Curvature Of Spectrum Research Articles

Closed universes, de Sitter space, and inflation

We present a new approach to constructing inflationary models in closed universes. Conformal embedding of closed-universe models in a de Sitter background suggests a quantization condition on the available conformal time. This condition implies that the universe is closed at no greater than the 10% level. When a massive scalar field is introduced to drive an inflationary phase this figure is reduced to closure at nearer the 1% level. In order to enforce the constraint on the available conformal time we need to consider conditions in the universe before the onset of inflation. A formal series around the initial singularity is constructed, which rests on a pair of dimensionless, scale-invariant parameters. For physically acceptable models we find that both parameters are of order unity, so no fine-tuning is required, except in the mass of the scalar field. For typical values of the input parameters we predict the observed values of the cosmological parameters, including the magnitude of the cosmological constant. The model produces a very good fit to the most recent cosmic microwave background (CMB) data. The primordial curvature spectrum provides a possible explanation for the low-$\ensuremath{\ell}$ falloff in the CMB power spectrum observed by Wilkinson microwave anisotropy probe. The spectrum also predicts a falloff in the matter spectrum at high $k$, relative to a power law. A further prediction of our model is a large tensor mode component, with $r\ensuremath{\approx}0.2$.

Outer-magnetospheric model for Vela-like pulsars: formation of sub-GeV spectrum

We investigate the γ-ray emission from an outer-gap accelerator, which is located in the outer magnetosphere of a pulsar. The charge depletion from the Goldreich–Julian density causes a large electric field along the magnetic field lines. The electric field accelerates electrons and positrons to high energies. We solve the formation of the electric field self-consistently with the radiation and the pair creation processes in one-dimensional geometry along the magnetic field lines, and calculate the curvature spectrum for comparison with the observations. We find that, because the particles escape from the accelerating region with Lorentz factors as large as 107.5, curvature radiation from outside of the gap, in which there is no electric field, still makes an important contribution to the spectrum. Including the outside emission, the model spectra are improved between 100 MeV and 1 GeV for the Vela-type pulsars. Since field line curvature close to the light cylinder affects the spectrum at several hundred MeV, there is a possibility to diagnose the field structure by using the spectrum in these bands. The difference found in the γ-ray spectra of the two pulsars, PSR B0833−45 (Vela) and B1706−44, is deduced from the dimensionless electric current running through the gap. The extension to three-dimensional modelling with self-consistent electrodynamics is necessary to understand the difference in the observed flux between the two pulsars.

Geometry of turbulent tangles of material lines

We show that a tangle of material lines in a purely structured, decaying turbulent flow presents flat scalings for the curvature and torsion spectra corresponding to both the inertial and viscous ranges in the energy spectrum and that it acquires a fractal dimension close to 2 resembling a material sheet.

The role of spectral curvature mapping in characterizing subsurface strain distributions

Abstract The curvature of structured geological surfaces can be used to assess the degree of strain they have undergone. In many hydrocarbon reservoirs, this strain is expressed as brittle fracturing that may significantly impact reservoir performance. Here we describe the development of an algorithm for measuring the curvature of gridded surfaces derived from seismic data. For any grid node, the algorithm calculates the magnitude and orientations of the two principal curvatures, K 1 and K 2 , from which other curvature measurements can be derived, such as Gaussian curvature and summed absolute curvature ( K 1 + K 2 ). The algorithm has also been used to generate plots of summed absolute curvature as a function of grid node separation ( k versus λ). These ‘spectral’ or k λ plots can be generated for each grid node and allow the definition of short-wavelength, high-amplitude noise cut-off lengths. They also deliver intermediate wavelength features such as fault drag or buckle folding and the identification of long-wavelength (basin-scale) curvatures. Portions of these data can be collapsed into single values by calculating the integral of the k λ curve. Further filters designed to screen the effects of background tectonic, or non-tectonic, curvatures can be applied to the k λ integral. This algorithm has been tested using data from several North Sea chalk fields. A range of alternative types of curvature and curvature spectra are compared with other approaches to curvature calculation and other factors relevant to the calibration of such techniques in terms of the distribution of brittle fractures in sedimentary rocks. The k λ integral provides a relatively simple approach to calculating the degree of multi-wavelength strain present at a particular grid node. Freeing algorithms from the restriction of the ‘arbitrarily’ selected minimum grid node spacing is a key step towards calibrating measured curvature against strain mechanisms. However, care must be taken to separate intrinsic and tectonic curvatures when generating and interpreting k λ plots and their integrals.

Kelvin waves cascade in superfluid turbulence.

We study numerically the interaction of four initial superfluid vortex rings in the absence of any dissipation or friction. We find evidence for a cascade of Kelvin waves generated by individual vortex reconnection events which transfers energy to higher and higher wave numbers k. After the vortex reconnections occur, the energy spectrum scales as k(-1) and the curvature spectrum becomes flat. These effects highlight the importance of Kelvin waves and reconnections in the transfer of energy within a turbulent vortex tangle.

Gravity–Capillary Wave Curvature Spectrum and Mean-Square Slope Retrieved from Microwave Radiometric Measurements (Coastal Ocean Probing Experiment)

Abstract Brightness temperature of the sea surface is determined by surface waves of different scales. Polarimetric measurements collected by microwave radiometers at several viewing angles and frequencies give an opportunity to convert the observed brightness temperatures to the mean-square slope of long waves and curvature spectral density in the gravity–capillary interval without a priori assumptions about the shape of spectrum. Such an inversion was made using radiometric data collected during the Coastal Ocean Probing Experiment (COPE’95) from a blimp. It is shown that at low and moderate wind speed conditions typical for COPE’95 (4–7 m s−1) the curvature spectrum has a maximum at wavenumber of about 4.5 rad cm−1, slightly higher than the wavenumber of the phase velocity minimum. The maximum corresponds to the wavenumber with the highest initial growth rate as predicted by theory for the growth rate of wind-induced gravity–capillary waves based on the Orr– Sommerfeld equation. The author’s estimation...

Mapping spatial variation in rock properties in relationship to scale-dependent structure using spectral curvature

Maps of the three-dimensional geometry of geologic surfaces show that structural curvature commonly varies with scale of observation: This fact can be viewed as superposition of structures at different wavelengths. Rock properties such as fracture density and orientation reflect the contribution of superimposed structures. For this reason, characterization of geologic surfaces is fundamentally different from purely geometrical characterization, for which local description of surface properties is sufficient. We show that measured curvature decays according to a power law with increasing size of measurement window, so short-wavelength curvatures do not obscure long-wavelength curvatures in the same data set. This property can be taken advantage of in a simple technique for automatically mapping multiwavelength curvatures. At each point on a surface, curvature is measured at a range of wavelengths. This curvature spectrum can be analyzed in map view or collapsed into a single value at each point in space. The results indicate that complex geologic surfaces can be characterized without any prior knowledge of structural wavelengths and orientation. The method should prove useful in applications requiring knowledge of spatial variation in rock properties from remotely sensed data, such as exploration for hydrocarbon reservoirs or nuclear waste repositories.

Dynamics of assisted inflation

We investigate the dynamics of the recently proposed model of assisted inflation. In this model an arbitrary number of scalar fields with exponential potentials evolve towards an inflationary scaling solution, even if each of the individual potentials is too steep to support inflation on its own. By choosing an appropriate rotation in field space we can write down explicitly the potential for the weighted mean field along the scaling solution and for fields orthogonal to it. This demonstrates that the potential has a global minimum along the scaling solution. We show that the potential close to this attractor in the rotated field space is analogous to a hybrid inflation model, but with the vacuum energy having an exponential dependence upon a dilaton field. We present analytic solutions describing homogeneous and inhomogeneous perturbations about the attractor solution without resorting to slow-roll approximations. We discuss the curvature and isocurvature perturbation spectra produced from vacuum fluctuations during assisted inflation.

Gamma-Ray Pulsars: Radiation Processes in the Outer Magnetosphere

We describe an emission model for gamma ray pulsars based on curvature radiation-reaction limited charges in the outer magnetosphere. We show how pair production on thermal surface flux can limit the acceleration zones. Estimates for the efficiency of GeV photon production eta gamma and the gamma-ray beaming fraction are derived, including their dependence on pulsar parameters. In general eta gamma increases with pulsar age, but is decreased for low magnetic fields and for small magnetic inclinations. We argue that this produces GeV pulse profiles, curvature spectra and detection statistics consistent with the observations. We also describe the optical through X-ray pulsar synchrotron spectrum and the spectral variations with pulsar phase. A test computation for Vela-like parameters reproduces phase-resolved GeV spectra consistent with those observed by EGRET. Finally we comment on very high energy pulsed emission and particle production and note extensions needed to allow a more complete pulsar model.

Shapes of aggregates from tilled layers of some Dutch and Australian soils

Five different methods were used to study the shapes of soil aggregates oriented in their planes of maximum projection. A measure of roundness was found to be relatively insensitive to shape features. The shortest aspect ratio (the ratio of the shortest to the longest diameters through the centroid) was found to be most sensitive of the “single value” shape measures. More comprehensive shape information was given by values of the radius spectra and the curvature spectra. The curvature spectra method seems to be slightly more sensitive than the radius spectra method and can also be applied to reentrant particles. Generally it was found that aggregates are less round with increasing clay content and decreasing organic matter content. There was also a hint that aggregates of the Dutch soils may become less round with increasing age since their reclamation. The Australian soil aggregates were slightly rounder than the Dutch aggregates.

Solutions of the exact time-dependent microbending equations

Bandwidth improvement factor and losses of two-dimensional fibers with power-law index profiles, submitted to microbends with power-law curvature spectra, are given here in a graphical form. The domain of validity of the analytical solutions is discussed.

Comparison of Quasi-specular radar scatter from the moon with surface parameters obtained from images

Quasi-specular radar data used to determine apparent surface roughness σ χ of geologic surfaces displays a variable wavelength λ dependence ranging between σ χ ∼ λ 0 and σ χ ∼ λ − 1 3 for 0.01 ⪝ λ ⪝ 1 m . The strongest changes in σ χ with wavelength are observed in lunar mare, while scatter from lunar highlands is nearly wavelength independent. Commonly used, gently undulating surface models for electromagnetic scatter predict no wavelength dependence. Wavelength dependence occurs whenever a significant fraction of the surface has local radii of curvature comparable to the observing wavelength. This condition can be determined by comparison of the value of the integrated surface curvature spectrum with the radar wavenumber, multiplied by a constant that depends on the geometry. Variations in curvature statistics calculated from photogrammetric reduction of lunar images are consistent with the observed variations in quasi-specular scatter between λ = 13 and 116 cm at the same locations. Variations in the strength of the wavelength dependence are correlated with the sizes of lunar craters that lie near the upper size limit for the local steady-state distribution. This correlation is also consistent with variations in the curvature spectrum calculated from crater size-frequency distributions.

Role of nonadjacent mode coupling in the theory of microbending

The error caused by the neglect of coupling between nonadjacent modes in the theory of microbending is calculated by comparison with the exact result. This comparison is made for two-dimensional fibers that have a power-law index profiles and various curvature spectra. For power-law spectra, closed from expressions for the microbending losses are given.

Ray theory of microbending

A ray theory is given for randomly bent (two-dimensional) optical fibers that have arbitrary index profiles and arbitrary curvature spectra. Simple closed form results are given for power-law profiles and spectra. No approximation is made besides the small bending approximation and the paraxial ray optics approximation. In particular, the coupling between all modes is effectively taken into account.

Curvature Spectra: Applications to Signal Detection and Estimation of the Resolving Power of Arrays

Summary We define the curvature spectrum as the two-dimensional Laplacian of a frequency-wavenumber spectrum, the derivatives being taken in the wave-number directions. Empirical results indicate that the method has the characteristics of a high-resolution f–k estimator, i.e. the spectral peaks are sharper than in ordinary f–k spectra. The curvature spectrum of an array response function can give useful information about the array capability; in particular, defining the differential array response as the (scalar) curvature of the array response at k= 0, we show that this differential response predicts the angular resolution capability of an array. We conclude that if two arrays have differential responses D1, and D2, then their angular resolution capabilities θ1, and θ2, are related by D1θ1, = D2θ2., It is shown that using a popular infrasonic signal analysis algorithm, the azimuth resolution of four- and five-element arrays with apertures as small as 20 km is about one-tenth of a degree. The capabilities of some microbarograph arrays in current use are compared.

Two methods for quantitative description of soil particle shape: P. F. Davis and A. R. DexterJ. Soil Sci.23 (4), 448–455 (1972)

It was required to be able to measure the shape of soil particles in order that the influence of shape on soil mechanical properties could be investigated. Two methods were developed, both of which can be applied to particle cross-sections or outlines. The first is a shape function which compares cord lengths with arc lengths around the particle outlines. The second method involves taking discrete Fourier transforms of the normalized curvatures of the outlines, and the production of curvature spectra. Both methods are illustrated with the use of the same data. The data were three sizes of particle from a sandy soil. Both methods illustrate the greater asphericity or angularity of the smaller particles, which is consistent with the Griffith theory of comminution. Change in shape with particle size may be useful for classifying soils according to their methods of formation.

TWO METHODS FOR QUANTITATIVE DESCRIPTION OF SOIL PARTICLE SHAPE

SummaryIt was required to be able to measure the shapes of soil particles in order that the influence of shape on soil mechanical properties could be investigated. Two methods were developed, both of which can be applied to particle cross‐sections or outlines. The first is a shape function which compares chord lengths with arc lengths around the particle outlines. The second method involves taking discrete Fourier transforms of the normalized curvatures of the outlines, and the production of curvature spectra. Both methods are illustrated with the use of the same data. The data were for three sizes of particle from a sandy soil. Both methods illustrate the greater asphericity or angularity of the smaller particles, which is consistent with the Griffith theory of comminution. Change in shape with particle size may be useful for classifying soils according to their methods of formation.