380 MACHTA ponential fallout rate is equivalent to the assumption of first-order kinetics. In an exponential formulation, the stratospheric content attime t is given by the simple expression Cit) = C(O)e™"”™ = C(O)[1 —t/Tm + (2!1)7! (t/Tm)? — (3!)7! (t/Tm)? +... where C = T,, = t= C(0) = stratospheric content Mean residence time time initial content The mean residence time, the time for the content to reduce to one-half its original content, is related to the half residence time, Ty, by T,, = T,/0.693. The exponential form is frequently expressed as the frac- tional removal from the stratosphere per year (or other time interval) when the mean residence time is small so that the square and higher- order terms in the exponential expansion are negligible: C(t) — C(0) cio) t «iT, When t equals one year and the T,, is expressed in years, the ratio t/T,, becomes 1/T,, and is the fractional removal per year. Thus a 10year mean residence time corresponds to a fractional removal rate of 0.1 per year. It should be noted that this approximation breaks down when the mean residence time becomes small, say, less than five years. The mathematical formulation in which a fixed fraction of the stratospheric content removes to the troposphere per unit of time and vice versa needs no physical rationalization. However, for realistically applying the mathematics, one may argue as follows: Both the stratosphere and troposphere are well mixed layers of the atmosphere, but the interface between the two layers, the tropopause, may be treated as a semipermeable membrane inhibiting the rapid exchange between the two layers. Such a physical model, as in chemical experiments, can be shown to lead to a simple exponential exchange rate. Unfortunately, the physical attributes of the exponential model do not realistically conform according to meteorological concepts. The troposphere may be well mixed on the time Scale of months but the stratosphere is not. Furthermore, the tropopause contains no special inhibiting properties; rather it simply marks the bottom of the stably stratified stratosphere and the top of the more turbulent troposphere. The reason for the longer residence time of debris in the stratosphere than in the troposphere is the lesser vertical mixing (to bring air quickly to the troposphere) and the absence of scavenging mechanisms;