2 = the index for the position of the nuclide in the decay chain (Rn™, Rn™ : i = 1; Po™, Po:i = 2; °°: *) Uz, Uy, Us = the components of the mean wind velocity. K = the turbulent diffusion coefficient. v, = the mean sedimentation velocity of the ith nuclide, according to sedimentation of their carrier particles (Rn™, Rn™ : v, = 0). A, = the radioactive decay constant of the ith nuclide. A, = the mean removal rate the ith nuclide, caused by washout and rainout (Rn™, Rn™ : A, = 0). ‘The terms on the right side of (1) refer, in this order, to turbulent mixing, convection, sedimentation of radioactive particles, radioactive buildup from the mother nuclide in the decay chain, radioactive decay, and removal by washout and rainout. Weare primarily interested in the mean vertieal distributions for steady-state conditions at constant exhalation rates of Rn™ and Rn™. These equilibrium profiles are obtained from (1) if dn./dt = 0. To solve (1) we makethefollow- ing additional assumptions: 1. dn,/dx = On,/dy = 0. This means a horizontal isotropic distribution, which will occur when the vertical turbulence profile and the Rn” and Rn™ exhalation rates are the same at each place. 2. u, = 0. A reasonable mean value of the vertical wind velocity cannot be given, but in most cases it will be small compared with the velocity of vertical turbulent diffusion. 8. uv, == 0. The mean radius of natural radioactive particles in the atmosphere is smaller than 1 uw; the corresponding sedimentation veloc- ity is smaller than about 1 m/h,ie., small com- pared with the transport velocity caused by turbulent mixing. 4. A, = constant = A. This meansthat the removal rate is independent of altitude and equal for all decay products. The independence of altitude will be approximately true in the troposphere, whereas above the tropopause A will be zero., This assumption is of importance only for the distribution of Pb™° and Po”, be- cause for the short-lived Rn™ decay products and all Rn™ decay products A, > A.. For Pb™ and Po™ nearly the same removal rate can be expected. With these assumptions the equilibrium vertical profiles of Rn™, Rn™, and their decay prod- ucts can be obtained from the following system of differential equations: d am) — “(x dz — —_— Aim = 0 &/= where (x dn) +n rv, + A)n; = 0 for t> 1 (2) To solve (2), we introduce the following boundary conditions: 1. f."X.ndz = E, where F, is the exhalation rate of Rn™ and Rn” atoms from the ground surface (z = 0). This condition means that the total activity of Rn™ and Rn™ in a vertical air column is equal to the exhalation rate of its ground-surface area. 2. E, = 0 for: > 1. This condition reflects the rapid diffusion of newly formed Po” or Po™ atoms in ground air and their deposition on the ground material. 3. m(z = 0) = 0 fori > 1. This takes into account the fact that all atoms and carrier particles of the decay products reaching the ground surface by diffusion will be deposited. 4. n(ze—> 0) > O for? = 1,2,3,.... This condition is a consequence of radioactive decay. The diffusion coefficient K is quite variable with altitude according to the vertical variations of wind velocity and atmospheric stability. Some characteristic air layers can be distinguished from the relative slope of the function K(z). In the boundary layer near the earth’s surface K increases rapidly with altitude, following an approximately linear or power law of z. Approaching the gradient wind height, dK/dz decreases and becomes nearly independent ofaltitude in the upper troposphere under normal conditions. Owing to the high stability of the stratosphere, K again decreases rapidly above the tropopause and is likely to approach a rather constant value in the lower stratosphere. The changes of wind velocity and temperature NOE ARCHIVES os gt JACOBI AND ANDRE 3800