A Perturbative Correction to the Quasi-Linear Approximation for Stratified Turbulence

Recently, Chini et al. (JFM vol. 933, 2022) demonstrated that a quasilinear (QL) reduction of the Boussinesq equations is formally justified in the limit of strong stable density stratification (when the Froude number Fr→0). In this limit, there is a separation between the time scale of the fastest-growing stratified shear instabilities and the advection/vertical-shear time scale of the large-scale, layer-like horizontal flow, causing the QL instabilities to self-tune to a state of near-marginal stability. These features enabled the authors to develop a hybrid eigenvalue/initial-value algorithm for efficiently simulating the resulting slow-fast QL system. For small but finite Fr, however, fast modes with horizontal wavenumbers differing from that of the dominant mode may be excited nonlinearly, leading to a spectrally-local downscale energy cascade. Here, we show that these additional modes can be perturbatively incorporated via a weakly nonlinear analysis about the emergent marginal stability manifold. Computation of the higher harmonics, which are coupled to the marginal fundamental mode, requires only three times the total operation count of the single-mode algorithm. We demonstrate the efficacy of the extended-QL algorithm for 2D stratified Kolmogorov flow via direct comparisons with DNS. The systematic extension of QL theory developed herein should prove advantageous for other shear flows for which the QL reduction provides a useful point of departure.