Abstract
The properties of the expected analysis and forecast error covariance matrices are explored using a novel method based on the tangent linearization and adjoint of a 4-dimensional variational (4D-Var) data assimilation system. The method is applied to the mesoscale circulation that develops in the presence of a baroclinically unstable mid-latitude ocean temperature front using a series of paternal twin experiments that employ both strong and weak constraint 4D-Var. Adopting the traditional view of Empirical Orthogonal Functions (EOFs) of a covariance matrix as the semi-major axes of a multi-dimensional hyper-ellipsoid, variations in the volume of the analysis and forecast error hyper-ellipsoids are explored which provides information about the flow of probability through state–space. The complementary variations in the expected total variance of the covariance matrix are also investigated. Two different kinds of behavior are identified that are associated with either the demise or growth of baroclinic instabilities. In both cases, the volume of the hyper-ellipsoid decreases during the 4D-Var analysis cycle. During the subsequent forecasts, the volume of the forecast error hyper-ellipsoid initially continues to collapse under both scenarios. During this time, the hyper-ellipsoid becomes increasingly elongated along some of the semi-major axes as forecast errors grow in preferential directions. Growth in these directions is controlled by the most unstable error modes, and projection of forecast error on to the precursors of these modes has been shown previously to be characterized by upscale energy transfer and non-normal processes. For the case of the growing wave, the forecast error hyper-ellipsoid continues to collapse through to the end of the forecast period. However, for the decaying wave, the hyper-ellipsoid may undergo expansion at longer forecast lead times.
•The adjoint of a data assimilation system is used to compute error covariances.•Some properties of the drift velocity of the probability density are quantified.•The growth and decay of baroclinic instabilities exhibit different error behaviors.