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

Spectral 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 satisfied by the basic equations. Several viscosity laws which correspond to different 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 different parameters in the viscosity laws affect 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 difference 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 first law. In both cases, we report time-dependent solutions that are greatly influenced 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 findings 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 effects 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 fluid 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 fixed Coriolis force, and hence an enhancement of the centrifugal buoyancy effects with significant dynamical consequences are described.