Diagnosis of Optimal Perturbation Evolution in the Eady Model

Abstract

The structure and evolution of Eady model singular vector (SV, also referred to as optimal perturbation) streamfunction perturbations are described using a combination of two different partitions of the vector subspace describing all possible streamfunction perturbations. A modal partitioning of the SV perturbation streamfunction (expressing the SV streamfunction as a linear combination of modal structures) is used to ascribe the roles and relative importance of the continuum modes (CMs) and the discrete normal modes (NMs) in SV initial structure and subsequent evolution. In addition, a potential vorticity (PV) partitioning of the SV perturbation streamfunction into parts attributed to the SV PV and the SV boundary thermal anomalies (BTAs) is employed. The structures of the CMs and NMs are described in terms of their characteristic perturbation PV and BTAs. Modal decomposition of the SVs reveals that for all zonal wavenumbers (k), the NMs have the largest projection coefficients (with magnitudes exceeding unity). Specifically, for k < kc, the growing NM has the largest magnitude projection coefficient, while for k > kc equally large projection coefficients are observed for the two neutral Eady modes. The fact that the magnitude of the NM projection coefficients exceeds unity necessitates the existence of structurally similar CMs to ?mask? the NMs at initial time. This initial masking, which has been previously reported, is interpreted from a PV perspective as resulting from the cancellation between the NM BTAs and the BTAs associated with the CMs. For all wavenumbers, the magnitude of these NM projection coefficients increases with increasing optimization time τopt before reaching a limiting value proportional to the mode's projectability as τopt ? ∞. For k < kc, the lower (upper) CM BTA is of the same (opposite) sign as the interior CM PV anomaly. For k > kc, for those CMs residing between the steering levels of the two neutral Eady modes, the lower (upper) BTAs are the same (opposite) sign as the CM PV anomaly, while for those CM modes residing at other levels, the signs of the lower and upper BTAs are reversed. For all wavenumbers, initial amplification of the SV is associated with the superposition of the interior PV anomalies. Concomitantly with the superposition of CM PV is the superposition of CM BTAs. Because of the aforementioned structure of the CM BTAs, for k < kc, the superposition of the CM BTAs represents a negative contribution to SV amplification. For k > kc, superposition of CM BTAs contributes positively to amplification, and the CM BTAs have a nondecaying streamfunction contribution nearly equivalent to the contribution from the edge waves.