Abstract
Optimal perturbations are constructed for a two-layer β-plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without β. The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution.
The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the f plane only the PVB at the surface and at the tropopause can be amplified. In the presence of β, however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer f-plane model, the inclusion of β slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface.
If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time.
The optimal perturbation bounds for the weighted Moore-Penrose inverse
