We have seen in Chapter 2 that the mass of an atmosphere in equilibrium with a reservoir of condensed substance (e.g. a water ocean) is not fixed. It increases with temperature in accordance with the dictates of the Clausius-Clapeyron relation. If the condensible substance is a greenhouse gas, then the optical thickness increases with temperature. This tends to reduce the OLR, offsetting or even reversing the tendency of rising temperature to increase the OLR. What are the implications of this for the dependence of OLR on surface temperature, and for planetary energy balance? The resulting phenomena are most commonly thought about in connection with the effects of a water ocean on evolution of a planet's climate, but the concept generalizes to any condensible greenhouse gas in equilibrium with a large condensed reservoir. We'll take a first look at this problem here, in the context of the grey-gas model.

In the general case, we'd like to consider an atmosphere in which the condensible greenhouse gas is mixed with a noncondensible background of fixed mass (which may also have a greenhouse effect of its own). This is the case for water vapor in the Earth's atmosphere, for methane on Titan, and probably also for water vapor in the early atmosphere of Venus. It could also have been the case for mixed nitrogen-CO2 atmospheres on Early Mars, with CO2 playing the role of the condensible component. We will eventually take up such atmospheres, but the difficulty in computing the moist adiabat for a two-component atmosphere introduces some distractions which get in the way of grasping the key phenomena. Hence, we'll start with the simpler case in which the atmosphere consists of a pure condensible component in equilibrium with a reservoir (an "ocean," or perhaps a glacier). In this case, the saturated moist adiabat is given by the simple analytic formula Eq. 2.27, obtained by solving the simplified form of the Clausius-Clapeyron relation for temperature in terms of pressure. We've already seen in Chapter 2 that a mixed atmosphere is dominated by the condensible component at large temperatures, so if we are primarily interested in the large-temperature behavior, the use of the one-component condensible atmosphere is not at all a bad approximation.

We write T(p) = To/(1 — ■RT0 ln pp), where (po,To) are a fixed reference temperature and pressure on the saturation curve, such as the triple point temperature and pressure. If the surface pressure is ps, then the surface temperature is Ts = T(ps). Hence, specifying surface pressure is equivalent to specifying surface temperature in this problem. To keep the algebra simple, we'll assume a constant specific absorption «. Then = Kps/g, which increases as Ts is made larger. Further, for constant specific absorption, p/po = (tto — t')/ro where To = «po/g. Now, the choice of the reference temperature and pressure (po,To) is perfectly arbitrary, and we'll get the same answer now matter what choice we make (within the accuracy of the approximate form of Clausius-Clapeyron we are using). Hence, we are free to set po = g/« so that To = 1. To then implicitly depends on «, and becomes larger as « gets smaller. To is the temperature at the level of the atmosphere where the optical depth measured relative to the top of the atmosphere is unity.

Substituting the one-component T(p) into the integral giving the solution to the Schwarzschild equation, and substituting for pressure in terms of optical thickness, we find

where we have in the second line defined a new dummy variable t1 = t0 — t' as before. The surface temperature enters the expression for OLR only through t0, which is proportional to surface pressure. In the optically thin limit, the integral on the right hand side of the expression is small (because t0 is small). This happens at low surface temperatures, because ps is small when the surface temperature is small. The OLR then reduces to the first term, which is approximately 1+(0),i.e. the unmodified upward radiation from the surface. In the optically thick limit, which occurs for high surface temperatures, the term proportional to 1+(0) is negligible, and the second term dominates. This term consists of the flux aT4 multiplied by a non-dimensional integral. Recall that To is a constant dependent on the thermodynamic and infrared optical properties of the gas making up the atmosphere; it does not change with surface temperature. Because of the decaying exponential in the integrand, the integral is dominated by the contribution from the vicinity of t1 = 0, and will therefore become independent of t0 for large t0 1. In the optically thick (high temperature) limit, then, the integral is a function of RTo/L alone. From this we conclude that the OLR becomes independent of surface temperature in the limit of large surface temperature (and hence large t0). This limiting OLR is known as the Kombayashi-Ingersoll limit. It was originally studied in connection with the long-term history of water on Venus, using a somewhat different argument than we have presented here. We shall use the term to refer to a limiting OLR arising from the evaporation of any volatile greenhouse gas reservoir, whether computed using a grey gas model or a more realistic radiation model.

It is readily verified that the integral multiplying aT^ approaches unity as RTo/L approaches zero. In fact, for typical atmospheric gases L/R is a very large temperature, on the order of several thousand Kelvins. Hence, unless the specific absorption is exceedingly small, RTo/L tends to be small, typically on the order of .1 or less. For RTo/L = .1, the integral has the value of .905. Thus, the limiting OLR is essentially aT^. Recalling that To is the temperature of the moist adiabat at one optical depth unit down from the top of the atmosphere, we see that the limiting OLR behaves very nearly as if all the longwave radiation were emitted from a layer one optical depth unit from the top of the atmosphere.

Figure 4.3 shows some results from a numerical evaluation of the integral in Eq. 4.38. For small surface temperatures, there is little atmosphere, and the OLR increases like aTs4. As the surface temperature is made larger, the atmosphere becomes thicker and the OLR eventually asymptotes to a limiting value, as predicted. In accordance with the argument given above, the limiting OLR should be slightly less than the blackbody flux corresponding to the temperature To found one optical depth down from the top of the atmosphere. This temperature depends on g/K, which is the pressure one optical depth down from the top. For g/K = 100Pa, solving the simplified Clausius-Clapeyron relation for T at 100Pa yields T0 = 250.3K, whence aTo4 = 222.6W/m2; for g/K = 1000Pa, To = 280.1K and aT04 = 349.0W/m2. These values are consistent with the numerical results shown in the graph.

Note that for a given atmospheric composition (which determines k) the Kombayashi-Ingersoll limit depends on the acceleration of gravity. A planet with stronger surface gravity will have a higher Kombayashi-Ingersoll limit than one with weaker gravity. An explicit formula for the dependence on g/K is obtained by substituting for To using the formula for the single-component saturated adiabat, Eq. 2.27. Thus, the limiting OLR can be written in the form

where A(L/R) is the order unity constant discussed previously and p* = pref exp(L/RTref). In technically, the integral diverges at extremely large r^, because the denominator of the integrand can vanish. This is an artifact of assuming a constant latent heat and has no physical significance.

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E 200

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Figure 4.3: OLR vs surface temperature for a one-component grey gas condensible atmosphere in equilibrium with a reservoir. Calculations were done for thermodynamic parameters L and R corresponding to water vapor. Results are shown for two different values of g/K, where k is the specific cross section of the gas and g is the acceleration of gravity.

the formula for p*, (pref, Tref) is any point on the saturation vapor pressure curve, for example the triple point temperature and pressure. p* is an enormous pressure (2.3 • 1011Pa for water vapor), so Eq. 4.39 predicts that the Kombayashi-Ingersoll limit increases as surface gravity increases, since increasing g makes the logarithm in the denominator smaller. The apparent singularity when Kp*/g = 1 is spurious, as the approximations we have made break down long before that value is reached.

We are now prepared to describe the runaway greenhouse phenomenon. Let S be the absorbed solar radiation per unit surface area of the planet, and let the limiting OLR computed above be OLRmax .If S < OLRmax the planet will come to equilibrium in the usual way, warming up until it loses energy by infrared radiation at the same rate as it receives energy from its star. But what happens if S > OLRmax ? In this case, as long as there is still an ocean or other condensed reservoir to feed mass into the atmosphere, the planet cannot get rid of all the solar energy it receives no matter how much it warms up; hence the planet continues to warm until the surface temperature becomes so large that the entire ocean has evaporated into the atmosphere. The temperature at this point depends on the mass and composition of the volatile reservoir. For example, the Earth's oceans contain enough mass to raise the surface pressure to about 100bars if dumped into the atmosphere in the form of water vapor. The ocean has been exhausted when the saturation vapor pressure reaches this value. Using the simplified exponential form of the Clausius-Clapeyron relation to extrapolate the vapor pressure from the sea level boiling point (1 bar at 373.15K), we estimate that this vapor pressure is attained at a surface temperature of about 550K. This estimate is inaccurate, because the latent heat of vaporization varies appreciably over the range of temperatures involved. A more exact value based on measurements of properties of steam is 584K, but the grim implications for survival or emergence of life as we know it are largely the same.

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Figure 4.4: Schematic picture of the termination of a runaway greenhouse upon depletion of the volatile reservoir.

At temperatures larger than that at which the ocean is depleted, the mass of the atmosphere becomes fixed and no longer increases with temperature. The greenhouse gas content of the atmosphere - which in the present case is the entire atmosphere - no longer increases with temperature. As a result, the OLR is once more free to increase as the surface becomes warmer, and the planet will warm up until it reaches an equilibrium at a temperature warmer than that at which the ocean is depleted. The additional warming required depends on the gap between the Kombayashi-Ingersoll limit and the absorbed solar radiation. This situation is depicted schematically in Figure 4.4. Once the ocean is gone, the lower atmosphere is unsaturated and air can be lifted some distance before condensation occurs. The resulting atmospheric profile is on the dry adiabat in the lower atmosphere, transitioning to the moist adiabat at the altitude where condensation starts. The situation is identical to that depicted for CO2 in Fig. 2.6. Rain will still form in the condensing layer. Much of it will evaporate in the lower noncondensing layer; some of it may reach the ground, but the resulting puddles would tend to rapidly evaporate back into the highly undersaturated lower atmosphere. As surface temperature is made larger, the altitude where condensation sets in moves higher, until at very large temperatures the atmosphere behaves like a noncondensing dry system (albeit one where the entire atmosphere may consist of water vapor).

The runaway greenhouse phenomenon may explain how Venus wound up with such a radically different climate from Earth, despite having started out in a rather similar state. The standard story goes something like this: Venus started with an ocean, and with most of it's CO2 bound up in rocks as is the case for Earth. However, it was just enough closer to the Sun to trigger a runaway greenhouse. Once the entire ocean had evaporated into the atmosphere, there was so much water vapor in the upper atmosphere that it could be broken apart by energetic solar ultraviolet rays, whereafter the light hydrogen could escape to space. The highly reactive oxygen left behind would react to form minerals at the surface. Once there was no more liquid water in play, the reactions that bind up carbon dioxide in rocks could no longer take place (as will be explained in Chapter 8), so all the planet's CO2 outgassed from volcanism and stayed in the atmosphere, leading to the hot, dry super-dense atmosphere of modern Venus.

Assuming habitability to require a reservoir of liquid water, the Kombayashi-Ingersoll limit for water determines the inner orbital limit for habitability, since if the Solar constant exceeds the limiting flux a runaway will ensue and any initial ocean will not persist. It also determines how long it takes before the planet's Sun gets bright enough to trigger a runaway, and thus sets the lifetime of a water-dependent biosphere (Earth's included). Accurate calculations of the Kombayashi-Ingersoll limit are therefore of critical importance to understanding the limits of habitability both in time and orbital position. The grey gas model is not good enough to determine the value of po appropriate to a given gas, and so cannot be used for accurate evaluations of the runaway greenhouse threshold. We can at least say that, all other things being equal, a planet with larger surface gravity will be less susceptible to the runaway greenhouse. This is so because po = g/«, whence larger g implies larger po, which implies in turn larger T(po) and hence a larger limiting OLR. This observation may be relevant to the class of extrasolar planets known as "Large Earths."

We will revisit the runaway greenhouse using more realistic radiation physics in Section 4.6. Some effects of the effects of the stratosphere and of clouds will be brought into the picture in Chapter 5.

The runaway greenhouse phenomenon is usually thought of in conjunction with water vapor, but the concept applies equally well to any situation where there is a volatile reservoir of greenhouse gas, whether it be in solid or liquid form. For example, one could have a runaway greenhouse in association with the sublimation of a large CO2 ice cap, or in association with the evaporation of a Methane or Ammonia ocean. In fact, the Kombayashi-Ingersoll limit determines whether a planet would develop a reservoir of condensed substance at its surface (a glacier or ocean), given sustained outgassing of that substance in the absence of any chemical sink. As the gas builds up in the atmosphere, the pressure increases and it would eventually tend to condense at the surface, preventig any further gaseous accumulation. However, the greenhouse effect of the gas warms the surface,which increases the saturation vapor pressure. The Kombayashi-Ingersoll limit tells us which effect wins out as surface pressure increases. Earth is below the threshold for water, so we have a water ocean. Venus is above the threshold for water and CO2, so both accumulate as gases in the atmosphere (apart from possible escape to space). When we revisit the problem with real-gas radiation, we will be able to say whether CO2 would form a condensed reservoir on Earth or Mars, or on Titan, given sustained outgassing in the absence of a chemical sink.

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