Linear evolution of error covariances in a quasigeostrophic model

The characteristics of forecast-error covariances, which are of central interest in both data assimilation and ensemble forecasting, are poorly known. This paper considers the linear dynamics of these covariances and examines their evolution from (nearly) homogeneous and isotropic initial conditions in a turbulent quasigeostrophic flow qualitatively similar to that of the midlatitude troposphere. The experiments use ensembles of 100 solutions to estimate the error covariances. The error covariances evolve on a timescale of O(1 day), comparable to the advective timescale of the reference flow. This timescale also defines an initial period over which the errors develop characteristics features that are insensitive to the chosen initial statistics. These include 1) scales comparable to those of the reference flow, 2) potential vorticity (PV) concentrated where the gradient of the reference-flow PV is large, particularly at the surface and tropopause, and 3) little structure in the interior of the troposphere. In the error covariances, these characteristics are manifest as a strong spatial correlation between the PV variance and the magnitude of the reference-flow PV gradient and as a pronounced enhancement of the error correlations along reference-flow PV contours. The dynamical processes that result in such structure are also explored; the key is the advection of reference-flow PV by the error velocity, rather than the passive advection of the errors by the reference flow.