The following diffusion equation was derived: 

 (1) |i=l±L.32l 



Ot r Or L 9r J 



where s is the concentration, r the distajice from the origin, and P 

 the constant mean velocity of diffusion. Solution of this equation gives; 



2 s = — ; exp 



2Tt Pt 2 ^ 



N] 



where s and M refer to the concentration and total amount of a given 

 isotope introduced. Note that concentration in this equation has the di- 

 mensions ML.*2 ; in applying the equation, volume concentrations can be 

 obtained by using an estimated layer thickness D as the third dimension. 



The diffusion velocity _P of this equation is related to the Fickian 

 coefficient of eddy diffusivity A by the equation 



Pr 

 A = — . 



Joseph and Sender examined quantitative descriptions of diffusion in a 

 variety of situations in the ocean and f ound ^ to be relatively constant 

 with a value of about 1 cm/ sec. 



This equation applies to a region of unrestricted horizontal di- 

 mensions. In restricted waterways, or near shore, the boundaries 

 would limit diffusion. The principle of reflection of the solution ob- 

 tained by Joseph and Sender, so that the angular range within which 

 diffusion can occur is limited in accordance with the existence of a 

 boundary, can be employed to obtain an approximate equation for use 

 in harbors and other restricted waterways. Thus diffusion of a sub- 

 stance released near an open shoreline would be limited to an arc of 

 180°. In such a case the reflection of the solution given by equation 2 

 would produce exactly the same equation, except that the right side 

 would be multiplied by a factor of 2 . In an elongated, restricted water- 

 way in which the boundaries restrict diffusion to an arc of, say, 30° 

 in the up- channel and 30°in the down- channel direction the approximate 

 equation for the concentration of a diffusing substance released as a 

 point source would be obtained simply by multiplying the right side of 

 equation 2 by the factor 360 °/2 x 30° = 6. Thus: 



, , 6M 



(3) s(r,t) 



2n(Pt)2 



exp 



{■i\ 



32 



