where the integral extends over the entire atmosphere. Note that this is analogous to the simple "replacement lifetime" discussed in the previous section (see Equation (2.27)).
One can also calculate a "stratospheric lifetime" where the integral in the denominator of Equation (2.30) extends only over the stratosphere. Many species, such as the chlorofluorocarbons (CFCs), are destroyed only in the stratosphere and above. Their global and stratospheric lifetimes will therefore be the same. Species with significant tropospheric sinks, such as CH4, have global lifetimes shorter than their stratospheric lifetimes. Table 2.1 shows the stratospheric lifetimes of several gases of importance in stratospheric chemistry.
Based on the CFC-11 lifetime of 45 ± 7 years. From Volk el al. , Table 5.
In order to analyze and interpret a set of atmospheric measurements, one must first put those measurements into a coordinate system. For example, if one is studying polar O, loss, then the measurements of most interest are those in the lower stratosphere and at high latitudes—and one must be able to select those measurements out of any data set of interest. In this case, pressure and latitude might be the coordinates of choice.
A quantity that is related to pressure is pressure altitude Z*. The pressure altitude of a given pressure p is calculated using the equation where H is the scale height, a constant with a typical value of ~7 km. pn is a reference pressure (often 1000 hPa, and in the same units as p).
Another commonly used vertical coordinate is potential temperature. The concept of potential temperature is based on an adiabatic process, one in which no heat flows into or out of the air parcel. If an air parcel moves adiabatically to a lower pressure, then the parcel will expand. Because no heat flows into the parcel, the work done to expand the parcel is provided by the internal energy of the parcel, and the temperature of the parcel decreases. Conversely, if an air parcel moves adiabatically to a higher pressure, the parcel is compressed. The work done on the parcel by the atmosphere during the compression increases the internal energy of the parcel, leading to a temperature increase.
The potential temperature of an air parcel is the temperature that the parcel would have if it were moved adiabatically to a reference pressure, usually the surface pressure (1000 hPa). It can be shown (for example, see Houghton , p. 22) that where 6 is the potential temperature (often referred to simply as "theta"), which has units of kelvin. T and p are the temperature (K) and pressure of the parcel, respectively. p0 is the reference pressure. Note that the units on p and p0 can be anything as long as they are the same. k. which is numerically equal to 2/7, is the ratio (cp-£'v)/ip, where cp and t\ are the heat capacities of air at constant pressure and volume, respectively. The potential temperature of the tropopause ranges from 320 K at high latitudes to 380 K over the equator. Potential temperature increases monotonically with altitude, and at the stratopause its value is -2000 K.
Figure 2.4 Time series of potential temperature, temperature, and pressure for a typical mid-latitude air mass at the equinox, as determined from a trajectory model [53J.
pressure on time-scales of 1 day. These short-term fluctuations are, however, adi-abalic. Thus, when the pressure of the parcel is increasing (decreasing), the temperature of the air parcel is increasing (decreasing) in such a way that the potential temperature remains constant. The result is that the potential temperature changes by only a few percent over the 15 day time series, despite large changes in pressure. In the parlance of stratospheric science, potential temperature is belter conserved than pressure. It is this quality that makes potential temperature a useful vertical coordinate.
This leads to an important question: over what time-scales are the potential temperature of an air parcel conserved? The assumption underlying the concept of potential temperature is that the air parcel is adiabatic: heat flow into or out of the air parcel is zero. As we will show in Chapter 5, however, air in the tropical stratosphere is being heated (i.e. the potential temperature of parcels increase with time), while the mid- and high-latitude stratosphere is being cooled (i.e. the potential temperature of parcels decreases with time) [54.1. The time series in Figure 2.4, for example, shows that potential temperature of the mid-latitude parcel is indeed decreasing with time, albeit slowly (dd/dt ~ 1 K day '). Figure 2.5 plots the zonally and annually averaged radiative damping time-scale, which is the time-scale over which a temperature perturbation is damped. This damping time can be considered the lifetime of potential temperature (much as the damping time for a chemical perturbation can be thought of as the lifetime). Over periods shorter than this damping time, potential temperature is conserved. Potential temperature is conserved for -30 and ~5 days in the lower and upper stratosphere, respectively.
This discussion brings up an interesting point. It is often said that flow in the stratosphere is generally horizontal. From the previous discussion, however, it should be clear that flow in the stratosphere actually occurs close to surfaces of constant potential temperature, also known as isentropic (constant entropy) surfaces. However, because the surfaces of constant potential temperature on average lie close to surfaces of constant geometric altitude, flow in the stratosphere is indeed approximately horizontal.