Nonlinear Dynamics in a TwoLayer Model of Baroclinic Instability and the Effects of Varying Sidewall Boundary Conditions.
Abstract
The behavior of twolayer, quasigeostrophic flow in a channel, which is subject to baroclinic instability, is investigated using a highresolution numerical model. Solutions are obtained for both freeslip sidewalls (which allow tangential velocities but zero stress) and rigid sidewalls (which enforce zero velocity). Results for the slippery model are presented first, and the physics underlying the observed behavior is examined. As the Froude number F is increased, the system exhibits a transition from steady flow to periodic, quasiperiodic, and finally chaotic behavior. As F is increased to about five times the linear critical value, the motion becomes chaotic, and for even larger values of F it moves toward a "geostrophic turbulence" regime. The route to chaos is determined to be the breakdown of a twotorus. A quasianalytic method for predicting the instabilities of finiteamplitude baroclinic waves is then formulated, and the results are compared with the numerical solutions. This approach predicts the locations (in parameter space) where the equilibrium points of the system are unstable, and these correlate well with transitions in the numerical model. Surrogate models of the slippery case, obtained by computing the empirical orthogonal functions (EOFs) and using these as basis functions, are also generated. Such loworder models are used to better interpret the results and to examine the link between partial differential equations and lowdimensional dynamical systems. The empirical models provide a quantitatively accurate approximation of the full flow in steady, periodic, and simple quasiperiodic regimes; for more complex quasiperiodic and chaotic flows, the approach yields qualitative agreement. As the system approaches the region of geostrophic turbulence, the number of EOFs required to accurately represent the flow rises rapidly. The EOF's are also shown to correlate well with the linear eigenfunctions of secondary instabilities in the slippery model. For the rigidwall geometry, the linear stability problem is first formulated and the results interpreted and compared to the freeslip case. The rigid case can be more or less stable than its slippery counterpart, depending on the magnitude of the bottom friction. In the region applicable to laboratory results, however, the neutral curves for both models are similar. Numerical solutions are then examined and compared to both the freeslip case and also to laboratory experiments. The onset of chaos occurs for very small or even negative supercriticality. The subcritical behavior observed in the model is determined to be due to wavemean interactions. In general, the rigid wall system is far more unstable and chaotic than the slippery model, counter to intuition but in better accord with laboratory (rigid wall) results. The route to chaos is again found to be the breakdown of a twotorus.
 Publication:

Ph.D. Thesis
 Pub Date:
 1993
 Bibcode:
 1993PhDT.......167M
 Keywords:

 Physics: Atmospheric Science; Physics: Fluid and Plasma