Earth Syst. Dynam., 4, 31-49, 2013
http://www.earth-syst-dynam.net/4/31/2013/ doi:10.5194/esd-4-31-2013 © Author(s) 2013. This work is distributed under the Creative Commons Attribution 3.0 License. |

Research article

25 Jan 2013

CSIRO Marine and Atmospheric Research, Canberra, ACT 2601, Australia

Received: 30 Jul 2012 – Discussion started: 19 Sep 2012

Revised: 30 Nov 2012 – Accepted: 22 Dec 2012 – Published: 25 Jan 2013
Abstract. Several basic ratios of responses to forcings in the carbon-climate system are observed to be relatively steady. Examples include the CO_{2} airborne fraction (the fraction of the total anthropogenic CO_{2} emission flux that accumulates in the atmosphere) and the ratio *T/Q*_{E} of warming (*T*) to cumulative total CO_{2} 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 CO_{2} emissions (Exp), then all ratios of responses to forcings are constant. Important cases are the CO_{2} airborne fraction (AF), the cumulative airborne fraction (CAF), other CO_{2} partition fractions and cumulative partition fractions into land and ocean stores, the CO_{2} sink uptake rate (*k*_{S}, the combined land and ocean CO_{2} sink flux per unit excess atmospheric CO_{2}), 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 CO_{2} emissions scenarios. However, *T/Q*_{E} remains approximately constant in typical scenarios, because of compensating interactions between CO_{2} emissions trajectories, carbon-climate nonlinearities (in land–air and ocean–air carbon exchanges and CO_{2} radiative forcing), and emissions trajectories for non-CO_{2} gases. This theory establishes a basis for the widely assumed proportionality between *T* and *Q*_{E}, and identifies the limits of this relationship.

**Citation:**
Raupach, M. R.: The exponential eigenmodes of the carbon-climate system, and their implications for ratios of responses to forcings, Earth Syst. Dynam., 4, 31-49, doi:10.5194/esd-4-31-2013, 2013.

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 CO

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