The generalized quasilinear approximation of two-dimensional Rayleigh–Bénard convection

In the generalized quasilinear approximation (GQLA) (Marston et al. , Phys. Rev. Lett. , vol. 116, 2016, 214501), a threshold wavenumber ( $k_0$ ) in the direction of translational symmetry segregates the total into large- ( $l$ ) and small-scale ( $h$ ) fields. While the governing equation for the large-scale field is fully nonlinear, that for the small scales is linearized with respect to the large-scale field. In addition, some nonlinear triad interactions are omitted in the GQLA. Herein, the GQLA is applied to two-dimensional planar Rayleigh–Bénard convection (RBC). A scale separation between the large-scale convection rolls and small-scale turbulent fluctuations is typical in RBC. The present work explores the efficacy of GQLA in capturing the scale-by-scale energy transfer processes in RBC. The initial condition for the GQLA simulations was either the statistically stationary state obtained in direct numerical simulation (DNS) or random fluctuations superimposed on the linear conductive temperature profile and $\boldsymbol {u}=0$ . The GQLA simulations can capture the convection rolls for $k_0$ larger than or equal to the dominant wavenumber for thermal driving of the flow ( $k_0 \ge k_{\hat {Q}}$ ). Additionally, the GQLA emulates the fully nonlinear dynamics for $k_0$ larger than or equal to the first harmonic of the convection-roll wavenumber ( $k_0 \ge 2k_{roll}$ ). In the intermediate regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$ , the dynamics captured in GQLA simulations is different from the DNS. In DNS, two primary energy transfer processes dominate: (i) the energy transfer to/from the convection rolls and (ii) the scale-by-scale inverse kinetic energy and forward thermal energy cascades mediated by the convection rolls. The fully nonlinear dynamics is emulated by GQLA when these energy transfer processes are faithfully reproduced. Utilizing the framework of altering the triad interactions in GQLA, an additional intrusive calculation, including target triad interactions, is performed here to study their influence. This intrusive calculation shows that the convection rolls are not captured in GQLA for $k_0 < k_{\hat {Q}}$ because of the exclusion of the $h \to l \to l$ and $l \to h \to h$ triad interactions in GQLA. The inclusion of these triad interactions in the intrusive calculation yields the convection rolls, and the reproduced dynamics is similar to that of the intermediate GQLA regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$ .