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- Pattern Formation in Rotating Fluid Systems under the Influence of Magnetic Fields (2004)
- Patterns are observed in many different systems in nature. They are seen in the cloud streets, in sand ripples, in the morphology of plants and animals, on weather maps, in chemical reactions. In all these cases one deals with open, continuous dissipative systems which are driven out of equilibrium by an external stress. If this stress is larger than a certain threshold value, the symmetry of the temporally and spatially homogeneous ground state is spontaneously broken. The resulting patterns show then periodicity in space and/or in time. One of the best studied examples is the convection instability when a fluid layer is subjected to a temperature gradient. For instance, in a horizontal fluid layer heated from below and cooled from above a striped patterns of convection rolls develop. This scenario describes the famous Rayleigh- Benard convection (RBC), as a standard paradigm of pattern formation. Many concepts and mathematical tools to analyze the patterns have been developed and tested for this case. This thesis deals with two different pattern forming systems, namely a particular example of a convection instability and the case of a shear flow driven instability. In the first part of the thesis, a variation of the standard RBC is investigated. We consider the problem of convection induced by radial buoyancy in an electrically conducting fluid contained in a rotating (angular frequency, Omega) cylindrical annulus which is cooled at the inner surface and heated from outside. In addition, an azimuthal magnetic field (B) is applied for instance by an electrical current through the cylinder axis. The motivation of this study has come originally from the geophysical context. This setup is hoped to capture some important features of convection patterns in rotating stars and planets near the equatorial regions. The problem is also of considerable interest from a more general point of view in that it is concerned with formations of patterns in the presence of two competing directional effects, in this case rotation and the magnetic field. The second part of the thesis is devoted to the the pattern formation by a shear flow between two rotating and infinitely electrically conducting plates with a magnetic field perpendicular to the plates. This geometry is called the magnetic Ekman-Couette layer and has been a basic model for magnetic activities at the boundary of the Earth's liquid core or at the tachocline in the Sun below the convection zone for a few decades. To analyze the forementioned problems, various codes and computational tools had to be developed, for instance, we were able to describe complex spatio-temporal patterns by the direct simulations of the underlying hydrodynamic equations for our problems. The discussion of the physical details of the systems are postponed to the introductory sections of the corresponding parts of the thesis. In Chapter 1, a general formulation of the linear and nonlinear analysis, methods, which are applicable to both pattern forming systems in this work will be presented. The investigation of thermal convection in a plane layer which is a geometry equivalent to the cylindrical annulus will be discussed in Chapter 2. The next chapter (Chapter 3) covers both the linear and nonlinear analyses in the case of magnetic Ekman-Couette layer problem. Finally, in Chapter 4, we will present the general conclusions on both of the systems.