10.1 The physical and chemical structure of the Atmosphere

10.1.1 Constituents of the atmosphere

In order to study the effect of the atmosphere on the outcoming longwave radiation, it is convenient to subdivide it into a ``clean dry'' component, water vapor, and aerosols (water droplets, as well as ice crystals, salt grains & dust particles, which serve as condensation seeds for water).

10.1.1.0.1 Abundances

Table 10.1: Main constituents of the dry air in the troposphere.

Name	Molec. mass	Normal abund.	Name	Molec. mass	Normal abund.
	amu	(% in volume)		amu	(% in volume)
N$_2$	28.013	78.084	He	4.003	0.0005
O$_2$	32.000	20.946	Kr	83.8	0.0001
Ar	39.948	0.934	CH$_4$	16.043	0.0002
CO$_2$	44.010	0.033$^v$	H$_2$	2.016	0.00005
Ne	20.183	0.0018	N$_2$O	44.013	0.00005

Table 10.1 gives the standard composition of the ``clean dry'' air in the troposphere (i.e at altitudes $\leq 20$ km). Except for CO$_2$, whose abundance at ground level may vary between day and night by up to a factor of 2, this composition is remarkably homogeneous and constant. Other trace components, most of which are unstable (SO$_2$, O$_3$, NO, CO, ...) have abundances (in volume) that never exceeds 10$^{-5}$.

The abundance of water is highly variable, but hardly exceeds 1% in mass, even locally. Most of the water in the air is in the form of vapor. Even inside the clouds, precipitation and turbulence insure that the mass of water droplets per cm$^{-3}$ seldom equals that of water vapor. In addition the water vapor mixing ratios above 15-20 km are under 10 ppmv making it just another trace gas there.

10.1.1.0.2 Overall picture of the atmospheric spectrum

Despite the above facts, water, which has a large absorption cross section in the IR and a large specific heat of vaporization, ( $L_w\sim 600$ cal/g), ozone, which has UV photodisociation bands playing a key role in the stratosphere, and carbon dioxide, which has large IR absorption cross sections, are the major actors of the thermal balance of the air.

Ozone and to much less extent molecular Oxygen are responsible for most of the absorption of the solar radiation in the UV, especially between 180 and 290 nm, thanks to these processes: $O_{3}+h\nu(\lambda<310\;nm) \rightarrow O_{2}(^{1}\Delta_{g})+O(^{1}D)$ (Hartley band), $O_{2}(^{3}\Sigma_{g}^{-})+h\nu(\lambda<\;175nm) \rightarrow O(^{3}P) + O(^{1}D)$ (Schuman-Runge band), and $O_{2}(^{3}\Sigma_{g}^{-})+h\nu(\lambda<\;242nm) \rightarrow O(^{3}P) + O(^{3}P)$ (Herzberg band),

In the visible, the air is fairly transparent except for scattering by aerosols, mostly water droplets, ice crystals and dust particles. In the infrared, H$_2$O, CO$_2$ (around 15 $\mu$m) and O$_3$ (around 10$\mu$m) are very efficient absorbers of the solar and ground radiation, to the extent that they prevent ground-based observations in large regions of the electromagnetic spectrum.

By clear weather, the atmospheric absorption at millimeter and submillimeter wavelengths is dominated by rotational and fine structure lines of molecules in their ground electronic and low vibrational states.

The strongest molecular rotational resonances appear in polar molecules (H$_{2}$O and O$_{3}$ being the most important of such molecules in the atmosphere) and are of the type E1 (electric dipole transitions). Intrinsically weaker M1 (magnetic dipole) transitions are of considerable practical importance in the atmosphere due to the high abundance of O$_2$. We will see that collision induced E2 (electric quadrupole) absorption involving N$_{2}$ and O$_{2}$ is measurable in the atmosphere. The different atmospheric hydrometeors (water droplets, snow, graupel, hale, ice cristals,...) scatter and absorb following different patterns across the longwave (radio to submm) spectral region. All the mechanisms involved in the radiative transfer of longwave radiation in the atmosphere will be described in this chapter.

10.1.2 Thermodynamics of the air

10.1.2.0.1 Gas mixture: Dalton's law

A mixture of ideal gases behaves like an ideal gas:

$$\mathrm{Partial pressures:} p_1 V = {\cal N}_1 k T, p_2 V= {\cal N}_2 k T,   ...$$ (10.1)

$$\mathrm{Total pressure: } p V = (p_1+p_2)V = ({\cal N}_1 + {\cal N}_2) k T+... = {\cal N} kT$$ (10.2)

Dry air is a mixture of N$_2$, O$_2$, ... molecules. It behaves indeed very much like an ideal gas: $R_a=c_{p_a}-c_{v_a}= 8.3143$ J/mol-deg (vs 8.3149 for an ideal gas), $\gamma_a$= 1.404 (vs 1.400 for ideal rigid molecules).

Wet air (without clouds) is a mixture of dry air + H$_2$O molecules. It is customary to denote by $e$ the partial pressure of water vapor, $p_a$ that of dry air, and $p'$ the wet air pressure. The specific heats of water vapor are not that different from those of ideal gases: $c_{v_{w}}= 25.3 + 2 10^{-3} (T-273)$; $\gamma_w= 1.37$, vs $c_v=3R= 24.9$ and $\gamma=\frac{4}{3}$ for a rigid asymmetric top.

Then, Dalton's law yields:

$$c'_v=(1-\frac{e}{p'})c_{v_a}+\frac{e}{p'}c_{v_w} \simeq (1+0.2\frac{e}{p'})c_{v_a}$$ (10.3)

The fractional abundance of water vapor and $\frac{e}{p}$ reaching seldom a few percent, the wet air constants are within a small correction term equal to the dry air constants. In particular, introducing the volume density $\rho$, it is customary to write:

$$p'= \frac{R\rho{'}T}{M'}= \frac{R\rho T'}{M_a}$$ (10.4)

where $T'=\frac{M' T}{M_a}= T (1-0.378\frac{e}{p'})^{-1}$, is the virtual temperature.

Then, for the adiabatic expansion of a wet air bubble, one has:

$$T'= Cst \times p^{m'}$$ (10.5)

$m'$ is equal to $m_a$ within few per mil, so, in practice, the adiabatic curves of dry air can be used for wet air (without clouds), provided one replaces $T$ by $T'$ (the difference could reach a few K and could be important near 0$^\circ$C). In the following, we drop the `prime' signs, except for the virtual temperature $T'$.

10.1.3 Hydrostatic equilibrium

At large scales, the air pressure and density depend essentially on the massive and slowly varying dry component and are well described by hydrostatic equilibrium. The air temperature, as we have seen, depends significantly on the abundance and distribution of water, CO$_2$ (and O$_3$ for the stratosphere).

At equilibrium:

$$dp/dh = -g\rho \;\;\;\;\; p = \frac{\rho R T'}{M_a}$$ (10.6)

where $\rho$ is the density at an altitude $h$, $p$ is the pressure, $T'\simeq T$ the air ``virtual'' temperature. $M_a\simeq{29}$ is the average molecular weight, and $g$ the local gravitational field.

$$\frac{dp}{p}=\frac{-g M_a}{R T'} dh$$ (10.7)

In the ``standard atmosphere'' model, $T'$, the temperature of the air varies linearly with altitude and is given throughout the troposphere (i.e. between $h$=0 and 11 km) by: $T'=T'_o-b(h-h_o)$, where b (in Kkm$^{-1}$) is a constant. Let us first consider a relatively small change in altitude: $h-h_o\leq 1$ km, $T'\simeq T'_{ave}= (T'(h)+T'(h_o))/2$; we find Laplace's hydrostatic formula:

$$\frac{\rho}{\rho_0} \simeq \frac{p}{p_0} \simeq e^{-\frac{g M_a h}{R T'_{ave}}} = e^{-\frac{h}{h_o}}$$ (10.8)

where $\rho_o$ is the density at sea level and $h_o=RT/M_ag=8.4 (T/288)$ km, the scale height. The gas column densities (expressed in g.cm$^{-2}$) along the vertical above sea level ($N_o$) and above a point at altitude h ($N$) are:

$$N_o M_a = \int \rho dh = \rho_o h_o\;\; ; \;\;\;\;\;\;\; N = N_o e^{-h/h_o}$$ (10.9)

For larger altitudes, from Eq.10.7 and $dh = -dT'/b$, then:

$$\frac{dp}{p}=\frac{ g M_a}{b R}\frac{dT'}{T'}\;\; ; \;\;\;\;\;\;\;\; \rho=\rho_o (\frac{T'}{T'_o})^{s-1}$$ (10.10)

with $s = g M_a /(b R)$.

Although the above equations represent fairly well the density and pressure throughout the troposphere, the temperature distribution can depart significantly from the above linear variation near the ground. This ground heats up faster than the transparent air during the day, and cools off more rapidly during the night. The temperature gradient at low altitudes (up to 1-2 km) can be thus steeper or shallower than described by $b$ (Kkm$^{-1}$). Occasionally, it can be inverted, the temperature increasing with altitude. The inversion layer usually stops briskly at 1 or 2 km altitude and the normal temperature gradient resumes above. Inversion layers are common during the night over bare land. They can also be caused by hot winds blowing from the sea.

The local temperature gradient determines stability of the air to vertical motions. A rising bubble of wet air expands adiabatically as long as the water vapor it contains does not condense. Expanding, it cools almost as an ideal gas with:

$$T \propto p^{m} \; {\rm and}\; m= (1-0.23 q)m_a \simeq m_a$$ (10.11)

The pressure is set by the surrounding air (Eq.10.8), and the bubble seen to cool down with an ``adiabatic'' gradient of

$$\frac{dT}{dh} = -g \frac{M_a}{C_p} = 9.8 \mathrm{K km}^{-1}$$

If the actual temperature gradient is smaller than the adiabatic gradient, the bubble becomes cooler, hence denser than the surrounding air and its ascent stops. The air is stable. If the local gradient is larger than the adiabatic gradient, the bubble becomes less dense than the surrounding air; the air is unstable and a thick convection layer develops, a situation likely to happen in a hot summer afternoon.

10.1.4 Water

The scale height of water, $h_w$, which results from a fast evaporation/condensation process, is small (typically $2$ km) compared to the equilibrium scale height $h_o= 8.4$ km. At $h = 2.5$ km, the altitude of the Plateau de Bure, the water vapor column density $N_w$ (or $w$, ``amount of precipitable water'', when expressed in g.cm$^{-2}$, or cm of water) is normally reduced by a factor of 3-4, with respect to sea level. This factor, as we have seen, is strongly modulated by the local temperature gradient. $w$ is lower in the presence of a low altitude inversion layer which reduces the vertical turbulence and traps most of the water well below the observatory.