The co-evolution of the global mean surface air temperature (T) and the net energy flux at the top of the atmosphere, in simulations of the response to a doubling of CO2 with GFDL’s CM2.1 model.
Slightly modified from Winton et al (2010).
Global climate models typically predict transient climate responses that are difficult to reconcile with the simplest energy balance models designed to mimic the GCMs’ climate sensitivity and rate of heat uptake. This figure helps define the problem.
Take your favorite climate model, instantaneously double the concentration of in the atmosphere, and watch the model return to equilibrium. I am thinking here of coupled atmosphere-ocean models of the physical climate system in which is an input, not models in which emissions are prescribed and the evolution of atmospheric is itself part of the model output.
Now plot the globally-averaged energy imbalance at the top of the atmosphere versus the globally-averaged surface temperature . In the most common simple energy balance models we would have where both , the radiative forcing, and , the strength of the radiative restoring, are constants. The result would be a straight line in the plane, connecting with as indicated in the figure above. The particular two-box model discussed in post #4 would also evolve along this linear trajectory; the different way in which the heat uptake is modeled in that case just modifies how fast the model moves along the line.
The figure at the top shows the behavior of GFDL’s CM2.1 model. The departure from linearity, with the model falling below the expected line, is common if not quite universal among GCMs, and has been discussed by Williams et al (2008) and Winton et al (2010) recently — these papers cite some earlier discussions of this issue as well. Our CM2.1 model has about as large a departure from linearity as any GCM that we are aware of, which is one reason why we got interested in this issue.
As indicated in the somewhat cryptic legend, we use two different types of simulations to make this plot. One is the instantaneous doubling of referred to above. We show annual means for the first 10 years (with each cross in the figure an average over 4 realizations to knock down the noise, branching off at different times from a control simulation) and then show 5-year means up till year 70, again averaging over 4 realizations. Because these integrations do not go out far enough to probe the slower long term evolution, we then append a single realization of the standard calculation in which is increased at 1%/year until the time of doubling (year 70) after which it is held fixed. We plot 5-year averages from this calculation, starting in year 70, so all points in the figure correspond to the same value of . 600 years still isn’t enough to equilibrate, but as long as something fundamentally new doesn’t happen in the model on longer time scales one can extrapolate to to get an estimate of the equilibrium temperature response. The two simulations match up nicely in year 70, as one expects if the 1%/yr case resides during its ramp-up phase in the intermediate regime (post #3). Because of the curvature of this trajectory, the temperature change at year 70, about 1.5-1.6K (the transient climate response (TCR)) is smaller than we might expect from the model’s equilibrium sensitivity and the model’s value of at that same time.
One’s first reaction might be to say — well, there is nonlinearity in the model in the sense that is effectively a function of . But I think there is agreement that the underlying dynamics is still best described as linear; it’s just that the global mean energy balance is not a function of the global mean surface temperature. A more general linear model assumes that the global mean energy balance is a linear functional of the surface temperature field, with different spatial structures in surface temperature perturbations, even if they have the same global mean, generating different perturbation to the global mean energy balance.
Think of some atmospheric model equilibrated over a prescribed surface temperature distribution. This temperature distribution is the input to the model. The model outputs climate statistics of interest, including the global mean energy balance. If the relation between input (surface temperatures) and output (global mean energy balance) is linear, we can write
Brackets denote a spatial average over the surface and is the position on the surface. The scalar radiative restoring constant has been replaced by . (By the way, I am not assuming here that the top-of-atmosphere energy balance in some small region is only a function of the surface temperature in that same region — the relation between these two is non-local due to mixing in the atmosphere.)
The simplest case is when temperature evolves in a self-similar manner, i.e., growing with a fixed spatial structure:
(I have normalized things so that ). The effective radiative forcing for temperature perturbations with this structure is
If temperatures perturbations have a different structure, , then we need to replace with . But suppose that the temperature perturbations are the sum of two patterns with relative contributions varying in time:
with . This gives us enough freedom to get evolution off the classic linear trajectory. But we haven’t learned anything yet about how and why the ratio of to is evolving in time.
One way of analyzing any linear system is through the frequency-dependence of the response to perturbations. Low frequency and high frequency forcing can result in different radiative restoring strengths if they result in different spatial structures in the response. Evidently, the low frequency component controlling the late time evolution in the response to doubling of CO2 is characterized by a structure that is restored less strongly than is the fast, early response. Why would that be?
The story seems to be something like this: The atmosphere tends to be most unstable to vertical mixing in the tropics, where the surface temperature are warmest, but the oceans are most unstable to vertical mixing at high latitudes, where the surface temperatures are the coldest. It is in the subpolar oceans that the mixing between surface and deeper waters is the strongest. One expects these regions to be a major source of the difference between fast and slow responses, with the slow responses having larger subpolar ocean warming. This effect tends to mix out to other high latitude regions, so the high latitude amplification of the response is typically larger in the slow response.
We now have to argue why a pattern with larger high latitude amplification is restored less strongly. This is more complicated. A part of the explanation seems to be that the surface is less strongly coupled to the atmosphere in high than in low latitudes, so the surface warming has a harder time affecting the radiation escaping to space. But a big part also seems to be played by different cloud feedbacks that come into play in the fast vs the slow responses, the clouds reacting to the different atmospheric conditions that occur when the subploar ocean warming is held back or is given time to respond.
One can still try to save the global mean perspective. Winton et al (2010) pursue this line of reasoning by referring to the “efficacy of ocean heat uptake”. The idea here is that the difference in spatial structure of the fast and slow responses can be attributed to the heat being transferred from shallow to deeper ocean layers. Putting aside the question of how this heat transfer is controlled, one can try to think of it as a different kind of “forcing” of the near-surface layer, alongside the radiative forcing. The response to heat uptake, being focused in high latitudes, naturally has a spatial structure that is more polar amplified than the response to CO2 (with the heat uptake fixed), so it experiences a smaller restoring strength. The effects of the surface cooling due to heat uptake by deeper layers is amplified, slowing down the initital fast warming more than one might otherwise expect. This picture has the nice feature that it ties the timing of the change in spatial structure directly to the saturation of the heat uptake. You may want to think about how to capture this effect with a simple modification of the two-box model described in earlier posts.
One moral of this story is that forcing a global mean perspective on the system can make things look more complicated than they actually are, making the response look superficially nonlinear when it is still quite linear.
Another moral is that the connection between transient and equilibrium responses may not be as straightforward as we might like, even when only considering the consequences of the physical equilibration of the deep ocean, leaving aside things like the slow evolution of ice sheets.
[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]