Borges, M. D., and P. D. Sardeshmukh, 1997: Application of perturbation theory to the stability analysis of realistic atmospheric flows. Tellus, 49A, 321-336.
The 1st-order perturbation technique is investigated as a tool for measuring the dynamical significance of a change in a background flow. The focus is on estimating the change in the behavior of the linear disturbance operator as manifest in both the most rapidly growing normal modes as well as initial disturbances that amplify the most over finite time intervals (also known as singular vectors). The notion of a "small change" to a flow can be made more precise by requiring, for instance, that the changes in the flow's stability properties be accurately estimated using first-order perturbation theory. The perturbation theory updates of the eigen- and singular-values are economical to compute, thus making this determination feasible even in a practical setting. Furthermore, a simple refinement of the basic algorithm enables an efficient updating of the singular vectors as well. The technique is illustrated and tested for a flow having a substantial number of degrees of freedom, the global 250-mb flow during northern hemisphere winter represented in a T31 spectral space. A complete generalized barotropic stability analysis of the observed long-term mean 250-mb flow is performed first. Various alterations to the flow are then considered, and the perturbation technique is applied to estimate both the asymptotic and finite-time stability of the altered flows. Finally, the accuracy of these estimates is checked against the complete stability analyses of the altered flows, and an assessment is made of what a "small change" is for these cases. It is shown that in many cases the perturbation theory estimates of the stability changes are accurate enough over a sufficiently large parameter range to be of practical use. Applications to the problems of: (1) anticipating variations in forecast skill associated with day-to-day variations in flow stability; and (2) anticipating the relevance and robustness of individual normal modes are discussed. The latter problem is closely related to the concept of pseudospectra, and perturbation theory can be used to estimate their coarse details.