Nonlinear Dynamics in a Two-Layer Model of Baroclinic Instability and the Effects of Varying Sidewall Boundary Conditions.
The behavior of two-layer, quasi-geostrophic flow in a channel, which is subject to baroclinic instability, is investigated using a high-resolution numerical model. Solutions are obtained for both free-slip 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, quasi-periodic, 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 two-torus. A quasi-analytic method for predicting the instabilities of finite-amplitude 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 low-order models are used to better interpret the results and to examine the link between partial differential equations and low-dimensional dynamical systems. The empirical models provide a quantitatively accurate approximation of the full flow in steady, periodic, and simple quasi-periodic regimes; for more complex quasi-periodic 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 rigid-wall geometry, the linear stability problem is first formulated and the results interpreted and compared to the free-slip 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 free-slip 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 wave-mean 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 two-torus.
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- Physics: Atmospheric Science; Physics: Fluid and Plasma