The graph shows three versions of the Impulse Response Function of atmospheric CO_{2}, perhaps the most important single entity in relating climate change to greenhouse gas emissions.

The dashed curve, first derived in the 1970s, used a so-called “**Diffusion Box Model**” of the atmosphere and various sinks of CO_{2} . The IRF curve, *G(t)*, was plotted from the formula given by equation (6) of Maier-Reimer and Hasselmann (1987). viz.:

This model has a number of different diffusion times, *τ _{j}*, and, most importantly, includes a constant term,

*A*, which means, according to this model, a

_{0}**residual fraction**of just under 20 percent of CO

_{2}remains in the atmosphere forever.

In the early 1960s atomic bomb tests in the atmosphere injected a quantity of radioactive carbon, ^{14}C, into the atmosphere which largely finished up as ^{14}CO_{2} well mixed over the entire globe. This was a wonderful accidental experiment, because the observed changes in the radioactive proportion of carbon, Δ^{14}C, enabled us to measure precisely how rapidly CO_{2} diffuses between planetary reservoirs in the way radioactive isotopes are used to to assess metabolic processes in nuclear medicine. The fraction of ^{14}CO_{2} remaining in the atmosphere as a function of time is shown as the dotted line in the above figure. It is known as *The Bomb Test Curve.*

The radioactive half-life of ^{14}C is 5730 years but measurements show that the half-time it takes to diffuse out of the atmosphere is only 11 years. Furthermore the decay is precisely exponential with a **single time constant** and with **no evidence of any residual fraction**.

Admittedly the Bomb Test Curve does not precisely emulate the behaviour of anthropogenic CO_{2 }in the atmosphere because it does not account for CO_{2 }which is returned to the atmosphere from the deep ocean in regions of upwelling, and from the biosphere by respiration, combustion and decay. Our thanks to Dr Roy W. Spencer for pointing this out.

We estimated the decay of an emission pulse of CO_{2 }statistically, using an ARX model (for *autoregressive with exogenous variable*). We found an exponential decay with a half time of 62.5 years, the solid line in the figure. See: Reid and Dengler (2021).

**Then why is the diffu****sion box model so wrong?**

A close examination of the assumptions underlying the diffusion box model are revealing. The model has only a small number of boxes, the biosphere, the mixed layer of the ocean (the top 100m or so) and below this the deep ocean which is modelled as a conveyor belt driven by the thermohaline circulation. It assumed that these boxes are well-mixed and that the diffusive fluxes into the different reservoirs can be added as a sum of exponential terms and that the diffusion rates into the deep ocean via the mixed layer is controlled by the physical chemistry of carbonate ions in the mixed layer.

These assumptions are wrong for the following reasons:

- Although the mixed layer is well mixed vertically (by definition), it is certainly not well mixed horizontally. It’s temperature varies from -2
^{o}C to 28^{o}C which greatly affects the physical chemistry from location to location. - The simplistic, conveyor-belt circulation model does not properly account for deep ocean turbulence and mixing.
- Diffusion from a single source reservoir into a number of sink reservoirs cannot be expressed as a sum of exponentials. Such a model has a
**single time constant**determined by the concentration of the source. - The physical chemistry model leads to an unrealistically saturated sink, so leading to the false conclusion that a residual fraction of CO
_{2}remains in the atmosphere forever. - Given the random inputs of volcanic CO
_{2}over geological time, the integrating effect of the constant term, A_{0}, would have lead to CO_{2}concentrations which were a random walk. This has not been observed.

The “diffusion box model” is not a box model because the boxes are not well-mixed and it is not a diffusion model because it is dominated by local carbonate chemistry. A brave “first guess” in the 1970s has somehow morphed into one of the pillars of Climate Science.

** It is time it was reviewed.**