Optimal growth of Eady edge waves

Publisher： E. Schweizerbart'sche Verlagsbuchhandlung

ISSN： 0941-2948

Source： Meteorologische Zeitschrift, Vol.15, Iss.4, 2006-08, pp. : 423-437

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Abstract

The classical Eady model with interior perturbation potential vorticity q′ = 0 and a modification with β ≠ 0 are treated in terms of edge waves, which are propagating on the oppositely directed pseudo-potential vorticity gradients induced by the rigid upper and lower lid. By this reformulation it is made possible to derive easily interpretable equations for optimal instantaneous growth. In the modified model the coupling between the amplitudes of the edge waves is not equal in contrast to the classical model. Nevertheless, it is proven that the optimally instantaneously growing disturbances (OIs) and their growth rates are identical in both models. The ability of the OIs to maintain their growth rates for longer time intervals is examined. This ability depends on the similarity between the normalmode (NM) structures and the OI structures and therefore differs between the two models. The singular vectors (SVs) of both models are computed, and it is shown that their growth rates and their evolution can be explained with the aid of the previous results, i.e. it is possible to deduce the SV growth rates qualitatively by modulating the OI growth rates with the ability of the SVs to maintain these optimal growth rates. Correspondingly, the SV evolution is determined by the OI and NM structures. The SVs evolve such that they transit exactly through the OI structures toward the growing NM structure or to its remnants near to the cutoffs, where a growing NM does not exist. The norm-dependency of the OIs and SVs is examined and on the basis of these results it is possible to classify all norms considered here into two groups with qualitatively different characteristics. The results suggest that the unequal coupling between the edge wave amplitudes, which is investigated here, might be relevant to rapid surface cyclone development and Petterssen type B development, in case the vertical windshear decreases with height.