Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).Comment: 12 pages, 1 figure; updated reference

Ricci flows with bursts of unbounded curvature

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor 1/2t ) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as t → ∞, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature. Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi’s flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations. All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate

Cloud condensation nucleus (CCN) behavior of organic aerosol particles generated by atomization of water and methanol solutions

Cloud condensation nucleus (CCN) experiments were carried out for malonic acid, succinic acid, oxalacetic acid, DL-malic acid, glutaric acid, DL-glutamic acid monohydrate, and adipic acid, using both water and methanol as atomization solvents, at three operating supersaturations (0.11%, 0.21%, and 0.32%) in the Caltech three-column CCN instrument (CCNC3). Predictions of CCN behavior for five of these compounds were made using the Aerosol Diameter Dependent Equilibrium Model (ADDEM). The experiments presented here expose important considerations associated with the laboratory measurement of the CCN behavior of organic compounds. Choice of atomization solvent results in significant differences in CCN activation for some of the compounds studied, which could result from residual solvent, particle morphology differences, and chemical reactions between the particle and gas phases. Also, significant changes in aerosol size distribution occurred after classification in a differential mobility analyzer (DMA) for malonic acid and glutaric acid. Filter analysis of adipic acid atomized from methanol solution indicates that gas-particle phase reactions may have taken place after atomization and before the methanol was removed from the sample gas stream. Careful consideration of these experimental issues is necessary for successful design and interpretation of laboratory CCN measurements

Sensitivities of the absorptive partitioning model of secondary organic aerosol formation to the inclusion of water

The predicted distribution of semi-volatile organic components between the gaseous and condensed phase as a function of ambient relative humidity and the specific form of the partitioning model used has been investigated. A mole fraction based model, modified so as not to use molar mass in the calculation, was found to predict identical RH dependence of component partitioning to that predicted by the conventional mass-based partitioning model which uses a molar mass averaged according to the number of moles in the condensed phase. A recently reported third version of the partitioning model using individual component molar masses was shown to give significantly different results to the other two models. Further sensitivities to an assumed pre-existing particulate loading and to parameterised organic component non-ideality are explored and shown to contribute significantly to the variation in predicted secondary organic particulate loading

A curved multi-component aerosol hygroscopicity model framework: Part 1 – Inorganic compounds

A thermodynamic modelling framework to predict the equilibrium behaviour of mixed inorganic salt aerosols is developed, and then coupled with a technique for finding a solution to the Kohler equation in order to create a diameter dependent hygroscopic aerosol model (Aerosol Diameter Dependent Equilibrium Model – ADDEM). The model described here provides a robust and accurate inorganic basis using a mole fraction based activity coefficient model and adjusted energies of formation for treating solid precipitation. The model framework can accommodate organic components, though this added complexity is considered in a companion paper, this paper describes the development of the modelling architecture to be used and predictions of an inorganic model alone. The modelling framework has been developed to flexibly use a combination of mixing rules and other potentially more accurate techniques where available to calculate the water content. Comparisons with other state-of-the-art general equilibrium models and experimental data are presented and show excellent agreement. The Kelvin effect can be considered in this scheme using a variety of surface tension models. Comparison of predicted diameter dependent phenomena, such as the increased relative humidity for onset of deliquescence with decreasing diameter, with another diameter dependent model is very good despite the different approach used. The model is subject to various sensitivities. For the inorganic systems studied here, the model is sensitive to choice of surface tension scheme used, which decreases for larger aerosol. Large sensitivities are found for the value of dry density used. It is thus likely that the history of the aerosol studied in a hygroscopic tandem differential mobility analyser (HTDMA), specifically the nature of the drying process that will influence the final crystalline form, will create systematic uncertainties upon comparisons with theoretical predictions. However, the magnitudes of all of the above sensitivities are potentially less than those introduced when using a semi ideal growth factor analogue for certain conditions

Existence of Ricci flows of incomplete surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction

New and extended parameterization of the thermodynamic model AIOMFAC: calculation of activity coefficients for organic-inorganic mixtures containing carboxyl, hydroxyl, carbonyl, ether, ester, alkenyl, alkyl, and aromatic functional groups

We present a new and considerably extended parameterization of the thermodynamic activity coefficient model AIOMFAC (Aerosol Inorganic-Organic Mixtures Functional groups Activity Coefficients) at room temperature. AIOMFAC combines a Pitzer-like electrolyte solution model with a UNIFAC-based group-contribution approach and explicitly accounts for interactions between organic functional groups and inorganic ions. Such interactions constitute the salt-effect, may cause liquid-liquid phase separation, and affect the gas-particle partitioning of aerosols. The previous AIOMFAC version was parameterized for alkyl and hydroxyl functional groups of alcohols and polyols. With the goal to describe a wide variety of organic compounds found in atmospheric aerosols, we extend here the parameterization of AIOMFAC to include the functional groups carboxyl, hydroxyl, ketone, aldehyde, ether, ester, alkenyl, alkyl, aromatic carbon-alcohol, and aromatic hydrocarbon. Thermodynamic equilibrium data of organic-inorganic systems from the literature are critically assessed and complemented with new measurements to establish a comprehensive database. The database is used to determine simultaneously the AIOMFAC parameters describing interactions of organic functional groups with the ions H^+, Li^+, Na^+, K^+, NH_(4)^+, Mg^(2+), Ca^(2+), Cl^−, Br^−, NO_(3)^−, HSO_(4)^−, and SO_(4)^(2−). Detailed descriptions of different types of thermodynamic data, such as vapor-liquid, solid-liquid, and liquid-liquid equilibria, and their use for the model parameterization are provided. Issues regarding deficiencies of the database, types and uncertainties of experimental data, and limitations of the model, are discussed. The challenging parameter optimization problem is solved with a novel combination of powerful global minimization algorithms. A number of exemplary calculations for systems containing atmospherically relevant aerosol components are shown. Amongst others, we discuss aqueous mixtures of ammonium sulfate with dicarboxylic acids and with levoglucosan. Overall, the new parameterization of AIOMFAC agrees well with a large number of experimental datasets. However, due to various reasons, for certain mixtures important deviations can occur. The new parameterization makes AIOMFAC a versatile thermodynamic tool. It enables the calculation of activity coefficients of thousands of different organic compounds in organic-inorganic mixtures of numerous components. Models based on AIOMFAC can be used to compute deliquescence relative humidities, liquid-liquid phase separations, and gas-particle partitioning of multicomponent mixtures of relevance for atmospheric chemistry or in other scientific fields

The sensitivity of secondary organic aerosol component partitioning to the predictions of component properties – Part 1: A systematic evaluation of some available estimation techniques

A large number of calculations of the absorptive partitioning of organic compounds have been made using a number of methods to predict the component vapour pressures, <i>p</i><sup>0</sup>, and activity coefficients, <i>γ</i><sub><i>i</i></sub>, required in the calculations. The sensitivities of the predictions in terms of the condensed component masses, volatility, O:C ratio, molar mass and functionality distributions to the choice of <i>p</i><sup>0</sup> and <i>γ</i><sub><i>i</i></sub> models and to the number of components to represent the organic mixture have been systematically compared. The condensed component mass was found to be highly sensitive to the vapour pressure model, and less sensitive to both the activity coefficient model and the number of components used to represent the mixture although the sensitivity to the change in property estimation method increased substantially with increased simplification in the treatment of the organic mixture. This was a general finding and was also clearly evident in terms of the predicted component functionality, O:C ratio, molar mass and volatility distributions of the condensed organic components. Within the limitations of the study, this clearly demonstrates the requirement for more accurate representation of the <i>p</i><sup>0</sup> and <i>γ</i><sub><i>i</i></sub> of the semi-volatile organic proxy components used in simplified models as the degree of simplification increases. This presents an interesting paradox, since such reduction in complexity necessarily leads to divergence from the complex behaviour of real multicomponent atmospheric aerosol

Size-resolved simulations of the aerosol inorganic composition with the new hybrid dissolution solver HyDiS-1.0: description, evaluation and first global modelling results

The dissolution of semi-volatile inorganic gases such as ammonia and nitric acid into the aerosol aqueous phase has an important influence on the composition, hygroscopic properties, and size distribution of atmospheric aerosol particles. The representation of dissolution in global models is challenging due to inherent issues of numerical stability and computational expense. For this reason, simplified approaches are often taken, with many models treating dissolution as an equilibrium process. In this paper we describe the new dissolution solver HyDiS-1.0, which was developed for the global size-resolved simulation of aerosol inorganic composition. The solver applies a hybrid approach, which allows for some particle size classes to establish instantaneous gas-particle equilibrium, whereas others are treated time dependently (or dynamically). Numerical accuracy at a competitive computational expense is achieved by using several tailored numerical formalisms and decision criteria, such as for the time- and size-dependent choice between the equilibrium and dynamic approaches. The new hybrid solver is shown to have numerical stability across a wide range of numerical stiffness conditions encountered within the atmosphere. For ammonia and nitric acid, HyDiS-1.0 is found to be in excellent agreement with a fully dynamic benchmark solver. In the presence of sea salt aerosol, a somewhat larger bias is found under highly polluted conditions if hydrochloric acid is represented as a third semi-volatile species. We present first results of the solver's implementation into a global aerosol microphysics and chemistry transport model. We find that (1) the new solver predicts surface concentrations of nitrate and ammonium in reasonable agreement with observations over Europe, the USA, and East Asia, (2) models that assume gas-particle equilibrium will not capture the partitioning of nitric acid and ammonia into Aitken-mode-sized particles, and thus may be missing an important pathway through which secondary particles may grow to radiation- and cloud-interacting size, and (3) the new hybrid solver's computational expense is modest, at around 10 % of total computation time in these simulations

Type-Decomposition of a Pseudo-Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.Comment: 18 page