Modeling of Atmospheric Chemistry
Page 7
Figure 2.2 Global annual mean energy budget of the Earth for the 2000–2004 period. Units are W m–2.
From Trenberth et al. (2009). Copyright © American Meteorological Society, used with permission.
2.3 Vertical Structure of the Atmosphere
Figure 2.3 shows the mean vertical profile of atmospheric temperature. Atmospheric scientists partition the atmosphere vertically on the basis of this thermal structure. The lowest layer, called the troposphere, is characterized by a gradual decrease of temperature with height due to solar heating of the surface. It typically extends to 16–18 km in the tropics and to 8–12 km at higher latitudes. It accounts for 85% of total atmospheric mass. Heating of the surface allows buoyant motions, called convection, to transport heat and chemicals to high altitude. During this rise the water cools and condenses, leading to the formation of clouds. The process of condensation releases heat, providing additional buoyancy to the rising air parcels that can result in thunderstorms extending to the top of the troposphere. The mean decrease of temperature with altitude (called the lapse rate) in the troposphere is 6.5 K km–1, reflecting the combined influences of radiation, convection, and the latent heat release from water condensation.
Figure 2.3 Mean vertical profile of air temperature and definition of atmospheric layers.
From Aguado and Burt (2013), Copyright © Pearson Education.
The top of the troposphere, defined by a temperature minimum (190–230 K), is called the tropopause. The layer above, called the stratosphere, is characterized by increasing temperatures with height to reach a maximum of about 270 K at the stratopause located at 50 km altitude. This warming is due to the absorption of solar UV radiation by ozone. A situation in which the temperature increases with altitude is called an inversion. Because heavier air is overlain by lighter air, vertical motions are strongly suppressed. The stratosphere is therefore very stable against vertical motions. Exchange of air with the troposphere is restricted, and vertical transport within the stratosphere is very slow. The residence time of tropospheric air against transport to the stratosphere is 5–10 years, and the residence time of air in the stratosphere ranges from a year to a decade. In summer, the zonal (longitudinal) mean temperature distribution in the stratosphere is determined primarily by radiative processes (solar heating by ozone absorption and terrestrial cooling by CO2, water vapor, and ozone emission to space). In winter, radiation is weaker and the radiative equilibrium is perturbed by the propagation of planetary waves. This generates a large-scale meridional (latitudinal) circulation, called the Brewer–Dobson circulation, transporting air from low to higher latitudes.
The mesosphere extends from 50 km to the mesopause located at approximately 90–100 km altitude, where the mean temperature is about 160 K (120 K at the summer pole, which is the lowest temperature in the atmosphere). In this layer, where little ozone is available to absorb solar radiation, but where radiative cooling by CO2 is still effective, the temperature decreases again with height. Turbulence is frequent and often results from the dissipation of vertically propagating gravity waves (see Section 2.11), when the amplitude of these waves becomes so large that the atmosphere becomes thermally unstable.
The thermosphere above 100 km is characterized by a dramatic increase in temperature with height resulting primarily from the absorption of strong UV radiation by molecular oxygen O2, molecular nitrogen N2, and atomic oxygen O. Collisions become rare so that a stable population of ions can be sustained, producing a plasma (ionized gas). The temperature above 200 km reaches asymptotic values of typically 500 to 2000 K, depending on the level of solar activity (Figure 2.4). This asymptotic behavior reflects the small heat content and the high heat conductivity of this low air density region. The corresponding altitude is called the thermopause and varies from 250 to 500 km altitude. Atmospheric pressure is sufficiently low above 100 km that vertical transport of atmospheric species occurs primarily by molecular diffusion. This process tends to separate with height the different chemical species according to their respective mass. As a result, the relative abundance of light species like atomic oxygen, helium, and hydrogen increases with height relative to species like molecular nitrogen and oxygen. Molecular nitrogen dominates up to 180 km, while the prevailing constituent between 180 and about 700 km is atomic oxygen. Helium is the most abundant constituent between 700 km and 1700 km, and atomic hydrogen at higher altitudes. Above the thermopause, atoms follow ballistic trajectories because of the rarity of collisions. In this region of the atmosphere, light atoms (hydrogen) can overcome the forces of gravity and escape to space if their velocity is larger than a threshold value (escape velocity). At that point the atmosphere effectively merges with outer space.
Figure 2.4 Vertical distribution of the mean temperature for two levels of solar activity with emphasis on the upper atmosphere layers.
Adapted from Banks and Kockarts (1973).
Air motions below 100 km are dominated by gravity and pressure forces, following the laws of hydrodynamics. Above 100 km, where ionization produces a plasma, the flow is affected by electromagnetic forces, and more complex equations from magneto-hydrodynamics must be applied. Aeronomy is the branch of science that describes the behavior of upper atmospheric phenomena (with emphasis on ionization and dissociation processes), while meteorology refers to the study of the lower levels of the atmosphere (with emphasis on dynamical and physical processes). The aeronomy literature has its own classification of atmospheric layers (see e.g., Prölss, 2004). For example, it refers to the troposphere as the lower atmosphere, to the stratosphere and mesosphere as the middle atmosphere, and to the thermosphere as the upper atmosphere. It defines the homosphere below 100 km as the region where vertical mixing is sufficiently intense to maintain constant the relative abundance of inert gases, and the heterosphere above 100 km as the region where gravitational settling becomes sufficiently important for the relative concentration of heavy gases to decrease more rapidly than that of lighter ones. The atmospheric region above 1700 km is often called the geocorona. It produces an intense glow resulting from the fluorescence of hydrogen excited by the solar Lyman-α radiation at 122 nm. In another nomenclature, one distinguishes between the barosphere, where air molecules are bound to the Earth by gravitational forces, and the exosphere in which the air density is so small that collisions can be neglected. The lower boundary of the exosphere, called the exobase, is located at 400–1000 km. Aeronomers refer to the ionosphere as the atmospheric region where ionization of molecules and atoms by extreme UV radiation (less than 100 nm) and energetic particle precipitation is a dominant process. Different ionospheric layers are distinguished: the D-region below 90 km altitude, the E-region between 90 and 170 km, the F-region between 170 and 1000 km, and the plasmasphere above 1000 km altitude. The region in which the magnetic field of the Earth controls the motions of charged particles is called the magnetosphere. Its shape is determined by the extent of the Earth’s internal magnetic field, the solar wind plasma, and the interplanetary magnetic field.
2.4 Temperature, Pressure, and Density: The Equation of State
The state of the atmosphere is described by pressure p [Pa], temperature T [K], and chemical composition. Atmospheric pressure is sufficiently low that the ideal gas law is obeyed within 1% under all conditions. The equation of state can therefore be expressed as
pV = RT
(2.2)
where V [m3 mol–1] represents the molar volume of air, and is the universal gas constant. When expressed as a function of the number density [molecules m–3], where molecules mol–1 is Avogadro’s number, this expression becomes
p = nakT
(2.3)
where is Boltzmann’s constant.
The equation of state can also be expressed as a function of the mass density of air
p = ρaRT
(2.4)
where is the specific gas constant for air and Ma [kg mol–1] is the molar mass of air.
The molar mass of air is the we
ighted average of the mass of its components
(2.5)
where Ci and Mi are respectively the mole fraction (commonly called molar or volume mixing ratio) and the molar mass of constituent i. Since dry air can be closely approximated as a mixture of nitrogen N2 (with CN2 = 0.78), oxygen O2 (with CO2 = 0.21) and argon Ar (with CAr = 0.01), the molar mass for dry air is Md = 28.97 × 10–3 kg mol–1. Water vapor, which can account for up to a few percent of air in the lower troposphere, will make air slightly lighter.
2.5 Atmospheric Humidity
Because of the high variability of water vapor in air, meteorologists like to use separate equations of state for dry air and water vapor. This is legitimate following Dalton’s law, which states that the total pressure of a mixture of gases is the sum of the partial pressures of its individual components. The equation of state for dry air is given by
pd = ρdRdT
(2.6)
where the specific gas constant for dry air is . A similar equation can also be applied to water vapor (or any chemical constituent). The partial pressure of water vapor is commonly noted e and the equation of state is expressed by
e = ρwRwT
(2.7)
where ρw is the water vapor mass density [kg m–3] and is the specific gas constant for water vapor with a molar mass Mw of 18.01 × 10–3 kg mol–1. Note that the total air pressure is p = pd + e. The volume mixing ratio Cw and mass mixing ratio μw of water vapor are expressed by
(2.8)
and
(2.9)
where nw and na are the number densities [molecules m–3] of water and moist (total) air, respectively, Mw = 18.01 × 10–3 kg mol–1 is the molar mass of water, and Ma is the molar mass of moist air: Ma = (1 – Cw)Md + CwMw. Meteorologists conventionally call μw the specific humidity (and write it q). They instead define the water vapor mass mixing ratio rw as the ratio between the water vapor density ρw and the dry air density ρd, where ρd = ρa – ρw:
(2.10)
The equation of state (2.6) for dry air can be applied to moist air if the temperature T is replaced by the virtual temperature Tv, the temperature at which dry air has the same pressure and density as moist air. Thus one writes
p = ρaRdTv
(2.11)
From the above equations, it follows that
or
(2.12)
with Rd/Rw = Mw/Md = 18/28.97 = 0.621. A good approximation to this expression is provided by
Tv ≈ (1 + 0.61 rw)T
(2.13)
Phase transitions of atmospheric water play a crucial role in meteorology. The relative humidity RH [percent] is expressed by
(2.14)
where es is the saturation pressure at which water vapor is in equilibrium with the condensed phase (liquid or ice). For a saturated atmosphere (e = es), condensation and evaporation are in balance. One shows easily that the water mass mixing ratio corresponding to saturation is
(2.15)
Its value is inversely proportional to the total pressure and is a function of temperature because the saturation pressure es varies with temperature (see later). An atmosphere with e < es is called subsaturated while one with e > es is called supersaturated. A supersaturated atmosphere leads to cloud formation, contingent on the presence of suitable aerosol particles to provide pre-existing surfaces for condensation and overcome the energy barrier from surface tension. These particles are called cloud condensation nuclei (CCN) for liquid-water clouds and ice nuclei (IN) for ice clouds. Water-soluble particles greater than 0.1 μm in size are adequate CCN, and are sufficiently plentiful that liquid cloud formation takes place at supersaturations of a fraction of a percent. Ice nuclei are solid particles such as dust that provide templates for ice formation and are present at much lower concentrations than CCN. Because of the paucity of IN, clouds may remain liquid or mixed ice–liquid at temperatures as low as –40 °C; one then refers to the metastable liquid phase as supercooled.
Phase equilibrium for water is defined by the phase diagram in Figure 2.5. Lines on this diagram represent equilibria between two phases. Equilibrium between vapor and condensed phases is expressed by the Clausius–Clapeyron equation
(2.16)
where L represents the latent heat of vaporization or sublimation [J kg–1]. Integration between a reference temperature T0 and temperature T yields
(2.17)
where L can be approximated as constant. For vaporization of liquid water, L has a value of 2.50 × 106 J kg–1 at 0 °C and varies with temperature TC (in degrees Celsius) as (Rogers and Yau, 1989):
(2.18)
For sublimation at 0 °C, L is 2.83 × 106 J kg–1. Latent heat is released to the atmosphere (warming) when clouds condense from the gas phase; conversely, latent heat is absorbed from the atmosphere (cooling) when clouds evaporate.
Figure 2.5 Phase diagram for water describing the stable phases present at equilibrium as a function of water vapor pressure and temperature. The thin line represents the metastable equilibrium between gas and liquid below 0 °C.
Reproduced from Jacob (1999).
The thin line in the phase diagram of Figure 2.5 represents the metastable phase equilibrium between water vapor and liquid water at temperatures below 0 °C. This equilibrium is relevant to the atmosphere because of supercooling of liquid clouds. When ice crystals do form in such clouds, the water vapor at equilibrium with the ice is lower than that at equilibrium with the supercooled liquid; thus, the liquid cloud droplets evaporate, transferring their water to the ice crystals. This transfer of water can also take place by collision between the supercooled liquid cloud droplets and the ice crystals (riming). In either case, the resulting rapid growth of the ice crystals promotes precipitation. The heat release associated with the conversion from liquid to ice also adds to the buoyancy of air parcels, fostering further rise and additional condensation and precipitation. Such precipitation formation in mixed-phase clouds is known as the Bergeron process.
2.6 Atmospheric Stability
2.6.1 The Hydrostatic Approximation
The vertical variation of atmospheric pressure can be deduced from hydrostatic equilibrium,
(2.19)
which expresses that the downward gravitational force acting on a fluid parcel is balanced by an upward force exerted by the vertical pressure gradient that characterizes the fluid. Here g = 9.81 m s–2 is the acceleration of gravity, ρa [kg m–3] the mass density of air, p [Pa] the atmospheric pressure, and z [m] is the altitude above the surface, often referred to as the geometric altitude. Equation (2.19) assumes that the fluid parcel is in vertical equilibrium between the gravitational and pressure-gradient forces, or more broadly that any vertical acceleration of the air parcel due to buoyancy is small compared to the acceleration of gravity. This is called the hydrostatic approximation. It is a good approximation for global models, but not for small-scale models attempting to resolve strong convective motions.
Because the atmosphere is thin relative to the Earth’s radius (6378 km), g can be treated as constant with altitude. From the ideal gas law, we can rewrite equation (2.19) as
(2.20)
where
(2.21)
is a characteristic length scale for the decrease in pressure with altitude z and is called the atmospheric scale height. Its value is 8 ± 1 km in the troposphere and stratosphere. Here k = 1.38 × 10–23 J K–1 is Boltzmann’s constant, is the mean molecular mass of air (28.97 × 10–3/6.022 × 1023 kg = 4.81 × 10–26 kg), and R = 287 J K–1 kg–1 is the specific gas constant for air. By integrating (2.20), one finds the vertical dependence of atmospheric pressure
(2.22)
where p(0) is the surface pressure. Approximating H as constant yields the simple expression
(2.23)
which states that the air pressure decreases exponentially with altitude. Equation (2.23) is called the barometric law.
The hydrodynamic equations of the atmosphere are often expressed by using the pressure p rather than the altitude z as
the vertical coordinate. It is then convenient to define the log-pressure altitude Z
(2.24)
as the vertical coordinate. Here is a constant “effective” scale height (specified to be 7 km) and p0 is a reference pressure (specified to be 1000 hPa). Thus Z depends solely on p. It is also convenient to introduce the geopotential Φ as the work required for raising a unit mass of air from sea level to geometric altitude z:
(2.25)
where g includes dependences on altitude and latitude, the latter due to the non-sphericity of the Earth (Section 2.7). g at Earth’s surface varies from 9.76 to 9.83 m s–2. The hydrostatic law relates Φ to . The geopotential height is defined as Φ/go, where go = 9.81 m s–2 is a constant called the standard gravity. Movement along a surface of uniform geopotential height involves no change in potential energy, i.e., no conversion between potential and kinetic energy. Meteorological weather conditions aloft are often represented as contour maps of geopotential heights at a given pressure. As we will see in Section 2.7, air motions tend to follow contour lines of geopotential heights, so this type of map is very useful.