A single‐column model of the dry, shear‐free, convective boundary layer is presented in which non‐local transports by coherent structures such as thermals are represented by the partitioning of the fluid into two components, updraught and environment, each with a full set of prognostic dynamical equations. Local eddy diffusive transport and entrainment and detrainment are represented by parametrizations similar to those used in eddy diffusivity mass flux schemes. The inclusion of vertical diffusion of the vertical velocity is shown to be important for suppressing an instability inherent in the governing equations. A semi‐implicit semi‐Lagrangian numerical solution method is presented and shown to be stable for large acoustic and diffusive Courant numbers, though it becomes unstable for large advective Courant numbers. The solutions are able to capture key physical features of the dry convective boundary layer. Some of the numerical challenges posed by sharp features in the solution are discussed, and areas where the model could be improved are highlighted.

Recent increases in computing power mean that atmospheric models for numerical weather prediction are now able to operate at grid spacings of the order of a few hundred meters, comparable to the dominant turbulence length scales in the atmospheric boundary layer. As a result, models are starting to partially resolve the coherent overturning structures in the boundary layer. In this resolution regime, the so‐called boundary layer “gray zone,” neither the techniques of high‐resolution atmospheric modeling (a few tens of meters resolution) nor those of traditional meteorological models (a few kilometers resolution) are appropriate because fundamental assumptions behind the parameterizations are violated. Nonetheless, model simulations in this regime may remain highly useful. In this paper, a newly formed gray zone boundary layer community lays the basis for parameterizing gray zone turbulence, identifies the challenges in high‐resolution atmospheric modeling and presents different gray zone boundary layer models. We discuss both the successful applications and the limitations of current parameterization approaches, and consider various issues in extending promising research approaches into use for numerical weather prediction. The ultimate goal of the research is the development of unified boundary layer parameterizations valid across all scales.

Operational high‐resolution numerical weather prediction models are now able to partially resolve turbulent motions due to increased computing power. The partitioning of resolved and parametrized fluxes becomes important in the representation of turbulent transfer that determines the state of the atmospheric boundary layer. In this study, successive simulations of a convective boundary layer using the Met Office Large Eddy Model from the large‐eddy simulation to the mesoscale limit are compared with the corresponding coarse‐grained fields to examine model behaviour over the grey zone. The differences in the turbulent kinetic energy partitioning between coarse‐grained (reference) and actual fields are identified and used to quantify sub‐grid diffusion in the grey zone under different atmospheric forcings and surface heat flux. It is shown that the excessive mixing due to the large values of mixing length at coarse resolutions results in the cut‐off of resolved turbulence in the grey zone. The damping of resolved motions comes earlier for wind shear runs. In contrast, coarse‐grained fields exhibit a smooth transition of the resolved turbulent kinetic energy across the scales. Decreasing numerical dissipation through the sub‐grid scheme leads to the increase of resolved turbulence, but fails to reproduce the reference transition pattern that imposes some physical limitations to the partially resolved turbulence simulations. Pragmatic blending of mixing length values maintains a more realistic turbulent kinetic energy transition from fine to coarse resolutions and potential temperature profiles in the grey zone. Bounding vertical diffusion to its effective values, an approach based on maintaining inherent properties of the flow across the scales, is able to match the coarse‐grained fields from the highly resolved to the almost unresolved state, regardless of the forcing. Finally, the complexity of modelling in the grey zone is exhibited in the dependence of turbulence onset on time and vertical resolution.

Numerical simulations of two cases of morning boundary layer development are conducted to investigate the impact of grid resolution on mean profiles and turbulent kinetic energy (TKE) partitioning from the large eddy simulation (LES) to the mesoscale limit. Idealized LES, using the 3‐D Smagorinsky scheme, is shown to be capable of reproducing the boundary layer evolution when compared against measurements. However, increasing grid spacing results in the damping of resolved TKE and the production of superadiabatic temperature profiles in the boundary layer. Turbulence initiation is significantly delayed, exhibiting an abrupt onset at intermediate resolutions. Two approaches, the bounding of vertical diffusion coefficient and the blending of the 3‐D Smagorinsky with a nonlocal 1D scheme, are used to model subgrid diffusion at grey zone resolutions. Simulations are compared against the coarse‐grained fields from the validated LES results for each case. Both methods exhibit particular strengths and weaknesses, indicating the compromise that needs to be made currently in high‐resolution numerical weather prediction. The blending scheme is able to reproduce the adiabatic profiles although turbulence is underestimated in favor of the parametrized heat flux, and the spin‐up of TKE remains delayed. In contrast, the bounding approach gives an evolution of TKE that follows the coarse‐grained LES very well, relying on the resolved motions for the nonlocal heat flux. However, bounding gives unrealistic static instability in the early morning temperature profiles (similar to the 3‐D Smagorinsky scheme) because model dynamics are unable to resolve TKE when the boundary layer is too shallow compared to the grid spacing.

No abstract

A scale-dependent Lagrangian-averaged dynamic Smagorinsky subgrid scheme with stratification effects is used to simulate the evolving convective boundary layer of the Wangara (Australia) case study in the gray-zone regime (specifically, for grid lengths from 25 to 400 m). The dynamic Smagorinsky and standard Smagorinsky approaches are assessed for first- and second-order quantities in comparison with results derived from coarse-grained large-eddy simulation (LES) fields. In the LES regime, the subgrid schemes produce very similar results, albeit with some modest differences near the surface. At coarser resolutions, the use of the standard Smagorinsky approach significantly delays the onset of resolved turbulence, with the delay increasing with coarsening resolution. In contrast, the dynamic Smagorinsky scheme much improves the spinup and so is also able to maintain consistency with the LES temperature profiles at the coarser resolutions. Moreover, the resolved part of the turbulence reproduces well the turbulence profiles obtained from the coarse-grained fields, especially in the near gray zone. The dynamic scheme does become somewhat overenergetic with further coarsening of the resolution, especially near the surface. The dynamic scheme reaches its limit in the current configuration when the test filter starts to sample at the unresolved scales, returning very small Smagorinsky coefficients. Sensitivity tests reveal that the dynamic model can adapt to changes in the imposed numerical or subgrid diffusion by adjusting the Smagorinsky constant to the changing flow field and minimizing the dissipation effects on the resolved turbulence structures.

The multifluid equations are derived from the compressible Euler equations (or any of the usual approximate equation sets used in meteorology) by conditional filtering. They have the potential to provide the basis for an improved representation of cumulus convection and its coupling to the boundary layer and larger scale flow in numerical models. The present article derives the prognostic equations for subfilter-scale turbulent second moments in the multifluid framework, along with certain systematic simplifications of them, thus providing a multifluid analogue of the well-known Mellor and Yamada hierarchy of turbulence closures. As well as enabling a more accurate calculation of subfilter-scale fluxes and the effects of subfilter-scale variability on cloud fraction, liquid water, and buoyancy, the second moment information can be used to obtain a more accurate parameterization of entrainment and detrainment. A subset of the turbulence equations derived here is employed in the two-fluid single-column model described in Part 2 and applied to the simulation of shallow cumulus cases in Part 3.

Numerical weather prediction (NWP) models are now capable of operating at horizontal resolutions in the 100-m to 1-km range, a grid spacing similar in scale to that of the turbulent eddies present in the atmospheric convective boundary layer (CBL). Known as the 'grey zone' of turbulence, this regime is characterized by significant contributions from both the resolved and subgrid components to represent the dominant motions of the system. This study examines how the initiation of resolved turbulence-a concept commonly referred to as 'spin-up'-can be delayed during the evolution of a simulated CBL in the grey zone. We identify the importance of imposed pseudo-random perturbations of potential temperature (θ) for the development of the resolved fields showing that without such perturbations, resolved turbulence does not become established at all. When the perturbations are organized, spin-up can develop more rapidly, and we find that the earliest spin-up times can be achieved by applying an idealized profile of variance to derive the θ perturbation values. The perturbation structures are shown to be most effective when applied at intervals following the mixed-layer time scale, t * , rather than perturbing only at the initial time. We also propose a modification to the three-dimensional Smagorinsky turbulence closure, in which the Smagorinsky constant is replaced by a scale-dependent coefficient. Both the approaches of: (1) applying structured θ perturbations, and (2) using a dynamically-evolving Smagorinsky coefficient are shown to encourage faster spin-up independently of each other, but the best results clearly emerge when the two methods are applied concurrently.