Dynamics and spectra of cumulant update closures for two-dimensional turbulence

Non-Markovian closure theories are compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large scale Reynolds numbers ranging from approximate to 50 to approximate to 4000. The closures, as well as DNS, are formulated for discrete wave numbers relevant to flows on the doubly periodic domain and are compared with the results of continuous wave number closures. The direct interaction approximation (DIA), self-consistent field theory (SCFT) and local energy-transfer theory (LET) closures are also compared with cumulant update versions of these closures (CUDIA, CUSCFT, CULET). The cumulant update closures are shown to have comparable performance to the standard closures but are much more efficient allowing long time integrations. The discrete wave number closures perform considerably better than continuous wave number closures as far as evolved energy and transfer spectra and skewness are concerned. The discrete wave number closures are in reasonable agreement with DNS in the energy containing range of the large scales for Reynolds numbers ranging from approximate to 50 to approximate to 4000. The closures tend to underestimate the enstrophy flux to high wave numbers, increasingly so with increasing Reynolds number, resulting in underestimation of small-scale kinetic energy.