Earth Syst. Dynam. Discuss., 3, 1107-1158, 2012
© Author(s) 2012. This work is distributed
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This discussion paper has been under review for the journal Earth System Dynamics (ESD). Please refer to the corresponding final paper in ESD.
The exponential eigenmodes of the carbon-climate system
M. R. Raupach
1CSIRO Marine and Atmospheric Research, Canberra, ACT 2601, Australia

Abstract. Several basic ratios describing the carbon-climate system are observed to adopt relatively steady values. 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 first-order linear system, 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 among fluxes and perturbation state variables 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. The Lin and Exp idealisations apply approximately (but not exactly) to the carbon-climate system in the period from the start of industrialisation (nominally 1750) to the present, consistent with the observed near-constancy of the AF, CAF and T/QE in this period.

A nonlinear carbon-climate model is used to explore how the likely 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 emissions trajectories, carbon-cycle dynamics and non-CO2 gases. This theory assists in establishing both the basis and limits of the widely-assumed proportionality between T and QE, at about 2 K per trillion tonnes of carbon.

Citation: Raupach, M. R.: The exponential eigenmodes of the carbon-climate system, Earth Syst. Dynam. Discuss., 3, 1107-1158, doi:10.5194/esdd-3-1107-2012, 2012.
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