Scaling Laws for Structure Functions in the Lowest Layers of the ABL

The development of simple models to describe the turbulent kinetic energy (TKE) distribution across scales remains an ongoing challenge in turbulence theory. The predictions from Kolmogorov for the scaling of the structure function within the inertial subrange have been one of the pillars of turbulence theory for over 70 years. However, in most practical applications of turbulence, the scales within the production range play a more dominant role than those within the inertial-subrange. For wall-bounded flows, dimensional analysis based on the attached eddy hypothesis yields a simple scaling for the energy spectrum suggesting that the spectral density scales with the inverse of the wavenumber (the so called “k^(-1)” range). This is equivalent to the log(r) scaling for the second-order structure function. Recent work conducted in rough-wall boundary layers suggests that the length scale produced with the friction velocity and the TKE dissipation rate (hereafter termed “dissipation-based length scale”), and not the distance from the wall, is the most fundamental length scale for the shear-production range of second-order streamwise structure function. Differences between the two length scales appear when there is imbalance between local shear production and dissipation of TKE.

In this work we revisit the scaling of the structure function in the production range. We show that the only assumptions required for the dimensional analysis results and the log(r) scaling to hold are: (i) the existence of a single velocity scale (i.e. the friction velocity) and (ii) the presence of large eddies that scale with an external length scale much larger than the local integral length scale. No assumptions are required about the integral length scale, which must be proportional to the dissipation-based length scale for Kolmogorov's theory to hold. Therefore, theoretical predictions should be applicable to flows in the atmospheric surface-layer and in the roughness sublayer above plant canopies, if buoyancy is weak enough that a second independent velocity scale is not needed.

The applicability of the theoretical predictions is demonstrated for the unstratified roughness sublayer above a plant canopy using large-eddy simulation (LES) results. Using the dissipation-based length scale, the production range of the longitudinal structure function at all heights above the canopy collapse to the log(r) prediction with the same constants of proportionality observed for wall-bounded flows in the laboratory.

The scaling is also assessed using atmospheric data measured in the surface layer during the AHATS experiment. The new length scale consistently outperforms the distance from the ground as the relevant scale in all cases where there is an imbalance between shear production and viscous dissipation. With the new length scale, the production range of the longitudinal structure function collapses for all heights and stability conditions. For stable and near-neutral conditions, the production range displays the well-known log(r) behavior, which smoothly approaches an r^2/3 power law as the atmospheric becomes increasingly unstable.