A Wiener-Hopf type factorization for the exponential functional of Levy processes

For a Lévy process ξ=(ξt)t≥0 drifting to −∞, we define the so-called exponential functional as follows: Formula Under mild conditions on ξ, we show that the following factorization of exponential functionals: Formula holds, where × stands for the product of independent random variables, H− is the descending ladder height process of ξ and Y is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of Iξ for a large class of Lévy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein–Uhlenbeck processes, which is itself of independent interest. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.

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

Inhomogeneous two-dimensional turbulence in the atmosphere

Faced with the strongly nonlinear and apparently random behaviour of the energy-containing scales in the atmosphere, geophysical fluid dynamicists have attempted to understand the synoptic-scale atmospheric flow within the context of two-dimensional homogeneous turbulence theory (e.g. FJØRTOFT [1]; LEITH [2]). However atmospheric observations (BOER and SHEPHERD [3] and Fig.1) show that the synoptic-scale transient flow evolves in the presence of a planetary-scale, quasi-stationary background flow which is approximately zonal (east-west). Classical homogeneous 2-D turbulence theory is therefore not strictly applicable to the transient flow. One is led instead to study 2-D turbulence in the presence of a large-scale (barotropically stable) zonal jet inhomogeneity.

Large-scale two-dimensional turbulence in the atmosphere

Global FGGE data are used to investigate several aspects of large-scale turbulence in the atmosphere. The approach follows that for two-dimensional, nondivergent turbulent flows which are homogeneous and isotropic on the sphere. Spectra of kinetic energy, enstrophy and available potential energy are obtained for both the stationary and transient parts of the flow. Nonlinear interaction terms and fluxes of energy and enstrophy through wavenumber space are calculated and compared with the theory. A possible method of parameterizing the interactions with unresolved scales is considered. Two rather different flow regimes are found in wavenumber space. The high-wavenumber regime is dominated by the transient components of the flow and exhibits, at least approximately, several of the conditions characterizing homogeneous and isotropic turbulence. This region of wavenumber space also displays some of the features of an enstrophy-cascading inertial subrange. The low-wavenumber region, on the other hand, is dominated by the stationary component of the flow, exhibits marked anisotropy and, in contrast to the high-wavenumber regime, displays a marked change between January and July.

The impracticality of a universal drought definition

This paper demonstrates the impracticality of a comprehensive mathematical definition of the term `drought' which formalises the general qualitative definition that drought is `a deficit of water relative to normal conditions'. Starting from the local water balance, it is shown that a universal description of drought requires reference to water supply, demand and management. The influence of human intervention through water management is shown to be intrinsic to the definition of drought in the universal sense and can only be eliminated in the case of purely meteorological drought. The state of `drought' is shown to be predicated on the existence of climatological norms for a multitude of process specific terms. In general these norms are either difficult to obtain or even non-existent in the non-stationary context of climate change. Such climatological considerations, in conjunction with the difficulty of quantifying human influence, lead to the conclusion that we cannot reasonably expect the existence of any workable generalised objective definition of drought.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.

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.

Threshold autoregressive and Markov switching models: an application to commercial real estate

Although financial theory rests heavily upon the assumption that asset returns are normally distributed, value indices of commercial real estate display significant departures from normality. In this paper, we apply and compare the properties of two recently proposed regime switching models for value indices of commercial real estate in the US and the UK, both of which relax the assumption that observations are drawn from a single distribution with constant mean and variance. Statistical tests of the models' specification indicate that the Markov switching model is better able to capture the non-stationary features of the data than the threshold autoregressive model, although both represent superior descriptions of the data than the models that allow for only one state. Our results have several implications for theoretical models and empirical research in finance.

The gravity wave momentum flux in hydrostatic flow with directional shear over elliptical mountains

Semi-analytical expressions for the momentum flux associated with orographic internal gravity waves, and closed analytical expressions for its divergence, are derived for inviscid, stationary, hydrostatic, directionally-sheared flow over mountains with an elliptical horizontal cross-section. These calculations, obtained using linear theory conjugated with a third-order WKB approximation, are valid for relatively slowly-varying, but otherwise generic wind profiles, and given in a form that is straightforward to implement in drag parametrization schemes. When normalized by the surface drag in the absence of shear, a quantity that is calculated routinely in existing drag parametrizations, the momentum flux becomes independent of the detailed shape of the orography. Unlike linear theory in the Ri → ∞ limit, the present calculations account for shear-induced amplification or reduction of the surface drag, and partial absorption of the wave momentum flux at critical levels. Profiles of the normalized momentum fluxes obtained using this model and a linear numerical model without the WKB approximation are evaluated and compared for two idealized wind profiles with directional shear, for different Richardson numbers (Ri). Agreement is found to be excellent for the first wind profile (where one of the wind components varies linearly) down to Ri = 0.5, while not so satisfactory, but still showing a large improvement relative to the Ri → ∞ limit, for the second wind profile (where the wind turns with height at a constant rate keeping a constant magnitude). These results are complementary, in the Ri > O(1) parameter range, to Broad’s generalization of the Eliassen–Palm theorem to 3D flow. They should contribute to improve drag parametrizations used in global weather and climate prediction models.

Instantaneous gelation in Smoluchowski’s coagulation equation revisited

We study the solutions of the Smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularized gelation time decreasesas M increases, consistent with the expectation that the gelation time is zero in the unregularized system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff. We also study the case when a source of monomers is introduced in the regularized system. In this case a stationary state is reached. We present a complete analytic description of this regularized stationary state for the model kernel, K(m1,m2)=max{m1,m2}ν, which gels instantaneously when M→∞ if ν>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ν for large cluster sizes. The total particle density in the stationary state slowly vanishes as [(ν−1)logM]−1/2 when M→∞. The approach to the stationary state is nontrivial: Oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.

The statistical cusp: a flux transfer event model

Traditionally, the cusp has been described in terms of a time-stationary feature of the magnetosphere which allows access of magnetosheath-like plasma to low altitudes. Statistical surveys of data from low-altitude spacecraft have shown the average characteristics and position of the cusp. Recently, however, it has been suggested that the ionospheric footprint of flux transfer events (FTEs) may be identified as variations of the “cusp” on timescales of a few minutes. In this model, the cusp can vary in form between a steady-state feature in one limit and a series of discrete ionospheric FTE signatures in the other limit. If this time-dependent cusp scenario is correct, then the signatures of the transient reconnection events must be able, on average, to reproduce the statistical cusp occurrence previously determined from the satellite observations. In this paper, we predict the precipitation signatures which are associated with transient magnetopause reconnection, following recent observations of the dependence of dayside ionospheric convection on the orientation of the IMF. We then employ a simple model of the longitudinal motion of FTE signatures to show how such events can easily reproduce the local time distribution of cusp occurrence probabilities, as observed by low-altitude satellites. This is true even in the limit where the cusp is a series of discrete events. Furthermore, we investigate the existence of double cusp patches predicted by the simple model and show how these events may be identified in the data.

Cities reducing their greenhouse gas emissions

The study asks how well are cities doing in reducing their greenhouse gas emissions. Data from six cities with repeat GHG emission inventories for the period 2004–2009 is examined: Berlin, Boston, Greater Toronto, London, New York City and Seattle. All of the cities are reducing their per capita GHG emissions, primarily through changes to stationary combustion. On average the cities are reducing per capita emissions by 0.27 t CO2e/capita per year; this is about the same average rate as the cities nation states, although the cities are reducing emissions faster in percentage terms.

Ice-sheet configuration in the CMIP5/PMIP3 Last Glacial Maximum experiments

We describe the creation of a data set describing changes related to the presence of ice sheets, including ice-sheet extent and height, ice-shelf extent, and the distribution and elevation of ice-free land at the Last Glacial Maximum (LGM), which were used in LGM experiments conducted as part of the fifth phase of the Coupled Modelling Intercomparison Project (CMIP5) and the third phase of the Palaeoclimate Modelling Intercomparison Project (PMIP3). The CMIP5/PMIP3 data sets were created from reconstructions made by three different groups, which were all obtained using a model-inversion approach but differ in the assumptions used in the modelling and in the type of data used as constraints. The ice-sheet extent in the Northern Hemisphere (NH) does not vary substantially between the three individual data sources. The difference in the topography of the NH ice sheets is also moderate, and smaller than the differences between these reconstructions (and the resultant composite reconstruction) and ice-sheet reconstructions used in previous generations of PMIP. Only two of the individual reconstructions provide information for Antarctica. The discrepancy between these two reconstructions is larger than the difference for the NH ice sheets, although still less than the difference between the composite reconstruction and previous PMIP ice-sheet reconstructions. Although largely confined to the ice-covered regions, differences between the climate response to the individual LGM reconstructions extend over the North Atlantic Ocean and Northern Hemisphere continents, partly through atmospheric stationary waves. Differences between the climate response to the CMIP5/PMIP3 composite and any individual ice-sheet reconstruction are smaller than those between the CMIP5/PMIP3 composite and the ice sheet used in the last phase of PMIP (PMIP2).

Interdecadal changes on the seasonal prediction of the western North Pacific summer climate around the late 1970s and early 1990s

Identifying predictability and the corresponding sources for the western North Pacific (WNP) summer climate in the case of non-stationary teleconnections during recent decades benefits for further improvements of long-range prediction on the WNP and East Asian summers. In the past few decades, pronounced increases on the summer sea surface temperature (SST) and associated interannual variability are observed over the tropical Indian Ocean and eastern Pacific around the late 1970s and over the Maritime Continent and western–central Pacific around the early 1990s. These increases are associated with significant enhancements of the interannual variability for the lower-tropospheric wind over the WNP. In this study, we further assess interdecadal changes on the seasonal prediction of the WNP summer anomalies, using May-start retrospective forecasts from the ENSEMBLES multi-model project in the period 1960–2005. It is found that prediction of the WNP summer anomalies exhibits an interdecadal shift with higher prediction skills since the late 1970s, particularly after the early 1990s. Improvements of the prediction skills for SSTs after the late 1970s are mainly found around tropical Indian Ocean and the WNP. The better prediction of the WNP after the late 1970s may arise mainly from the improvement of the SST prediction around the tropical eastern Indian Ocean. The close teleconnections between the tropical eastern Indian Ocean and WNP summer variability work both in the model predictions and observations. After the early 1990s, on the other hand, the improvements are detected mainly around the South China Sea and Philippines for the lower-tropospheric zonal wind and precipitation anomalies, associating with a better description of the SST anomalies around the Maritime Continent. A dipole SST pattern over the Maritime Continent and the central equatorial Pacific Ocean is closely related to the WNP summer anomalies after the early 1990s. This teleconnection mode is quite predictable, which is realistically reproduced by the models, presenting more predictable signals to the WNP summer climate after the early 1990s.

Evaluating glucose and xylose as cosubstrates for lipid accumulation and c-linolenic acid biosynthesis of Thamnidium elegans

Aims: To study the biotechnological production of lipids containing rich amounts of the medically and nutritionally important c-linolenic acid (GLA), during cultivation of the Zygomycetes Thamnidium elegans, on mixtures of glucose and xylose, abundant sugars of lignocellulosic biomass. Methods and Results: Glucose and xylose were utilized as carbon sources, solely or in mixtures, under nitrogen-limited conditions, in batch-flask or bioreactor cultures. On glucose, T. elegans produced 31.9 g/L of biomass containing 15.0 g/L lipid with significantly high GLA content (1014 mg/L). Xylose was proved to be an adequate substrate for growth and lipid production. Additionally, xylitol secretion occurred when xylose was utilized as carbon source, solely or in mixtures with glucose. Batch-bioreactor trials on glucose yielded satisfactory lipid production, with rapid substrate consumption rates. Analysis of intracellular lipids showed that the highest GLA content was observed in early stationary growth phase, while the phospholipid fraction was the most unsaturated fraction of T. elegans. Conclusions: Thamnidium elegans represents a promising fungus for the successful valorization of sugar-based lignocellulosic residues into microbial lipids of high nutritional and pharmaceutical interest.