Radon as an Indicator of Nocturnal Atmospheric Stability: A Simplified Theoretical Approach

Abstract

Nocturnal evolution of radon concentration and the height of a box model, which is determined from radon concentration and local radon flux at the ground, are used as indicators of nocturnal atmospheric stability in single-height observations. However, quantitative relationships between these indicators and meteorological conditions, including the turbulent diffusion coefficient, have not yet been well established. Here, we construct a simple model based on the heat exchange process of the lower atmosphere to relate these parameters. The model neglects radiative flux divergence and assumes a uniform constant radon flux, making it most applicable to low wind conditions at sites well-removed from coastal influences, when advective effects are minimal. The model shows that the box height (equivalent mixing height) can be determined from near-surface parameters including sensible heat flux and the decrease in potential temperature after sunset. For these parameters, static stability and mechanical mixing components are incorporated. In addition, the constructed equations suggest the equivalent mixing height is proportional to the inversion layer height with a slope that depends on the vertical profile of potential temperature. The equivalent mixing height can be also related to the turbulent diffusion intensity. We demonstrate that radon observations at a single height are useful for monitoring nocturnal atmospheric stability.

Appendix 1

Dimension analysis using the Pi theorem (Buckingham 1914) is applied to a system governing the variation of radon concentration through heat exchange in a layer. An air layer with two parallel surfaces at a distance of $\Delta z$ is considered where the lower and the upper surfaces correspond to the ground and inversion layer height in the environment, respectively. The whole layer is characterized by the density $\rho$ and the specific heat $C_{\mathrm{p}}$ of air. In addition, radon and heat are supplied to the layer only at the lower surface and their rates are expressed as E(t) and H(t), respectively. Under these conditions, we investigate the relationship between vertical gradients of radon concentration $\Delta C(\Delta z, t)/\Delta z (\Delta C(\Delta z, t)=C(\Delta z, t) - C(0,t))$ and potential temperature $\Delta \theta (\Delta z, t)/\Delta z (\Delta \theta (\Delta z, t) = \theta (\Delta z, t) - \theta (0,t))$. Table 1 lists the fundamental dimensions of seven physical quantities in the considered system. In this study, we identify six fundamental dimensions: mass (M), length (L), area (S), volume (V), time (T), and temperature (K), although area and volume scales may be converted into the length dimension in ordinary studies. Based on these quantities, we may construct the functional relation in dimensional form: $f(\Delta C(\Delta z, t)/\Delta z, \Delta \theta (\Delta z, t)/\Delta z, E(t), H(t), \rho , C_{\mathrm{p}}, \Delta z) = 0$. Based on the Pi theorem, only one dimensionless product is determined and, therefore, the following dimensionless expression is given as

$$\begin{aligned} \phi \left( {\frac{H(t)\frac{\Delta C(\Delta z,t)}{\Delta z}\Delta z}{\rho C_{\mathrm{p}} E(t)\frac{\Delta \theta (\Delta z,t)}{\Delta z}\Delta z}}\right) =0, \end{aligned}$$

(12)

or

$$\begin{aligned} \frac{H(t)\Delta C(\Delta z,t)}{\rho C_{p}E(t)\Delta \theta (\Delta z,t)}=A, \end{aligned}$$

(13)

where A is a constant. We assume from the discussions in Sect. 2 that the dimensionless constant A represents the ratio of the turbulent diffusion coefficient of heat $(K_{\mathrm{H}})$ to the turbulent diffusion coefficient of radon $(K_{\mathrm{R}})$. For $A = 1$, the linear relationship between radon concentration and potential temperature is obtained in a form similar to Eq. 6.