Perturbation growth in terms of baroclinic alignment properties

Mechanisms leading to perturbation growth in complex time-dependent quasi-geostrophic (QG) flows are addressed in this paper. The dynamics of small three-dimensional (3D) perturbations are studied for the complete set of the linearized local QG equations. An analytical diagnostic of these equations shows that, at each spatial location, the preferred 3D structures of the perturbations are related to the eigenvector directions of a matrix, denoted A, which is the 3D generalization of the basic-state strain-rate tensor. The matrix A has a degenerate form and depends on both the horizontal deformation and the vertical shear of the unperturbed reference flow. By using a nonlinear Monte-Carlo technique, the 3D structures related to A's eigenvectors are shown to be the most probable ones for perturbation growth. We also provide simple analytical expressions for quantifying the barotropic and baroclinic energy extraction from the reference flow by the perturbations. In particular, our analytical expression for baroclinic energy extraction is found to be more relevant than the Eady index widely used in the literature. An interesting outcome of the Monte-Carlo simulations is that the maxima of the total-energy error field are found to be localized in regions where the norm of Ā is large.