where Tskin is the skin temperature in the absence of Solar absorption. The formula shows that Solar absorption always increases the temperature of the skin layer. The temperature increases as the ratio of shortwave absorption to infrared emissivity is made larger. So long as the temperature remains less than the Solar blackbody temperature, the system does not violate the Second Law of Thermodynamics, since the radiative transfer is still acting to close the gap between the cold atmospheric temperature and the hot Solar temperature. As the atmospheric temperature approaches that of the Sun, however, it would no longer be appropriate to use the infrared emissivity, since the atmosphere would then be radiating in the shortwave range. Kirchoff's Law would come into play, requiring a/e = 1. This would prevent the atmospheric temperature from approaching the photospheric temperature.
If the shortwave absorptivity is small, the skin layer can be divided into any number of sublayers, and the argument applies to determine the temperature of each one individually. This is so because the small absorptivity of the upper layers do not take much away from the Solar beam feeding absorption in the lower layers. We can then infer that the temperature of an absorbing stratosphere will increase with height if the absorption increases with height, making asw/eir increase with height.
Armed with our new understanding of the optically thin outer portions of planetary atmospheres, let's take another look at a few soundings. The skin temperature, defined in Eq 3.24, provides a point of reference. It is shown for selected planets in Table 3.3. Except for the Martian case, these values were computed from the global mean OL—, either observed directly (for Jupiter) or inferred from the absorbed Solar radiation. In the case of present Mars, the fast thermal response of the atmosphere and surface makes the global mean irrelevant. Hence, assuming the atmosphere to be optically thin, we compute the skin temperature based on the upwelling infrared from a typical daytime summer surface temperature corresponding to the Martian soundings of Figure 2.2. The tropical Earth atmosphere sounding shown in Fig. 2.1 shows that the temperature increases sharply with height above the tropopause. This suggests that solar absorption is important in the Earth's stratosphere. For Earth, the requisite solar absorption is provided by ozone, which strongly absorbs Solar ultraviolet. This is the famous "ozone layer," which shields life on the surface from the sterilizing effects of deadly Solar ultraviolet rays. However, it is striking and puzzling that virtually the entire stratosphere is substantially colder than the skin temperature based on the global mean radiation budget. The minimum temperature in the sounding is 188K, which is fully 26K below the skin temperature. If anything, one might have expected the tropical temperatures to exceed the global mean skin temperatures, because the local tropospheric temperatures are warmer than the global mean. A reasonable conjecture about what is going on is that high, thick tropical clouds reduce the local OL—, thus reducing the skin temperature. However, the measured tropical OL— In Fig. 3.7 shows that at best clouds reduce the tropical OL— to 240W/m2, which yields the same 214K skin temperature computed from the global mean budget. Apart from possible effects of dynamical heat transports, the only way the temperature can fall below the skin temperature is
Mars (255K sfc)
Table 3.3: Computed skin temperatures of selected planets.
if the infrared emissivity becomes greater than the infrared absorptivity. This is possible, without violating Kirchoff's law, if the spectrum of upwelling infrared is significantly different from the spectrum of infrared emitted by the skin layer. We will explore this possibility in the next chapter.
Referring to Fig. 2.2 we see that the temperature of the Martian upper atmosphere declines steadily with height, unlike Earth; this is consistent with Mars' CO2 atmosphere, which has only relatively weak absorption in the Solar near infrared spectrum. The Martian upper atmosphere presents the same quandary as Earth's though, in that the temperatures fall well below the skin temperature estimates. Just above the top of the Venusian troposphere, there is an isothermal layer with temperature 232K, just slightly higher than the computed skin temperature. However, at higher altitudes, the temperature falls well below the skin temperature, as for Mars.
Between 500mb and 100mb, just above Titan's troposphere, Titan has an isothermal layer with temperature of 75K, which is very close to the skin temperature. Above 100mb, the atmosphere warms markedly with height, reaching 160K at 10mb. The solar absorption in Titan's stratosphere is provided mostly by organic haze clouds. Jupiter, like Titan, has an isothermal layer just above the troposphere, whose temperature is very close to the skin temperature. Jupiter's atmosphere also shows warming with height; its upper atmosphere becomes nearly isothermal at 150K, which is 44K warmer than the skin temperature. This indicates the presence of solar absorbers in Jupiter's atmosphere as well, though the solar absorption is evidently more uniformly spread over height on Jupiter than it is on Earth or Titan.
We have been using the term "stratosphere" rather loosely, without having attempted a precise definition. It is commonly said, drawing on experience with Earth's atmosphere, that a stratosphere is an atmosphereic layer within which temperature increases with height. This would be an overly restrictive and Earth-centric definition. The dynamically important thing about a stratosphere is that it is much more stably stratified than the troposphere, i.e. that its temperature goes down less steeply than the adiabat appropriate to the planet under consideration. The stable stratification of a layer indicates that convection and other dynamical stirring mechanisms are ineffective or absent in that layer, since otherwise the potential temperature would become well mixed and the temperature profile would become adiabatic. An isothermal layer is stably stratified, because its potential temperature increases with height; even a layer like that of Mars' upper atmosphere, whose temperature decreases gently with height, can be stably stratified. We have shown that an optically thin stratosphere is isothermal in the absence of solar absorption. Indeed, this is often taken as a back-of-the envelope model of stratospheres in general, in simple calculations. In the next chapter, we will determine the temperature profile of stratospheres that are not optically thin.
In a region that is well mixed in the vertical, for example by convection, temperature will decrease with height. Dynamically speaking, such a mixed layer constitutes the troposphere. By contrast the stratosphere may be defined as the layer above this, within which vertical mixing plays a much reduced role. Note, however, that the temperature minimum in a profile need not be coincident with the maximum height reached by convection; as will be discussed in Chapter 4, radiative effects can cause the temperature to continue decreasing with height above the top of the convectively mixed layer. Yet a further complication is that, in midlatitudes, large scale winds associated with storms are probably more important than convection in carrying out the stirring which establishes the tropopause.
We conclude this chapter with a few comparisons of observed tropopause heights with the predictions of the optically thin limit. We'll leave Venus out of this comparison, since its atmosphere is about as far from the optically thin limit as one could get. On Mars, using the dry adiabat for CO2 and a 5mb surface pressure puts the tropopause at 2.4mb, which is consistent with the top of the region of steep temperature decline seen in the daytime Martian sounding in Fig. 2.2. For Titan, we use the dry adiabat for N2 and predict that the tropopause should be at 816mb,which is again consistent with the sounding. If we use the methane/nitrogen moist adiabat instead of the nitrogen dry adiabat, we put the tropopause distinctly higher, at about 440mb. Because the moist adiabatic temperature decreases less rapidly with height than the dry adiabat, one must go to greater elevations to hit the skin temperature (as in Fig. 3.14). The tropopause height based on the saturated moist adiabat is distinctly higher than seems compatible with the sounding, from which we infer that the low levels of Titan must be undersaturated with respect to Methane. Using R/cp = 7 for Earth air and 1000mb for the surface pressure, we find that the Earth's tropopause would be at 545mb in the optically thin, dry limit. This is somewhat higher in pressure (lower in altitude) that the actual midlatitude tropopause, and very much higher in pressure than the tropical tropopause. Earth's real atmosphere is not optically thin, and the lapse rate is less steep than the dry adiabat owing to the effects of moisture. The effects of optical thickness will be treated in detail in Chapter 4, but we can already estimate the effect of using the moist adiabat. Using the computation of the water-vapor/air moist adiabat described in Chapter 2, the tropopause rises to 157mb, based on a typical tropical surface temperature of 300K and the skin temperature estimated in Table 3.3. This is much closer to the observed tropopause (defined as the temperature minimum in the sounding), with the remaining mismatch being accounted for by the fact that the minimum temperature is appreciably colder than the skin temperature.
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