Dynamics of short-term univariate forecast error covariances

Abstract

The covariance equation can be solved approximately, to any desired accuracy, by solving instead in auxiliary system of PDEs in just n dimensions. The first of these is a dynamical equation for the variance field. Successive equations describe, to increasingly high order, the dynamics of the shape of either the covariance function or the correlation function for points separated by small distances. The second-order equation, for instance, describes the evolution of the correlation length (turbulent microscale) field. Each auxiliary equation is coupled only to the preceding, lower-order equations if the governing dynamics are hyperbolic, but is weakly coupled to the following equation in the presence of diffusion. -from Author