Markov Chain Stochastic Parametrizations of Essential Variables

We analyze the performance of the novel Markov chain stochastic modeling technique for derivation of effective equations for a set of essential variables. This technique is an empirical approach where the right-hand side of the essential variables is modeled by a Markov chain. We demonstrate that the Markov chain modeling approach performs well in a prototype model without scale separation between the essential and the nonessential variables. Moreover, we utilize the truncated Burgers–Hopf model to show that the Markov chain should be properly conditioned on the essential variables to reproduce the structure of two-point statistical quantities. On the other hand, the conditioning can be rather straightforward and unsophisticated.