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The problem of shear dispersion in the atmospheric boundary layer (ABL) is revisited. The aim is to improve understanding of how and why the behaviour of state-of-the-art 'random flight' Lagrangian particle dispersion models (RFMs) can differ from that of simpler 'random displacement' models (RDMs or eddy diffusivity models). First an asymptotic analysis is used to obtain a formula, valid for quite general profiles of turbulent statistics and the mean wind, for the effective horizontal diffusivity of a tracer in the ABL. Second, with 'poison gas release' problems in mind, a large-deviation approach is used to understand in greater detail the behaviour of the concentration in the tails of the distribution. Results are verified by solving the RFM equations numerically for a large ensemble of particles. Turbulent statistics relevant to stable and neutral boundary-layer conditions are considered, as is the effect of non-uniqueness in the RFM equations. The importance of three-dimensional effects such as the effect of an Ekman spiral in the mean wind are then considered, and criteria determining whether plume widths are controlled by direct horizontal diffusion or by secondary shear dispersion effects are obtained. Finally, a quantitative account of 'plume bending' in the stable ABL is presented. [ABSTRACT FROM AUTHOR]