An Analytical Approach to the Determination of Optimal Perturbations in the Eady Model

Abstract

Abstract Within the framework of the baroclinic instability Eady model, an analytical approach to the determination of optimal perturbations with a maximum of the energy growth rate or the ratio of the final and initial energies is considered. This approach is based on the energy balance equation and explicit expressions for the energy functionals resulting from the perturbation representation by means of the superposition of the edge Rossby waves (ERWs). The corresponding expressions are functions of the parameters of the initial perturbation, and the determination of optimal parameters reduces to the study of these functions on an extremum. For perturbations with zero potential vorticity (PV), the amplitudes of the initial buoyancy distributions at the boundaries of the atmospheric layer and the phase shift between these distributions serve as parameters. Analytical formulas are obtained for the optimal phase shift and the maximum of the energy ratio, which determine their dependence on the wavenumber and optimization time. It is also shown that the optimal perturbations always have equal boundary amplitudes. The parameters of the optimal perturbations are compared with the parameters of the growing normal modes. It is established that there exists only one exponentially growing normal mode, which is the optimal perturbation. Along with the instability, the ERWs can be excited by their interaction with the initial vortex perturbations (PV ≠ 0). The optimal regime of ERWs excitation by the initial singular distribution of PV is investigated.