Abstract. S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier–Stokes equations (referred to as the NS system). Although this restricted nonlinear system (referred to as the RNL system) is greatly simplified in its dynamics in comparison to the associated NS, it nevertheless self-sustains a turbulent state in wall-bounded shear flow with structures and dynamics comparable to those observed in turbulence. Moreover, RNL turbulence can be analysed effectively using theoretical methods developed to study the closely related S3T system. In order to better understand RNL turbulence and its relation to NS turbulence, an extensive comparison is made of diagnostics of structure and dynamics in these systems. Although quantitative differences are found, the results show that turbulence in the RNL system closely parallels that in NS and suggest that the S3T/RNL system provides a promising reduced complexity model for studying turbulence in wall-bounded shear flows.
Two-dimensional laminar roll convection is capable of generating energetic horizontal mean flows via a well-understood process known as the tilting instability. Less wellunderstood is the physical mechanism behind the strong dependence of this effect on the horizontal lengthscale of the convection pattern. Mean flows of this type have been found to form for sufficiently large Rayleigh number in periodic domains with a small aspect ratio of horizontal length to vertical height, but not in large aspect ratio domains.We demonstrate that the elimination of the tilting instability for large aspect ratio is due to a systematic eddy-eddy advectionmechanism intervening at linear order in the tilting instability, and that this effect can be captured in a model retaining two nonlinearly interacting horizontal wavenumber components of the convection field. Several commonly used low-order models of convection also exhibit a shutdown of the tilting instability for large aspect ratio, even though thesemodels do not contain the eddy-eddy advection mechanism. Instability suppression in such models is due to a different mechanism involving vertical advection.We showthat this vertical advection mechanism is excessively strong in the low-order models due to their low resolution, and that the instability shutdown in such models vanishes when they are appropriately extended.