This paper shows that a very simple model can reproduce two connected facts about the atmosphere since the industrial era: the rise in carbon dioxide (CO2) and the fall in the 13C/12C isotope ratio of atmospheric carbon. The isotope change, called the Suess effect, happens because fossil fuels are depleted in the heavier isotope 13C. The authors find that their model fits the observed CO2 concentrations and the observed drop in the isotope delta value (delta‑13C) from preindustrial times to the present.

The model uses only two key parameters and treats the atmosphere’s carbon like a reservoir that exchanges carbon with the land and oceans. Mathematically it is written as simple linear differential equations, analogous to Newton’s law of cooling. One parameter controls how quickly extra CO2 above the preindustrial level relaxes back to the old level. The other controls how quickly the product of concentration and isotope delta returns to its preindustrial value. The authors fit the model to published fossil‑fuel emission data and to measured CO2 and delta‑13C records.

With numbers from the paper, the preindustrial CO2 concentration is taken as 280 parts per million (ppm). The model reproduces observed trends while using an atmospheric CO2 relaxation time of 60 years. It also finds that the concentration × delta‑13C quantity relaxes much faster, with an 8‑year time constant. The paper gives concrete data points: modern fossil fuel emissions near 10.5 petagrams of carbon per year in 2024, a preindustrial carbon input to the atmosphere of about 9.9 Pg/year, and a preindustrial atmospheric delta‑13C of about −6.5 per mil (parts per thousand). The typical delta‑13C of fossil fuels used in the model is about −28 per mil, and a recent atmospheric value quoted is −8.7 per mil.

Why this matters: the two different decay times show that CO2 amount and the isotopic fingerprint return to preindustrial values on very different schedules if human emissions were to stop. The faster 8‑year time scale for the isotope product is a consequence of isotopic exchange processes, largely photosynthesis and respiration on land, which shuffle isotopes more quickly than they remove excess carbon. This helps explain why the isotope ratio responds faster than the total CO2 concentration.