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## The Water Vapor Feedback and Runaway Greenhouse

In previous posts, I have outlined in a few steps the interplay between the radiation balance of a planet, its temperature structure, and the feedback mechanisms that gives freedom for the climate to depart from its reference norm. In the ice-albedo post for example, we have seen a bifurcation structure in which removing enough CO2 (or lowering the sunlight a planet receives) can plunge the planet into a runaway ice-covered state. In this post, I’ll consider the “hot end” of a similar type of bifurcation, although now we enter a regime in which no ice exists and a significant fraction of the atmosphere is composed of water vapor. This would be typical of any ocean planet that becomes sufficiently hot to make water vapor a dominant constituent of the air. More traditional applications of the “water vapor feedback” to the global warming issue will be discussed.

A common reason that water vapor is treated differently from CO2 in the present Earth-based climate is that water exists in multiple phases (i.e., between liquid, solid, and vapor) and the partitioning between water in those various forms on Earth is a function of temperature. For our purposes, we are interested in the atmospheric component, which is predominately vapor. We define two quantities that will be useful for this post: the first, is the saturation vapor pressure, which is the maximum partial pressure of water vapor that can build up in an air mass. This quantity is a rapidly increasing function of temperature. For an Earth-like background pressure of ~1000 mb, then above about 350 K, water vapor would change from a trace constituent to one of the dominant atmospheric molecules by volume (like diatomic nitrogen or oxygen). We also define relative humidity, as the ratio of the actual vapor pressure to its saturation value. A relative humidity of one (or commonly “100%”) means the pressure of water vapor can be fully described by its saturation value, $e_{s}(T)$), which is given by the Clausius-Clapeyron law (the below is an integrated form assuming constant latent heat of vaporization):

$\displaystyle e_{s}(T)=A \; exp (-\frac{T^*}{T})$

where A and T* are constants (with dimensions of pressure and temperature, respectively) that depend on the gas in consideration. T* is 5419 K for condensation into liquid and 6148 K for condensation into ice, while A is a large constant for water vapor (~2.56 x 1011 Pa). Note that 1 Pa = 0.01 mb. This formula thus predicts that $e_{s}$ is a rapidly increasing function of temperature.

It is easy to visualize the distinction between water vapor and the more “permanent” constituents of the atmosphere. Shown in the figure below, is a model result showing the evolution of water vapor concentration following an instantaenous removal or doubling of water vapor concentration. Convergence to the original equilibrium is very fast for doubled water vapor since relative humidity above 100% leads to rapid condensation and rainout. Evaporation is also fast-acting when starting with zero relative humidity, but the vertical and horizontal transport of moisture requires about a week to approach its equilibrium amount. Thus, on timescales of days to weeks, the “virtual forcing” of changing water vapor is no longer felt by the climate system. Thus, it is appropriate to think of water vapor as a feedback response to a climate change that must be initiated externally, rather than an independent forcing. This would be true for any condensing substance in a fluid dynamical setting, such as with methane on Saturn’s moon Titan.

Atmospheric moisture response to an instantaneous doubling and zeroing of atmospheric water vapor. Panels show global mean water vapor at the 299 and 974 mb level converging to control run equilibrium amounts. Figure from Andy Lacis, Bert Bolin Symposium, May 2012

Because water vapor is a greenhouse gas, its concentration increasing with temperature means that atmospheric opacity increases with temperature, thus providing a positive feedback to climate change. The physics of feedbacks were discussed in the last post (if the reader has not been initiated into the language of feedbacks, it may be worth visiting that post). The rapid adjustment of water vapor to the climate state also means one must be cautious with popular statements like “water vapor is the most important greenhouse gas.” This is true in the sense it makes up the bulk of infrared opacity, but its concentration is a slave to the current climatic boundary conditions. Aside from the solar energy received by the planet, CO2 concentration is the next most fundamental long-term determinant of Earth’s climate, and the most readily available “knob” that can be turned to shift Earth’s climate on decadal to geologic timescales.

A common remark one sometimes hears is that the water vapor feedback can be traced back to increased surface evaporation as the climate warms. This is a misguided line of thought, which mistakes the flux of water into the atmosphere with a reservoir. In fact, in a warming climate, water vapor could accumulate to its new (higher) saturation value even if evaporation remained constant. Evaporation (and precipitation) are largely controlled by the energy constraints at the surface and troposphere, and the changes in those variables in a warming climate do not scale with the amount of water vapor in the atmosphere (e.g., Held and Soden, 2006; O’Gorman et al (2012)).

Another common misconception is that Clausius-Clapeyron alone inherently means that the water vapor concentration of the atmosphere must increase with temperature. In fact, Clausius-Clapeyron is only an upper bound, and one needs to consider the large-scale dynamics of the atmosphere in order to determine the degree to which an air column will remain sub-saturated, and how that might change in a new climate (the Sahara desert is a clear example of where hot temperatures will not necessarily lead to abundant water vapor amounts). The distribution of relative humidity and the factors governing subtropical dryness is therefore a first-order determinant to Earth’s climate, as the behavior of outgoing radiation (OLR) is intimately connected with the humidity field.

The underlying physics has been explored at the interface of theory, e.g., the “advection‐condensation” paradigm based on trajectory calculations. Results such as this indicate that the observed humidity distribution in the troposphere can, to first order, be reconstructed by considering the large-scale circulation and temperature structure. Basically, this circulation advects a passive tracer that is allowed to precipitate its condensate when it exceeds 100% relative humidity returning the parcel to saturation. Within this framework, saturation occurs as air ascends and cools; upon descent, the parcel of air will conserve its water vapor mixing ratio, and so the parcel’s humidity will equal the lowest saturation value it has experienced since leaving the boundary layer. The humidity can be understood in terms of the point at which air parcels last experienced saturation (the saturation vapor pressure at the temperature of the last saturation point). This is for a point. For a volume, one can treat water vapor in a volume as the linear combination of all locations of last saturation. The relative humidity in a location is thus a numerator which accounts for the circulation and non-local temperature influences, divided by a saturation humidity corresponding to a local temperature. This simplistic view, despite limitations such as ignoring the evaporation of falling condensate (as occurs in areas of deep convection), can boast immense explanatory ability in reconstructing the observed humidity field. It has been applied in applications ranging from idealized models of the PDF of relative humidity to Lagrangian and Eulerian-based computations. This type of thinking has also impacted isotopic-based studies examining how the heavy and light isotopes of water are transported through the atmosphere, which has implications for the interpretation of natural archives used in paleoclimate reconstructions.

In a last‐saturation framework, a global warming situation translates to an increase in the temperature of last saturation and an associated increase in saturation specific humidity at the target point. So Clausius-Clapeyron does in fact enter into the heart of things, but one must consider its usage in proper context (i.e., one cannot just say that water vapor increases because the surface temperature goes up). Instead, it is linked to C-C through the statistics of last saturations. That water vapor feedback is substantially positive, and that global relative humidity changes are quite small when compared to the Clausius-Clapeyron dependence, is now a remarkably robust and uncontroversial result that has been established in many observational studies, and in models of varying complexity.

As it turns out, a great starting point for thinking about the water vapor feedback in climate changes is to consider a relative humidity which is invariant with temperature, such that the actual vapor pressure scales with Clausius-Clapeyron. This does not, however, mean that the distribution of relative humidity is constant. Consider the following figure, which shows the relative humidity difference on a pressure-latitude grid (averaged zonally) between 2xCO2 and control runs in several models. This emergent structure is robust amongst GCMs and to simpler models and within the simple advection-condensation philosophy.

Figure 1: Zonally (east-west) averaged relative humidity change as a function of latitude and height. See Wright et al (2010)

Note that relative humidity increases at the tropopause outside the tropics, in the tropical deep convective zone, in the stratosphere, and decreases in the subtropics (near the positions of the mean-climate subtropical relative humidity minima). One typically sees an upward shift in the relative humidity field and poleward expansion of the descending (drying) cells of the Hadley circulation in global warming experiments. The circulation physics that leads to the low humidity conditions in the subtropics has been explored in a great many studies, including those elaborating on the simplistic view above and in more comprehensive models (see e.g., Galewsky et al., 2005 ; Pierrehumbert et al. 2007; Sherwood et al., 2010a ; Sherwood et al., 2010b). Paul O’Gorman and Tapio Schneider also have several good papers on this matter.

The fundamental determinant to climate sensitivity is to ask how the surface temperature is connected to the slope of the net radiation flux as a function of temperature, as explored in the last post. If, for example, OLR increases only weakly with temperature, than sensitivity will be higher than if the OLR increases very efficiently with temperature (and thus is quicker at restoring a new equilibrium). A question to ask concerning the water vapor feedback is what locations in the atmosphere contribute what to the perturbations in the radiative balance. Summing over all k locations,

$\displaystyle \delta OLR \: = \sum_{k=1}^{N}[\frac{\delta OLR}{\delta T_{k}} \delta T_{k} + \frac{\delta OLR}{\delta e_{k}} \delta e_{k}]$

where for constant RH,

$\displaystyle \delta e = (RH)\frac{de_{s}}{dT}\delta T$

The spatial contribution to water vapor feedback can be understood from this diagram in Soden et al (2008), which shows max sensitivity to water vapor in the mid to upper troposphere in the intertropical regions. The dry free troposphere is important, and clouds can mask the the effects of lower level moistening even in the tropics, while shifting peak contributions to the tropical upper atmosphere. Most of the water vapor feedback occurs above 800 millibars, since the reduction in OLR depends on the temperature gradient between the surface and layers aloft. See “ScienceofDoom” for a qualitative introduction into this figure and a broader discussion of the “radiative kernel” approach in climate sensitivity studies, which decomposes feedbacks into components determined by how a climate variable changes with temperature, and how that climate variable then modifies the radiative budget of the planet.

The Runaway Greenhouse

In the last post on feedbacks, I defined the feedback factor as:

$\displaystyle f = - \left (\frac{\partial R}{\partial r}\frac {\partial r}{\partial T} \right) / \frac{\partial R}{\partial T}$

with the terms defined in the previous post, with $R$ being the net radiation budget (OLR-absorbed shortwave) and $r$ in this case being the water vapor concentration. It was also demonstrated that climate sensitivity was proportional to 1/(1-f). Water vapor is a positive feedback since the above expression for f is positive. As the planet gets warmer, we can expect that the outgoing radiation to space should increase in order to restore the climate to a new equilibrium point. This equilibrium will be established as the absorbed shortwave radiation equals the outgoing thermal radiation, which in the optically thick limit, is “originating” from the upper troposphere, since all the exiting radiation from the surface will be absorbed before it reaches the optically thin regions where the optical depth becomes less than unity.

It is reasonable to ask what the temperature structure in a high-temperature, and water vapor enriched atmosphere would look like. Shown below, I have plotted a family of curves for the temperature as a function of pressure (decreasing with height). Curves are shown for surface temperatures of 250, 300, 350, 425, and 475 K. The temperature structure typically behaves like a moist adiabat, which is determined by the cooling of air parcels as they rise and expand, and the warming as the parcels release their latent heat. The net effect is for cooling with height.

However, once water vapor becomes a dominant constituent of the atmosphere, the moist adiabatic temperature profile simplifies to the saturation vapor pressure curve. In essence, on can re-arrange the first equation in this post, in order to solve for the temperature structure of a saturated atmosphere in which water vapor is the prevailing component.

Temperature Structure of the atmosphere at different surface temperatures. Atmospheric mass is allowed to increase with higher water vapor concentration, and with a background non-condensing gas exerting a pressure of 1000 mb. Calculations were done with the “moistadiabat.py” script available in the Ch. 2 scripts here

In this limit, we see that the temperature structure of the upper atmosphere becomes fixed toward the saturation vapor pressure curve. Increasing the surface temperature by whatever means (and thus adding more and more water vapor via feedback) can then be viewed as increasing the total mass of the atmosphere and extending the warmer temperatures down to new (higher) pressures that did not previously exist. For example, at close to 500 K the total atmospheric pressure has now increased to over 10 times its modern value of 1000 mb. However, the temperature profile has become fixed aloft. Because all the radiation to space is emanating from the high, optically thin regions of the atmosphere, and because the temperature structure of this region is now fixed, we can intuit that the outgoing radiation must also be capped at a limiting value. We call this the Nakajima-Simpson limit (see Nakajima et al (1992) .

We can now see how the runaway greenhouse effect operates. In essence, there is competition between the OLR increasing with temperature, and OLR decreasing with water vapor content, as in the preceding equation for f. Eventually the latter term wins, and OLR no longer goes up (or even decreases) with temperature. It is possible to overshoot the limiting OLR values by different processes, such as relative humidity decreasing, or mixing in other gases into the atmosphere. The following figure is from Sugiyama et al (2005) (also see a more descriptive review of various radiation limits that have been found in the review by Goldblatt and Watson, 2012). The physics here has a long history extending at least back to the 1969 paper by Ingersoll, and evidently even in a 1920’s paper by Simpson (which I haven’t read). Kasting (1988) studied the problem with a non-grey (i.e., where absorption is spectrally resolved) model in connection with the evolution of Venus, where a runaway may have occurred.

(left) OLR as a function of surface temperature and constant but different relative
humidity, each line representing H = 0%, 10%, . . . , 100%. Each line is extended to the maximum temperature allowed for each value of relative humidity. (right) OLR as a function of surface temperature when relative humidity is allowed
to change interactively. Lines from the left panel are also shown.

The criteria for a runaway greenhouse to sustain itself is that the absorbed solar radiation exceeds the limiting OLR. In this case, there is no equilibrium solution, and no radiative balance. The planet becomes locked into a state in which it continues to absorb more solar radiation than can be emitted to space, and temperatures can runaway into excess of 1000 K. This state of affairs can eventually be escaped, either by terminating the water vapor feedback (e.g., when all the oceans are in the atmosphere, and the atmospheric mass no longer increases with temperature) or when the planet becomes hot enough to emit light near the visible or near-infrared, where atmospheric opacity is weaker.

Some claims have surfaced (e.g., by NASA’s Jim Hansen) that a runaway greenhouse is possible if we burned all the CO2. Unfortunately, there is no evidence in the planetary science literature to support the claim, and it can be dismissed based on the fairly trivial fact that the amount of sunlight that Earth absorbs does not even come close to the limiting OLR values typically found in the literature (usually > 300 W/m2). Thus, even if the temperature is shifted to higher values, there will always be a radiative equilibrium. Carbon dioxide can cause a runaway in some of the “extreme” situations where you have overshot the OLR limit, as in the lower relative humidity cases above. CO2 will remove this hump, and can push you into a runaway, but you still need enough solar radiation to sustain it. That said, in principle, CO2 could still raise surface temperatures many hundreds of Kelvin if you continued to increase its concentration (a runaway wouldn’t occur at the modern boiling point of 373 K, since atmospheric pressure has increased quite a bit…one would need to wait until the critical point, 647 K, was reached for an Earth-sized ocean to be lost). As the Goldblatt and Watson review paper pointed out, clouds are a typical ? factor that could change the “no runaway possible” argument, but it is quite farfetched to think that the net effect of clouds could come close to making up the difference in energy needed to trigger a runaway.