Surface Thermal Heterogeneities and the Atmospheric Boundary Layer: The Thermal Heterogeneity Parameter

Abstract

Representing land–atmosphere exchange processes at the ground surface of numerical-weather-prediction models remains a challenge in spite of the recent advances in computing. Previous studies investigating the effects of spatial surface heterogeneities have been viewed from a turbulence perspective, mostly assuming the existence of a blending length scale above which surface-induced perturbations are modelled using an ad hoc bulk surface parameter representing a pseudo-equivalent surface condition. While these types of approaches can generate reasonable results, they fail to account for the long-lasting spatial perturbations that modify the mean flow. In this work, the interactions between the characteristic scales of surface thermal heterogeneities and the mean resolved fluid dynamics are investigated for a broad range of unstable atmospheric conditions. Thermal dispersive fluxes, which naturally appear as a means to account for persistent-in-time advection fluxes generated by unresolved spatial heterogeneities, provide a quantification of the interaction between surface thermal heterogeneities and the atmospheric boundary-layer mean flow. Hence, they also provide a deterministic approach for including the effect of unresolved processes on the mean flow. We introduce a new non-dimensional number (i.e., the heterogeneity parameter) that can be used to identify the flow conditions and surface configurations in which heterogeneity effects become important. The heterogeneity parameter can be used to distinguish cases with high and low dispersive-flux contributions based on the mean flow and characteristics of the thermal heterogeneities. These results suggest that under weak geostrophic forcing, surface heterogeneity effects should be accounted for in numerical-weather-prediction models.

Appendix: Large-Eddy Simulation Framework

The different ABL flows used are modelled with an LES framework where the non-dimensional, filtered, incompressible Navier–Stokes equations are solved using a pseudo-spectral approach (Moeng 1984; Albertson and Parlange 1999). In this framework, the rotational form of the momentum equations is used to ensure conservation of mass and energy in the inertial terms (Kravchenko and Moin 1997). The filtered potential temperature is modelled with an advection–diffusion equation. The buoyancy forces are computed with the Boussinesq approximation to couple the temperature to the momentum equation (Tritton 1988). This method makes it possible to account for the fluctuations in density through the fluctuations of the temperature field. The subgrid-scale stresses and heat fluxes are computed using the Lagrangian scale-dependent dynamic Smagorinsky models for momentum and scalars (Bou-Zeid et al. 2005; Calaf et al. 2011). The viscous and molecular diffusive effects are neglected because of the high Reynolds number of the ABL flow. The active Coriolis effects drive the flow through the pressure gradient induced by the geostrophic forcing. To integrate the equations in time, the Chorin’s projection fractional-step method (Chorin 1968) alongside a second-order Adams–Bashforth scheme is used. Spatially, the equations are discretized on a vertically staggered grid, where the horizontal derivatives are computed using discrete Fourier transforms, and the vertical derivatives are computed with second-order centred finite differences. The aliasing errors on the non-linear convective terms of the momentum and temperature equations are treated using the 3/2 rule (Canuto et al. 1988). Because of the spectral methods, the side boundary conditions are inherently periodic. The vertical velocity uses the non-penetration condition at the top and bottom of the domain, a stress-free lid condition is used as the top boundary conditions for the horizontal velocities, and a constant flux is imposed for the temperature top boundary condition. A wall model is used to prescribe the bottom wall-shear stress and surface flux through the vertical derivatives at the first staggered grid point (Moeng 1984; Bou-Zeid et al. 2004; Hultmark et al. 2013). This model uses Monin–Obukhov similarity theory to capture the effects of the thermal stratification on the ABL flow (Monin and Obukhov 1954) and applies Brutsaert’s formulation of the atmospheric stability correction functions (Brutsaert 1982). Also, the scalar roughness length is set at one tenth of the aerodynamic roughness length (Brutsaert et al. 1989). Note that although the stability correction functions were originally developed for flat homogeneous surfaces, these are used here because there is no better alternative, and this methodology has been shown to generate acceptable results over heterogeneous surface conditions as long as there exists a pseudo-equilibrium within the heterogeneities (Stoll and Porté-Agel 2008; Basu and Lacser 2017).

Finally, the numerical code is parallelized using a two-dimensional pencil decomposition where the domain is partitioned into squared cylinders (Sullivan and Patton 2011; Margairaz et al. 2018). The code uses the 2DECOMP & FFT open-source library to implement the two-dimensional pencil decomposition (Li and Laizet 2010). Further details on the numerical code used here can be found in Margairaz et al. (2018, (2020).