On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

The omega equation and potential vorticity

Two fundamental perspectives on the dynamics of midlatitude weather systems are provided by potential vorticity (PV) and the omega equation. The aim of this paper is to investigate the link between the two perspectives, which has so far received very little attention in the meteorological literature. It also aims to give a quantitative basis for discussion of quasi-geostrophic vertical motion in terms of components associated with system movement, maintaining a constant thermal structure, and with the development of that structure. The former links with the isentropic relative-flow analysis technique. Viewed in a moving frame of reference, the measured development of a system depends on the velocity of that frame of reference. The requirement that the development should be a minimum provides a quantitative method for determining the optimum system velocity. The component of vertical velocity associated with development is shown to satisfy an omega equation with forcing determined from the relative advection of interior PV and boundary temperature. The analysis carries through in the presence of diabatic heating provided the omega equation forcing is based on the interior PV and boundary thermal tendencies, including the heating effect. The analysis is shown to be possible also at the level of the semi-geostrophic approximation. The analysis technique is applied to a number of idealized problems that can be considered to be building blocks for midlatitude synoptic-scale dynamics. They focus on the influences of interior PV, boundary temperature, an interior boundary, baroclinic instability associated with two boundaries, and also diabatic heating. In each case, insights yielded by the new perspective are sought into the dynamical behaviour, especially that related to vertical motion. Copyright © 2003 Royal Meteorological Society

Empirical evidence for a nonlinear effect of galactic cosmic rays on clouds

Galactic cosmic ray (GCR) changes have been suggested to affect weather and climate, and new evidence is presented here directly linking GCRs with clouds. Clouds increase the diffuse solar radiation, measured continuously at UK surface meteorological sites since 1947. The ratio of diffuse to total solar radiation-the diffuse fraction, (DF)-is used to infer cloud, and is compared with the daily mean neutron count rate measured at Climax; Colorado from 1951-2000, which provides a globally representative indicator of cosmic rays. Across the UK, oil days of high cosmic ray flux (above 3600 X 10(2) neutron counts h(-1), which occur 87% of the time on average) compared with low cosmic ray flux, (i) the chance of an overcast day increases by (19 +/- 4)%; and (ii) the diffuse fraction increases by (2 +/- 0.3)%. During sudden transient reductions in cosmic rays (e.g. Forbush events), simultaneous decreases occur in the diffuse fraction. The diffuse radiation changes are; therefore; unambiguously due to cosmic rays. Although the statistically significant nonlinear cosmic ray effect is small, it will have a considerably larger aggregate effect on longer timescale (e.g. centennial) climate variations when day-to-day variability averages out.