Energy spectra of vortex distributions in two-dimensional quantum turbulence

We theoretically explore key concepts of two-dimensional turbulence in a homogeneous compressible superfluid described by a dissipative two-dimensional Gross-Pitaeveskii equation. Such a fluid supports quantized vortices that have a size characterized by the healing length $\xi$. We show that for the divergence-free portion of the superfluid velocity field, the kinetic energy spectrum over wavenumber $k$ may be decomposed into an ultraviolet regime ($k\gg \xi^{-1}$) having a universal $k^{-3}$ scaling arising from the vortex core structure, and an infrared regime ($k\ll\xi^{-1}$) with a spectrum that arises purely from the configuration of the vortices. The Novikov power-law distribution of intervortex distances with exponent -1/3 for vortices of the same sign of circulation leads to an infrared kinetic energy spectrum with a Kolmogorov $k^{-5/3}$ power law, consistent with the existence of an inertial range. The presence of these $k^{-3}$ and $k^{-5/3}$ power laws, together with the constraint of continuity at the smallest configurational scale $k\approx\xi^{-1}$, allows us to derive a new analytical expression for the Kolmogorov constant that we test against a numerical simulation of a forced homogeneous compressible two-dimensional superfluid. The numerical simulation corroborates our analysis of the spectral features of the kinetic energy distribution, once we introduce the concept of a {\em clustered fraction} consisting of the fraction of vortices that have the same sign of circulation as their nearest neighboring vortices. Our analysis presents a new approach to understanding two-dimensional quantum turbulence and interpreting similarities and differences with classical two-dimensional turbulence, and suggests new methods to characterize vortex turbulence in two-dimensional quantum fluids via vortex position and circulation measurements.Comment: 19 pages, 8 figure

A Novel Method of Solution for the Fluid Loaded Plate

We study the Cauchy problem associated with the equations governing a fluid loaded plate formulated on either the line or the half-line. We show that in both cases the problem can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990s. The problem on the full line was analysed by Crighton et. al. using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in $L^1_{dt}(R^+)$ and furthermore yields a simpler solution representation which immediately implies the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, but this formula involves two unknown functions. The main difficulty with the half-line problem is the characterisation of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex valued integral equation of the convolution type. This equation can be solved in closed form using the Laplace transform. By prescribing the initial data $\eta_0$ to be in $H^3(R^+)$, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed.Comment: 17 pages, 3 figures. Minor adjustments made to the introductio

Snell's Law for a vortex dipole in a Bose-Einstein condensate

A quantum vortex dipole, comprised of a closely bound pair of vortices of equal strength with opposite circulation, is a spatially localized travelling excitation of a planar superfluid that carries linear momentum, suggesting a possible analogy with ray optics. We investigate numerically and analytically the motion of a quantum vortex dipole incident upon a step-change in the background superfluid density of an otherwise uniform two-dimensional Bose-Einstein condensate. Due to the conservation of fluid momentum and energy, the incident and refracted angles of the dipole satisfy a relation analogous to Snell's law, when crossing the interface between regions of different density. The predictions of the analogue Snell's law relation are confirmed for a wide range of incident angles by systematic numerical simulations of the Gross-Piteavskii equation. Near the critical angle for total internal reflection, we identify a regime of anomalous Snell's law behaviour where the finite size of the dipole causes transient capture by the interface. Remarkably, despite the extra complexity of the surface interaction, the incoming and outgoing dipole paths obey Snell's law.Comment: 16 pages, 7 figures, Scipost forma

Ground Water: Alaska's Hidden Resource: Proceedings

Surface water quality -- Surface/ground water interactions -- Ground water monitoring, modeling, and data management -- Transport and removal of contaminants in soil and ground wate

Effects of competition variables and prior attempts on approach run velocity during a pole vaulter\u27s final clearance 2018

Approach run velocity of a vaulter is strongly correlated to the highest height a vaulter clears in pole vault competition and the number of attempts taken throughout a competition influences pole vault strategy. Since approach run velocity greatly affects the crossbar height cleared and number of attempts affects time spent in the competition, perhaps a better approach to determine optimal competition strategy is to first identify how competition variables influence approach run velocity. The purpose of this study was to determine if the approach run velocity during a pole vaulter’s last clearance can be predicted by: (1) the number of previous attempts by the vaulter in the competition, (2) the range of approach run velocities in the vaulter’s previous attempts, and/or (3) the time elapsed from the vaulter’s first attempt to the vaulter’s final clearance. It was hypothesized that the total number of attempts, range of approach run velocity, and total time elapsed from first attempt to final clearance can adequately predict approach run velocity for a pole vaulter’s final clearance. Number of attempts was the lone statistically significant variable for predicting the Z-score of final clearance velocity. The prediction equation for the Z-score of the final clearance velocity using number of attempts is: VFclearance = 0.124 (Attempts) - 0.676. A second prediction equation formulated from the Z-score final clearance equation can predict real clearance velocities (m/s). The prediction equation for real clearance velocity is: Vpredicted = [0.124 (SD)](Attempts) - 0.676(SD) + VRavg. However, number of attempts only explains a very small percentage of variance in final clearance approach run velocity (6.3%). National caliber coaches and athletes may use the formulated Z-score prediction equation and/or real velocity prediction equation to estimate approach run velocity and make decisions regarding competition strategies to maximize performance