Nonlinear growth of singular-vector-based perturbations

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Abstract

The nonlinearity of singular-vector-based (SV-based) perturbation growth is examined within the context of a global atmospheric forecast model. The characteristics of these nonlinearities and their impact on the utility of SV-based diagnostics are assessed both qualitatively and quantitatively. Nonlinearities are quantified by examining the symmetry of evolving positive and negative ‘twin’ perturbations. Perturbations initially scaled to be consistent with estimates of analysis uncertainty become significantly nonlinear by 12 h. However, the relative magnitude of the nonlinearities is a strong function of scale and metric. Small scales become nonlinear very quickly while synoptic scales can remain significantly linear out to three days. Phase differences between the positive and negative perturbation patterns relative to each other can result in significant nonlinearities even when the basic anomaly patterns are quite similar. Thus, SVs may be qualitatively useful even when nonlinearities are large.

Post-time forecast corrections using SV-based pseudo-inverse perturbations show that, despite significant nonlinear perturbation growth, the nonlinear forecast corrections are similar to the expected linear corrections, even at 72 h. When the nonlinear correction does differ significantly from the expected linear correction, the nonlinear correction is usually better, indicating that in some cases the pseudo-inverse correction effectively suppresses error growth outside the subspace defined by the leading (dry) SVs. Because a significant portion of the nonlinear growth occurs outside of the dry SV subspace, an a priori relative nonlinearity index based on the full perturbations is not a good predictor of when pseudo-inverse-based corrections will be ineffective. However, one can construct a reasonable predictor of pseudo-inverse ineffectiveness by considering the nonlinearities in the synoptic scales or in the SV subspace only. Copyright © 2003 Royal Meteorological Society

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