An Analytical Approach to Shear Dispersion and Tracer Age

Abstract

Solutions to the sheared Fickian advection–diffusion equation in a half-space with arbitrary surface source are given using a ‘transfer function’ method. The method uses Fourier transforms in two horizontal coordinates and time, along with complex Airy functions in the vertical coordinate. Surface deposition and tracer decay are included in the formulation. ‘Puff’ and steady ‘plume’ solutions are compared with Saffman’s moment formulae. The inclusion of a decay rate factor (α) allows the average tracer age to be computed from steady state solutions for concentration C(x, y, z) according to Age = − dln C/dα. A comparison between the puff centroid formula of Saffman and plume Age computations confirms that shear causes tracer puffs to accelerate horizontally as they diffuse upward into a different wind regime. In forward shear, tracer ages are younger than in unsheared flow but the range of ages is greater due to the existence of a high fast pathway and a low slow pathway. In reverse shear, concentrations, ages and the range of ages all rise markedly near the source. Large tracer age suggests that some tracer has taken a very distant path involving a low-level outbound trip and a high-level return. The effect of surface deposition is to reduce the influence of the distant path. In the case of reverse shear, deposition makes the tracer younger. In a turning wind, the time needed to reach a given radius increases due to the curved path of the plume.