Balance dynamics and stability of vortical flows

grantor: University of TorontoThis thesis reports results obtained in an effort to understand why the essentially inviscid atmospheric flow at the largest scales stays very nearly balanced, or in other words, is dominated by the vortical component of the dynamics. Many of these results have been made possible by the Hamiltonian structure present in most descriptions of inviscid fluid and by the associated conservation laws. In the first part of this work, it is shown in simple dynamical systems with separation of timescale how balanced behaviour can be explained by the slow drift of the fast action variable. This drift is likely the generic property of any model with more than two degrees of freedom, and the exponential scaling of the drift rate predicted by Nekhoroshev's theorem is confirmed numerically even for modest values of the timescale separation parameter. We also obtained numerical evidence for similar scaling (i) in the drift of the system from elliptic equilibria (of the fast variables) in cases where no fast action can be constructed due to resonances in the fast frequencies, and (ii) in the magnitudes of the smaller Lyapunov exponents which are subsequently identified with the weak chaos arising from the fast-slow coupling (the largest Lyapunov exponent, which is O (1), is associated with the chaos in the slow dynamics). The second part of this work seeks to find how this behaviour generalises to fluid models with a large (typically infinite) number of degrees of freedom; this is done in the context of a weak-wave model and its Hamiltonian truncation. The exponential scaling was found to deteriorate quickly with increasing system size for arbitrary initial conditions. The infinite number of Casimir invariants in this system are generally ineffective to prevent thermalisation, and thus the breakdown of balance, with a notable exception being nonlinearly stable flows and those nearby. The identification of stable flow profiles is therefore important in the process of finding long-time balance; to this end, we report two theorems on nonlinear stability of non-divergent flows and discuss their consequences for the weak-wave model.Ph.D