A Semi-Implicit Scheme for Stationary Statistical Properties of the Infinite Prandtl Number Model

We propose a semisecret in time semi-implicit numerical scheme for the infinite Prandtl model for convection. Besides the usual finite time convergence, this scheme enjoys the additional highly desirable feature that the stationary statistical properties of the scheme converge to those of the infinite Prandtl number model at vanishing time stop. One of the key characteristics of the scheme is that it preserves the dissipativity of the infinite Prandtl number model uniformly in terms of the time stop. So far as wo know, this is the first rigorous result on convergence of stationary statistical properties of numerical schemes for infinite dimensional dissipative complex systems. © 2008 Society for Industrial and Applied Mathematics

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations

We investigate the long tim behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box: \frac{3\omega^{n+1}-4\omega^n+\omega^{n-1}}{2k} + \nabla^\perp(2\psi^n-\psi^{n-1})\cdot\nabla(2\omega^n-\omega^{n-1}) - \nu\Delta\omega^{n+1} = f^{n+1}, \quad -\Delta \psi^n = \om^n. The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme in \dL2, \dH1 and $\dot{H}^2_{per}$ provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed

Approximation of Stationary Statistical Properties of Dissipative Dynamical Systems: Time Discretization

We consider temporal approximation of stationary statistical properties of dissipative infinite-dimensional dynamical systems. We demonstrate that stationary statistical properties of the time discrete approximations, i.e., numerical scheme, converge to those of the underlying continuous dissipative infinite-dimensional dynamical system under three very natural assumptions as the time step approaches zero. the three conditions that are sufficient for the convergence of the stationary statistical properties are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval [0,1] uniformly with respect to initial data from the union of the global attractors; (3) uniform continuity of the solutions to the continuous dynamical system on the unit time interval [0,1] uniformly for initial data from the union of the global attractors. the convergence of the global attractors is established under weaker assumptions. an application to the infinite Prandtl number model for convection is discussed. © 2009 American Mathematical Society

A Uniformly Dissipative Scheme for Stationary Statistical Properties of the Infinite Prandtl Number Model

The purpose of this short communication is to announce that a class of numerical schemes, uniformly dissipative approximations, which uniformly preserve the dissipativity of the continuous infinite dimensional dissipative complex (chaotic) systems possess desirable properties in terms of approximating stationary statistics properties. in particular, the stationary statistical properties of these uniformly dissipative schemes converge to those of the continuous system at vanishing mesh size. the idea is illustrated on the infinite Prandtl number model for convection and semi-discretization in time, although the general strategy works for a broad class of dissipative complex systems and fully discretized approximations. as far as we know, this is the first result on rigorous validation of numerical schemes for approximating stationary statistical properties of general infinite dimensional dissipative complex systems. © 2008 Elsevier Ltd. All rights reserved

Approximating Stationary Statistical Properties

It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. Many times these statistical properties of the system must be approximated numerically. the main contribution of this manuscript is to provide simple and natural criterions on numerical methods (temporal and spatial discretization) that are able to capture the stationary statistical properties of the underlying dissipative chaotic dynamical systems asymptotically. the result on temporal approximation is a recent finding of the author, and the result on spatial approximation is a new one. Applications to the infinite Prandtl number model for convection and the barotropic quasi-geostrophic model are also discussed. © Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2009

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

Stochastic Flux-Freezing and Magnetic Dynamo

We argue that magnetic flux-conservation in turbulent plasmas at high magnetic Reynolds numbers neither holds in the conventional sense nor is entirely broken, but instead is valid in a novel statistical sense associated to the "spontaneous stochasticity" of Lagrangian particle tra jectories. The latter phenomenon is due to the explosive separation of particles undergoing turbulent Richardson diffusion, which leads to a breakdown of Laplacian determinism for classical dynamics. We discuss empirical evidence for spontaneous stochasticity, including our own new numerical results. We then use a Lagrangian path-integral approach to establish stochastic flux-freezing for resistive hydromagnetic equations and to argue, based on the properties of Richardson diffusion, that flux-conservation must remain stochastic at infinite magnetic Reynolds number. As an important application of these results we consider the kinematic, fluctuation dynamo in non-helical, incompressible turbulence at unit magnetic Prandtl number. We present results on the Lagrangian dynamo mechanisms by a stochastic particle method which demonstrate a strong similarity between the Pr = 1 and Pr = 0 dynamos. Stochasticity of field-line motion is an essential ingredient of both. We finally consider briefly some consequences for nonlinear MHD turbulence, dynamo and reconnectionComment: 29 pages, 10 figure