Abstract. Several basic ratios of responses to forcings in the carbon-climate system are observed to be relatively steady. Examples include the CO2 airborne fraction (the fraction of the total anthropogenic CO2 emission flux that accumulates in the atmosphere) and the ratio T/QE of warming (T) to cumulative total CO2 emissions (QE). This paper explores the reason for such near-constancy in the past, and its likely limitations in future. The contemporary carbon-climate system is often approximated as a set of first-order linear systems, for example in response-function descriptions. All such linear systems have exponential eigenfunctions in time (an eigenfunction being one that, if applied to the system as a forcing, produces a response of the same shape). This implies that, if the carbon-climate system is idealised as a linear system (Lin) forced by exponentially growing CO2 emissions (Exp), then all ratios of responses to forcings are constant. Important cases are the CO2 airborne fraction (AF), the cumulative airborne fraction (CAF), other CO2 partition fractions and cumulative partition fractions into land and ocean stores, the CO2 sink uptake rate (kS, the combined land and ocean CO2 sink flux per unit excess atmospheric CO2), and the ratio T/QE. Further, the AF and the CAF are equal. Since the Lin and Exp idealisations apply approximately to the carbon-climate system over the past two centuries, the theory explains the observed near-constancy of the AF, CAF and T/QE in this period. A nonlinear carbon-climate model is used to explore how future breakdown of both the Lin and Exp idealisations will cause the AF, CAF and kS to depart significantly from constancy, in ways that depend on CO2 emissions scenarios. However, T/QE remains approximately constant in typical scenarios, because of compensating interactions between CO2 emissions trajectories, carbon-climate nonlinearities (in land–air and ocean–air carbon exchanges and CO2 radiative forcing), and emissions trajectories for non-CO2 gases. This theory establishes a basis for the widely assumed proportionality between T and QE, and identifies the limits of this relationship.
Formulae of partial reduction for linear systems of first order operator equations
