  Direct statistical simulation (DSS) of nonlinear dynamical systems bypasses
the traditional route of accumulating statistics by lengthy direct numerical
simulations (DNS) by solving the equations that govern the statistics
themselves. DSS suffers, however, from the curse of dimensionality as the
statistics (such as correlations) generally have higher dimension than the
underlying dynamical variables. Here we investigate two approaches to reduce
the dimensionality of DSS, illustrating each method with numerical experiments
with the Lorenz-96 dynamical system. The forms of DSS chosen here involve
approximate closures at second and third order in the equal time cumulants. We
demonstrate significant reduction in computational effort that can be achieved
without sacrificing the accuracy of DSS. The methods developed here can be
applied to turbulent fluid and magnetohydrodynamical systems.
