Carbon budget concept and its deviation through the pulse response lens

Abstract. The carbon budget concept states that the global mean temperature (GMT) increase is roughly linearly dependent on cumulative emissions of CO2. The proportionality is measured as the transient climate response to cumulative emissions of carbon dioxide (TCRE). In this paper, the deviations of the carbon budget from the strict linear relationship implied by the TCRE are examined through the lens of a temperature response to an emission pulse (i.e., pulse response) and its relationship with a nonlinear TCRE. Hereby, two sources of deviation are distinguished: emission scenario and climate state dependence. The former stems from the scenario choice, i.e., the specific emission pathway for a given level of cumulative emissions and the latter from the change in TCRE with changing climatic conditions. Previous literature argues for scenario independence using a stylized set of emission scenarios, and offers a way to fit a nonlinear carbon budget equation. This paper shows how the pulse response, viewed as a Green's function, gives a unifying perspective on both scenario and state dependence. Moreover, it provides an optimization program that tests the scenario independence under the full range of emission pathways for a given set of constraints. In a setup chosen in this paper, the deviations stemming from emission pathway choices are less than 10 % of the overall temperature increase and gradually diminish. Moreover, using the pulse response as a Green's function, the scenario-dependent effects of a reduced-complexity climate model were replicated to a high degree, confirming that the behavior of scenario-dependent deviations can be explained and predicted by the shape of the pulse response. Additionally, it is shown that the pulse response changes with climatic conditions, through which the carbon budget state dependency is explained. Using a pulse response as an approximation for a state-dependent TCRE, an alternative method to derive a nonlinear carbon budget equation is provided. Finally, it is shown how different calibrations of a model can lead to different degrees of carbon budget nonlinearities. The analysis is done using FaIRv2.0.0, a simple climate emulator model that includes climate feedback modifying the carbon cycle, along with a one-box model used for comparison purposes. The Green's function approach can be used to diagnose both models' carbon budget scenario dependency, paving the way for future investigations and applications with other and more complex models.