Effects of channel cross-sectional geometry on long wave generation and propagation

Joint theoretical and experimental studies are carried out to investigate the effects of channel cross-sectional geometry on long wave generation and propagation in uniform shallow water channels. The existing channel Boussinesq and channel KdV equations are extended in the present study to include the effects of channel sidewall slope at the waterline in the first-order section-mean equations. Our theoretical results show that both the channel cross-sectional geometry below the unperturbed water surface (characterized by a shape factor kappa) and the channel sidewall slope at the waterline (represented by a slope factor gamma) affect the wavelength (lambda) and time period (Ts) of waves generated under resonant external forcing. A quantitative relationship between lambda, Ts, kappa, and gamma is given by our theory which predicts that, under the condition of equal mean water depth and equal mean wave amplitude, lambda and Ts increase with increasing kappa and gamma. To verify the theoretical results, experiments are conducted in two channels of different geometries, namely a rectangular channel with kappa[equivalent]1, gamma=0 and a trapezoidal channel with kappa=1.27, gamma=0.16, to measure the wavelength of free traveling solitary waves and the time period of wave generation by a towed vertical hydrofoil moving with critical speed. The experimental results are found to be in broad agreement with the theoretical predictions

Propagation of solitary waves through signicantly curved shallow water channels

Propagation of solitary waves in curved shallow water channels of constant depth and width is investigated by carrying out numerical simulations based on the generalized weakly nonlinear and weakly dispersive Boussinesq model. The objective is to investigate the effects of channel width and bending sharpness on the transmission and reflection of long waves propagating through significantly curved channels. Our numerical results show that, when travelling through narrow channel bends including both smooth and sharp-cornered 90°-bends, a solitary wave is transmitted almost completely with little reflection and scattering. For wide channel bends, we find that, if the bend is rounded and smooth, a solitary wave is still fully transmitted with little backward reflection, but the transmitted wave will no longer preserve the shape of the original solitary wave but will disintegrate into several smaller waves. For solitary waves travelling through wide sharp-cornered 90°-bends, wave reflection is seen to be very significant, and the wider the channel bend, the stronger the reflected wave amplitude. Our numerical results for waves in sharp-cornered 90°-bends revealed a similarity relationship which indicates that the ratios of the transmitted and reflected wave amplitude, excess mass and energy to the original wave amplitude, mass and energy all depend on one single dimensionless parameter, namely the ratio of the channel width b to the effective wavelength [lambda][sub]e. Quantitative results for predicting wave transmission and reflection based on b/[lambda][sub]e are presented

Regression Depth and Center Points

We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(n/(d+1)). as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.Comment: 14 pages, 3 figure

A Supernova Factory in the Merger System Arp 299

We have imaged the nearby galaxy merger Arp 299 at arcsecond and milliarcsecond resolution, using both the Very Large Array and the Very Long Baseline Array. The large-scale radio emission from the merger contains 5 bright, compact radio sources embedded in diffuse emission, with diameters less than 200 pc. Supernova rates of 0.1 to 1 per year are required to produce the VLA-detected radio emission in these sources. Two of the compact VLA radio sources, designated Source A and Source D, also have been detected and imaged at milliarcsecond scales. Source A, which is associated with the nucleus of one of the merging galaxies, contains five milliarcsecond-scale sources, each with a radio power between 100 and 1000 times that of the Galactic supernova remnant Cassiopeia A. Four of these have flat or inverted spectra and appear to be young supernovae. Three of the VLBI-scale sources are located within 10 pc (projected) of one another, and two are separated by less than 3 pc, indicating that they all may be within the same super starcluster or complex of such clusters. The brightest VLBI-scale source, A0, has an extremely inverted pectrum, with alpha larger than +2 at gigahertz frequencies. It seems to be the youngest supernova, which has not yet broken out of its circumstellar shell. The milliarcsecond radio sources within Source A appear to constitute a upernova factory, confirming the presence of an extreme starburst that peaked at least a few million years ago.Comment: Accepted for the Astrophysical Journal, 22 pages, 10 figure

Evolution of long water waves in variable channels

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data availabl

Half-Megasecond Chandra Spectral Imaging of the Hot Circumgalactic Nebula around Quasar Mrk 231

A deep 400-ksec ACIS-S observation of the nearest quasar known, Mrk 231, is combined with archival 120-ksec data obtained with the same instrument and setup to carry out the first ever spatially resolved spectral analysis of a hot X-ray emitting circumgalactic nebula around a quasar. The 65 x 50 kpc X-ray nebula shares no resemblance with the tidal debris seen at optical wavelengths. One notable exception is the small tidal arc 3.5 kpc south of the nucleus where excess soft X-ray continuum emission and Si XIII 1.8 keV line emission are detected, consistent with star formation and its associated alpha-element enhancement, respectively. An X-ray shadow is also detected at the location of the 15-kpc northern tidal tail. The hard X-ray continuum emission within 6 kpc of the center is consistent with being due entirely to the bright central AGN. The soft X-ray spectrum of the outer (>6 kpc) portion of the nebula is best described as the sum of two thermal components with T~3 and ~8 million K and spatially uniform super-solar alpha element abundances, relative to iron. This result implies enhanced star formation activity over ~10^8 yrs accompanied with redistribution of the metals on large scale. The low-temperature thermal component is not present within 6 kpc of the nucleus, suggesting extra heating in this region from the circumnuclear starburst, the central quasar, or the wide-angle quasar-driven outflow identified from optical IFU spectroscopy on a scale of >3 kpc. Significant azimuthal variations in the soft X-ray intensity are detected in the inner region where the outflow is present. The soft X-ray emission is weaker in the western quadrant, coincident with a deficit of Halpha and some of the largest columns of neutral gas outflowing from the nucleus. Shocks created by the interaction of the wind with the ambient ISM may heat the gas to high temperatures at this location. (abridged)Comment: 43 pages, 11 figures, accepted for publication in the Astrophysical Journa

Exploring Oxidation in the Remote Free Troposphere: Insights from Atmospheric Tomography (ATom)

Earth's atmosphere oxidizes the greenhouse gas methane and other gases, thus determining their lifetimes and oxidation products. Much of this oxidation occurs in the remote, relatively clean free troposphere above the planetary boundary layer, where the oxidation chemistry is thought to be much simpler and better understood than it is in urban regions or forests. The NASA airborne Atmospheric Tomography study (ATom) was designed to produce cross sections of the detailed atmospheric composition in the remote atmosphere over the Pacific and Atlantic Oceans during four seasons. As part of the extensive ATom data set, measurements of the atmosphere's primary oxidant, hydroxyl (OH), and hydroperoxyl (HO₂) are compared to a photochemical box model to test the oxidation chemistry. Generally, observed and modeled median OH and HO₂ agree to with combined uncertainties at the 2σ confidence level, which is ~±40%. For some seasons, this agreement is within ~±20% below 6 km altitude. While this test finds no significant differences, OH observations increasingly exceeded modeled values at altitudes above 8 km, becoming ~35% greater, which is near the combined uncertainties. Measurement uncertainty and possible unknown measurement errors complicate tests for unknown chemistry or incorrect reaction rate coefficients that would substantially affect the OH and HO₂ abundances. Future analysis of detailed comparisons may yield additional discrepancies that are masked in the median values