Analytical solution for the effect of increasing CO₂ on global mean temperature

Increasing atmospheric carbon dioxide concentration is expected to cause substantial changes in climate¹. Recent model studies suggest that the equilibrium warming for a CO₂ doubling (Δ T _2×) is about 3-4°C^2-4. Observational data show that the globe has warmed by about 0.5°C over the past 100 years ^5,6. Are these two results compatible? To answer this question due account must be taken of oceanic thermal inertia effects, which can significantly slow the response of the climate system to external forcing. The main controlling parameters are the effective diffusivity of the ocean below the upper mixed layer (κ) and the climate sensitivity (defined by Δ T_2×). Previous analyses of this problem have considered only limited ranges of these parameters. Here we present a more general analysis of two cases, forcing by a step function change in CO₂ concentration and by a steady CO₂ increase. The former case may be characterized by a response time which we show is strongly dependent on both κ and Δ T_2×. In the latter case the damped response means that, at any given time, the climate system may be quite far removed from its equilibrium with the prevailing CO₂ level. In earlier work this equilibrium has been expressed as a lag time, but we show this to be misleading because of the sensitivity of the lag to the history of past CO₂ variations. Since both the lag and the degree of disequilibrium are strongly dependent on κ and Δ T_2×, and because of uncertainties in the pre-industrial CO₂ level, the observed global warming over the past 100 years can be shown to be compatible with a wide range of CO ₂-doubling temperature changes.