Linear and Nonlinear Perturbation Growth in a Simple Atmospheric Model

Leading singular vectors (SVs) are the fastest growing perturbations to a given trajectory in a linear sense. As such, they have been used for a variety of applications such as basic atmospheric instability and predictability studies, targeted observing, ensemble design, and the identification of key components of the analysis error. Of course, in operational forecasting, perturbation/error growth is nonlinear. In this study, the impact of nonlinearities on singular-vector calculations is investigated within an idealized framework using a global quasi-geostrophic potential vorticity model. It is shown here that, as the size of the perturbation increases and nonlinearities become more important, the effectiveness of the leading singular vectors for describing perturbation growth does decreases. However, it is also found that SV-based pseudo-inverse calculations can still be very relevant to the nonlinear forecasting problem. In fact, the nonlinear corrections are often better than the expected linear corrections, indicating that the pseudo-inverse perturbations are suppressing error growth outside of the SV subspace through nonlinear interactions. This rarely occurs if the pseudo-inverse perturbation has degraded the analysis. Therefore, linear-nonlinear differences can be used to provide information about the impact of the pseudo-inverse on the initial error. The impact of model error on the effectiveness of singular vectors is also examined.