Posted on March 5th, 2011 in Isaac Held's Blog
An estimate of the forced response in global mean surface temperature, from simulations of the 20th century with a global climate model, GFDL’s CM2.1, (red) and the fit to this evolution with the simplest one-box model (black), for two different relaxation times. From Held et al (2010).
When discussing the emergence of the warming due to increasing greenhouse gases from the background noise, we need to clearly distinguish between the forced response and internal variability, and between transient and equilibrium forced responses. But there is another fundamental, often implicit, assumption that underlies nearly all such discussions: the simplicity of the forced response. Without this simplicity, there is little point in using concepts like “forcing” or “feedback” to help us get our minds around the problem, or in trying to find simple observational constraints on the future climatic response to increasing CO2. The simplicity I am referring to here is “emergent”, roughly analogous to that of a macroscopic equation of state that emerges, in the thermodynamic limit, from exceedingly complex molecular dynamics.
I’ll begin by looking at some results from a climate model. The model (GFDL’s CM2.1) is one that I happen to be familiar with; it is described in Delworth et al (2006). This model simulates the time evolution of the state of the atmosphere, ocean, land surface, and sea ice, given some initial condition. The complexity of the evolution of the atmospheric state is qualitatively similar to that shown in the videos in posts #1 and #2, although the atmospheric component of CM2.1 has lower horizontal spatial resolution (roughly 200km). The input to CM2.1 includes prescribed time-dependent values for the well-mixed greenhouse gases (carbon dioxide, methane, nitrous oxide, CFCs) and other forcing agents (volcanoes, solar irradiance, aerosol and ozone distributions, and land surface characteristics). The model then attempts to simulate the evolution of atmospheric winds, temperatures, water vapor, and clouds; oceanic currents, temperature, and salinity; sea ice concentration and thickness; and land temperatures and ground water. It does not attempt to predict glaciers, land vegetation, the ozone distribution, or the distribution of aerosols; all of these are prescribed. Different classes of models prescribe and simulate different things; when reading about a climate model it is always important to try to get a clear idea of what the model is prescribing and what it is simulating.
Holding all of the forcing agents fixed at values thought to be relevant for the latter part of the 19th century and integrating for a while, the model settles into a statistically steady state, with assorted spontaneously generated variability, including mid-latitude weather, ENSO, and lower frequency variations on decadal and longer time scales. Now perturb this control climate by letting the forcing agents evolve in time according to estimates of what occurred in the 20th century. Do this multiple times, with the same forcing evolution in each case, but selecting different states from the control integration as initial conditions. Average enough of these realizations together to define the forced response of whatever climate statistic one is interested in. (One might want to call this the mean forced response. There is more information than this in the ensemble of forced runs, but let’s not worry about that here.) Each realization from a particular initial condition consists of this forced response plus internal variability, but I want to focus here on the forced response. Observations are not expected to closely resemble this forced response unless the internal variability in the quantity being examined is small compared to the variations in the forced response.
The red curve in the figure is an average over 4 of these realizations of the annual mean and global mean surface temperature. A bigger ensemble would be needed to fully wash out the model’s internal variability (CM2.1 has the interesting problem that its ENSO is too strong). Volcanoes are the only part of the forcing that has rapid variations; besides these impulsive events, the impression is that the forced response would be smooth if estimated with a much bigger ensemble.
The black curve is a solution to the simplest one-box model of the global mean energy balance:
where is the radiative forcing, is the strength of relaxation of surface temperatures back to equilibrium, and a heat capacity. The global mean temperature is the perturbation from the control climate. Where does come from? Here we follow the approach labeled in Hansen et al (2005); it is the net energy flowing in at the top of the model atmosphere, in response to changes in the forcing agents, after allowing the atmosphere (and land) to equilibrate while holding ocean temperatures and sea ice extent fixed. That is, we use calculations with another configuration of the same model, constrained by prescribing ocean temperatures and sea ice, to tell us what “radiative forcing” it feels as a function of time. This estimate is sometimes referred to as the “radiative flux perturbation”, or RFP, rather than “radiative forcing”, especially in the aerosol literature, but I think it is the most appropriate way of defining the to be used in this kind of energy balance emulation of the full model. (Why do we fix only ocean and sea ice surface boundary conditions and not land conditions? This is an interesting point that I will need to come back to in another post.) This estimate of the forcing felt by this particular model increases by about 2.0 W/m2 over the time period shown.
The relaxation time, , is set at 4 years for the plot in the upper panel, a number that was actually obtained by fitting another calculation in which CO2 is instantaneously doubled, which isolates this fast time scale a bit more simply. Not surprisingly, being this short, decreasing this time scale by reducing the heat capacity, or even setting it to zero, has little effect on the overall trend over the century; all that happens is that the response to the volcanic forcing has larger amplitude and a shorter recovery time (conserving the integral over time of the volcanic cooling — as one can see from the lower panel, where the black line is simply ). Other than for the volcanic response, the important parameter is . The best fit to the GCM’s evolution is obtained with 2.3 W/(m2 K). (To get a time scale of 4 years with this value of , the heat capacity needs to be that of about 70 m of water.)
If we compute the forcing due to doubling of CO2 with the same method that we use to compute above, we get 3.5 W/m2, so the response to doubling using this value of would be roughly 1.5 K. However, if we double the CO2 in the CM2.1 model and integrate long enough so that it approaches its new equilibrium, we find that the global mean surface warming is close to 3.4 K.. Evidently, the simple one-box model fit to CM2.1 does not work on the time scales required for full equilibration. Heat is taken up by the deep ocean during this transient phase, and the effects of this heat uptake are reflected in the value of in the one-box fit. Longer time scales, involving a lot more that 70 meters of ocean, come into play as the heat uptake saturates and the model equilibrates. I will be discussing this issue in the next few posts.
Emulalting GCMs with simpler models has been an ongoing activity over decades. Most of these simple models are more elaborate than that used here and typically try to do more than just emulate the global mean temperature evolution of GCMs (MAGICC is a good example). Not all GCMs are this easily fit with simple energy balance models. In particular, different forcing agents can have different efficacies, that is, they force different global mean surface temperature for the same global mean radiative forcing (Hansen et al (2005)).
Additionally, there exist components of the oceanic circulation with decadal to multi-decadal time scales that have the potential to impact the evolution of the forced response over the past century. (This is a different question than whether this internal variability contributes significantly to individual realizations.) I would like to clarify in my own mind whether the ability to fit the 20th century evolution in this particular GCM with the simplest possible energy balance model, with no time scales longer than a few years, is typical or idiosyncratic among GCMs. Other GCMs may require simple emulators with more degrees of freedom to achieve the same quality of fit. There is no question that more degrees of freedom are needed to describe the full equilibration of these models to perturbed forcing, as already indicated by the difference in CM2.1’s transient and equilibrium responses described above, but my question specifically refers to simulations of the past century. I would be very interested if this is discussed somewhere in the literature on GCM emulators. The problem seems to be that accurate computations of the RFP’s felt by individual models are not generally available.
Forced, dissipative dynamical systems can certainly do very complicated things. But you can probably find a dynamical system to make just about any point that you want (there may even be a theorem to that effect); it has to have some compelling relevance to the climate system to be of interest to us here. We will have to return to this issue of linearity-complexity-structural stability, and the critique of climate modeling that we might call “the argument from complexity”, the essence of which is often simply “Who are you kidding? — the system is far too complicated to model with any confidence”.
In the meantime, the goal here has been to try to convince you that the transient forced response in one climate model has a certain simplicity, despite the complexity in the model’s chaotic internal variability. (Admittedly, we have only talked up to this point about global mean temperature.) But is there observational evidence for this emergent simplicity in nature? In the limited context of fitting simple energy balance models to the global mean temperature evolution, convincing quantitative fits are more difficult to come by due to uncertainties in the forcing and the fact that we have only one realization to work with. Fortunately, we have other probes of the climate system. The seasonal cycle on the one hand and the orbital parameter variations underlying glacial-interglacial fluctuations on the other are wonderful examples of forced responses that nature has provided for us, straddling the time scales of interest for anthropogenic climate change. In both cases the relevant change in external forcing involves the Earth-Sun configuration, and we know precisely how this configuration changes. Both have a lot to teach us about the simplicity and/or complexity of climatic responses. Some of the lessons taught by the seasonal cycle are especially simple and important. Watch out for a future post entitled “Summer is warmer than winter”.
[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.]