Non-linear forcing singular vector of a two-dimensional quasi-geostrophic model

We propose a non-linear forcing singular vector (NFSV) approach to infer the effect of non-linearity on the predictability associated with model errors. The NFSV is a generalisation of the forcing singular vector (FSV) to non-linear fields and acts as a tendency perturbation that results in a significantly large perturbation growth. In predictability studies, the NFSV, as a tendency error, may provide useful information about model errors that cause severe prediction uncertainties. In this article, a two-dimensional quasi-geostrophic (QG) model is used to study NFSVs and make a comparison between NFSVs and FSVs. We choose two basic flows: the first is a zonal steady flow (Ref-1), and the second is a meridional steady flow (Ref-2). The results demonstrate that the corresponding NFSVs contain a phase where the stream function tends to be contracted around regions of strong velocity shear. Furthermore, the NFSVs for the Ref-1 tend to have a meridional asymmetric spatial structure. Due to the absence of non-linearity, FSVs tend to have a larger spatial extension than NFSVs; in particular, the FSVs for the Ref-1 are almost symmetric in the stream function component. The prediction errors caused by FSVs in the non-linear QG model are generally smaller than those caused by FSVs in the linearised QG model; therefore, the non-linearity in the QG model would significantly saturate the perturbation growth. Nevertheless, the prediction errors caused by NFSVs (especially for the Ref-1) in the non-linear QG model are larger than those caused by FSVs, which further implies that the tendency errors of NFSV structures tend to reduce the damping effect of the non-linearity on the perturbation growth and are more applicable than those of FSV structures to describing the optimal mode of the model errors. The differences between NFSVs and FSVs demonstrate the usefulness of NFSVs in revealing the effects of non-linearity on predictability. The NFSV may be a useful non-linear technique for exploring the predictability problems introduced by model errors.