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... 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...