Readers have recently discussed the correlation through time between global temperature on the one hand, and CO 2 concentration on the other. Close examination shows that the correlation is stronger during some time intervals, weaker during others, and although it’s strong overall, there seems to be a lot happening to temperature other than mimicry of the CO 2 changes.

One suggestion was to study the relationship, not with CO 2 concentration, but with its logarithm. This is because climate forcing — a measure of the ultimate climate-changing impact — is proportional to the logarithm of the CO 2 concentration, not to the concentration itself. The idea is to look for correlation between temperature and climate forcing — and it makes sense.

The fascinating thing is: there are many different climate forcings. A lot more than just CO 2 .



For one thing, there are other greenhouse gases beside CO 2 . Methane (CH 4 ) has a surprisingly large effect, and bringing up the rear is a host of others, including N 2 O, chloroflourocarbons, hydroflourocarbons, and others. And, these gases can interact with each other; the climate forcing of methane, e.g., depends on the concentration of N 2 O, and vice versa. NASA has estimated the total greenhouse gas forcing, from all gases combined, and came up with this:

An important thing to note is that although water vapor (H 2 O) is a potent greenhouse gas, and the most abundant in our atmosphere, it’s not a climate forcing. That’s because it is controlled by temperature; if we tried to increase or decrease global water vapor artificially, it would quickly rain back out of or evaporate back in to the atmosphere. Water vapor doesn’t hang around long enough be a climate forcing, it adjusts to temperature (and other things) too quickly. It is, however, a climate feedback (we’ll leave that discussion for another day).

Greenhouse gases just get us started. Ozone is it’s own thing, working differently high in the stratosphere than lower in the troposphere. When the sun gets hotter, so does Earth, when the sun cools off we do too, so changes in solar output make another climate forcing. Land use is crucial; when we replace forest with cropland (or cities) it affects the climate. Albedo is the reflectivity of Earth’s surface, and when it decreases because ice and snow are replaced by open land or water, more solar energy is absorbed and we heat up — albedo is another forcing. There are even small forcings due to changes in Earth’s orbit. Then there are the aerosols, which, low in the troposphere, have both a direct and an indirect effect, and which, high in the stratosphere (from volcanic eruptions) make one of the strongest cooling effects. NASA has estimated all these forcings (data through the year 2012):

The two strongest forcings are the thick black line on top, for well-mixed greenhouse gases, and the thin black line on the bottom, stratospheric aerosols from volcanic eruptions. NASA also provides estimates of the total climate forcing, from 1850 through 2012:

Rather than look for the correlation between global temperature and just CO 2 concentration, let’s see how it relates to total climate forcing.

Climate forcing means more energy coming in to the system, which heats things up, and it takes time for the heat to accumulate, so it’s just the laws of physics that the climate doesn’t respond to a forcing instantaneously. The simplest physical model is the energy balance model. It’s quite simple really, and can’t (in my opinion) compete with the fancy supercomputer models that run the details, but it does reflect some of the basic physics that’s happening.

It tells us to seek correlation between temperature and exponentially smoothed climate forcing. An exponential smooth has a “time scale” — short for a “fast” smooth”, long for a “slow” smooth. If the climate system has low thermal inertia then it will respond quickly — maybe with a time scale as short as one year. Let’s correlate global temperature with a 1-year exponentially smoothed climate forcing, by making a model of temperature (we’ll call it the “fast-response model”). I’ll use the temperature data from Berkely Earth, and the model looks like this:

The first thing we notice is that the large downward spikes (from volcanic eruptions) in the model actually correspond to downward spikes in the temperature data — but the model spikes are much too large. It’s like the effect of climate forcing is being magnified too much by this model.

But the overall pattern is actually quite close, so the scaling factor for forcing looks about right in the long term, but far off in the short-term stuff.

The land surface, and the atmosphere, might respond to climate forcing that quickly, but the oceans take longer with so much thermal inertia. Just the upper ocean, let alone the deep abyss, is sluggish compared to land and air. Let’s try a model using a more long-term smooth, say a 22-year exponential smooth (I’ll call it the “slow-response model”):

It follows the long-term pattern excellently, and the unrealistic huge downward spikes are now gone. We still see drops when volcanic eruptions happened, but nothing like a spike, and not as deep a dive as is seen in real data. Even so, this model is at least realistic, and avoids nonsense features.

Those energy balance models only use one “compartment” to the system. You can treat it like a land-atmosphere system and get fast response, or like an upper-ocean system with slow response, but the former exaggerates the fast response (to match the slow) while the latter subdues it too much. A more realistic model, both mathematically and physically, is to use two (or more) “compartments” for a model — a two-box energy balance model.

Then the math tells us to model something like surface temperature as a combination of two different exponential smooths with two different time scales. I found that the best combination of time scales was 1 and 22 years, leading to this two-box model:

Now we have a rather good model, based on extremely simple physics (it certainly omits a lot of detail!). It matches the global data well, matches the response to volcanic eruptions well, and even has a plausible physical interpretation. The land+atmosphere system responds quickly (time scale 1 year), the upper ocean more slowly (time scale 22 years), and observed surface temperature is a manifestation of both (like the physics says).

Efforts to correlation global temperature with CO 2 alone, are often interesting, sometimes persuasive, sometimes informative. I encourage getting to know how these things are related to each other. But let’s not lose sight of the fact that with so many other forcings, and with nature’s never-ceasing fluctuation even in the most constant of times, this is one of those cases where deeper understanding deserves closer examination.

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