Singular Vectors, Finite-Time Normal Modes, and Error Growth during Blocking

Abstract

The evolution of finite-time singular vectors growing on four-dimensional space?time basic states is studied for cases of block development over the Gulf of Alaska and over the North Atlantic, using a two-level tangent linear model. The initial singular vectors depend quite sensitively on the choice of norm with the streamfunction norm characterized by small-scale baroclinic disturbances, the kinetic energy norm giving intermediate-scale baroclinic disturbances, and the enstophy norm typified by large-scale disturbances with large zonal flow contributions. In all cases, the final evolved singular vectors consist of large-scale equivalent barotropic wave trains across the respective blocking regions. There are close similarities between the evolved singular vectors in each of the norms, particularly for the longer time periods considered, and with corresponding evolved finite-time adjoint modes and evolved maximum sensitivity perturbations. For the longer time periods considered, each of these evolved perturbations also closely resembles some of the dominant finite-time normal mode disturbances, which are norm independent. For periods between about two weeks and a month, the convergence of the evolved leading singular vector and leading finite-time normal mode toward the leading left Lyapunov vector has been examined. The evolution of errors, represented by singular vectors, is also considered in the space of finite-time normal modes. In all cases the evolved error dynamics contracts onto a low-dimensional subspace characterized by the dominant finite-time normal modes. The growth of norms based on streamfunction, kinetic energy, or enstrophy is compared with the growth of a norm based on the projection coefficients of the disturbance onto the dominant finite-time normal modes. The prospect of ensemble prediction schemes in which the control initial conditions are perturbed by superpositions of the dominant finite-time normal modes is discussed.