Conditional nonlinear optimal perturbation of a T21L3 quasi‐geostrophic model

Abstract

AbstractConditional nonlinear optimal perturbation (CNOP) is defined as an initial perturbation that makes the cost function attain the maximum at a prescribed forecast time with physical constraint conditions, which is a natural generalization of the linear singular vector (LSV) into the nonlinear regime. In this paper, CNOPs of a T21L3 quasi‐geostrophic (QG) spectral model in three kinds of norms are obtained numerically by solving nonlinear optimization problems, and further compared with their linear counterparts, namely LSVs.Results reveal that CNOPs do depend, as LSVs do, on the norm chosen. The stream‐function‐squared norm yields small‐scale disturbances; the results obtained by total‐energy norm are characterized by intermediate‐scale disturbances; and in the case of the enstrophy norm, CNOPs are typified by large‐scale disturbances with large zonal flow contribution. If the linear approximation is valid, CNOP shows great resemblance to LSV, both of which are much localized in cases of the stream‐function‐squared and total‐energy norms. However, with the increasing of the optimization time interval and/or the magnitude of the initial constraint condition, CNOP has less localized structures than the corresponding LSV in these two norms. What is more, the wave train structures of CNOP may even be found in the whole zonal direction in the Northern Hemisphere. The evolutions of perturbations, represented by CNOPs and LSVs in the total‐energy norm, are also investigated in detail. The similarities and dissimilarities between CNOPs and LSVs are revealed not only from the growth rate but also from the similarity index of spatial patterns.The numerical results imply that the method of CNOP may be a more appropriate tool for the study of stability and sensitivity problems when nonlinearity is of importance. Furthermore, the proper norm should be chosen for different physical problems. Copyright © 2008 Royal Meteorological Society