Water vapor feedback

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For a planet like the Earth which has a substantial reservoir of condensed water at the surface (be it ocean or glacier), if left undisturbed for a sufficiently long time water vapor would enter the atmosphere until the atmosphere reached a state where the water vapor pressure was equal to the saturation vapor pressure at all points. In a case like this, if anything happens to increase the temperature of the atmosphere, then the water vapor content will eventually increase; since water vapor is a greenhouse gas, the additional water vapor will lead to an additional greenhouse gas, warming the planet further beyond the initial warming. Amplification of this sort is known as water vapor feedback, and works to amplify cooling influences as well. In this section we'll examine some quantitative models of real-gas water vapor feedback.

It turns out that atmospheric motions have a drying effect which keeps the atmosphere from reaching saturation. This is a very active subject of current research, but suffice it to say for the moment that comprehensive simulations of the Earth's atmosphere suggest that the situation can be reasonably well represented by keeping the relative humidity fixed at some subsaturated value as the climate warms or cools. That is the approach we shall adopt here, and it still yields an atmosphere whose water vapor content increases roughly exponentially with temperature. At present, there is no generally valid theory which allows one to determine the appropriate value of

relative humidity a priori, so one must have recourse to fully dynamic general circulation models or observations. Indeed, the relative humidity is not uniform but varies considerably both in the vertical and horizontal, so there is no single value that can be said to characterize the global humidity field. Similar considerations would apply to any condensible substance, for example methane on Titan.

In these all-troposphere models, we shall also assume that the temperature profile is given by the moist adiabat. Observations of the Earth's tropics show this to be a good description of the tropospheric temperature profile even where the atmosphere is unsaturated and not undergoing convection. (See Problem ??). Evidently, the regions that are undergoing active moist convection control the lapse rate throughout the tropical troposphere; there are also theoretical reasons for believing this to be the case, but they require fluid dynamical arguments that are beyond the scope of the present volume. The situation in the midlatudes is rather less clear, but we will use the moist adiabat there as well, because it is hard to come up with something better in a model without any atmospheric circulation in it. Results are presented for Earth surface pressure and gravity, though we will make a few remarks on how the results scale to planets with greater or lesser gravity.

Since most of the big questions of Earth climate and climate of habitable Earthlike planets involve water vapor feedback in conjunction with one or more other greenhouse gases, it is in this section that we will for the first time get fairly realistic answers regarding the Faint Young Sun problem, and so forth.

The case of a saturated pure water vapor atmosphere will be treated in Section 4.6 as part of our treatment of the runaway greenhouse for real gas atmospheres. Here will will begin our discussion with water vapor mixed with Earth air, with the temperature profile on the moist adiabat for an air-water mixture. The OLR curve for this case is shown in Figure 4.31. Since there is no other greenhouse gas, the OLR for the dry case (zero relative humidity) is just gT4. For ground temperatures below 240K there is so little water vapor in the air that the exact amount of water vapor has little effect on the OLR. At larger temperatures, the curves for different relative humidity begin to diverge. Even for relative humidity as low as 10%, the water vapor greenhouse effect is sufficient to nearly cancel the upward curvature of the dry case; at 320K the OLR is reduced by over 130W/m2 compared to the dry case. At larger relative humidites, the curvature reverses, and the OLR as a function of temperature shows signs of flattening at high temperature, in a fashion reminiscent of that we saw in our discussion of the Kombayashi-Ingersoll limit for grey gases. This is our first acquaintance with the essential implication of water vapor feedback: the increase of water vapor with temperature reduces the slope of the OLR vs. temperature curve, making the climate more sensitive to radiative forcing of all sort - whether it be changes in the Solar constant, changes in surface albedo, or changes in the concentration of CO2. To take but one example, increasing the absorbed solar radiation from 346W/m2 to 366W/m2 increases the ground temperature from 280K to 283K in the dry case. When the relative humidity is 50%, it takes only 290W/m2 of absorbed solar energy to maintain the same 280K ground temperature, but now increasing the solar absorption by 20W/m2 increases the surface temperature to 288K. Water vapor feedback has approximately doubled the climate sensitivity, which is a typical result for Earthlike conditions. The increase in climate sensitivity due to water vapor feedback plays a part in virtually any climate change phenomenon that can be contemplated on a planet with a liquid water ocean.

The influence of water vapor is strongest at tropical temperatures, but is still significant even at temperatures near freezing. It only becomes negligible at temperatures comparable to the polar winter.

Ground Temperature (K)

Figure 4.31: OLR vs. surface temperature for water vapor in air, with relative humidity held fixed. The surface air pressure is 16ar, and Earth gravity is assumed. The temperature profile is the water/air moist adiabat. Calculations were carried out with the ccm radiation model.

One sometimes hears it remarked cavalierly that water vapor is the "most important" greenhouse gas in the Earth's atmosphere. The misleading nature of such statements can be inferred directly from Figure 4.31. Let's suppose the Earth's climate to be in equilibrium with 256W/m2 of absorbed solar radiation, averaged over the Earth's surface. This corresponds to an albedo of 25%, which we take to be somewhat smaller than the actual observed albedo to account crudely for the fact that part of the cloud albedo effect is canceled by cloud greenhouse effects which we do not take into account in the Figure. If water vapor were the only greenhouse gas in the Earth's atmosphere, the temperature would be a chilly 268K, and that's even before taking ice albedo feedback into account, which would most likely cause the Earth to fall into a frigid Snowball state. We saw earlier that the Earth would also be uninhabitably cold if CO2 were the only greenhouse gas in the atmosphere. With regard to Earth's habitability, it takes two to tango. In order to maintain a habitable temperature on Earth without the benefit of CO2, the Sun would have to be 13% brighter. It will take well over a billion more years before the Sun will become this bright.

Now let's add some CO2 to the atmosphere. Figure 4.32 shows how the OLR curve changes if we add in 300ppmv of CO2 - slightly more than the Earth's pre-industrial value. The general pattern is similar to the water-only case, but shifted downward by an amount that varies with temperature. At 50% relative humidity, the addition of CO2 reduces the OLR at 280K by a further 36W/m2 below what it was with water vapor alone. Because of the additional greenhouse effect of CO2, the same 256W/m2 of absorbed solar radiation we considered previously can now support a temperature of 281K when the relative humidity is 50%. Note that without the action of CO2, the atmosphere would be too cold to have much water vapor in it, so one would lose much of the greenhouse effect of water vapor as well. It appears that the actual surface temperature of Earth can be satisfactorily accounted for on the basis of the CO2 greenhouse effect supported by water vapor feedback.

Exercise 4.5.2 For the four moisture conditions in Figure 4.32, determine how much absorbed solar radiation would be needed to support a surface temperature of 280K. For each case, use the graph to estimate how much the surface temperature would increase if the absorbed solar radiation were increased were increased by 20W/m2 over the original value. How does the amplication due

Ground Temperature (K)

Figure 4.32: As in Figure 4.31, but with 300ppmv of CO2 included. Note that the "Dry" case excludes only the radiative effects of water vapor; the moist adiabat is still employed for the temperature profile.

to water vapor feedback compare with the results obtained without CO2 in the atmosphere?

Next, let's take a look at how the OLR curve varies as a function of CO2, with relative humidity held fixed at 50%. The results are shown in Fig. 4.33. The addition of CO2 to the atmosphere lowers all the curves, and the more CO2 you add, the lower the curves go. CO2 is planetary insulation: adding CO2 to a planet reduces the rate at which it loses energy, for any given surface temperature, just as adding fiberglass insulation to a house reduces the rate at which the house loses energy for a fixed interior temperature (thus reducing the fuel that must be burned in order to maintain the desired temperature). Another thing we note is that when the CO2 concentration becomes very large, the curve loses its negative curvature and becomes concave upward, like <rT4. This happens because the CO2 greenhouse effect starts to dominate the water vapor greenhouse effect, so that the flattening of the OLR curve due to the increase of water vapor with temperature becomes less effective over the temperature range shown. Even for very high CO2, water vapor eventually would assert its dominance as temperatures are raised in excess of 320K, causing the curve once more to flatten as the Kombayashi-Ingersoll limit is approached.

As an example of the use of the information in Fig. 4.33, let's start with a planet with 100ppmv of CO2 in its atmosphere, together with a sufficient supply of water to keep the atmosphere 50% saturated in the course of any climate change. From the graph, we see that absorbed solar radiation of 257 W/m2 would be sufficient to maintain a mean surface temperature of 280K. From the graph we can also see that if the absorbed solar radiation is held fixed, increasing CO2 tenfold to 1000ppmv would increase the temperature to 285K once equilibrium is re-established, and increasing it another tenfold to 10000ppmv (about 1% of the atmosphere) would increase the temperature to 293K. A further increase to 100000ppmv (10% of the atmosphere) increases the temperature to 309K. These represent substantial climate changes, but not nearly so extreme as they would have been if CO2 were a grey gas. As another example of what the graph can tell us, let's ask how much CO2 increase is needed to maintain the same 280K surface temperature with a dimmer sun. Reading vertically from the intersection with the line Tg = 280K, we find that this temperature can be maintained with an absorbed solar radiation of 195W/m2 if the CO2 concentration is 100000ppmv. Thus, an increase in CO2 by a factor of 1000 can make up for a Sun

Ground Temperature (K)

Figure 4.33: OLR vs surface temperature for various CO2 concentrations, at a fixed relative humidity of 50%. Other conditions are the same as for Figure 4.31.

which is 25% dimmer than the base case.

Finally, we'll take a look at how the OLR varies with CO2 for a fixed surface temperature (Fig. 4.34). This figure is a more Earthlike version of the results in Fig. 4.29, in that the effects of water vapor on radiation and the adiabat have been taken into account. As in the dry case, there is broad range of CO2 concentrations - about 5ppmv to 5000ppmv - within which the OLR decreases very nearly like the logarithm of CO2 concentration. The slope depends only weakly on the relative humidity, especially if one leaves out the completely dry case. This suggests that the effects of water vapor and CO2 on the OLR are approximately additive in this range. We note further that the logarithmic slope of OLR vs CO2 becomes steeper at very high CO2, since one begins to engage more of the outlying absorption features of the CO2 spectrum; again, CO2 becomes an increasingly effective greenhouse gas at high concentrations. Conversely, at very low concentrations the logarithmic slope is reduced, as CO2 absorption comes to be dominated by a relatively few dominant,narrow absorption features.

For any given CO2 value, increasing the moisture content reduces the OLR, as would be expected from the fact that water vapor is a greenhouse gas. A little bit of water goes a long way. With 100ppmv of CO2 in the atmosphere, going from the dry case to 10% relative humidity reduces the OLR by 36 W/m2. To achieve the same reduction through an increase of CO2, one would have to increase the CO2 concentration all the way from 100ppmv to 10000ppmv. Clearly, water vapor is a very important player in the radiation budget, though we have already seen that because of the thermodynamic control of water vapor in Earthlike conditions, CO2 nonetheless remains important. As the atmosphere is made moister, the further effects of water vapor are less dramatic. Increasing the relative humidity from 10% to 50% only brings the OLR down by 17W/m2, and going all the way to a saturated atmosphere only brings OLR down by a further 12 W/m2. Further, at very high CO2 concentrations the OLR becomes somewhat more insensitive to humidity, as CO2 begins to dominate the greenhouse effect.

The information presented graphically in Figures 4.32, 4.33 and 4.34 amount to a miniature climate model, allowing many interesting questions about climate to be addressed quantitatively without the need to perform detailed radiative and thermodynamic calculations; it's the kind of climate model that could be printed on a wallet-sized card and carried around everywhere.

100 1000 10 CO2 (ppmv)

Figure 4.34: OLR vs CO2 for a fixed surface temperature Tg = 280K, for various values of the relative humidity. Other conditions are the same as for Figure 4.31.































Table 4.2: Coefficients for polynomial fit OLR Calculation carried out with rh = .5.

Table 4.2: Coefficients for polynomial fit OLR Calculation carried out with rh = .5.

Calculations using this information can be simplified by presenting the data as polynomial fits, which eliminates the tedium and inaccuracy of measuring quantities off graphs. Then, what we have is a miniature climate model that can be programmed into a pocket calculator. Polynomial fits allowing the OLR to be calculated as a function of temperature, CO2 and relative humidity are tabulated in Tables 4.2 and 4.3. Values of OLR for parameters intermediate between the tabulated one can easily be obtained by interpolation.

Using these polynomial fits, we can put numbers to some of our old favorite climate questions by solving OLR(T, CO2) = S for T, given various assumptions about CO2 and the absorbed solar radiation S. To wit -

• Global Warming: If we assume an albedo of 22.5%, the absorbed solar radiation is 265

rh ao ai a2

Saturated 249.1 -5.183 -0.32187 -0.017367

Table 4.3: Coefficients for polynomial fit OLR = ao + aix + a2x2 + a3x3, where x = ln(CO2/100), with CO2 in ppmv. Calculation carried out with Tg = 280K for the indicated moisture conditions.

W/m2. For 50% relative humidity, and a pre-industrial CO2 concentration of 280ppmv, the corresponding equilibrium temperature is 285K, which is close to the pre-industrial global mean temperature. The albedo we needed to assume to get this base case is somewhat smaller than the Earth's observed albedo, because a portion of the cloud albedo is offset by the cloud greenhouse effect. Now, if we double the CO2 to 560ppmv, the new temperature is 287K - a two degree warming. This is essentially the same answer as obtained by Manabe and Wetherald in their pioneering 1967 calculation, and was obtained by essentially the same kind of calculation we have employed. If we double CO2 once more, to 1120ppmv, then the temperature rises to 289K, a further two degrees of warming. The fact that each doubling of CO2 gives a fixed additional increment of warming reflects the logarithmic dependence of OLR on CO2; until one gets to extremely high concentrations, each doubling reduces the OLR by approximately 4W/m2.

• Pleistocene Glacial-Interglacial Cycles: In the depths of an ice age, the CO2 drops to 180ppmv. Using the same base case as in the previous example, the temperature drops to 284K, about a one degree cooling relative to the base case. This by no means accounts for the full amount of ice age cooling, but it is significant enough to imply that CO2 is a major player. In the Southern Hemisphere midlatitudes, away from the direct influence of the growth of major Northern Hemisphere ice sheets, the CO2 induced cooling is a half to a third of the total, indicating that either the ice sheet influence is propagated into the Southern Hemishere through the atmosphere or ocean, or that the cooling we have calculated has been further enhanced by feedbacks due to clouds or sea ice.

• The PETM warming: How much CO2 would you have to dump into the ocean-atmosphere system to account for the PETM warming discussed in Section 1.9.1? One answers this by first deciding how much the atmospheric CO2 concentration needs to increase, and then making use of information about the partitioning of carbon between the atmosphere and ocean. The PETM warming has been conservatively estimated at 4C, and has nearly the same magnitude in the tropics as in the Arctic. The PETM event starts from an already warm hothouse climate, so for the sake of argument let's assume that the CO2 starts at four times the pre-industrial value, yielding a starting temperature of 289K. We need to increase the CO2 from 280ppmv to 1090ppmv (just under two doublings) to achieve a warming to 293K. This amounts to an addition of 1744 gigatonnes of C to the atmosphere as CO2, which is comfortably within the limits imposed by the 13C data, but not all carbon added to the atmosphere stays in the atmosphere. Over the course of a thousand years, approximately 80% of atmospheric carbon will be absorbed into the ocean, and over longer periods the ocean may be able to take up even more. Thus, to sustain the warming more than a millennium, one would need to add at least 5 times the nominal value, or 8720 gigatonnes to the ocean-atmosphere system. Can such a large addition of carbon be reconciled with the carbon isotopic record? This is the essential puzzle of the PETM, and may call for some kind of strong destabilizing feedback in the climate system.

• Deglaciation of Snowball Earth: If we increase the Earth's albedo to 60% (in accord with the reflectivity of ice) and reduce the solar constant by 6% (in accord with the Neoproterozoic value) the absorbed solar radiation is only 128 W/m2. With CO2 of 280 ppmv, the equilibrium global mean temperature is a chilly 228K, more or less independent of what we assume about relative humidity. To determine the deglaciation threshold, we'll assume generously that the Equator is 20K warmer than the global mean, so that we need to warm the global mean to 254K to melt the tropics. Assuming 50% relative humidity, increasing CO2 all the way to 200,000 ppmv (about 20% of the atmosphere) still only brings the global mean up to 243K, which is not enough to deglaciate. From this we conclude that without help from some other feedback in the system, CO2 would have to be increased to values in excess of 20% of the atmosphere to deglaciate. In fact, detailed climate model calculations indicate that it is even harder to deglaciate a snowball than this calculation suggests.

• Temperature of the post-Snowball hothouse Let's assume that somehow or other the Snowball does deglaciate when CO2 builds up to 20%. After deglaciation, the albedo will revert to 22.5%, and the absorbed solar radiation to 249 W/m2. When the planet re-establishes equilibrium, the temperature will have risen to 311K. This is hot, and the tropics will be hotter than the global mean. However, the planet does not enter a runaway greenhouse and these temperature are well within the survival range of heat-tolerant organisms, especially since the polar regions would probably be no warmer than today's tropics.

• Faint Young Sun: Let's consider a time when the Sun was 25% fainter than today, reducing the absorbed solar radiation by 66 W/m2. How much would CO2 have to increase relative to pre-industrial values in order to keep the global mean temperature at 280K and prevent a freeze-out? We'll address this question by using the fit in Table 4.3. For 280 ppmv of CO2 the OLR is 253 W/m2 assuming 50% relative humidity. We need to bring this down by 66 W/m2 to make up for the faint Sun. Using the fit, this can be done by increasing CO2 to 240,000 ppmv (24% of the atmosphere) if we keep the relative humidity at 50%. If we on the other hand assume that for some reason the atmosphere becomes saturated with water vapor, then it is only necessary to increase the CO2 to 15% of the atmosphere.

• The Earth in one billion years: According to Eq. 1.1, the solar constant will have increased to 1497 W/m2, increasing the absorbed solar radiation per unit surface area to 290 W/m2. If CO2 is held fixed at the pre-industrial value, the Earth will warm to a global mean temperature of 296 K if relative humidity is held fixed at 50%. In order for silicate weathering to restore a temperature of 287K, the weathering would have to bring the CO2 all the way down to 10ppmv, at which point photosynthesis as we know it would probably become impossible.

• Temperature of Gliese 581c and 581d: The planets Gliese 581c and 581d are in close orbits around a dim M-dwarf star. The redder spectrum of an M-dwarf would have some effect on the planetary energy balance, through changes in the proportion of solar energy absorbed directly in the atmosphere. Neglecting this effect, though we can estimate the temperatures of these planets assuming them to have an Earthlike atmosphere consisting of water vapor, CO2 and N2/O2. Gliese 581c is in an orbit where it would absorb about 583 W/m2, assuming the typical albedo of a rocky planet with an ocean. Gliese 581d would absorb only 50 W/m2. Even if CO2 were 20% of the atmosphere, the OLR would be 82 W/m2, so Gliese 581d is likely to be an icy Snowball. On the other hand even with only 1ppmv of CO2 in the atmosphere the OLR at 330K would be 351 W/m2, far below the absorbed solar radiation for Gliese 581c. Thus, if Gliese 581c has an ocean, it is very likely to be in a runaway state - something we'll confirm when we re-examine the runaway greenhouse for real-gas atmospheres. There's an additional wrinkle to the Gliese system, though, in that these planets are more massive than Earth and have higher surface gravity. The higher gravity somewhat reduces the water vapor greenhouse effect, since for a given vapor pressure the corresponding amount of mass in the atmosphere is lower, according to the hydrostatic relation. This turns out to cool Gliese 581c somewhat, but still not enough to save it from a runaway.

Exercise 4.5.3 About how much carbon would need to be added to the atmosphere to achieve a 4K PETM warming if the initial CO2 at the beginning of the event were only twice the pre-industrial value? If the initial CO2 were eight times the pre-industrial value? (Note that in the first case we are implicitly assuming some unknown process keeps the late Paleocene warm even with relatively little help from extra CO2)

The above calculations include the effects of water vapor feedback, and also include the effects of changes in albedo due to ice cover, where explicitly mentioned. However, they do not incorporate any feedbacks due to changing cloud conditions. Cloud changes could either amplify or damp the climate change predicted on the basis of clear-sky physics, according to whether changes in the cloud greenhouse effect or the cloud albedo effect win out. Unfortunately, there is no simple thermodynamic prescription that does for cloud feedbacks what the assumption of fixed relative humidity does for water vapor feedbacks. We will learn more about the factors governing cloud radiative forcing in Chapter 5, but idealized conceptual models for prediction of cloud feedbacks remain elusive.

The problem of whether elevated CO2 can account for hothouse climates such as the Cretaceous and Eocene is considerably more challenging than the other problems we have discussed above, since it requires one to answer a regional climate question: Under what conditions can we suppress the formation of polar ice? We already saw in Chapter 3 that a rather small change in radiation balance can make the difference between a planet with a small polar ice sheet and a planet which is globally ice-free. Suffice it to say at this point that an increase of CO2 to 16 times pre-industrial values - the upper limit of what is plausibly consistent with proxy data - would yield a global mean warming of 10K. Would this be enough to suppress formation of sea ice in the Arctic, and keep the mean Arctic temperatures around 10C? Would the associated tropical temperatures be too hot to be compatible with available proxy data? We'll have to return to these questions in Chapter 7, where we discuss the regional and seasonal variations of climate.

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