Application of the generalized quasilinear approximation in Rayleigh-Bénard convection

In the generalized quasilinear (GQL) approximation (Marston et al., Phys. Rev. Lett., 116, 214501, 2016), a flow field is decomposed into large-scale (l) and small-scale (h) components by applying a spectral cutoff filter. In this work, we investigate the applicability of the GQL approximation to the two-dimensional planar Rayleigh-Bénard convection (RBC). Recently, regimes have been identified in RBC where the flow can attain different turbulent roll states based on initial roll conditions defined in numerical simulations (Wang et al., Phys. Rev. Lett., 125 (7), 074501, 2020). The GQL approximation is tested in this flow regime. A direct numerical simulation (DNS) is performed first starting from an initial temperature field composed of random perturbations superimposed on a linear conductive profile. The GQL simulations are initiated once a statistically stationary state is reached in the DNS. The GQL approximation is able to capture the convection roll states if the l-subspace contains more wavenumbers than a critical limit. Interestingly, the GQL approximation leads to different states depending on the threshold wavenumber segregating the l and h–subspaces. An increase in the threshold wavenumber yields states corresponding to higher number of convection rolls as in the DNS. Additionally, the GQL methodology is applied as an intrusive technique to study the non-linear triadic scale interactions in the planar RBC system. It is found that the h → l → h scattering is essential for the sustenance of the convection rolls.