Towards reducing continuous symmetry of baroclinic flows

The concept of baroclinic instability is perhaps one of the harder ones to grasp in geophysical fluid mechanics. However, it is also one of the most fundamental concepts on this field, as it is the main driver for the large scale circulations in the atmosphere and important circulations in the ocean. It is by this mechanism that the atmosphere redistributes heat from low latitudes to high latitudes, that it sustains synoptic weather system and by which some oceanic eddies develop. Its importance can not be overstated. On the other hand, one could argue that the study of baroclinic flows remains in its early stages. Although there exist a very well developed linear theory for the onset of this type of instability, there still much room to explore the nonlinear regimes. A way to approach this studies is given by recent advances in the dynamical approaches to study of turbulence [1, 2, 3]; making use of the symmetries of the system, and finding periodic orbits and fixed points, as a way to understand the manifold of this type of setups. In this study we introduce the physics and present some of the nonlinear theory methods that might be used to analyze baroclinic flows. We will briefly mention insights from stability theory, but the emphasis would not be on them. Great papers and books have already been written about it. The work is divided as follows. In sect. 2 we introduce the problem in a qualitative matter, hoping that this simple approach illuminates the underlying physical principles. In sect. 3 we use Navier-Stokes equation to derive the vorticity equation and explicitly expose the the baroclinic term which is the cause of this instability. In sect. 4 we