On calculating the transfer of carbon-13 in reservoir models of the carbon cycle

An approach to calculating the transfer of isotopic tracers in reservoir models is outlined that takes into account the effects of isotopic fractionation at phase boundaries without any significant approximations. Simultaneous variations in both the rare isotopic tracer and the total elemental (the sum of its isotopes) concentration are considered. The proposed procedure is applicable to most models of the carbon cycle and a four-box model example is discussed. Although the exact differential equations are non-linear, a simple linear approximation exists that gives insight into the nature of the solution. The treatment will be in terms of isotopic ratios which are the directly measured quantities.