We've identified two types of studies for which we use computer modeling, process studies and assessment studies. The former attempts to recreate existing observations, while the latter makes predictions based on past, current, and possible future trends in one or more atmospheric variable. In this section, we review the physical and chemical processes that are at the core of all atmospheric models. Next, we present three different types of models used by atmospheric scientists today: the photochemical box model, the trajectory model, and the zonal mean or two dimensional model. Each of these models is applied to some set of problems. In this Chapter series, we are concerned about stratospheric ozone and those compounds like CFCs that affect its concentration.

3.1 Physical and Chemical Processes

There are four basic processes which form the components of a stratospheric photochemical ozone model. We refer to these as modules or parts of the model. These are

Depending on the problem under consideration, atmospheric models will include the theoretical framework from one or more of these processes in varying amounts of detail. We review each of these processes in turn.

3.1.1 Radiation and the Atmosphere -- When solar radiation enters Earth's atmosphere, it interacts with individual molecules. The nature of the interaction depends on the wavelength of the radiation and on the particular molecule encountered. The sun radiates most strongly at those wavelengths that fall in the visible light portion of the energy spectrum. The details of this interaction are discussed in Chapter 4. Solar radiation is absorbed by Earth and reemitted as infrared (thermal) radiation. Stratospheric ozone absorbs ultraviolet light directly, and reemits it too as infrared radiation. The thermal radiation emitted by the surface heats the atmosphere, providing energy to drive atmospheric motions. The thermal radiation emitted by stratospheric ozone warms this layer of the atmosphere, imparting its temperature and dynamical characteristics (see Chapter 2).

Solar radiation (visible and ultraviolet) can also directly impact the chemistry of the atmosphere itself by providing enough energy to a molecule to break its molecular bonds. The process of breaking molecules apart by sunlight to form other molecules is known as photolysis. Figure 12.01 is a schematic diagram of how ozone is produced through the photolysis of oxygen molecules (O2) by highly energetic UV radiation and its reaction with free oxygen atoms.

3.1.2 Gas phase chemistry -- A substantial number of chemical species are present in the atmosphere as gases. These gases sometimes participate in chemical reactions that can alter the concentration of ozone either directly or indirectly. Chemical species can be categorized in one of three ways: source gases, reservoir species (chemical compounds that store a particularly species in a nonreactive form for some period of time), and radical species (reactive gases, also called "active species," that readily participate in chemical reactions).

1. Source gases -- In the stratosphere, source gases are those gases transported from the troposphere (see Chapter 5 and 6) into the stratosphere. In order to survive the transport processes and attain stratospheric altitudes, source gases must have long lifetimes. (In order to understand what is a "long lifetime" for a trace gas, recall from Chapter 6 that trace gases are transported meridionally from equator to pole by the Brewer-Dobson Circulation. To say that a specific trace gas has a "long lifetime" means that if we followed an individual molecule of the trace gas as it was transported by this slow circulation, the molecule would survive on the order of years before reacting with some other molecule or being photolyzed by solar radiation.) In addition, source gases are not water soluble. If they were water soluble, they would likely be captured by water vapor in the troposphere, eventually falling back to the surface in rain water. As noted, source gases eventually break down through reactions or photolysis.

The molecules that are formed from reactions of the source gases when they break down typically are not long lived and are water soluble, so we know they could not have been transported from the troposphere. The source gases represent the only possible mechanism for introducing these molecules in the stratosphere. An example of source gases are the family of compounds known as chlorofluorocarbons (CFCs). These are the manmade chemical compounds developed in the 1920s and used for decades as refrigerants and spray can propellants. CFCs are stable molecules that are not easily broken down in the lower atmosphere, but instead are lofted into the tropical stratosphere and then poleward by the Brewer-Dobson Circulation. Once in the middle to upper stratosphere, CFCs are broken down by energetic photochemical reactions. They provide a source of reactive chlorine in the stratosphere, which in turn destroys ozone (see Chapter 5 and 11). Figure 12.02 shows how the concentration of one such CFC compound, CFC-12, varies with height. The arrows drawn on the image show schematically how CFC-12 is transported meridionally (north-south) and vertically (up-down) by the Brewer-Dobson Circulation (see Chapter 6 for full discussion).

From the figure, we get a sense of how CFC-12 is distributed with height and latitude. The importance of CFC-12 as a source gas for chlorine in the stratosphere is made evident by the image.

2. Reservoir species -- Reservoir species are those molecules that do not participate in active chemistry and hence their alternate description as inactive species. These molecules are typically formed through chemical reactions involving radical species. For example, NO2 (nitrogen dioxide) and ClO (chlorine monoxide) both participate in a series of reactions that destroy ozone and recreate themselves. Such a series of reactions is called a catalytic cycle (see Chapter 5). When NO2 and ClO react with one another, however, they form ClONO2 (chlorine nitrate), a reservoir species that does not destroy ozone. ClONO2 instead locks up or sequesters the reactive species that destroy ozone. The more ClONO2 that's around, the less ozone destruction there is. When such a molecule breaks apart either in another reaction or by photolysis, the radicals are again free to react with ozone. Reservoir species such as ClONO2 therefore act as "warehouses" of radicals.

3. Radical species -- A radical species is technically a molecule with an odd number of valence electrons. This electronic state makes the molecule highly reactive. As such, radicals are short lived. In the stratosphere they are involved in catalytic reaction cycles that destroy ozone but preserve the concentration of the radical species. Chapter 5 gives an overview of the different families of catalytic cycles that occur in the stratosphere. Sometimes radicals react with each other or other molecules to form reservoir species. In the example above, CFCs are the source gases, while the reactive chlorine species liberated in the photochemical reactions are the radical species. Other radical species include NO2 and ClO. Both participate in catalytic cycles that destroy ozone.

3.1.3 Heterogeneous chemistry -- Heterogeneous chemical reactions occur on surfaces (liquid or solid). In the stratosphere, such surfaces are found on particles known as aerosols. Several types of aerosols exist in the lower stratosphere. These include liquid droplets of sulfuric acid, nitric acid, and water. They also include solid crystals of nitric acid trihydrate and water ice particles. These particles typically result from the condensation of gas phase constituents at sufficiently low temperatures. Such aerosols range in size from a few tenths of a micron (10-6 m) up to a few microns in size. Particles of such a small size can persist in the stratosphere for weeks to years.

Heterogenous reactions are extremely important in the development of the Antarctic ozone hole (see Chapter 11). Before such reactions were hypothesized, scientists had tried to explain observed ozone measurements in Antarctica with gas phase chemistry. Such efforts, however, always failed to explain the dramatic seasonal ozone losses that are observed. Atmospheric chemical models of the lower stratosphere where aerosols are found (e.g., winter polar vortex) need to include heterogeneous reactions.

3.1.4 Dynamics -- The dynamics of a model refer to the way the model represents the real-world behavior of air circulation and mixing. Because the lifetime of ozone in the lower stratosphere is relatively long (several months), the distribution of ozone in these regions is highly dependent on transport or dynamic processes. Such processes are also responsible for the introduction of chemicals into the stratosphere from the troposphere (see Chapter 6). A good dynamical simulation is therefore very important to any stratospheric photochemical model if it would accurately capture the observed distribution of ozone in the lower stratosphere.

3.2 Three Stratospheric Ozone Photochemical Models

We are now ready to examine the three main stratospheric ozone photochemical models used today by atmospheric scientists as part of the continuing research into issues of atmospheric chemistry, especially stratospheric ozone destruction. Each model is applied to a different set of problems, and each model emphasizes different aspects of the four basic processes outlined in section 3.1. The overall goal is, as stated, to make improved model assessments that allow us to project more accurately future trends in ozone.

3.2.1 The photochemical box model -- a. Description -- You can think of a photochemical box model as a sealed fish tank. Inside the tank, there are a variety of plants, fish, snails, and chemicals in the water. Nothing is allowed either to enter or leave the tank, except light from an overhead lamp. The lamp light provides the energy that drives the biological and chemical activity inside the tank. A hypothetical computer model that calculated what happens to the temperature, chemical concentrations, and reaction rates within the sealed tank as a function of time would be a "box model." The only external aspect to the model would be the lamp light entering the model, which we refer to as the "radiative flux." Figure 12.03 is a schematic diagram of five main pieces of the box model. These pieces are each represented by the appropriate sets of time dependent equations.

(1) Isolated parcel or "box" of air -- In the atmosphere, our fish tank is replaced by a box that consists of a parcel of air that is assumed to be isolated from the surrounding air mass. The "lamp light" is sunlight: shortwave solar radiation. The photochemical box model consists of all the relevant chemical reactions and rates of photolysis necessary to calculate the concentration of each of the chemical species in the box as a function of time. In order to make these calculations, the model must know the radiative flux (amount of solar radiation) at the location of the "box" as a function of wavelength and time, the temperature of air within the box, and initial amounts of the various constituents within the box.

It is also important to note that our fish tank metaphor is based on the assumption that there is no outside flow through the parcel of air. The parcel can be moved by the wind or by some larger circulation pattern, but the integrity of the parcel is unchanged. This assumption is actually a reasonable one, as a parcel of air can maintain its characteristics -- e.g., potential temperature, moisture content, and trace gas components -- for a period of many days even outside its source region. For example, in the winter time during an Arctic outbreak across the Eastern U.S., the air remains cold and dry for days even in places as far south as Georgia and the Carolinas. A parcel of Arctic air can be "tagged" and monitored as it moved from its source region in far northern Canada all the way to the Southeastern U.S.

(2) Photochemical equilibrium inside the box -- Once an isolated box of air is set up in a model, the modeler then defines the chemical mechanism for the model. The chemical mechanism is the set of constituent (trace) gases and their reactions that are assumed to occur within the box. If the amount and distribution of solar radiation is held fixed, running the chemical box model will result in an output known as photochemical equilibrium. Equilibrium is achieved within a model when all the constituents reach a constant value with time. That is, if we look at how the concentration of any gas within the model is changing as a function of time, we will observe no change with time after the model has reached equilibrium.

The notion of equilibrium is an important one in understanding and interpreting the output of chemical models. Let's say we have a model that is in photochemical equilibrium. We then force a change in the concentrations of radical species within the box. Examples of radical species include ClO and NO2. In our example, we force a change by moving 0.5 ppbv (parts per billion by volume) of ClO and NO2 into ClONO2 (chlorine nitrate, a reservoir species). Such a change is known as a perturbation to the model, that is, a small change away from chemical equilibrium. If we now start our model running again with the new concentrations of these three species, the model will redistribute the constituents via photochemical reactions away from ClONO2 and back into the radical forms ClO and NO2. The model always tends toward equilibrium. That is, changes away from equilibrium conditions will tend to be countered by the model. Figure 12.04 demonstrates this using a box model and trace species HCl (hydrochloric acid).

In Figure 12.04, we have a demonstration of how a box model run eventually brings the concentration of HCl into photochemical equilibrium. HCl is one of the reservoir species that locks up reactive chlorine. We see that HCl concentrations (given in ppbv) gradually reach an equilibrium state over the course of 3-week model run. The fine structure of the curve (the minor fluctuations on a diurnal basis) are explained in the next subsection.

(3) Diurnal cycles and photochemical equilibrium -- Because the amount of solar radiation changes constantly in the real atmosphere through the course of the day, the usefulness of strict equilibrium calculations, such as the one discussed above, is clearly limited. More frequently, box models are run through a daily or diurnal cycle of time-varying solar radiation input in order to make predictions about how the concentrations of different species will change throughout the course of a day. Using a diurnal cycle for solar radiation input, the notion of photochemical equilibrium is modified somewhat. Equilibrium in such a model is said to occur when the diurnal cycle in a species concentration repeats itself. For example, we observe a species increase from 5 ppmv (parts per million by volume) to 8 ppmv between sunrise and noon, then decrease to 6 ppmv by sunset and 5 ppmv by sunrise the next morning. If for the next day of the model run, we see exactly the same concentrations of these species at the same times of the day, we say the model has achieved a diurnal cycle equilibrium state. Figure 12.05 demonstrates such a diurnal equilibrium state for ClO (chlorine monoxide), a radical species that exists only when there is solar radiation present (i.e., during the daylight hours), using a box model calculation.

Figure 12.05 shows the box model calculation for the concentration of ClO, which exists only in the presence of shortwave solar radiation and the resulting photochemistry. ClO concentrations fall to zero at night as the photochemistry "shuts off" and ClO is converted entirely into ClONO2 (chlorine nitrate). That is, ClONO2 is a chlorine reservoir species that locks up reactive ClO at night. This explains the diurnal variations in ClO concentration. Beneath this diurnal cycle is a slower cycle that over the course of about a one month model run eventually brings ClO concentration into equilibrium. Note that the period chosen is identical to the period of time in Figure 12.04. This is no coincidence.

Since our box model is a closed system, the total supply of chlorine is fixed. Like chlorine nitrate, HCl is another chlorine reservoir species. During the day, HCl is a source of free chlorine atoms in the stratosphere due to ultraviolet photochemistry. These liberated Cl atoms are rapidly interconverted back and forth from Cl to ClO during the day in the presence of reactive nitrogen and oxygen (which also only exist during the day).

The diurnal cycle in both ClO and HCl concentrations exist on top of a slower cycle that brings both species into a steady state equilibrium. In a 1-month calculation in our box model, ClO concentration slowly climbs to a diurnal cycle steady state at the same rate that HCl concentration drops to a diurnal cycle steady state. The directly inverse relationship between HCl and ClO is expected inside our closed system: ClO increases over time at the expense of HCl.

As for the "bumps" in HCl concentration in Figure 12.04, these are due to the fact that reactive chlorine (both ClO molecules and Cl atoms) production and concentration falls to zero at night. These radicals are converted into both ClONO2 and HCl, both of which increase in concentration at night.

b. Uses -- The most important current use of a box model is to aid in the interpretation of observations of a wide variety of species. These observations are made simultaneously and are referred to as a "suite" of observations. A good example of such a suite of observations are those taken by the NASA ER-2 aircraft. The ER-2 aircraft measures many species important in the natural (photochemical) regulation of stratospheric ozone levels. The box model can be initialized with the actual measurements of the longer lived species. Since the long lived species do not show significant changes over the course of a day, the model, when run, can predict the diurnal variation of the shorter lived species. Good agreement between the model predictions and the ER-2 observations indicates that the photochemistry of the region of the atmosphere studied by that flight of the ER-2 is well understood. In addition to testing our understanding of atmospheric chemistry, the model also provides us with predictions of the concentration and variability of species other than those measured by the ER-2.

A photochemical box model is limited in that "dynamics" or transport processes in the real atmosphere are not considered. For chemical reactions that proceed quickly, it is okay to disregard these atmospheric motions and think of an air parcel as being effectively isolated. However, for long-lived trace species, we do not want to completely discount the fact that transport processes are advecting (moving) such species into and out of the box. That is, the "integrity" of the air parcel is not maintained over long periods of time.

Figure 12.06 shows a comparison of a photochemical box model's diurnal variability of stratospheric free radicals with data on these radical species gathered by the ER-2 instruments. The figure shows the diurnal variation of the radical species OH (hydroxide) and HO2 (peroxide), as well as the ratios of the ClO, NO2, and NO radicals to their total family concentrations. Also plotted on the figure are predictions from a constrained photochemical box model, where the box model has been run using several different assumptions about the heterogeneous chemistry.

As the figure shows, only one of the model runs shows very good agreement with all of the observations, providing a strong clue to guide our investigation of stratospheric photochemistry.

3.2.2 The trajectory model -- a. Description -- The idea of a trajectory model is very simple. Consider a box or parcel of air located at some point in the atmosphere at a particular time. If you knew which way and how fast the wind was blowing, you could predict where that air parcel would be in the future. For example, if you were told that the wind was blowing due east at 10 m/sec, one minute (60 seconds) from now, you would expect to find the air parcel 600 m east of its current location. A trajectory model takes inputs of initial parcel locations and the magnitude and direction of the winds (known as wind field analyses) and uses this information to determine whence the air parcels originated and where they are going to go. Figure 12.07 is a schematic diagram of a trajectory model map. It shows the same latitude-longitude grid for three separate times. The colored dots represent air parcels at a given time. The colored dashed lines indicate how far the parcels have moved by the next run.

Figure 12.07 consists of three latitude-longitude grids each showing the same region (0°E to 3°E and 0°N to 2°N), but at different times. The top is time 12:00Z, the middle is 13:26Z (an hour and 26 minutes later), and the bottom is 15:12Z (an hour and 46 minutes later still, or three hours and 12 minutes after the start time). The 12:00Z panel shows three red dots, each representing a parcel of air at a particular latitude and longitude on the grid. The 13:26Z panel shows the three red dots in new locations, with dashed red lines stretching back to each of their original positions. Three blue dots indicate the location of three other parcels of air, initialized at 13:26Z. They are located on the same lines of latitude as the 12:00Z red dots, but each are positioned at a different longitude. The zonal displacement is the same in each case. In the 15:12Z panel, the three red dots have moved even more, and the red dashed lines trace their paths back to their initial points. The blue dots have also moved and blue dashed lines trace their paths back to their initial locations. Three green dots are initialized along the same lines of latitude, but again removed by some constant zonal (longitudinal) amount. In all of this, we have created air parcel trajectory maps that let us trace back different air parcels to some origin. Though different air parcels are initialized at different times, the locations are chosen so that the new parcels are on the same lines of latitude displaced by some constant zonal distance. This allows us to keep track of different air parcels originating at different times along the same latitude but displaced by some constant longitude at each time step.

(1) Mixing time scales and air parcel integrity -- Like the photochemical box model, the trajectory model assumes that the integrity of the air parcel is maintained throughout its travels along its projected path. We know that over sufficiently long time scales, the atmosphere certainly mixes. Think about how long you can detect a puff of smoke after it leaves the smoker's mouth. Why does it eventually disappear? Mixing processes in the air around the smoker's head eventually reduce the local concentration of smoke to the point at which it is no longer visibly detectable. If we ran this example through the trajectory model, the model would predict we could follow the smoke along a specific path for a very much longer time, clearly inconsistent with what our eyes tell us. It is therefore important to realize that small scale atmospheric motions (like mixing) will limit the length of time a trajectory calculation provides meaningful information. Fortunately, the spatial scales over which ozone varies and the dynamics of the stratosphere permit these models to be used for substantially longer periods of time than the few seconds over which the smoke ring will be visible.

(2) Spatial scales and air parcel integrity -- The spatial scales under consideration also influence the accuracy of the trajectory calculations. Previous studies have shown that small scale variability can lead to an almost immediate uncertainty in trajectory calculations. However, large scale features -- like the winter polar vortex or planetary long waves (see Chapter 2) -- are considerably more resilient in trajectory calculations, and maintain their integrity for periods of months. Simulating the motions and chemistry inside such large scale features gives the model output more credibility as a predictive tool. Like any tool used to study the atmosphere, trajectory models have their limitations. It is important to be aware of these limitations before applying them to gain an understanding of real atmospheric problems.

b. Uses -- Trajectory models are useful in several ways when studying atmospheric photochemical processes and transport. Trajectory models are used to

(1) Back trajectories: history of an air parcel -- Knowing the past history of an air parcel (i.e., knowing where it came from) is an important component in interpreting the observations that have been made of that parcel. For instance, observations of ClO were routinely made by an instrument on board the NASA ER-2 aircraft during a mission to the northern polar region in the winter of 1991-92. This radical species plays a key role in the catalytic cycle of polar stratospheric ozone depletion (see Chapters 5 and 11). Sometimes the ER-2 observations showed low amounts of ClO, while at other times the observations showed high ClO. In order to explain the ClO variability, air parcel histories were found using the trajectory model. Back trajectories are the calculated path an air parcel followed over the previous several days as calculated backward in time from the time of the observation. These were calculated for both low ClO and high ClO air parcels. The study indicated the important role of temperature in determining the amount of ClO observed. Figure 12.08 shows observations of ClO concentration plotted against a temperature difference. The temperature difference incorporates a model-calculated minimum temperature experienced by the parcel along its back trajectory over the course of 10 days. The data was measured by the ER-2 flight.

The figure demonstrates that in most of the parcels with low amounts of ClO, the parcels over the previous ten days had never experienced temperatures colder than 195 Kelvin. The parcels with high ClO values had experienced temperatures colder than 195 K. This result is an important one in that polar stratospheric clouds (PSCs) first form at such temperatures. These clouds provide surfaces on which the heterogeneous chemical reactions can occur. Thus, the back trajectory calculation suggests that the formation of PSCs in the Arctic led to elevated levels of ClO by providing the catalytic surface necessary for the chemical reactions.

(2) Mixing and transport properties -- A second important use of trajectory models is in the study of atmospheric mixing and transport properties. A good example is the study of the interaction between the winter polar vortex and the surrounding midlatitude air. The boundary between the vortex and the midlatitudes can be defined in terms of potential vorticity (PV), as discussed in Chapter 2. PV is a quantity that can be thought of as a tracer of atmospheric motion, since it tends to be well conserved along an air parcel's trajectory. Parcels with very high PV magnitudes are typically found within the polar vortex, while parcels with low PV are found outside of the vortex. The edge of the polar vortex is usually found where the zonal and/or meridional gradient of PV is high.

It is possible to follow the evolution of the vortex by tracking the motion of air parcels that are initially located along the vortex edge. When connected together, this collection of air parcels, all containing the same PV, form a contour around the polar vortex. Tracking the evolution of this PV contour is known as contour advection, since the contours move (advect) in time. Figure 12.09 shows the evolution of the Arctic winter vortex between February 28, 1993 and March 10, 1993 in terms of contours of constant PV at the 850 K level or 10 mb surface (about 32 km altitude) in the middle stratosphere. As this is a back trajectory, the model shows how filaments of air trace backwards in time from March 10 to February 28.

As the calculation proceeds, filaments of the polar vortex are stretched into midlatitude air, while midlatitude air becomes wrapped up into the vortex. This application of the trajectory model allows us to watch atmospheric mixing processes at work.

(3) Synoptic versus asynoptic maps -- Atmospheric scientists often wish to examine the global distribution of an atmospheric constituent like ozone. Ozone observations, however, are not made the same way as the images of cloud distributions that you see on your local weather forecast, such as the example found in Figure 12.10.

Images such as that in Figure 12.10 are described as providing synoptic views of some atmospheric quantity, in this case, cloud distribution, because they provide a picture of what is going on over a large region at one instant of time. The picture can be a weather map or a contour map or even a photograph, such as the satellite picture of cloud distribution in the figure.

Producing a synoptic map of the global distribution of a trace species in the atmosphere is relatively complicated. Current technology does not permit us to produce 3-D images of the distribution of trace species from geostationary satellites. Instead, data on trace species are usually taken by low orbiting satellites that circle Earth many times per day. Over the course of a day (24 hour period), such satellites may be able to measure ozone over most of the globe, but in order to compile such a set of observations, it takes 24 hours. This is not the instantaneous synoptic view we would like to have. In addition, these observations are relatively narrow in scope geographically (typically covering less than 400 km on the ground). Figure 12.11 shows a map of such satellite measurements of ozone distribution near 30 km altitude (800 K surface) in the stratosphere. About 1200 observations of ozone were made over the 24-hour period on February 15, 1992, by the SBUV instrument. The observations have been put on the map in the location at which they were made.

As stated, this map was made over the course of a 24 hour period. The measurements do not take into account any of the meteorological, dynamical, or photochemical processes that are ceaselessly occurring in the atmosphere. Such maps as shown in Figure 12.11 are called asynoptic maps: maps that consist of data gathered over an extended period of time without concern for the dynamic variation in the atmosphere.

(4) Transforming asynoptic maps into synoptic maps using trajectory models -- We would like to transform the asynoptic ozone data in Figure 12.11 into synoptic data, like the cloud distribution found in Figure 12.10. Fortunately, trajectory models provide a good approach to this problem. The procedure is straightforward. For each ozone observation, an air parcel is initialized in the trajectory model. The model is run forward until the time of the next observation, at which point another air parcel is added to the model (as was done in Figure 12.07, see above description for trajectory models). In addition, the model is run backward in time until the time of the previous observation, at which point another air parcel is added to the model. If we want the map for a particular day, D, we can run the model forward from D-N and the model backward from D+N. The process is repeated until the desired synoptic map product is produced for day D. This is what is done in the next figure.

Figure 12.12 shows the resulting ozone distribution for the same 800 K surface or about the 30 km altitude for February 15, 1992, after having run a trajectory model on the SBUV ozone observations from February 12-18, 1992. We are interested in a map for February 15, 1992. Observations from February 12 are run forward three days, observations from February 13 are run forward two days, and observations from February 14 are run forward one day. The observations from February 15 (shown in Figure 12.11) are then added. Next, the observations from February 18 are run backward three days, the observations from February 17 are run backward two days, and the observations from February 16 are run backward one day. Thus, the map was produced by running a trajectory model forward and backward, so that all the ozone data points are at their 1200Z February 15, 1992 position, the time of the asynoptic map shown in Figure 12.11.

The resulting map is a synoptic map: it shows what is going on in ozone distribution at the 800 K level globally at one instant of time (1200Z February 15, 1992).

The resulting data shows that high values of ozone are generally found in the tropics, low values in the polar regions, and middle values in the midlatitudes. In addition, intrusions of tropical ozone values to high latitudes in the northern hemisphere and the comma shaped vortex of low ozone air extending as an elongated filament across the North Atlantic into northern Europe provide clues about the nature of northern hemisphere transport and mixing at this altitude.

3.2.3 The zonal mean or 2-D model -- a. Description -- The zonal mean of some variable (such as temperature, ozone concentration, wind speed, et al.) is derived by taking an average of the variable around a circle of constant latitude. That is, the latitude is fixed and we average through all longitudes for which we have available data for the variable. For many stratospheric trace gases, their zonal (east-west) variations in concentrations are smaller than their vertical or meridional (north-south) variation. Owing to this fact, the zonal mean value of the constituent is a good approximation to its actual value at a particular longitude (though significant departures from the zonal mean field in certain variables exist; see below). Therefore, it is reasonable to develop a model which neglects zonal (longitudinal) variability and only calculates constituent concentrations as a function of latitude and altitude. Such a model is referred to as a zonal mean or two-dimensional (2-D) model.

(1) Radiation and chemistry modules of a 2-D model -- The radiation module incorporated by a 2-D model calculates the interaction of sunlight with the atmosphere to determine the radiation field throughout the model. In this way, the model determines the photolysis coefficients for the constituents it describes. It includes a chemistry module to calculate the time evolution of all the species in its chemical mechanism (i.e., the set of chemical reaction equations built into the model).

(2) Dynamics module of a 2-D model -- The 2-D model also includes a dynamics module to transport the gas constituents both meridionally and vertically. Because a 2-D model is defined in the vertical and meridional plane, a reasonable representation of atmospheric transport processes in these planes had to be developed. A considerable amount of effort over the past thirty or so years has been devoted to this task.

Both the gas constituents and the wind fields in the atmosphere can be described as the sum of two components: the zonal mean (which is a function of latitude and altitude) and the deviation from the zonal mean (which is three-dimensional). A deviation in the wind field from the zonal mean value is referred to as an eddy. Both the zonal mean and eddy components of the wind field transport constituents from place to place in the atmosphere. While constituent deviations from the zonal mean values are small and can be neglected, eddy driven transport is significant and thus cannot be neglected. The central problem in developing two-dimensional transport fields is determining how to represent the eddy transport terms.

(3) Representing eddy transport processes: diffusion versus advection -- The first 2-D models attempted to represent eddy transport entirely as a diffusive process. Diffusive transport is a slow, molecular-level process in which individual molecules pass along heat and momentum one to another. Diffusive transport of a trace gas depends on how the trace gas concentration changes over some region. This change over a distance is referred to as a gradient. Thus, the trace gas gradient describes how the concentration of a particular trace gas changes in space.

Trace gas transport by diffusion is characterized by a motion (called a flux) that is proportional to the gradient of the trace gas concentration. The coefficient of proportionality is known as the diffusion coefficient. The steeper the gradient, the larger the flux of the trace gas from a region of high concentration to a region of low concentration.

Diffusive transport can be contrasted with advective transport, where the trace gas flux is proportional to its concentration, not to its concentration gradient. This means that in advection, there is no requirement that a trace gas be transported from a region of high concentration into a region of low concentration. Advection is a fundamentally different process than diffusion.

The early zonal mean circulation models were unsuccessful in representing transport processes in the atmosphere because they did not satisfactorily account for eddy motion. Eddy motions in the real atmosphere transport physical quantities, including heat, moisture, momentum, and trace gas constituents both advectively and diffusively. This was recognized in the early 1980s and 2-D transport models were devised which accounted for the advective component of eddy motions. The result is something called a residual circulation. (This is an example of a conceptual breakthrough. It has nothing to do with the computational power of the computer running the model.) The Residual Circulation models produced much better results than the earlier models (known as Eulerian Mean models) and have since replaced them as tools of atmospheric scientists. The most modern 2-D models are still Residual Circulation models. 

b. Uses -- Two-dimensional mean circulation models have several important applications in atmospheric science research. They are used to

(1) Comparisons of observations to model-predicted CFC-12 distribution -- A 2-D model can be used to predict the zonal mean structure of all of the constituents in its chemical mechanism. A basic use of this type of model thus involves comparison of the model-predicted constituent distributions with any existing observations. For instance, Figure 12.13 compares observations of the manmade chlorofluorocarbon CFC-12 (whose chemical representation is given by CF2Cl2) with predictions made by the Goddard Space Flight Center (GSFC) 2-D model. The CFC-12 measurements were made by the CLAES instrument aboard the UARS satellite. Recall that chlorofluorocarbons are source gases that provide the chlorine which destroys stratospheric ozone.

As Figure 12.13 shows, the model predicted distribution CFC-12 shows many similarities with the observed distribution, as well as some differences. The good agreement between the modeled and measured distributions suggests that the GSFC 2-D model is a reasonable model representation of both the photochemical loss and atmospheric transport processes which together determine the CFC-12 distribution. As noted, CFC-12 is a primary source of stratospheric inorganic chlorine (Cly). If the model were unable to correctly describe CFC-12, then it would be similarly unable to correctly describe Cly concentrations and hence the effect of Cly on ozone.

(2) Limitation of 2-D models: the ClO overprediction problem -- Sometimes comparisons of model predictions and observations show significant discrepancies. For instance, Figure 12.14 shows model predicted, zonal mean distribution with height of ClO (chlorine monoxide) concentrations compared with actual ClO observations. The observations were made by the UARS satellite MLS instrument for August-October, 1992.

The figure shows a significant model overprediction of ClO in the 1 to 10 millibar region. Overprediction of ClO in this region is a well-known modeling problem, and is due to the omission in the models of a particular reaction that converts ClO to HCl. The existence of this reaction was shown in the past couple years by researchers (Lipson, et al., 1997). Future models will take this reaction into account.

(3) Consequences of overprediction -- The consequences of this overprediction are very important. This is because in overpredicting ClO, a key control to stratospheric ozone levels, the model predicted ozone distribution will be in error. Figure 12.15 compares the model predicted ozone distribution as a function of latitude and altitude with observations from the SBUV-2 satellite.

As we see in the bottom panel of Figure 12.15 where the differences in the satellite measurements and the model predictions are plotted, the comparison is quite good. Differences between the observed and modeled values not exceeding 15%. However, note that the modeled ozone mixing ratio is smaller than the observed values in the same 1 to 10 mb location that the model overpredicts ClO. When adjustments are made to the model chemistry to correct the overprediction of ClO, the discrepancy between modeled and measured ozone in this region disappears. The use of 2-D models in the manner shown above both tests the theory of stratospheric ozone photochemistry and elucidates how the various atmospheric processes combine to produce the concentrations of ozone observed in the stratosphere.

(4) Advantages of the 2-D model -- One of the primary advantages of a 2-D model is its computational economy compared to trajectory and 3-D global models (i.e. ones in which longitudinal variability in trace gas constituent concentrations is not neglected). Because 2-D models are inexpensive to run, they can be run multiple times or for many year predictions. Thus, 2-D models are often used to study the sensitivity of the model to changes in assumptions about atmospheric processes that are not well known. In summary, the 2-D models allows us to make sensitivity studies, make assessment studies, and to determine long-term trends.