Reduced-Order Direct Statistical Simulation of Planetary Boundary-Layer Flows for a Novel Turbulence Model

Direct statistical simulation (DSS) using an expansion in equal-time cumulants is a novel approach to subgrid modeling that avoids the common simplifying assumptions of isotropy, homogeneity, and the locality of correlations. This work introduces a reduced-order model (RM) in which a second-order closure (CE2) is Galerkin projected onto a truncated basis obtained by proper orthogonal decomposition. Langmuir turbulence and Rayleigh-Bénard convection are investigated using the CE2 RM in addition to fully resolved and reduced-order nonlinear (NL) direct numerical simulation (DNS) for reference. Quasilinear (QL) DNS, the statistics of which are exactly described by CE2, is also used as a diagnostic tool. Vertical turbulent transports and turbulent kinetic energy production are compared. Results indicate that the CE2 RM can accurately reproduce the NL DNS vertical profiles of some of these quantities with fewer than 0.1% of modes retained. Modified versions of these turbulent statistical quantities that are projected onto the truncated basis are demonstrated to be more consistently represented over time and ensemble instance in the reduced model than the original quantities, revealing limitations of the statistics of such RMs and suggesting possible strategies for optimization of the reduced basis to capture specific statistical quantities. It is shown that the RM of CE2 is able to approximate ensemble-averaged statistics in a single execution by initializing the second cumulant diagonally with ensemble-averaged noise. Reduced CE2 is also shown to scale significantly better with problem size than NL models under the constraint that the number of retained modes remains small compared to that of the full system.