But the fall rate of the maximum point is governed only by the gravitational term of the flow rate equation, since ac 9x = 0 for the maximum point. From the considerations of the previous paragraph, it is also evident that the fall rate of the constant profile may be analytically computed, since, from equation (1), the flow rate at the maximum point is K af = (2) ec=yv Cc. (8) In this equation, Vmax denotes the instantaneous velocity of the maximum point and is given by Vv max dx dt K, p = = = -\-——-}] KL x (-*) Py CxP x = ———— , (9) where Py is the density of the atmosphere at x = 0(i.e., sea level) and A = mm is the e-fold distance for t=- Integrating (9) gives pr —° i, exp (. *). ~) exp ( x )| _ = x u average air particles. pr —s_ K, (* *) S exp\—\ | - lly (10) where x is the position of the maximum point at t = 0 and P . is the density of the atmosphere at xo: tion (10) expresses how long it will take the constant profile to fall from x, to x. x=x s -A RY pir In (A) ’ Equa- Or from (10), > (11) which expresses where the profile will be at time t after starting at Xs for time zero. Equations (10) and (11) govern the fall rate of the constant profile, and diffusion has no effect on the fall rate of the constant profile. The only effect of diffusion is to keep the profile constant. In fact, if one injects any profile of particles of mass greater than 100 atomic units at high altitudes and uses equation (11) to calculate the position of later profiles, an error of only a few kilofeet is made. Semiquantitatively, this means that one does not have to do numerical calculations in order to determine the positions of later profiles for impurity profiles injected into our model at high altitudes. That is, one makes a relatively small error by completely neg- lecting diffusion in calculating the fall rate of impurity particles injected into our model of the upper atmosphere. To give a feeling for the fall rate of various particle masses, we have used equation (11) to calculate the po- sitions of later profiles for particles injected into our model at 305 kilofeet. These results are plotted in Figures 3 and 4, It should be remembered that all of the discussion so far has been for a stagnant atmosphere. desirable to take turbulence into account in such calculations. mathematically. It would be Unfortunately, turbulence is difficult to describe However, we have considered a very simple model of layer turbulence and will proceed to show that such a simple model of turbulence does not appreciably affect the profile fall rate. We assume a layer of the atmosphere which is b kilofeet deep, with b being much greater than the profile width of the impurity 21