Stable Numerical Implementation of a Turbulence Scheme with Two Prognostic Turbulence Energies

In this paper, we present a new and more stable numerical implementation of the two-energy configuration of the Third Order Moments Unified Condensation and N-dependent Solver (TOUCANS) turbulence scheme. The original time-stepping scheme in TOUCANS tends to suffer from spurious oscillations in stably stratified turbulent flows. Because of their high frequency, the oscillations resemble the so-called fibrillations that are caused by the coupling between turbulent exchange coefficients and the stability parameter. However, our analysis and simulations show that the oscillations in the two-energy scheme are caused by the usage of a specific implicit–explicit temporal discretization for the relaxation terms. In TOUCANS, the relaxation technique is used on source and dissipation terms in prognostic turbulence energy equations to ensure numerical stability for relatively long time steps. We present both a detailed linear stability analysis and a bifurcation analysis, which indicate that the temporal discretization is oscillatory for time steps exceeding a critical time-step length. Based on these findings, we propose a new affordable time discretization of the involved terms that makes the scheme more implicit. This ensures stable solutions with enough accuracy for a wider range of time-step lengths. We confirm the analytical outcomes in both idealized 1D and full 3D model experiments.