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

Abstract

The multifluid equations are derived from the compressible Euler equations (or any of the usual approximate equation sets used in meteorology) by conditional filtering. They have the potential to provide the basis for an improved representation of cumulus convection and its coupling to the boundary layer and larger scale flow in numerical models. The present article derives the prognostic equations for subfilter-scale turbulent second moments in the multifluid framework, along with certain systematic simplifications of them, thus providing a multifluid analogue of the well-known Mellor and Yamada hierarchy of turbulence closures. As well as enabling a more accurate calculation of subfilter-scale fluxes and the effects of subfilter-scale variability on cloud fraction, liquid water, and buoyancy, the second moment information can be used to obtain a more accurate parameterization of entrainment and detrainment. A subset of the turbulence equations derived here is employed in the two-fluid single-column model described in Part 2 and applied to the simulation of shallow cumulus cases in Part 3.

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

“…The governing equations are adapted from the conditionally filtered compressible Euler equations (Thuburn et al, 2018) for a single-column setting, where horizontal gradients are neglected. Additional equations for higher-order moment terms were derived in Thuburn et al, (2022a). Each of the two fluids (with subscripts i ∈ {1, 2}) has prognostic equations for the mass per unit volume (m i ), entropy (𝜂 i ), specific humidity (q i ), velocity components (u i , v i , w i ), and turbulent kinetic energy (TKE, k i ), respectively:…”

“…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.…”

“…b INS 𝜂,21 = 1 is used because the correct fluid 2 surface buoyancy is produced-using the LES-diagnosed value of b INS 𝜂,21 ≈ 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 INS 𝜂,21 and b INS q,21 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.…”

“…This article is the third in a three-part series that documents progress in the development of a two-fluid single-column model and its application to simulating shallow cumulus convection. Part I (Thuburn et al, 2022a) derived the equations for subfilter-scale turbulent second moments. Part II (Thuburn et al, 2022b) presented a new model formulation, including new prognostic equations, closures, and numerics.…”

“…With the aim of further advancing the multi-fluid model, this article focusses on using the new multi-fluid higherorder-moment terms from Thuburn et al (2022a), and the new closures and numerics from Thuburn et al (2022b), to simulate shallow convection over land (ARM case) and ocean (BOMEX case) using a two-fluid single-column model (2FSCM). Brown et al (2002) performed a Large Eddy Simulation (LES) of the formation and evolution of shallow cumulus over the Atmospheric Radiation Measurement (ARM) Southern Great Plains site in June 1997.…”