Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. ChandlerWilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

Solvothermal synthesis of one-dimensional chalcogenides containing group 13 elements

The solvothermal synthesis and characterisation of [C6H16N2][GaS2]2 (1), [C6H16N2][Ga2Se3(Se2)] (2), and mixed-metal phases with composition [C6H16N2][Ga2–xInxSe3(Se2)] (0 < x < 2)(3–5), is described. These materials have been characterised by single-crystal and powder X-ray diffraction, thermogravimetric analysis and UV/Vis diffuse reflectance spectroscopy. The materials contain one-dimensional anionic chains. In 1, these chains consist of edge-linked GaS4 tetrahedra, whilst in 2–5, the chains contain perselenide (Se2)2– units and comprise alternating four-membered [M2Se2] and five-membered [M2Se3] rings (where M = Ga, In). Compounds 3–5 represent the first examples of ternary mixed-metal [M2Se3(Se2)]2– chains.

Hydrothermal synthesis of [C6H16N2][In2Se3(Se2)]: a new one-dimensional indium selenide

A new organically templated indium selenide, [C6H16N2][In2Se3(Se2)], has been prepared hydrothermally from the reaction of indium, selenium and trans-1,4-diaminocyclohexane in water at 170 °C. This material was characterised by single-crystal and powder X-ray diffraction, thermogravimetric analysis, UV–vis diffuse reflectance spectroscopy, FT-IR and elemental analysis. The compound crystallises in the monoclinic space group C2/c (a=12.0221(16) Å, b=11.2498(15) Å, c=12.8470(17) Å, β=110.514(6)°). The crystal structure of [C6H16N2][In2Se3(Se2)] contains anionic chains of stoichiometry [In2Se3(Se2)]2−, which are aligned parallel to the [1 0 1] direction, and separated by diprotonated trans-1,4-diaminocyclohexane cations. The [In2Se3(Se2)]2− chains, which consist of alternating four-membered [In2Se2] and five-membered [In2Se3] rings, contain perselenide (Se2)2− units. UV–vis diffuse reflectance spectroscopy indicates that [C6H16N2][In2Se3(Se2)] has a band gap of 2.23(1) eV

Synthesis, characterisation and magnetic properties of a one-dimensional Iron(II) coordination polymer

A new iron(II) coordination polymer, [FeCl2(NC7H9)2(N2C12H12)], has been synthesized under solvothermal conditions and structurally characterized by single-crystal X-ray diffraction. This material crystallizes in the monoclinic space group C2/c, with a = 11.2850(6), b = 13.8925(7), c = 17.0988(9) Å and β = 94.300(3)º (Z = 4). The crystal structure consists of neutral zig-zag chains, in which the iron(II) ions are octahedrally coordinated. The infinite polymer chains are packed into a three-dimensional structure through C–H···Cl interactions. Magnetic susceptibility measurements reveal the existence of weak antiferromagnetic interactions between the iron(II) ions. The effective magnetic moment, μ eff = 5.33 μ B , is consistent with a high-spin iron(II) configuration.

Coupling an aerosol box model with one-dimensional flow: a tool for understanding observations of new particle formation events

Field observations of new particle formation and the subsequent particle growth are typically only possible at a fixed measurement location, and hence do not follow the temporal evolution of an air parcel in a Lagrangian sense. Standard analysis for determining formation and growth rates requires that the time-dependent formation rate and growth rate of the particles are spatially invariant; air parcel advection means that the observed temporal evolution of the particle size distribution at a fixed measurement location may not represent the true evolution if there are spatial variations in the formation and growth rates. Here we present a zero-dimensional aerosol box model coupled with one-dimensional atmospheric flow to describe the impact of advection on the evolution of simulated new particle formation events. Wind speed, particle formation rates and growth rates are input parameters that can vary as a function of time and location, using wind speed to connect location to time. The output simulates measurements at a fixed location; formation and growth rates of the particle mode can then be calculated from the simulated observations at a stationary point for different scenarios and be compared with the ‘true’ input parameters. Hence, we can investigate how spatial variations in the formation and growth rates of new particles would appear in observations of particle number size distributions at a fixed measurement site. We show that the particle size distribution and growth rate at a fixed location is dependent on the formation and growth parameters upwind, even if local conditions do not vary. We also show that different input parameters used may result in very similar simulated measurements. Erroneous interpretation of observations in terms of particle formation and growth rates, and the time span and areal extent of new particle formation, is possible if the spatial effects are not accounted for.

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.

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

An efficient numerical scheme for the two-dimensional shallow water equations using arithmetic averaging

A finite difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gas dynamics is defined, and a scheme, based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem, and incorporates the technique of operator splitting. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging. This is in contrast to usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. An extension to the two-dimensional equations with source terms is included. The scheme is applied to the one-dimensional problems of a breaking dam and reflection of a bore, and in each case the approximate solution is compared to the exact solution of ideal fluid flow. The scheme is also applied to a problem of stationary bore generation in a channel of variable cross-section. Finally, the scheme is applied to two other dam-break problems, this time in two dimensions with one having cylindrical symmetry. Each approximate solution compares well with those given by other authors.

Using Doppler radar with a simple explicit microphysics model to diagnose problems with ice sublimation depth-scales in forecast models

Several previous studies have attempted to assess the sublimation depth-scales of ice particles from clouds into clear air. Upon examining the sublimation depth-scales in the Met Office Unified Model (MetUM), it was found that the MetUM has evaporation depth-scales 2–3 times larger than radar observations. Similar results can be seen in the European Centre for Medium-Range Weather Forecasts (ECMWF), Regional Atmospheric Climate Model (RACMO) and Météo-France models. In this study, we use radar simulation (converting model variables into radar observations) and one-dimensional explicit microphysics numerical modelling to test and diagnose the cause of the deep sublimation depth-scales in the forecast model. The MetUM data and parametrization scheme are used to predict terminal velocity, which can be compared with the observed Doppler velocity. This can then be used to test the hypothesis as to why the sublimation depth-scale is too large within the MetUM. Turbulence could lead to dry air entrainment and higher evaporation rates; particle density may be wrong, particle capacitance may be too high and lead to incorrect evaporation rates or the humidity within the sublimating layer may be incorrectly represented. We show that the most likely cause of deep sublimation zones is an incorrect representation of model humidity in the layer. This is tested further by using a one-dimensional explicit microphysics model, which tests the sensitivity of ice sublimation to key atmospheric variables and is capable of including sonde and radar measurements to simulate real cases. Results suggest that the MetUM grid resolution at ice cloud altitudes is not sufficient enough to maintain the sharp drop in humidity that is observed in the sublimation zone.

Four-dimensional variational data assimilation for high resolution nested models

Four-dimensional variational data assimilation (4D-Var) is used in environmental prediction to estimate the state of a system from measurements. When 4D-Var is applied in the context of high resolution nested models, problems may arise in the representation of spatial scales longer than the domain of the model. In this paper we study how well 4D-Var is able to estimate the whole range of spatial scales present in one-way nested models. Using a model of the one-dimensional advection–diffusion equation we show that small spatial scales that are observed can be captured by a 4D-Var assimilation, but that information in the larger scales may be degraded. We propose a modification to 4D-Var which allows a better representation of these larger scales.

From a discrete to a continuum model of cell dynamics in one dimension

Multiscale modeling is emerging as one of the key challenges in mathematical biology. However, the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions. For example, at the cellular scale, how can the appropriate discrete cell-level model be identified in a given context? Additionally, how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior? In order to begin to address such questions, we consider a discrete one-dimensional cell-based model in which cells are assumed to interact via linear springs. From the discrete equations of motion, the continuous Rouse [P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)] model is obtained. This formalism readily allows the definition of a cell number density for which a nonlinear "fast" diffusion equation is derived. Excellent agreement is demonstrated between the continuum and discrete models. Subsequently, via the incorporation of cell division, we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail. In the limit of stiff springs, where cells can be considered to be incompressible, we show that cell velocity can be directly related to cell production. This assumption is frequently made in the literature but our derivation places limits on its validity. Finally, the model is compared with a model of a similar form recently derived for a different discrete cell-based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models.

Full-dimensional quantum calculations of ground-state tunneling splitting of malonaldehyde using an accurate ab initio potential energy surface

Quantum calculations of the ground vibrational state tunneling splitting of H-atom and D-atom transfer in malonaldehyde are performed on a full-dimensional ab initio potential energy surface (PES). The PES is a fit to 11 147 near basis-set-limit frozen-core CCSD(T) electronic energies. This surface properly describes the invariance of the potential with respect to all permutations of identical atoms. The saddle-point barrier for the H-atom transfer on the PES is 4.1 kcal/mol, in excellent agreement with the reported ab initio value. Model one-dimensional and "exact" full-dimensional calculations of the splitting for H- and D-atom transfer are done using this PES. The tunneling splittings in full dimensionality are calculated using the unbiased "fixed-node" diffusion Monte Carlo (DMC) method in Cartesian and saddle-point normal coordinates. The ground-state tunneling splitting is found to be 21.6 cm(-1) in Cartesian coordinates and 22.6 cm(-1) in normal coordinates, with an uncertainty of 2-3 cm(-1). This splitting is also calculated based on a model which makes use of the exact single-well zero-point energy (ZPE) obtained with the MULTIMODE code and DMC ZPE and this calculation gives a tunneling splitting of 21-22 cm(-1). The corresponding computed splittings for the D-atom transfer are 3.0, 3.1, and 2-3 cm(-1). These calculated tunneling splittings agree with each other to within less than the standard uncertainties obtained with the DMC method used, which are between 2 and 3 cm(-1), and agree well with the experimental values of 21.6 and 2.9 cm(-1) for the H and D transfer, respectively. (C) 2008 American Institute of Physics.

Prediction of supercritical flow in open channels

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow water equations in open channels. A linearised problem, analogous to that of Riemann for gas dynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids non-physical, spurious oscillations. The scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow.