Pressure‐potential‐temperature covariance in convection with rotation

The pressure‐potential‐temperature covariance in free and rotating turbulent convection with no mean velocity shear is analysed, using a dataset generated with a large‐eddy simulation (LES) model. The pressure field is resolved into turbulence‐turbulence, buoyancy and Coriolis components, and the contributions from these components to the pressure‐gradient‐potential‐temperature covariance in the budget equation for the potential‐temperature flux are examined. In non‐rotating convection, the buoyancy contribution compensates for about one half of the buoyant production term in the flux budget equation, and the turbulence‐turbulence contribution is well approximated by the Rotta‐type return‐to‐isotropy model with the relaxation time‐scale set proportional to the turbulence energy dissipation time‐scale. In convection with rotation, neither the simplest Rotta‐type model with the relaxation time‐scale proportional to the energy dissipation time‐scale nor the more sophisticated two‐component‐limit (TCL) nonlinear model are able to accurately describe the LES data. A somewhat better agreement is found when a limitation is imposed on the relaxation time‐scale due to the background rotation. The simplest model for the buoyancy contribution, where it is set proportional to the buoyant production term in the flux budget equation, fares poorly. The TCL model shows better agreement with LES data although some uncertainties remain. However, the relative importance of the buoyancy contribution to the pressure‐gradient‐potential‐temperature covariance decreases with increasing rotation rate. In contrast, the Coriolis contribution becomes more important as the rotation rate increases. Neither the simplest linear model for the Coriolis contribution nor the much more complex nonlinear TCL model are found to be adequate. Neither model appropriately accounts for the component of the angular velocity of rotation that is parallel to the component of the pressure gradient in question. In the seemingly simplest case considered in the present paper, when the rotation vector is aligned with the vector of gravity, no Coriolis contribution to the vertical‐pressure‐gradient‐potential‐temperature covariance is predicted by these models. This results in a strong underestimation of the pressure term in the flux budget equation and may lead to an erroneous prediction of the vertical potential‐temperature flux in convection with rotation. In an attempt to remedy the situation, an extension of the TCL model that contains only one extra empirical coefficient is developed and checked against LES data.