Now we will examine how the absorption of solar radiation within an atmosphere affects the temperature structure of the atmosphere in radiative equilibrium. The prime application of this calculation is to understand the thermal structure of stratospheres. Under what circumstances does the temperature of a stratosphere increase with height? The effect of solar absorption on gas giant planets like Jupiter is even more crucial. There being no distinct surface to absorb sunlight, all solar driving of the atmosphere for gas giants comes from deposition of solar energy within the atmosphere. In this case, the profile of absorption determines in large measure where, if anywhere, the radiative equilibrium atmosphere is unstable to convection, and therefore where a troposphere will tend to form. The answer determines whether convection on gas giants is driven in part by solar heating as opposed to ascent of buoyant plumes carrying heat from deep in the interior of the planet.
In the Earth's stratosphere, solar absorption is largely due to the absorption of ultraviolet by ozone. On Earth as well as other planets having appreciable water in their atmospheres, absorption of solar near-infrared by water vapor and water clouds is important. CO2 also has significant near-infrared absorption, which is relatively unimportant at present-day CO2 concentrations on Earth, but becomes significant on the Early Earth when CO2 concentrations were much higher; solar near-infrared absorption by CO2 is of course important in the CO2 dominated atmospheres of Mars (present and past) and Venus. Solar absorption by dust is important to the Martian thermal structure throughout the depth of the atmosphere. On Titan, it is solar absorption by organic haze clouds that control the thermal structure of the upper atmosphere. Solar absorption is also crucial to the understanding of the influence of greenhouse gases like CH4 and SO2, which strongly absorb sunlight in addition to being radiatively active in the thermal infrared. Strong solar absorption also would occur in the high-altitude dust and soot cloud that would be lofted in the wake of a global thermonuclear war or asteroid impact (the "Nuclear Winter" problem).
Since the Schwartzschild equations in this chapter are used to describe the infrared flux alone, the addition of solar heating does not change these equations. The heating due to solar absorption only alters the condition for local equilibrium, which now involves the deposition of solar as well as infrared flux. We write the solar heating rate per unit optical depth in the form Qq = dFQ/dT, where Fq is the net downward solar flux as a function of infrared optical depth. At the top of the atmosphere, Fq = (1 — a)S, where a is the planetary albedo - that is, the albedo measured at the top of the atmosphere. Since atmospheres at typical planetary temperatures do not emit significantly in the solar spectrum, there is no internal source of solar flux and therefore Fq must decrease monotonically going from the top of the atmosphere to the ground.
The net radiative heating at a given position is now the sum of the infrared and solar term, i.e.
Integrating this equation and requiring that the top of atmosphere energy budget be in balance with the local absorbed solar radiation, we find
At the top of the atmosphere, this reduces to OLR — (1 — a)S = 0, which is the requirement for top of atmosphere energy balance. Because the solar absorption does not change the infrared Schwartzschild equations, Eq. 4.41 is unchanged from the case of pure radiative equiibrium without solar absorption. Substituting Eq 4.51 and integrating, we obtain
In writing this expression we have made use of the boundary condition 1+ — 1— = OLR = (1 — a)S at the top of the atmosphere. The heat balance equation 4.40 needs to be slightly modified, since the infrared cooling now balances the solar heating, instead of being set to zero. Thus, dTFQ = dT (1+ — 1-) = —(1+ + 1-) + 2ffT4 (4.53)
from which we infer d /"T~
This gives the vertical profile of temperature in terms of the vertical profile of the solar flux; the previous case (without solar absorption) can be recovered by setting F0 = const = (1 — a)S. At the top of the atmosphere, the integral in Eq. 4.53 vanishes, and the temperature becomes identical to the temperature of a skin layer heated by solar absorption, derived in Chapter 3 (Eq. 3.27).
Taking the derivative with respect to t yields d d2
This equation provides a simple criterion determining when the solar absorption causes the temperature to increase with height. When there is no absorption, F0 is a constant and since it is positive the temperature decreases with height. The quantity J|F0 1 d2F0/dT2| is local solar flux decay rate, expressed in units of infrared optical depth. Where the local solar decay rate is less than unity - meaning solar flux is attenuated at a lower rate than infrared - the radiative equilibrium temperature decreases with height. Where the local solar decay rate is greater than unity, the temperature increases with height. Note that it is the solar extinction rate relative to the infrared extinction rate that counts. One can make the temperature increase with height either by increasing the solar opacity or decreasing the infrared opacity (generally by decreasing the greenhouse gas concentration). The profile of solar absorption is also sensitive to the vertical distribution of solar absorbers. Where solar absorbers increases sharply with height, as is the case for ozone on Earth or organic haze on Titan, the stratospheric temperature increases with height.
By way of illustration, let's suppose that the net downward solar flux decays exponentially as it penetrates the atmosphere. Specifically, let Fq = (1 — a)Sexp( —(t0 — t)/ts), where ts is a constant. ts is the decay rate of solar radiation, measured in infrared optical depth units. When ts is large, solar absorption is weak compared to infrared absorption, and one must go a great distance before the solar beam is appreciably attenuated. Conversely, when ts is small, solar absorption is strong and the solar beam decays to zero over a distance so short that infrared is hardly attenuated at all. With the assumed form of the solar flux, the temperature profile is given
If ts > 1 the temperature decreases with height, and if ts < 1 the temperature increases with height. Defining the skin temperature as Tskin = (^ (1-a)S))1/4 the temperature at the top of the atmosphere is (1 + 1/ts)Tsfcm, which reduces to the skin temperature when ts is large and becomes much greater than the skin temperature when ts is small. If the atmosphere is deep enough that essentially all solar radiation is absorbed before reaching the ground, then the exponential term vanishes in the deep atmosphere and the deep atmosphere becomes isothermal with temperature (1 + ts)Tskin. Thus, when ts is small, all the solar radiation is absorbed within a thin layer near the top of the atmosphere. The temperature increases rapidly with height in this layer, but the bulk of the atmosphere below is approximately isothermal at the skin temperature. The strong solar absorption causes the deep atmosphere, and the ground (if there is one) , to be colder than it would have been in the absence of an atmosphere. This anti-greenhouse effect arises because the deep atmosphere is heated only by downwelling infrared emitted by the solar-absorbing layer. This downward radiation equals the upward radiation loss to space, which must equal (1 — a)S to satisfy the top of atmosphere balance. The deep atmosphere falls to the skin temperature because it is being illuminated by this flux from one side, but is radiating from both its top and bottom. This limit is relevant to the nuclear winter phenomenon, in which energetic explosions and fires loft a global or hemispheric solar-absorbing soot and dust cloud to high altitudes. The same situation would occur in the aftermath of a large bolide (asteroid or comet) impact. In either case, the atmosphere below the soot layer would become as frigid as the depth of winter, but moreover the relaxation to a uniform temperature state would shut off the convection which in large measure drives the hydrological cycle.
It can happen that the atmosphere is deep enough to absorb all solar radiation before it reaches the ground, even if the rate of solar absorption is weak and ts >> 1. This would happen if the atmosphere is so optically thick in the infrared that r/¿n/ty/rS >> 1 despite ts being large. In this case, the deep atmosphere is still isothermal, but it becomes much hotter than the skin temperature - indeed it becomes hotter without bound as ts is made larger. In this case, it is the top of the atmosphere which equilibrates at the relatively cold skin temperature, while the deep atmosphere exhibits a strong greenhouse effect. Because the deep atmosphere is isothermal, it is by
stable and will not generate a troposphere. This is a possible model for the internal state of a gas or ice giant with little internal heat flux, whose atmosphere is optically thick in the infrared but for which there is only weak solar absorption in the deep atmosphere.
The solution given in Eq. 4.56 shows that one can account for the temperature increase in the Earth's stratosphere if the upper atmosphere strongly absorbs solar radiation. As a model of the Earth's atmosphere, however, it has the shortcoming that if one makes the stratospheric absorption strong enough to yield a temperature increase with height, essentially all the solar beam is depleted in the stratosphere, leaving an isothermal lower atmosphere that won't convect and create a troposphere. What happens in Earth's real stratosphere is that ozone is a good absorber only in the ultraviolet part of the Solar spectrum. Once the ultraviolet is depleted, the remaining flux making its way into the lower atmosphere is only weakly absorbed by the atmosphere. In terms of Eq. 4.55, the solar decay rate ^J|F(_1 d2Fq^/d,T2| is large in the upper atmosphere, where there is still plenty of ultraviolet to absorb; in consequence, the temperature increases with height there. In the lower atmosphere the solar decay rate becomes small, so that the radiative equilibrium temperature decreases with height. There is also plenty of solar radiation left to heat the ground, destabilize the atmosphere, and create a troposphere.
The solution in Eq. 4.56 also explains why human-caused increases in CO2 over the past century have led to tropospheric warming but stratospheric cooling, as illustrated in Figure 1.17. Increasing the greenhouse gas concentration is equivalent to increasing tto. If one plots temperature as a function of pressure for a sequence of increasing tto, the phenomenon is immediately apparent in cases where the upper level solar absorption is sufficiently strong. The behavior is explored in Problems ??. Without solar absorption, increasing warms the atmosphere at every level, though the amount of warming decreases with height as the temperature asymptotes to the skin temperature. With solar absorption, however, the increased infrared cooling of the upper atmosphere offsets more and more of the warming due to solar absorption, leading to a cooling there. In the real atmosphere convection modifies the temperature profile in the lower atmosphere. Further, one must take into account real gas infrared and solar absorption in order to quantitatively account for the observed temperature trends. Nonetheless, the grey-gas pure radiative equilibrium calculation captures the essence of the mechanism.
The general lesson to take away from this discussion is that solar absorption near the top of the atmosphere stabilizes the atmosphere, reduces the greenhouse effect, and cools the lower portion of the atmosphere and also the ground. This is important in limiting the effectiveness of greenhouse gases like CH4 and SO2, which significantly absorb solar radiation when their concentration becomes very high. It is also the way high altitude solar-absorbing haze clouds on Titan and perhaps Early Earth act to cool the troposphere. The soot and dust clouds lofted by an asteroid impact act similarly. In contrast, solar absorption concentrated near the ground has an effect which is not much different from simply reducing the albedo of the ground itself.
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