John Reid

The relation between CO₂ emissions and atmospheric CO₂ concentration has traditionally been treated with more or less complex models with several boxes. Our approach is motivated by the question of how much CO₂ must necessarily be absorbed by sinks. This is determined by accepted measurements and the global carbon budget. Observations lead to the model assumption that carbon sinks, similar to oceans or the biosphere, are linearly dependent on CO₂ concentration on a decadal scale. In particular, this implies the falsifiable hypothesis that oceanic and biological CO₂ buffers have not significantly changed in the past 70 years and are not saturated in the foreseeable future. A statistical model with two parameters is built from the global carbon budget and two testable assumptions. This model explains the relation between CO₂ emission and historical CO₂ concentration data very well. The model gives estimates of the natural emissions, the pre-industrial CO₂ equilibrium concentration levels, the half-life time of an emission pulse, and the future CO₂ concentration level from a given emission scenario. It is validated by an ex-post forecast of the last 20 years. The important result is that, with the stated polices emission scenario of the International Energy Agency (IEA), the future CO₂ concentrations will not rise above 475 ppm. The model is compared with the carbon module of the Bern model, mapping their complex impulse response functions (IRFs) to a single time variant absorption parameter.

The relation between CO2 emissions and atmospheric CO2 concentration has traditionally been treated with more or less complex models with several boxes. Our approach is motivated by the question of how much CO2 must necessarily be absorbed by sinks. Observations lead to the model assumption, that carbon sinks like oceans or biosphere are linearly dependent on CO2 concentration on a decadal scale. In particular this implies the falsifiable hypothesis that oceanic and biological CO2 buffers have not significantly changed in the past 50 years and are not saturated in the forseeable future. The simple model with 2 parameters explains very well the CO2 emission and historical CO2 concentration data. The model gives estimates of the natural emissions, the pre-industrial CO2 equilibrium concentration levels, the half-life time of an emission pulse, and the future CO2 concentration level from a given emission scenario. This is validated by an ex-post forecast of the last 20 years. The important result is that with the stated polices scenario of the IEA future CO2 concentrations will not rise above 475 ppm. The model is compared with the Carbon modul of the Bern model, mapping their complex IRFs to a single time variant parameter.

The impulse response of CO2 concentration due to variation in CO2 emissions was estimated from observed time series using the ARX method augmented by testing residuals for self-correlation. It was found to be a simple exponential with a half-time of 43 years with no remnant component. The longer half times and large remnant fraction of the impulse response derived from ocean circulation models is attributed to the failure of these models to account for turbulent mixing in the deep ocean.

The impulse response of CO2 concentration due to variation in CO2 emissions was estimated from observed time series using the ARX method augmented by testing residuals for self-correlation. It was found to be a simple exponential with a half-time of 43 years with no remnant component. The longer half times and large remnant fraction of the impulse response derived from ocean circulation models is attributed to the failure of these models to account for turbulent mixing in the deep ocean.

The successful application of simple statistical methods in economic modelling suggest that similar methods could provide insight into global climate. We apply an autoregressive method to observed time series to determine the dependence of atmospheric CO2 concentration on carbon emissions and, in turn, the dependence of globally averaged temperature on atmospheric CO2 concentration. We ascribe physical meaning to the regression parameters in terms of first order differential equations describing the diffusion of CO2 and temperature between reservoirs, viz.: the diffusion of CO2 between the atmosphere and the deep ocean and the transport of heat from the mixed layer. A strong feature of regression models is their built-in mechanism for deciding when a model provides an adequate description of the given data. Two implications of this statistically robust approach are that CO2 diffuses from the atmosphere within a time scale of decades and that global average temperatures are unlikely to exceed 2 deg C above pre-industrial values.

Two unintended experiments provide insight into the role of anthropogenic emissions in the global carbon cycle. One was the temporary reduction in emissions during the COVID-19 lock-down. The other was the emission of a radioactive isotope of carbon during the testing of nuclear weapons in the 1950s and the abrupt cessation of these tests in 1963. Together they imply the existence of two distinct reservoirs which exchange carbon with the atmosphere, viz.: the mixed layer and the deep ocean. Exchanges with the former are noisy because they are influenced by sea surface temperature which, in turn, depends on local weather conditions. They completely mask any variations caused by the COVID-19 lock-down. Exchanges with the latter are steady, long-term and uninfluenced by the weather. They result in half the carbon content of the atmosphere and mixed layer being replaced every eleven years.

The HadCRUT4 time series of 166 annual values of global average temperature was analysed both deterministically and stochastically and the results compared. The deterministic model comprised the sum of a linear trend and a multi-decadal oscillation fitted by ordinary least squares regression. The stochastic model was an ARMA(1,2) model with a drift term included. The deterministic model showed a linear trend of 0.5℃ per century while the stochastic model showed no significant drift. In both cases, the residuals were tested for self-correlation using standard statistical tests. The residuals from the deterministic model were significantly self-correlated whereas those from the stochastic model were not. We conclude that the stochastic model was a much better fit to the data and that the apparent linear trend of the deterministic model was spurious and a consequence of performing a regression in which time was the explanatory variable.