Master of Science / Department of Mathematics / Nathan Albin / This work utilizes finite differences to approximate the first derivative of non-periodic smooth functions. Math literature indicates that stabilizing Partial Differential Equation solvers based on high order finite difference approximations of spatial derivatives of a non-periodic function becomes problematic near a boundary. Hagstrom and Hagstrom have discovered a method of introducing additional grid points near a boundary, which has proven to be effective in stabilizing Partial Differential Equation solvers. Hagstrom and Hagstrom demonstrated their method for the case of the one-dimensional advection equation using spatial derivative approximations of even orders up to twenty-second order. In this dissertation, we explore the efficacy of the Hagstrom and Hagstrom method for the same Partial Differential Equation with spatial derivative approximations of odd orders and orders higher than twenty-two and report the number and locations of additional grid points required for stability in each case.

The stably stratified nocturnal boundary layer (SBL) can be classified into two distinct regimes: one with moderate to strong winds, weak stratification and mechanically sustained turbulence (wSBL) and the other one with moderate to weak wind conditions, strong stratification and collapsed turbulence (vSBL). With the help of a hidden Markov model (HMM) analysis of the three dimensional state variable space of stratification, mean wind speeds, and wind shear the SBL can be classified in these two regimes in both the Reynolds-averaged as well as turbulence state variables. The two-regime SBL is a generic structure at different tower sites around the world independent of the location specific conditions. Besides clustering the data the HMM analysis calculates the most likely regime occupation sequence which allows for detailed analysis of the structure of the meteorological state variables in conditions of very persistent nights. Conditioning on these very persistent nights clear influences of external drivers (such as pressure gradient force and low level cloud cover) are found. As the HMM analysis captures regime transitions accurately changes of state variables and external drivers across transitions can easily be assessed. Different meteorological state variables behave in times of turbulence collapse (wSBL to vSBL transition) and turbulence recovery (vSBL to wSBL transitions) as expected physically. The results reveal further that clear precursors for transitions in the state variable profiles or external drivers are cannot be determined and that on observed timescales regime transitions are relatively sharp. The absence of clear precursors suggests that parameterisations of SBL regime behaviour and turbulence in the two regimes in weather and climate models have to be stochastic. As regime statistics are relatively insensitive to changes in the stochastic properties of the HMM analysis observed regime statistics are compared to ’freely-running’ Markov chains. The SBL regime statistics do not follow a simple Markov process and more complex parameterisations are necessary. A possible approach of parameterising SBL regime behaviour stochastically using climatological results from this analysis is presented. / Graduate / 2019-12-17

<p>In this thesis various descriptions of the near surface atmospheric flow over a high latitude glacier is used in an effort to increase our understanding of the basic flow dynamics there.</p><p>Through their contribution to sea-level change, mountain glaciers play a significant role in Earth’s climate system. Properties of the near surface atmospheric flow are important for understanding glacier response to climate change.</p><p>Here, the near surface atmospheric flow is studied from several perspectives including the effects of both rotation and slope. Rotation is an important aspect of most atmospheric flows and its significance for mesoscale flows have gained recognition over the last years. Similarly, the very stable boundary layer (VSBL) has lately gained interest. Within a VSBL over sloping terrain katabatic flow is known to be usual and persistent. For the present thesis a combination of numerical and simple analytical models as well as observations from the Vatnajökull glacier on Iceland have been used. The models have continuously been compared to available observations. Three different approaches have been used: linear wave modeling, analytic modeling of katabatic flow and of the Ekman layer, and numerical simulations of the katabatic flow using a state of the art mesoscale model. The analytic models for the katabatic flow and the Ekman layer used in this thesis both utilizes the WKB method to allow the eddy diffusivity to vary with height. This considerably improves the results of the models. Among other findings it is concluded that: a large part of the flow can be explained by linear theory, that good results can be obtained for surface energy flux using simple models, and that the very simple analytic models for the katabatic flow and the Ekman layer can perform adequately if the restraint of constant eddy diffusivity is relieved.</p>

In this thesis various descriptions of the near surface atmospheric flow over a high latitude glacier is used in an effort to increase our understanding of the basic flow dynamics there. Through their contribution to sea-level change, mountain glaciers play a significant role in Earth’s climate system. Properties of the near surface atmospheric flow are important for understanding glacier response to climate change. Here, the near surface atmospheric flow is studied from several perspectives including the effects of both rotation and slope. Rotation is an important aspect of most atmospheric flows and its significance for mesoscale flows have gained recognition over the last years. Similarly, the very stable boundary layer (VSBL) has lately gained interest. Within a VSBL over sloping terrain katabatic flow is known to be usual and persistent. For the present thesis a combination of numerical and simple analytical models as well as observations from the Vatnajökull glacier on Iceland have been used. The models have continuously been compared to available observations. Three different approaches have been used: linear wave modeling, analytic modeling of katabatic flow and of the Ekman layer, and numerical simulations of the katabatic flow using a state of the art mesoscale model. The analytic models for the katabatic flow and the Ekman layer used in this thesis both utilizes the WKB method to allow the eddy diffusivity to vary with height. This considerably improves the results of the models. Among other findings it is concluded that: a large part of the flow can be explained by linear theory, that good results can be obtained for surface energy flux using simple models, and that the very simple analytic models for the katabatic flow and the Ekman layer can perform adequately if the restraint of constant eddy diffusivity is relieved.

Fourier-based stochastic simulation of wind fields commonly used in wind turbine loads computations is unable to account for contrasting states of atmospheric stability. Flow fields in the stable boundary layer (SBL), for instance, have characteristics such as enhanced wind shear and veering wind direction profiles; the influence of such characteristics on utility-scale wind turbine loads has not been studied. To investigate these influences, we use large-eddy simulation (LES) to generate inflow wind fields and to estimate load statistics for a 5-MW wind turbine model. In the first part of this thesis, we describe a procedure employing LES to generate SBL wind fields for wind turbine load computations. In addition, we study how large-scale atmospheric conditions affect the characteristics of wind fields and turbine loads. Next, in the second part, we study the contrasting characteristics of LES-SBL and stochastic NBL flow fields and their influences on wind turbine load statistics by isolating effects of the mean wind (shear) profile and of variation in wind direction and turbulence levels over the rotor sept area. Among large-scale atmospheric conditions, the geostrophic wind speed and surface cooling rate have the greatest influence on flow field characteristics and, thus, on wind turbine loads. Higher geostrophic winds lead to increased mean and standard deviation values of the longitudinal wind speed at hub height. Increased surface cooling rates lead to steeper shear profiles and appear to also increase fatigue damage associated with out-of-plane blade root moments. In summary, our studies suggest that LES may be effectively used to model wind fields in the SBL, to study characteristics of turbine-scale wind fields, and to assess turbine loads for conditions that are not typically examined in design standards. / text

Alpine valleys are rarely closed systems, implying that the atmospheric boundary layer of a particular valley section is influenced by the surrounding terrain and large-scale flows. A detailed characterisation and quantification of these effects is required in order to design appropriate parameterisation schemes for complex terrains. The focus of this work is to improve the understanding of the effects of surrounding terrain (plains, valleys or tributaries) on the heat and mass budgets of the stable boundary layer of a valley section, under dry and weak large-scale wind conditions. Numerical simulations using idealised and real frameworks are performed to meet this goal. Several idealised terrains (configurations) were considered: an infinitely long valley (i.e. two-dimensional), and upstream valleys opening either on a plain (valley-plain), on a wider valley (draining) or on a narrower valley (pooling). In three-dimensional valleys, two main regimes can be identified for all configurations: a transient regime, before the down-valley flow develops, followed by a quasi-steady regime, when the down-valley flow is fully developed. The presence of a downstream valley reduces the along-valley temperature difference, therefore leading to weaker down-valley flows. As a result, the duration of the transient regime increases compared to the respective valley-plain configuration. Its duration is longest for the pooling configuration. For strong pooling the along-valley temperature difference can reverse, forcing up-valley flows from the narrower towards the wider valley. In this regime, the average cooling rate at the valley-scale is found to be a maximum and its magnitude is dependent on the configuration considered. Therefore pooling and draining induce colder and deeper boundary layers than the respective valley-plain configurations. In the quasisteady regime the cooling rate is smaller than during the transient regime, and almost independent of the configuration considered. Indeed, as the pooling character is more pronounced, the warming contribution from advection to the heat budget decreases because of weaker down-valley flows, and so does the cooling contribution from the surface sensible heat flux. The mass budget of the valley boundary layer was found to be controlled by a balance between the convergence of downslope flows at the top of the boundary layer and the divergence of the down-valley flow along the valley axis, with negligible contributions of subsidence far from the valley sidewalls. The mass budget highlighted the importance of the return current above the down-valley flow, which may contribute significantly to the inflow of air at the top of the boundary layer. A case-study of a persistent cold-air pool event which occurred in February 2015 in the Arve River Valley during the intensive observation period 1 (IOP1) of the PASSY- 2015 field campaign, allowed us to quantify the effects of neighbouring valleys on the heat and mass budgets of a real valley atmosphere. The cold-air pool persisted as a result of warm air advection at the valley top, associated with the passage of an upper-level ridge over Europe. The contributions from each tributary valley to the mass and heat budgets of the valley atmosphere were found to vary from day to day within the persistent stage of the cold-air pool, depending on the large-scale flow. Tributary flows had significant impact on the height of the inversion layer and the strength of the cold-air pool, transporting a significant amount of mass within the valley atmosphere throughout the night. The strong stratification of the near-surface atmosphere prevented the tributary flows from penetrating down to the valley floor. The evolution of the large-scale flow during the episode had a profound impact on the near-surface circulation of the valley. The channelling of the large-scale flow at night, can lead to the decrease of the horizontal temperature difference driving the near-surface down-valley flow, favouring the stagnation of the air close to the ground.

Turbulent collapse and recovery are both observed to occur abruptly in the atmospheric stable boundary layer (SBL). The understanding and predictability of turbulent recovery remains limited, reducing numerical weather prediction accuracy. Previous studies have shown that regime occupation is the result of the net effect of highly variable processes, from turbulent to synoptic scales, making stochastic methods a compelling approach. Idealized stable boundary layer models have shown that under some circumstances, regimes can be related to the stable branches of model equilibria, and an additional unstable equilibrium is predicted. This work seeks to determine the extent to which the SBL regime occupation can be explained using a one-dimensional stochastic differential equation (SDE). The drift and diffusion coefficients of the SDE of an input time series are approximated from the statistics of its averaged time tendencies. These approximated coefficients are fit using Gaussian Process Regression. Probabilistic estimates of the system's equilibrium points are then found and used to create an empirical bifurcation diagram without making any prior assumptions on the dynamical form of the system. This data driven bifurcation diagram is compared to modelled predictions. The analysis is repeated on several meteorological towers around the world to assess the influence of local meteorological settings. This work provides empirical insights into the nature of regime dynamics and the extent to which the SBL displays hysteresis. / Graduate

Améliorer la représentation de la couche limite stable constitue un des grands challenges de la prévision numérique du temps et du climat. Sa représentation est clé pour la prévision du brouillard, du gel des surfaces, des inversions de température, du jet de basse couche et des épisodes de pollution. De plus, à l'échelle climatique, la hausse de la température moyenne globale de l'air en surface impacte davantage les régions polaires : améliorer la représentation de la couche limite stable est un enjeu important pour réduire les incertitudes autour des projections climatiques. Depuis une quinzaine d'années, les exercices d'intercomparaison de modèles GABLS ont montré que le mélange turbulent dans la couche limite stable est généralement surestimé par les modèles de prévision du temps. En effet, de nombreux modèles intensifient artificiellement l'activité de leur schéma de turbulence afin d'éviter une décroissance inévitable du mélange lorsque la stabilité dépasse un seuil critique en terme de nombre de Richardson gradient. Ce problème numérique et théorique n'est pas en accord avec de nombreuses observations et simulations à haute résolution qui montrent une activité turbulente séparée en deux régimes : un régime faiblement stable dans lequel l'atmosphère est turbulente de manière continue et intense, et un régime très stable dans lequel la turbulence est très intermittente, anisotrope et faible en intensité. Ces travaux de thèse s'articulent autour de deux parties dont l'objectif principal est d'améliorer la paramétrisation de la turbulence dans le modèle atmosphérique de recherche Méso-NH développé conjointement par Météo-France et le Laboratoire d'Aérologie, et dans le modèle opérationnel AROME. Cette étude utilise une méthodologie communément employée dans le développement de paramétrisations qui consiste à comparer des simulations à très haute résolution qui résolvent les structures turbulentes les plus énergétiques (LES) à des simulations uni-colonnes d'un modèle méso-échelle. Plusieurs simulations 3D couvrant différents régimes de stabilité de l'atmosphère sont réalisées avec Méso-NH. Les limites du modèle LES en stratification stable sont documentées. Une première partie répond à la problématique de la surestimation du mélange dans le régime faiblement stable. Une expression originale pour la longueur de mélange est formulée. La longueur de mélange est un paramètre clé pour les schémas de turbulence associés à une équation pronostique pour l'énergie cinétique turbulente. Cette longueur de mélange non-locale combine un terme de cisaillement vertical du vent horizontal à une formulation existante qui repose sur la flottabilité. Le nouveau schéma est évalué dans des simulations 1D par rapport aux LES d'une part ; et dans le modèle opérationnel AROME par rapport aux observations de l'ensemble du réseau opérationnel de Météo-France d'autre part. Une deuxième partie apporte des éléments d'évaluation d'un schéma combinant deux équations pronostiques pour les énergies cinétiques et potentielles turbulentes. En condition stable, le flux de chaleur négatif contribue à la production d'énergie potentielle turbulente. L'interaction entre les deux équations d'évolution permet, via une meilleure prise en compte de l'anisotropie et d'un terme à contre gradient dans le flux de chaleur, de limiter la destruction de l'énergie turbulente dans les modèles. Dans les cas simulés, cette nouvelle formulation ne montre pas un meilleur comportement par rapport à un schéma à une équation pour l'énergie cinétique turbulente car le mécanisme d'auto-préservation n'est pas dominant par rapport au terme de dissipation. Il conviendra d'améliorer la paramétrisation du terme de dissipation dans le régime très stable. / The modeling of the stable atmospheric boundary layer is one of the current challenge faced by weather and climate models. The stable boundary layer is a key for the prediction of fog, surface frost, temperature inversion, low-level jet and pollution peaks. Furthermore, polar regions, where stable boundary layer predominates, are one of the region with the largest temperature rise : the stable boundary layer modeling is crucial for the reduction of the spread of climate predictions. Since more than 15 years, the GABLS models intercomparison exercices have shown that turbulent mixing in the stable boundary layer is overestimated by numerical weather prediction models. Numerous models artificially strengthen the activity of their turbulence scheme to avoid a laminarization of the flow at a critical value of the gradient Richardson number. The existence of this threshold is only a theoretical and a numerical issues. Numerous observations and high-resolution numerical simulations do not support this concept and show two different regimes : the weakly stable boundary layer that is continuously and strongly turbulent; and the very stable boundary layer globally intermittent with a highly anisotropic and very weak turbulence. This thesis aims at improving the turbulence scheme within the atmospheric research model Méso-NH developped by Météo-France and the Laboratoire d'Aérologie, and the operational weather forecast model AROME. We use a traditional methodology based on the comparison of high-resolution simulations that dynamically resolve the most energetic turbulent eddies (Large-Eddy Simulations) to single-column simulations. Several LES covering the weakly and the very stable boundary layer were performed with Méso-NH. The limits of applicability of LES in stratified conditions are documented. The first part of the study deals with the overmixing in the weakly stable boundary layer. We propose a new diagnostic formulation for the mixing length which is a key parameter for turbulence schemes based on a prognostic equation for the turbulent kinetic energy. The new formulation adds a local vertical wind shear term to a non-local buoyancy-based mixing length currently used in Méso-NH and in the French operational model AROME. The new scheme is evaluated first in single-column simulations with Méso-NH and compared to LES, and then in the AROME model with respect to observations collected from the operational network of Météo-France. The second part presents a theoretical and numerical evaluation of a turbulence scheme based on two prognostic equations for the turbulent kinetic and potentiel energies. In stratified conditions, the heat flux contributes to the production of turbulent potential energy. The laminarization of the flow is then limited by a reduction of the destruction of the turbulent kinetic energy by a better representation of the anisotropy and a counter-gradient term in the heat flux. On the simulated cases, this new formulation behaves similarly than the scheme with one equation for the turbulent kinetic energy because the self-preservation mechanism is not dominant compared to the dissipation term. Further research should improve the turbulent kinetic energy dissipation closure in the very stable regime.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The appropriate estimation of turbulent fluxes in very stable conditions is a great challenge for numerical models that simulate the average behavior of the stable boundary layer. Although many features of the flow are usually reproduced, the intermittent variation turbulence is not simulated by most atmospheric models that use K theory for the determination of turbulent fluxes in the stable boundary layer. Moreover, the transition between the very stable and weakly stable regimes of the stable boundary layer is not well understood and described by numerical models. Therefore, in this work a one-and-a-half order numerical model is proposed to represent the average behavior of the nocturnal boundary layer. The model is based in the one proposed by Costa et al. (2011). In the presently proposed scheme, both the sensible heat flux and temperature variance are solved by prognostic equations in order to add degrees of freedom and physical detail to the model. Throughout the work, comparisons are made among the solutions varying different parameters. Results are also compared to the solutions obtained using the model proposed by Costa et al. (2011). The results show that the present model is able to reproduce the transition between the coupled and decoupled stable boundary layer regimes, in a manner similar to what is observed in nature. It also reproduces the occurrence of intermittent non periodical events and the formation of shallow mixing in weak wind conditions. The inclusion of prognostic equations for the sensible heat flux and for the temperature variance provides transitions between regimes at larger winds than those obtained when these quantities are parameterized, and closer to the observed values. The model provides a dependence of the potential temperature scale, θ∗ and of the sensible heat flux on the wind speed that is similar to observations. It also reproduces abrupt transitions between the stable boundary layer regimes, something not observed in the model proposed by Costa et al. (2011). The turbulent kinetic energy balance obtained by the model is closer to the observed by Acevedo et al. (2016) than was obtained by Costa et al. (2011). Dissipation is the dominat mechanism of turbulence destruction in very stable conditions, a role played by buoyant destruction in the model by Costa et al. (2011). The results sustain the hypothesis proposed by Van Weil et al. (2012), that the very stable regime happens when turbulence is not capable of sustaining turbulent heat fluxes large enough to accompany the long wave radiative loss. / A estimativa adequada dos fluxos turbulentos em condições muito estáveis ainda é um grande desafio para os modelos numéricos que descrevem o comportamento médio da CLE. Embora, muitas características do escoamento sejam simuladas, a variação intermitente da turbulência não é reproduzida por grande parte dos modelos atmosféricos que utilizam teoria K na estimativa dos fluxos turbulentos na CLE. Além disso, a transição entre os regimes muito estável e fracamente estável da CLE, ainda, não é bem compreendida e descrita pelos modelos numéricos. No presente trabalho é proposto um modelo numérico, de uma ordem e meia, para descrever o comportamento médio de uma camada limite noturna. O modelo é baseado no que foi proposto por COSTA et al. (2011). O modelo aqui proposto tanto o fluxo de energia na forma de calor sensível como a variância de temperatura são estimados através de equações prognósticas, afim de acrescentar graus de liberdade ao sistema e de modo a acrescentar detalhamento físico à solução. Ao longo do trabalho, foram realizadas comparações entre as soluções obtidas no modelo variando diferentes parâmetros. As soluções também são comparadas com as obtidas utilizando o modelo proposto por Costa et al. (2011). Os resultados mostram que o presente modelo é capaz de reproduzir as transições entre os estados acoplado e desacoplado da camada limite estável, de forma semlhante à observada na natureza. São simulados eventos intermitentes não periódicos que se assemelham aos observados na camada limite. O modelo também reproduz a ocorrência de mistura rasa em condições de vento fraco. A inclusão de equações prognósticas para o fluxo de energia na forma de calor sensível e para a variância de temperatura proporciona transições entre os estados e em condições de vento mais intensas que as obtidas quando essas quantidades são parametrizadas, e mais próximo dos valores observados. Ao observar a dependêndia das quantidades turbulentas, tais como a escala de temperatura potencial, θ∗, e o fluxo de energia na forma de calor sensível em função da velocidade do vento, o presente modelo modelo apresentou os resultados mais semelhante aos observados, se mostrando capaz de reproduzir as transições entre os estados de forma abrupta, diferente do que ocorre no modelo proposto por Costa et al. (2011). O balanço de ECT obtido pelo presente modelo, se aproxima mais do observado por Acevedo et al. (2016) que o obtido por Costa et al. (2011). A dissipação é o mecanismo dominante de destruição de turbulência no estado desacoplado, papel que era ocupado pela destruição térmica no modelo de Costa et al. (2011). Os resultados coroboram a hipótese de Van de Weil et al. (2012), de que o regime muito estável ocorre quando a turbulência não é capaz de produzir fluxos de energia na forma de calor sensível capazes de acompanhar a perda de energia por radiação.

This work is about to verify the performance of the boundary layer budgeting method as alternative for measuring the surface turbulent fluxes in conditions that allow the validation of the measurements with data taken from a tower equipped with sensors, which were processed through the eddy correlation method. A research is made on the knowledge generated to the present, which reveals the limitations of the eddy correlation method and the promises of the boundary layer budgeting in relation to measuring the surface turbulent fluxes. This work describes its theoretical underpinnings, regarding the energy budget in the atmosphere and the role of turbulence in the transport phenomena and in energy exchanges. These concepts are interpreted physically and described in the form of ruling equations and hypotheses used for the analysis of the behavior of such phenomena. The eddy correlation is described as a method that directly measures the sensible heat and latent heat fluxes in function of the data picked up by the sensors. The boundary layer budgeting method, by its turn, is described from the general formulation of scalar accumulation, on which are taken the hypotheses of elimination of the horizontal advection and of any sources or drains in the atmosphere. The context of the measuring work is described in the following, from the presentation of the project, its places, and its campaigns and to the day and times in which the measurements took place. The instruments used for data acquisition are also described, as well as the processing systematics of the raw data. The obtained results reveal significant differences in the sensible heat flux estimates, which increase with the progress of the night and smaller differences in the latent heat flux, which remain constant in the time. / Este trabalho trata de verificar o desempenho do método de balanço da camada limite como alternativa para a medição dos fluxos turbulentos superficiais em condições que permitem a validação das medidas com dados tomados a partir de uma torre equipada com sensores, os quais foram processados com o método de correlação de vórtices. É feita uma pesquisa do conhecimento gerado até o presente, que revela as limitações da sistemática de medição por correlação de vórtices e as promessas do método de balanço em relação à medição dos fluxos turbulentos. É descrita neste trabalho a fundamentação teórica necessária, a respeito do balanço energético na atmosfera e do papel da turbulência nos fenômenos de transporte e de troca de energia. Tais conceitos são interpretados fisicamente e descritos na forma de equações governantes e hipóteses utilizadas para a análise do comportamento de tais fenômenos. A correlação de vórtices é descrita como um método que mede diretamente os fluxos de calor sensível e de calor latente em função dos dados recolhidos pelos sensores. O método de balanço da camada limite, por sua vez é descrito a partir da formulação geral de acumulação de escalares, sobre a qual são tomadas as hipóteses de eliminação da advecção horizontal e de quaisquer fontes ou sumidouros na atmosfera. O contexto das medições é descrito a seguir, a partir da apresentação do projeto, dos locais, das campanhas e do dia e dos horários em que as medições foram realizadas. A instrumentação utilizada para a aquisição dos dados é também descrita, assim como a sistemática de processamento dos dados brutos. Os resultados obtidos revelam diferenças significativas na estimativa do fluxo de calor sensível, que aumentam com o avanço da noite e diferenças menores no fluxo de calor latente, que permanecem praticamente constantes no tempo.