The windfield over a canopy covered hill

We use the following notation Ub(z) is the undisturbed upwind flow (or the areally averaged flow in a region of continuous topography) while Ap(x, z), Ar(x, z) and Au(x, z) are the perturbations caused by the hill in the mean kinematic pressure, kinematic shear stress and streamwise velocity, respectively. Hence, 

The model of Finnigan and Belcher, 2004 [189] considers the perturbations caused by the hill to a background flow that consists of a logarithmic profile above the canopy 

4In this chapter Ca is defined by the expression Fo = CddU ii, where Fp is the aerodynamic drag and so differs by a factor of 2 from some definitions. 

The analyses of Hunt, Liebovich and Richards, 1988 [287] and of Finnigan and Belcher, 2004 [189] divide the flow in the canopy and in the free boundary layer above into a series of layers with essentially different dynamics. The dominant terms in the momentum balance in each layer are determined by a scale analysis and the eventual solution to the flow held is achieved by asymptotically matching solutions for the flow in each layer. The model apphes in the limit that H/L 1. By adopting this limit, Hunt, Liebovich and Richards [287] were able to make the important simplification of calculating the leading order perturbation to the pressure held using potential how theory. This perturbation to the mean pressure, A p x, z), can then be taken to drive the leading order (i.e. 0(II/I.) ]) velocity and shear stress perturbations over the hill. 

Higher order corrections to the leading order terms can then be obtained using standard methods of perturbation analysis. See for example Van Dyke, 1975 [629], 

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