Mesoscale Predictability and the Spectrum of Optimal Perturbations

Abstract

The spectrum of finite-time most unstable structures, also referred to as singular vectors (SVs), is computed for a regional, mesoscale primitive-equation model. The number of growing SVs present in this spectrum is of interest for investigating mesoscale predictability since it provides an estimate of the dimension of the unstable subspace of the model phase space. This dimension is used to critically assess the contrasting conclusions that have been reached by different authors in mesoscale predictability studies. Computations are carried out for two different synoptic cases (explosive cyclogenesis over the North Atlantic and Alpine lee cyclogenesis) using two different norms. The first is loosely related to total perturbation energy and the second measures the energy of rotational normal modes only. The latter is designed to reduce the influence of geostrophic adjustment on the measure of growth. The models used are the tangent-linear and adjoint components of the dry-adiabatic version of the primitive-equation regional model denoted as the National Center for Atmospheric Research Mesoscale Adjoint Modeling System version 1. Spectra are obtained for a 24-hour optimization time interval by partially solving the relevant eigenproblems in an iterative fashion using a Lanczos algorithm. The spectra relevant for the first norm are found to possess a very large number of growing SVs, a considerable part of which, however, is growing purely due to adjustment processes. The number of growing structures relevant for the second norm is between 150 and 200, which is approximately 3% of the total degrees of freedom allowed for the perturbations in this situation. This percentage becomes as small as 0.25% if all degrees of freedom are taken into account. The first few SVs amplify by factors between 5 and 10 and are strongly related to the synoptic situation under consideration, being quite localized and possessing baroclinic structure. Also, similar characteristics are found for the first few SVs, independent of which norm is being used. Based on the small number of growing perturbations, it is concluded that it is quite likely that a randomly chosen perturbation will decay because its projection on the growing part of the spectrum is small. Nevertheless, this does not necessarily imply that mesoscale circulations are more, predictable than synoptic-scale circulations due to neglected larger growing scales, the neglect of physical processes like convection, and the short timescale considered. Implications of the results regarding the role of SVs for data assimilation and ensemble prediction are briefly discussed.