An Adjoint Sensitivity Study of the Efficacy of Modal and Nonmodal Perturbations in Causing Model Block Onset*

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Abstract
With a blocking index as the response function, the adjoint sensitivity formalism is used to assess the impact of normal modes, adjoint modes, and regional singular vectors on prediction of block onset in a two-layer model. The authors focus on three blocks excited by perturbing the model’s state vector at times preselected using the maximal perturbation that defines the direction in phase space associated with the largest possible change in the response function. The sets of normal modes, adjoint modes, and regional singular vectors (using the total energy or the L2 norm) are computed on instantaneous basic-state flows for the preselected times and sensitivity results are presented for a time window of 3 days. When ordered by decreasing values of the growth rates of the normal modes, the authors find that some distant normal modes and adjoint modes can produce larger changes in the response function than some of their leading counterparts. In contrast, the sets of regional singular vectors contain easily identifiable subsets of structures associated with relatively large changes in the response function. The largest changes are produced by less than the first 20 regional singular vectors. Some of these individual regional singular vectors capture the onset of the block when used as perturbations to the initial condition in a nonlinear model integration, a result of the importance for ensemble forecasting. It is found that the first five most explosive regional singular vectors of the energy (L2) norm explain over 20% (60%) of the norm contained in the maximal perturbation at initial time. Despite the failure of all individual normal modes to excite the block, as opposed to adjoint modes and regional singular vectors, the authors argue that, paradoxically, the normal mode concept remains a viable tool to explain the dynamics of block onset.

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