Sedimentology and kinematics of a large, retrogressive growth-fault system in Upper Carboniferous deltaic sediments, western Ireland

Growth faulting is a common feature of many deltaic environments and is vital in determining local sediment dispersal and accumulation, and hence in controlling the resultant sedimentary facies distribution and architecture. Growth faults occur on a range of scales, from a few centimetres to hundreds of metres, with the largest growth faults frequently being under-represented in outcrops that are often smaller than the scale of feature under investigation. This paper presents data from the exceptionally large outcrops of the Cliffs of Moher, western Ireland, where a growth-fault complex affects strata up to 60 m in thickness and extends laterally for 3 km. Study of this Namurian (Upper Carboniferous) growth-fault system enables the relationship between growth faulting and sedimentation to be detailed and permits reconstruction of the kinematic history of faulting. Growth faulting was initiated with the onset of sandstone deposition on a succession of silty mudstones that overlie a thin, marine shale. The decollement horizon developed at the top of the marine shale contact for the first nine faults, by which time aggradation in the hangingwall exceeded 60 m in thickness. After this time, failure planes developed at higher stratigraphic levels and were associated with smaller scale faults. The fault complex shows a dominantly landward retrogressive movement, in which only one fault was largely active at any one time. There is no evidence of compressional features at the base of the growth faults, thus suggesting open-ended slides, and the faults display both disintegrative and non-disintegrative structure. Thin-bedded, distal mouth bar facies dominate the hangingwall stratigraphy and, in the final stages of growth-fault movement, erosion of the crests of rollover structures resulted in the highest strata being restricted to the proximity of the fault. These upper erosion surfaces on the fault scarp developed erosive chutes that were cut parallel to flow and are downlapped by the distal hangingwall strata of younger growth faults

A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (p.d.f.) of the random shear tensor at a general point in the lens plane due to point masses in the limit of an infinite number of stars. Up to this order, the p.d.f. depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the stars' masses. As a consequence, the p.d.f.s of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic p.d.f. of the shear magnitude in the limit of an infinite number of stars is also presented. Extending to general random distributions of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of {\it global} expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.Comment: To appear in JM

Impact of numerical integration on gas curtain simulations

In recent years, we have presented a less than glowing experimental comparison of hydrodynamic codes with the gas curtain experiment (e.g., Kamm et al. 1999a). Here, we discuss the manner in which the details of the hydrodynamic integration techniques may conspire to produce poor results. This also includes some progress in improving the results and agreement with experimental results. Because our comparison was conducted on the details of the experimental images (i.e., their detailed structural information), our results do not conflict with previously published results of good agreement with Richtmyer-Meshkov instabilities based on the integral scale of mixing. New experimental and analysis techniques are also discussed

The Littlewood-Gowers problem

We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - \epsilon} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.Comment: 31 pp. Corrected typos. Updated references

On Sub-linear Convergence for Linearly Degenerate Waves in Capturing Schemes

A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t{sup 1/2}, yielding a convergence rate of 1/2. Self-similar heuristics for higher order discretizations produce a growth rate for the layer thickness of {Delta}t{sup 1/(p+1)} which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations

Large droplet impact on water layers

The impact of large droplets onto an otherwise undisturbed layer of water is considered. The work, which is motivated primarily with regard to aircraft icing, is to try and help understand the role of splashing on the formation of ice on a wing, in particular for large droplets where splash appears, to have a significant effect. Analytical and numerical approaches are used to investigate a single droplet impact onto a water layer. The flow for small times after impact is determined analytically, for both direct and oblique impacts. The impact is also examined numerically using the volume of fluid (VOF) method. At small times there are promising comparisons between the numerical results, the analytical solution and experimental work capturing the ejector sheet. At larger times there is qualitative agreement with experiments and related simulations. Various cases are considered, varying the droplet size to layer depth ratio, including surface roughness, droplet distortion and air effects. The amount of fluid splashed by such an impact is examined and is found to increase with droplet size and to be significantly influenced by surface roughness. The makeup of the splash is also considered, tracking the incoming fluid, and the splash is found to consist mostly of fluid originating in the layer

Measurement and Compensation of Horizontal Crabbing at the Cornell Electron Storage Ring Test Accelerator

In storage rings, horizontal dispersion in the rf cavities introduces horizontal-longitudinal (xz) coupling, contributing to beam tilt in the xz plane. This coupling can be characterized by a "crabbing" dispersion term {\zeta}a that appears in the normal mode decomposition of the 1-turn transfer matrix. {\zeta}a is proportional to the rf cavity voltage and the horizontal dispersion in the cavity. We report experiments at the Cornell Electron Storage Ring Test Accelerator (CesrTA) where xz coupling was explored using three lattices with distinct crabbing properties. We characterize the xz coupling for each case by measuring the horizontal projection of the beam with a beam size monitor. The three lattice configurations correspond to a) 16 mrad xz tilt at the beam size monitor source point, b) compensation of the {\zeta}a introduced by one of two pairs of RF cavities with the second, and c) zero dispersion in RF cavities, eliminating {\zeta}a entirely. Additionally, intrabeam scattering (IBS) is evident in our measurements of beam size vs. rf voltage.Comment: 5 figures, 10 page