Appropriate Boundary Conditions and Near-Wall Treatment in Stratified Atmospherics Boundary Layer Using the Explicit Algebraic Reynolds-Stress Model

In the models for the atmospheric boundary layer (ABL) the near-wall treatment has a profound effect on the solution in the surface layer, however this has not been a main focus of the atmospheric community research. The common approach is to set up the log-layer boundary conditions for neutrally stratified atmosphere and estimate surface fluxes from the vertical profiles, that correspond to the stably or unstably stratified atmosphere, using the Monin-Obukhov similarity theory (MOST). This leads to an inconsistency between boundary conditions, surface fluxes and turbulent fluxes. The aim of the present study is to construct a clean and consistent boundary condition treatment for a recently derived nonlinear turbulence model, the so-called explicit algebraic Reynolds-stress model (EARSM), see [1], in order to improve the behavior of the model in the surface layer. This turbulence model is considered to be one of the more advanced ABL models that take into an account, aside from the mean atmospheric variables, many atmospheric properties related to turbulence (e.g., turbulent kinetic energy (TKE), shear and buoyancy production as well as the dissipation of TKE) and where the turbulent fluxes do not necessarily have to be aligned with the mean gradients. It is derived from the turbulence model for full prognostic equation (differential Reynolds-stress model, Mellor and Yamada level 4 model) so it does not require modeling of shear and buoyancy production terms. Furthermore, it contains a weak-equilibrium assumption which makes it explicit and somewhat similar to the Mellor and Yamada level 3 model. This means that it is more complex than the standard eddy-viscosity/eddy-diffusivity approach and contains transport equation for temperature variance beside the equations for TKE and dissipation, and at the same time does not depend on any empirical function nor ad-hoc correction, see [1,2,3]. It is obvious that due to the complexity of the model it requires careful formulations for consistency in the near-wall region. In our near-wall treatment the model uses a vertical grid that is fine enough to capture the wind shear effects in the first several meters of the surface layer, where the buoyancy effects are not prominent and neutral log-layer relations are valid, in order to avoid predefining the surface fluxes with MOST. This allows the use of log-layer relations in calculating the surface momentum flux from the logarithmic wind speed profile, as well as the equality between the shear production and dissipation of TKE for setting up the boundary conditions. It is also shown that calculating the surface heat flux from the surface scaling ensures that momentum and heat surface flux are consistent. Transport equations are modified in the first cell center point in order to makes sure that the turbulence model is consistent with the assumption of the neutral atmospheric condition used at the surface. With the new modifications for the near-wall behavior the model was investigated for three different levels of grid refinement for the case of GEWEX Atmospheric Boundary Layer Study (GABLS2), based on the recent intercomparison study in [4], which is an ideal test-case for the near-wall treatment in order to see its behavior in stable, unstable and transitioning ABL. Since the focus of the study is to understand the influence of boundary conditions on the solution in the surface layer, results were considered only in the lower part of the ABL. It is shown that the highly consistent near-wall behavior of the model gives a grid-independent solution for the mean horizontal wind speed, wind direction, mean potential temperature and surface fluxes. Furthermore, since MOST is not used in setting up the surface fluxes it is justified to express the results in the dimensionless form given by MOST. It is shown that these results collapse on a single curve and that way form a clear functional dependence and confirm that the physical part of the turbulence model is well described. Comparison between the model results and atmospheric data from [5,6] shows good agreement for the stable part of the day where as in the unstable part there are some differences. In the free-convection limit, for high values of stability parameter -ξ, the model results seem to scale properly but there are slight deviations for potential temperature variances and more pronounced deviations for vertical velocity variances compared to the theoretically predicted values. Interestingly enough, scaling coefficients found for the highest stability parameter are very close to the ones that are reported in [7], only considering that EARSM belongs to the family of RANS models this is a positive and unexpected result. Additionally, aside from the standard form of Obukhov length a new scaling length was introduced, a physical Obukhov length, which follows from the physical definition of Obukhov length and is estimated by comparing the shear with buoyancy production term in the equation for turbulent kinetic energy. To our knowledge this is a new way of calculating the Obukhov length scale which could be an interesting alternative for large-eddy simulations [7,8]. In summary, a more consistent formulation of the interaction between the ground surface and atmosphere is given which can be implemented also in other eddy-viscosity/eddy-diffusivity models. This way of forming near-wall treatment enables a full potential of the EARSM in the surface region and shows that without the use of MOST it is possible to reproduce the scaling behavior in the atmosphere.

[1] Lazeroms et al. 2016 Bound.-Layer Meteor. 161:19
[2] Lazeroms et al. 2015 Int. J. Heat Fluid Fl. 15:28
[3] Lazeroms et al. 2013 J. Fluid Mech. 91:125
[4] Svensson et al. 2011 Bound.-Layer Meteor. 140:177
[5] Businger et al. 1971 Bound.-Layer Meteor. 2:3
[6] Högström 1988 Bound.-Layer Meteor. 42:263
[7] Maronga, Reuder 2017 J. Atmospheric Sci. 989:1010
[8] Sullivan et al. 2016 J. Atmospheric Sci. 1815:1840

Figure caption: Mean horizontal wind speed profile (left) and potential temperature profile (right) scaled according to MOST. Model results scaled by the physical Obukhov length (blue) and by classical Obukhov length (yellow); Experimental data: (dashed line) from [5] and (solid line) from [6].