Multi-fluid single-column modelling of Rayleigh-Bénard convection

Atmospheric convection remains one of the weakest parts of weather and climate models, especially in the tropics. As model resolutions increase, the assumptions underlying traditional convection parametrisations break down; however, we are still far from fully resolving all convective processes, showing a need for convection parametrisation well into the future.

A multi-fluid framework for parametrising convection has been proposed, based on conditionally filtering the Navier-Stokes equations. This results in a set of equations for multiple fluids, where each fluid has its own dynamic and thermodynamic fields. The approach is fully 3D and time-dependent, allowing for both convective memory and net mass transport due to convection. However, the approach differs from higher-order turbulence closures in that it attempts to capture the important coherent structures of convection via the partitioning into multiple fluids. In addition to the usual sub-filter fluxes, the equations contain terms involving the exchange of momentum, entropy, moisture, and tracers between different fluids. The problem of parametrising convection then becomes the problem of parametrising these exchange terms. This means that within this framework the convection is fundamentally a part of the dynamics: there is no separate “convection scheme" which is called by the dynamical core.

As a first step towards using this framework to parametrise atmospheric convection, we consider a highly simplified model: dry, 2D Rayleigh-Bénard convection in the Boussinesq limit. This model captures the essentials of buoyant convection, with additional symmetry constraints which help with building a parametrisation. In the single-column limit of the single-fluid case, no circulation can exist. This leads to a very poor solution, in particular vastly underestimating the heat transport.

In a two-fluid model, the separate dynamical fields for each fluid mean that a circulation can exist even at the coarsest resolutions. We show that a simple two-fluid single-column model can capture all the essentials of the horizontally-averaged time-mean high resolution solution, including the buoyancy, vertical velocity, and pressure profiles, as well as much better representation of the heat flux. We explore the consequences of different choices for the parametrisations of the exchange terms, showing that a good representation of volume, momentum, and buoyancy exchange, and of the pressure difference between the fluids, is required. For this simple case, this is achieved entirely without parametrisation of subfilter terms, showing that the multi-fluid approach is capturing the coherent structures of convection well.