Relationships between lightning and properties of convective cloud clusters

Satellite observations of convective system properties and lightning flash rate are used to investigate the ability of potential lightning parameterizations to capture both the dominant land-ocean contrast in lightning occurrence and regional differences between Africa, the Amazon and the islands of the maritime continent. As found in previous studies, the radar storm height is tightly correlated with the lightning flash rate. A roughly second order power-law fit to the mean radar echo top height above the 0C isotherm is shown to capture both regional and land-ocean contrasts in lightning occurrence and flash rate using a single set of parameters. Recent developments should soon make it possible to implement a parameterization of this kind in global models. Parameterizations based on cloud top height, convective rain rate and convective rain fraction all require the use of separate fits over land and ocean and fail to capture observed differences between continental regions.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

Molecular structures of trimethylchlorogermane, (CH3)(3)GeCl, and trimethylbromogermane, (CH3)(3)GeBr, obtained by gas-phase electron diffraction and theoretical calculations

The structures of trimethylchlorogermane ((CH3)(3)GeCl) and trimethylbromogermane ((CH3)(3)GeBr) have been determined by gas-phase electron diffraction (GED), augmented by the results from ab initio calculations employing second-order Moller-Plesset (MP2) level of theory and the 6-311+G(d) basis set. All the electrons were included in the correlation calculation. The results from the ab initio calculations indicated that these molecules have C-3v symmetry, and models with this symmetry were used in the electron diffraction analysis. The results for the principal distances (r(g)) and angles (angle(alpha)) from the combined GED/ab initio study of trimethylchlorogermane (with estimated 2sigma uncertainties) are: r(Ge-C) = 1.950(4) Angstrom, r(Ge-Cl) = 2.173(4) Angstrom, r(C-H) = 1.090(9) Angstrom, angleCGeC = 112.7(7)degrees, angleCGeCl = 106.0(8)degrees, angleGeCH = 107.8(12)degrees. The results for the principal distances (r(g)) and angles (angle(alpha)) from the combined GED/ab initio study of trimethylbromogermane (with estimated 2sigma uncertainties) are: r(Ge-C) = 1.952(7) Angstrom, r(Ge-Br) = 2.325(4) Angstrom, r(C-H) = 1. 140(28) Angstrom, angleCGeC = 114.2(11)degrees, angleCGeBr = 104.2(13)degrees, angleGeCH 106.9(43)degrees. Local C-3v symmetry and staggered conformation were assumed for the methyl groups.

A comprehensive investigation of the synthesis of prebiotic galactooligosaccharides by whole cells of Bifidobacterium bifidum NCIMB 41171

The synthesis of galactooligosaccharides (GOS) by whole cells of Bifidobacterium bifidum NCIMB 41171 was investigated by developing a set of mathematical models. These were second order polynomial equations, which described responses related to the production of GOS constituents, the selectivity of lactose conversion into GOS, and the relative composition of the produced GOS mixture, as a function of the amount of biocatalyst, temperature, initial lactose concentration, and time. The synthesis reactions were followed for up to 36 h. Samples were withdrawn every 4 h, tested for β-galactosidase activity, and analysed for their carbohydrate content. GOS synthesis was well explained by the models, which were all significant (P < 0.001). The GOS yield increased as temperature increased from 40 °C to 60 °C, as transgalactosylation became more pronounced compared to hydrolysis. The relative composition of GOS produced changed significantly with the initial lactose concentration (P < 0.001); higher ratios of tri-, tetra-, and penta-galactooligosaccharides to transgalactosylated disaccharides were obtained as lactose concentration increased. Time was a critical factor, as a balanced state between GOS synthesis and hydrolysis was roughly attained in most cases between 12 and 20 h, and was followed by more pronounced GOS hydrolysis than synthesis.

Fractional order modeling of large three-dimensional RC networks

An incidence matrix analysis is used to model a three-dimensional network consisting of resistive and capacitive elements distributed across several interconnected layers. A systematic methodology for deriving a descriptor representation of the network with random allocation of the resistors and capacitors is proposed. Using a transformation of the descriptor representation into standard state-space form, amplitude and phase admittance responses of three-dimensional random RC networks are obtained. Such networks display an emergent behavior with a characteristic Jonscher-like response over a wide range of frequencies. A model approximation study of these networks is performed to infer the admittance response using integral and fractional order models. It was found that a fractional order model with only seven parameters can accurately describe the responses of networks composed of more than 70 nodes and 200 branches with 100 resistors and 100 capacitors. The proposed analysis can be used to model charge migration in amorphous materials, which may be associated to specific macroscopic or microscopic scale fractal geometrical structures in composites displaying a viscoelastic electromechanical response, as well as to model the collective responses of processes governed by random events described using statistical mechanics.

Time-resolved gas-phase kinetic, quantum chemical, and RRKM studies of reactions of silylene with alcohols

Time-resolved kinetic studies of silylene, SiH2, generated by laser flash photolysis of 1-silacyclopent-3-ene and phenylsilane, have been carried out to obtain rate constants for its bimolecular reactions with methanol, ethanol, 1-propanol, 1-butanol and 2-methyl-1-butanol. The reactions were studied in the gas phase over the pressure range 1-100 Torr in SF6 bath gas, at room temperature. In the study with methanol several buffer gases were used. All five reactions showed pressure dependences characteristic of third body assisted association reactions. The rate constant pressure dependences were modelled using RRKM theory, based on Eo values of the association complexes obtained by ab initio calculation (G3 level). Transition state models were adjusted to fit experimental fall-off curves and extrapolated to obtain k∞ values in the range 1.9 to 4.5 × 10-10 cm3 molecule-1 s-1. These numbers, corresponding to the true bimolecular rate constants, indicate efficiencies of between 16 and 67% of the collision rates for these reactions. In the reaction of SiH2 + MeOH there is a small kinetic component to the rate which is second order in MeOH (at low total pressures). This suggests an additional catalysed reaction pathway, which is supported by the ab initio calculations. These calculations have been used to define specific MeOH-for-H2O substitution effects on this catalytic pathway. Where possible our experimental and theoretical results are compared with those of previous studies.

Free gravity waves and balanced dynamics

It is shown how a renormalization technique, which is a variant of classical Krylov–Bogolyubov–Mitropol’skii averaging, can be used to obtain slow evolution equations for the vortical and inertia–gravity wave components of the dynamics in a rotating flow. The evolution equations for each component are obtained to second order in the Rossby number, and the nature of the coupling between the two is analyzed carefully. It is also shown how classical balance models such as quasigeostrophic dynamics and its second-order extension appear naturally as a special case of this renormalized system, thereby providing a rigorous basis for the slaving approach where only the fast variables are expanded. It is well known that these balance models correspond to a hypothetical slow manifold of the parent system; the method herein allows the determination of the dynamics in the neighborhood of such solutions. As a concrete illustration, a simple weak-wave model is used, although the method readily applies to more complex rotating fluid models such as the shallow-water, Boussinesq, primitive, and 3D Euler equations.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 2. pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy. A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states. The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

Reduced-order modeling of light transport in tissue for real-time monitoring of brain hemodynamics using diffuse optical tomography

This paper proposes a new reconstruction method for diffuse optical tomography using reduced-order models of light transport in tissue. The models, which directly map optical tissue parameters to optical flux measurements at the detector locations, are derived based on data generated by numerical simulation of a reference model. The reconstruction algorithm based on the reduced-order models is a few orders of magnitude faster than the one based on a finite element approximation on a fine mesh incorporating a priori anatomical information acquired by magnetic resonance imaging. We demonstrate the accuracy and speed of the approach using a phantom experiment and through numerical simulation of brain activation in a rat's head. The applicability of the approach for real-time monitoring of brain hemodynamics is demonstrated through a hypercapnic experiment. We show that our results agree with the expected physiological changes and with results of a similar experimental study. However, by using our approach, a three-dimensional tomographic reconstruction can be performed in ∼3  s per time point instead of the 1 to 2 h it takes when using the conventional finite element modeling approach

Satellite-supported flood forecasting in river networks: a real case study

Satellite-based (e.g., Synthetic Aperture Radar [SAR]) water level observations (WLOs) of the floodplain can be sequentially assimilated into a hydrodynamic model to decrease forecast uncertainty. This has the potential to keep the forecast on track, so providing an Earth Observation (EO) based flood forecast system. However, the operational applicability of such a system for floods developed over river networks requires further testing. One of the promising techniques for assimilation in this field is the family of ensemble Kalman (EnKF) filters. These filters use a limited-size ensemble representation of the forecast error covariance matrix. This representation tends to develop spurious correlations as the forecast-assimilation cycle proceeds, which is a further complication for dealing with floods in either urban areas or river junctions in rural environments. Here we evaluate the assimilation of WLOs obtained from a sequence of real SAR overpasses (the X-band COSMO-Skymed constellation) in a case study. We show that a direct application of a global Ensemble Transform Kalman Filter (ETKF) suffers from filter divergence caused by spurious correlations. However, a spatially-based filter localization provides a substantial moderation in the development of the forecast error covariance matrix, directly improving the forecast and also making it possible to further benefit from a simultaneous online inflow error estimation and correction. Additionally, we propose and evaluate a novel along-network metric for filter localization, which is physically-meaningful for the flood over a network problem. Using this metric, we further evaluate the simultaneous estimation of channel friction and spatially-variable channel bathymetry, for which the filter seems able to converge simultaneously to sensible values. Results also indicate that friction is a second order effect in flood inundation models applied to gradually varied flow in large rivers. The study is not conclusive regarding whether in an operational situation the simultaneous estimation of friction and bathymetry helps the current forecast. Overall, the results indicate the feasibility of stand-alone EO-based operational flood forecasting.

Connections between clouds, radiation, and midlatitude dynamics: a review

We review the effects of dynamical variability on clouds and radiation in observations and models and discuss their implications for cloud feedbacks. Jet shifts produce robust meridional dipoles in upper-level clouds and longwave cloud-radiative effect (CRE), but low-level clouds, which do not simply shift with the jet, dominate the shortwave CRE. Because the effect of jet variability on CRE is relatively small, future poleward jet shifts with global warming are only a second-order contribution to the total CRE changes around the midlatitudes, suggesting a dominant role for thermodynamic effects. This implies that constraining the dynamical response is unlikely to reduce the uncertainty in extratropical cloud feedback. However, we argue that uncertainty in the cloud-radiative response does affect the atmospheric circulation response to global warming, by modulating patterns of diabatic forcing. How cloud feedbacks can affect the dynamical response to global warming is an important topic of future research.

Rossby wave propagation on potential vorticity fronts with finite width

The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here an approximate correction is derived for the case of a finite width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.

Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Structure of turbulent flow over arrays of cubical roughness

The structure of turbulent flow over large roughness consisting of regular arrays of cubical obstacles is investigated numerically under constant pressure gradient conditions. Results are analysed in terms of first- and second-order statistics, by visualization of instantaneous flow fields and by conditional averaging. The accuracy of the simulations is established by detailed comparisons of first- and second-order statistics with wind-tunnel measurements. Coherent structures in the log region are investigated. Structure angles are computed from two-point correlations, and quadrant analysis is performed to determine the relative importance of Q2 and Q4 events (ejections and sweeps) as a function of height above the roughness. Flow visualization shows the existence of low-momentum regions (LMRs) as well as vortical structures throughout the log layer. Filtering techniques are used to reveal instantaneous examples of the association of the vortices with the LMRs, and linear stochastic estimation and conditional averaging are employed to deduce their statistical properties. The conditional averaging results reveal the presence of LMRs and regions of Q2 and Q4 events that appear to be associated with hairpin-like vortices, but a quantitative correspondence between the sizes of the vortices and those of the LMRs is difficult to establish; a simple estimate of the ratio of the vortex width to the LMR width gives a value that is several times larger than the corresponding ratio over smooth walls. The shape and inclination of the vortices and their spatial organization are compared to recent findings over smooth walls. Characteristic length scales are shown to scale linearly with height in the log region. Whilst there are striking qualitative similarities with smooth walls, there are also important differences in detail regarding: (i) structure angles and sizes and their dependence on distance from the rough surface; (ii) the flow structure close to the roughness; (iii) the roles of inflows into and outflows from cavities within the roughness; (iv) larger vortices on the rough wall compared to the smooth wall; (v) the effect of the different generation mechanism at the wall in setting the scales of structures.