Fast-response concentration data from dispersion experiments with continuous surface releases has been analysed, and striking self similarities in the spatial profiles of concentration fluctuations are identified. The cross-wind mean profile, analysed in a moving frame of reference following the centre of the meandering plume, deviates from the expected Gaussian shape. This deviation is now theoretically explained by a model, in which the diffusion coefficient is defined by two length scales. With appropriate scaling of widths and centre-line values, the observed profiles of higher-order statistical moments of concentration fluctuations are shown to coincide with the mean profile. An empirical relation between the scaling widths for these profiles is presented. It is shown that, with moving-frame analysis, the centre-line fluctuation intensity has little dependence on friction velocity or atmospheric stability.