Thu, 07/22/2010 - 07:55 — apsmith

Since various people thought some of my recent comments trying to explain certain basic properties of the greenhouse effect were educational, I thought I'd repost them here organized in more narrative fashion, as a follow-on to the previous "climate change basics" post. The questions this time regarded the definitions of radiative forcing and feedbacks, the magnitude of various feedbacks, and the relation of surface energy fluxes (the subject of the previous post) to different forcings. Once again the Trenberth-Fasullo-Kiehl diagram of Earth's energy flows is a useful reference:

First, the term "radiative forcing" is defined by the IPCC (TAR and AR4 - see sec 2.2) as

the change in net (down minus up) irradiance (solar plus longwave; in W m–2) at the tropopause after allowing for stratospheric temperatures to readjust to radiative equilibrium, but with surface and tropospheric temperatures and state held fixed at the unperturbed values

Note first that the definition specifies a *change*, not the *total* (such as that for all incoming solar radiation) irradiance. Second note that the definition actually involves the *difference* between incoming and outgoing radiation. Since a good fraction of incoming solar energy is reflected, a change in incoming solar of, say, 2%, would have a radiative forcing of 2% of the *net* average solar radiation absorbed by the Earth - i.e. of (341 - 79 - 23) or 239 W/m^2 from the above diagram. That is, a 2% change in the sun means a radiative forcing of about 4.8 W/m^2, by the IPCC definition.

This is quite a large forcing on the scale of things, even though it's a relatively small number compared to many of the fluxes in the above figure. Actual observed changes in solar forcing from year to year are much smaller - 0.1% to 0.2% over the typical 11-year solar cycle, and likely even smaller (though we don't have precise calibrated measurements going back very far) over decades. The 2% number has been used in models to compare with the results of doubling CO2 - this was done from one of Hansen's earliest papers, and also for example in this post on tropospheric amplification.

The IPCC definition of radiative forcing (net change in radiative flux at the tropopause with the state of the surface and lower atmosphere unchanged) is a useful simplification that avoids the complexity of convective and latent heat flow, since convective and latent heat flow through the tropopause (the top of the troposphere, the boundary between troposphere and stratosphere) is essentially zero. Recognition that the response to different types of perturbations of the climate depends largely on just this one number probably dates back to V. Ramanathan and J.A. Coakley Jr., Reviews of Geophysics and Space Physics, Vol. 16, No. 4, p. 465 (1978) - "Climate Modeling Through Radiative-Convective Models".

Now, doubling CO2 does not change incoming solar energy, but if the lower atmosphere is otherwise left unchanged, it does reduce the level of outgoing radiation. This comes about from two effects: first the quantity of surface radiation that goes directly to space is reduced due to the closure of the "atmospheric window" in spectrum. Second, the radiation that leaves to space from the atmosphere is emitted on average at higher average altitudes, and therefore from atmospheric regions at lower temperature, so that means less radiation leaving. Original calculations of this effect were close to the 2% solar (4.8 W/m^2) level, but since at least the second IPCC report it has been well-established that the forcing from doubling CO2 is closer to 3.7 W/m^2. So a 2% solar increase is about 30% more than doubling CO2; they're not quite equal (Gavin Schmidt used the term "comparable"). They indeed are close enough to make comparisons of those two canonical changes reasonable.

Now the existence of a positive radiative forcing means the planet is receiving more energy than it is emitting - it is out of balance and should be warming up (or at least melting ice). To return to balance from a radiative forcing change of X W/m^2, there must be changes in the surface and the troposphere to result in an outgoing additional X W/m^2 through the tropopause. All radiative forcing changes, whether from a change in incoming solar radiation, GHG changes, aerosols, clouds, etc. have that same forcing-response balance and will in large measure show the same averaged behavior at the surface and throughout the troposphere, although regional and local-in-time detailed responses may differ somewhat.

The central question of climate sensitivity concerns exactly that question - what is the change in surface temperature that returns the planetary system to balance?

First note that we can make some simple conclusions on the changes of fluxes in balance just from the above diagram of average fluxes. 78 W/m^2 of the incoming 239 W/m^2 total is absorbed by the atmosphere (and clouds), and the remaining 161 W/m^2 by the surface, so one might expect a similar 1/3 - 2/3 split in the absorption of any radiative forcing change as well. If solar input increases by 4.8 W/m^2, then about 1.6 W/m^2 of that would be absorbed by the atmosphere and clouds, and 3.2 W/m^2 would make it to the surface.

At the surface, total upward energy flux is 396 W/m^2 radiated plus 97 W/m^2 in convective and latent heat flow. However, as pointed out in the previous post, that upward radiative flux is somewhat artificial, because it is immediately countered by 333 W/m^2 of downward "back radiation" from the atmosphere, much of it from the nearby lower atmosphere, effectively acting as resistance to the radiative flow. In the response to a forcing, one would expect at least a similar ratio of upward and back-radiation, i.e. only 63/396 or about 1/6 of the initial upward radiative flux getting through as a net energy flow away from the surface. Warming would also increase convective and latent heat flows - if that continued to be split roughly 60% (convective + latent heat)/40% (radiative), then to get an increase of 3.2 W/m^2 from the surface to balance an incoming 2% solar forcing would require about 1.3 W/m^2 of net radiative change, which multiplied by 6 gives 7.8 W/m^2 total upward radiative flux (the 1.3 is the net increase after back-radiation). That is, a solar forcing of 4.8 W/m^2 at the tropopause implies an increase of 3.2 W/m^2 in downward surface flux, and to balance that the total surface radiative flux would have to increase by at least 7.8 W/m^2 - a considerable amplification. But that is an extremely rough model of the problem - getting all those numbers right of course is why climate modeling is a little hard.

In summary on this point, as surface and lower atmosphere temperatures increase, there's a big increase in back-radiation to the surface. So the effect of an increase in solar input, or the increase in net flux due to GHG's, is greatly enhanced at the surface once things start to warm up.

Now this is so far just considering the "bare" response of the climate system; an additional issue is feedbacks. In particular, warming the surface puts more water vapor into the air, which has various effects, but most directly it increases the greenhouse effect since H2O is itself an important greenhouse gas. The discussion of feedback in the latest IPCC report (section 8.6) is pretty precise, and based on for example the review by Bony et al, Journal of Climate 19:3445 (2006) - in particular Appendix A, "How are Feedbacks defined?". Here input is a radiative forcing at the tropopause (some number of W/m^2) and the response is evaluated similarly as a radiative change at the tropopause. Balance is restored when the total radiative change at the tropopause (forcing + response) is equal to zero.

That is, the strongest response has to be a negative response that comes close to matching the forcing - then other feedbacks modify that strong response. That first strong ("bare") response can be identified as the "Planck response", simply assuming a uniform temperature increase through the troposphere. It is that response which stabilizes things, and other feedbacks are generally measured in terms of their relationship to that one. But the Planck response can be considered a very large negative feedback on the forcing in the first place - the one that does most of the work to restore balance.

Calculating the Planck response is not a trivial matter of finding the black-body change in emission by the surface though, because the response has to be calculated at the tropopause, where forcings are determined, and where balance must be restored.

Note however that the Planck response is intimately related to temperature. If surface (and atmospheric) temperature does not change, the Planck response is zero. It is the increase in surface temperature, and the correlated increase in tropospheric temperatures, that produces the large negative Planck feedback. Those are thermal radiative response components that explicitly are determined by the temperature of the surface and atmosphere. The W/m^2 of the stabilizing Planck response requires a change in temperature. All other feedbacks are then measured relative to that, which is where the natural sensitivity measure of K/(W/m^2) is relevant.

If surface temperature does not increase, the Planck response is not initiated and the net radiative forcing simply cannot be balanced - Earth will continue absorbing energy until its temperature eventually does rise to balance. Temperature increase is key, not some after-thought.

Now so far in discussing solar forcing I've referred to longer-term averages. However, solar forcing does change quite dramatically within a given year. This results from two effects: the angle at which incoming sunlight strikes Earth's surface, which changes with the seasons (and the differences between Earth's northern and southern hemisphere's particularly in relative land fraction make that effectively a forcing change) and the distance between Earth and Sun which results in a solar "constant" variation of almost 6% from peak to peak. The response of Earth's climate to the seasonal variation is obviously strong, but the actual average surface temperature doesn't change all that much. As with any such periodic forcing you will always see a phase shift in the response due to the time dependence of the response components. The average temperature response and yearly flux variation are out of phase in this case which indicates there are large response components that take more than 6 months to come into effect, and the actual magnitude of the response is going to be considerably less than a longer-term flux change would impose. The largest component is ocean inertia. 70% of Earth's surface is ocean; the first 100 m or so is reasonably well mixed. (Note that 100 m is far less than the "entire ocean". Water between the surface and top of the thermocline is what participates (at least in the short term) in the thermal response. As the ocean thermocline varies between about 50 and 1000 m, 100 m isn't a bad estimate.) To drop by 1 degree over 6 months requires a real heat loss of about 3 x10^8 J/m^2 (from heat capacity of water x 70%) or 20 W/m^2 continuously - so unless the forcing change is very large, 6 months is simply insufficient to see much response. The problem with GHG changes is they persist in providing added heat year after year, decade after decade. Of course seasonal variations can have well over 100 W/m^2 in local difference between winter and summer incoming radiation, so seeing substantial (local) temperature changes with the seasons isn't surprising. Global averages are a different matter because they do not have this regional seasonal component in total incoming solar energy, and so are naturally more stable (but of course strongly affected by the seasonal element due to land/ocean distribution differences between the hemispheres).

Response to the flux change associated with volcanic eruptions like Pinatubo can allow for longer-term components to respond, and those largely agree with climate models that produce the standard sensitivity values; even longer-term changes associated with forcing changes deduced from geological history provide further evidence. Assessments of our understanding of sensitivity from many different lines of argument of this sort have been done, for example, by Knutti and Hegerl, Nature Geoscience 1, 735 (2008).

Returning to the mathematics of the feedback process, the definitions in Bony et al make the mathematics of the energy balance problem perfectly clear and completely independent of any control-theory concepts, whether or not they are relevant - the result is simple mathematics. Here's a re-cap of the argument just show how simple it is:

(1) Radiative balance R is defined as incoming minus outgoing radiation at the tropopause (since there should be negligible convective or other heat transport effects across the tropopause, radiation balance means full balance). R = 0 in balanced steady state

(2) Radiative forcing Q is change in R due to some system change (solar, GHG, clouds, etc) with surface and lower atmosphere otherwise held constant (no response).

(3) The largest (and quickest) response to radiative imbalance, Q != 0, is the Planck response, associated with a uniform change in surface and troposphere temperatures delta T = delta T_s (surface).

(4) The Planck response (like other responses) isn't entirely linear (Stefan-Boltzmann is T^4!) but linearization isn't far off for small forcings Q, so the resulting value of R can be written as

R = Q + lambda delta T_s

where lambda is the ratio between the change in radiative balance at the tropopause and the surface temperature change.

(5) Balance is then restored when R = 0, i.e.

delta T_s = - Q/ lambda

(note the minus sign). -1/lambda is then a measure of the sensitivity of the climate system to radiative forcings Q.

(6) The largest response term, again, the Planck response, has a negative value of lambda. If lambda were zero or positive, the temperature response would be infinite - so it's a good thing that the Planck response dominates. All analyses find the Planck value to be very close to -3.2 W/m^2/K.

(7) Other feedbacks are represented by their ratio to the Planck feedback g_x = - lambda_x/lambda_P, with sign reversed so that positive g_x means an increase in sensitivity. If the sum of the g_x's ever gets as large as 1, you have overwhelmed the basic Planck response and can get runaway (at least for a time). Very few people believe that's actually possible for our planet right now.

And that's it - forcings and feedbacks, with surface temperature as the central reference term because it characterizes the dominant Planck response so well. If you have some other measure based on energy fluxes that captures the physics, feel free to try introducing it, but the temperature-based analysis makes physical and logical sense in the climate system, which is why it's so widely used.

We can return now to our estimate above regarding the "base" response to a forcing of, say, 4.8 W/m^2 from a 2% increase in solar input. The basic Planck response results in a surface temperature increase of 1/3.2 K times the radiative forcing in W/m^2. Using the average surface temperature of 288 K, 4.8 W/m^2 of solar forcing at the tropopause results in an increase to 289.5 K, or an increase of 8.1 W/m^2 of surface thermal radiation, an ("open loop") amplification ratio of 1.7. As I mentioned before, most of that increase in surface thermal radiation doesn't go very far because it is matched by increases in thermal back-radiation from the lower troposphere; that's why you get an open-loop gain of that size.

Now the Planck response is just a bare response; water-vapor and other feedbacks need to included to get the full response. Climate models and other lines of argument show that they, with some uncertainty, result in double or even slightly more the response. So the final surface power, after feedbacks, is in the area of 3 to 5 times the radiative forcing. Some of the response is slow (ocean inertia in particular) so that full amplification won't be there for short-term variations.

But aside from the time-dependence, that same amplification applies to all sources of forcing - solar, GHG, volcanic aerosols, whatever you look at. Climate models show this similarity in response all the time - though the detailed breakdown in space and time for different types of forcings are slightly different.

One more interesting facet from the 2009 Trenberth, Fasullo, Kiehl paper (Figure 1 above): the total energy flux absorbed by the atmosphere (including clouds) is 78 + 356 + 97 = 531 W/m^2. On the emission side, 333 W/m^2 is emitted to the surface, and 199 W/m^2 to space, i.e. about 63% returned to the surface, 37% to space. If the atmosphere behaved like a single layer, on the other hand, you would expect the same radiative emission up as down (any given region of atmosphere must emit radiation isotropically - the same in all directions). The reason it is not 50/50 in the real atmosphere is because in the regions of peak absorption, the initial absorption is close to the surface (at a relatively high temperature) and power in that spectral region is then emitted and reabsorbed several times up through the atmosphere, at colder and colder temperatures.

There are a vast array of climate models out there, from simple zero- or one-dimensional approximations to sophisticated 3-dimensional detailed earth system models - for example the NCAR Community Earth System model which is openly available and well-documented. All of the modern models out there roughly agree on the basics of the energy balance of this problem, roughly agreeing with the Trenberth-Fasullo-Kiehl diagram above (which is derived more from observation than models). Any alternative model must be very clear on how it differs and why, if it finds any significant difference from these established constraints.