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.

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

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.

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective.

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.

Virtual landmarking for locality aware peer IDs

In Peer-to-Peer (P2P) networks, it is often desirable to assign node IDs which preserve locality relationships in the underlying topology. Node locality can be embedded into node IDs by utilizing a one dimensional mapping by a Hilbert space filling curve on a vector of network distances from each node to a subset of reference landmark nodes within the network. However this approach is fundamentally limited because while robustness and accuracy might be expected to improve with the number of landmarks, the effectiveness of 1 dimensional Hilbert Curve mapping falls for the curse of dimensionality. This work proposes an approach to solve this issue using Landmark Multidimensional Scaling (LMDS) to reduce a large set of landmarks to a smaller set of virtual landmarks. This smaller set of landmarks has been postulated to represent the intrinsic dimensionality of the network space and therefore a space filling curve applied to these virtual landmarks is expected to produce a better mapping of the node ID space. The proposed approach, the Virtual Landmarks Hilbert Curve (VLHC), is particularly suitable for decentralised systems like P2P networks. In the experimental simulations the effectiveness of the methods is measured by means of the locality preservation derived from node IDs in terms of latency to nearest neighbours. A variety of realistic network topologies are simulated and this work provides strong evidence to suggest that VLHC performs better than either Hilbert Curves or LMDS use independently of each other.

Mixed copper, silver, and gold cyanides, (MxM′1–x)CN: tailoring chain structures to influence physical properties

Binary mixed-metal variants of the one-dimensional MCN compounds (M = Cu, Ag, and Au) have been prepared and characterized using powder X-ray diffraction, vibrational spectroscopy, and total neutron diffraction. A solid solution with the AgCN structure exists in the (CuxAg1–x)CN system over the range (0 ≤ x ≤ 1). Line phases with compositions (Cu1/2Au1/2)CN, (Cu7/12Au5/12)CN, (Cu2/3Au1/3)CN, and (Ag1/2Au1/2)CN, all of which have the AuCN structure, are found in the gold-containing systems. Infrared and Raman spectroscopies show that complete ordering of the type [M–C≡N–M′–N≡C−]n occurs only in (Cu1/2Au1/2)CN and (Ag1/2Au1/2)CN. The sense of the cyanide bonding was determined by total neutron diffraction to be [Ag–NC–Au–CN−]n in (Ag1/2Au1/2)CN and [Cu–NC–Au–CN−]n in (Cu1/2Au1/2)CN. In contrast, in (Cu0.50Ag0.50)CN, metal ordering is incomplete, and strict alternation of metals does not occur. However, there is a distinct preference (85%) for the N end of the cyanide ligand to be bonded to copper and for Ag–CN–Cu links to predominate. Contrary to expectation, aurophilic bonding does not appear to be the controlling factor which leads to (Cu1/2Au1/2)CN and (Ag1/2Au1/2)CN adopting the AuCN structure. The diffuse reflectance, photoluminescence, and 1-D negative thermal expansion (NTE) behaviors of all three systems are reported and compared with those of the parent cyanide compounds. The photophysical properties are strongly influenced both by the composition of the individual chains and by how such chains pack together. The NTE behavior is also controlled by structure type: the gold-containing mixed-metal cyanides with the AuCN structure show the smallest contraction along the chain length on heating.

A model of the three-dimensional evolution of Arctic melt ponds on first-year and multiyear sea ice

During winter the ocean surface in polar regions freezes over to form sea ice. In the summer the upper layers of sea ice and snow melts producing meltwater that accumulates in Arctic melt ponds on the surface of sea ice. An accurate estimate of the fraction of the sea ice surface covered in melt ponds is essential for a realistic estimate of the albedo for global climate models. We present a melt-pond–sea-ice model that simulates the three-dimensional evolution of melt ponds on an Arctic sea ice surface. The advancements of this model compared to previous models are the inclusion of snow topography; meltwater transport rates are calculated from hydraulic gradients and ice permeability; and the incorporation of a detailed one-dimensional, thermodynamic radiative balance. Results of model runs simulating first-year and multiyear sea ice are presented. Model results show good agreement with observations, with duration of pond coverage, pond area, and ice ablation comparing well for both the first-year ice and multiyear ice cases. We investigate the sensitivity of the melt pond cover to changes in ice topography, snow topography, and vertical ice permeability. Snow was found to have an important impact mainly at the start of the melt season, whereas initial ice topography strongly controlled pond size and pond fraction throughout the melt season. A reduction in ice permeability allowed surface flooding of relatively flat, first-year ice but had little impact on the pond coverage of rougher, multiyear ice. We discuss our results, including model shortcomings and areas of experimental uncertainty.

Estimating interchannel observation-error correlations for IASI radiance data in the Met Office system

The optimal utilisation of hyper-spectral satellite observations in numerical weather prediction is often inhibited by incorrectly assuming independent interchannel observation errors. However, in order to represent these observation-error covariance structures, an accurate knowledge of the true variances and correlations is needed. This structure is likely to vary with observation type and assimilation system. The work in this article presents the initial results for the estimation of IASI interchannel observation-error correlations when the data are processed in the Met Office one-dimensional (1D-Var) and four-dimensional (4D-Var) variational assimilation systems. The method used to calculate the observation errors is a post-analysis diagnostic which utilises the background and analysis departures from the two systems. The results show significant differences in the source and structure of the observation errors when processed in the two different assimilation systems, but also highlight some common features. When the observations are processed in 1D-Var, the diagnosed error variances are approximately half the size of the error variances used in the current operational system and are very close in size to the instrument noise, suggesting that this is the main source of error. The errors contain no consistent correlations, with the exception of a handful of spectrally close channels. When the observations are processed in 4D-Var, we again find that the observation errors are being overestimated operationally, but the overestimation is significantly larger for many channels. In contrast to 1D-Var, the diagnosed error variances are often larger than the instrument noise in 4D-Var. It is postulated that horizontal errors of representation, not seen in 1D-Var, are a significant contributor to the overall error here. Finally, observation errors diagnosed from 4D-Var are found to contain strong, consistent correlation structures for channels sensitive to water vapour and surface properties.

A three-dimensional open-framework indium selenide: [C7H10N][In9Se14]

An open-framework indium selenide, [C7H10N][In9Se14], has been prepared under solvothermal conditions in the presence of 3,5-dimethylpyridine, and characterized by single crystal diffraction, thermogravimetry, elemental analysis, FTIR spectroscopy and UV-Vis diffuse reflectance. The crystal structure of [C7H10N][In9Se14] contains an unusual building unit, in which corner-linked and edge-linked InSe45- tetrahedra coexist. The presence of one-dimensional circular channels, of ca. 6 Å diameter, results in approximately 25% of solvent accessible void space.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

The equivalent-weights particle filter in a high-dimensional system

In general, particle filters need large numbers of model runs in order to avoid filter degeneracy in high-dimensional systems. The recently proposed, fully nonlinear equivalent-weights particle filter overcomes this requirement by replacing the standard model transition density with two different proposal transition densities. The first proposal density is used to relax all particles towards the high-probability regions of state space as defined by the observations. The crucial second proposal density is then used to ensure that the majority of particles have equivalent weights at observation time. Here, the performance of the scheme in a high, 65 500 dimensional, simplified ocean model is explored. The success of the equivalent-weights particle filter in matching the true model state is shown using the mean of just 32 particles in twin experiments. It is of particular significance that this remains true even as the number and spatial variability of the observations are changed. The results from rank histograms are less easy to interpret and can be influenced considerably by the parameter values used. This article also explores the sensitivity of the performance of the scheme to the chosen parameter values and the effect of using different model error parameters in the truth compared with the ensemble model runs.

Analysis of convectively-generated gravity waves in mesoscale model simulations and wind profiler observations

The characteristics of convectively-generated gravity waves during an episode of deep convection near the coast of Wales are examined in both high resolution mesoscale simulations [with the (UK) Met Oce Unified Model] and in observations from a Mesosphere-Stratosphere-Troposphere (MST) wind profiling Doppler radar. Deep convection reached the tropopause and generated vertically propagating, high frequency waves in the lower stratosphere that produced vertical velocity perturbations O(1 m/s). Wavelet analysis is applied in order to determine the characteristic periods and wavelengths of the waves. In both the simulations and observations, the wavelet spectra contain several distinct preferred scales indicated by multiple spectral peaks. The peaks are most pronounced in the horizontal spectra at several wavelengths less than 50 km. Although these peaks are most clear and of largest amplitude in the highest resolution simulations (with 1 km horizontal grid length), they are also evident in coarser simulations (with 4 km horizontal grid length). Peaks also exist in the vertical and temporal spectra (between approximately 2.5 and 4.5 km, and 10 to 30 minutes, respectively) with good agreement between simulation and observation. Two-dimensional (wavenumber-frequency) spectra demonstrate that each of the selected horizontal scales contains peaks at each of preferred temporal scales revealed by the one- dimensional spectra alone.

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.

An adaptive finite element water column model using the Mellor-Yamada level 2.5 turbulence closure scheme

A one-dimensional water column model using the Mellor and Yamada level 2.5 parameterization of vertical turbulent fluxes is presented. The model equations are discretized with a mixed finite element scheme. Details of the finite element discrete equations are given and adaptive mesh refinement strategies are presented. The refinement criterion is an "a posteriori" error estimator based on stratification, shear and distance to surface. The model performances are assessed by studying the stress driven penetration of a turbulent layer into a stratified fluid. This example illustrates the ability of the presented model to follow some internal structures of the flow and paves the way for truly generalized vertical coordinates. (c) 2005 Elsevier Ltd. All rights reserved.