Pure radiative equilibrium for real gas atmospheres

Pure radiative equilibrium amounts to an all-stratosphere model of an atmosphere, and is a counterpoint to the all-troposphere models we have been discussing. Real atmospheres sit between the two extremes, sometimes quite near one of the idealizations. In this section we will focus on pure infrared radiative equilibrium. The effects of solar absorption in real gases will be taken up in

Chapter 5.

From simple analytic solutions, we know essentially all there is to know about pure radiative equilibrium for grey gases. It is important to understand these things because the structure of atmospheres results from an interplay of convection and pure radiative equilibrium. A thorough understanding of pure radiative equilibrium provides the necessary underpinning for determining where the Stratosphere starts, and its thermal structure. We will now examine how the key elements of the behavior of pure radiative equilibrium differ for real gases. The specific issues to be addressed are:

• For grey gases in radiative equilbrium, the minimum temperature is the skin temperature based on OLR, and is found in the optically thinnest part of the atmosphere in the absence of atmospheric solar absorption. For real gases, can the radiative equilibrium temperature be much lower than the skin temperature?

• For a grey gas atmosphere with a given vertical distribution of absorbers, the radiative equilibrium temperature profile is uniquely determined once the OLR is specified. Specifically, one can determine the temperature profile of the radiative-equilibrium stratosphere without needing to know anything about the troposapheric temperature structure. To what extent is this also true for real gases?

• For a grey gas atmosphere with a given vertical distribution of absorbers, the normalized temperature profile T(p)/Tg is independant of the ground temperature. This means that the radiative equilibrium temperature profile has the same shape, regardless of the magnitude of the solar radiation with which the atmosphere is in balance. How much does this result change for real gases?

• For a grey gas, the temperature jump at the ground is greatest when the atmosphere is optically thin, and vanishes as the atmosphere becomes optically thick. For real gases, the atmosphere is optically thick in some parts of the spectrum, and optically thin in others. What determines the temperature jump in these circumstances?

• Grey gases are most unstable to convection where they are optically thick. In the optically thick limit, the slope dln T/d lnp equals 4 without pressure broadening, and 1 with pressure broadening. Hence, a dry adiabat can go unstable near the ground only if R/cp < 4 without pressure broadening, or R/cp < 1 with pressure broadening. How do these thresholds differ for a real gas?

All of the issues except for the behavior of the static stability near the ground are well illustrated by the semi-grey model, which we referred to more poetically as "one-band Oobleck" in Section 4.4.1. In this model, we assume k to be constant within a band of width A centered on frequency vo, and zero elsewhere. To keep the algebra simple we will make the additional assumption that A is small enough that B can be considered essentially frequency-independent within the band; this assumption can be easily dispensed with at the cost of a slightly more involved calculation. What makes the semigrey model tractable 9 is that the infrared heating is due solely to the flux within the absorbing band, so that one can still deal with a single optical thickness without any need to sum or integrate over frequency to get the net heating.

First, let's consider the semigrey radiative equilibrium in the optically thin limit. From the results in Section 4.2.2, the infrared radiative cooling at any level is simply 2nB(vo, T)A, whereas

9Approximate solutions for the semigrey ("window-grey") model were presented by Sagan in 1969 (Icarus 10 290-300.). The model has been rediscovered independently a number of times since.

the heating by absorption of upwelling infrared from the ground is nB(vo,Tg)A. Balancing the two, gives the equation determining the atmospheric temperature:

Since the equation for T is independent of level, we conclude that in the optically thin case, the atmosphere is isothermal in infrared radiative equilibrium, just as it is for the optically thin grey case. The resulting atmospheric temperature is always lower than the ground temperature, and may be called the semigrey skin temperature. For a grey gas, Tskin/Tg = 1/21/4 « .84, but for the semigrey case, the ratio depends on the frequency of the absorbing band. From the form of the Planck function, it can easily be shown that Tskin/Tg depends on frequency only through the ratio hvo/kT. A simple analytic calculation shows that Tskin/Tg ^ 1 for small values of hvo/kTg and Tskin/Tg ^ 1 when hvo/kTg becomes large; a simple numerical solution shows that the ratio increases monotonically between the two limiting cases (see Problem ??). The grey-gas skin temperature ratio sits in the middle of this range, and indeed for frequencies near the peak of the Planck function, the semigrey ratio does not differ greatly from the grey value. For example, at 650cm-1 and a ground temperature of 280K, the semigrey skin temperature is 233K, whereas the grey skin temperature is 235K

When deriving the radiative equilibrium for grey gases, we specified the OLR and used that as a boundary condition for determining the thermal structure; the ground temperature is then computed from the resulting lower boundary fluxes. For real gases, it proves more convenient to specify the ground temperature Tg, and find the resulting temperature structure and OLR. We adopt this approach not only in our analytic solution of the semigrey case, but also in our numerical solutions for actual gases. When fixing Tg and increasing optical thickness, the OLR goes down, and so the amount of absorbed solar radiation needed to maintain the stated Tg decreases.

The derivation of the full infrared radiative equilibrium solution for the semigrey case is identical to that we used in the grey case, with the following substitutions: (1) The optical thickness t is based on the value of k in the absorbing band alone,(2) aT4 is replaced by nB(vo, T) (3) The fluxes I+ and I- represent the flux integrated over the band A alone, and (4) The OLR appearing in the top boundary condition is no longer the net OLR emitted by the planet, but only the portion of OLR (call it OLRa) emitted in the absorbing band. The assumption of infrared equilibrium still implies that the flux I+ —1_ is constant and equal to OLR&, but since this flux is only the portion of OLR in the band, it is no longer determined a priori by planetary energy balance. Instead, it must be determined by making use of both boundary conditions, that I- = 0 at the top of the atmosphere, and I+ = nB(vo, Tg) at the bottom of the atmosphere (assuming the ground to have unit emissivity). The result of applying both boundary conditions is

This is equal to the ground emission in the optically thin case, and approaches zero as the atmosphere is made more optically thick. Note, however, that it is only the OLR in the band that approaches zero; the net emission approaches a finite, and possibly quite large, lower bound, because the emission from the rest of the spectrum where the atmosphere is transparent is simply the ground emission. By substituting the expression for the net upward flux into the expression for I+ +1-, we find the following expression determining T(t)

where t is the optical thickness within the band where k is nonzero. At the top of the atmosphere, this reduces to B(vT) = B(vo,Tg)/(2 + tto) Note that this is always less than the semi-grey skin temperature, and becomes progressively colder in comparison the the skin temperature as is made large. For the semigrey case, then, the stratospheric temperature differs from the grey gas case in two important ways. First, as the atmosphere is made optically thick in the absorbing band, the stratospheric temperature approaches zero even though the net OLR remains finite (owing to the emission through the transparent part of the spectrum). Thus, the stratosperic temperature can be much colder than the grey-gas skin temperature, resolving the quandary raised in Chapter 3. Moreover, as the optical thickness of the atmosphere is increased, the stratosphere actually becomes colder than even the semigrey skin temperature; this contrasts with the grey case, where the temperature of the uppermost part of the atmosphere always approaches the skin temperature, regardless of how optically thick the rest of the atmosphere is. This important difference arises because, in the semigrey case, the optical thickness of the lower part of the atmosphere causes the upwelling spectrum illuminating the stratosphere to be depleted in those frequencies where the stratosphere absorbs best. Nonetheless, the stratosphere continues to emit effectively at those frequencies, leading to very cold temperature. This depletion also has the important consequence that ths stratospheric temperature is no longer independent of the existence of a troposphere, or of the tropopause height and thermal structure of the troposphere. This has the potential to affect the calculation of the tropopause height when we put radiative equilibrium together with convection.

The temperature jump at the ground is determined by B(vo, Tsa) = ((1+rTO)/(2+rTO))B(vo, Tg), which has the same form as the corresponding expression for the grey gas case, save for the appearance of the Planck function in place of <rT4. As in the grey gas case, the jump is a maximum for optically thin atmospheres, and vanishes in the optically thick limit. The main difference here is that the jump can be made to vanish by making the atmosphere optically thick in just a limited part of the spectrum, even though the atmosphere is optically thin (in this case absolutely transparent) elsewhere. This feature will reappear in our discussion of radiative equilibrium for an atmosphere in which infrared absorption is provided by CO2. As the atmosphere becomes optically thick in the absorption band, the unstable temperature jump at the ground diminishes, but what happens to the interior static stability of the air near the ground? To determing this, we take the derivative of Eq. 4.97 and multiply by ps/TSa to obtain the logarithmic radiative equilibrium lapse rate at the ground:

In the optically thick limit, the second factor is unity without pressure broadening, or 2 with pressure broadening. When the atmosphere is optically thick, the first factor can be evaluated with Tsa = Tg. With a little algebra, the first term can be re-written as (exp(u) — 1)/(u exp(u)), where u = hvo/(kTg); this term has a maximum value of unity at u = 0 (high frequencies or low temperatures) and decays to zero like 1/u at large u (low frequencies or high temperatures). For u = 1 which puts the absorption band near the maximum of the Planck function, the value is about .63. For the semi-grey model then, we conclude that the degree of instability of radiative equilibrium in the optically thick limit is bounded; for example, with u = 1 and incorporating pressure broadening, the radiative equilibrium near the ground is statically unstable when R/cp > . 315 (vs. a threshold value of .5 for a grey gas). Hence, the semigrey case has a somewhat enhanced instability at the ground as compared to the grey case, but the difference is not great. We'll see shortly that this is one regard in which the qualitative behavior of a real gas differs significantly from the semigrey case.

Finally, let's look at how the radiative equilibrium profile for the semigrey atmosphere changes if we change Tg and leave everything else fixed. For a grey gas, the function T/Tg is invariant because both the surface emission and the interior atmospheric emission increase by a d ln T _ B(v0,rg) w(ps )ps/g d ln p lps _ TsadB (Vo, Tsa) 2 +

T dB sa dT

-1 band Oobleck -2 band Oobleck --<S--300ppm CO2 -e -3000ppm CO2

1 105

-1 band Oobleck -2 band Oobleck --<S--300ppm CO2 -e -3000ppm CO2

120 160

120 160

1 105

0.2 0.4 0.6 0.8 Logarithmic Slope

Figure 4.41: Left panel: Infrared radiative equilibrium temperature profiles for one and two-band Oobleck, and for a CO2-air mixture at 300ppmv and 3000ppmv, subject to a fixed 280K ground temperature. For the Oobleck cases, the absorption coefficients were chosen such that the optical depth of the atmosphere as a whole is 10 in the strong absorption band (650 — 700cm-1) and 1 in the weak bands on the flanks (600 — 650cm-1 and 700 — 750cm-1). Right panel: The corresponding logarathmic slope, d ln T/dlnp factor b4 if we replace Tg by b ■ Tg. For the semigrey case, the emission is given by B(vo, T), which is no longer a simple power of T. This means that the ratio T/Tg is no longer independent of Tg. The nature of the dependance is left for the reader to explore in Problem ??.

Working our way up the ladder to reality, we'll now present some numerical solutions for 2-band Oobleck and for mixtures of CO2 in air. The latter are computed using our homebrew exponential sum radiation code, incorporating the effect of pressure broadening but without taking temperature scaling into account. This is sufficient to show how the extreme range of absorption coefficients in a typical real gas affect the radiative equilibrium. The equilibria were found by a simple time stepping method with fixed Tg. For any given initial temperature profile T(p) one can calculate the infrared fluxes, and hence, by differencing in the vertical, the infrared heating rates. These are used to update T(p), and the whole process is repeated until equilibrium is achieved, where the infrared heating is zero and the temperature no longer changes. Since we are only interested in the equilibrium and not the time course of the approach to equilibrium, we can afford to be somewhat sloppy in our time-stepping method, so long as it is stable enough to yield an equilibrium at the end. Figure 4.41 shows the resulting equilibrium profiles for two-band Oobleck with the secondary-band absorption coefficient one tenth the value of the primary, and for a CO2-air mixture in an Earthlike atmosphere, at 300ppmv and 3000ppmv. The left panel shows the temperature profile, while the right panel gives the logarithmic slope of temperature, which determines the static stability of the atmosphere. For comparison, the analytically derived results for one-band pressure-broadened Oobleck (the semigrey model) are also shown. These calculations were carried out with a ground temperature Tg = 280K. Some key characteristics of the results are summarized in Table 4.4.

Comparing the curves for 1-band and 2-band Oobleck shows that adding in the weakly absorbing bands reduces the vertical temperature gradient except in a thin layer near the ground. In this sense, most of the atmosphere acts as if it were made more optically thin, despite the fact that we have actually made the atmosphere more opaque to infrared by adding new absorption in additional bands without taking away absorption in the original band. Indeed, the OLR goes down from 333 W/m2 in the 1-band case to 309 W/m2 in the 2-band case, despite the warmer temperatures aloft in the latter. The key to this behavior was pointed out back at the end of Section 4.2.2: in an atmosphere which is optically thin in some parts of the spectrum but optically thick in others, the heating rate (which in turn determines the radiative equilibrium) is dominated by the parts of the spectrum where the optical thickness is nearest unity. This is the reason the weaker absorption bands control the behavior of the temperature in the interior of the atmosphere. The associated reduction in temperature gradient warms the atmosphere aloft, at pressures below 700mb. In essence, the weak bands allow the upper reaches of the atmosphere to capture more of the infrared upwelling from below; spreading the absorption over a somewhat broader range of the spectrum in essence makes the problem a bit more like a grey gas, and makes the upper air temperature somewhat closer to the grey gas skin temperature.

While the addition of the weak bands reduces the temperature gradient aloft and hence stabilizes the atmosphere against convection, this comes at the expense of increasing the gradient near the ground, and destabilizing the layer there. This stabilization/destabilization pattern shows up clearly in the plot of the logarithmic temperature derivative shown in the right panel of Figure 4.41. Based on R/cp = 2/7, the one-band Oobleck profile is unstable for pressures higher than 450mb, whereas the two-band case is only unstable for pressures higher than 760mb, though within the unstable layer the 2-band case is considerably more unstable than the one-band case. In the two-band case, more of the net vertical temperature contrast of the atmosphere is concentrated in a thin layer near the ground. Overall, the atmosphere acts as if it were optically thick near the ground, but relatively optically thin aloft. The behavior near the ground results from the extremely strong absorption near the center of the CO2 absorption band. This spectral region captures the upwelling radiation from the ground, which is not yet depleted in the strongly absorbing wavenumbers. If there were much temperature discontinuity near the ground, the absorption would lead to strong radiative heating of the low level air; hence the only way to be in radiative equilibrium is for the air temperature to approach the ground temperature. More mathematically speaking, the phenomenon arises because the low level heating is controlled largely by the boundary terms in the flux integral in Eq 4.9, which in turn is dominated by the strongly absorbing spectral regions.

The real-gas CO2 results have many features in common with 2-band Oobleck, notably the weak temperature gradient in the interior of the atmosphere and the enhancement of temperature gradient near the ground. The strong destabilization near the ground is even more pronounced for CO2, because it has a far stronger peak absorption than the Oobleck case we considered, and because the absorption varies over a greater range of values. The associated optical thickness near the ground also keeps the unstable temperature jump between the ground and the overlying air small. An optically thin grey-gas would have a large unstable temperature jump at the ground. The strong absorption bands in a real gas smooth out this discontinuity and move it into a finite width layer in the interior of the atmosphere. From the standpoint of the convection produced, there is little physical difference between the two cases.

The main differences between the CO2 cases and the Oobleck cases shows up in the upper atmosphere, where the CO2 cases show steep declines of temperature with height - though not so steep as to destabilize the upper atmosphere. As at the bottom boundary, the culprit is the spectral region where absorption coefficients peak. These regions lead to very strong emission in the upper layers of the atmosphere, which are poorly compensated by absorption since the upwelling infrared

OLR,W/m2

Tskin

T (0)

T (P/Ps = .5)

Tg - T(Ps)

1-band Oobleck

333

232K

161K

206K

6.5K

2-band Oobleck

309

228K

182K

215K

12.75K

CO2/air 300ppmv

295

226K

126K

214K

2.8K

CO2/air 3000ppmv

275

222K

115K

214K

1.75K

Mars, CO2,ps = 7mb

303

227K

128K

215K

2.9K

Mars, CO2,ps = 7mb,Tg =

250K

192

203K

126K

196K

4.4K

Mars, CO2,ps = 2bar

86

166K

102K

208K

1.3K

Venus, CO2,ps = 90bar,Tg

= 700K

55

148K

77K

500K

.1K

Table 4.4: Summary properties of infrared radiative equilibrium solutions. Calculations were done

Table 4.4: Summary properties of infrared radiative equilibrium solutions. Calculations were done with Tg = 280K unless otherwise noted.

reaching the upper atmosphere has been depleted in most wavenumbers that absorb at all well. The strong emission causes the temperature near the top to fall far below the skin temperature. Based on the net OLR, the grey gas skin temperature for the two CO2 cases is somewhat above 220K, whereas the actual temperature at the 1mb level is 126K in the 300ppmv case and 115K in the more optically thick 3000ppmv case. Finally, we note that increasing CO2 by a factor of 10 has relatively little effect on the radiative equilibrium temperature profile, despite the fact that the increase lowers the OLR by nearly 20W/m2. The changes are principally seen near the ground, where the increased optical thickness in the wings has reduced the surface temperature jump. While the two-band Oobleck model reproduces the near-ground destablization present in the real CO2 calculation, it is unable to simultaneously represent the temperature jump at the ground. Lacking the extremely strong peak absorption of CO2, the addition of the weak absorption wings in 2-band Oobleck makes the surface budget act like an optically thinner atmosphere, increasing the surface jump.

To illustrate how the radiative equilibrium solution scales with Tg, we show the profiles of T/Tg for surface temperatures ranging from 240K to 320K. Only the case of 300ppmv CO2 is shown, though the other atmospheres treated in Figure 4.41 yield similar results. For a grey gas, all the curves for a given atmospheric composition would collapse onto a single universal profile; for the reasons discussed in the semigrey case, this is no longer true for real gases. However, while the temperature aloft does not scale precisely with the ground temperature, the deviations are modest enough that one can still get useful intuition about the behavior of the system by assuming that radiative equilibrium temperature scales with the ground temperature. For CO2, the actual temperature aloft is always somewhat colder than that which one would estimate by proportionately scaling the temperature upward from a colder to warmer ground temperature. Recall that these calculations are done without temperature scaling of the absorption coefficient, so the effect shown is purely due to the shape of the Planck function.

The general features encountered in the terrestrial calculations discussed above carry over to the pure CO2 atmospheres characteristic of Present and (possibly) Early Mars. Martian radiative equilibrium solutions with a 7mb thin atmosphere or 2bar thick atmosphere are shown in Figure 4.43. The most striking feature of these solutions is that increasing the mass of the atmosphere by a factor of nearly 300 causes very little increase in the vertical temperature contrast. This stands in sharp contrast to the grey gas case, for which the enormous increase in optical depth going from the 7mb case to the 2bar case would cause the temperature in the latter to drop to nearly zero within a short distance of the ground. As before, the reason for the relative insensitivity of the temperature profile is that the radiative heating is determined largely by the part of the spectrum where the optical depth is of order unity. For the present Mars case, this occurs in the near wings

Figure 4.42: Variation of the shape of the temperature profile as a function of ground temperature. Results are shown for 300ppmv of CO2 in air.

Figure 4.43: Infrared radiative equilibrium for pure CO2 atmospheres with Martian gravity. Results are shown for an Early Mars case with a 2 bar surface pressure and 280K ground temperature, and for Present Mars cases with 700mb surface pressure and 250K and 280K ground temperatures. To make it possible to compare the cases, the temperatures have been plotted as a function of p/ps. The 2bar case includes the CO2 continuum absorption.

of the principle absorption peak, whereas for Early Mars it occurs within the continuum window region. The shift allows both cases to act roughly like a case with order unity optical depth, apart from the thin radiative boundary layers near the ground and the top. The temperature profiles are similar despite the fact that it would require 303W/m2 of absorbed Solar radiation to maintain a surface temperature of 280K in the thin present Martian atmosphere, but only 86W/m2 in a 2bar atmosphere. For the Present Mars case, we have also included a calculation with with a realistically cold daytime surface temperature, the equilibrium temperature aloft is too cold in comparison with observations. This suggests once more an important role for solar absorption in determining the temperature structure of the present Martian atmosphere.

It is only when we go to massive atmospheres like that of Venus that the atmosphere becomes optically thick throughout the spectrum, allowing the vertical temperature contrast to increase dramatically. A simplified calculation without temperature scaling, and ignoring emission beyond 2300cm-1, shows that with a 700K ground temperature, the radiative equilibrium temperature drops to 500K at the midpoint of the atmosphere and all the way to 80K at the 100mb level. This yields a very strong greenhouse effect: it takes only a trickle of 55W/m2 of absorbed solar radiation to maintain the torrid ground temperature. Note, however, that it is important that a fair amount of this trickle actually be absorbed at the ground, and not in the upper reaches of the atmosphere; otherwise the deep atmosphere becomes isothermal, and can in fact become as cold as the skin temperature in extreme cases, as discussed in Section 4.3.5. So far as the maintainence of its thermal structure is concerned, the troposphere of Venus is more like the Antarctic glacier than it is like the Earth's troposphere. The trickle of heat escaping the Earth's interior beneath the glacier - a mere 30 mW/m2 - is sufficient to raise the basal ice temperature to the melting point and create subglacial Lake Vostok precisely because the diffusivity of heat in ice is so small. So it is, too, with the atmosphere of Venus; the extremely optically thick tropophere renders the radiative diffusivity of heat very small, and allows the tiny trickle of solar radiation reaching the surface to accumulate in the lower atmosphere and raise the temperature to extreme values. Unlike the glacier, however, when the lower atmosphere becomes hot enough, it can start to convect. Convection supplants the radiative heat flux, but also establishes the adiabat, allowing the surface to be much hotter than the radiating level.

Different noncondensible greenhouse gases differ somewhat in details of their radiative-equilibrium profiles, but the general picture does not differ greatly from what we have learned by looking at CO2. When the greenhouse gas can condense near the ground, however,the situation becomes quite different. The case of water vapor in air provides a prime example. If the ground temperature is high enough that the amount of water vapor present in saturation makes the lower atmosphere optically thick, then the temperature will decline rapidly with height above the ground, because that is what optically thick atmospheres do in radiative equilibrium. Water vapor exhibits this effect particularly strongly, since it easily makes the atmosphere optically thick everywhere outside the window regions of the spectrum, and even the windows close off above 300K. As the temperature decreases, however, the water vapor content decreases in accordance with the limits imposed by Clausius-Clapeyron. Within a small distance above the ground, the air is so cold that there is little water vapor left, and the atmosphere further aloft becomes optically thin. As a result, most of the variation in optical thickness of the atmosphere is concentrated into a thin, radiative boundary layer near the ground, and the optical thickness (and hence the temperature) varies greatly within this layer. Because of the strong temperature gradient and high optical thickness of the boundary layer, a strong greenhouse effect is generated entirely within the boundary layer, leading to low OLR. If one imposes equilibrium with an Earthlike absorbed solar radiation, the ground temperature must increase to temperature well in excess of 320K to achieve balance, and the steep increase of saturation vapor pressure with temperature further exacerbates the high temperature gradient in the radiative boundary layer. It is a bit as if the

entire optically thick atmosphere of Venus were squeezed into a boundary layer having a depth of a kilometer or less. Adding a noncondensing gas like CO2 to the mix alters the temperature profile above the boundary layer, but does not eliminate the basic pathology of the situation. The equilibrium profile in this case is of little physical consequence because the slightest convection or other turbulent mixing would mix away the thin radiative boundary layer, warming and moistening a much deeper layer of the atmosphere The radiative equilibrium solutions we have been studying for the noncondensing case are worthy of protracted consideration because they provide some useful insight as to the stratospheric temperature for more realistic atmospheres in which there is some low level convection. The same cannot be said for pure radiative equilibrium in the condensing water-vapor/air system, which is an exercise in pathology having little or no bearing on the operation of real atmospheres.

Though real gas radiative equilibrium is not amenable to the kind of complete solution we enjoyed for the grey gas case, its behavior can be reasonably well captured by a few generalities, summarized in Figure 4.44. For real gases as for grey gases heated by infrared emission from the lower boundary, the temperature decreases with distance from the boundary. This can be viewed as a kind of thermal diffusion in the sense that heat transfer is down the temperature gradient, though the process is only described by a true local diffusion in the optically thick limit. Real gases behave as if they are optically thick near the ground, exhibiting strong convectively unstable temperature gradients there and little temperature jump between the ground and overlying air. The temperature gradient weakens in the interior, but there is generally a region of strong, though stable, temperature decline near the top of the atmosphere. The upper atmosphere is considerably colder than the grey gas skin temperature, since (by Kirchoff's law) the atmosphere radiates efficiently in the strongly absorbing parts of the spectrum, but the radiation illuminating the upper atmosphere is depleted in this portion of the spectrum. In contrast to the grey gas case, the contrast in temperature across the depth of the atmosphere is relatively insensitive to the amount of greenhouse gas in the atmosphere. As long as there are some spectral regions where the atmosphere is optically thick, and some where the atmosphere is optically thin, the radiative cooling tends to be dominated by the intermediate spectral regions. As a result, temperature tends to drop by a factor of two to three between the ground and the upper atmosphere, with only slight increases even when the greenhouse gas content is increased by many orders of magnitude. This behavior persists until there is so much greenhouse gas present that the atmosphere becomes very optically thick throughout the thermal infrared spectrum, as is the case on Venus.

The upshot of all this is that atmospheres whose temperature is maintained by absorption of upwelling infrared from a blackbody surface will never exhibit pure radiative equilibrium. There will always be a layer near the surface which is unstable to convection. If the atmosphere is optically thin, the instability is generated by a temperature jump at the surface. If the atmosphere is optically thick and subject to pressure broadening, the instability is generated by strong temperature gradients in the interior of the atmosphere near the surface. This remark even applies qualitatively to gas giants which have no surface, as the deep atmosphere is dense enough that it can begin to act like a blackbody even though there is no distinct surface. An atmosphere can be stabilized throughout its depth, however, if it is subject to atmospheric solar heating which increases with altitude in a suitable fashion.

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