Diffusion in the Upper Atmosphere

Abstract

FUNDAMENTALS OF THE THEORY OF DIFFUSION 1. General Remarks. The kinetic theory of gases defines diffusion as the average motion of selected molecules relative to other molecules. The physical units of diffusion are number per square centimeter per second, that is, diffusion velocity times number of selected molecules per cubic centimeter. Diffusion depends on the composition of the gaseous mixture. A classical model considers two sets of molecules of different mass, effective diameter, and velocity distribution. The theory of diffusion results in complicated equations, even in the simple case of binary mixtures in a closed system when the physical state is well defined. Pure, dry air is more complex than a binary mixture. Water (vapor, liquid, solid) and particulate matter (nuclei, smoke, dust) complicate the composition of the atmosphere. The atmosphere is nota closed system and the physical state beyond the scope of soundings is uncertain in many respects [35]. Sources 1 and sinks 2 of constituents must be considered. Large-and small-scale atmospheric motions produce eddy diffusion. Diffusion equilibrium results when molecular and eddy diffusion balance the effects of forces acting on the molecules or the effects of sources and sinks so that the composition is a steady function of height or of height and the horizontal coordinates. Time variations of composition result from variations of the physical state-especially of motion and turbulence-and from intensity variations of sources and sinks of certain constituents. Because of the scarcity of direct observations, the problems of diffusion in the upper atmosphere were developed theoretically and on the hasis of conjecture. In this article, the author has tried to point out the inadequacies of the field by thoroughly outlining the assumptions necessary for a mathematical analysis of atmospheric diffusion. In general, applications of the theory must be confined to the lowest 200 km. Future work will be more promising when theoretical research can be supported by more and better observations from the stratosphere and ionosphere. 2. Molecular Diffusion. The study of nonuniform gases by Chapman and Cowling [4] resulted in the following general equation of diffusion velocity in a binary gas mixture: c1-c2 =-dtz {\7 111-C.u1-.u2)V' In p 1/j 112 (1) + mlm2 (Fl-F2)}-dT V' In r. mkT v1v2 1. Gas and particle production of the lithosphere and hydro-sphere, volcanic activity, photochemical reactions, industrial The subscript s denotes o ne constituent of the mixture · for a binary mixture, s = 1 and 2. The rectangular coordinate~ x, y, and zare oriented so that zis positive towards the zenith. The unit vectors i, j, and k are in the directions x, y, and z. Constituent gases c, = c,xi + c.,yj + c"k = vector of mean molecular motion n, = number of molecules per cma m, = mass of the molecule p, = hydrostatic partial pressure F, = externa! acceleration acting on the molecules dT = coefficient of thermal diffusion d,. = coefficien t of self-diffusion Gaseous mixture d12 = d21 = coefficient of mutual diffusion n = ~n, = total number of molecules per cm3 (Loschmidt's number) = 2.705 X 10 19 at NTP NTP = normal temperature and pressure p = hydrostatic pressure; normal pressure = 1013 mb 1' = absolute tempera ture; normal tempera ture = 273K v, = n. • /n = number concentration m = fv,m, = mass of a fictitious average molecule p = nm = density of the mixture p., = (m. •-m)/m = relative weight factor k = 1.372 X 10-16 ergs per degree = Boltzmann's constant Fundamental equations are Dalton's law: P = L Ps' (2) s and the equation of state: Ps = n,kT or p = nkT. (3) Equation (1) shows that the vector of diffusion velocity c1-c2 has four constituents, C1-C2 = Ca + Cp + CF + CT , where ca is ordinary diffusion velocity, Cp is pressure diffusion velocity, cF is forced diffusion velocity, cT is thermal diffusion velocity. Experiments are usually based on ordinary diffusion due to initial nonuniform composition in a closed system , V'v, ~ O, when Cp , cF , and cT are neglected. In the atmosphere, pressure diffusion (an indirect effect of gravity since the direct effect of gravitational acceleration on all molecules is the same) must be considered; Cp is due to the gradient of In p set up by gravity and/or rotation in compressible media. The most important example of forced diffusion is the effect of electric fields on the motion of ionized gases. Thermal diffusion results when different parts of the mixture are maintained at different temperatures; however, the degree of separation produced by cT is small; experience has proven that dT/d12 ~ 0.1 and decreases when ndn1 320 H. R. Byers et al., Compendium of Meteorology