ATMOSPHERIC CARBON DIOXIDE AND RADIOCARBON: II 97 



NUMERICAL CALCULATIONS 



The inactive and radiocarbon equations given in Appendix C were stepped 

 using a fourth-order Runge— Kutta procedure with two steps per year until 

 1969, then four steps per year. The starting year was 1700 for the inactive 

 carbon equations. In those models which included a heliomagnetic effect, as 

 discussed below, the radiocarbon equations were started with the year 1500 so 

 that 14 C levels in the model reservoirs in 1700 would be in approximate secular 

 adjustment with the long-term influences of this effect. The interval from 1500 

 to 1700 was deemed to be adequate because the longest adjustment time in the 

 model (excepting the 8267-year radioactive decay time) was found to be 

 approximately 200 years by solving a linear algebraic five-reservoir model as 

 described by Keeling. 5 The Runge — Kutta method was checked against this 

 algebraic model. 



The polynomial equation for hydrogen-ion concentration, which arises when 

 evaluating £, was solved by Newton's method (see Appendix D). 



EFFECT OF CHANGES FROM THE MODEL OF 

 BOLINAND ERIKSSON 



It is useful to add details to a model in small steps so that one can learn 

 what is important. Figure 4 shows such a progression, beginning with the results 

 of Bolin and Eriksson for a three-reservoir model (atmosphere, ocean surface 

 layer, and deep ocean) and ending with our five-reservoir model (former 

 reservoirs plus short- and long-lived biota) with a constant ' C source. If the 

 average C/C ratios of surface and deep-ocean water are as quoted earlier, the 

 air-to-sea transfer time, r am (the reciprocal of the transfer coefficient k am 

 defined in Appendix A), is about 5 years, and the deep-ocean to surface-layer 

 transfer time, Tj m (reciprocal of kj m ), is about 1500 years. The C0 2 records at 

 Mauna Loa and the South Pole for 1959 to 1969, when compared with the total 

 record of industrial C0 2 production, indicate that about 45% of the accumu- 

 lated industrial C0 2 should have remained in the atmosphere in 1954. This is 

 equivalent to an increase of 5% in atmospheric C0 2 abundance. As has been 

 discussed, the Suess effect in 1954 relative to the preindustrial era was 

 approximately —2.6%. We see that the curves labelled 1 in Fig. 4, the 

 Bolin — Eriksson result, are quite far from agreeing with either the C0 2 increase 

 or the Suess effect. 



Bolin and Eriksson, in their mathematical solution, neglected a term that 

 dies out after only a few years. Their solution should be a good approximation. 

 However, they neglected the term before matching the solution to the initial 

 conditions, and their result is not the same as it would have been had they first 

 fit the initial conditions and then neglected the term. The exact solution (both 

 curves 2) differs little in C0 2 increase in the atmosphere but predicts a 

 significantly smaller Suess effect. Using the annual C0 2 production data (see 



