On the foundation of the α-β-γ approach to carbon-climate feedbacks

Abstract. The α-β-γ approach used to quantify the size of the feedbacks between climate and carbon cycle consists of two elements: the α-β-γ formalism expressing the feedback strength by the sensitivities α, β, and γ, and an experimental practice to determine these sensitivities from Earth system model simulations using a transient scenario where CO₂ is forced to rise far above its pre-industrial value. There are several reasons to be unsatisfied with this approach: the α, β, and γ sensitivities are introduced as linear expansion coefficients into the forcing and thus should be characteristics of the considered model as such, but they are known to be non-constant in time and to depend on the simulation scenario used to determine their values. Moreover, being linear, the whole approach should be valid only for sufficiently small forcing, so that the practice to calculate the sensitivities at maximum forcing reached in the simulations is rather questionable. Finally, the definition of the sensitivities as linear expansion coefficients into the forcing turns out to be inconsistent with the practice to apply the formalism to transient simulations: we demonstrate that, because of the internal memory of the Earth system, by such a definition all sensitivities are mathematically zero and thus not well defined. But as we show here, the whole approach can be justified when introducing the α, β, and γ sensitivities from the outset not as differential, but as difference quotients. In this way a linearization is not needed and one obtains a fully non-linear description of the feedbacks. Moreover, thereby the formalism can be extended to include also the synergy between the feedbacks so that it gets even exact. Nevertheless, the scenario and time dependence remain, being a necessary consequence of the application of the formalism to transient simulations. In this respect the α-β-γ approach to climate-carbon feedbacks differs from the well-known description of atmospheric feedbacks: in the latter case not transient, but equilibrium states are employed to quantify the feedbacks, a practice consistent with a linearization into the forcing; accordingly, the obtained sensitivities, as well as the feedback strengths calculated from them, are proper characteristics of the system, independently of how the equilibrium was reached. This would also be the case for the calculation of climate-carbon feedbacks by the α-β-γ formalism if one used equilibrium instead of transient simulations to compute the sensitivities. In the light of these results we discuss in the outlook the pros and cons of various options for future research on the size climate-carbon feedbacks, including also the application of the generalized α-β-γ framework to obtain insight into the memory structure of the climate-carbon system.