The Bomb Test Curve

The natural logarithm of Delta14C values recorded at Fruholmen, Norway as a function of time. The dashed regression line was fitted between January 1966 and June 1993.

The testing of nuclear weapons during the 1950s and 1960s injected significant amounts of the radioactive ^{14}C isotope of carbon into the atmosphere. More importantly, the abrupt cessation of atmospheric testing following the Nuclear Test Ban Treaty of 5 August 1963, meant that the rate of production of the ^{14}C isotope reverted to the constant natural background level. This allows the movement of carbon dioxide between natural reservoirs to be assessed in much the same way that radioactive isotopes are used to assess the rates of metabolic processes in nuclear medicine.

The decrease in Δ14C is known as “The Bomb Test Curve”. Numerous observations were made in the decades following the cessation of testing. Here we look at a single high quality data set from Fruholmen, Norway shown in the figure. The natural logarithm, ln(Δ14C), is plotted on the vertical axis rather than Δ14C itself so that exponential behaviour becomes linear. A regression line was fitted between January 1966 and the end of the data set in June 1993.

The fit is remarkably good and accounts for 98.8 percent of the variance. Hence, with a high degree of accuracy:

    \begin{equation*} \text{ln}(\Delta^{14}C)=-\frac{t}{\tau}+\text{ln}(A) \end{equation*}

where τ and A are constants. Thus

    \begin{equation*} \Delta^{14}C=Ae^{-t/\tau} \end{equation*}

where A is the value of Δ14C at t=0 and τ is the time constant given by

τ  =  -1/slope = 15.9 \pm  0.085 years.

The time for Δ 14C to decay to half its initial value is given by t{1/2} = τ ln(0.5) = 11.02 ±  .059 years.

Thus half of the bomb test 14CO2 disappears from the atmosphere every 11 years. This is the solution of the classic diffusion equation:

    \begin{equation*} \frac{dc}{dt}+\frac{c}{\tau}=F(t) \end{equation*}

where c is the concentration of the quantity being diffused (Δ14CO2 in this case), τ is the diffusion time or time constant and F(t) specifies the rate at which concentration increases due to new material being introduced into the reservoir. In this case following the cessation of nuclear testing F(t) is the constant background rate associated with the bombardment of upper atmospheric Nitrogen by cosmic rays.

Carbon dioxide reacts with water to form carbonate and bicarbonate ions. Hence the diffusion rate of carbon \textit{per se} involves reaction rates and diffusion rates for each of these three species. These are almost completely independent of atomic mass and so all the isotopes of carbon, 12C, 13C, 14C, in the form of CO2 and its radicals, diffuse through water at the same rate and the time constant, τ, applies equally to all isotopic species of CO2.

It is therefore reasonable to assume that CO2 diffuses from the atmosphere into other reservoirs or sinks.