Figure 3.7: The Earth's observed zonal-mean OLR for January, 1986. The observations were taken by satellite instruments during the Earth Radiation Budget Experiment (ERBE), and are averaged along latitude circles. The figure also shows the radiation that would be emitted to space by the surface (aTf) if the atmosphere were transparent to infrared radiation.
condensed substances absorb infrared as well as water does. Liquid methane (imporant on Titan) and CO2 ice (important on present and early Mars) are comparatively poor infrared absorbers. They affect OLR in a fundamentally different way, through reflection instead of absorption and emission. This will be discussed in Chapter 5.
In a nutshell, then, here is how the greenhouse effect works: From the requirement of energy balance, the absorbed solar radiation determines the effective blackbody radiating temperature Trad. This is not the surface temperature; it is instead the temperature encountered at some pressure level in the atmosphere prad, which characterizes the infrared opacity of the atmosphere, specifically the typical altitude from which infrared photons escape to space. The pressure prad is determined by the greenhouse gas concentration of the atmosphere. The surface temperature is determined by starting at the fixed temperature Trad and extrapolating from prad to the surface pressure ps using the atmosphere's lapse rate, which is approximately governed by the appropriate adiabat. Since temperature decreases with altitude over much of the depth of a typical atmosphere, the surface temperature so obtained is typically greater than Trad, as illustrated in Figure 3.6. Increasing the concentration of a greenhouse gas decreases prad, and therefore increases the surface temperature because temperature is extrapolated from Trad over a greater pressure range. It is very important to recognize that greenhouse warming relies on the decrease of atmospheric temperature with height, which is generally due to the adiabatic profile established by convection. The greenhouse effect works by allowing a planet to radiate at a temperature colder than the surface, but for this to be possible, there must be some cold air aloft for the greenhouse gas to work with.
For an atmosphere whose temperature profile is given by the dry adiabat, the surface temperature is
With this formula, the Earth's present surface temperature can be explained by taking prad/ps = .67, whence prad « 670mb. Earth's actual radiating pressure is somewhat lower than this estimate, because the atmosperic temperature decays less strongly with height than the dry adiabat. The high surface temperature of Venus can be accounted for by taking prad/ps = .0095, assuming that the temperature profile is given by the noncondensing adiabat for a pure CO2 atmosphere. Given Venus' 93bar surface pressure, the radiating level is 880mb which, interestingly, is only slightly less than Earth's surface pressure. Earth radiates to space from regions quite close to its surface, whereas Venus radiates only from a thin shell near the top of the atmosphere. Note that from the observed Venusian temperature profile in Fig. 2.2, the radiating temperature (253K) is encountered at p = 250mb rather than the higher pressure we estimated. As for the Earth, our estimate of the precise value prad for Venus is off because the ideal-gas noncondensing adiabat is not a precise model of the actual temperature profile. In the case of Venus, the problem most likely comes from the ideal-gas assumption and neglect of variations in cp, rather than condensation.
The concept of radiating level and radiating temperature also enables us to make sense of the way energy balance constrains the climates of gas giants like Jupiter and Saturn, which have no distinct surface. The essence of the calculation we have already done for rocky planets is to use the top of atmosphere energy budget to determine the parameters of the adiabat, and then extrapolate temperature to the surface along the adiabat. For a non-condensing adiabat, the atmospheric profile compatible with energy balance is T(p) = Trad(p/prad)R/Cp. This remains the appropriate temperature profile for a (noncondensing) convecting outer layer of a gas giant, and the only difference with the previous case is that, for a gas giant, there is no surface to act as a natural lower boundary for the adiabatic region. At some depth, convection will give out and the adiabat must be matched to some other temperature model in order to determine the base of the convecting region, and to determine the temperature of deeper regions. There is no longer
Observed OLR (W/m2) Absorbed Solar Flux (W/m2) Trad (actual) Trad (Solar only)
Jupiter 14.3 12.7 126K 110K
Saturn 4.6 3.8 95K 81K
Uranus .52 .93 55K 58K
Table 3.1: The energy balance of the gas giant planets, with inferred radiating temperature. The solar-only value of Trad is the radiating temperature that would balance the observed absorbed solar energy, in the absence of any internal heat source.
any distinct surface to be warmed by the greenhouse effect, but the greenhouse gas concentration of the atmosphere nonetheless affects T(p) through prad. For example, adding some additional greenhouse gas to the convecting outer region of Jupiter's atmosphere would decrease prad, and therefore increase the temperature encountered at, say, the 1 bar pressure level.
The energy balance suffices to uniquely determine the temperature profile because the non-condensing adiabat is a one-parameter family of temperature profiles. The saturated adiabat for a mixture of condensing and noncondensing gases is also a one parameter family, defined by Eq. 2.33, and can therefore be treated similarly. If the appropriate adiabat for the planet had more than one free parameter, additional information beyond the energy budget would be needed to close the problem. On the other hand, a single component condensing atmosphere such as described by Eq. 2.27 yields a temperature profile with no free parameters that can be adjusted so as to satisfy the energy budget. The consequences of this quandary will be taken up as part of our discussion of the runaway greenhouse phenonenon, in Chapter 4.
Using infrared telescopes on Earth and in space, one can directly measure the OLR of the planets in our Solar System. In the case of the gas giants, the radiated energy is substantially in excess of the absorbed solar radiation. Table 3.1 compares the observed OLR to the absorbed solar flux for the gas giants. With the exception of Uranus, the gas giants appear to have a substantial internal energy source, which raises the radiating temperature to values considerably in excess of it would be if the planet were heated by solar absorption alone. Uranus is anomalous, in that it actually appears to be emitting less energy than it receives from the sun. Uncertainties in the observed OLR for Uranus would actually allow the emission to be in balance with solar absorption, but would still appear to preclude any significant internal energy source. This may indicate a profound difference in the internal dynamics of Uranus. On the other hand, the unusually large tilt of Uranus' rotation axis means that Uranus has an unusually strong seasonal variation of solar heating, and it may be that the hemisphere that has been observed so far has not yet had time to come into equilibrium, which would throw off the energy balance estimate.
Because it is the home planet, Earth's radiation budget has been very closely monitored by satellites. Indirect inferences based on the rate of ocean heat uptake indicate that the top of atmosphere radiation budget is currently out of balance, the Earth receiving about 1W/m2 more from Solar absorption than it emits to space as infrared 3. This is opposite from the imbalance that would be caused by an internal heating. It is a direct consequence of the rapid rise of CO2 and other greenhouse gases, caused by the bustling activities of Earth's human inhabitants. The rapid greenhouse gas increase has cut down the OLR, but because of the time required to warm up the oceans and melt ice, the Earth's temperature has not yet risen enough to restore the energy balance.
3At the time of writing, top-of-atmosphere satellite measurements are not sufficiently accurate to permit direct observation of this imbalance
Exercise 3.3.2 A typical well-fed human in a resting state consumes energy in the form of food at a rate of 100W, essentially all of which is put back into the surroundings in the form of heat. An astronaut is in a spherical escape pod of radius r, far beyond the orbit of Pluto, so that it receives essentially no energy from sunlight. The air in the escape pod is isothermal. The skin of the escape pod is a good conductor of heat, so that the surface temperature of the sphere is identical to the interior temperature. The surface radiates like an ideal blackbody.
Find an expression for the temperature in terms of r, and evaluate it for a few reasonable values. Is it better to have a bigger pod or a smaller pod? In designing such an escape pod, should you include an additional source of heat if you want to keep the astronaut comfortable?
How would your answer change if the pod were cylindrical instead of spherical? If the pod were cubical?
Bodies such as Mercury or the Moon represent the opposite extreme from the uniform-temperature limit. Having no atmosphere or ocean to transport heat, and a rocky surface through which heat is conducted exceedingly slowly, each bit of the planet is, to a good approximation, thermally isolated from the rest. Moreover, the rocky surface takes very little time to reach its equilibrium temperature, so the surface temperature at each point is very nearly in equilibrium with the instantaneous absorbed solar radiation, with very little day-night or seasonal averaging. In this case, averaging the energy budget over the planet's surface gives a poor estimate of the temperature, and it would be more accurate to compute the instantaneous equilibrium temperature for each patch of the planet's surface in isolation. For example, consider a point on the planet where the Sun is directly overhead at some particular instant of time. At that time, the rays of sunlight come in perpendicularly to a small patch of the ground, and the absorbed solar radiation per unit area is simply (1 — a)LQ; the energy balance determing the ground temperature is then <rT4 = (1 — a)Lg, without the factor of 4 we had when the energy budget was averaged over the entire surface of an isothermal planet. For Mercury, this yields a temperature of 622K, based on the mean orbital distance and an albedo of .1. This is similar to the observed maximum temperature on Mercury, which is about 700K (somewhat larger than the theoretical calculation because Mercury's highly elliptical orbit brings it considerably closer to the Sun than the mean orbital position). The Moon, which is essentially in the same orbit as Earth and shares its Solar constant, has a predicted maximum temperature of 384K, which is very close to the observed maximum. In contrast, the maximum surface temperature on Earth stays well short of 384K, even at the hottest time of day in the hottest places. The atmosphere of Mars in the present epoch is thin enough that this planet behaves more like the no-atmosphere limit than the uniform-temperature limit. Based on a mean albedo of .25,the local maximum temperature should be 297K, which is quite close to the observed maximum temperature.
More generally speaking, when doing energy balance calculations the temperature we have in mind is the temperature averaged over an appropriate portion of the planet and over an appropriate time interval, where what is "appropriate" depends on the response time and the efficiency of the heat transporting mechanisms of the planet under considerations. Correspondingly, the appropriate incident solar flux to use is the incident solar flux per unit of radiating surface, averaged consistently with temperature. We will denote this mean solar flux by the symbol S. For an isothermal planet S = 4 Lq, while at the opposite extreme S = Lq for the instantaneous response at the subsolar point. In other circumstances it might be appropriate to average along a latitude circle, or over a hemisphere. A more complete treatment of geographical, seasonal and diurnal temperature variations will be given in Chapter 7.
Exercise 3.3.3 Consider a planet which is tide-locked to its Sun, so that it always shows the same
Clean new H2O snow Bare Sea ice Clean H2O glacier ice Deep Water Sahara Desert sand Martian sand Basalt (any planet) Granite Limestone Grassland Deciduous forest Conifer forest Tundra
Table 3.2: Typical values of albedo for various surface types. These are only representative values. Albedo can vary considerably as a function of detailed conditions. For example, the ocean albedo depends on the angle of the solar radiation striking the surface (the value given in the table is for near-normal incidence), and the albedo of bare sea ice depends on the density of air bubbles.
face to the Sun as it proceeds in its orbit (just as the Moon always shows the same face to the Earth). Estimate the mean temperature of the day side of the planet, assuming the illuminated face to be isothermal, but assuming that no heat leaks to the night side.
Albedo is not a static quantity determined once and for all time when a planet forms. In large measure, albedo is determined by processes in the atmosphere and at the surface which are highly sensitive to the state of the climate. Clouds consist of suspended tiny particles of the liquid or solid phase of some atmospheric constituent; such particles are very effective reflectors of visible and ultraviolet light, almost regardless of what they are made of. Clouds almost entirely control the albedos of Venus, Titan and all the gas giant planets, and also play a major role in Earth's albedo. In addition, the nature of a planet's surface can evolve over time, and many of the surface characteristics are strongly affected by the climate. Table 3.2 gives the albedo of some common surface types encountered on Earth. The proportions of the Earth covered by sea-ice, snow, glaciers, desert sands or vegetation of various types are determined by temperature and precipitation patterns. As climate changes, the surface characteristics change too, and the resulting albedo changes feed back on the state of the climate. It is not a "chicken and egg" question of whether climate causes albedo or albedo causes climate; rather it is a matter of finding a consistent state compatible with the physics of the way climate affects albedo and the way albedo affects climate. In this sense, albedo changes lead to a form of climate feedback. We will encounter many other kinds of feedback loops in the climate system.
Among all the albedo feedbacks, that associated with the cover of the surface by highly reflective snow or ice plays a distinguished role in thinking about the evolution of the Earth's climate. Let's consider how albedo might vary with temperature for a planet entirely covered by a water ocean - a reasonable approximation to Earth, which is | ocean. We will characterize the climate by the global mean surface temperature Ts, but suppose that, like Earth, the temperature
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