An Adjoint Trajectory Model of the Perturbed Carbon Cycle

Here we present a new model framework for an adjoint trajectory model of the carbon cycle, which can be used to constrain carbon emission rates over a specific time interval given only a path constraint on the fraction of CO₂ that remains airborne over that interval. Earth's ocean and terrestrial biosphere act to quickly remove a large portion of anthropogenic CO₂ emissions from the atmosphere, thus partially reducing the climate change-inducing radiative forcing of the emissions. Knowing the fraction of anthropogenic carbon emissions that remain airborne between early industrialization years and the next few hundred years is a goal shared by many researchers and policy designers, but is limited primarily by our uncertainty in the projected CO₂ emission schedule that humans will follow. With this adjoint trajectory model, we can, without having any information about the emission schedule, invoke the use of mathematical inverse methods to computationally search for an emission scenario that best reproduces a time series constraint of observed (or desired) atmospheric CO₂ concentrations. A simple yet powerful 3-box transfer model is used to partition CO₂ emissions between the atmosphere and ocean. The transfer functions that govern this 3-box model are also utilized to provide a physical constraint on the acceptable set of emission scenario solutions. We will present the governing mathematical framework of the adjoint trajectory model and provide a general algorithm for determining the unknown emission scenario that optimally reproduces the observed (or desired) atmospheric trend. We will show examples of the model being used to constrain both past and future emission scenarios, over multiple periods in Earth's history.