Convection parameterizations such as eddy-diffusivity mass-flux (EDMF) schemes require a consistent closure formulation for the perturbation pressure, which arises in the equations for vertical momentum and turbulence kinetic energy (TKE). Here we derive an expression for the perturbation pressure from approximate analytical solutions for 2D and 3D thermal bubbles. The new closure combines modified pressure drag and virtual mass effects with a new momentum advection damping term. This advection damping is an important source in the lower half of the thermal bubble and at cloud base levels in convective systems. It represents the effect of the perturbation pressure to ensure the non-divergent properties of the flow. The new formulation represents the pressure drag to be inversely proportional to updraft depth. This is found to significantly improve simulations of the diurnal cycle of deep convection, without compromising simulations of shallow convection. It is thus a key step toward a unified scheme for a range of convective motions. By assuming that the pressure only redistributes TKE between updrafts and the environment laterally, a closure for the velocity pressure-gradient correlation is obtained from the perturbation pressure closure. This novel pressure closure is implemented in an extended EDMF scheme and is shown to successfully simulate a rising bubble as well as shallow and deep convection in a single column model.

Most machine learning applications in Earth system modeling currently rely on gradient-based supervised learning. This imposes stringent constraints on the nature of the data used for training (typically, residual time tendencies are needed), and it complicates learning about the interactions between machine-learned parameterizations and other components of an Earth system model. Approaching learning about process-based parameterizations as an inverse problem resolves many of these issues, since it allows parameterizations to be trained with partial observations or statistics that directly relate to quantities of interest in long-term climate projections. Here we demonstrate the effectiveness of Kalman inversion methods in treating learning about parameterizations as an inverse problem. We consider two different algorithms: unscented and ensemble Kalman inversion. Both methods involve highly parallelizable forward model evaluations, converge exponentially fast, and do not require gradient computations. In addition, unscented Kalman inversion provides a measure of parameter uncertainty. We illustrate how training parameterizations can be posed as a regularized inverse problem and solved by ensemble Kalman methods through the calibration of an eddy-diffusivity mass-flux scheme for subgrid-scale turbulence and convection, using data generated by large-eddy simulations. We find the algorithms amenable to batching strategies, robust to noise and model failures, and efficient in the calibration of hybrid parameterizations that can include empirical closures and neural networks.

Because of their limited spatial resolution, numerical weather prediction and climate models have to rely on parameterizations to represent atmospheric turbulence and convection. Historically, largely independent approaches have been used to represent boundary layer turbulence and convection, neglecting important interactions at the subgrid scale. Here we build on an eddy-diffusivity mass-flux (EDMF) scheme that represents all subgrid-scale mixing in a unified manner, partitioning subgrid-scale fluctuations into contributions from local diffusive mixing and coherent advective structures and allowing them to interact within a single framework. The EDMF scheme requires closures for the interaction between the turbulent environment and the plumes and for local mixing. A second-order equation for turbulence kinetic energy (TKE) provides one ingredient for the diffusive local mixing closure, leaving a mixing length to be parameterized. A new mixing length formulation is proposed, based on constraints derived from the TKE balance. It expresses local mixing in terms of the same physical processes in all regimes of boundary layer flow. The formulation is tested at a range of resolutions and across a wide range of boundary layer regimes, including a stably stratified boundary layer, a stratocumulus-topped marine boundary layer, and dry convection. Comparison with large eddy simulations (LES) shows that the EDMF scheme with this diffusive mixing parameterization accurately captures the structure of the boundary layer and clouds in all cases considered.

We demonstrate that an extended eddy-diffusivity mass-flux (EDMF) scheme can be used as a unified parameterization of subgrid-scale turbulence and convection across a range of dynamical regimes, from dry convective boundary layers, over shallow convection, to deep convection. Central to achieving this unified representation of subgrid-scale motions are entrainment and detrainment closures. We model entrainment and detrainment rates as a combination of turbulent and dynamical processes. Turbulent entrainment/detrainment is represented as downgradient diffusion between plumes and their environment. Dynamical entrainment/detrainment are proportional to a ratio of buoyancy difference and vertical velocity scale, partitioned based on buoyancy sorting approaches and modulated by a function of relative humidity difference in cloud layer to represent buoyancy loss owing to evaporation in mixing. We first evaluate the closures offline against entrainment and detrainment rates diagnosed from large-eddy simulations (LES) in which tracers are used to identify plumes, their turbulent environment, and mass and tracer exchanges between them. The LES are of canonical test cases of a dry convective boundary layer, shallow convection, and deep convection, thus spanning a broad range of regimes. We then compare the LES with the full EDMF scheme, including the new closures, in a single column model (SCM). The results show good agreement between the SCM and LES in quantities that are key for climate models, including thermodynamic profiles, cloud liquid water profiles, and profiles of higher moments of turbulent statistics. The SCM also captures well the diurnal cycle of convection and the onset of precipitation.