A Natural Nuclear Reactor — Question 8

Question 8

Spatial confinement places a bound on long-term diffusion

Assume a stable tracer’s one-dimensional rms displacement follows $\ell=\sqrt{2Dt}$. A fossil pattern remains confined to an rms scale at most 1.00 m over $2.00\times10^9$ yr. Use $1\,\mathrm{yr}=3.156\times10^7$ s and constant D.

1. Find the upper bound on D.

2. If D were four times this bound, find the predicted rms displacement and explain one geological limitation of the inference.

Question 8 - Solution

Spatial confinement places a bound on long-term diffusion

(a) Method and calculation

$$t=(2.00\times10^9)(3.156\times10^7)=6.312\times10^{16}\,\mathrm s.$$$$\boxed{D\le\ell^2/(2t)=7.92142\times10^{-18}\,\mathrm{m^2/s}}.$$

(b) Interpretation and further calculation

$$\boxed{\ell\prime=\sqrt{2(4D_{\max})t}=2.00\,\mathrm m}.$$$$\ell^2=2\int_0^tD(t\prime)dt\prime\quad\Rightarrow\quad\text{the constant-D bound is time averaged}.$$

Physical check

Advection, chemical trapping, changing temperature and measurement geometry can invalidate a simple free-diffusion interpretation.