abstract: The formation and stability of a slowly evolving zonal jet in 2-D flow with beta effect is analyzed using the model developed by Manfroi and Young in which the final governing equation was derived by means of a perturbation analysis of a barotropic vorticity equation with sinusoidal meridional mean flow. However in the original study the term β0, that represents the effect of large-scale Rossby waves, was dropped and was proceeded on a path of finding solutions for a simplified 1-D flow. The idea of this study is to understand the effects of the dropped term on the overall dynamics of the zonal jet evolution. For this purpose the system that is entirely deterministic with no additional forcing is solved by means of a standard finite difference scheme. The Numerical solutions are found for varying β0 and μ values where μ represents the bottom drag. In addition to this the criteria for the formation of zonal jets developed originally for the 1-D system is verified for the 2-D system as well. The study reveals the similarity in some of the results of the 1-D and the 2-D system like the merging of jets in the absence of bottom drag, formation of steady jets in presence of a non-zero bottom drag and the adherence to the boundary criteria for the formation of zonal jets. But when it comes to the formation of steady jets, a finite β0 value is required above which the solution is similar to the 1-D system. Also the jets formed under the presence of non-zero bottom drag seem wavy in nature which is different from the steady horizontal jets produced in the 1-D system. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2017

The giant planets exhibit banded zonal jet streams that have maintained theirstructures over decades. There are long-standing questions: how deep the windstructures extend? What mechanisms generate and maintain the observed winds?Why are the wind structures so stable? To answer these questions, we performedthree-dimensional numerical simulations of the atmospheric flow using the primitiveequations.First, we use a simple Newtonian cooling scheme as a crude approach to gener-ate atmospheric latitudinal temperature differences that could be caused by latentheating or radiation. Our Jupiter-like simulations show that shallow thermal forcingconfined to pressures near the cloud tops can produce deep zonal winds from thetropopause all the way down to the bottom of the simulated atmosphere (a fewhundred bars). These deep winds can attain speeds comparable to the zonal jetspeeds within the shallow, forced layer; they are pumped by Coriolis accelerationacting on a deep meridional circulation driven by the shallow-layer eddies.Next, we explicitly include the transport of water vapor and allow condensationand latent heating to occur whenever the water vapor is supersaturated. Our simu-lations show that large-scale moist convection associated with condensation of watervapor can produce multiple zonal jets similar to those on the gas giants (Jupiterand Saturn) and ice giants (Uranus and Neptune). For plausible water abundances(3-5 times solar on Jupiter/Saturn and 30 times solar on Uranus/Neptune), oursimulations produce about 20 zonal jets for Jupiter and Saturn and 3 zonal jetson Uranus and Neptune. Moreover, these Jupiter/Saturn cases produce equatorialsuperrotation whereas the Uranus/Neptune cases produce equatorial subrotation,consistent with the observed equatorial jet direction on these planets. Sensitiv-ity tests show that the water abundance is the controlling factor; modest waterabundances favor equatorial superrotation, whereas large water abundances favorequatorial subrotation. This provides a possible mechanism for the existence ofequatorial superrotation on Jupiter and Saturn and the lack of superrotation onUranus and Neptune.

Cette thèse porte sur la dynamique des grandes échelles des écoulements géophysiques turbulents, en particulier sur leur organisation en écoulements parallèles orientés dans la direction est-ouest (jets zonaux). Ces structures ont la particularité d'évoluer sur des périodes beaucoup plus longues que la turbulence qui les entoure. D'autre part, on observe dans certains cas, sur ces échelles de temps longues, des transitions brutales entre différentes configurations des jets zonaux (multistabilité). L'approche proposée dans cette thèse consiste à moyenner l'effet des degrés de liberté turbulents rapides de manière à obtenir une description effective des grandes échelles spatiales de l'écoulement, en utilisant les outils de moyennisation stochastique et la théorie des grandes déviations. Ces outils permettent d'étudier à la fois les attracteurs, les fluctuations typiques et les fluctuations extrêmes de la dynamique des jets. Cela permet d'aller au-delà des approches antérieures, qui ne décrivent que le comportement moyen des jets.Le premier résultat est une équation effective pour la dynamique lente des jets, la validité de cette équation est étudiée d'un point de vue théorique, et les conséquences physiques sont discutées. De manière à décrire la statistique des évènements rares tels que les transitions brutales entre différentes configurations des jets, des outils issus de la théorie des grandes déviations sont employés. Des méthodes originales sont développées pour mettre en œuvre cette théorie, ces méthodes peuvent par exemple être appliquées à des situations de multistabilité. / This thesis deals with the dynamics of geophysical turbulent flows at large scales, more particularly their organization into east-west parallel flows (zonal jets). These structures have the particularity to evolve much slower than the surrounding turbulence. Besides, over long time scales, abrupt transitions between different configurations of zonal jets are observed in some cases (multistability). Our approach consists in averaging the effect of fast turbulent degrees of freedom in order to obtain an effective description of the large scales of the flow, using stochastic averaging and the theory of large deviations. These tools provide theattractors, the typical fluctuations and the large fluctuations of jet dynamics. This allows to go beyond previous studies, which only describe the average jet dynamics. Our first result is an effective equation for the slow dynamics of jets, the validityof this equation is studied from a theoretical point of view, and the physical consequences are discussed. In order to describe the statistics of rare events such as abrupt transitions between different jet configurations, tools from large deviation theory are employed. Original methods are developped in order to implement this theory, those methods can be applied for instance in situations of multistability.

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

Cette thèse considère quatre systèmes physiques complexes pour lesquels il est exceptionnellement possible d’identifier des variables lentes qui contrôlent l'évolution à temps long du système complet. La séparation d'échelle de temps entre ces variables lentes et les autres variables permet d'utiliser la technique de moyennisation stochastique pour obtenir une dynamique effective pour les variables lentes. Cette thèse considère la possibilité de prédire les événements rares dans le système solaire. Nous avons étudié deux types d’événements rares. Le premier est un renversement possible de l'axe de rotation de la Terre en l'absence des effets de marée de la Lune. Le second est la désintégration de l'ensemble du système solaire interne suite à une instabilité dans l'orbite de Mercure. Pour chacun des deux problèmes, il existe des variables lentes non triviales, qui ne sont pas données par des variables physiques naturelles. La moyennisation stochastique a permis de découvrir le mécanisme physique qui conduit à ces événements rares et de donner, par une approche purement théorique, l'ordre de grandeur de la probabilité de ces phénomènes. Nous avons également montré que la déstabilisation de Mercure sur un temps inférieur à l'âge du système solaire obéit à un mécanisme d'instanton bien décrit par la théorie des grandes déviations. Le travail effectué dans cette thèse ouvre donc un nouveau champ d'action pour l'utilisation d'algorithmes de calcul d'événements rares. Nous avons utilisé pour la première fois les théorèmes de moyennisation stochastique dans le cadre de la mécanique céleste pour quantifier l'effet stochastique des astéroïdes sur la trajectoire des planètes. Enfin, une partie du travail porte sur un problème de turbulence géophysique: dans l'atmosphère de Jupiter, on peut observer des structures zonales (jets) à grande échelles évoluant beaucoup plus lentement que les tourbillons environnants. Nous montrons qu'il est pour la première fois possible d'obtenir explicitement le profil de ces jets par moyennisation des degrés de liberté turbulents rapides. / The present thesis describes four complex dynamical systems. In each system, the long-term behavior is controlled by a few number of slow variables that can be clearly identified. We show that in the limit of a large timescale separation between the slow variables and the other variables, stochastic averaging can be performed and leads to an effective dynamics for the set of slow variables. This thesis also deals with rare events predictions in the solar system. We consider two possible rare events. The first one is a very large variation of the spin axis orientation of a Moonless Earth. The second one is the disintegration of the inner solar system because of an instability in Mercury’s orbit. Both systems are controlled by non-trivial slow variables that are not given by simple physical quantities. Stochastic averaging has led to the discovery of the mechanism leading to those rare events and gives theoretical bases to compute the rare events probabilities. We also show that Mercury’s short-term destabilizations (compared to the age of the solar system) follow an instanton mechanism, and can be predicted using large deviation theory. The special algorithms devoted to the computation of rare event probabilities can thus find surprising applications in the field of celestial mechanics. We have used for the first time stochastic averaging in the field of celestial mechanics to give a relevant orders of magnitude for the long-term perturbation of planetary orbits by asteroids. A part of the work is about geophysical fluid mechanics. In Jupiter atmosphere, large scale structures (jets) can be observed, the typical time of evolution of which is much larger than that of the surrounding turbulence. We show for the first time that the mean wind velocity can be obtained explicitly by averaging the fast turbulent degrees of freedom.