## OLR AaT p14 AaLR4494

where p* is defined as before and A' is an order unity constant which depends on L/R An examination of the g dependence of the calculated Kombayashi-Ingersoll limit in Figure 4.37 shows that over the range 1m/s2 < g < 100m/s2, the numerically computed dependence can be fit almost exactly with this formula if we take A' = .7344 and ko = .055 (assuming po = 104Pa). Though the pressure dependence of absorption causes the limiting OLR to vary more slowly with g than was the case for constant k, the limit in the real gas case otherwise behaves very much like an equivalent grey gas with ko = .055. This is a surprising result, given the complexity of the real-gas absorption spectrum. The fact that the equivalent absorption is similar to that characterizing the 2500 cm-1 continuum suggests that the limiting OLR is being controlled primarily by this continuum. Thus, the behavior of this continuum is crucial to the runaway greenhouse phenomena (see Problem ??). It cannot be ruled out that other continua may affect the OLR as temperature is increased to very high temperatures. For example, the total blackbody radiation at wavenumbers greater than 5000cm-1 is only 1.14W/m2 at 500K, so it matters little what the absorption properties are in that part of the spectrum at 500K or cooler. However, when the temperature is reaised to 600K the shortwave emission is 15W/m2 and so the shortwave absorption begins to matter; by the

320 300

pi eR 280

260 240

260 280 300 320 340 360 380 400 Surface Temperature (K)

Figure 4.38: As for Fig. 4.37 but for a mixture of water vapor in N2 on the saturated moist adiabat. Calculations were carried out with a surface gravity of 20m/s2, for the indicated values of N2 partial pressure at the ground.

time T reaches 700K the shortwave blackbody emission is 106W/m2 and the shortwave emission properties are potentially important. On the other hand, at such temperatures there is so much water vapor in the atmosphere that a very feeble absorption would be sufficient to eliminate the contribution to the OLR.

Exercise 4.6.1 Verify the shortwave blackbody emission numbers given in the preceding paragraph by using numerical quadrature applied to the Planck function.

Exercise 4.6.2 Compute the Kombayashi-Ingersoll limit for water vapor on Mars, which has g = 3.71 m/s2. Compute the limit for Titan, which has g = 1.35m/s2, and Europa which has g = 1.31m/s2.

Now let's generalize the calculation, and introduce a noncondensing background gas which is transparent in the infrared; we use N2 in this example, though the results are practically the same if we use any other diatomic molecule. The background gas affects the Kombayashi-Ingersol limit in two ways: First, the pressure broadening increases absorption, which should lower the limit. Secondly, the background gas shifts the lapse rate toward the dry adiabat, which is much steeper than the single-component saturated adiabat. The increase in lapse rate in principle could enhance the greenhouse effect, but given the condensible nature of water vapor, it actually reduces the greenhouse effect, because the low temperatures aloft sharply reduce the amount of water vapor there. If the background gas were itself a greenhouse gas, this effect might play out rather differently. At sufficiently high temperatures, water vapor will dominate the background gas and so the limiting OLR at high temperature will approach the pure water vapor limit shown in Figure 4.37. However, for intermediate temperatures, the background gas can modify the shape of the OLR curve.

Results for various amounts of N2 are shown in Fig. 4.38. As expected., at large temperatures the limiting OLR asymptotes to the value for a pure water vapor atmosphere. This can be seen especially clearly for the case with only 100MB of N2 in the atmosphere; with more N2, one has to go to higher temperatures before the water vapor completely dominates the OLR, but the

260 280 300 320 340 360 380 400 Surface Temperature (K)

Figure 4.38: As for Fig. 4.37 but for a mixture of water vapor in N2 on the saturated moist adiabat. Calculations were carried out with a surface gravity of 20m/s2, for the indicated values of N2 partial pressure at the ground.

trend is clear. A very important qualitative difference with the pure water vapor case is that the OLR curve for a binary mixture shows a distinct maximum at intermediate temperatures. This maximum arises because the foreign broadening of water vapor absorption features is relatively weak, while the presence of a noncondensing background gas steepens the lapse rate and reduces the amount of water vapor aloft. The hump in the curve means that the surface temperature exhibits multiple equlibria for a given absorbed solar radiation. For example, with 100mb of N2, if the absorbed solar radiation is 320W/m2 there is a cool equilibrium with Tg = 288K and a hot equilibrium with Tg = 360K. The latter is an unstable equilibrium; displacing the temperature in the cool direction will cause water vapor to condense and OLR to decrease, cooling the climate further until the system falls into the cool equilibrium. Conversely, displacing the temperature slightly to the warm side of the hot equilibrium will cause the climate to go into a runaway state. For these atmospheric parameters, the planet is in a metastable runaway state. The climate will persist in the cooler non-runaway state unless some transient event warms the planet enough to kick it over into the runaway regime. It is only when the absorbed solar radiation is increased to the maximum OLR at the peak (328 W/m2) that a runaway becomes inevitable. For future use, we'll note that a calculation with g = 10m/s2 and 1 bar of N2 has a peak OLR of 310 W/m2 at Tg = 325K, while a case with the same gravity and 3 bars of N2 likewise has a peak at 310W/m2 but the position of the peak is shifted to 360K. The corresponding parameters for the slightly lower surface gravity of Venus differ little from these numbers. Both cases asymptote to the OLR for pure water vapor when the temperature is made much larger than the temperature at which the peak OLR occurs.

• Runaway greenhouse on Earth: With present absorbed solar radiation (adjusted for net cloud effects) of 265 W/m2, the Earth at present is comfortably below the Kombayashi- Ingersoll limit for a planet of Earth's gravity. According to Eq. 1.1, as the solar luminosity continues to increase, the Earth will pass the 291 W/m2 threshold where a runaway becomes possible in about 700 million years. In 1.7 billion years, it will pass the 310W/m2 threshold where a runaway becomes inevitable for an atmosphere with 1 bar of N2 and no greenhouse gases other than water vapor.

• Venus: The present high albedo of Venus is due to sulfuric acid clouds that would almost certainly be absent in a less dry atmosphere. If we assume an Earthlike albedo of 30%, then very early in the history of the Solar System, the absorbed solar radiation of Venus would be 327 W/m2. This is just barely in excess of the mandatory runaway threshold of 310W/m2 for a planet of Venus' surface gravity, assuming a bar or two of N2 in the atmosphere and no greenhouse gases other than water vapor. It is thus possible that neglected effects (clouds, subsaturation, a higher albedo surface) could allow Venus to exist for a while in a hot, steamy but non-runaway state with a liquid ocean. The high water vapor content of the upper atmosphere would still allow an enhanced rate of photo-dissociation and escape of water to space. If Venus indeed started life with an ocean, however, it is plausible that it eventually succumbed to a runaway state, since with the present solar constant the absorbed solar radiation without sulfuric acid clouds would be 457W/m2, well in excess of the runaway threshold.

• Gliese 581c: We can now improve our earlier estimates of the conditions on the extrasolar planet Gliese 581c, which has an absorbed solar radiation of 583 W/m2 assuming a rocky surface. This flux is well above the threshold of 334W/m2 for a mandatory runaway for a planet with twice Earth's surface gravity, even allowing for 2 bar of N2 in the atmosphere. Thus, if Gliese 581c ever had an ocean it is likely to have gone into a runaway state; if the composition of the planet included a substantial amount of carbonate in the interior, subsequent outgassing is likely to have turned it into a planet rather like Venus. It still remains, however, to assess the implications of the increased atmospheric absorption due to the greater proportion of infrared output of the M-dwarf host star.

• Evaporation of icy -moons in Earthlike orbit: It has been suggested that icy moons like Europa or Titan could become habitable if the host planet were in an orbit implying Earthlike solar radiation. The low Kombayashi-Ingersoll limit for bodies with low surface gravity puts a severe constraint on this possibility, however. With the albedo of ice, such bodies could exist as Snowballs in an Earthlike orbit, but if the surface ever thawed, or failed to freeze in the firt place, the absorbed solar radiation corresponding to an albedo of 20% would be 274W/m2 -well above the runaway threshold of 232 W/m2 for a body with surface gravity of 1 m/s2. Small icy moons in Earthlike orbits are thus likely to evaporate away, unless they are locked in a Snowball state.

• Lifetime of a post-impact steam atmosphere: Suppose that in the Late Early Bombardment stage, enough asteroids and comets hit the Earth to evaporate 10 bars worth of the ocean and give the Earth a 10 bar atmosphere consisting of essentially pure water vapor (and a surface temperature in excess of 440K, according to Clausius-Clapeyron). How long would it take for the steam atmosphere to rain out and the temperature to recover to normal? To do this problem, we assume that the atmosphere remains saturated as it cools, and loses heat at the maximunm rate given by the Kombayashi-Ingersoll limit for Earth; we also need to subtract the absorbed solar radiation from the heat loss. For Early Earth conditions, the net heat loss is about 100W/m2. On the other hand, the latent heat per square meter of the Earth's surface in a steam atmosphere with surface pressure ps is Lps/g, or 2.5 • 1011 J/m2 for the stipulated atmosphere. To remove this amount of energy at a rate of 100W/m2 would take 2.5 • 109s, or 80 years. The rainfall rate during this time would be warm but gentle: 3.5(kg/m2)/day, or a mere 3.5mm/day based on the water density of 1000kg/m3. This is the average rainfall rate constrained by the rate of radiative cooling, but it is likely that at places the local rainfall rate could be orders of magnitude greater, owing to the lifting and condensation in storms and other large scale atmospheric circulations.

• Freeze-out time of a magma ocean:In Chapter 1 we introduced the problem of the freeze-out time of a magma ocean on the Early Earth, and estimated the time assuming a transparent atmosphere in Problem ??. How long does it take for the magma ocean to freeze out if the planet is sufficiently water-rich that the atmosphere consists of essentially pure water vapor in saturation? The time is estimated in the same way as in Problem ?? except that the rate of heat loss is again taken to be the difference between the Kombayashi-Ingersoll limit -giving the maximum OLR - and the rate of absorption of solar energy. For the Early Earth this would be about 100W/m2, which is far smaller than the transparent atmosphere case, where the energy loss is nearly 100,000W/m2 based on <rT4 for the 2000K temperature of molten magma. As a result, the freeze-out time (using the same assumptions as in Problem ??) increases to 3.5 million years.

The latter two estimates follow the line of thinking introduced by Norman Sleep of Stanford

University, and again illustrate the principle that Big Ideas come from simple models.

Exercise 4.6.3 Estimate the lifetime of a post-impact pure water vapor atmosphere on Mars assuming that the planet absorbs 90W/m2 of solar radiation, per unit surface area. Estimate the precipitation rate, in mm of liquid water per day.

P4 J

Figure 4.39: Qualitative influence of a noncondensible greenhouse gas on the shape of the OLR curves. The upper curve gives the OLR for an atmosphere consisting of a mixture of a saturated condensible greenhouse gas with a noncondensing transparent background gas, as in the N2/H2O case shown in Fig. 4.38, while the lower curve illustrates how the behavior would chance if a large amount of noncondensible greenhouse gas were added. The intermediate curve gives the OLR for a one-component saturated greenhouse gas atmosphere as in Fig. 4.37.

### Surface Temperature

Figure 4.39: Qualitative influence of a noncondensible greenhouse gas on the shape of the OLR curves. The upper curve gives the OLR for an atmosphere consisting of a mixture of a saturated condensible greenhouse gas with a noncondensing transparent background gas, as in the N2/H2O case shown in Fig. 4.38, while the lower curve illustrates how the behavior would chance if a large amount of noncondensible greenhouse gas were added. The intermediate curve gives the OLR for a one-component saturated greenhouse gas atmosphere as in Fig. 4.37.

The concepts of runaway greenhouse and the Kombayashi-Ingersoll limit generalize to gases other than water vapor. For example, consider a planet with a reservoir of condensed CO2 at the surface, which may take the form of a CO2 glacier or a CO2 ocean, according to the temperature of the planet. Specifically, if the surface temperature is above the triple point of 216.5K the condensible reservoir takes the form af a CO2 ocean; otherwise it takes the form of a dry-ice glacier. If the atmosphere is in equilibrium with the surface reservoir and has no other gases in it besides the CO2 which evaporates from the surface, then one can use the one-component adiabat with the homebrew radiation code to compute an OLR curve for the saturated CO2 atmosphere which is analogous to the water vapor result shown in Fig. 4.37. Results, for various surface gravity, are shown in Fig. 4.40. The general behavior is very similar to that we saw for water vapor, but the whole system operates at a lower temperature and the OLR reaches its limiting value at a much lower temperature than was the case for water vapor.

The CO2 runaway imposes some interesting contraints on the form in which CO2 could exist on Mars, both present and past. For Martian surface gravity, the Kombayashi-Ingersoll limit for CO2 is a bit over 63W/m2. In consequence, when the absorbed solar radiation exceeds this value, a permanent reservoir of condensed CO2 cannot exist at the surface of the planet; it will sublimate or evaporate into the atmosphere, and continue to warm the planet until all the condensed reservoir has been converted to the gas phase. At present, the globally averaged solar absorption is about 110 W/m2, so the planet is well above the runaway threshold for CO2. From this we can conclude that Mars cannot have an appreciable permanent reservoir of condensed CO2 which can exchange with the atmosphere. Note, however, that this does not preclude the temporary buildup of CO2 snow at the surface. Such deposits can and do form near the winter poles, but sublimate back into the atmosphere as spring approaches. This situation can be thought of as arising from the fact that the local absorbed solar radiation near the winter pole is below the Kombayashi-Ingersoll limit for CO2. The local reasoning applies because the thin atmosphere of present Mars cannot effectively transport heat from the summer hemisphere.

Even without the albedo due to a thick CO2 atmosphere, Early Mars would have an absorbed solar radiation of only 77W/m2. This is still somewhat above the Kombayashi-Ingersoll limit for a pure CO2 atmosphere, but allowing for the scattering effects of the atmosphere and perhaps also the influence of nitrogen in the atmosphere, Early Mars could well have sustained permanent CO2 glaciers, given a sufficient supply of CO2. Because the planet is so near a threshold, a more detailed calculation - probably involving consideration of atmospheric heat transports - would be needed to resolve the issue.

One can similarly compute a Kombayashi-Ingersoll limit for methane, using the continuum absorption properties described in Section 4.4.8. This calculation would determine whether a body could have a permanent methane ocean, swamp or glacier at the surface.

Any gas becomes condensible at sufficiently low temperatures or high pressures, and it is in fact the Kombayashi-Ingersoll limit that determines whether a volatile greenhouse gas outgassing from the interior of the planet accumulates in the atmosphere, or accumulates as a massive condensed reservoir (which may be a glacier or ocean) 8. In the latter case, additional outgassing goes into the condensed reservoir, and the amount of volatile remaining in the atmosphere in the gas phase is determined by the the temperature of the planet. The condensed reservoir can form only if the absorbed solar radiation is below the Kombayashi-Ingersoll limit for the gas in question, and even then only if the total mass of volatiles available is sufficient to bring the atmosphere to a state of saturation. As an example of the latter constraint, let's suppose that Mars were in a

8There are other places an outgassed atmosphere can go; water can go into hydration of minerals, and CO2 can be bound up as carbonate rocks.

Surface Pressure (bar)

Surface Pressure (bar)

Surface Temperature (K)

Figure 4.40: OLR vs surface temperature for a saturated pure CO2 atmosphere. Calculations were performed with the values of surface gravity indicated on each curve. The scale at the top gives the surface pressure corresponding to the temperature on the lower scale.

Surface Temperature (K)

Figure 4.40: OLR vs surface temperature for a saturated pure CO2 atmosphere. Calculations were performed with the values of surface gravity indicated on each curve. The scale at the top gives the surface pressure corresponding to the temperature on the lower scale.

more distant orbit, where the absorbed solar radiation were only 40W/m2. Then, according to Fig. 4.40, the equilibrium surface temperature would be 165K in saturation, and the corresponding surface pressure would be 5600 Pa. In order to reach this surface pressure it is necessary to outgas 5600/g, or 1509 kg of CO2 per square meter of the planet's surface. On Earth, outgassed water vapor accumulates in an ocean because the Earth is below the Kombayashi-Ingersoll limit for water vapor. With present solar luminosity Venus (without clouds) is well above the limit, so any outgassed water vapor would accumulate in the atmosphere (apart from the leakage to space). For CO2, Earth, Mars and Venus are all above the Kombayashi-Ingersoll limit, so outgassed CO2 accumulates in the atmosphere (apart from whatever gets bound up in mineral form). Even if you took away the water that allows CO2 to be bound up as carbonate, Earth would not develop a CO2 ocean; it would become a hot Venus-like planet instead, with a dense CO2 atmosphere.

In parting, we must mention two serious limitations of our discussion of the runaway greenhouse phenomenon. First, in computing the Kombayashi-Ingersoll limit, it was assumed that the atmosphere was saturated with the condensible greenhouse gas. However, real atmospheres can be substantially undersaturated, though the dynamics determining the degree of undersaturation is intricate and difficult to capture in simplified models. Undersaturation is likely to raise the threshold solar radiation needed to trigger a runaway state. The second limitation is that the calculations were carried out for clear sky conditions. Clouds exert a cooling influence through their shortwave albedo, and a warming influence through their effect on OLR, and the balance is again hard to determine by means of any idealized calculation. Whether clouds have an inhibitory effect on the runaway greenhouse is one of the many remaining Big Questions.