Co-spectral budget model for eddy-viscosity and law of the wall adjustment above the Amazon Forest

The significance of the flow within the roughness sublayer (RSL) above tall vegetated canopies to a plethora of physical, chemical, and biological processes is not in dispute. It suffices to note that numerical weather predictions (NWP) and Earth Systems Models (ESM) require a handshake between the land surface and the atmosphere. Above tall forests such as the Amazon, this handshake occurs in the RSL whose effects on the flow are usually ignored within NWP and ESM. The RSL delineates a region where the flow statistics are impacted by the presence of roughness elements but is below the much-studied inertial sublayer (ISL). For vegetated canopy flows, this region spans z/h between 1.1 and 5 with the lower bound associated with momentum exchange and the upper bound associated with scalar exchange, where z is the distance from the ground and h is the mean canopy height. For the mean velocity (U), the RSL effects are operationally accommodated using a dimensionless roughness sublayer correction function (φ) to the law-of-the wall. This correction function φ measures the ratio of the eddy viscosity from attached eddies to a zero-plane displacement (d) and the actual eddy viscosity in the RSL at (z-d). The most common formulation uses the vorticity thickness (Ls) as a canonical length scale for the actual eddy viscosity in the RSL making φ dependent on (z-d)/Ls. The mathematical form of φ is selected so as to ensure that when (z-d)/Ls >> 1, φ ~ 1. This formulation remains adhoc with the choice of Ls being only plausible for dense vegetated canopies so that the mixing layer analogy at the canopy top holds. Transitioning from sparse-to-dense canopies in such φ formulation remains a daunting task. An alternative formulation to φ is proposed here based on a scale-wise co-spectral budget model that balances mechanical production to pressure-decorrelation terms in the co-spectral budget. The key variable impacting the eddy-viscosity in the RSL is shown to be related to the shape of the vertical velocity spectrum. An immediate outcome of this analysis is the emergence of a macro-scale dissipation length Ld = u3*/ε given that ε is conserved across the energy cascade. Here, u* is the friction velocity at the canopy top and ε is the mean turbulent kinetic energy dissipation rate at z. The newly proposed approach to inferring φ from the vertical velocity energy spectrum is discussed using high frequency measurements at two sites in Amazonia (Brasil): The Amazon Tall Tower Observatory (ATTO) and the tower at the Cuieiras Biological Reserve (ZF2) that was part of the GoAmazon experiment. The moving equilibrium hypothesis was also used to interpreted the turbulence statistics given the complexity in terrain at these two sites. At both sites, modeled φ from the vertical velocity spectrum agreed with independent estimates of φ using measured eddy viscosity (determined from measured turbulent momentum flux and mean velocity gradient). The data sets also show that Ldbetter agrees with the peaks in the pre-multiplied vertical velocity. spectra when compared to Ls.