The growth of disturbances in barotropic models of the atmosphere

An eigenvalue analysis of a divergent barotropic model on a sphere is extended to the formulation of a global optimization problem, whose solution selects an initial perturbation that experiences the greatest possible growth over a finite time interval, tau. The evolution of this optimal perturbation is obtained from companion linear and nonlinear spectral time-dependent models. The two methods of analysis are compared using idealized nonzonal flows and three zonally asymmetric flows defined by time-mean global 300 mb analyses. The results using the idealized flows illustrate the application and insights provided by the optimization problem. For finite time intervals of ten days or less, the optimal perturbations grow much more rapidly than the eigenmodes. This is particularly true for basic flows whose meridional absolute vorticity gradient is negative over a limited sector of the hemisphere. These latter idealized flows support the development of quasi-stationary dipole structures. The initial growth is dominated by the shear in the basic state, while the less rapid growth and persistence at later times is associated with the diffluence of the basic flow. The observed flows are used to investigate the applicability of the eigenanalysis and optimization problem to large amplitude, long duration anomalies in the midlatitude flow that are known to develop quickly. The optimal perturbations to these more complicated flows evolve into large anomalies that look very much like the observed anomalies in both linear and nonlinear models. Some of the optimal perturbations grow rapidly in the strong shear of the jet stream, propagate along the jet axis, slow down and subsequently form dipole structures in the jet exit region of diffluence. Reasonable finite amplitude perturbations are found to be affected by nonlinearity after a relatively short time, supporting the focus on the initial growth. The dependence of the eigenanalysis and optimal perturbations on the model parameters is examined.