An Analytical model of Airflow over and within a Tall Canopy on a Low Hill

The majority of forest canopies studied by micrometeorologists are on hilly terrain with consequent two and three dimensionality of the flow fields. For example, the analysis of many long-term eddy flux studies is complicated by the advective and eddy flux divergence that their location ensures. Despite numerical modeling efforts and some rare experiments, significant questions remain about these flow fields. For instance it is observed that the modeled pressure drag over hills increases significantly when the momentum absorbing capacity of the surface is represented as a model canopy rather than a surface roughness length, z0 even though z0 is chosen so as to absorb the same amount of momentum as the model canopy on flat ground.

To address some of these questions we have constructed an analytical model of flow in a canopy on a hill which is valid in the limit of a low hill and a canopy sufficiently dense that there is negligible shear stress on the underlying soil surface. The canopy model is constructed using the method of matched asymptotic expansions and forms the lowest layers of a Hunt, Leibovich and Richards (1988) type linear hill flow model.

We find that the canopy must be represented by two layers: an upper layer, where the perturbation momentum balance to first order is between streamwise pressure gradient, foliage drag and shear stress divergence, and a lower layer, where the balance to first order contains only streamwise pressure gradient and foliage drag. Although velocity perturbations in the upper canopy layer can be linearised about the upstream velocity profile, in the lowest layer the perturbation must be treated non-linearly.

Matching the canopy solutions with the layers above, we observe that the within-canopy velocity perturbations are roughly in phase with the streamwise pressure gradient while those above follow (minus) the pressure itself. This has several consequences: · Velocities within the canopy peak well upwind of the hill crest and are falling rapidly by the crest. The asymmetry in streamline displacement about the crest that this causes is significantly larger than in the absence of a canopy. This is probably responsible for the increased pressure drag and also for the early separation observed on hills covered with tall roughness. · The peak in momentum absorbed by the canopy is displaced upwind relative to the peak in surface stress on a hill covered with low roughness. · The phase difference between the response of velocity perturbations within and above the canopy strongly modulates the canopy-top shear, reducing it on the upwind slope and strongly reinforcing it on the crest. All of these features can be observed in the wind tunnel model study of Finnigan and Brunet (1995) although that experiment did not conform to the limiting assumptions required to construct the analytical model. The flow field predicted by the model has obvious implications for constructing mass and energy balances from single towers in complex terrain.