Finite departure from convective quasi-equilibrium: periodic cycle and discharge-recharge mechanism

A simple self–contained theory is proposed for describing life cycles of convective systems as a discharge–recharge process. A closed description is derived for the dynamics of an ensemble of convective plumes based on an energy cycle. The system consists of prognostic equations for the cloud work function and the convective kinetic energy. The system can be closed by intro ducing a functional relationship between the convective kinetic energy and the cloud–base mass flux. The behaviour of this system is considered under a bulk simplification. Previous cloud–resolving mo delling as well as bulk statistical theories for ensemble convective systems suggest that a plausible relationship would be to assume that the convective kinetic energy is linearly proportional to the cloud–base mass flux. As a result, the system reduces to a nonlinear dynamical system with two dependent variables, the cloud–base mass flux and the cloud work function. The fully nonlinear solution of this system always represents a periodic cycle regardless of the initial condition under constant large–scale forcing. Importantly, the inclusion of energy dissipation in this model does not in itself lead the system to an equilibrium.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Some numerical experiments on the effect of the variation of static stability in two-layer quasi-geostrophic models

A method to solve a quasi-geostrophic two-layer model including the variation of static stability is presented. The divergent part of the wind is incorporated by means of an iterative procedure. The procedure is rather fast and the time of computation is only 60–70% longer than for the usual two-layer model. The method of solution is justified by the conservation of the difference between the gross static stability and the kinetic energy. To eliminate the side-boundary conditions the experiments have been performed on a zonal channel model. The investigation falls mainly into three parts: The first part (section 5) contains a discussion of the significance of some physically inconsistent approximations. It is shown that physical inconsistencies are rather serious and for these inconsistent models which were studied the total kinetic energy increased faster than the gross static stability. In the next part (section 6) we are studying the effect of a Jacobian difference operator which conserves the total kinetic energy. The use of this operator in two-layer models will give a slight improvement but probably does not have any practical use in short periodic forecasts. It is also shown that the energy-conservative operator will change the wave-speed in an erroneous way if the wave-number or the grid-length is large in the meridional direction. In the final part (section 7) we investigate the behaviour of baroclinic waves for some different initial states and for two energy-consistent models, one with constant and one with variable static stability. According to the linear theory the waves adjust rather rapidly in such a way that the temperature wave will lag behind the pressure wave independent of the initial configuration. Thus, both models give rise to a baroclinic development even if the initial state is quasi-barotropic. The effect of the variation of static stability is very small, qualitative differences in the development are only observed during the first 12 hours. For an amplifying wave we will get a stabilization over the troughs and an instabilization over the ridges.

Nonlinear stability of continuously stratified quasi-geostrophic flow

Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-geostrophic flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-geostrophic model and the modified quasi-geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

Non-competitive markets

Among all the paradigms in economic theory, the theoretical predictions of oligopoly were the first to be examined in the laboratory. In this chapter, instead of surveying all the experiments with few sellers, we adopt a narrower definition of the term “oligopoly”, and focus on the experiments that were directly inspired by the basic oligopolistic models of Cournot, Bertrand, Hotelling, Stackelberg, and some extensions. Most of the experiments we consider in this chapter have been run in the last three decades. This literature can be considered as a new wave of experimental works aiming at representing basic oligopolistic markets and testing their properties. The chapter is divided into independent sections referring to different parts of the oligopolistic theory, including both monopoly as well as a number of extensions of the basic models, which have been chosen with the aim of providing a representative list of the relevant experimental findings.

Generalized convective quasi-equilibrium principle

A generalization of Arakawa and Schubert's convective quasi-equilibrium principle is presented for a closure formulation of mass-flux convection parameterization. The original principle is based on the budget of the cloud work function. This principle is generalized by considering the budget for a vertical integral of an arbitrary convection-related quantity. The closure formulation includes Arakawa and Schubert's quasi-equilibrium, as well as both CAPE and moisture closures as special cases. The formulation also includes new possibilities for considering vertical integrals that are dependent on convective-scale variables, such as the moisture within convection. The generalized convective quasi-equilibrium is defined by a balance between large-scale forcing and convective response for a given vertically-integrated quantity. The latter takes the form of a convolution of a kernel matrix and a mass-flux spectrum, as in the original convective quasi-equilibrium. The kernel reduces to a scalar when either a bulk formulation is adopted, or only large-scale variables are considered within the vertical integral. Various physical implications of the generalized closure are discussed. These include the possibility that precipitation might be considered as a potentially-significant contribution to the large-scale forcing. Two dicta are proposed as guiding physical principles for the specifying a suitable vertically-integrated quantity.

A review of the theoretical basis for bulk mass flux convective parameterization

Most parameterizations for precipitating convection in use today are bulk schemes, in which an ensemble of cumulus elements with different properties is modelled as a single, representative entraining-detraining plume. We review the underpinning mathematical model for such parameterizations, in particular by comparing it with spectral models in which elements are not combined into the representative plume. The chief merit of a bulk model is that the representative plume can be described by an equation set with the same structure as that which describes each element in a spectral model. The equivalence relies on an ansatz for detrained condensate introduced by Yanai et al. (1973) and on a simplified microphysics. There are also conceptual differences in the closure of bulk and spectral parameterizations. In particular, we show that the convective quasi-equilibrium closure of Arakawa and Schubert (1974) for spectral parameterizations cannot be carried over to a bulk parameterization in a straightforward way. Quasi-equilibrium of the cloud work function assumes a timescale separation between a slow forcing process and a rapid convective response. But, for the natural bulk analogue to the cloud-work function (the dilute CAPE), the relevant forcing is characterised by a different timescale, and so its quasi-equilibrium entails a different physical constraint. Closures of bulk parameterization that use the non-entraining parcel value of CAPE do not suffer from this timescale issue. However, the Yanai et al. (1973) ansatz must be invoked as a necessary ingredient of those closures.

A calibrated, non-invasive method for measuring the internal interface height field at high resolution in the rotating, two-layer annulus

We describe a remote sensing method for measuring the internal interface height field in a rotating, two-layer annulus laboratory experiment. The method is non-invasive, avoiding the possibility of an interaction between the flow and the measurement device. The height fields retrieved are accurate and highly resolved in both space and time. The technique is based on a flow visualization method developed by previous workers, and relies upon the optical rotation properties of the working liquids. The previous methods returned only qualitative interface maps, however. In the present study, a technique is developed for deriving quantitative maps by calibrating height against the colour fields registered by a camera which views the flow from above. We use a layer-wise torque balance analysis to determine the equilibrium interface height field analytically, in order to derive the calibration curves. With the current system, viewing an annulus of outer radius 125 mm and depth 250 mm from a distance of 2 m, the inferred height fields have horizontal, vertical and temporal resolutions of up to 0.2 mm, 1 mm and 0.04 s, respectively.

Does reactive surface area depend on grain size? Results from pH 3, 25 circle C far-from-equilibrium flow-through dissolution experiments on anorthite and biotite

Laboratory determined mineral weathering rates need to be normalised to allow their extrapolation to natural systems. The principle normalisation terms used in the literature are mass, and geometric- and BET specific surface area (SSA). The purpose of this study was to determine how dissolution rates normalised to these terms vary with grain size. Different size fractions of anorthite and biotite ranging from 180-150 to 20-10 mu m were dissolved in pH 3, HCl at 25 degrees C in flow through reactors under far from equilibrium conditions. Steady state dissolution rates after 5376 h (anorthite) and 4992 h (biotite) were calculated from Si concentrations and were normalised to initial- and final- mass and geometric-, geometric edge- (biotite), and BET SSA. For anorthite, rates normalised to initial- and final-BET SSA ranged from 0.33 to 2.77 X 10(-10) mol(feldspar) m(-2) s(-1), rates normalised to initial- and final-geometric SSA ranged from 5.74 to 8.88 X 10(-10) mol(feldspar) m(-2) s(-1) and rates normalised to initial- and final-mass ranged from 0.11 to 1.65 mol(feldspar) g(-1) s(-1). For biotite, rates normalised to initial- and final-BET SSA ranged from 1.02 to 2.03 X 10(-12) mol(biotite) m(-2) s(-1), rates normalised to initial- and final-geometric SSA ranged from 3.26 to 16.21 X 10(-12) mol(biotite) m(-2) s(-1), rates normalised to initial- and final-geometric edge SSA ranged from 59.46 to 111.32 x 10(-12) mol(biotite) m(-2) s(-1) and rates normalised to initial- and final-mass ranged from 0.81 to 6.93 X 10(-12) mol(biotite) g(-1) s(-1). For all normalising terms rates varied significantly (p <= 0.05) with grain size. The normalising terms which gave least variation in dissolution rate between grain sizes for anorthite were initial BET SSA and initial- and final-geometric SSA. This is consistent with: (1) dissolution being dominated by the slower dissolving but area dominant non-etched surfaces of the grains and, (2) the walls of etch pits and other dissolution features being relatively unreactive. These steady state normalised dissolution rates are likely to be constant with time. Normalisation to final BET SSA did not give constant ratios across grain size due to a non-uniform distribution of dissolution features. After dissolution coarser grains had a greater density of dissolution features with BET-measurable but unreactive wall surface area than the finer grains. The normalising term which gave the least variation in dissolution rates between grain sizes for biotite was initial BET SSA. Initial- and final-geometric edge SSA and final BET SSA gave the next least varied rates. The basal surfaces dissolved sufficiently rapidly to influence bulk dissolution rate and prevent geometric edge SSA normalised dissolution rates showing the least variation. Simple modelling indicated that biotite grain edges dissolved 71-132 times faster than basal surfaces. In this experiment, initial BET SSA best integrated the different areas and reactivities of the edge and basal surfaces of biotite. Steady state dissolution rates are likely to vary with time as dissolution alters the ratio of edge to basal surface area. Therefore they would be more properly termed pseudo-steady state rates, only appearing constant because the time period over which they were measured (1512 h) was less than the time period over wich they would change significantly. (c) 2006 Elsevier Inc. All rights reserved.

Mesoscale simulations of organized convection: importance of convective equilibrium

The validity of convective parametrization breaks down at the resolution of mesoscale models, and the success of parametrized versus explicit treatments of convection is likely to depend on the large-scale environment. In this paper we examine the hypothesis that a key feature determining the sensitivity to the environment is whether the forcing of convection is sufficiently homogeneous and slowly varying that the convection can be considered to be in equilibrium. Two case studies of mesoscale convective systems over the UK, one where equilibrium conditions are expected and one where equilibrium is unlikely, are simulated using a mesoscale forecasting model. The time evolution of area-average convective available potential energy and the time evolution and magnitude of the timescale of convective adjustment are consistent with the hypothesis of equilibrium for case 1 and non-equilibrium for case 2. For each case, three experiments are performed with different partitionings between parametrized and explicit convection: fully parametrized convection, fully explicit convection and a simulation with significant amounts of both. In the equilibrium case, bulk properties of the convection such as area-integrated rain rates are insensitive to the treatment of convection. However, the detailed structure of the precipitation field changes; the simulation with parametrized convection behaves well and produces a smooth field that follows the forcing region, and the simulation with explicit convection has a small number of localized intense regions of precipitation that track with the mid-levelflow. For the non-equilibrium case, bulk properties of the convection such as area-integrated rain rates are sensitive to the treatment of convection. The simulation with explicit convection behaves similarly to the equilibrium case with a few localized precipitation regions. In contrast, the cumulus parametrization fails dramatically and develops intense propagating bows of precipitation that were not observed. The simulations with both parametrized and explicit convection follow the pattern seen in the other experiments, with a transition over the duration of the run from parametrized to explicit precipitation. The impact of convection on the large-scaleflow, as measured by upper-level wind and potential-vorticity perturbations, is very sensitive to the partitioning of convection for both cases. © Royal Meteorological Society, 2006. Contributions by P. A. Clark and M. E. B. Gray are Crown Copyright.

Thermal physics of the atmosphere

Thermal Physics of the Atmosphere offers a concise and thorough introduction on how basic thermodynamics naturally leads on to advanced topics in atmospheric physics. The book starts by covering the basics of thermodynamics and its applications in atmospheric science. The later chapters describe major applications, specific to more specialized areas of atmospheric physics, including vertical structure and stability, cloud formation, and radiative processes. The book concludes with a discussion of non-equilibrium thermodynamics as applied to the atmosphere. This book provides a thorough introduction and invaluable grounding for specialised literature on the subject. Introduces a wide range of areas associated with atmospheric physics Starts from basic level thermal physics Ideally suited for readers with a general physics background Self-assessment questions included for each chapter Supplementary website to accompany the book

Perturbed Rankine vortices in surface quasi-geostrophic dynamics

An analytical dispersion relation is derived for linear perturbations to a Rankine vortex governed by surface quasi-geostrophic dynamics. Such a Rankine vortex is a circular region of uniform anomalous surface temperature evolving under quasi-geostrophic dynamics with uniform interior potential vorticity. The dispersion relation is analysed in detail and compared to the more familiar dispersion relation for a perturbed Rankine vortex governed by the Euler equations. The results are successfully verified against numerical simulations of the full equations. The dispersion relation is relevant to problems including wave propagation on surface temperature fronts and the stability of vortices in quasi-geostrophic turbulence.

Omega-3 fatty acids, cardiovascular disease and stability of atherosclerotic plaques

Long-chain n-3 polyunsaturated fatty acids are found in oily fish and in fish oils and similar preparations. Substantial evidence from epidemiological and case-control studies indicates that consumption of fish, oily fish and long-chain n-3 fatty acids reduces risk of cardiovascular mortality. Secondary prevention studies using long-chain n-3 fatty acids in patients post-myocardial infarction have shown a reduction in total and cardiovascular mortality with an especially potent effect on sudden death. Long-chain n-3 fatty acids have been shown to beneficially modify a range of cardiovascular risk factors, which may result in primary cardiovascular prevention. However, reduced non-fatal and fatal events and a reduction in sudden death probably involve other mechanisms. Reduced thrombosis following long-chain n-3 fatty acids may play a role. A decrease in arrhythmias is a favoured mechanism of action of long-chain n-3 fatty acids and is supported by cell culture and animal studies. However human trials using implantable cardiac defibrillators have produced inconsistent findings and a recent meta-analysis does not support this mechanism of action. An alternative mechanism of action may be stabilisation of atherosclerotic plaques by long-chain n-3 fatty acids. This is suggested by one published human study which showed that incorporation of long-chain n-3 fatty acids into plaques collected at carotid endarterectomy resulted in fewer macrophages in the plaque and a morphology indicative of increased stability. These findings are supported from observations in an animal model and suggest that the primary effect of long-chain n-3 fatty acids might be on macrophages within the plaque.