Extending the Cospectral Budget Model to a Stratified Roughness Sublayer

A plethora of physical, chemical, and biological processes are influenced by the flow structure in the roughness sublayer layer (RSL). Further, above tall and dense canopies, the handshake between the land and the atmosphere in numerical Weather Predictions and Earth Systems Models occurs in the RSL given the limited extent of the inertial sublayer. For the mean velocity, the RSL effects are operationally accommodated using a dimensionless roughness sublayer correction function (ϕ) to the law-of-the wall. In neutral stratification, the mixing-layer analogy is assumed for dense canopies, and the correction function ϕ depends on the vorticity thickness, L_s. However, this formulation remains incomplete when extending this theory to sparse and urban canopies. The experimental determination and modeling of ϕ may also be challenging in forest topography that introduce z-dependent mean pressure gradients that then lead to variability in second-order flow velocity statistics with z. Here, measurements collected at two experimental sites, the Amazon Tall Tower Observatory (ATTO) and the Station for Measuring Ecosystem-Atmosphere Relations (SMEARII) were used to explore deviations from the law-of-the wall for the RSL above tall forests in the presence of non-neutral stratification. In this work, an original formulation of ϕ is proposed based on a scale-wise co-spectral budget model that balances mechanical production to pressure-decorrelation and longitudinal buoyancy production (or destruction) in the co-spectral budget for momentum. Because the turbulent kinetic energy dissipation rate (ϵ) is conserved across the vertical velocity energy spectrum, the co-spectral budget model reveals a novel link between ϕ, a macro-scale dissipation length, L_d=u_*^3/ϵ and the turbulent horizontal heat flux. A co-spectrum budget model for the turbulent vertical heat flux is also presented.