820 BOWDEN [sect. 6 



number k, within the range given hy d< Hk<l, were effective in influencing 

 the dispersion. 



Observations reported by Hanzawa (1953) on the spreading of pumice stones, 

 produced by a volcanic eruption in the Pacific, provide a large-scale example 

 of the dispersion of discrete indicators. After drifting 600 km from the source 

 at an estimated mean speed of 36 cm/sec, stones were observed at distances up 

 to 120 km from the centroid of their distribution. The value of Kn estimated 

 from the maximum dispersion was 4.3 x 10'^ cm^/sec. 



It seems fairly clear that Richardson's neighbour separation theory is in good 

 agreement with observations and has a sound physical basis. Most oceano- 

 graphic problems, however, deal with a continuous distribution of the con- 

 centration of a certain property, which bears no simple relation to the spacing 

 of pairs of particles. Richardson (1926) gave the following transformation from 

 the concentration v{x) to the "neighbour concentration" q{l) : 



r*oo 

 q{l) = v{x)v{x + l)dx. (31) 



j-oo 



q then has to satisfy the equation 



When the problem has been solved for q{l, t), a transformation back to v{x, t) is 

 necessary to obtain the solution in terms of the concentration, Richardson 

 gave some attention to such transformations in his 1926 paper and returned 

 to the problem later (Richardson, 1952). Although it may be possible, theo- 

 retically, to carry out the transformations in certain cases, the practical 

 difficulties have prevented the method being applied to observational data 

 hitherto. Problems concerning the horizontal spreading of a patch of pollutant 

 have been treated by the eddy diffusion method, with the coefficients Kx and 

 Ky considered either as constants or as functions of position. 



The case of radial diffusion in two dimensions of a patch of pollutant initially 

 concentrated at one point will be considered assuming, for simplicity, that the 

 diffusion is the same in all directions. If the concentration of the pollutant is S, 

 and the distance from the origin is-r, the general diffusion equation becomes 



If K is constant, as in diffusion according to Fick's law, the solution is 



S{r, t) = {SQJ4.7TKt)e-^'I^Kt^ (34) 



Sverdrup proposed in 1946 (unpublished, quoted by Stommel, 1949) that K 

 should be taken as proportional to r, i.e. K{r) = Pr. Joseph and Sendner (1958) 

 introduced postulates which led to the same form for K. The solution then 

 becomes 



S{r, t) = {Sol27TPH^)e-r/Pt. (35) 



