Dynamics of Singular Vectors in the Semi-Infinite Eady Model: Nonzero β but Zero Mean PV Gradient

Abstract

Abstract 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.