An analytical calculation of two-dimensional dispersion

The methods of Okuboet al. (1976a) are used to calculate the Lagrangian deformations and diffusivities of a cluster of drifters. A solution of the two-dimensional first-order advection-diffusion equation (Okuboet al., 1983a) is then used to calculate the dimensions and orientation of the cluster from these Lagrangian deformations and diffusivities. The solution is shown to be internally consistent (to give cluster areas that are consistent with the observed cluster areas) to within a 0.5% error. As time progresses a larger portion of the dispersion is caused by the diffusivities rather than the deformations. In the experiments analyzed the Lagrangian deformations and diffusivities are generally observed to increase at a constant rate over time intervals of about one hour. Dimensional arguments suggest that Lagrangian diffusivities increase proportional tot ² and the deformations proportional tot ^1,5 for time intervals large compared to the period required to spread from a point source to the initial cluster dimensions. Small quadratic velocity gradients cause the solution of the first order advection-diffusion equation to overestimate cluster spreading. Most of the displacement (once motion due to the mean velocity and linear deformations is extracted) is caused by scales of motion much smaller than the cluster. This explains the relatively small magnitude of the errors caused by parameterizing quadratic and other statistically significant nonlinear shears as a component of the eddy-diffusivity.