Regimes of tropical low-level clouds are commonly identified according to large-scale subsidence and lower tropospheric stability (LTS). This definition alone is insufficient for the distinction between regimes and limits the comparison of low-level clouds from CloudSat radar observations and the ECHAM5 GCM run with the COSP radar simulator. Comparisons of CloudSat radar cloud altitude-reflectivity histograms for stratocumulus and shallow cumulus regimes, as defined above, show nearly identical reflectivity profiles, because the distinction between the two regimes is dependent upon atmospheric stability below 700 hPa and observations above 1.5 km. Regional subsets, near California and Hawaii, for example, have large differences in reflectivity profiles than the dynamically defined domain; indicating different reflectivity profiles exist under a given large-scale environment. Regional subsets are better for the evaluation of low-level clouds in CloudSat and ECHAM5 as there is less contamination between 2.5 km and 7.5 km from precipitating hydrometeors which obscured cloud reflectivities.

We propose a mathematical procedure to derive a stochastic parameterization for the bulk warm cloud micro-physical properties of collision and coalescence. Unlike previous bulk parameterizations, the stochastic parameterization does not assume any particular droplet size distribution, all parameters have physical meanings which are recoverable from data, all equations are independently derived making conservation of mass intrinsic, the auto conversion parameter is finely controllable, and the resultant parameterization has the flexibility to utilize a variety of collision kernels. This new approach to modelling the kinetic collection equation (KCE) decouples the choice of a droplet size distribution and a collision kernel from a cloud microphysical parameterization employed by the governing climate model. In essence, a climate model utilizing this new parameterization of cloud microphysics could have different distributions and different kernels in different climate model cells, yet employ a single parameterization scheme. This stochastic bulk model is validated theoretically and empirically against an existing bulk model that contains a simple enough (toy) collision kernel such that the KCE can be solved analytically. Theoretically, the stochastic model reproduces all the terms of each equation in the existing model and precisely reproduces the power law dependence for all of the evolving cloud properties. Empirically, values of stochastic parameters can be chosen graphically which will precisely reproduce the coefficients of the existing model, save for some ad-hoc non-dimensional time functions. Furthermore values of stochastic parameters are chosen graphically. The values selected for the stochastic parameters effect the conversion rate of mass cloud to rain. This conversion rate is compared against (i) an existing bulk model, and (ii) a detailed solution that is used as a benchmark. The utility of the stochastic bulk model is extended to include hydrodynamic and turbulent collision kernels for both clean and polluted clouds. The validation and extension compares the time required to convert 50\% of cloud mass to rain mass, compares the mean rain radius at that time, and used detailed simulations as benchmarks. Stochastic parameters can be chosen graphically to replicate the 50\% conversion time in all cases. The curves showing the evolution of mass conversion that are generated by the stochastic model with realistic kernels do not match corresponding benchmark curves at all times during the evolution for constant parameter values. The degree to which the benchmark curves represent ground truth, i.e. atmospheric observations, is unknown. Finally, among alternate methods of acquiring parameter values, getting a set of sequential values for a single parameter has a stronger physical foundation than getting one value per parameter, and a stochastic simulation is preferable to a higher order detailed method due to the presence of bias in the latter. / Graduate / 0725 0608 0405 / davidc@uvic.ca

Cette thèse s'intéresse à la modélisation numérique du couplage entre l'océan et l'atmosphère. Bien que présentant un certain nombre de caractéristiques communes, ces deux milieux physiques sont suffisamment dissemblables pour être numériquement simulés par des modèles distincts, incluant chacun des spécificités propres. Par conséquent, leurs interactions sont prises en compte via des algorithmes de couplage multiphysique.La mise en place de tels algorithmes nécessite une bonne compréhension des modélisations des milieux océanique et atmosphérique, en particulier au voisinage de leur interface commune. C'est pourquoi une partie conséquente de la présente thèse dissèque, analyse et complète les paramétrisations turbulentes, qui sont des mécanismes numériques définis au niveau continu, traitant la couche limite turbulente au voisinage de la surface océanique. Les travaux entrepris ont permis d'identifier deux sources d'erreurs, théoriquement et numériquement significatives, dans la modélisation numérique standard de l'interface océan-atmosphère.La première source d'erreur se manifeste dans les formulations continues des paramétrisations turbulentes: celles-ci sont actuellement utilisées de manière incomplète, ce qui se traduit par le caractère mathématiquement irrégulier des solutions qu'elles génèrent. En revenant aux fondements de la théorie dont les paramétrisations découlent, la présente thèse étend leur domaine d'application, permettant de générer des profils de solution réguliers, dans un cadre théorique uniforme et bi-domaine. Les effets d'une telle extension sont numériquement évalués sur des cas tests physiquement réalistes: celle-ci peut mener à des biais considérables (de l'ordre de 20%) dans les flux échangés entre océan et atmosphère. D'un point de vue théorique, cette extension permet de définir des critères simples sous lesquels le couplage océan-atmosphère peut être considéré comme cohérent par rapport aux deux domaines physiques, et surtout aux paramétrisations turbulentes.La seconde source d'erreur est de nature algorithmique: elle concerne la discrétisation temporelle des mécanismes de couplage. Les méthodes actuelles, dites ad hoc, ne garantissent pas une complète cohérence des flux d'un modèle à l'autre. Les algorithmes de Schwarz globaux en temps, issus de thématiques liées à la décomposition de domaine, constituent une piste intéressante pour traiter ces aspects. La mise en place de tels algorithmes sur des modèles physiquement réalistes représente un défi considérable. Leur impact numérique sur des cas tests simplifiés est évalué. L'étude préalable des paramétrisations turbulentes permet de donner des pistes quant au développement d'algorithmes de couplage, concernant à la fois la cohérence du couplage précédemment introduite, et l'incorporation graduelle d'effets physiques plus complexes. / This thesis focuses on the numerical modelling of the air-sea coupling. Although they share some common features, these two physical environments are sufficiently dissimilar for their numerical treatment to be carried out by distinct models, each including their own specificities. The interactions between these two components are thus taken into account through coupling algorithms.Implementing such algorithms requires proper understanding of the oceanic and atmospheric modelling, most importantly in the vicinity of their common interface. Therefore a substantial part of this thesis dissects, analyzes and completes turbulent parameterization schemes, which are the numerical mechanisms, defined at a continuous level, through which the turbulent surface layer at the vicinity of the sea surface is treated. Two theoretically and numerically meaningful sources of errors in the standard numerical modelling of the air-sea interface have been isolated.The first source of error lies in the continuous formulation of the turbulent parameterizations, which are currently used in an incomplete manner, leading to mathematically irregular solution profiles. By carefully studying their theoretical bases, this thesis extends the parameterizations, allowing them to generate regular profiles within a standardized, bi-domain framework. Numerical investigations on physically relevant test cases show that including such an extension can result in considerable bias (of the order of 20%) in air-sea fluxes evaluations. From a theoretical perspective, carrying this extension leads to establishing simple criteria under which the air-sea coupling can be considered as coherent with respect to the two physical environments, and more importantly, to the turbulent parameterizations.The second source of error is algorithmic in essence: it is linked to the temporal discretization of the coupling mechanisms. Existing ad hoc methods do not guarantee perfect coherence of the air-sea fluxes from one model to the other. Global in time Schwarz algorithms, which have first been developed as domain decomposition methods, are good candidates for correcting these flaws, although their implementation to the air-sea context is a considerable challenge, given the complexity of this problem. Investigations on the numerical impact of such algorithms are carried out on simplified test cases. Thanks to the undertaken work on turbulent parameterizations, perspectives on the development of coupling algorithms are given, regarding both their coherence as per the aforementioned conditions, and the gradually increasing complexity of physical effects that are accounted for.

A census of 24 coupled (CMIP) and 13 uncoupled (AMIP) models from the Intergovernmental Panel on Climate Change (IPCC) fourth assessment report (AR4) were compared with observations and reanalysis to show varied ability of the models to simulate Caribbean precipitation and mechanisms related to precipitation in the region. Not only were errors seen in the annual mean, with CMIP models underestimating both rainfall and sea surface temperature (SST) and AMIP models overestimating rainfall, the annual cycle was also incorrect. Large overestimates of precipitation at all SSTs (and particularly above 28 degrees C) and at vertical circulations less than -10 hPa/day (the deep convective regime) were inherent in the atmospheric models with models using spectral type convective parameterizations performing best. In coupled models, however, errors in the frequency of occurrence of SSTs (the distribution is cold biased) and deep convective vertical circulations (reduced frequency) lead to an underestimation of Caribbean mean precipitation. On daily timescales, the models were shown to produce too frequent light rainfall amounts (especially less than 1 mm/day) and dry extremes and too few heavy rainfall amounts and wet extremes. The simulation of the mid-summer drought (MSD) proved a challenge for the models, despite their ability to produce a Caribbean low-level jet (CLLJ) in the correct location. Errors in the CLLJ, such as too strong magnitude and weak semi-annual cycle, were worse in the CMIP models and were attributed to problems with the location and seasonal evolution of the North Atlantic subtropical high (NASH) in both CMIP and AMIP models. Despite these discrepancies between models and observations, the ability of the models to simulate the correlation between the CLLJ and precipitation varied based on season and region, with the connection with United States precipitation particularly problematic in the AMIP simulations. An observational study of intraseasonal precipitation in the Caribbean showed an explicit connection between the Madden-Julian oscillation (MJO) and Caribbean precipitation for the first time. Precipitation anomalies up to 50 percent above (below) the annual mean are observed in phases 1 and 2 (5 and 6) of the MJO and are related to changes in the CLLJ, that is also modulated by the MJO. Considerable progress has been made on identifying both problems and successes in the simulation of Caribbean climate in general circulation models, but many areas still require investigation.

The topic of this thesis is motivated by recent developments in Architectural Geometry, namely Eike Schling’s asymptotic gridshells and progress in solutions for paneling freeform facades. An asymptotic gridshell is fabricated from flat straight lamellas of bendable material such as sheet metal. These strips are then arranged in a grid-like spatial structure, such that the lamellas are orthogonal to a reference surface, which however is not materialized. Differential geometry then tells us that the strips must follow asymptotic curves of that reference surface. The actual construction is simplified if angles at nodes are constant. If that angle is a right angle, one gets minimal surfaces as reference surfaces. If the angle is constant, one obtains negatively curved surfaces with a constant ratio of principal curvatures (CRPC surfaces). Their characteristic parameterizations are equi-angular asymptotic parameterizations. We are also interested in the positively curved CRPC surfaces. Due to the relation between curvatures, they have a one-parameter family of curvature elements, which facilitates cost-effective paneling solutions through mold-reuse. Our approach to positively curved CRPCS surfaces is again based on equi-angular characteristic parameterizations. These characteristic parameterizations are conjugate and symmetric with respect to the principal curvature directions. After a review of the required results from classical surface theory, we first present a derivation of rotational CRPC surfaces. By simple geometric considerations one can find their characteristic parameterizations. In this way we add some new insight to this known class of surfaces. However, it turns out to be very hard to come up with explicit results on non-rotational CRPC surfaces. This is in big contrast to the special case of minimal surfaces which are characterized be the constant principal curvature ratio -1. Due to the difficulties in handling smooth CRPC surfaces, we turn to discrete models in form of constrained quad meshes. The discrete models belong to the area of Discrete Differential Geometry. There, one does not discretize equations from the smooth theory, but fundamental concepts of the theory. We introduce the basic structures needed in this context: asymptotic nets, conjugate nets and principal symmetric nets. The latter are a recent development in discrete differential geometry and characterized by spherical vertex stars. This means that a vertex of the quad mesh and its four connected neighbors lie on a sphere. If that sphere degenerates to a plane at all vertices, one has the classical discrete asymptotic parameterization as an A-net. Several ways to discretize the constant intersection angle are presented. The actual computation of discrete CRPC surfaces is performed with numerical optimization with an appropriately regularized Gauss-Newton algorithm for solving a nonlinear least squares problem. Optimization requires initial configurations. Those can come from the known classes of CRPC surfaces such as rotational surfaces of minimal surfaces. The latter case yields some surprising results on negatively curves CRPC surfaces of nontrivial topology. In general, such discrete models can serve as a guiding line for future research on the theoretical side. This is briefly indicated in the final discussion on future research directions.

Regimes of tropical low-level clouds are commonly identified according to large-scale subsidence and lower tropospheric stability (LTS). This definition alone is insufficient for the distinction between regimes and limits the comparison of low-level clouds from CloudSat radar observations and the ECHAM5 GCM run with the COSP radar simulator. Comparisons of CloudSat radar cloud altitude-reflectivity histograms for stratocumulus and shallow cumulus regimes, as defined above, show nearly identical reflectivity profiles, because the distinction between the two regimes is dependent upon atmospheric stability below 700 hPa and observations above 1.5 km. Regional subsets, near California and Hawaii, for example, have large differences in reflectivity profiles than the dynamically defined domain; indicating different reflectivity profiles exist under a given large-scale environment. Regional subsets are better for the evaluation of low-level clouds in CloudSat and ECHAM5 as there is less contamination between 2.5 km and 7.5 km from precipitating hydrometeors which obscured cloud reflectivities.

Over the years, global climate modelling has advanced, aiming for realistic and precise models by increasing their complexity. An integral component of climate models, the physics parameterizations, are a major limitation, but are required due to limited computational power. Grid adaptivity is an avenue that is being explored to mitigate these challenges, but comes with its own difficulties. For example, the question of whether the physics should be ``scale-aware’’, by adjusting according to the resolution and the fact that parameterizations are optimized for specific grid ranges. To research these challenges, test cases that work in both the adaptive and non-adaptive cases are required. This thesis concentrates on physics parameterizations of Atmospheric Global Climate Models (AGCMs) presenting the current hierarchy of idealized physics parameterizations found in the literature. It focuses on and provides a comprehensive explanation of a simplified dry physics model for AGCMs, exploring where it is situated in the current hierarchy and its steady states in the uncoupled case. A coupling of the physics model to the adaptive dynamical core wavetrisk is explained and explored. This includes characterizing the results in the non-adaptive case for time convergence, grid convergence, and the effects of the soil, while also benchmarking the climatology of the coupling. The simplified dry physics model introduces another level of complexity in the current dry physics hierarchy and is stable in the coupled and uncoupled cases. A decreasing temperature trend with height is observed, however warmer surface temperatures and cooler upper atmosphere temperatures, than that of Earth, are produced in the steady states. Additionally a linear rate of convergence in space is noted and an improvement in parallel efficiency with resolution is required. Overall these results can be used as a benchmark for future coupling in the adaptive case. / Thesis / Master of Science (MSc)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Weather Research and Forecasting (WRF) model is used in the simulation of 26 cases of cyclogenesis in southeastern South America, on the cyclogenetic region near the La Plata River mouth. The simulations comprised 26 cases of cyclogenesis in which rainfall was observed over the region, especially on the state of Rio Grande do Sul. Given the important role of convective parameterization in the production of precipitation into a numerical weather prediction model, this study aims to evaluate the precipitation produced in the simulations of cases of cyclogenesis using three different convection parameterization schemes: Betts-Miller-Janjic (BMJ), Grell-Dévényi (GD) e Kain-Fritsch (KF), keeping the other physical options, to isolate the impact of using different parameterizations for convection. For the evaluation of simulated rainfall, verification skill scores were used in order to make an objective assessment of simulated rain field from the observed data set (TRMM and MERGE), also allowing a comparison between simulations with different schemes. The results of the skill scores showed that, in general, the WRF best represents the rain of lower thresholds, with indexes based positioning (ETS, POD and RAF) with values closer to the ideal. The BIAS score indicated that the area of light rain is usually overestimated, while moderate and heavy rain showed greater differences between simulations with KF and BMJ schemes and verified with the two sets of observed data. Differences in the verification scores showed that the way how the convective flows are treated by schemes exert great influence on the precipitation produced, with the largest differences being observed between the BMJ and KF schemes. The simulations with the BMJ, wich is a convective adjustment based scheme, produced clearly smaller areal coverage and rainfall volume, especially if compared to the simulations with the KF scheme. Along with lower production of precipitation, the simulations with the BMJ showed, on average, smaller partition of convective rainfall compared to others. The sea level pressure field showed no significant differences, with the WRF simulating consistently with the GFS-FNL analysis, the position and central pressure of the cyclones. / O modelo Weather Research and Forecasting (WRF) é utilizado na simulação de 26 casos de ciclogênese no sudeste da América do Sul, na região ciclogenética próxima da foz do Rio da Prata. As simulações compreenderam 26 casos de ciclogênese nos quais foi observada precipitação sobre a região, especialmente sobre o estado do Rio Grande do Sul. Tendo em vista o importante papel das parametrizações convectivas na produção de precipitação em um modelo numérico de previsão de tempo, este estudo visa avaliar a precipitação produzida nas simulações dos casos de ciclogênese utilizando três diferentes esquemas de parametrização da convecção: Betts-Miller-Janjic (BMJ), Grell-Dévényi (GD) e Kain-Fritsch (KF), mantendo as outras opções físicas constantes, de modo a isolar o impacto do uso das diferentes parametrizações para a convecção. Para a avaliação da precipitação simulada foram utilizados índices estatísticos, com o objetivo de avaliar objetivamente o campo de chuva simulado a partir do conjunto de dados observados (TRMM e MERGE), possibilitando também uma comparação entre as simulações com os diferentes esquemas. Os resultados dos índices mostraram que em geral, o WRF representa melhor a chuva dos limiares mais fracos, com índices baseados no posicionamento (ETS, POD e RAF) tendo valores mais próximos dos ideais. O índice BIAS indicou que a área de chuva fraca é, em geral, superestimada enquanto que a de chuva moderada a forte apresentou diferenças maiores entre as simulações com os esquemas KF e BMJ e na verificação com os dois conjuntos de dados observados. As diferenças nos índices mostraram que a forma como os fluxos convectivos são tratados pelos esquemas exerce grande influência na precipitação produzida, com as maiores diferenças sendo observadas entre os esquemas KF e BMJ. As simulações com o esquema BMJ, que é baseado no ajuste convectivo dos perfis, claramente produziram menor área e volume de chuva, principalmente se comparadas às simulações com o KF. Juntamente com menor produção de precipitação, as simulações com o BMJ apresentaram, em média, menor partição de precipitação convectiva se comparada às outras. O campo de pressão ao nível do mar não apresentou diferenças muito significativas, com o WRF simulando de forma consistente com as análises FNL-GFS, o posicionamento e pressão central do ciclone.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This thesis aims to present aspects of parametric equations and planar a ne transformations that can be exploited in basic education. With respect to parameterizations we present a succession of elementary examples and make the comparison between the parametric and Cartesian equations - highlighting the advantages of using one over the other. Furthermore, we discussed the process of obtaining the Cartesian equations from parametric and importance of this type of equations for physics. With respect to a ne transformations our interest is to look at them from the perspective of Felix Klein program, in which a geometry is classi ed as a set of objects on the action of a group set. We emphasize some special transformations and their importance in the generation of the a nity group and the implementation of coordinated of change process. We emphasize that we have not aimed this work to be fully applied as teaching materials for elementary education, what we want is it to be a provocateur to the teacher researcher instinct . / A presente disserta c~ao tem como objetivo apresentar aspectos das equa c~oes param etricas e das transforma c~oes a ns planares que podem ser explorados no ensino b asico. No que diz respeito as parametriza c~oes apresentamos uma sucess~ao de exemplos elementares e fazemos a compara c~ao entre as equa c~oes param etricas e as cartesianas - destacando as vantagens de usar uma em detrimento da outra. Al em disso, discutimos sobre o processo de obter as equa c~oes cartesianas a partir das param etricas e a import^ancia dessa modalidade de equa c~oes para a f sica. No que se refere as transforma c~oes a ns nosso interesse e olhar para elas segundo a perspectiva do programa de Felix Klein, onde uma geometria e classi cada como um conjunto de objetos sobre a a c~ao de um grupo xado. Enfatizamos algumas transforma c~oes especiais e a import^ancia das mesmas na gera c~ao do grupo de a nidades e na implementa c~ao do processo de mudan ca de coordenadas. Ressaltamos que n~ao temos como objetivo que essa material seja totalmente aplicado como material did atico para o ensino b asico, o que desejamos e que ele seja um provocador ao instinto pesquisador do professor.

Appropriate calculation of surface turbulent fluxes between the atmosphere and the underlying surface is one of the major challenges in geosciences. In practice, the surface turbulent fluxes are estimated from the mean surface meteorological variables based on the bulk transfer model combined with Monnin-Obukhov Similarity (MOS) theory. Few studies have been done to examine the extent that to which this flux parameterization framework can be applied to different weather and surface conditions. A novel validation method is developed in this thesis research, which is applied to evaluate the surface flux parameterization using in-situ observations. The main findings are: (a) the theoretical prediction that uses MOS theory does not match well with those directly computed from the observations collected in the coastal region. (b) Large spread in exchange coefficients mainly occurs in the calm wind regime with strong stability. (c) Large turbulent eddies, which depend largely on the mean flow and surface conditions, tend to break the constant flux assumption in the surface layer.