ALBERT

Description: 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.

Description: A detailed investigation has been performed of the role of the different growth mechanisms (resonance, potential vorticity unshielding, and normal-mode baroclinic instability) in the evolution of optimal perturbations constructed for a two-layer Eady model and a kinetic energy norm. The two-layer Eady model is obtained by replacing the conventional upper rigid lid by a simple but realistic stratosphere. To make an unambiguous discussion possible, generally applicable techniques have been developed. At the heart of these techniques lies a description of the linear dynamics in terms of a variable number of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically localized sheets of potential vorticity. If the optimal perturbation is composed of only one PVB, the rapid surface cyclogenesis can be attributed to the growth of the surface PVB (the edge wave), which is excited by the tropospheric PVB via a linear resonance effect. If the optimal perturbation is constructed using multiple PVBs, this simple picture is modified only in the sense that PV unshielding dominates the surface amplification for a short time after initialization. The unshielding mechanism rapidly creates large streamfunction values at the surface, as a result of which the resonance effect is much stronger. A similar resonance effect between the tropospheric PVBs and the tropopause PVB acts negatively on the surface streamfunction amplification. The influence of the stratosphere to the surface development is negligible. In all cases reported here, the growth due to traditional normal-mode baroclinic instability contributes either negative or only little to the surface development up to the optimization time of two days. It takes at least four days for the flow to become fully dominated by normal-mode growth, thereby confirming that finite-time optimal perturbation growth differs in many aspects fundamentally from asymptotic normal-mode baroclinic instability.

Description: A nonmodal approach based on the potential vorticity (PV) perspective is used to compute the singular vector (SV) that optimizes the growth of kinetic energy at the surface for the β-plane Eady model without an upper rigid lid. The basic-state buoyancy frequency and zonal wind profile are chosen such that the basic-state PV gradient is zero. If the f-plane approximation is made, the SV growth at the surface is dominated by resonance, resulting from the advection of basic-state potential temperature (PT) by the interior PV anomalies. This resonance generates a PT anomaly at the surface. The PV unshielding and PV–PT unshielding contribute less to the final kinetic energy at the surface. The general conclusion of the present paper is that surface cyclogenesis (of the 48-h SV) is stronger if β is included. Three cases have been considered. In the first case, the vertical shear of the basic state is modified in order to retain the zero basic-state PV gradient. The increased shear enhances SV growth significantly first because of a lowering of the resonant level (enhanced resonance), and second because of a more rapid PV unshielding process. Resonance is the most important contribution at optimization time. In the second case, the buoyancy frequency of the basic state is modified. The surface cyclogenesis is stronger than in the absence of β but less strong than if the shear is modified. It is shown that the effect of the modified buoyancy frequency profile is that PV unshielding occurs more efficiently. The contribution from resonance to the SV growth remains almost the same. Finally, the SV is calculated for a more realistic buoyancy frequency profile based on observations. In this experiment the increased value of the surface buoyancy frequency reduces the SV growth significantly as compared to the case in which the surface buoyancy frequency takes a standard value. All growth mechanisms are affected by this change in the surface buoyancy frequency.

Description: Using a nonmodal decomposition technique based on the potential vorticity (PV) perspective, the optimal perturbation or singular vector (SV) of the Eady model without upper rigid lid is studied for a kinetic energy norm. Special emphasis is put on the role of the continuum modes (CMs) in the structure of the SV, and on the importance of resonance to the SV evolution. The basis for the SV is formed by a number of nonmodal structures, each consisting of a superposition of one CM and one edge wave, such that the initial surface potential temperature (PT) is zero. These nonmodal structures are used as PV building blocks to construct the SV. The motivation for using a nonmodal approach is that no attempt has been made so far to include the CM residing at the steering level of the surface edge wave in the perturbation, although it is known that this CM is in linear resonance with the surface edge wave. Experiments with one PV building block in the initial disturbance show that the SV growth is dominated by the resonance effect except for small optimization times (less than 1 day), in which case the unshielding of PV and surface PT dominates the growth of the SV. The PV–PT unshielding provides additional growth to the SV and this explains the observation that the PV resides above the resonant level. More PV building blocks are added to include PV unshielding as a third growth mechanism. Which of the three mechanisms dominates during the SV evolution depends on the region of interest (interior or surface), as well as on the optimization time and on the number of building blocks used. At the surface, resonance plays a dominant role even when a large number of building blocks is used and relatively small optimization times are used. For the interior of the domain, PV unshielding becomes the dominant growth mechanism when more than two PV building blocks are used. With increasing optimization times, the PV distribution of the SV becomes increasingly more concentrated near the steering level of the edge wave. This concentration of PV is explained by the enhanced importance of resonance for long optimization times as compared to short optimization times.