Singular vectors, finite-time normal modes, and error growth during blocking

The evolution of finite-time singular vectors growing on four-dimensional s pace-time basic states is studied for cases of block development over the G ulf 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 baroclin ic disturbances, the kinetic energy norm giving intermediate-scale baroclin ic disturbances, and the enstophy norm typified by large-scale disturbances with large zonal Row contributions. In all cases, the final evolved singul ar vectors consist of large-scale equivalent barotropic wave trains across the respective blocking regions. There are close similarities between the e volved singular vectors in each of the norms, particularly for the longer t ime periods considered, and with corresponding evolved finite-time adjoint modes and evolved maximum sensitivity perturbations. For the longer time pe riods considered, each of these evolved perturbations also closely resemble s some of the dominant finite-time normal mode disturbances, which are norm independent. For periods between about two weeks and a month, the converge nce of the evolved leading singular vector and leading finite-time normal m ode toward the leading left Lyapunov vector has been examined. The evolution of errors, represented by singular vectors, is also considere d in the space of finite-time normal modes. In all cases the evolved error dynamics contracts onto a low-dimensional subspace characterized by the dom inant 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 co nditions are perturbed by superpositions of the dominant finite-time normal modes is discussed.