Flow over a hill covered with a plant canopy

We develop an analytical model for atmospheric boundary-layer flow over a hill that is covered with a vegetation canopy. The slope of the hill is assumed to be small enough that the flow above the canopy can be treated within the linear framework of Hunt. Perturbations to the flow within the canopy are driven by the pressure gradient associated with the flow over the hill. In the upper canopy this pressure gradient is balanced by downwards turbulent transport of momentum and the canopy drag. The flow there can be calculated from linearized dynamics, which show that the maximum streamwise winds are where the perturbation pressure is at a minimum, i.e. near the crest of the hill. Deep within the canopy the pressure gradient associated with the flow over the hill is balanced by the canopy drag, here the nonlinear canopy drag. This nonlinear balance shows how the streamwise winds are largest where the perturbation pressure gradient is largest, i.e. on the upwind slope of the hill. In the lee of the hill this nonlinear solution shows how the pressure gradient decelerates the wind deep within the canopy, leading to separation with a region of reversed flow when the canopy is sufficiently deep. Coupling between the out-of-phase flows within and above the canopy means that the maximum velocity is further upwind of the hill crest than in flow over a rough hill, while the extra turbulent mixing caused by the canopy significantly reduces the magnitude of the velocity speed-up over the hill. Finally, we find that there is no formal limit process where the solutions with a canopy yield the well-known solutions for flow over a rough hill. This finding calls into question the very use of a roughness length in accelerating or decelerating turbulent boundary layers.