by Chris Colose
There has been a lot of blog interest recently on feedback theory and climate sensitivity (e.g., Isaac Held, ClimateAudit, Science of Doom, Nick Stokes, one on control theory here at Climate Etc.).
A lot of this interest surrounds the debate between Roy Spencer and Andrew Dessler on cloud feedbacks, of which I have little to add. However, this general topic stands out as perhaps the most important focus of study in climate research, and my impression from reading the various entries (and a lot of comments) is that a lot of people are thinking about the issue in a very different manner. I’d therefore like to take a step back from some of the discussions on how well observations (such as the co-variability of the net radiation and global temperature, accounting for noise, ocean heat uptake, etc) constrain climate sensitivity. While the previous article on control theory by Richard Saumarez was enlightening, I’d like to do a climate sensitivity/feedback primer from the perspective of how a climate scientist thinks, and on how this framework can be applied to larger planetary climate problems. Various perspectives are always insightful, and physics is universal, but a lot of the definitional framework differs across disciplines.
For climate sensitivity, we are interested primarily in how a coarse variable such as global mean temperature responds to a radiative perturbation. This is a useful metric since we can treat the surface temperature in terms of the top of atmosphere energy balance in a convecting troposphere (and this balance is relatively simple, being purely radiative), and the surface temperature change is relatively uniform field over the globe- with the usual caveat that land will warm more than oceans, or Poles more than the tropics.
Consider a parameter G which represents the net top of atmosphere energy balance (outgoing thermal radiation – incoming absorbed radiation) so if G > 0, the planet cools, and if G < 0, it warms. Now, think about applying a radiative forcing to the system, such as an increase in carbon dioxide or sunlight. Climate sensitivity is then related to the slope of a line in a G vs. T plane (where T is the surface temperature). In particular, higher climate sensitivity implies a weak slope (see figure 1), so that the planet is more sluggish at changing its outgoing radiation in response to perturbations. This allows for larger temperature changes.
Figure 1: Net TOA energy balance (G) vs. surface temperature. For a low climate sensitivity (solid lines) the slope is large. For large climate sensitivity, the slope is more shallow (dashed lines). Starting at T0 in either case, applying an increase of CO2 decreases the outgoing longwave radiation, shifting the blue curves down to the corresponding red curve positions. The new intersection point (equilibrium) is T1 for the low sensitivity, or T2 for a higher sensitivity. From Climate Stabilization Targets: Emissions, Concentrations, and Impacts over Decades to Millennia (NAS, 2011)
In order to think about feedbacks, it is necessary to define a baseline (or reference system). One reasonable reference system would be to consider a naked planet (with no atmosphere) which radiates energy at a rate of σT4 (so that if the planet became twice as hot, it would radiate sixteen times as much energy away, allowing it to relax to this new equilibrium temperature). In models, this is actually calculated with a full radiative transfer code for each gridpoint on the Earth; this so-called “Planck feedback” amounts to about an extra 3.2 Watts per square meter (W/m2) power emitted for every 1 K temperature increase (Soden and Held, 2006). Climate sensitivity is often taken as the ratio of the temperature change to the radiative forcing; so for example, a doubling of CO2 which constitutes a forcing of ~3.7 W/m2 would produce a temperature response of 3.7/3.2 = 1.2 K. You can get a similar answer with a simple back of the envelope calculation. Note that while the Planck restoring mechanism might be thought of as a “negative feedback” it is simply a baseline “no-feedback response” in climate lingo.
Now, consider a positive feedback, for example the increased water vapor associated with a warming climate. What impact does this have on Earth’s radiation budget? From the perspective of Figure 1, the increased infrared absorption from water vapor makes the outgoing radiation response more linear than a T4 dependence, so it takes a greater surface temperature rise in order to accommodate the necessary infrared emission change required to re-establish equilibrium. Note, however, that the slope of outgoing radiation is still positive and so it is possible to re-establish equilibrium without “running away.” In the case where we double CO2, feedbacks being net positive simply means the global temperature rise is larger than 1.2 K.
Non-linearities, bifurcations, and extreme atmospheres
Armed with this definitional framework, we can now think about some larger applications in the context of climate. The reason positive feedbacks don’t cause runaway warming on Earth is often explained in terms of a converging power series which can easily be shown to take the form ΔT=ΔT0/1-f (where ΔT0 is the Planck restoring response defined before, and f is a feedback parameter that depends on how a particular feedback variable like water vapor changes with temperature, and on how much that change impacts Earth’s radiant energy balance). In a linear worldview, f being greater than zero but less than one implies a positive feedback, yet still one that allows the system to relax at a stable temperature. f being less than zero means the feedback is negative. The table below shows estimated values of f for individual feedbacks.
What does it mean when f = 1, or for that matter, what if f is greater than one?
It is sometimes argued that this scenario mandates a “runaway greenhouse/icehouse” scenario; in reality, when f gets close to one, this linear analysis breaks down. There are certainly many feedbacks that are not linear over a large enough range, so rather than thinking of f =1 as a runaway point, we can think of it as a bifurcation point (or loosely some sort of “tipping point”) but what resides on the other end of the bifurcation requires more information than just a local analysis around the base climate. This point is argued further in a great talkby Dr. Ray Pierrehumbert at the Society for Industrial and Applied Mathematics (SIAM) conference this year, entitled “Climate Sensitivity, Feedback and Bifurcation: From Snowball Earths to the Runaway Greenhouse.” See slides 24-31 for some examples of this, including energy budget structures that have various regimes of “f” values. Non-linearity appears to be of small significance for the modern climate change problem (Roe and Baker, 2011, Nonlin. Processes Geophys)
The extreme hot end of a planetary bifurcation is indeed a runaway greenhouse, provided the planet in question has a large enough liquid ocean. A runaway greenhouse requires a large enough supply of solar radiation to sustain (some speculations, some of which Jim Hansen provided in his “Storms of my Grandchildren” book, that a large amount of CO2 could trigger a Venus-like syndrome, are not actually possible).
Current climate sensitivity estimates generally yield a coherent picture of ~2-4.5 C per doubling of CO2, with very little chance of sensitivity at the low end of this. Current methods of estimating climate sensitivity do not completely rule out high-end values of climate sensitivity for various reasons (Knutti and Hegerl., 2008); perturbed physics ensembles tend to exhibit distributions of sensitivity with a skewed tail toward higher values (e.g., Stainforth et al., 2005), much in the same way as thinking about climate sensitivity being proportional to 1/1-f would yield a skewed distribution (assuming a symmetric and broad uncertainty distribution in the total feedback strength, e.g., Roe and Baker, 2007). Moreover, if one is interested in the forecast out to hundreds of years or longer, climate sensitivity is likely even higher than these estimates when including “slow feedback” processes, like the decay of ice sheets (Lunt et al., 2010; Pagani et al., 2010; Royer et al., 2011).
Whether or not this is “significant” is ultimately a social, economical, and political question although it is very likely a geologist a million years from now would view the “Anthropocene” in the same light as a glacial-interglacial change, except on a much more rapid timescale, and with settled global populations nearing 7 billion.
Take-Home Points
1) Talking about “positive” and “negative” feedbacks requires defining the reference system, in this case, the Planck radiative response. Changing the reference system changes the feedback.
2) In the end, it is possible to have positive feedbacks without causing a runaway warming/cooling. In this case, these feedbacks reduce the efficiency of the Planck restoring effect, so the system can accommodate a larger temperature change before a new equilibrium is established.
3) Climate sensitivity, after accounting for “fast feedback processes” (like water vapor, sea ice, clouds, etc) is probably within a range of ~2-4.5 C, and higher when including slower feedbacks (ice sheets, vegetation)
JC note: My own essay on climate feedbacks was posted [here].