A two‐fluid single‐column model of turbulent shallow convection. Part II: Single‐column model formulation and numerics

Abstract

The two‐fluid single‐column model of Thuburn et al. (Quart. J. R. Meteorol. Soc., 2019, 145, 1535–1550) is extended to include moisture and horizontal wind shear. Turbulent kinetic energy is introduced as a prognostic variable, dependence on a diagnosed boundary‐layer height is removed, and subfilter fluxes are approximated using a two‐fluid version of a Mellor–Yamada scheme. Three mechanisms for entrainment and detrainment processes are introduced, which represent entrainment of unstable air at the surface, forced detrainment of air at the top of the boundary/cloud layers, and turbulent mixing that relaxes the convective fluid to a reference profile. A semi‐implicit Eulerian discretization replaces the semi‐implicit semi‐Lagrangian implementation of Thuburn et al. (Quart. J. R. Meteorol. Soc., 2019, 145, 1535–1550) to improve numerical stability and conservation. The equations for the implicit time step are solved using a quasi‐Newton method, which is shown to perform well in numerical tests for conservation and convergence. The two‐fluid single‐column model presented in this article will be applied to simulations of shallow cumulus convection in Part III.

A two‐fluid single‐column model of turbulent shallow convection. Part III: Results and parameter sensitivity

“…This study closes the entrainment and detrainment formulations of the two‐fluid single‐column model from Thuburn et al, (2022a; 2022b) by using LES data to diagnose the expected properties of entrained and detrained air. Other coefficients, such as the Mellor–Yamada parameters and the mixing rates, were guided using a gradient‐descent algorithm that minimized the root‐mean‐square error of the cloud fraction and other mean profiles of the convective fluid.…”

“…$$$ {N}_i\approx \sqrt{\frac{g}{\theta_i}\frac{\partial {\theta}_i}{\partial z}} $$$ is the Brunt–Väisälä frequency for fluid $$$ i $$$. $$$ {L}_i^{\mathrm{PLM}} $$$ is the plume length‐scale, which is calculated from the height, TKE, and Brunt–Väisälä frequency (Thuburn et al, 2022b). $$$ {W}^{\mathrm{CLD}} $$$ is a weighting that transitions smoothly between cloudy ($$$ {W}^{\mathrm{CLD}}=1 $$$) and noncloudy regions ($$$ {W}^{\mathrm{CLD}}=0 $$$).…”

“…$$$ {b}_{\eta, 21}^{\mathrm{INS}}=1 $$$ is used because the correct fluid 2 surface buoyancy is produced—using the LES‐diagnosed value of $$$ {b}_{\eta, 21}^{\mathrm{INS}}\approx 0.5 $$$ (Figure 4) results in double the magnitude of the diagnosed buoyancy and unrealistically high vertical velocities at all heights. It should be noted that a smaller coefficient for both $$$ {b}_{\eta, 21}^{\mathrm{INS}} $$$ and $$$ {b}_{q,21}^{\mathrm{INS}} $$$ should be feasible with a different partitioning of the surface fluxes between fluid 1 and 2 (see Thuburn et al, 2022b). This will be investigated further in future studies.…”

“…The equations for the subfilter terms are based on a level‐2.5 multifluid Mellor–Yamada scheme (see Part I for full details; Thuburn et al, 2022a) and thus neglect transience, advection, and third‐order turbulent fluxes (except in the TKE equation). Full details of the chosen higher‐order terms and closures are detailed in Thuburn et al, (2022b), and a summary of all parameters is provided in Table 1.…”

A two‐fluid single‐column model of turbulent shallow convection. Part 1: Turbulence equations in the multifluid framework

“…The main developments, relative to the model described by Thuburn et al (2019), are the inclusion of moist processes and horizontal wind components, the prediction or diagnosis of various subfilter-scale turbulence quantities, which are used in the parameterization of several processes in the model, and improved numerical methods, making the model more accurate, stable, and robust. The new model formulation is detailed in Part 2 (Thuburn et al, 2022) and example results are presented in Part 3 (McIntyre et al, 2022). The present article derives the equations for subfilter-scale turbulent second moments in the multifluid framework, providing the basis for the turbulence component of the model as well as its possible future extensions and generalizations.…”