The counter-propagating Rossby-wave perspective on baroclinic instability. Part IV: Nonlinear life cycles

Pairs of counter-propagating Rossby waves (CRWs) can be used to describe baroclinic instability in linearized primitive-equation dynamics, employing simple propagation and interaction mechanisms at only two locations in the meridional plane—the CRW ‘home-bases’. Here, it is shown how some CRW properties are remarkably robust as a growing baroclinic wave develops nonlinearly. For example, the phase difference between upper-level and lower-level waves in potential-vorticity contours, defined initially at the home-bases of the CRWs, remains almost constant throughout baroclinic wave life cycles, despite the occurrence of frontogenesis and Rossby-wave breaking. As the lower wave saturates nonlinearly the whole baroclinic wave changes phase speed from that of the normal mode to that of the self-induced phase speed of the upper CRW. On zonal jets without surface meridional shear, this must always act to slow the baroclinic wave. The direction of wave breaking when a basic state has surface meridional shear can be anticipated because the displacement structures of CRWs tend to be coherent along surfaces of constant basic-state angular velocity, U. This results in up-gradient horizontal momentum fluxes for baroclinically growing disturbances. The momentum flux acts to shift the jet meridionally in the direction of the increasing surface U, so that the upper CRW breaks in the same direction as occurred at low levels

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.

Synthesis and characterization of thermally stable second-order nonlinear optical side-chain polyimides containing thiazole and benzothiazole push-pull chromophores

Push-pull nonlinear optical (NLO) chromophores containing thiazole and benzothiazole acceptors were synthesized and characterized. Using these chromophores a series of second-order NLO polyimides were Successfully prepared from 4,4'-(hexafluoroisopropylidene) diphthalic anhydride (6FDA), pyromellitic dianhydride (PMDA) and 3,3'4,4'-benzophenone tetracarboxylic dianhydride (BTDA) by a standard condensation polymerization technique. These polyimides exhibit high glass transition temperatures ranging from 160 to 188 degrees C. UV-vis spectrum of polyimide exhibited a slight blue shift and decreases in absorption due to birefringence. From the order parameters, it was found that chromophores were aligned effectively. Using in situ poling and temperature ramping technique, the optical temperatures for corona poling were obtained. It was found that the optimal temperatures of polyimides approach their glass transition temperatures. These polyimides demonstrate relatively large d(33) values range between 35.15 and 45.20 pm/V at 532 nm. (C) 2008 Elsevier B.V. All rights reserved.

Stochastic resonance in a nonlinear model of a rotating, stratified shear flow, with a simple stochastic inertia-gravity wave parameterization

We report on a numerical study of the impact of short, fast inertia-gravity waves on the large-scale, slowly-evolving flow with which they co-exist. A nonlinear quasi-geostrophic numerical model of a stratified shear flow is used to simulate, at reasonably high resolution, the evolution of a large-scale mode which grows due to baroclinic instability and equilibrates at finite amplitude. Ageostrophic inertia-gravity modes are filtered out of the model by construction, but their effects on the balanced flow are incorporated using a simple stochastic parameterization of the potential vorticity anomalies which they induce. The model simulates a rotating, two-layer annulus laboratory experiment, in which we recently observed systematic inertia-gravity wave generation by an evolving, large-scale flow. We find that the impact of the small-amplitude stochastic contribution to the potential vorticity tendency, on the model balanced flow, is generally small, as expected. In certain circumstances, however, the parameterized fast waves can exert a dominant influence. In a flow which is baroclinically-unstable to a range of zonal wavenumbers, and in which there is a close match between the growth rates of the multiple modes, the stochastic waves can strongly affect wavenumber selection. This is illustrated by a flow in which the parameterized fast modes dramatically re-partition the probability-density function for equilibrated large-scale zonal wavenumber. In a second case study, the stochastic perturbations are shown to force spontaneous wavenumber transitions in the large-scale flow, which do not occur in their absence. These phenomena are due to a stochastic resonance effect. They add to the evidence that deterministic parameterizations in general circulation models, of subgrid-scale processes such as gravity wave drag, cannot always adequately capture the full details of the nonlinear interaction.

Quantifying uncertainty in critical loads: (A) literature review

Critical loads are the basis for policies controlling emissions of acidic substances in Europe. The implementation of these policies involves large expenditures, and it is reasonable for policymakers to ask what degree of certainty can be attached to the underlying critical load and exceedance estimates. This paper is a literature review of studies which attempt to estimate the uncertainty attached to critical loads. Critical load models and uncertainty analysis are briefly outlined. Most studies have used Monte Carlo analysis of some form to investigate the propagation of uncertainties in the definition of the input parameters through to uncertainties in critical loads. Though the input parameters are often poorly known, the critical load uncertainties are typically surprisingly small because of a "compensation of errors" mechanism. These results depend on the quality of the uncertainty estimates of the input parameters, and a "pedigree" classification for these is proposed. Sensitivity analysis shows that some input parameters are more important in influencing critical load uncertainty than others, but there have not been enough studies to form a general picture. Methods used for dealing with spatial variation are briefly discussed. Application of alternative models to the same site or modifications of existing models can lead to widely differing critical loads, indicating that research into the underlying science needs to continue.

Quantifying uncertainty in critical loads: (B) acidity mass balance critical loads on a sensitive site

This paper reports an uncertainty analysis of critical loads for acid deposition for a site in southern England, using the Steady State Mass Balance Model. The uncertainty bounds, distribution type and correlation structure for each of the 18 input parameters was considered explicitly, and overall uncertainty estimated by Monte Carlo methods. Estimates of deposition uncertainty were made from measured data and an atmospheric dispersion model, and hence the uncertainty in exceedance could also be calculated. The uncertainties of the calculated critical loads were generally much lower than those of the input parameters due to a "compensation of errors" mechanism - coefficients of variation ranged from 13% for CLmaxN to 37% for CL(A). With 1990 deposition, the probability that the critical load was exceeded was > 0.99; to reduce this probability to 0.50, a 63% reduction in deposition is required; to 0.05, an 82% reduction. With 1997 deposition, which was lower than that in 1990, exceedance probabilities declined and uncertainties in exceedance narrowed as deposition uncertainty had less effect. The parameters contributing most to the uncertainty in critical loads were weathering rates, base cation uptake rates, and choice of critical chemical value, indicating possible research priorities. However, the different critical load parameters were to some extent sensitive to different input parameters. The application of such probabilistic results to environmental regulation is discussed.

Estimating uncertainty in terrestrial critical loads and their exceedances at four sites in the UK

Critical loads are the basis for policies controlling emissions of acidic substances in Europe and elsewhere. They are assessed by several elaborate and ingenious models, each of which requires many parameters, and have to be applied on a spatially-distributed basis. Often the values of the input parameters are poorly known, calling into question the validity of the calculated critical loads. This paper attempts to quantify the uncertainty in the critical loads due to this "parameter uncertainty", using examples from the UK. Models used for calculating critical loads for deposition of acidity and nitrogen in forest and heathland ecosystems were tested at four contrasting sites. Uncertainty was assessed by Monte Carlo methods. Each input parameter or variable was assigned a value, range and distribution in an objective a fashion as possible. Each model was run 5000 times at each site using parameters sampled from these input distributions. Output distributions of various critical load parameters were calculated. The results were surprising. Confidence limits of the calculated critical loads were typically considerably narrower than those of most of the input parameters. This may be due to a "compensation of errors" mechanism. The range of possible critical load values at a given site is however rather wide, and the tails of the distributions are typically long. The deposition reductions required for a high level of confidence that the critical load is not exceeded are thus likely to be large. The implication for pollutant regulation is that requiring a high probability of non-exceedance is likely to carry high costs. The relative contribution of the input variables to critical load uncertainty varied from site to site: any input variable could be important, and thus it was not possible to identify variables as likely targets for research into narrowing uncertainties. Sites where a number of good measurements of input parameters were available had lower uncertainties, so use of in situ measurement could be a valuable way of reducing critical load uncertainty at particularly valuable or disputed sites. From a restricted number of samples, uncertainties in heathland critical loads appear comparable to those of coniferous forest, and nutrient nitrogen critical loads to those of acidity. It was important to include correlations between input variables in the Monte Carlo analysis, but choice of statistical distribution type was of lesser importance. Overall, the analysis provided objective support for the continued use of critical loads in policy development. (c) 2007 Elsevier B.V. All rights reserved.

Linear and nonlinear generalized Fourier transforms

This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ?.( b(u)? 2u), where generically b(u) := |u|? for any given ? ? (0,?). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ? 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.