Abstract
The covariance equation based on second-order closure for dynamics governed by a general scalar nonlinear partial differential equation (PDE) is studied. If the governing dynamics involve n space dimensions, then the covariance equation is a PDE in 2n space dimensions. Solving this equation for n = 3 is therefore computationally infeasible. This is a hindrance to stochastic-dynamic prediction as well as to novel methods of data assimilation based on the Kalman filter.
It is shown that the covariance equation can be solved approximately, to any desired accuracy, by solving instead an 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.
Analysis of these equations reveals some of the qualitative behavior of their solutions. It is shown that the variance equation, through nonlinear coupling with the mean equation, describes the nonlinear effect of saturation of variance as well as the internal and external growth of variance. Further, it is shown that, in the presence of model error, the initial correlation field is transient, being damped as the influence of the model error correlation grows, while in the absence of model error the initial correlation is simply advected. There is also a critical correlation length, depending on the internal dynamics and on the model error, toward which the forecast error correlation length generally tends.
