Laplace transform analysis of the carbon cycle

A Laplace transform representation is used to describe the changes in atmospheric CO2 in response to emissions. The formalism gives an explicit representation of generic relations that are less clear when model results are presented as numerical integrations with particular parameter values. In particular, the Laplace transform formalism clarifies some issues involved in inversion of ice-core data and analysis of geosequestration. The airborne fraction is expressed as the emission growth rate multiplied by the Laplace transform of the atmospheric response function, evaluated at the growth rate. This representation emphasises that historical data only capture carbon cycle dynamics over a limited range of timescales. The Laplace transform formalism provides a basis for expressing uncertainties in the response function in terms of the Pade´eLaplace transformation used for fitting sums of exponentials.