Prediction of the Evolution of the Variance in a Barotropic Model

Abstract

The Schmidt decomposition is applied to the evolution operator of the linearized barotropic equation on a sphere (in the following referred to as the barotropic propagator) to study the evolution of the variance, that is, of the collective evolution of a cloud of trajectories centered around the initial condition. The variance can give reliable information on the tendency that some initial conditions may have to generate large spreads in the subsequent time evolution, especially when many modes with similarly large amplifying rates exist. It appears rather arbitrary, under these circumstances, to pick a particular mode just because it happens to have the largest rate for that particular numerical formulation and resolution setting. It is also shown that the Golden-Thompson generalized inequality and other indicators can be used to estimate the linear variance from the analysis of the initial condition itself, without the need for performing the costly explicit calculation of the propagator. Numerical experiments performed on a set of initial conditions obtained from a simulation experiment and from observations show that in a barotropic model a spread index based on an indicator of non-self-adjointness, as the Golden-Thompson index, is capable of detecting with good reliability initial conditions with a tendency to produce large spreads.