In the simple context of the Eady model, we consider a classical cyclogenetic perturbation, consisting of an upper-level wave with no potential vorticity perturbation in the troposphere, as a typical analysis error that may then be decomposed by its projection onto each energy SV. Viewed in terms of the energy SVs, the smooth vertical structure of this perturbation, as well as its evolution, results from strong cancellation between the growing and decaying SVs, both of which are highly structured and tilted in the vertical.
This result may be understood by noting that the use of the energy norm implicitly assumes initial perturbations with second-order statistics given by the covariance matrix whose inverse defines the energy norm. Unlike the "typical" perturbation we chose above, perturbations with those statistics have large variance of potential vorticity in the troposphere and have fine vertical structure. We will also present evidence from analysis differences and from experiments with an ensemble Kalman filter in a simple GCM, that the "typical" perturbation is, in fact, broadly representative of analysis errors.