An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.

The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis

It is shown that Bretherton's view of baroclinic instability as the interaction of two counter-propagating Rossby waves (CRWs) can be extended to a general zonal flow and to a general dynamical system based on material conservation of potential vorticity (PV). The two CRWs have zero tilt with both altitude and latitude and are constructed from a pair of growing and decaying normal modes. One CRW has generally large amplitude in regions of positive meridional PV gradient and propagates westwards relative to the flow in such regions. Conversely, the other CRW has large amplitude in regions of negative PV gradient and propagates eastward relative to the zonal flow there. Two methods of construction are described. In the first, more heuristic, method a ‘home-base’ is chosen for each CRW and the other CRW is defined to have zero PV there. Consideration of the PV equation at the two home-bases gives ‘CRW equations’ quantifying the evolution of the amplitudes and phases of both CRWs. They involve only three coefficients describing the mutual interaction of the waves and their self-propagation speeds. These coefficients relate to PV anomalies formed by meridional fluid displacements and the wind induced by these anomalies at the home-bases. In the second method, the CRWs are defined by orthogonality constraints with respect to wave activity and energy growth, avoiding the subjective choice of home-bases. Using these constraints, the same form of CRW equations are obtained from global integrals of the PV equation, but the three coefficients are global integrals that are not so readily described by ‘PV-thinking’ arguments. Each CRW could not continue to exist alone, but together they can describe the time development of any flow whose initial conditions can be described by the pair of growing and decaying normal modes, including the possibility of a super-modal growth rate for a short period. A phase-locking configuration (and normal-mode growth) is possible only if the PV gradient takes opposite signs and the mean zonal wind and the PV gradient are positively correlated in the two distinct regions where the wave activity of each CRW is concentrated. These are easily interpreted local versions of the integral conditions for instability given by Charney and Stern and by Fjørtoft.

Convectively coupled equatorial waves: A new methodology for identifying wave structures in observational data

Convectively coupled equatorial waves are fundamental components of the interaction between the physics and dynamics of the tropical atmosphere. A new methodology, which isolates individual equatorial wave modes, has been developed and applied to observational data. The methodology assumes that the horizontal structures given by equatorial wave theory can be used to project upper- and lower-tropospheric data onto equatorial wave modes. The dynamical fields are first separated into eastward- and westward-moving components with a specified domain of frequency–zonal wavenumber. Each of the components for each field is then projected onto the different equatorial modes using the y structures of these modes given by the theory. The latitudinal scale yo of the modes is predetermined by data to fit the equatorial trapping in a suitable latitude belt y = ±Y. The extent to which the different dynamical fields are consistent with one another in their depiction of each equatorial wave structure determines the confidence in the reality of that structure. Comparison of the analyzed modes with the eastward- and westward-moving components in the convection field enables the identification of the dynamical structure and nature of convectively coupled equatorial waves. In a case study, the methodology is applied to two independent data sources, ECMWF Reanalysis and satellite-observed window brightness temperature (Tb) data for the summer of 1992. Various convectively coupled equatorial Kelvin, mixed Rossby–gravity, and Rossby waves have been detected. The results indicate a robust consistency between the two independent data sources. Different vertical structures for different wave modes and a significant Doppler shifting effect of the background zonal winds on wave structures are found and discussed. It is found that in addition to low-level convergence, anomalous fluxes induced by strong equatorial zonal winds associated with equatorial waves are important for inducing equatorial convection. There is evidence that equatorial convection associated with Rossby waves leads to a change in structure involving a horizontal structure similar to that of a Kelvin wave moving westward with it. The vertical structure may also be radically changed. The analysis method should make a very powerful diagnostic tool for investigating convectively coupled equatorial waves and the interaction of equatorial dynamics and physics in the real atmosphere. The results from application of the analysis method for a reanalysis dataset should provide a benchmark against which model studies can be compared.

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.

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.

Tropospheric planetary wave dynamics and mixture modeling: Two preferred regimes and a regime shift

Investigation of preferred structures of planetary wave dynamics is addressed using multivariate Gaussian mixture models. The number of components in the mixture is obtained using order statistics of the mixing proportions, hence avoiding previous difficulties related to sample sizes and independence issues. The method is first applied to a few low-order stochastic dynamical systems and data from a general circulation model. The method is next applied to winter daily 500-hPa heights from 1949 to 2003 over the Northern Hemisphere. A spatial clustering algorithm is first applied to the leading two principal components (PCs) and shows significant clustering. The clustering is particularly robust for the first half of the record and less for the second half. The mixture model is then used to identify the clusters. Two highly significant extratropical planetary-scale preferred structures are obtained within the first two to four EOF state space. The first pattern shows a Pacific-North American (PNA) pattern and a negative North Atlantic Oscillation (NAO), and the second pattern is nearly opposite to the first one. It is also observed that some subspaces show multivariate Gaussianity, compatible with linearity, whereas others show multivariate non-Gaussianity. The same analysis is also applied to two subperiods, before and after 1978, and shows a similar regime behavior, with a slight stronger support for the first subperiod. In addition a significant regime shift is also observed between the two periods as well as a change in the shape of the distribution. The patterns associated with the regime shifts reflect essentially a PNA pattern and an NAO pattern consistent with the observed global warming effect on climate and the observed shift in sea surface temperature around the mid-1970s.

A comparative method to evaluate and validate stochastic parametrizations

There is a growing interest in using stochastic parametrizations in numerical weather and climate prediction models. Previously, Palmer (2001) outlined the issues that give rise to the need for a stochastic parametrization and the forms such a parametrization could take. In this article a method is presented that uses a comparison between a standard-resolution version and a high-resolution version of the same model to gain information relevant for a stochastic parametrization in that model. A correction term that could be used in a stochastic parametrization is derived from the thermodynamic equations of both models. The origin of the components of this term is discussed. It is found that the component related to unresolved wave-wave interactions is important and can act to compensate for large parametrized tendencies. The correction term is not proportional to the parametrized tendency. Finally, it is explained how the correction term could be used to give information about the shape of the random distribution to be used in a stochastic parametrization. Copyright © 2009 Royal Meteorological Society

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are Investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode 'mild-slope' approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Equatorial waves in the lower stratosphere. I: A novel detection method

Wavenumber-frequency spectral analysis and linear wave theory are combined in a novel method to quantitatively estimate equatorial wave activity in the tropical lower stratosphere. The method requires temperature and velocity observations that are regularly spaced in latitude, longitude and time; it is therefore applied to the ECMWF 15-year re-analysis dataset (ERA-15). Signals consistent with idealized Kelvin and Rossby-gravity waves are found at wavenumbers and frequencies in agreement with previous studies. When averaged over 1981-93, the Kelvin wave explains approximately 1 K-2 of temperature variance on the equator at 100 hPa, while the Rossby-gravity wave explains approximately 1 m(2)s(-2) of meridional wind variance. Some inertio-gravity wave and equatorial Rossby wave signals are also found; however the resolution of ERA-15 is not sufficient for the method to provide an accurate climatology of waves with high meridional structure.

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.

Wave functions for the methane molecule

If the potential field due to the nuclei in the methane molecule is expanded in terms of a set of spherical harmonics about the carbon nucleus, only the terms involving s, f, and higher harmonic functions differ from zero in the equilibrium configuration. Wave functions have been calculated for the equilibrium configuration, first including only the spherically symmetric s term in the potential, and secondly including both the s and the f terms. In the first calculation the complete Hartree-Fock S.C.F. wave functions were determined; in the second calculation a variation method was used to determine the best form of the wave function involving f harmonics. The resulting wave functions and electron density functions are presented and discussed