This paper describes a model of the effects of low hills upon scalar fields, particularly the temperature and humidity fields, in the planetary boundary layer under near-neutral conditions. The approach is to use linearized equations and two-layer asymptotic matching, similar to methods which are now well-established for the velocity field. In the inner region near the surface the scalar fields are affected by turbulent transfer from the surface and are represented using an eddy diffusivity. In the outer region the response of both the flow and the scalar fields to the hill is essentially inviscid. By matching these layers, hill-induced perturbations in the concentrations and fluxes of an arbitrary scalar can be determined in terms of upwind and surface conditions, and the calculated mean and turbulent wind fields over the hill. The analysis separates these perturbations into components associated with (1) the convergence and divergence of the streamlines, (2) vertical stress gradients, and (3) changes in boundary conditions (surface stress and scalar flux density) over the hill. In order to describe coupled heat and water-vapour transfer with this model, it is necessary to decouple the scalars by using two new scalar entities, essentially energy per unit volume and potential saturation deficit. This is a flow-independent technique previously applied to sensible- and latent-heat fluxes in local advection and in convective boundary layers. An extension given here accounts for the effects of change in elevation on the evaporation process. Perturbations in the temperature and humidity fields and fluxes over the hill arise physically from slope-induced radiative effects, aerodynamic effects induced by changes in the mean wind and stress fields, and elevation effects induced by adiabatic cooling on ascent. Three dimensionless parameters control these effects: the hill slope, H/L; the unperturbed Priestley-Taylor ratio, alpha(1) (the ratio of surface latent-heat flux density to available energy flux density, far from the hill); and an 'elevation parameter', P(elev), proportional to the change in saturation deficit induced by adiabatic cooling as air ascends through the hill height, H, normalized by the turbulence specific-humidity scale. For the surface latent-heat flux, radiatively-induced perturbations scale on (H/L)alpha(1)-1; aerodynamic-induced perturbations on (H/L)(alpha(1)-1 - alpha(eq)-1) (where alpha(eq) is the thermodynamic equilibrium value of alpha(1), 0.688 at 20-degrees-C); and elevation-induced perturbations on P(elev). In general terms, the strongest influences on the perturbations of heat and water-vaopour fields and fluxes are aerodynamic effects associated with changes in surface shear stress, and elevation effects. For a hill of height 100 m, surface resistance 50 s m-1 and slope H/L = 0.2, typical changes in potential temperature and specific humidity are of order 1 degC and 1 g kg-1, respectively, while typical perturbations in sensible- and latent-heat fluxes are of order 20 per cent. The linear analysis predicts that spatial averages of heat and water-vapour fluxes are independent of low-terrain undulations, except for adiabatic elevation effects.
