Barotropic Instability and Optimal Perturbations of Observed Nonzonal Flows

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 evolves into the most energetic structure at a finite time interval, τ. The evolution of this perturbation is obtained from companion linear and nonlinear global spectral time-dependent models, and the optimization prediction of perturbation size at time τ is verified. Two zonally asymmetric flows defined by time-mean ECMWF global 300-mb analyses during winter 1985/86 are used to illustrate the application and insights provided by the optimization problem. The dependence of the optimal perturbations on the parameter τ is examined. The optimal perturbations become increasingly localized as τ is decreased to periods on the order of three days. The initial growth rates of these perturbations greatly exceed that of the most unstable normal mode, and also exceed the growth rate of a disturbance with maximum projection onto the most unstable mode (i.e., the adjoint structure). Furthermore, the development of the optimal perturbations in the nonlinear model is in reasonable agreement with the available observations. The optimal perturbations may thus be more important than either the eigenmode or adjoint structure for determining the stability and expected behavior of anomalies to some time-mean flows.