66 



EKDAHL AND KEELING 



TIME- 

 VARYING 

 SOURCE 

 T(t) 



LOSS TO SINKS 



PLUS 



DECAY IN ATMOSPHERE 



Fig. 11 Single reservoir model of atmospheric radiocarbon response to a 

 time-varying source. The mass of atmospheric radiocarbon is *N a (t); *k e ff is 

 the effective net-transfer coefficient. 



by adjacent reservoirs (Fig. 11). This model allows us to develop the mathe- 

 matical approach of using a transfer function without requiring much analytical 

 complexity. 



The time-dependent mass of atmospheric ' C, *N a (t), produced by the 

 variable C source *T(t) is described by the equation 



d*N t 

 dt 



= -*k e ff*N a + *T 



(3) 



where *T e ff (= *k e ff ) is the effective transfer time for the combined loss of 

 C bv decay in the atmosphere and the net transfer to adjacent reservoirs. To 

 solve Fig. 3, we decompose both the atmospheric ' C mass and the source 

 function into a steady-state term and a perturbation: 



*N a (t)=*N a0 + *n a (t) 



T(t) = *r + *7(0 



(4) 

 (5) 



yielding a steadv-state equation 



d*N a0 

 dt 



- n - _ * L- 







k e ff* N aO + *To 



(6) 



and a perturbation equation 



d*n a 

 dt 



: k e ff*n a + *7(t) 



(7) 



Grey and Damon, ' using Lingenfelter's ' C production equation and the 

 observed annual sunspot record to establish *7(t), approximateh - solved these 



