Linear and non-linear energy balance model calibration across consecutive abrupt CO2 doubling experiments

In climate sensitivity literature, simple energy balance models provide insight into how energy moves throughout the climate system. The first-order approximation of these models assumes a linear relationship between the forcing, ocean heat uptake, and radiative response, including a constant feedback parameter. However, these linear assumptions have been shown to inaccurately estimate the effective (or equilibrium) global mean temperature response across consecutive CO₂ doubling experiments, with second-order approximations required to capture climate system non-linearities such as CO₂-temperature (state) dependence or the pattern effect. It is common to express these non-linearities in an energy balance model using an inconstant feedback, ocean heat uptake efficacy, or forcing efficacy factor. While these climate system non-linearities are well studied, no research has systematically assessed whether individual parameterisations differ in their ability to capture the temperature response across multiple CO₂-doubling experiments – that is, whether non-linearities acting on specific components of the climate system are more effective at reproducing responses across successive forcing scenarios. Using 12 CMIP6 models for which abrupt CO₂ doubling and quadrupling experiments are available (nine of which also include abrupt halving), we calibrate a two-layer energy balance model simultaneously to the surface air temperature time series from each experiment for each model. We perform multiple calibrations under both linear and non-linear assumptions. Preliminary results indicate that, for most models, a first-order approximation with a constant feedback parameter is sufficient to capture the surface air temperature response across multiple CO₂ doublings. Where a constant feedback parameter is not sufficient, initial findings suggest that a state-dependent forcing is the most effective correction. Future work will consider how this work can be reconciled with the temporal evolution of the feedback parameter seen in many observation-based historical CMIP6 simulations and the implications of our findings for projections of future climate.