Analytical insights into the transient climate response

The temperature in the transient climate response is lower than the equilibrium temperature for the same amount of forcing. The degree of disequilibrium is not constant in time and depends on various climate parameters. We derive intuition for this by solving the heat equation with a surface temperature feedback for linearly increasing forcing. The surface temperature initially evolves at a slower rate than the corresponding steady state (SS) temperature and it accelerates until quasi-steady state (QSS), when the SS and QSS temperatures evolve in parallel with a constant offset. The offset depends on the rate of forcing and total heat capacity of the system divided by the square of the climate feedback. The timescale over which the climate system approaches QSS depends also on the effective ocean mixing and is order of thousands of years. Over societally relevant timescales (around 100 years), the top-of-atmosphere energy imbalance increases, and the actual temperature moves farther from the steady-state temperature expected for the same forcing.