Closure theories with non-Gaussian restarts for truncated two-dimensional turbulence

Abstract

NonMarkovian closure theories, with and without non-Gaussian restarts, are compared with ensemble averaged direct numerical simulations (DNS) for severely truncated two-dimensional Navier–Stokes flows. Both the closures and DNS are formulated for discrete spectra relevant to flows on the doubly periodic domain allowing unambiguous comparisons between the closure and DNS results. We examine the performance of the direct interaction approximation (DIA), self-consistent field theory (SCFT) and local energy-transfer theory (LET) closures and are particularly interested in the reliability of cumulant update versions of these closures (CUDIA, CUSCFT, and CULET). In the latter, the potentially long time-history integrals are periodically truncated and the closures are restarted using a three-point cumulant as the new non-Gaussian initial conditions, thus yielding computationally much more efficient closures. In 80-day integrations, the DIA replicates the DNS results most faithfully in inviscid, viscous decay and forced dissipative experiments. With an update time of T=10 days, the CUDIA is particularly promising performing nearly as well but with some extra oscillations at intermediate times. The SCFT and particularly LET, have spurious oscillations in inviscid and viscous decay experiments; this is also the case, but to a greater degree, for the CUSCFT and CULET closures.