A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Stratosphere-troposphere transport in a numerical simulation of midlatitude convection

The transport of stratospheric air deep into the troposphere via convection is investigated numerically using the UK Met Office Unified Model. A convective system that formed on 27 June 2004 near southeast England, in the vicinity an upper level potential vorticity anomaly and a lowered tropopause, provides the basis for analysis. Transport is diagnosed using a stratospheric tracer that can either be passed through or withheld from the model’s convective parameterization scheme. Three simulations are performed at increasingly finer resolutions, with horizontal grid lengths of 12, 4, and 1 km. In the 12 and 4 km simulations, tracer is transported deeply into the troposphere by the parameterized convection. In the 1 km simulation, for which the convective parameterization is disengaged, deep transport is still accomplished but with a much smaller magnitude. However, the 1 km simulation resolves stirring along the tropopause that does not exist in the coarser simulations. In all three simulations, the concentration of the deeply transported tracer is small, three orders of magnitude less than that of the shallow transport near the tropopause, most likely because of the efficient dilution of parcels in the lower troposphere.

A multiwave range test for obstacle reconstructions with unknown physical properties

We develop a new multiwave version of the range test for shape reconstruction in inverse scattering theory. The range test [R. Potthast, et al., A ‘range test’ for determining scatterers with unknown physical properties, Inverse Problems 19(3) (2003) 533–547] has originally been proposed to obtain knowledge about an unknown scatterer when the far field pattern for only one plane wave is given. Here, we extend the method to the case of multiple waves and show that the full shape of the unknown scatterer can be reconstructed. We further will clarify the relation between the range test methods, the potential method [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Inverse Problems (Oberwolfach, 1986), Internationale Schriftenreihe zur Numerischen Mathematik, vol. 77, Birkhäuser, Basel, 1986, pp. 93–102] and the singular sources method [R. Potthast, Point sources and multipoles in inverse scattering theory, Habilitation Thesis, Göttingen, 1999]. In particular, we propose a new version of the Kirsch–Kress method using the range test and a new approach to the singular sources method based on the range test and potential method. Numerical examples of reconstructions for all four methods are provided.

A new Rossby Wave-breaking interpretation of the North Atlantic Oscillation

This paper proposes the hypothesis that the low-frequency variability of the North Atlantic Oscillation (NAO) arises as a result of variations in the occurrence of upper-level Rossby wave–breaking events over the North Atlantic. These events lead to synoptic situations similar to midlatitude blocking that are referred to as high-latitude blocking episodes. A positive NAO is envisaged as being a description of periods in which these episodes are infrequent and can be considered as a basic, unblocked situation. A negative NAO is a description of periods in which episodes occur frequently. A similar, but weaker, relationship exists between wave breaking over the Pacific and the west Pacific pattern. Evidence is given to support this hypothesis by using a two-dimensional potential-vorticity-based index to identify wave breaking at various latitudes. This is applied to Northern Hemisphere winter data from the 40-yr ECMWF Re-Analysis (ERA-40), and the events identified are then related to the NAO. Certain dynamical precursors are identified that appear to increase the likelihood of wave breaking. These suggest mechanisms by which variability in the tropical Pacific, and in the stratosphere, could affect the NAO.