The instability of the atmosphere places an upper bound on the predictability of instantaneous weather patterns. The lack of complete periodicity in the atmosphere's behavior is sufficient evidence for instability (Lorenz, 1963), but it does not reveal the range at which the uncertainty in prediction must become large. Most estimates of this range have been based on numerical integrations of systems of equations of varying degrees of complexity, starting from two or more rather similar initial states. It has become common practice to measure the error which would be made by assuming one of these states to be correct, when in fact another is correct, by the root-mean-square difference between the two fields of wind, temperature, or some other element, and to express the rate of amplification of small errors in terms of a doubling time (Lorenz, 1963).
The purpose of this thesis is to build tools with rigorous support useful for studying predictability of average temperatures. We apply our tools to a simple Earth-like example and make use of the Bred Vector algorithm to generate initial perturbations. The numerical model used is that of the Natural Convection problem. The analysis is done in steps, first by analyzing the turbulent natural convection problem then by introducing a fast calculation of an ensemble of solutions of the Navier-Stokes equations coupled with the temperature equation. Complete stability and convergence analysis of the methods are presented. The turbulent Earth model and its stability conditions are introduced at the end of the thesis
