Comparison of some different approximations in the statistical theory of relative dispersion

The effect on the theory of relative dispersion of some different approximations to the two‐particle Lagrangian correlation function is examined. Two of these, one due to Taylor, the other to Smith and Hay, are treated in detail. Through numerical solution of the dispersion equations, the influence of the initial cluster size, a number of simple variations of the spatial separation argument of the two‐particle correlation function, the number of particles in the cluster and the ratio of Lagrangian to Eulerian integral scales are examined. With the exception of the initial cluster size, which is important in the early stage of growth, these parameters are relatively unimportant, particularly compared to the overall difference between the two approximations. In qualitative agreement with Batchelor's inertial range theory, both the Taylor and Smith‐Hay approximations show linear growth at small time with an accelerated growth region at intermediate time. However, between these regions, the Smith‐Hay solution shows a region of less‐than‐linear growth for which there appears to be no observational or theoretical support. This regime is more pronounced, and the difference between the two approximations greater, for initially small clusters. Comparison with suitably documented observations, while not entirely definitive, shows a degree of consistency and suggests the Taylor approximation to be the more appropriate.