Guest post by Chris Colose (e-mail: colose-at-wisc.edu)
This post is a more technical version of Part 1, meant to quantify and expand upon some of the feedback concepts laid out previously. Additionally, the role of the water vapor feedback in planetary climate is discussed.
By convention, climate scientists define a feedback parameter, λ, to encompass the effects of the various feedback processes discussed in Part 1. In that post, it was noted that the ‘reference’ sensitivity parameter (λ0) was about 0.3 °C/(Wm-2). If we perturb the climate, this is the equilibrium temperature response (per unit forcing) that would be experienced after neglecting all feedbacks and only allow the planet to come back into radiative equilibrium. Creating this reference system (or baseline) is critical to discussing climate sensitivity, since positive and negative feedbacks are defined relative to how they modify this so-called Planck response.
The above value can be obtained by taking the derivative of the Stefan-Boltzmann equation with respect to the emission temperature (which is ~255 K on Earth).
A more realistic value of λ0 obtained in models is slightly higher than this back-of-envelope calculation (~0.3-0.31 °C/(W m-2)) due to atmospheric longwave absorption. The no feedback response is ΔT0 =λ0*RF (RF is the radiative forcing). For a doubling of CO2, which has a radiative forcing of 3.7 W m-2 in the modern atmosphere (Myhre et al 1998), one obtains an estimate of about ~1.2°C temperature change.
In the real world of course, we expect various components of the climate system to respond and further modify the temperature. This works by taking some fraction of the system output and feeding it back into the input, creating an additional term proportional to the temperature response (kiΔT for the ith feedback of interest). For n number of feedbacks, this can be formally written as (Roe, 2009),
These additional terms serve to modify the initial radiative forcing, making the full perturbation dependent on the response. Each individual feedback has an associated feedback factor (f) as defined in part 1. This is proportional to the system output which is then fed back and amplifies the original forcing (f = ki*λ0). The associated climate response is
This connection between feedback factors and the final temperature response has been known for some time (see review by Roe 2009 ;Hansen et al 1984 lay out the basic mathematical framework and definitions in feedback discussions: but note the Hansen paper reverses the standard definitions of a gain and a feedback factor used in electronics, and which is used here).
The importance of the above relationship is that the climate has a non-linear response which depends on the sum of the feedback factors. For instance, with one feedback of f=1/3 the sensitivity is increased by a factor of 1.5 above the base (Planck) response. For two feedbacks of f=1/3 each, the sensitivity is increased by a factor of 3. Furthermore, the uncertainty in the temperature response due to small uncertainties in the total feedback factor depends on the true mean value of the feedback parameter.
In figure 2 note that the slope of the temperature response curve increases at higher values of f close to one. Therefore the same uncertainty in the feedback parameter range has a stronger projection on uncertainties in the response if positive feedbacks dominate over negative ones, as appears to be the case in the present day climate.
It should be kept in mind that feedbacks are, mathematically, just a Taylor Series. The often-cited divergence point to infinity as f=1 is really an artifact of assuming a linear system and ignoring higher order terms (see Zaliapin and Ghil, 2010 in discussion of the Roe and Baker work). Really, this limit corresponds to a bifurcation (loosely, a tipping point); the state on the the other side of this point could be a runaway effect or it could be just a small change in temperature.
The Water Vapor Feedback
As the globe warms from anthropogenic forcing, the water vapor feedback is the strongest amplifier of global temperature change. Despite this, some of the very popular descriptions of how water vapor feedback operates are incorrect, or at least incomplete. One rather common fallacy is to simply say “the Clausius-Clapeyron equation means warmer air will hold more water vapor” or to assert that a warmer planet must mean more evaporation, which will in turn increase the moisture content of the atmosphere.
The argument that increased evaporation is central to the water vapor feedback does not turn out to be very useful. For one thing, evaporation describes a flux of vapor from the wet surface into the atmosphere. This is a physical quantity that doesn’t have much direct relation to the amount of water vapor that is left behind, and indeed, is measured in completely different units. Evaporation (and precipitation) is also tied to the surface energy balance and depends on the wind speed and boundary layer relative humidity, so it is entirely possible for evaporation to go down in a warmer world or increase in a colder one. The surface energy balance reads as:
Ssfc is the shortwave radiation incident on the surface, weighted by the albedo α. P is the precipitation, L is the latent heat of vaporization, SH is the net upward sensible heat flux, and R is the net upward longwave radiative flux at the surface. Solar radiation absorbed at the surface is a large constraint on the amount of evaporation and precipitation that can occur. Even if you add CO2 to the atmosphere, evaporation and precipitation eventually level off even as the surface temperature increases in warm climates.
In general, evaporation and precipitation go up much less rapidly than the water vapor content, resulting in an increase of the mean residence time of water vapor in the atmosphere. This has strong implications for circulation patterns, including a weakening of the zonal (Walker) circulation and to some extent, the Hadley circulation (Held and Soden, 2006). In the warm limit, the latent heat flux is much larger than the sensible and net radiation terms, so precipitation asymptotes to a value like Pmax = S(1-alb)/L (Equation 6 in O’Gorman & Schneider, 2008), which is about 6 mm/day in the global mean.
Clausius-Clapeyron provides only an upper bound on the pressure of water vapor that can build up in a parcel of air at some temperature before it reaches saturation. This thermodynamic law says that the saturation pressure increases nearly exponentially with temperature (in the current climate, ~7% per °C). It is given in approximate form as
where A=2.53×1011 Pascals and B=5420 K (Petty,2008). However, this doesn’t tell us how or if that upper bound will be reached. For example, within the tropical high level atmosphere, conditions depart significantly from saturation.
Still further, simply adding more column integrated water vapor in the atmosphere is not enough to get a strong water vapor feedback, even if it matters for precipitation. In order to get a strong greenhouse effect you must increase the infrared opacity of the high, cold atmosphere where you can reduce the emission of longwave radiation to space. That is, water vapor will have much more impact at high altitudes than near the surface. As with CO2, the radiative effects of water vapor depend on the fractional change rather than the absolute change. Increasing the humidity of the free troposphere from 3 to 6% has nearly the same impact as increasing the humidity from 30 to 60%, so it is important to understand how the actual humidity changes at these levels scale with the upper bound set by Clausius-Clapeyron. This requires knowledge of the relative humidity distribution and how it may change in a warmer world.
Unfortunately, there are no simple constraints on the maintenance of relative humidity in the upper atmosphere (Couhert et al., 2010; Schneider et al 2010), and testing simulations of the water vapor feedback with observations is critical. In observations of seasonal variations, responses to volcanic eruptions (Soden et al 2002), ENSO, and in trends from satellites, the relative humidity tends to change very little on the global scale. This provides important evidence that the water vapor scales pretty well with Clausius-Clapeyron (Manabe and Wetherald, 1967; Held and Soden 2000; Dessler and Sherwood, 2009). This is also a result that all of the models produce fairly closely (Sherwood et al., 2010, JGR).
Understanding why this is the case is not straightforward. There is no fundamental reason that relative humidity cannot change. In the subtropics for example, relative humidity does change in twenty-first century simulations and considerably so over even broader range of climate change scenarios (O’Gorman and Schneider, 2008).
Over the last decade, a leading framework that has emerged is that the tropospheric water vapor distribution can be quantitatively predicted by taking into account the large-scale wind and temperature fields, without the need to consider small-scale cloud microphysics. Simple models examining the statistics of water vapor ‘trajectories’ in time, and keeping track of parcel transport and saturation/condensation events, can simulate the large scale humidity fields well. In such a model, the specific humidity is determined approximately by the most recently experienced saturation value. Changes in RH owing to the change in the temperature of last saturation are quite small in the present-day climate (e.g., Pierrehumbert et al., 2007 Equation 6-19; see also Pierrehumbert and Roca 1998; Dessler and Minschwaner 2007; Sherwood et al., 2010; Reviews of Geophysics).
In addition to the direct radiative impacts discussed above, the water vapor distribution is tied to various cloud feedbacks that may occur in a changing climate. For instance, the troposphere is heated by convection only where it is cooled down by radiation. The profile of tropospheric infrared cooling is expected to shift upward in a warmer world primarily due to the increase in upper level humidity. It follows that the vertical extent of convective overturning should increase and the temperature at the tops of tropical anvil clouds should remain about the same during climate change, which would decouple the emission temperature from the surface temperature (Hartmann and Larson, 2002). This could be a mechanism by which virtually all models tend to simulate positive longwave feedbacks (Zelinka and Hartmann, 2010). The bulk of uncertainty behind the cloud feedback actually comes from the shortwave component, which is dominated by low clouds that have a high albedo.
The Runaway Greenhouse
The simplest energy balance model that provides a boundary condition which constrains the climate of all Earth-like planets is obtained by equating the absorbed solar radiation with the outgoing terrestrial radiation.
What water vapor does as a greenhouse gas is to smooth out the outgoing radiation curve as a function of temperature, making it more linear than the T4 Stefan-Boltzmann dependence. Just as with CO2, water vapor makes the atmosphere a poorer emitter of radiation to space and enhances the surface temperature (Figure 3) .
The water vapor feedback in this diagram can thought of as the warming difference between b’-b and a’-a, where the prime terms are the new climate state as the solar radiation is increased. In other words, the range of temperature change becomes larger for similar changes in the OLR for a planet with higher relative humidity.
The extreme warm end of a feedback scenario, popularly known as the runaway greenhouse, can arise when the atmosphere is composed of a greenhouse gas that is in vapor pressure equilibrium with a large, surface volatile reservoir. When you add greenhouse gases to the atmosphere, warm surface emission is preferentially replaced by emission from high, cold regions of the atmosphere. Since the water vapor feedback means the IR opacity is dependent on the temperature itself, eventually the emission to space can be decoupled from the surface temperature completely.
As the specific humidity continues to climb, the limiting infrared cooling that can occur for a planet comes at a threshold known as the Simpson-Kombayasi-Ingersoll (SKI) limit. If the incoming absorbed solar radiation exceeds this radiation threshold then the surface temperature must rise until the oceans are either depleted or the body becomes hot enough to radiate in significantly shorter wavelengths to which the air is rather transparent. The SKI limit therefore sets the inner edge of the habitable zone (where a planet can evolve with liquid water). This standard scenario is very important for understanding the evolution of other climates (in particular Venus, or exoplanets close to their host star). The runaway greenhouse scenario is likely the harsh fate Earth will encounter in the geologically distant future, as the sun gradually brightens over time.
Contrary to some popular impressions, CO2 plays only a very small role in setting the limit at which a runaway greenhouse is obtained. Once an ocean’s worth of vapor is in the atmosphere, CO2 is relatively unimportant unless it exists at extreme concentrations. One (numerically) calculated upper limit on the outgoing radiation for a planet with Earth’s gravity is about 309 W m-2 or about 1.4 times the modern solar constant, noting that the albedo of an H2O atmosphere is higher than the modern case (Kasting, 1988).
In conclusion, water vapor is the dominant feedback on our planet and is also important for understanding the evolution of planetary climate. Observations and models support a roughly 7% increase in specific humidity per degree warming, consistent with scaling of the Clausius-Clapeyron relation, although the nuances of tropospheric humidity are still a topic of worthy research.