The Physics of Energy, page 144
In Earth’s actual atmosphere, greenhouse gases such as water vapor and CO2 are distributed through the atmosphere. These greenhouse gases are not perfect absorbers of all infrared radiation. Rather, they absorb certain frequencies very effectively and allow other frequencies of infrared radiation to pass through. The dependence on the number of levels in the simplified n-level model given in Example 34.2 is unrealistically strong because the gaps in the real absorption spectrum allow certain frequency ranges of infrared radiation to escape freely through the atmosphere (Problem 34.3). To obtain a more realistic picture of the radiative equilibrium of the atmosphere, we should include information about the details of the atmospheric constituents and their absorption spectra as well as their distribution in the atmosphere. This distribution, in turn – as well as the flow of radiation from the surface – depends upon the temperature and pressure profiles of the atmosphere, which vary from place to place on the globe. In particular, the distribution of the most significant greenhouse gas, water vapor, also depends crucially on non-radiative heat transfer through convection, giving a complex coupling between the radiative trapping of heat through greenhouse gases and other heat transport mechanisms.
We therefore would like to have a more detailed model of Earth’s atmosphere and the large-scale distribution and transport of thermal energy in Earth’s climate system. We proceed in §34.2 to develop some of the basic principles of atmospheric physics. This will enable us to describe in more detail the flow of energy in the vertical direction, governing Earth’s radiative equilibrium. In §34.3 we describe some basic aspects of large-scale global energy flow in the horizontal directions across the surface of the planet. Any detailed understanding of Earth’s climate and temperature must incorporate all of these aspects of global energy flow.
34.2Atmospheric Physics
In this section we analyze the vertical structure of the atmosphere, treating the atmosphere as a one-dimensional system. We describe the equilibrium pressure and density profiles of the atmosphere and the role of various atmospheric constituents in the vertical structure of the atmosphere in radiative equilibrium. The vertical temperature gradient, called the lapse rate, and its connection to convective instability are discussed. This leads to a one-dimensional model of the radiative–convective equilibrium of the atmosphere. The notion of radiative forcing is developed as a key concept for comparing quantitatively the effects of various phenomena on Earth’s net radiative equilibrium.
34.2.1 Hydrostatic Equilibrium
The pressure of the planetary atmosphere arises from the weight of the column of gas above the point where the pressure is measured.2 In equilibrium, the vertical change in pressure over a small height differential dz (Figure 34.3) is given by the gravitational force on the volume of gas contained within that height differential
(34.5)
where ρ is the atmospheric mass density. To proceed we make the simplifying assumption that the atmosphere is an ideal gas composed of identical molecules of mass . This is a relatively good approximation, because the atmosphere is mostly oxygen () and nitrogen () with similar masses and the relative concentration of N and O is close to 4:1 throughout the atmosphere, so that most properties – such as density – can be accurately estimated by taking a weighted average over these primary atmospheric constituents. Assuming identical molecules of mass m, the ideal gas relation can be used to relate pressure to density through
(34.6)
where . It follows that
(34.7)
Combining eq. (34.5) and eq. (34.7) gives
(34.8)
where is the (temperature-dependent) scale height of the atmosphere.
Figure 34.3 In hydrostatic equilibrium, the change in pressure over a height interval dz must compensate the force of gravity on the mass of air in that interval.
Equation 34.8 can be solved for the pressure at altitude z in terms of the pressure at height ,
(34.9)
In a region of constant temperature, H is constant and we have
(34.10)
In this case, the scale height is the height over which the atmospheric pressure drops by a factor of . Near Earth’s surface, estimating for example K, we have km. The density can similarly be obtained from eq. (34.7),
(34.11)
in a region of constant temperature. This is precisely the density profile that would be obtained from the Boltzmann distribution for the molecules in a gas at constant temperature T in a gravitational potential (see §8.6.1). In general, the temperature T – and therefore the scale height H – depends on z, and we must use the more general formula (34.9).
Hydrostatic Equilibrium
In hydrostatic equilibrium the pressure at any point in the atmosphere is determined by the weight of the column of air above a unit area parallel to the surface. In the approximation that the atmosphere consists of a single molecular species with mass m in hydrostatic equilibrium, the pressure varies with the height z as
where is the scale height. In a region of constant temperature, pressure varies as
From eq. (34.8), it follows that pressure is a decreasing function of height. Since many physical properties of the atmosphere depend primarily upon pressure, p is often used as a convenient proxy for height.
34.2.2 Atmospheric Temperature and the Lapse Rate
The temperature of the atmosphere is not constant but, at least in the lower atmosphere, generally decreases with increasing altitude. The rate at which T decreases is known as the lapse rate Γ, defined as
(34.12)
Depending upon the lapse rate, the vertical temperature profile of the atmosphere can lead to convective instability. In general, if the temperature drops too quickly with altitude, the warmer air from below tends to rise and reduce the lapse rate. To understand this in a quantitative fashion, consider a parcel of air that moves upward adiabatically, i.e. without heat exchange with the surrounding air. Then the first law of thermodynamics states that
(34.13)
Since p decreases with height, if a packet of air moves upward, it expands and dV is positive, as is . Thus from eq. (34.13), the change in temperature dT is negative. Suppose that the decrease in temperature of the packet is less than the actual temperature change of the atmosphere. From the ideal gas law, a warmer packet of air at the same pressure as its surroundings will have lower density, and therefore will continue to rise, so convection will continue. If, on the other hand, the atmosphere cools more slowly with height than an adiabatically rising air packet, then the atmosphere will be stable against convection. The adiabatic lapse rate is defined to be the lapse rate such that the temperature change in a parcel of air moving upward adiabatically matches the change in temperature of the surrounding air. This determines the maximum rate at which the temperature can decrease vertically without convective instability.
To compute the adiabatic lapse rate, first take the derivative of the ideal gas law
(34.14)
then combine this with eq. (34.13) and eq. (34.5) to obtain
(34.15)
The dry adiabatic lapse rate (we have ignored water vapor) is then
(34.16)
Near Earth’s surface at K the specific heat capacity of dry air is
(34.17)
so the dry adiabatic lapse rate is
(34.18)
When the surface of Earth is heated by solar radiation, thermal energy is transferred to the air just above the surface. This produces a temperature gradient in the vertical direction. When this temperature gradient is larger than the adiabatic lapse rate , convection occurs and thermal energy is transferred upward until the temperature gradient is reduced to (or below) the adiabatic value. Thus, in the lower part of Earth’s atmosphere called the troposphere – generally up to a height of 9 to 17 km – the temperature generally falls with altitude,3 and convection makes an important contribution (roughly 60% [173]) to the upward transport of energy. The energy transported in this fashion includes not only the thermal energy of dry air but also latent heat in water vapor. Thus, a quantitative understanding of vertical energy flow in Earth’s lower atmosphere involves not only radiative heat transfer but also convection. Convection of energy and water vapor give rise to the rich variety of weather phenomena that characterize the troposphere. The troposphere ends rather abruptly at the tropopause, where, for reasons explained below, the temperature begins to increase with altitude, so that convection and weather diminish dramatically in the region above the tropopause known as the stratosphere.
The adiabatic lapse rate given in eq. (34.18) is the value for dry air. When the air includes water vapor, the situation is slightly more complicated, and convective effects generally determine the vertical distribution. If the air is saturated, then as it rises and the temperature decreases, some of the water vapor will condense, releasing latent heat. This raises the temperature of the parcel of air, and can lead to convective instability at a smaller value of the lapse rate. Thus, for air saturated with water vapor the moist adiabatic lapse rate is smaller. Since convection occurs on shorter time scales than the convergence to radiative equilibrium, the lapse rate in radiative–convective equilibrium is close to the moist adiabatic lapse rate. In the lower atmosphere, the observed lapse rate typically varies between 3℃/km and 10℃/km, with a mean value of around 6.5℃/km.
Lapse Rate
The lapse rate
describes the rate of decrease of temperature with altitude in the atmosphere.
For dry air, the maximum lapse rate at which the atmosphere is stable to convection is the dry adiabatic lapse rate
The observed lapse rate in the lower atmosphere ranges from 3℃/km to 10℃/km, with a mean value around 6.5℃/km.
34.2.3 Atmospheric Composition and Radiative Absorption
While Earth’s atmosphere is primarily composed of nitrogen and oxygen, many other types of molecules appear in trace amounts. The absorption properties of many of these other molecules play an important role in the flow of energy through absorption and radiation within the atmosphere. Some of the most significant atmospheric constituents are listed in Table 34.1.
Table 34.1 Some relevant atmospheric constituents as of 2015 (ppm/ppb = parts per million/billion by volume, see Box 34.3).
Molecule Fraction by volume
Nitrogen (N) 78.08%
Oxygen (O) 20.95%
Argon (Ar) 0.934%
Water vapor (HO) variable
Carbon dioxide (CO2) 400 ppm
Methane (CH) 1.83 ppm
Nitrous oxide (NO) 328 ppb
Ozone (O) variable
In the preceding subsection, we made the approximation that the atmosphere consists of a single type of molecule with a mass equal to the weighted average of the masses of nitrogen and oxygen. Since they make up 99% of the atmosphere this is a reasonable first approximation. In principle, the relative density of different molecules in the atmosphere could depend upon altitude. For an atmosphere in equilibrium without convection and at constant temperature T, the density of each type of molecule would vary as the Boltzmann factor (eq. (34.11)), so heavier molecules generally would be more prevalent in the lower atmosphere. The time period over which convection mixes Earth’s atmosphere is so short, however, even above the tropopause, that the atmospheric composition is actually quite homogeneous for most molecules up to an altitude of about 100 km. Thus, the approximation of treating the atmosphere as a single (ideal) gas with effective mass is an excellent one. Certain constituents such as ozone and water vapor that are added to and removed from the atmosphere by local processes on time scales short compared to the atmospheric mixing time are exceptions to this result. They are generally out of equilibrium over large (~100 km) distance scales and their density distribution is generally not described by eq. (34.11). The density of water vapor in the atmosphere, in particular, is affected by many factors, and depends upon rates of evaporation, convection, and condensation as well as atmospheric circulation.
The absorption spectra of the minority constituents in the atmosphere affect both the downward flow of solar radiation and the upward propagation of infrared radiation. As discussed in §23.4, the diatomic molecules N2 and O2 cannot easily absorb single photons and are effectively transparent to most incoming solar radiation and to outgoing infrared radiation. Oxygen can, however, be dissociated by UV photons of wavelength below 246 nm,
(34.19)
This occurs in the stratosphere at altitudes of ~15 km or more; below this altitude the level of UV radiation is significantly decreased. Ozone is produced when a single oxygen atom resulting from this dissociation combines with an oxygen molecule through a three-body collision with a third molecule M that conserves momentum and energy,
(34.20)
Little ozone is produced above about 60 km since the density of material is so low that three-body collisions of this type are rare.
Ozone in turn is readily dissociated by UV photons in the continuous Hartley band between approximately 200 nm and 300 nm,
(34.21)
as discussed in §23.5. Because ozone is created and then destroyed within the stratosphere, the density of ozone has a local maximum at high altitude, around 25 km, and is far from hydrostatic equilibrium. The absorption of high-energy UV photons in the stratosphere by ozone and oxygen dissociation deposits significant solar energy in the middle atmosphere, heating the stratosphere, so that the temperature actually rises through the stratosphere. This absorption of UV radiation also helps to protect life on Earth, which is easily damaged by UV radiation.
In the troposphere, below roughly 15 km altitude, water vapor, CO2, and other greenhouse gases absorb both incoming and outgoing infrared radiation and can influence the temperature profile of the lower atmosphere, though generally in radiative–convective equilibrium the lapse rate is determined by convection. Water vapor is the most significant atmospheric constituent for radiative absorption in the IR; it has a number of absorption bands that capture incoming solar radiation as well as outgoing thermal radiation from Earth. Because the distribution of water vapor in the atmosphere is not uniform, and is affected by many aspects of climate and weather over short time scales, incorporating water vapor accurately into atmospheric models is complicated. Water vapor enters the atmosphere when it evaporates from the ocean or land surface, or is transpired by plants. Roughly 87% of water vapor comes from the ocean surface; because the rates of evaporation and plant transpiration are difficult to evaluate separately, generally the combined rate of evapotranspiration is measured. Most water vapor stays within the bottom 3 km of the atmosphere, and individual water molecules are resident in the atmosphere for on the order of 10 days before precipitating back to the surface. Some of the large absorption bands of water vapor in the solar wavelength spectrum can be seen in Figure 23.9. The infrared absorption spectra of water and a selection of other atmospheric gases, as well as the aggregate absorption, are depicted in more detail in Figure 34.4. This graph shows the percentage of solar radiation that has been absorbed from radiation that reaches sea level for wavelengths between 700 nm and 70 μm. These absorption bands are associated with vibrational excitations combined with rotational excitations at closely spaced frequencies, which with line broadening (see Example 23.4) produce effectively continuous absorption bands. Water vapor has a number of (primarily vibrational, see §9.3.1) absorption bands in the near-infrared part of the spectrum with wavelengths between 1 μm and 4 μm that absorb incoming solar radiation, as well as a large vibration–rotation band around 6.3 μm in the infrared (associated with the symmetric bend mode in Table 9.4). There is a window from about 8 μm to 12 μm in which solar radiation passes relatively intact through the atmosphere (except for an ozone absorption band around 9.6 μm). Beyond 12 μm, there is a densely spaced set of pure rotation bands for water. Around 15 μm there is a significant vibration–rotation absorption band for CO2 (associated with the vibrational bending mode), which we discuss further in §34.4.
Figure 34.4 Infrared absorption for various atmospheric constituents. (Credit: Robert A. Rohde, Global Warming Art Project, reproduced under CA-BY-SA 3.0 license via Wikimedia Commons)
As discussed in §23.4, radiation of a specific frequency ν passing through a gas or other medium is absorbed at a rate
(34.22)
The absorption coefficient for radiation of a particular wavelength by a particular material or atmospheric constituent can be expressed as the product
(34.23)
where ρ is the density of the absorber and k is the absorption cross section (which has the same units – area per unit mass – and a similar physical interpretation as the cross sections encountered in §18.2). To describe absorption of radiation in the atmosphere, it is therefore sufficient to know the absorption cross section of each atmospheric constituent (as a function of frequency), and the density profile of each as a function of altitude, which can be determined as described in §34.2.1. The optical depth (see §23.4) of the atmosphere in a particular frequency range is then given by
(34.24)
Thus, for example, when the Sun is directly overhead, the flux of solar radiation in a frequency range around ν that penetrates to a height z is
