High performance polyester-based materials

Biaxially oriented films produced from semi-crystalline, semi-aromatic polyesters are utilised extensively as components within various applications, including the specialist packaging, flexible electronic and photovoltaic markets. However, the thermal performance of such polyesters, specifically poly(ethylene terephthalate) (PET) and poly(ethylene-2,6-naphthalate) (PEN), is inadequate for several applications that require greater dimensional stability at higher operating temperatures. The work described in this project is therefore primarily focussed upon the copolymerisation of rigid comonomers with PET and PEN, in order to produce novel polyester-based materials that exhibit superior thermomechanical performance, with retention of crystallinity, to achieve biaxial orientation. Rigid biphenyldiimide comonomers were readily incorporated into PEN and poly(butylene-2,6-naphthalate) (PBN) via a melt-polycondensation route. For each copoly(ester-imide) series, retention of semi-crystalline behaviour is observed throughout entire copolymer composition ratios. This phenomenon may be rationalised by cocrystallisation between isomorphic biphenyldiimide and naphthalenedicarboxylate residues, which enables statistically random copolymers to melt-crystallise despite high proportions of imide sub-units being present. In terms of thermal performance, the glass transition temperature, Tg, linearly increases with imide comonomer content for both series. This facilitated the production of several high performance PEN-based biaxially oriented films, which displayed analogous drawing, barrier and optical properties to PEN. Selected PBN copoly(ester-imide)s also possess the ability to either melt-crystallise, or form a mesophase from the isotropic state depending on the applied cooling rate. An equivalent synthetic approach based upon isomorphic comonomer crystallisation was subsequently applied to PET by copolymerisation with rigid diimide and Kevlar®-type amide comonomers, to afford several novel high performance PET-based copoly(ester-imide)s and copoly(ester-amide)s that all exhibited increased Tgs. Retention of crystallinity was achieved in these copolymers by either melt-crystallisation or thermal annealing. The initial production of a semi-crystalline, PET-based biaxially oriented film with a Tg in excess of 100 °C was successful, and this material has obvious scope for further industrial scale-up and process development.

On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.

The stability of a two-dimensional vorticity filament under uniform strain

The quantitative effects of uniform strain and background rotation on the stability of a strip of constant vorticity (a simple shear layer) are examined. The thickness of the strip decreases in time under the strain, so it is necessary to formulate the linear stability analysis for a time-dependent basic flow. The results show that even a strain rate γ (scaled with the vorticity of the strip) as small as 0.25 suppresses the conventional Rayleigh shear instability mechanism, in the sense that the r.m.s. wave steepness cannot amplify by more than a certain factor, and must eventually decay. For γ < 0.25 the amplification factor increases as γ decreases; however, it is only 3 when γ e 0.065. Numerical simulations confirm the predictions of linear theory at small steepness and predict a threshold value necessary for the formation of coherent vortices. The results help to explain the impression from numerous simulations of two-dimensional turbulence reported in the literature that filaments of vorticity infrequently roll up into vortices. The stabilization effect may be expected to extend to two- and three-dimensional quasi-geostrophic flows.

Understanding the dimensional change card sort: Perspectives from task success and failure

Four experiments consider some of the circumstances under which children follow two different rule pairs when sorting cards. Previous research has repeatedly found that 3-year-olds encounter substantial difficulties implementing the second of two conflicting rule sets, despite their knowledge of these rules. One interpretation of this phenomenon [Cognitive Complexity and Control (CCC) theory] is that 3-year-olds have problems establishing an appropriate hierarchical ordering for rules. The present data suggest an alternative account of children's card sorting behaviour, according to which the cognitive salience of test card features may be more important than inflexibility with respect to rule representation.

Shrubs, granivores and annual plant community stability in an arid ecosystem

Compensatory population dynamics among species stabilise aggregate community variables. Inter-specific competition is thought to be stabilising as it promotes asynchrony among populations. However, we know little about other inter-specific interactions, such as facilitation and granivory. Such interactions are also likely to influence population synchrony and community stability, especially in harsh environments where they are thought to have relatively strong effects in plant communities. We use a manipulative experiment to test the effects of granivores (harvester ants) and nurse plants (dwarf shrubs) on annual plant community dynamics in the Negev desert, Israel. We present evidence for weak and inconsistent effects of harvester ants on plant abundance and on population and community stability. By contrast, we show that annual communities under shrubs were more species rich, had higher plant density and were temporally less variable than communities in the inter-shrub matrix. Species richness and plant abundance were also more resistant to drought in the shrub under-storey compared with the inter-shrub matrix, although population dynamics in both patch types were synchronised. Hence, we show that inter-specific interactions other than competition affect community stability, and that hypothesised mechanisms linking compensatory dynamics and community stability may not operate to the same extent in arid plant communities.

A data forest: multi-dimensional visualization

As Terabyte datasets become the norm, the focus has shifted away from our ability to produce and store ever larger amounts of data, onto its utilization. It is becoming increasingly difficult to gain meaningful insights into the data produced. Also many forms of the data we are currently producing cannot easily fit into traditional visualization methods. This paper presents a new and novel visualization technique based on the concept of a Data Forest. Our Data Forest has been designed to be used with vir tual reality (VR) as its presentation method. VR is a natural medium for investigating large datasets. Our approach can easily be adapted to be used in a variety of different ways, from a stand alone single user environment to large multi-user collaborative environments. A test application is presented using multi-dimensional data to demonstrate the concepts involved.

Bistable systems with stochastic noise: virtues and limits of effective one-dimensional Langevin equations

The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dimensional Langevin models are fitted to reproduce the observed statistical properties, and used to investing-ate the projected dynamics of the observable. This is of great relevance for studying potential catastrophic changes in the properties of the underlying system or resonant behaviours like those related to stochastic resonance-like mechanisms. In this paper, we propose a framework for encasing this kind of studies, using simple box models of the oceanic circulation and choosing as observable the strength of the thermohaline circulation. We study the statistical properties of the transitions between the two modes of operation of the thermohaline circulation under symmetric boundary forcings and test their agreement with simplified one-dimensional phenomenological theories. We extend our analysis to include stochastic resonance-like amplification processes. We conclude that fitted one-dimensional Langevin models, when closely scrutinised, may result to be more ad-hoc than they seem, lacking robustness and/or well-posedness. They should be treated with care, more as an empiric descriptive tool than as methodology with predictive power.

Standardized test to evaluate numerical weather prediction algorithms

In order to assist in comparing the computational techniques used in different models, the authors propose a standardized set of one-dimensional numerical experiments that could be completed for each model. The results of these experiments, with a simplified form of the computational representation for advection, diffusion, pressure gradient term, Coriolis term, and filter used in the models, should be reported in the peer-reviewed literature. Specific recommendations are described in this paper.

Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain [0,2π]x[0,2π/α], where α ϵ(0,1], with doubly periodic boundary conditions. For the linear problem we employ the classical energy{enstrophy argument to derive some fundamental properties of unstable eigenmodes. From this it is shown that forces of pure χ2-modes having wavelengths greater than 2π do not give rise to linear instability of the corresponding primary stationary solutions. For the nonlinear problem, we prove the equivalence of nonlinear stability with respect to the energy and enstrophy norms. This equivalence is then applied to derive optimal conditions for nonlinear stability, including both the high-and low-Reynolds-number limits.

Nonlinear stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

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.

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

Stability of an ice sheet on an elastic bed

The stability of stationary flow of a two-dimensional ice sheet is studied when the ice obeys a power flow law (Glen's flow law). The mass accumulation rate at the top is assumed to depend on elevation and span and the bed supporting the ice sheet consists of an elastic layer lying on a rigid surface. The normal perturbation of the free surface of the ice sheet is a singular eigenvalue problem. The singularity of the perturbation at the front of the ice sheet is considered using matched asymptotic expansions, and the eigenvalue problem is seen to reduce to that with fixed ice front. Numerical solution of the perturbation eigenvalue problem shows that the dependence of accumulation rate on elevation permits the existence of unstable solutions when the equilibrium line is higher than the bed at the ice divide. Alternatively, when the equilibrium line is lower than the bed, there are only stable solutions. Softening of the bed, expressed through a decrease of its elastic modulus, has a stabilising effect on the ice sheet.