The relative dispersion process for clouds of contaminant in generic atmospheric flow is considered. The properties of the separation distance for pairs of particles are simplified by implicitly averaging over the spatial domain of the dispersing cloud. A Lagrangian stochastic model of relative dispersion at high Reynolds numbers is derived. The model uses a new formulation for the acceleration of separation, satisfies the criterion of conserving a well-mixed distribution of particle separations and accounts explicitly for non-Gaussian statistics of the turbulence velocity differences. The results are in very good agreement with the similarity theory in the inertial range and are consistent with uncorrelated velocities at length scales larger than the turbulence integral scale. The model is applied to the estimation of internal concentration fluctuations relevant for meandering plume and puff approaches. The dependence of the concentration statistics on the source size is eliminated via a new scaling law for the time, which in fact determines a universal behaviour for the concentration field. Simple formulae are derived that are consistent with previous theories and are successfully tested against our numerical simulations.