Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence

We examine the probability distribution function (PDF) of the energy injection rate (power) in numerical simulations of stationary two-dimensional (2D) turbulence in the Lagrangian frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating 2D turbulence in the laboratory. In our simulations, the forcing and velocity fields are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured PDFs with the theoretical calculations and briefly discuss how the power PDF might change with other forcing mechanisms

Force Chain Evolution in a Two-Dimensional Granular Packing Compacted by Vertical Tappings

We experimentally study the statistics of force-chain evolution in a vertically-tapped two-dimensional granular packing by using photoelastic disks. In this experiment, the tapped granular packing is gradually compacted. During the compaction, the isotropy of grain configurations is quantified by measuring the deviator anisotropy derived from fabric tensor, and then the evolution of force-chain structure is quantified by measuring the interparticle forces and force-chain orientational order parameter. As packing fraction increases, the interparticle force increases and finally saturates to an asymptotic value. Moreover, the grain configurations and force-chain structures become isotropically random as the tapping-induced compaction proceeds. In contrast, the total length of force chains remains unchanged. From the correlations of those parameters, we find two relations: (i) a positive correlation between the isotropy of grain configurations and the disordering of force-chain orientations, and (ii) a negative correlation between the increasing of interparticle forces and the disordering of force-chain orientations. These relations are universally held regardless of the mode of particle motions with/without convection

Grid-scale Fluctuations and Forecast Error in Wind Power

The fluctuations in wind power entering an electrical grid (Irish grid) were analyzed and found to exhibit correlated fluctuations with a self-similar structure, a signature of large-scale correlations in atmospheric turbulence. The statistical structure of temporal correlations for fluctuations in generated and forecast time series was used to quantify two types of forecast error: a timescale error ($e_{\tau}$) that quantifies the deviations between the high frequency components of the forecast and the generated time series, and a scaling error ($e_{\zeta}$) that quantifies the degree to which the models fail to predict temporal correlations in the fluctuations of the generated power. With no $a$ $priori$ knowledge of the forecast models, we suggest a simple memory kernel that reduces both the timescale error ($e_{\tau}$) and the scaling error ($e_{\zeta}$)

Nonparametric Stochastic Volatility

Using recent advances in the nonparametric estimation of continuous-time processes under mild statistical assumptions as well as recent developments on nonparametric volatility estimation by virtue of market microstructure noise-contaminated high-frequency asset price data, we provide (i) a theory of spot variance estimation and (ii) functional methods for stochastic volatility modelling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and diffusion functions, nonlinear leverage effects, jumps in returns and volatility with possibly state-dependent jump intensities, as well as nonlinear risk-return trade-offs. Our identification approach and asymptotic results apply under weak recurrence assumptions and, hence, accommodate the persistence properties of variance in finite samples. Functional estimation of a generalized (i.e., nonlinear) version of the square-root stochastic variance model with jumps in both volatility and returns for the S&P500 index suggests the need for richer variance dynamics than in existing work. We find a linear specification for the variance's diffusive variance to be misspecified (and inferior to a more flexible CEV specification) even when allowing for jumps in the variance dynamics.Spot variance, stochastic volatility, jumps in returns, jumps in volatility, leverage effects, risk-return trade-offs, kernel methods, recurrence, market microstructure noise.

On angled bounce-off impact of a drop impinging on a flowing soap film

Small drops impinging angularly on thin flowing soap films frequently demonstrate the rare emergence of bulk elastic effects working in-tandem with the more common-place hydrodynamic interactions. Three collision regimes are observable: (a) drop piercing through the film, (b) it coalescing with the flow, and (c) it bouncing off the film surface. During impact, the drop deforms along with a bulk elastic deformation of the film. For impacts that are close-to-tangential, the bounce-off regime predominates. We outline a reduced order analytical framework assuming a deformable drop and a deformable three-dimensional film, and the idealization invokes a phase-based parametric study. Angular inclination of the film and the ratio of post and pre impact drop sizes entail the phase parameters. We also perform experiments with vertically descending droplets impacting against an inclined soap film, flowing under constant pressure head. Model predicted phase domain for bounce-off compares well to our experimental findings. Additionally, the experiments exhibit momentum transfer to the film in the form of shed vortex dipole, along with propagation of free surface waves. On consulting prior published work, we note that for locomotion of water-walking insects using an impulsive action, the momentum distribution to the shed vortices and waves are both significant, taking up respectively 2/3-rd and 1/3-rd of the imparted streamwise momentum. In view of the potentially similar impulse actions, this theory is applied to the bounce-off examples in our experiments, and the resultant shed vortex dipole momenta are compared to the momenta computed from particle imaging velocimetry data. The magnitudes reveal identical order ($10^{-7}$ N$\cdot$s), suggesting that the bounce-off regime can be tapped as a simple analogue for interfacial bio-locomotion relying on impulse reactions