Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Modelling metal cutting using modern ductile fracture mechanics: Quantitative explanations for some longstanding problems

The assumption that negligible work is involved in the formation of new surfaces in the machining of ductile metals, is re-examined in the light of both current Finite Element Method (FEM) simulations of cutting and modern ductile fracture mechanics. The work associated with separation criteria in FEM models is shown to be in the kJ/m2 range rather than the few J/m2 of the surface energy (surface tension) employed by Shaw in his pioneering study of 1954 following which consideration of surface work has been omitted from analyses of metal cutting. The much greater values of surface specific work are not surprising in terms of ductile fracture mechanics where kJ/m2 values of fracture toughness are typical of the ductile metals involved in machining studies. This paper shows that when even the simple Ernst–Merchant analysis is generalised to include significant surface work, many of the experimental observations for which traditional ‘plasticity and friction only’ analyses seem to have no quantitative explanation, are now given meaning. In particular, the primary shear plane angle φ becomes material-dependent. The experimental increase of φ up to a saturated level, as the uncut chip thickness is increased, is predicted. The positive intercepts found in plots of cutting force vs. depth of cut, and in plots of force resolved along the primary shear plane vs. area of shear plane, are shown to be measures of the specific surface work. It is demonstrated that neglect of these intercepts in cutting analyses is the reason why anomalously high values of shear yield stress are derived at those very small uncut chip thicknesses at which the so-called size effect becomes evident. The material toughness/strength ratio, combined with the depth of cut to form a non-dimensional parameter, is shown to control ductile cutting mechanics. The toughness/strength ratio of a given material will change with rate, temperature, and thermomechanical treatment and the influence of such changes, together with changes in depth of cut, on the character of machining is discussed. Strength or hardness alone is insufficient to describe machining. The failure of the Ernst–Merchant theory seems less to do with problems of uniqueness and the validity of minimum work, and more to do with the problem not being properly posed. The new analysis compares favourably and consistently with the wide body of experimental results available in the literature. Why considerable progress in the understanding of metal cutting has been achieved without reference to significant surface work is also discussed.

Modelling methodology and forecast failure

We analyse by simulation the impact of model-selection strategies (sometimes called pre-testing) on forecast performance in both constant-and non-constant-parameter processes. Restricted, unrestricted and selected models are compared when either of the first two might generate the data. We find little evidence that strategies such as general-to-specific induce significant over-fitting, or thereby cause forecast-failure rejection rates to greatly exceed nominal sizes. Parameter non-constancies put a premium on correct specification, but in general, model-selection effects appear to be relatively small, and progressive research is able to detect the mis-specifications.

High-precision measurements of the co-polar correlation coefficient: non-Gaussian errors and retrieval of the dispersion parameter µ in rainfall

The co-polar correlation coefficient (ρhv) has many applications, including hydrometeor classification, ground clutter and melting layer identification, interpretation of ice microphysics and the retrieval of rain drop size distributions (DSDs). However, we currently lack the quantitative error estimates that are necessary if these applications are to be fully exploited. Previous error estimates of ρhv rely on knowledge of the unknown "true" ρhv and implicitly assume a Gaussian probability distribution function of ρhv samples. We show that frequency distributions of ρhv estimates are in fact highly negatively skewed. A new variable: L = -log10(1 - ρhv) is defined, which does have Gaussian error statistics, and a standard deviation depending only on the number of independent radar pulses. This is verified using observations of spherical drizzle drops, allowing, for the first time, the construction of rigorous confidence intervals in estimates of ρhv. In addition, we demonstrate how the imperfect co-location of the horizontal and vertical polarisation sample volumes may be accounted for. The possibility of using L to estimate the dispersion parameter (µ) in the gamma drop size distribution is investigated. We find that including drop oscillations is essential for this application, otherwise there could be biases in retrieved µ of up to ~8. Preliminary results in rainfall are presented. In a convective rain case study, our estimates show µ to be substantially larger than 0 (an exponential DSD). In this particular rain event, rain rate would be overestimated by up to 50% if a simple exponential DSD is assumed.

Statistical inference of OH concentrations and air mass dilution rates from successive observations of non-methane hydrocarbons in single air masses

Bayesian inference has been used to determine rigorous estimates of hydroxyl radical concentrations () and air mass dilution rates (K) averaged following air masses between linked observations of nonmethane hydrocarbons (NMHCs) spanning the North Atlantic during the Intercontinental Transport and Chemical Transformation (ITCT)-Lagrangian-2K4 experiment. The Bayesian technique obtains a refined (posterior) distribution of a parameter given data related to the parameter through a model and prior beliefs about the parameter distribution. Here, the model describes hydrocarbon loss through OH reaction and mixing with a background concentration at rate K. The Lagrangian experiment provides direct observations of hydrocarbons at two time points, removing assumptions regarding composition or sources upstream of a single observation. The estimates are sharpened by using many hydrocarbons with different reactivities and accounting for their variability and measurement uncertainty. A novel technique is used to construct prior background distributions of many species, described by variation of a single parameter . This exploits the high correlation of species, related by the first principal component of many NMHC samples. The Bayesian method obtains posterior estimates of , K and following each air mass. Median values are typically between 0.5 and 2.0 × 106 molecules cm−3, but are elevated to between 2.5 and 3.5 × 106 molecules cm−3, in low-level pollution. A comparison of estimates from absolute NMHC concentrations and NMHC ratios assuming zero background (the “photochemical clock” method) shows similar distributions but reveals systematic high bias in the estimates from ratios. Estimates of K are ∼0.1 day−1 but show more sensitivity to the prior distribution assumed.

Non-commutative symbolic coding

We give a non-commutative generalization of classical symbolic coding in the presence of a synchronizing word. This is done by a scattering theoretical approach. Classically, the existence of a synchronizing word turns out to be equivalent to asymptotic completeness of the corresponding Markov process. A criterion for asymptotic completeness in general is provided by the regularity of an associated extended transition operator. Commutative and non-commutative examples are analysed.

A revised Tobit procedure for mitigating bias in the presence of non-zero censoring with an application to milk-market participation in the Ethiopian highlands

Fixed transactions costs that prohibit exchange engender bias in supply analysis due to censoring of the sample observations. The associated bias in conventional regression procedures applied to censored data and the construction of robust methods for mitigating bias have been preoccupations of applied economists since Tobin [Econometrica 26 (1958) 24]. This literature assumes that the true point of censoring in the data is zero and, when this is not the case, imparts a bias to parameter estimates of the censored regression model. We conjecture that this bias can be significant; affirm this from experiments; and suggest techniques for mitigating this bias using Bayesian procedures. The bias-mitigating procedures are based on modifications of the key step that facilitates Bayesian estimation of the censored regression model; are easy to implement; work well in both small and large samples; and lead to significantly improved inference in the censored regression model. These findings are important in light of the widespread use of the zero-censored Tobit regression and we investigate their consequences using data on milk-market participation in the Ethiopian highlands. (C) 2004 Elsevier B.V. All rights reserved.

Model selection approaches for non-linear system identification: a review

The identification of non-linear systems using only observed finite datasets has become a mature research area over the last two decades. A class of linear-in-the-parameter models with universal approximation capabilities have been intensively studied and widely used due to the availability of many linear-learning algorithms and their inherent convergence conditions. This article presents a systematic overview of basic research on model selection approaches for linear-in-the-parameter models. One of the fundamental problems in non-linear system identification is to find the minimal model with the best model generalisation performance from observational data only. The important concepts in achieving good model generalisation used in various non-linear system-identification algorithms are first reviewed, including Bayesian parameter regularisation and models selective criteria based on the cross validation and experimental design. A significant advance in machine learning has been the development of the support vector machine as a means for identifying kernel models based on the structural risk minimisation principle. The developments on the convex optimisation-based model construction algorithms including the support vector regression algorithms are outlined. Input selection algorithms and on-line system identification algorithms are also included in this review. Finally, some industrial applications of non-linear models are discussed.

Non-testability of equal weights spatial dependence

We show that any invariant test for spatial autocorrelation in a spatial error or spatial lag model with equal weights matrix has power equal to size. This result holds under the assumption of an elliptical distribution. Under Gaussianity, we also show that any test whose power is larger than its size for at least one point in the parameter space must be biased.

Simplified parameter adaptive control

A simple parameter adaptive controller design methodology is introduced in which steady-state servo tracking properties provide the major control objective. This is achieved without cancellation of process zeros and hence the underlying design can be applied to non-minimum phase systems. As with other self-tuning algorithms, the design (user specified) polynomials of the proposed algorithm define the performance capabilities of the resulting controller. However, with the appropriate definition of these polynomials, the synthesis technique can be shown to admit different adaptive control strategies, e.g. self-tuning PID and self-tuning pole-placement controllers. The algorithm can therefore be thought of as an embodiment of other self-tuning design techniques. The performances of some of the resulting controllers are illustrated using simulation examples and the on-line application to an experimental apparatus.

Production of pH-responsive microparticles by spray drying: investigation of experimental parameter effects on morphological and release properties

During spray drying, emphasis is placed on process optimisation to generate favourable particle morphological and flow properties. The effect of the initial feed solution composition on the drug release from the prepared microparticles is rarely considered. We investigated the effects of solvent composition, feed solution concentration and drug-loading on sodium salicylate, hydrocortisone and triamcinolone release from spray dried Eudragit L100 microparticles. Eudragit L100 is a pH-responsive polymer whose dissolution threshold is pH 6 so dissolution testing of the prepared microparticles at pH 5 and 1.2 illustrated non-polymer controlled burst release. Increasing the water content of the initial ethanolic feed solution significantly reduced hydrocortisone burst release at pH 5, as did reducing the feed solution concentration. These findings caution that changes in feed solution concentration or solvent composition not only affect particles’ morphological characteristics but can also negatively alter their drug release properties. This work also illustrate that drug-free microparticles can have different morphological properties to drug-loaded microparticles. Therefore, process optimisation needs to be carried out using drug-loaded systems. Depending on the physicochemical properties of the encapsulated API, drug-loading can affect the polymer solubility in the initial feed solution with consequent impact on microparticles morphological and release properties.

The importance of friction in mountain wave drag amplification by Scorer parameter resonance

A mechanism for amplification of mountain waves, and their associated drag, by parametric resonance is investigated using linear theory and numerical simulations. This mechanism, which is active when the Scorer parameter oscillates with height, was recently classified by previous authors as intrinsically nonlinear. Here it is shown that, if friction is included in the simplest possible form as a Rayleigh damping, and the solution to the Taylor-Goldstein equation is expanded in a power series of the amplitude of the Scorer parameter oscillation, linear theory can replicate the resonant amplification produced by numerical simulations with some accuracy. The drag is significantly altered by resonance in the vicinity of n/l_0 = 2, where l_0 is the unperturbed value of the Scorer parameter and n is the wave number of its oscillation. Depending on the phase of this oscillation, the drag may be substantially amplified or attenuated relative to its non-resonant value, displaying either single maxima or minima, or double extrema near n/l_0 = 2. Both non-hydrostatic effects and friction tend to reduce the magnitude of the drag extrema. However, in exactly inviscid conditions, the single drag maximum and minimum are suppressed. As in the atmosphere friction is often small but non-zero outside the boundary layer, modelling of the drag amplification mechanism addressed here should be quite sensitive to the type of turbulence closure employed in numerical models, or to computational dissipation in nominally inviscid simulations.

(Non-)ergodicity of a degenerate diffusion modeling the fiber lay down process

We analyze the large time behavior of a stochastic model for the lay down of fibers on a moving conveyor belt in the production process of nonwovens. It is shown that under weak conditions this degenerate diffusion process has a unique invariant distribution and is even geometrically ergodic. This generalizes results from previous works [M. Grothaus and A. Klar, SIAM J. Math. Anal., 40 (2008), pp. 968–983; J. Dolbeault et al., arXiv:1201.2156] concerning the case of a stationary conveyor belt, in which the situation of a moving conveyor belt has been left open.

Complex dynamics in simple systems with periodic parameter oscillations

We study systems with periodically oscillating parameters that can give way to complex periodic or nonperiodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of ϵ-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative nontrivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.

Effects of non-uniform crosswind fields on scintillometry measurements

The effects of a non-uniform wind field along the path of a scintillometer are investigated. Theoretical spectra are calculated for a range of scenarios where the crosswind varies in space or time and compared to the ‘ideal’ spectrum based on a constant uniform crosswind. It is verified that the refractive-index structure parameter relation with the scintillometer signal remains valid and invariant for both spatially and temporally-varying crosswinds. However, the spectral shape may change significantly preventing accurate estimation of the crosswind speed from the peak of the frequency spectrum and retrieval of the structure parameter from the plateau of the power spectrum. On comparison with experimental data, non-uniform crosswind conditions could be responsible for previously unexplained features sometimes seen in observed spectra. By accounting for the distribution of crosswind, theoretical spectra can be generated that closely replicate the observations, leading to a better understanding of the measurements. Spatial variability of wind speeds should be expected for paths other than those that are parallel to the surface and over flat, homogenous areas, whilst fluctuations in time are important for all sites.