Scanning Superfluid-Turbulence Cascade by Its Low-Temperature Cutoff

On the basis of recently proposed scenario of the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, we develop a theory of low-temperature cutoff. The theory predicts a specific behavior of the quantized vortex line density, $L$, controlled by the frictional coefficient, $\alpha(T) \ll 1$, responsible for the cutoff. The curve $\ln L(\ln \alpha)$ is found to directly reflect the structure of the cascade, revealing four qualitatively distinct wavenumber regions. Excellent agreement with recent experiment by Walmsley {\it et al.} [arXiv:0710.1033]--in which $L(T)$ has been measured down to $T \sim 0.08$K--implies that the scenario of low-temperature superfluid turbulence is now experimentally validated, and allows to quantify the Kelvin-wave cascade spectrum.Comment: 4 pages, 2 figures, v2: extended introduction, the controversy with the scenario by L'vov et al. [13] is discussed in conclusio

Geometric Symmetries in Superfluid Vortex Dynamics

Dynamics of quantized vortex lines in a superfluid feature symmetries associated with the geometric character of the complex-valued field, $w(z)=x(z)+iy(z)$, describing the instant shape of the line. Along with a natural set of Noether's constants of motion, which---apart from their rather specific expressions in terms of $w(z)$---are nothing but components of the total linear and angular momenta of the fluid, the geometric symmetry brings about crucial consequences for kinetics of distortion waves on the vortex lines---the Kelvin waves. It is the geometric symmetry that renders Kelvin-wave cascade local in the wavenumber space. Similar considerations apply to other systems with purely geometric degrees of freedom.Comment: 4 REVTeX pages, minor stylistic changes, references to recent related preprints adde

Local Asymmetry and the Inner Radius of Nodal Domains

Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue \lambda. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/\lambda^n. We apply this result to prove that each nodal domain contains a ball of radius > C/\lambda^n.Comment: 12 pages, 1 figure; minor corrections; to appear in Comm. PDE

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum

Heat kernels on curved cones

A functorial derivation is presented of a heat-kernel expansion coefficient on a manifold with a singular fixed point set of codimension two. The existence of an extrinsic curvature term is pointed out.Comment: 4p.,sign errors corrected and a small addition,uses JyTeX,MUTP/94/0

The Heat Kernel Expansion on a Cone and Quantum Fields Near Cosmic Strings

An asymptotic expansion of the trace of the heat kernel on a cone where the heat coefficients have a delta function behavior at the apex is obtained. It is used to derive the renormalized effective action and total energy of a self-interacting quantum scalar field on the cosmic string space-time. Analogy is pointed out with quantum theory with boundaries. The surface infinities in the effective action are shown to appear and are removed by renormalization of the string tension. Besides, the total renormalized energy turns out to be finite due to cancelation of the known non-integrable divergence in the energy density of the field with a counterterm in the bare string tension.Comment: 20 pages, JINR preprint August, 1993, E2-93-291, LATEX fil

Assessing the effects of the first 2 years of industry-led badger culling in England on the incidence of bovine tuberculosis in cattle in 2013–2015

Culling badgers to control the transmission of bovine tuberculosis (TB) between this wildlife reservoir and cattle has been widely debated. Industry-led culling began in Somerset and Gloucestershire between August and November 2013 to reduce local badger populations. Industry-led culling is not designed to be a randomised and controlled trial of the impact of culling on cattle incidence. Nevertheless, it is important to monitor the effects of the culling and, taking the study limitations into account, perform a cautious evaluation of the impacts. A standardised method for selecting areas matched to culling areas in factors found to affect cattle TB risk has been developed to evaluate the impact of badger culling on cattle TB incidence. The association between cattle TB incidence and badger culling in the first two years has been assessed. Descriptive analyses without controlling for confounding showed no association between culling and TB incidence for Somerset, or for either of the buffer areas for the first two years since culling began. A weak association was observed in Gloucestershire for Year 1 only. Multivariable analysis adjusting for confounding factors showed that reductions in TB incidence were associated with culling in the first two years in both the Somerset and Gloucestershire intervention areas when compared to areas with no culling (IRR: 0.79, 95%CI: 0.72-0.87, p<0.001 and IRR: 0.42, 95%CI: 0.34-0.51, p<0.001 respectively). An increase in incidence was associated with culling in the 2 km buffer surrounding the Somerset intervention area (IRR: 1.38, 95%CI: 1.09-1.75, p=0.008), but not in Gloucestershire (IRR: 0.91, 95%CI: 0.77-1.07, p=0.243). As only two intervention areas with two years’ of data are available for analysis, and the biological cause-effect relationship behind the statistical associations is difficult to determine, it would be unwise to use these findings to develop generalisable inferences about the effectiveness of the policy at present

Dissipative dynamics of superfluid vortices at non-zero temperatures

We consider the evolution and dissipation of vortex rings in a condensate at non-zero temperature, in the context of the classical field approximation, based on the defocusing nonlinear Schr\"odinger equation. The temperature in such a system is fully determined by the total number density and the number density of the condensate. A vortex ring is introduced into a condensate in a state of thermal equilibrium, and interacts with non-condensed particles. These interactions lead to a gradual decrease in the vortex line density, until the vortex ring completely disappears. We show that the square of the vortex line length changes linearly with time, and obtain the corresponding universal decay law. We relate this to mutual friction coefficients in the fundamental equation of vortex motion in superfluids.Comment: 4 pages, 3 figure

Heat-kernel Coefficients and Spectra of the Vector Laplacians on Spherical Domains with Conical Singularities

The spherical domains $S^d_\beta$ with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with such a kind of singular points. In this paper the vector Laplacian on $S^d_\beta$ is considered and its spectrum is calculated exactly for any dimension $d$. This enables one to find the Schwinger-DeWitt coefficients of this operator by using the residues of the $\zeta$-function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on $S^d_\beta$ and its generalization to arbitrary manifolds is found. As an application of this result, the standard renormalization of the one-loop effective action of gauge fields is demonstrated to be sufficient to remove the ultraviolet divergences up to the first order in the conical deficit angle.Comment: plain LaTeX, 23 pp., revised version, a misprint in expressions (1.8) and (4.38) of the second heat coefficient for the vector Laplacian is corrected. No other change