Composite control of non-linear singularly perturbed systems: a geometric approach

Using a geometric approach, a composite control—the sum of a slow control and a fast control—is derived for a general class of non-linear singularly perturbed systems. A new and simpler method of composite control design is proposed whereby the fast control is completely designed at the outset. The slow control is then free to be chosen such that the slow integral manifold of the original system approximates a desired design manifold to within any specified order of ε accuracy.

Composite control of non-linear singularly perturbed systems: a geometric approach

Using a geometric approach, a composite control—the sum of a slow control and a fast control—is derived for a general class of non-linear singularly perturbed systems. A new and simpler method of composite control design is proposed whereby the fast control is completely designed at the outset. The slow control is then free to be chosen such that the slow integral manifold of the original system approximates a desired design manifold to within any specified order of ε accuracy.

Precipitation distributions for explicit versus parameterized convection in a large-domain high-resolution tropical case study

Global climate and weather models tend to produce rainfall that is too light and too regular over the tropical ocean. This is likely because of convective parametrizations, but the problem is not well understood. Here, distributions of precipitation rates are analyzed for high-resolution UK Met Office Unified Model simulations of a 10 day case study over a large tropical domain (∼20°S–20°N and 42°E–180°E). Simulations with 12 km grid length and parametrized convection have too many occurrences of light rain and too few of heavier rain when interpolated onto a 1° grid and compared with Tropical Rainfall Measuring Mission (TRMM) data. In fact, this version of the model appears to have a preferred scale of rainfall around 0.4 mm h−1 (10 mm day−1), unlike observations of tropical rainfall. On the other hand, 4 km grid length simulations with explicit convection produce distributions much more similar to TRMM observations. The apparent preferred scale at lighter rain rates seems to be a feature of the convective parametrization rather than the coarse resolution, as demonstrated by results from 12 km simulations with explicit convection and 40 km simulations with parametrized convection. In fact, coarser resolution models with explicit convection tend to have even more heavy rain than observed. Implications for models using convective parametrizations, including interactions of heating and moistening profiles with larger scales, are discussed. One important implication is that the explicit convection 4 km model has temperature and moisture tendencies that favour transitions in the convective regime. Also, the 12 km parametrized convection model produces a more stable temperature profile at its extreme high-precipitation range, which may reduce the chance of very heavy rainfall. Further study is needed to determine whether unrealistic precipitation distributions are due to some fundamental limitation of convective parametrizations or whether parametrizations can be improved, in order to better simulate these distributions.

Groundwater residence time distributions in peatlands: implications for peat decomposition and accumulation

Peat soils consist of poorly decomposed plant detritus, preserved by low decay rates, and deep peat deposits are globally significant stores in the carbon cycle. High water tables and low soil temperatures are commonly held to be the primary reasons for low peat decay rates. However, recent studies suggest a thermodynamic limit to peat decay, whereby the slow turnover of peat soil pore water may lead to high concentrations of phenols and dissolved inorganic carbon. In sufficient concentrations, these chemicals may slow or even halt microbial respiration, providing a negative feedback to peat decay. We document the analysis of a simple, one-dimensional theoretical model of peatland pore water residence time distributions (RTDs). The model suggests that broader, thicker peatlands may be more resilient to rapid decay caused by climate change because of slow pore water turnover in deep layers. Even shallow peat deposits may also be resilient to rapid decay if rainfall rates are low. However, the model suggests that even thick peatlands may be vulnerable to rapid decay under prolonged high rainfall rates, which may act to flush pore water with fresh rainwater. We also used the model to illustrate a particular limitation of the diplotelmic (i.e., acrotelm and catotelm) model of peatland structure. Model peatlands of contrasting hydraulic structure exhibited identical water tables but contrasting RTDs. These scenarios would be treated identically by diplotelmic models, although the thermodynamic limit suggests contrasting decay regimes. We therefore conclude that the diplotelmic model be discarded in favor of model schemes that consider continuous variation in peat properties and processes.

Axonal velocity distributions in neural feld equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Padé approximants for the acoustical properties of rigid frame porous media with pore size distributions

Expressions for the viscosity correction function, and hence bulk complex impedance, density, compressibility, and propagation constant, are obtained for a rigid frame porous medium whose pores are prismatic with fixed cross-sectional shape, but of variable pore size distribution. The lowand high-frequency behavior of the viscosity correction function is derived for the particular case of a log-normal pore size distribution, in terms of coefficients which can, in general, be computed numerically, and are given here explicitly for the particular cases of pores of equilateral triangular, circular, and slitlike cross-section. Simple approximate formulae, based on two-point Pade´ approximants for the viscosity correction function are obtained, which avoid a requirement for numerical integration or evaluation of special functions, and their accuracy is illustrated and investigated for the three pore shapes already mentioned

Constructing the frequency and wave normal distribution of whistler-mode wave power

We introduce a new methodology that allows the construction of wave frequency distributions due to growing incoherent whistler-mode waves in the magnetosphere. The technique combines the equations of geometric optics (i.e. raytracing) with the equation of transfer of radiation in an anisotropic lossy medium to obtain spectral energy density as a function of frequency and wavenormal angle. We describe the method in detail, and then demonstrate how it could be used in an idealised magnetosphere during quiet geomagnetic conditions. For a specific set of plasma conditions, we predict that the wave power peaks off the equator at ~15 degrees magnetic latitude. The new calculations predict that wave power as a function of frequency can be adequately described using a Gaussian function, but as a function of wavenormal angle, it more closely resembles a skew normal distribution. The technique described in this paper is the first known estimate of the parallel and oblique incoherent wave spectrum as a result of growing whistler-mode waves, and provides a means to incorporate self-consistent wave-particle interactions in a kinetic model of the magnetosphere over a large volume.

A closer look at chaotic advection in the stratosphere: part I: geometric structure

The relevance of chaotic advection to stratospheric mixing and transport is addressed in the context of (i) a numerical model of forced shallow-water flow on the sphere, and (ii) a middle-atmosphere general circulation model. It is argued that chaotic advection applies to both these models if there is suitable large-scale spatial structure in the velocity field and if the velocity field is temporally quasi-regular. This spatial structure is manifested in the form of “cat’s eyes” in the surf zone, such as are commonly seen in numerical simulations of Rossby wave critical layers; by analogy with the heteroclinic structure of a temporally aperiodic chaotic system the cat’s eyes may be thought of as an “organizing structure” for mixing and transport in the surf zone. When this organizing structure exists, Eulerian and Lagrangian autocorrelations of the velocity derivatives indicate that velocity derivatives decorrelate more rapidly along particle trajectories than at fixed spatial locations (i.e., the velocity field is temporally quasi-regular). This phenomenon is referred to as Lagrangian random strain.

Model simulations and aircraft measurements of vertical, seasonal and latitudinal O3 and CO distributions over Europe

During a series of 8 measurement campaigns within the SPURT project (2001-2003), vertical profiles of CO and O3 have been obtained at subtropical, middle and high latitudes over western Europe, covering the troposphere and lowermost stratosphere up to ~14 km altitude during all seasons. The seasonal and latitudinal variation of the measured trace gas profiles are compared to simulations with the chemical transport model MATCH. In the troposphere reasonable agreement between observations and model predictions is achieved for CO and O3, in particular at subtropical and mid-latitudes, while the model overestimates (underestimates) CO (O3 in the lowermost stratosphere particularly at high latitudes, indicating too strong simulated bi-directional exchange across the tropopause. By the use of tagged tracers in the model, long-range transport of Asian air masses is identified as the dominant source of CO pollution over Europe in the free troposphere.

Evaluating the Survey of Professional Forecasters probability distributions of expected inflation based on derived event probability forecasts

Techniques are proposed for evaluating forecast probabilities of events. The tools are especially useful when, as in the case of the Survey of Professional Forecasters (SPF) expected probability distributions of inflation, recourse cannot be made to the method of construction in the evaluation of the forecasts. The tests of efficiency and conditional efficiency are applied to the forecast probabilities of events of interest derived from the SPF distributions, and supplement a whole-density evaluation of the SPF distributions based on the probability integral transform approach.

Probability distributions or point predictions? Survey forecasts of US output growth and inflation

We consider whether survey respondents’ probability distributions, reported as histograms, provide reliable and coherent point predictions, when viewed through the lens of a Bayesian learning model. We argue that a role remains for eliciting directly-reported point predictions in surveys of professional forecasters.

Pollen-based reconstructions of biome distributions for Australia, Southeast Asia and the Pacific (SEAPAC region) at 0,6000 and 18,000 14C yr B.P.

Aim  This paper documents reconstructions of the vegetation patterns in Australia, Southeast Asia and the Pacific (SEAPAC region) in the mid-Holocene and at the last glacial maximum (LGM). Methods  Vegetation patterns were reconstructed from pollen data using an objective biomization scheme based on plant functional types. The biomization scheme was first tested using 535 modern pollen samples from 377 sites, and then applied unchanged to fossil pollen samples dating to 6000 ± 500 or 18,000 ± 1000 14C yr bp. Results  1. Tests using surface pollen sample sites showed that the biomization scheme is capable of reproducing the modern broad-scale patterns of vegetation distribution. The north–south gradient in temperature, reflected in transitions from cool evergreen needleleaf forest in the extreme south through temperate rain forest or wet sclerophyll forest (WSFW) and into tropical forests, is well reconstructed. The transitions from xerophytic through sclerophyll woodlands and open forests to closed-canopy forests, which reflect the gradient in plant available moisture from the continental interior towards the coast, are reconstructed with less geographical precision but nevertheless the broad-scale pattern emerges. 2. Differences between the modern and mid-Holocene vegetation patterns in mainland Australia are comparatively small and reflect changes in moisture availability rather than temperature. In south-eastern Australia some sites show a shift towards more moisture-stressed vegetation in the mid-Holocene with xerophytic woods/scrub and temperate sclerophyll woodland and shrubland at sites characterized today by WSFW or warm-temperate rain forest (WTRF). However, sites in the Snowy Mountains, on the Southern Tablelands and east of the Great Dividing Range have more moisture-demanding vegetation in the mid-Holocene than today. South-western Australia was slightly drier than today. The single site in north-western Australia also shows conditions drier than today in the mid-Holocene. Changes in the tropics are also comparatively small, but the presence of WTRF and tropical deciduous broadleaf forest and woodland in the mid-Holocene, in sites occupied today by cool-temperate rain forest, indicate warmer conditions. 3. Expansion of xerophytic vegetation in the south and tropical deciduous broadleaf forest and woodland in the north indicate drier conditions across mainland Australia at the LGM. None of these changes are informative about the degree of cooling. However the evidence from the tropics, showing lowering of the treeline and forest belts, indicates that conditions were between 1 and 9 °C (depending on elevation) colder. The encroachment of tropical deciduous broadleaf forest and woodland into lowland evergreen broadleaf forest implies greater aridity. Main conclusions  This study provides the first continental-scale reconstruction of mid-Holocene and LGM vegetation patterns from Australia, Southeast Asia and the Pacific (SEAPAC region) using an objective biomization scheme. These data will provide a benchmark for evaluation of palaeoclimate simulations within the framework of the Palaeoclimate Modelling Intercomparison Project.

Autobiographical memory distributions for negative self-images: memories are organised around negative as well as positive aspects of identity

The relationship between developmental experiences, and an individual’s emerging beliefs about themselves and the world, is central to many forms of psychotherapy. People suffering from a variety of mental health problems have been shown to use negative memories when defining the self, however little is known about how these negative memories might be organised and relate to negative self-images. In two online studies with middle-aged (N = 18; Study 1) and young (N = 56; Study 2) adults, we found that participants’ negative self-images (e.g., I am a failure) were associated with sets of autobiographical memories that formed clustered distributions around times of self-formation, in much the same pattern as for positive self-images (e.g., I am talented). This novel result shows that highly organised sets of salient memories may be responsible for perpetuating negative beliefs about the self. Implications for therapy are discussed.

Towards the computer-aided diagnosis of dementia based on the geometric and network connectivity of structural MRI data

We present an intuitive geometric approach for analysing the structure and fragility of T1-weighted structural MRI scans of human brains. Apart from computing characteristics like the surface area and volume of regions of the brain that consist of highly active voxels, we also employ Network Theory in order to test how close these regions are to breaking apart. This analysis is used in an attempt to automatically classify subjects into three categories: Alzheimer’s disease, mild cognitive impairment and healthy controls, for the CADDementia Challenge.