# Shear stress partitioning in large patches of roughness in the atmospheric inertial sublayer

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## Abstract

Drag partition measurements were made in the atmospheric inertial sublayer for six roughness configurations made up of solid elements in staggered arrays of different roughness densities. The roughness was in the form of a patch within a large open area and in the shape of an equilateral triangle with 60 m long sides. Measurements were obtained of the total shear stress (τ) acting on the surfaces, the surface shear stress on the ground between the elements (τ_{S}) and the drag force on the elements for each roughness array. The measurements indicated that τ_{S} quickly reduced near the leading edge of the roughness compared with τ, and a τ_{S} minimum occurs at a normalized distance (*x*/*h*, where *h* is element height) of
\(\approx -42\) (downwind of the roughness leading edge is negative), then recovers to a relatively stable value. The location of the minimum appears to scale with element height and not roughness density. The force on the elements decreases exponentially with normalized downwind distance and this rate of change scales with the roughness density, with the rate of change increasing as roughness density increases. Average τ_{S} : τ values for the six roughness surfaces scale predictably as a function of roughness density and in accordance with a shear stress partitioning model. The shear stress partitioning model performed very well in predicting the amount of surface shear stress, given knowledge of the stated input parameters for these patches of roughness. As the shear stress partitioning relationship within the roughness appears to come into equilibrium faster for smaller roughness element sizes it would also appear the shear stress partitioning model can be applied with confidence for smaller patches of smaller roughness elements than those used in this experiment.

## Keywords

Atmospheric inertial sublayer Drag partition Roughness arrays Shear stress partitioning## List of Symbols

*A*_{f}frontal area of roughness elements (m

^{2})*A*_{u}unit area over which surface shear stress associated with a roughness element is distributed (m

^{2})*b*element breadth (m)

*Cd*surface drag coefficient

*Cd*_{e}roughness element drag coefficient

*Cd*_{r}rough surface drag coefficient

*Cd*_{s}smooth surface drag coefficient

- cv
coefficient of variation

*d*displacement height (m)

*F*force on a roughness element (N)

*g*acceleration due to gravity (m s

^{−2})*h*element height (m)

- IBL
internal boundary layer

- ISL
inertial sublayer

*m*empirical constant between 0 and 1

*n*number of roughness elements occupying the ground area of the roughness array

- NDD
normalized downwind distance (

*x*/*h*)- NED
normalized element drag

*R*average friction velocity ratio

*R*_{l}local friction velocity ratio at different positions in a roughness array

*Re*Reynolds number

*R*_{t}threshold wind friction velocity ratio

- SD
standard deviation of a mean value

*u*wind speed (m s

^{−1})*u*_{*}wind friction velocity (m s

^{−1})*u*_{* tR}threshold wind friction velocity with roughness elements (m s

^{−1})*u*_{* tS}threshold wind friction velocity of bare surface (m s

^{−1})*x*downwind distance (m)

*z*reference height above surface (m)

*z*_{w}roughness sublayer height (m)

*z*_{o}aerodynamic roughness length (m)

- β
ratio of element to surface drag coefficients

- φ
_{m} dimensionless wind speed gradient

- κ
von Kármán constant (0.4)

- λ
roughness density

- μ
molecular viscosity (N s m

^{−2})- ρ
_{a} air density (kg m

^{−3})- σ
roughness element basal area to frontal area ratio

- τ
total surface shear stress (N m

^{−2})- τ
_{S} surface shear stress on the area not covered by the roughness elements (N m

^{−2})- τ
_{R} surface shear stress attributed to the roughness elements (N m

^{−2})

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