CSIRO Marine and Atmospheric Research, Canberra, ACT 2601, Australia Abstract. Several basic ratios of responses to forcings in the carbonclimate 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 nearconstancy in the past, and its likely limitations in future. The contemporary carbonclimate system is often approximated as a set of firstorder linear systems, for example in responsefunction 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 carbonclimate 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 carbonclimate system over the past two centuries, the theory explains the observed nearconstancy of the AF, CAF and T/Q_{E} in this period. A nonlinear carbonclimate 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, carbonclimate nonlinearities (in landâ€“air and oceanâ€“air carbon exchanges and CO_{2} radiative forcing), and emissions trajectories for nonCO_{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 carbonclimate system, and their implications for ratios of responses to forcings, Earth Syst. Dynam., 4, 3149, doi:10.5194/esd4312013, 2013. 
