The exponential eigenmodes of the carbon-climate system, and their implications for ratios of responses to forcings

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/Q_E of warming (T) to cumulative total CO₂ emissions (Q_E). 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 (k_S, the combined land and ocean CO₂ sink flux per unit excess atmospheric CO₂), and the ratio T/Q_E. 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/Q_E 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 k_S to depart significantly from constancy, in ways that depend on CO₂ emissions scenarios. However, T/Q_E 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 Q_E, and identifies the limits of this relationship.