5/3 and a new "3-scaling law" E k ∝ k −3 , each in a well-separated range of k.PACS numbers: 43.37.+q,47.32.Cc, 67.40.Vs, 67.57.Fg Superfluid consists of mutually penetrating components -viscous normal component and one or several frictionless superfluid components. This explains why different types of turbulent motion are possible depending on whether the normal and the superfluid components move together or separately. Here, we are interested in the most simple case when the dynamics of the normal component can be neglected. This occurs, for example, in the superfluid phases of 3 He where the normal component is so viscous that it is practically clamped to the container walls. The role of the normal component in this case is to provide the preferred heat-bath reference frame, where the normal component, and thus the heat bath, are at rest. Dissipation takes place when the vortices move with respect to this reference frame. Turbulence in such a superfluid component with the normal component at rest will be called here superfluid turbulence. Recent experiments in3 He-B [1] demonstrated that the fate of a few vortices injected into a rapidly moving superfluid depends on a dimensionless intrinsic temperaturedependent parameter q rather than on the flow velocity. At q ∼ 1, a rather sharp transition is observed between laminar evolution of the injected vortices and a turbulent many-vortex state of the whole superfluid. In this Letter we describe how the celebrated Kolmogorov-Obukhov 5 3 -law for the turbulent energy spectrum in normal fluid, E k ∝ k −5/3 , gets modified in the superfluid turbulence, giving rise the much steeper decrease, E k ∝ k −3 .As a starting point we utilize a coarse-grained hydrodynamic equation for the superfluid dynamics with distributed vortices. In this equation the parameter q characterizes the friction force between the superfluid and the normal components of the liquid, which is mediated by quantized vortices. According to this equation, turbulence develops only if the friction is relatively small compared to the inertial term, i.e. when q < 1. Here, we will study the case of developed turbulence which must occur at q ≪ 1.An important feature of superfluid turbulence is that the vorticity of the superfluid component is quantized in terms of the elementary circulation quantum κ (in 3 He-B, κ = π /m where m is the mass of 3 He atom). Thus, superfluid turbulence is a chaotic motion of welldetermined and well-separated vortex filaments [3]. Using this as starting point we can simulate the main ingredients of classical turbulence -the chaotic dynamics of the vortex degrees of freedom of the liquid. H...

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“…We consider the simplest case of classically generated turbulence in quantum liquids such as He II or 3 He-B that can be thought of as isothermal, homogeneous and isotropic. We work within a framework of the two-fluid model, building on ideas first introduced by Volovik 9 and Vinen 10 , by further developing our previous work 11,12 . We stress that our approach does not directly apply to quantum turbulence generated in superfluid helium by the thermal counterflow 2 , which is anisotropic, being generated thermally, by the temperature gradient in the channel.…”