Some consequences of non-proportionality between fluxes and reservoir contents in natural systems

We study the effect on simple reservoir systems of non-proportionality between fluxes and reservoir contents. In particular, we assume that the flux F out of a reservoir is related to the reservoir content M by F/F₀ = (M/M₀)^α, where a may be greater than, equal to or less than one. It is demonstrated that adjustment rates and equilibrium distributions are critically dependent upon the value of the parameter α

A simple model of the global carbon cycle is used as an illustration. We show among other things that, because of non-proportionalities in the fluxes between the ocean surface layer, the atmosphere and the terrestrial system, the decay time of a man-made carbon emission into the atmosphere is much longer than the turn-over time with respect to exchange with the deep ocean layers.