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In this thesis I discuss a series of anelastic approximations and detail the assumptions used in the derivation. I derive an entropy and temperature formulation of the anelastic approximation along with a simplification to the entropy formulation introduced by Lantz (1992) and independently by Braginsky & Roberts (1995). I assess range of applicability of the anelastic approximation, which is often used in describing the dynamics of geophysical and astrophysical flows. I consider two linear problems: magneto convection and magnetic buoyancy and compare the fully compressible solutions with those determined by solving the anelastic problem. I further compare the Lantz-Braginsky simplification with the full anelastic formulation which I find to work well if and only if the atmosphere is nearly adiabatic. I find that for the magnetoconvection problem the anelastic approximation works well if the departure from adiabaticity is small (as expected) and determine where the approximation breaks down. When the magnetic field is large then the anelastic approximation produces results which are markedly different from the fully compressible results. I also investigate the effects of altering the boundary conditions from isothermal to isentropic and the effect of stratification on how some of the parameters scale with the Chandrasekhar number. The results for magnetic buoyancy are less straight-forward, with the accuracy of the approximation being determined by the growth rate of the instability. I argue that these results make it difficult to assess a priori whether the anelastic approximation will provide an accurate approximation to the fully compressible system for stably stratified problems. Thus, unlike the magnetoconvection problem, for magnetic buoyancy it is difficult to provide general rules as to when the anelastic approximation can be used. When the instability grows quickly or the magnetic field is large the results do not compare well with the fully compressible equations. I outline a method for a two-dimensional non-linear time-stepping computer program and explain some problems with current non-linear programs.

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Bibliography : leaves 179-186 / 186 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1992

Les études récentes de la déformation plastique à l'aide de techniques expérimentales à haute résolution témoignent que les processus de déformation sont souvent caractérisés par des effets collectifs qui émergent à une échelle mésoscopique, intermédiaire entre celle de défauts cristallins et celle d'une éprouvette macroscopique. Notamment, la méthode de l'émission acoustique (EA) révèle, dans divers conditions expérimentales, l'intermittence de la déformation plastique, qui se manifeste par une propriété de l'invariance d'échelle, caractéristique de phénomènes d'auto-organisation. L'objectif de la thèse a été d'étudier la structure inhérente de l'EA pour différents mécanismes de déformation plastique, d'examiner sa dépendance à la vitesse de déformation et à l'écrouissage du matériau, et d'appréhender les liens entre les petites échelles de temps, liées à l'organisation des défauts, et celles qui relèvent de l'approche continue de la plasticité. L'étude a été réalisée sur des alliages AlMg et des alliages base Mg, dont la déformation plastique est accompagnée d'une forte activité acoustique et contrôlée par différents mécanismes physiques : l'effet Portevin-Le Chatelier (PLC) dans les premiers et une combinaison du maclage et du glissement des dislocations dans les deuxièmes. L'utilisation de la technique d'enregistrement continue de l'EA ("data streaming") a permis de montrer que le comportement apparent - discrète ou continue - de l'EA accompagnant l'effet PLC dépend de l'échelle de temps d'observation et du paramètre physique étudié. Cependant, contrairement à une vision traditionnelle, il se trouve que l'EA a un caractère intermittent pendant l'écoulement macroscopiquement lisse tant que pendant l'instabilité macroscopique de la déformation plastique. Grace aux méthodes d'analyse issues de la théorie des systèmes dynamiques non linéaires, telles que l'analyse multifractale, une tendance à la transition entre la dynamique invariante d'échelle et les comportements caractérisés par des échelles intrinsèques a été trouvée lors de l'écrouissage des matériaux. Enfin, nous avons prouvé que les distributions statistiques en loi puissance persistent dans des larges intervalles de variation des paramètres, conventionnellement utilisés pour individualiser les événements acoustiques. Ce résultat est d'une importance générale car il s'applique à tous les processus avalancheux émergeant dans différents systèmes dynamiques / Recent studies of plastic deformation using high-resolution experimental techniques testify that deformation processes are often characterized by collective effects that emerge on a mesoscopic scale, intermediate between the scale of individual crystal defects and that of the macroscopic sample. In particular, the acoustic emission (AE) method reveals intermittency of plastic deformation in various experimental conditions, which is manifested by the property of scale invariance, a characteristic feature of self-organized phenomena. The objective of the dissertation was to study the inherent structure of AE for different mechanisms of plastic deformation, to examine its dependence on the strain rate and strain hardening of the material, and to understand the relationships between short time scales related to organization of defects and those relevant to the continuous approach of plasticity. The study was performed on AlMg and Mg-based alloys, the plastic deformation of which is accompanied by a strong acoustic activity and controlled by different physical mechanisms: the Portevin-Le Chatelier (PLC) effect in the first case and a combination of twinning and dislocation glide in the second case. Application of a technique of continuous AE recording ("data streaming") allowed proving that the apparent behavior, discrete or continuous, of AE accompanying the PLC effect depends on the time scale of observation and the physical parameters surveyed. However, unlike the traditional view, it appears that AE has an intermittent character during both stress serrations and macroscopically smooth flow. Using methods of the theory of nonlinear dynamical systems, such as the multifractal analysis, a tendency to a transition between the scale-invariant dynamics and the behaviors characterized by intrinsic scales was detected during work hardening. Finally, we proved that the power-law statistical distributions persist in wide ranges of variation of parameters conventionally used to individualize acoustic events. This result is of general importance because it applies to all avalanche-like processes emerging in dynamical systems