92 BACASTOW AND KEELING 



MODEL EQUATIONS 



The equations for the model (Appendixes A, B, and C) express material 

 balances for each reservoir. The fluxes between reservoirs are, with the 

 exceptions noted in this paragraph, taken to be proportional to the total carbon 

 in the reservoir from which they originate. The exchange of carbon between the 

 ocean surface layer and the deep ocean takes place by both water exchange and 

 a constant downward gravitational transport of particulate carbonaceous matter. 

 The return flux to the atmosphere from the ocean surface layer is proportional 

 to the partial pressure of C0 2 exerted by the surface layer in accordance with 

 the law of gas exchange. 2 3 The exchange of carbon between the atmosphere and 

 the land biota also forms an exception, as discussed in a later section. 



Each flux of I4 C is obtained by multiplying the corresponding flux of 

 inactive carbon by the isotopic mass or pressure fraction of C in the 

 originating reservoir and by an isotopic fractionation factor that differs from 

 unity by no more than a few percent. The transfer coefficients are always 

 treated as input parameters. The preindustrial 14 C/C ratios are consequently 

 derived by the model (see Eqs. B.4 to B.9 of Appendix B). 



Industrial C0 2 production, entered into the rate equations as a carbon source 

 for the lower atmosphere beginning with the year 1700, drives the entire model. 

 The sources that cause variations in the 14 C system are the inactive-carbon 

 perturbations, particularly the increase in total carbon in the ocean surface layer. 

 (In Appendixes B and C, these are called virtual sources.) For all calculations not 

 otherwise described, we assume, in addition to the steady-state production of 

 14 C, a variable 14 C source in the upper atmosphere to represent a heliomagnetic 

 effect correlated with sunspot numbers, beginning with the year 1500 

 (Appendix C). This latter source is expressed as a perturbation and is required to 

 sum to zero over the period 1500 to 1960 for which sunspot data are available. 

 It introduces small fluctuations with periods of approximately 11 and 85 years, 

 as discussed in the preceding paper (Ekdahl and Keeling ). 



In the following sections we explain the basis for the nonlinear equations of 

 the model. A detailed explanation of the linear equations appearing in the 

 appendixes is given by Keeling. 



EVASION FACTOR 



The mass-balance equations which connect the atmosphere and surface 

 ocean water involve a relationship between the partial pressure of 

 C0 2 exerted by the ocean surface water, P m , and the total inorganic carbon in 

 this water, 2C. A mathematically convenient form of this relation is represented 

 by the evasion factor, £, 



I 



m °mo)'°mo 



C — ^Cq )/wCq 



(1) 

 constant alkalinity 



