where iH2 is the mass path of Hydrogen in the box, in kg/m2, and p0 is a standard pressure. The coefficient kH2 has dimension m2/kg and can be used in precisely the same way as the absorption

N2-N2 continuum at 100mb N2

N2-N2 continuum at 100mb N2

8 10

4 10

A2 10

50 100 150

Wavenumber (cm ')

50 100 150

Wavenumber (cm ')

N2-H2 continuum at 100mb N2

N2-H2 continuum at 100mb N2

0 100 200 300 400 500 600 700 800 Wavenumber (cm ')

0 100 200 300 400 500 600 700 800 Wavenumber (cm ')

8 10

4 10

A2 10

Figure 4.23: The N2 — N2 and N2 — H2 collision-induced continuum absorption coefficients as a function of temperature (indicated on curves). The coefficients are given at an N2 partial pressure of 100mb.

coefficients we defined earlier for use with the line spectrum. To relate this to the binary absorption coefficient commonly defined in the spectroscopy literature, let z be the length of the path in meters, so that ¿H2 = z • pH2/(RH2T). Then

Po RH2J

which defines the binary coefficient in the case where the collider amounts are specified as partial pressures. It is more common to specify the collider amounts in terms of densities or molar densities, but the alternate forms can be readily derived from the preceding by use of the ideal gas law.

Continuum absorption may be difficult to understand from a priori physical principles, and difficult to measure accurately in the laboratory, but by definition the absorption coefficient for the continuum is a smoothly varying function of wavenumber. Therefore, it is relatively easy to incorporate into radiative transfer models. One only needs to determine the absorption coefficients and their pressure and temperature scaling in a set of relatively broad bands, and multiply the transmission computed from the line absorption (if any) by the corresponding exponential decay factor.

The continuum arising from diatomic molecule collisions becomes particularly important for dense, cold, massive atmospheres, of which Titan's is probably the best studied example. Figure 4.23 shows the N2 — N2 and N2 — H2 collision induced absorption coefficients in the temperature range prevailing in Titan's atmosphere. These coefficients are based on laboratory measurements made at somewhat higher temperatures, extrapolated to colder values using a theoretical model with a few empirical coefficients fit to the data. (See the paper by Courtin et al. listed in the Further Readings section of this chapter). The equivalent path for Titan based on N2 partial pressure is about 106kg/m2, which yields a peak optical thickness of 40 for temperatures near those prevailing at Titan's surface. The H2 content of Titan's atmosphere is less well constrained, but plausible estimates suggest that this gas, too, can contribute significantly to the infrared opacity of Titan's atmosphere. Note that the absorption decreases sharply with increasing temperature; this is partly due to the decrease in density with temperature, but is also affected by the shorter duration of high-velocity collisions, which apparently are less effective at inducing a dipole moment. The N2 — N2 continuum is unimportant for Earthlike collisions because of the higher temperatures on Earth, and because Earth's atmosphere is much less massive than Titan's,per unit surface area.

N2-CH4 and CH4-CH4 continuum at 100mb collider pressure

N2-CH4 and CH4-CH4 continuum at 100mb collider pressure

H2 also has a significant self-broadened continuum, which provides a great deal of the infrared opacity on the gas giant planets.

Methane also significantly affects the infrared opacity of Titan's atmosphere, though its effects on OLR are rather less than N2 or H2 since methane is concentrated in the warmer, lower layers of the atmosphere. Essentially all of the infrared absorption due to methane on Titan comes from a collision-induced continuum. The N2-induced and self-induced absorption coefficients at 100K are shown in Figure 4.24. Like the other continuum coefficients, these too become weaker with increasing temperature. Assuming Methane to be in saturation with temperature given by the Methane-Nitrogen moist adiabat in Titan conditions, the pressure-weighted path for self-induced absorption is in excess of 40000%/m2, while the equivalent path for N2-induced absorption is over 170000%/m2. The self-absorption yields a peak optical thickness of about 5, while the foreign absorption gives a peak optical thickness of about 20, dropping to 10 in the higher frequency shoulder near 200cm-1

Given the importance of the CO2 window regions to the high-CO2 climates of Venus and Early Mars, it is rather surprising that the CO2 continuum has been so little studied. The coefficients in use in most models stem from limited experiments and there is little agreement on the theoretical basis for this continuum or its temperature scaling. At the time of writing, it appears that the subject has not been re-examined in laboratory experiments since the late 1970's. The discussion below is based on absorption coefficients reported in the literature cited in the Further Readings section of this chapter.

The measured CO2 continuum absorption, rescaled to 100mb is shown in Figure 4.25. The values shown are for collisions of CO2 in air; the self-induced continuum absorption is generally assumed to be 1.3 times that of the foreign-induced continuum. Referring to the equivalent paths in Table 4.1, we see that the continuum absorption is large enough to make the top one bar of

the atmosphere of Venus optically thick throughout most of the window region. The continuum absorption is strong enough to be important in the thick atmosphere of Early Mars, but only marginally so for the more moderate CO2 levels present on Early Earth.

When incorporating the continuum in radiation models, it may be more convenient to work from a curve-fit rather than tabulated data. The absorption coefficient for the CO2 continuum can be fit with the function

KCO2(v, 300K, 100mb) = exp(-8.853+0.028534v-0.00043194v2 + 1.4349• 10-6v3 -1.5539• 10-9v4)

from 25 to 450 cm-1 and by

KCO2(v, 300K, 100mb) = exp(-537.09 + 1.0886v-0.0007566v2 + 1.8863• 10-7v3 -8.2635 • 10-12v4)

from 1150 to 1800 cm-1 where v is measured in cm-1. The continuum absorption coefficients weaken with increasing temperature, according to the empirical power law (300./T)n , with n = 1.7 for wavenumbers greater than 190cm-1, increasing to 1.9 at 130 cm-1, 2.2 at 70 cm-1 and 3.4 at 20 cm-1.

The CO2 continuum is poorly characterized experimentally, and not well understood theoretically. The continuum fits provided above are based on rather dated experiments, and there are some indications that a part of the above continuum may actually represent line spectra from asymmetric isotopologues of CO2 such as C16O18O. This is an area where more experiments using modern techniques are badly needed.

Since water vapor has absorption lines throughout the spectrum, it is hard to unambiguously define the continuum. Laboratory measurements clearly show, however, that in the window regions indicated in Figure 4.19 the net absorption is far in excess of what can be accounted for by the contributions of nearby lines. The prevailing view currently is that this excess absorption is due to the very far tails of the stronger absorption bands flanking the window regions, rather than dimers, forbidden transitions, or collision-induced dipole moments. The theoretical and observational basis for this viewpoint is exceedingly weak, however. In the following we will confine ourselves to empirical descriptions of the laboratory measurements, without reference to underlying mechanisms. Comparisons with direct measurements of transmission in the Earth's atmosphere have confirmed that the laboratory measurements provide an adequate basis for modelling water vapor absorption in the window regions. The laboratory measurements show that the air-broadened or N2 broadened water vapor continuum is very weak, so that the window region absorption is by far dominated by self-collisions of water vapor. The following discussion will therefore be limited to self-induced absorption; the characterization of foreign-induced absorption by CO2 appears to be an open question at present, though it is potentially of importance to water-CO2 atmospheres such as might have occurred on Early Mars.

There is some ambiguity in the spectroscopic literature as to how to define the water vapor continuum, given that in analyzing measurements one must be careful not to be thrown off by the strong absorption near the centers of individual lines in the window regions. Most useful definitions of the continuum amount to reading the absorption at the minima "between the lines." The results of such a measurement of the self-induced continuum are shown in Figure 4.26. The measurements were made for water vapor in saturation at 296K, with water vapor partial pressure of about 28mb, but have been scaled to a standard water vapor partial pressure of 100mb for the sake of discussion. We'll focus on the lower frequency of the two window regions, since that is by far the most important for planetary climate calculations. Similar data exists for the higher frequency window. From the figure, we see that the measured continuum absorption is several orders of magnitude stronger than the typical line contribution. To get an idea of the significance of the water vapor continuum, let's consider a layer of air of depth z, with uniform temperature T, within which the water vapor is at the saturation vapor pressure corresponding to T. Since the water vapor continuum is dominated by self-collisions, it matters little what the background air pressure is in this layer. The equivalent path for this layer is (psat(T)/p0)(psat(T)/(RwT))z; the first factor gives the degree of pressure-induced enhancement of absorption relative to the standard, while the secondd factor is the density of water vapor in the layer. Note that the equivalent path is quadratic in the water vapor partial pressure. For this reason, the optical thickness in the continuum region grows very rapidly with temperature. At 300K, then, with a layer depth of 1km the equivalent path is 9.3kg/m2. Since the minimum absorption coefficient in the window regionis about .1m2/kg, this path gives the layer an optical thickness of unity or more in the window region. Since the absorption is even stronger outside the window region, at the lowest layer of the Earth's atmosphere acts practically like an ideal blackbody at tropical temperatures. At 310K the equivalent path increases to 27kg/m2, so the window region closes off even more. This has profound consequences for the runaway greenhouse. In fact, there would be essentially no prospect for a runaway greenhouse even in Venusian conditions were it not for the water vapor continuum.

Over the range of wavenumbers shown in the figure, the water vapor continuum absorption can be fit by the polynomial

KH2O(v, 296K, 100mb) = exp(12.167—0.050898v+8.3207-10-5v2—7.0748-10—8v3+2.326M0-iiv4)

Like the other continua, the water vapor continuum absorption becomes weaker as temperature increases. Data on the temperature dependence is sparse, but suggests a temperature dependence of the form (296/T)4 25. For temperatures much colder than 300K, the saturation vapor pressure is so low that the details of the temperature dependence are unimportant. As temperature increases beyond 300K, the exponential growth of saturation vapor pressure is far more important to the optical thickness than the rather mild decline of the continuum absorption coefficient with temperature.

Figure 4.26: The water vapor self-induced continuum near 1000 cm-1, compared with the median absorption coefficient computed from the Lorentz line contribution within 1000 line widths of the line centers in the HITRAN database. The continuum curve given is based on laboratory observations in saturation at 296K, scaled up to what they would be at a water vapor partial pressure of 100mb. See citations in the Further Readings section of this chapter for data sources.

Figure 4.26: The water vapor self-induced continuum near 1000 cm-1, compared with the median absorption coefficient computed from the Lorentz line contribution within 1000 line widths of the line centers in the HITRAN database. The continuum curve given is based on laboratory observations in saturation at 296K, scaled up to what they would be at a water vapor partial pressure of 100mb. See citations in the Further Readings section of this chapter for data sources.

Water vapor has another continuum region at shorter wavelengths, in the vicinity of 2500cm-1. This is not important for Earthlike temperatures, but it is a very significant factor for the hotter temperatures encountered in runaway greenhouse calculations. At temperatures much in excess of 320K, there is enough emission in this region that it accounts for a significant part of the infrared cooling if the continuum is not included. The 2500cm-1 continuum, covering 2100cm-1 < v < 3000cm 1 can be represented by the polynomial fit

KH2O(v, 296K, 100mb) = exp(—6.0055—0.0021363x+6.472340-7x2— 1.49340—8x3+2.562M0-11x4+7.32840-14x5)

where x = v — 2500cm-1. The scaling in pressure and temperature can be taken to be the same as for the longer wavelength continuum, though the experimental support for the temperature dependence is somewhat weak.

The importance of the water vapor continuum to climate when temperatures approach or exceed 300K is demonstrated in Figure 4.27. Here we present calculations of the spectrum of OLR and of surface back-radiation using the exponential sum radiation code described previously, but modified to take into account the vertical variation of water vapor concentration and the continuum. With the continuum included, the low layer atmosphere radiates to the surface practically like a blackbody; in fact, if one increases the surface air temperature slightly, to 310K, the back radiation becomes indistinguishable from the blackbody spectrum corresponding to the surface air temperature. In contrast, without the continuum, there is essentially no back-radiation in the window region, allowing the surface to cool strongly through the window. Likewise, without the continuum, the atmosphere can radiate to space very strongly through the window, whereas the cooling to space is very much reduced if the continuum absorption is included. These calculations were carried out for an Earthlike water-air atmosphere on a planet with g = 9.8m/s2. On a planet with weaker surface gravity, the water vapor continuum would become important at lower temperatures, because a given partial pressure corresponds to a greater mass of water. Conversely, on a planet with stronger surface gravity, the water vapor window closes at higher temperatures.

Wavenumber ( cm 1) Wavenumber ( cm 1)

Figure 4.27: The spectra of surface back radiation and OLR computed using an exponential sum radiation code including the effects of the longwave water vapor continuum (left panel) and excluding the continuum (right panel). The calculation was done with the temperature profile on the water-air moist adiabat corresponding to 300K surface temperature, assuming the water vapor partial pressure to be saturated at all levels. This calculation does not take into account the temperature variation of absorption coefficients or the enhancement of self-induced absorption in the line contribution.

Wavenumber ( cm 1) Wavenumber ( cm 1)

Figure 4.27: The spectra of surface back radiation and OLR computed using an exponential sum radiation code including the effects of the longwave water vapor continuum (left panel) and excluding the continuum (right panel). The calculation was done with the temperature profile on the water-air moist adiabat corresponding to 300K surface temperature, assuming the water vapor partial pressure to be saturated at all levels. This calculation does not take into account the temperature variation of absorption coefficients or the enhancement of self-induced absorption in the line contribution.

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