Numerical solution of the conditionally averaged equations for representing net mass flux due to convection

Abstract

The representation of subgrid‐scale convection is a weak aspect of weather and climate prediction models and the assumption that no net mass is transported by convection in parametrizations is increasingly unrealistic as models enter the grey zone, partially resolving convection. The solution of conditionally averaged equations of motion (multifluid equations) is proposed in order to avoid this assumption. Separate continuity, temperature, and momentum equations are solved for inside and outside convective plumes, which interact via mass‐transfer terms, drag, and by a common pressure. This is not a convection scheme that can be used with an existing dynamical core—this requires a whole new model. This article presents stable numerical methods for solving the multifluid equations, including large transfer terms between the environment and plume fluids. Without transfer terms, the two fluids are not sufficiently coupled and solutions diverge. Two transfer terms are presented, which couple the fluids together in order to stabilize the model: diffusion of mass between the fluids (similar to turbulent entrainment) and drag between the fluids. Transfer terms are also proposed to move buoyant air into the plume fluid and vice versa as would be needed to represent initialization and termination of subgrid‐scale convection. The transfer terms are limited (clipped in size) and solved implicitly in order to achieve bounded, stable solutions. Results are presented for a well‐resolved warm bubble with rising air being transferred to the plume fluid. For stability, equations are formulated in advective rather than flux form and solved using bounded finite‐volume methods. Discretization choices are made to preserve boundedness and conservation of momentum and energy when mass is transferred between fluids. The formulation of transfer terms in order to represent subgrid convection is the subject of future work.

Multifluids for Representing Subgrid‐Scale Convection

“…The final stage of the time step iteration is to reconstruct u i from u i following Weller and McIntyre (2019). The stages calculating P , p i , and u i are repeated twice per time step iteration.…”

“…Thuburn et al (2019) and Weller and McIntyre (2019) used the same pressure for both fluids. This led to unstable equations that Thuburn et al (2019) stabilized using diffusion of vertical velocity and Weller and McIntyre (2019) stabilized using mass exchanges or drag between the fluids.…”

“…Pressure differences between the fluids can lead to form drag, which is parameterized following Simpson and Wiggert (1969), Romps and Charn (2015), and Weller and McIntyre (2019) as where C D is a drag coefficient and r c is a cloud radius. Sensitivity to C D / r c is explored in section 4.…”

Numerical methods for entrainment and detrainment in the multi‐fluid Euler equations for convection

“…In order to test the properties of the various transfer schemes on a staggered grid, we have implemented them into the multi‐fluid fully compressible Euler equation solver from Weller and McIntyre () using operator splitting. We will run test cases adapted from the single‐fluid rising bubble test case (defined in Bryan and Fritsch, ) where an initially stationary temperature anomaly rises and generates resolved circulations (Figure ).…”

“…Subsequent studies have built upon these foundations including Thuburn and Vallis () who investigate the conservation properties and normal modes of the equations, Thuburn et al . () who use the method for a two‐fluid single‐column convective boundary‐layer scheme, and Weller and McIntyre () who formulate a numerical solution of the multi‐fluid compressible Euler equations. Thus far, little is known about the numerical properties of solutions to the multi‐fluid equations.…”