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

Abstract

Convection schemes are a large source of error in global weather and climate models, and modern resolutions are often too fine to parametrize convection, but are still too coarse to fully resolve it. Recently, numerical solutions of multi-fluid equations have been proposed for a more flexible and consistent treatment of subgrid-scale convection, including net mass transport by convection and non-equilibrium dynamics. The technique involves splitting the atmosphere into multiple fluids. For example, the atmosphere could be divided into buoyant updraughts and stable regions. The fluids interact through a common pressure, drag and mass transfers (entrainment and detrainment). Little is known about the numerical properties of mass transfer terms between the fluids. We derive mass transfer terms which relabel the fluids and derive numerical properties of the transfer schemes, including boundedness, momentum conservation and energy conservation on a co-located grid. Numerical simulations of the multi-fluid Euler equations using a C-grid are presented using stable and unstable treatments of the transfers on a well-resolved two-fluid dry convection test case. We find two schemes which are conservative, stable and bounded for large time steps, and maintain their numerical properties on staggered grids. K E Y W O R D S convection, multi-fluid equations, numerical analysis 1This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Two-fluid single-column modelling of Rayleigh-Bénard convection as a step towards multi-fluid modelling of atmospheric convection

“…We choose to model the exchanges of momentum and buoyancy from fluid i to j as a characteristic value, u T i j or b T i j , times the rate of transfer of fluid fraction from i to j , σ i S i j . This aligns with the modelling approach taken in other recent works on multi-fluid modelling (Thuburn et al, 2018(Thuburn et al, , 2019Weller and McIntyre, 2019;Weller et al, 2020;McIntyre et al, 2020).…”

“…The transfer terms S i j are handled explicitly, while the momentum and buoyancy transfers are implicit and operatorsplit, as in (Weller and McIntyre, 2019;McIntyre et al, 2020;Weller et al, 2020).…”

Two-fluid single-column modelling of Rayleigh-Bénard convection as a step towards multi-fluid modelling of atmospheric convection

“…We choose to model the exchanges of momentum and buoyancy from fluid i to j as a characteristic value, u T ij or b T ij , times the rate of transfer of fluid fraction from i to j, σ i S ij . This aligns with the modelling approach taken in other recent works on multi-fluid modelling (Thuburn et al 2018;Thuburn et al 2019;Weller and McIntyre 2019;Weller et al 2020;McIntyre et al 2020).…”

Two‐fluid single‐column modelling of Rayleigh–Bénard convection as a step towards multi‐fluid modelling of atmospheric convection

“…(2020). This handling of the transfer terms guarantees boundedness and conservation of energy and momentum (McIntyre et al ., 2020).…”

“…We choose to model the exchanges of momentum and buoyancy from fluid i to j as a characteristic value, or , times the rate of transfer of fluid fraction from i to j , . This aligns with the modelling approach taken in other recent works on multi‐fluid modelling (Thuburn et al ., 2018, 2019; Weller and McIntyre, 2019; Weller et al ., 2020; McIntyre et al ., 2020).…”