Research Article

25 Jan 2013

The exponential eigenmodes of the carbon-climate system, and their implications for ratios of responses to forcings M. R. RaupachCSIRO Marine and Atmospheric Research, Canberra, ACT 2601, Australia

Received: 30 Jul 2012 – Published in Earth Syst. Dynam. Discuss.: 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.