14 C CYCLE AND ITS IMPLICATIONS FOR MIXING RATES 7 



molecules, n, originally present in the reservoir at t = is continuously decreasing 

 with time. The number of these molecules, dn, removed over an interval dt is 

 proportional to the flux and relative concentration n/N of these particular 

 molecules, i.e., dn = — (n/N)0 dt. Integration yields n = Ne — ^'N. 



In many instances the transfer process between reservoirs depends on the 

 total amount of material in the reservoirs. Thus in most models the rate of 

 removal depends only on the total number N of the kind of molecule under 

 consideration. The outgoing flux is then given as the product of N and an 

 exchange-rate constant, k. The exchange-rate constant has no clearly defined 

 physical basis but is a useful parameter for box-model studies. With = kN, the 

 number of molecules n in the reservoir is decreasing according to n = Ne — . 



The decrease in the number of molecules originally present follows an 

 exponential law similar to radioactive decay, with the decay constant A 

 equivalent to k. For radioactive decay the average life of an atom is 1/A years. In 

 a similar way the average life, or mean residence time, r, of our particular 

 molecules in the reservoir is 1/k. The average life r is, of course, also the time 

 required for the number of original molecules to be reduced to 1/e times the 

 number at t = 0. 



The relation dn = — kn dt for a reservoir exchanging with only one other 

 reservoir can be enlarged to dn = — (kj + k 2 + . . . + kjvi)n dt for a multi- 

 component system where this reservoir exchanges with M other reservoirs. The 

 average life r equals, in this instance, l/(k x + k 2 + . . . + kjvi); the inverse of this 

 relation yields 1/r = (1/Tj) + (l/r 2 ) +(...)+ (1/tm). The equations given here 

 greatly simplify the analysis of box-model systems. 



The mean residence time of C in a specific reservoir is not identical to the 

 stable-carbon residence time. Because of radioactivity, the removal rate includes 

 both the physical removal to other reservoirs and the 14 C decay rate, — AN. 

 Mean residence times of the 14 C molecules may also differ slightly from the 

 stable species because of isotope fractionation in the removal processes. The 

 small isotope-fractionation effects depend on isotopic-mass differences only and 

 can occasionally be neglected in the calculations. 



The development of box models of increasing complexity has resulted in an 

 understanding of the mean residence times of carbon in the atmosphere and in 

 the various regions of the sea and has also provided information on the exchange 

 rate between atmosphere and sea. It is easy to get lost in the mathematical 

 details and nomenclature of these box models. We can become familiar with the 

 general procedures by discussing here the calculation of the residence time of 

 carbon dioxide in a simple box model involving the atmosphere and the mixed 

 layer of the oceans. For the more complicated models, only the results are given; 

 the technical details can be found in the original publications. 



In a steady-state condition, the net transfer of l4 C to the oceans equals the 

 total C decay in the large oceanic reservoirs. If one assumes that the exchange 

 of carbon is uniform over the oceans, the following equation holds: 



