Designing LES of the atmospheric boundary layer to capture law-of-the-wall: SFS stress model and grid

Large-eddy simulation (LES) of the atmospheric boundary layer has been plagued by an inability to capture the law-of-the-wall (LOTW) in mean wind through the surface layer. Often LES systematically over-predicts the mean velocity gradient in the shear-dominated inertial surface layer, and even when this "overshoot" is minimized, the mean velocity gradient normalized on LOTW scales is rarely constant throughout the surface layer, roughly the lower 20% of the boundary layer. Whereas the overshoot is especially apparent in LES of high Reynolds number rough surface boundary layers it is also apparent in the inertial part of the surface layer in hybrid RANS-LES such as "Discrete Eddy Simulations" (where it is referred to as the "log layer mismatch"). The over prediction of mean velocity gradient leads to a wide range of negative effects, such as a strongly enhanced stream-wise coherence and incorrect local eddy structure. In the presence of vertical buoyancy-driven motions, as in the moderately unstable ABL, the error infects the entire thermal structure of the boundary layer and affects vertical transport of heat, humidity and scalar. Furthermore, the overshoot precludes a grid-independent prediction for mean velocity.

The current discussion is limited to LES which does not externally impose LOTW on the simulation. The only modeling elements that can be included in the transition from the exact continuous dynamical system to the discretized dynamical system are models for the SFS stress tensor within the computational domain and the horizontal total stress components at a rough surface, with a roughness length introduced by specifying the ratio of surface friction velocity to mean wind at the first grid level. We develop a theory to show that the overshoot is a consequence of a spurious length scale that is created in the transition to the discretized system advanced from the modeling of the SFS stress divergence as frictional when, physically, it is fully inertial. When sufficiently strong, the spurious scale interferes with LOTW scaling. The strength of the spurious scale, however, is controlled by a combination of SFS model constant and grid aspect ratio that combine into a single parameter to adjust the resolved to the SFS shear stress. When this ratio exceeds a critical value, the overshoot is removed. However, the removal of the overshoot does not guarantee LOTW scaling within the computational domain; further details of the grid must be taken into account. In particular, the theory shows that the vertical resolution of the grid enters directly into the ratio of inertia to friction within the discretized dynamical system. This "LES Reynolds number" must also exceed a critical value for LES to capture LOTW, leading to a third criterion, the need for sufficient vertical grid resolution to resolve the surface layer. We will describe the process of designing LES on a parameter space where one systematically moves the three parameters of the simulation to supercritical values required for the simulation to capture LOTW within the surface layer, and we shall demonstrate the process with over 120 large-eddy simulation of the neutral atmospheric boundary layer. The process is independent of SFS stress closure.