^{1}CSIRO Marine and Atmospheric Research, Canberra, ACT 2601, Australia Abstract. Several basic ratios describing the carbonclimate system are observed to adopt relatively steady values. 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 firstorder linear system, 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 among fluxes and perturbation state variables 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. The Lin and Exp idealisations apply approximately (but not exactly) to the carbonclimate system in the period from the start of industrialisation (nominally 1750) to the present, consistent with the observed nearconstancy of the AF, CAF and T/Q_{E} in this period. A nonlinear carbonclimate model is used to explore how the likely 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 emissions trajectories, carboncycle dynamics and nonCO_{2} gases. This theory assists in establishing both the basis and limits of the widelyassumed proportionality between T and Q_{E}, at about 2 K per trillion tonnes of carbon. Citation: Raupach, M. R.: The exponential eigenmodes of the carbonclimate system, Earth Syst. Dynam. Discuss., 3, 11071158, doi:10.5194/esdd311072012, 2012. 
