56 REDFIELD, KETCHUM AND KrCHARD9 [CHAP. 2 



that the water immediately below the surface and above the bottom was in- 

 cluded in volume units of one-half this length (see Fig. 10). Characteristic 

 values for the temperature and phosphorus concentration based on hydro- 

 graphic data obtained at stated intervals were assigned to the mid-depth of 

 each segment, and to the water at the surface and bottom. These values were 

 considered to represent the mean temperature, T, and mean phosphorus 

 concentration, N, of the segment as a whole. It was assumed that horizontal 

 movements of water were negligible in their effect on this distribution of 

 properties. 



In the case of the full segments the rate of change in N in the interval At 

 depends on the balance between the rate of change due to biological activity, jR, 

 and that due to vertical diffusion across the upper and lower boundaries. 

 Using the subscripts z and —2 to distinguish properties of lower and upper 

 boundaries respectively, the equilibrium may be expressed as 



AN _, 1 /. AN Z . AN-A 



_— = R + -r \A Z — A A- z - — - A — • (4) 



At Az \ Az Az J v ' 



In the case of the segments immediately below the surface and above the 

 bottom, exchange takes place through only one surface and the appropriate 

 diffusion term in (4) is omitted. Rearranging so that each term expresses a 

 quantity rather than a concentration, (4) becomes 



AN-Az n . 4 AN Z , x 



-2AT = RAz+A ^ (4a) 



for the sub-surface segment, and 



AN-Az _ . , AN- Z 



— — ;— = RAz-A-z— 7— 4b 



2At Az ' 



for the segment above the bottom. 



In solving these equations for a given segment, AN /At is obtained from the 

 difference in the mean concentration of N in the segment determined at two 

 times separated by the interval At. AN/Az is the average of the values of the 

 gradients in concentration across the boundary in question at these two times, 

 each of which is given by the difference in N assigned to the adjacent segments 

 divided by the distance between their centers. 



Before these equations could be used to evaluate R for each segment, it was 

 necessary to know the value of the diffusion coefficient at the depth of each 

 bounding surface. These values were estimated from the distribution of tem- 

 perature, T. The flux of heat across each boundary was expressed by a modifica- 

 tion of (4b) in which N is replaced by T, R is omitted because the temperature 

 is uninfluenced by biological activity, and Az is replaced by A, to express the 

 length of the water column between the bounding surface in question and 

 the bottom. Thus 



AT-h AT 



~AT ~~ ~ ' z '~Az" 



