On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization.Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program. John Simon Guggenheim Memorial Foundation

Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data

Funder: Einstein Stiftung Berlin; doi: http://dx.doi.org/10.13039/501100006188Funder: John Simon Guggenheim Memorial Foundation; doi: http://dx.doi.org/10.13039/100005851This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order $\mathcal O(\varepsilon)$, as $\varepsilon \rightarrow 0^+$, where $\varepsilon$ represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows

Approaches for the improvement of physical transport processes in numerical models of coastal oceans

In this thesis three approaches to improve the simulated transport processes in coastal ocean models are discussed. The first approach deals with the discretisation of the governing equations and provides a diagnostic tool to assess the accuracy of a numerical transport scheme. The second approach considers the validity of the governing equations itself and suggests an alternative inclusion of missing nonhydrostatic dynamics. The third approach presents the inclusion of unresolved wind wave effects into a coastal ocean model

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

DEVELOPMENT OF CLOSURE RELATIONSHIPS FOR ADVANCED TWO-PHASE FLOW ANALYSIS

During this doctoral activity, developed at TEA Sistemi SpA with the contribution of the University of Pisa in the context of an R&D project funded by ENI E&P, the formulation of a new set of liquid-wall and gas-liquid interfacial friction factor correlations was performed. The attention was focused on the improvement of existing correlations when applied to the design of long transportation pipelines. In this aim, a new set of data related to nitrogen-water flow in a 80 mm pipe operating at pressures in the range 5-25 bar has been used along with data published in the open literature (mainly concerning air-water flows at atmospheric pressure). These data were used to develop new correlations for friction factors in horizontal stratified gas-liquid flow conditions. Moreover a new multi-field model called MAST (Multiphase Analysis and Simulation of Transition), recently developed at TEA Sistemi SpA with the support of ENI E&P and addressing the Oil&Gas field, was presented in detail during this activity and validated against experimental measurements for the investigation of the long slug flow sub-regime

Iterative solvers for modeling mantle convection with strongly varying viscosity

Die Dissertation beschreibt Verbesserungen der FEM-Diskretisierung und des Lösers der Stokes-Gleichungen im sphärischen Mantelkonvektionsmodell Terra. Zunächst wurde in einem zweidimensionalen quadratischen Gitter mit jeweils stückweise linearen Ansatzfunktionen für Druck und Geschwindigkeit eine stabilisierte Diskretisierung nach Dohrmann & Bochev (2004) mit Projektionen auf stückweise konstante Druckfunktionen implementiert. Deren spektrale Eigenschaften wurden systematisch untersucht. Die Stabilisierung bewirkt eine Gitterunabhängigkeit des Spektrums des Schurkomplements S. Die Viskositätsunabhängigkeit wird durch Präkonditionierung von S mit einer viskositätsabhängigen Massenmatrix Mη bzw. durch Skalierung mit deren Diagonale erreicht. Damit wurden drei Krylov-Unterraumverfahren hinsichtlich ihrer Robustheit gegenüber Viskositätsvariationen und Lösertoleranzen untersucht: Druckkorrektur- (PC), Minimierte Residuen- (MINRES) und ein konjugiertes Gradientenverfahren (BPCG) mit einem von Bramble and Pasciak (1988) entwickelten Blockpräkonditionierer. PC und BPCG wurden in einer äußeren Schleife mit aus Eigenwertabschätzungen berechneten Abbruchkriterien mehrfach gestartet. In der Rechenzeit unterscheiden sich die Löser um weniger als Faktor 2. Bei starken Viskositätskontrasten ist PC der einfachste und schnellste Löser. In Terra kann die o.g. Stabilisierung ohne Einschränkung auf Gittern mit mindestens 85 Millionen Knoten verwendet werden. Für gröbere Gitter wurde eine adaptive Wichtung entwickelt. Das PC-Verfahren in Terra wurde gemäß der o.g. Ergebnisse optimiert. Die Diagonalskalierung von S mit Mη bewirkt eine Rechenzeitreduktion um Faktor 4 bei starken lateralen Viskositätsvariationen. Bei Verwendung eines optimalen Multigrid-Lösers für den Impulsoperator wäre es Faktor 30. Diese Verbesserungen sind wesentliche Schritte zur Verwendung realitätsnäherer Erdmantelmodelle

Introductory Lectures on Turbulence: Physics, Mathematics and Modeling

From Chapter 1: The understanding of turbulent behavior in flowing fluids is one of the most intriguing, frustrating— and important—problems in all of classical physics. The problem of turbulence has been studied by many of the greatest physicists and engineers of the 19th and 20th Centuries, and yet we do not understand in complete detail how or why turbulence occurs, nor can we predict turbulent behavior with any degree of reliability, even in very simple (from an engineering perspective) flow situations. Thus, study of turbulence is motivated both by its inherent intellectual challenge and by the practical utility of a thorough understanding of its nature.https://uknowledge.uky.edu/me_textbooks/1001/thumbnail.jp

Instabilities in geophysical fluid dynamics: the influence of symmetry and temperature dependent viscosity in convection

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 25-04-2014Spectral numerical methods are proposed to solve the time evolution of a convection problem in a 2D domain with viscosity strongly dependent on temperature. We have considered periodic boundary conditions along the horizontal coordinate which introduce the O(2) symmetry into the setting. This motivates the use of spectral methods as an approach to the problem. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, and systematic method for gaining insight into the possible stationary solutions satis ed by the basic equations. Several viscosity laws which correspond to di erent dependences of the viscosity with the temperature are investigated. Numerous examples are found along the branching diagrams, in which stable stationary solutions become unstable through a Hopf bifurcation. In the neighborhood of these bifurcation points, the scope of our techniques is examined by exploring transitions from stationary regimes towards time dependent regimes. Our study is mainly focused on viscosity laws that model an abrupt transition of viscosity with temperature. In particular, both a smooth and a sharp transition are explored. Regarding the stationary solutions, the way in which di erent parameters in the viscosity laws a ect the formation and morphology of thermal plumes is discussed. A variety of shapes ranging from spout to mushroom shaped are found. Some stationary stable patterns that break the plume symmetry along their vertical axis are detected, as well as others that correspond to non-uniformly distributed plumes. The main di erence between the solutions observed for the smooth and sharp transition laws is the presence in the latter case of a stagnant lid, which is absent in the rst law. In both cases, we report time-dependent solutions that are greatly in uenced by the presence of the symmetry and which have not previously been described in the context of temperature-dependent viscosities, such as travelling waves, heteroclinic connections and chaotic regimes. Notable solutions are found for the sharp transition viscosity law in which time-dependent solutions alternate an upper stagnant lid with plate-like behaviors that move either towards the right or towards the left. This introduces temporary asymmetries on the convecting styles. This kind of solutions are also related to the presence of the O(2) symmetry and constitute an example of a plate-like convective style which is not linked to a subduction process. These ndings provide an innovative approach to the understanding of convection styles in planetary interiors and suggest that symmetry may play a role in describing how planets work. Finally, the centrifugal and viscosity e ects in a rotating cylinder with large Prandtl number are numerically studied in a regime where the Coriolis force is relatively large. Our focus is on aqueous mixtures of glycerine with mass concentration in the range of 60%-90%, and Rayleigh number values that extend from the onset, where thermal convection is in the so-called wall modes regime, in which pairs of hot and cold thermal plumes ascend and descend in the sidewall boundary layer, to values in which the bulk uid region is also convecting. The mean viscosity, which varies faster than exponentially with variations in the percentage of glycerine, leads to a faster than exponential increase in the Froude number for a xed Coriolis force, and hence an enhancement of the centrifugal buoyancy e ects with signi cant dynamical consequences are described.En esta tesis proponemos métodos numéricos espectrales, para resolver la evolución temporal de un problema de convección en un dominio 2D con viscosidad fuertemente dependiente de la temperatura. Las condiciones de contorno periódicas a lo largo de la coordenada horizontal introducen la simetría O(2) en el problema lo que motiva el uso de métodos espectrales en este contexto. Realizamos un análisis de las soluciones mediante técnicas propias de la teoría de bifurcaciones, y constatamos que son un método útil y sistemático para describir el panorama de las soluciones estacionarias que satisfacen las ecuaciones básicas. Investigamos varias leyes de viscosidad que corresponden a diferentes dependencias de ésta con la temperatura. A lo largo de los diagramas de bifurcación se encuentran numerosos ejemplos en los que la solución estacionaria estable se vuelve inestable a través de una bifurcación Hopf. En las proximidades de esos puntos examinamos el alcance de nuestras técnicas, explorando la transición desde regímenes estacionarios a regímenes dependientes del tiempo. Nuestro estudio se centra principalmente en las leyes de la viscosidad que modelan una transición abrupta de la viscosidad con la temperatura. En particular, se exploran tanto una transición suave como una brusca. En cuanto a las soluciones estacionarias, se discute como los diferentes pará metros en las leyes de viscosidad afectan a la formación y la morfología de las plumas térmicas. Se encuentran una variedad de la formas que van desde forma de protuberancia (\spout") a la forma de seta. Se detectan algunos patrones de soluciones estacionarias estables que rompen la simetría de la pluma a lo largo de su eje vertical y otros que se corresponden con plumas distribuidas de manera no uniforme. La principal diferencia entre las soluciones observadas para las leyes de transición suave y brusca es la presencia, con esta última ley, de una capa estancada que no está presente con la primera. En ambos casos mostramos soluciones dependientes del tiempo que están muy influenciadas por la presencia de la simetría y que no se han descrito previamente en el contexto de convección con viscosidad dependiente de la temperatura. Estas soluciones son por ejemplo ondas viajeras, conexiones heteroclínicas y regímenes caótico. Para transiciones bruscas de la ley de viscosidad destacan soluciones dependientes del tiempo, en las que se alternan una capa superior estancada, con una capa o placa que se mueve rígidamente hacia la derecha o la izquierda. Esto introduce estilos de convección que son asimétricos en el tiempo. Este tipo de soluciones también están relacionadas con la presencia de la simetría O(2) y constituyen un ejemplo de convección en forma de placa que no est a vinculada a un proceso de subducción. Estos resultados aportan un enfoque innovador para la comprensión de estilos de convección en el interior de planetas y sugieren que la simetría puede desempeñar un papel importante en la descripción de como funcionan. Por último, se estudian numéricamente los efectos centrífugos en un cilindro que rota, en un régimen en el que la fuerza de Coriolis es relativamente grande y en el que el fluido tiene un número de Prandtl alto. Nuestra atención se centra en mezclas acuosas de glicerina con concentraciones de masa en el intervalo de 60 %-90% y valores de número de Rayleigh que se extienden desde el inicio de la convección térmica; que son el denominado régimen de modos de pared, donde pares de plumas calientes y frías ascienden y descienden en la capa límite de la pared lateral; hasta valores en los que la convección está completamente desarrollada en toda la celda. El aumento de la viscosidad media, que varía con el porcentaje de glicerina considerado, conduce, para una fuerza de Coriolis ja, a un aumento en el n mero de Froude y por lo tanto, a un incremento de los efectos centrífugos para los que describimos su impacto en la dinámica