October 8, 2013

Radiative Forcing and Global Warming Potential

Sometimes science seems to use English words, but on closer inspection you find that the meaning of those words are not what you're used to.

"Positive Feedback" is a good example. At school, or at work, positive feedback is when you're complimented for a job done well. In the context of climate science, positive feedback often denotes a vicious cycle or self-reinforcing global warming, as in when waming leads to melting of permafrost, which releases methane (a powerful greenhouse gas), which in turn leads to further warming.

"Stampede Loop" by John Trevithick, a great example of a positive feedback loop.

I could think of a bunch of racier examples, but this is a general-audience blog, so I will refrain. Suffice it to say that sometimes scientists, with a straight face, will bandy about terms that in an ordinary non-scientific conversation would make you blush.

But back to global warming. The following are notes on my reading up about several terms that have been used a lot - but I realised I didn't quite understand. I learned that the fearfully named "radiative forcing" is simply the imbalance on the planet's energy balance sheet. That the term "global warming potential" is fraught with details, caveats and snags that nobody ever talks about. And I found out the reason for why people talk of the temperature rise associated with a "doubling of the CO2 concentration", rather than an increase by some amount.

There's a bit of math. That's okay: an equation is worth a thousand words. And math was the gateway through which I found out that some of the words used in the climate change discussion don't mean quite what I thought they did. So here goes.


The term "Radiative Forcing" gets used a lot in discussions on climate change. At first hearing, it might sound like the light emanating from the sun is pushing the earth into a larger orbit. But it's nothing of the kind.

Radiative Forcing (RF) is the difference between the radiant energy (per unit area, per unit time) received by the earth - mostly from the sun - and the energy radiated by the earth, back into space. The units are Watts per square meter (Wm-2).

Before the Industrial Revolution, the total RF=0, that is, the outgoing radiation exactly balanced the incoming radiation, so the earth was not accumulating excess energy, and did not warm (at least over the holocene, the last 10,000 or so years).

In the context of climate change, we are interested in the imbalance: anthropogenic CO2 emissions have caused the atmosphere to be more retentive of heat, so the earth is radiating less energy than it receives: this accumulation of energy is what causes global warming.


For the global warming due to a particular component of our planet, such as albedo, land use or any greenhouse gases, the IPCC defines radiative forcing as follows (my italics):

"Radiative forcing is a measure of the influence a factor has in altering the balance of incoming and outgoing energy in the Earth-atmosphere system and is an index of the importance of the factor as a potential climate change mechanism. In this report radiative forcing values are for changes relative to preindustrial conditions defined at 1750 (AD) and are expressed in Watts per square meter (W/m2)."

Here is an example for carbon dioxide (CO2). The earth radiates energy because its surface is warm, and the radiation has a spectrum that follows roughly the black body radiation spectrum, but modified by the atmospheric absorption. The absorption shows up as dips in the spectrum. The entire spectrum is in the far infrared (visible light would be at 19,000 - 28,000 cm-1).

The green line in the graph is the earth's irradiance for CO2 levels of 300 ppm. Wavenumber is the inverse of wavelength, measured in cm-1. Irradiance (on the y-axis) is the radiative forcing found over an interval of the spectrum; when graphed as a function of wavenumber, its units are Wm-2cm. The big dip around 660 cm-1 is the absorption band of CO2.

The blue line, for 600 ppm CO2, shows a slightly deeper dip at 660 cm-1, that is, the earth radiates less heat at the higher CO2 concentration. The difference doesn't look like much (more on that below), but integrating over the spectrum gives a total change in outward irradiance of ΔF=3.39 W/m2. This is the radiative forcing due to the additional 300 ppm of CO2.

[Incidentally, the sun's radiation also has a black-body spectrum. But because the sun's surface is much hotter than the earth's, the peak of the solar irradiance occurs in the visible, around 21000 cm-1 (or if you prefer thinking about wavelengths, around λ=590nm: yellow). This why the sun's incoming radiation doesn't depend on the concentration of greenhouse gases]

Carbon dioxide happens to be rather a difficult case: doubling the CO2 concentration does not make the absorption band at 660 cm-1 twice as deep. This is because CO2 is such a strong absorber that even at 300ppm it's well into the saturation region. The saturation effects cause the radiative forcing to have a logarithmic, rather than a linear, dependence on the CO2 concentration:

ΔF = 5.4 × ln (C/C0)

Here C is the CO2 concentration in ppm, and C0 is the reference concentration in the year 1750. This is why there is so much talk about "climate sensitivity", which is the amount of global warming following a "doubling" of CO2 concentrations: because of the logarthmic dependence, a doubling of the concentration will give a certain amount of radiative forcing that is independent of the starting concentration.

The current CO2 concentration is 400ppm, only 120ppm above pre-industrial values, so right now ΔF=1.9 Wm-2. The total anthropogenic radiative forcing, 2.3 Wm-2, is only 1.5% of the total average solar flux (150Wm-2), yet is enough to cause the global warming that represents such a threat to our species. Maybe it's that the planet is large: another measure of global warming is that our planet is accumulating the equivalent of four Hiroshima bombs worth of energy every second.

NOTE: the logarithmic dependence holds only for CO2. In contrast, ΔF for methane and N2O have a square-root dependence on concentration, while ΔF of a number of CFC gases have a linear dependence: these are far from the saturation limit. These are listed, with the coefficients, in IPCC's AR3.

Nature is just marvellous in its complexity. But it's not easy to talk about it in the kind of soundbites that the media is used to feeding us.


In case RF wasn't complicated enough, now let's talk about Global Warming Potential (GWP).

Remember, radiative forcing is the rate at which heat is retained in the earth's atmosphere instead of being radiated out into space. That's why its units are W/m2. But it doesn't tell you about the total heat retained by the earth because of an emissions of some greenhouse gas.

[In everyday terms: your hair dryer might use 1500 W of power. But the energy consumed (and the amount of heat you release in your bathroom) depends on how long you have the hair dryer turned on. If you've used it for half an hour, that's 750Wh, or 0.75kWh. If your teenage daughter busies herself with it for a whole hour, she will contribute 1.5kWh to the household energy use stated on your electricity bill (and you may want to talk to her about the risk of hearing damage) ].

Energy is power multiplied by the time it is applied. SI units doesn't use kWh, but Joules: 1J=1Ws=1 Watt-second. Heat is one particular kind of energy, and is also measured in Joules.

I'm going on about this because knowing the radiative forcing due to some of greenhouse gas (GHG) is not enough to tell you how much total warming that gas will cause: for that you need to integrate the radiative forcing over the lifetime of the gas. Each gas has its own lifetime in the atmosphere, ranging from hundreds of years in the case of CO2, to a few weeks for tropospheric ozone. For methane it's about 12 years.

So while methane may be a much feistier greenhouse gas than carbon dioxide, it has much less time to wreak its havoc. The concept of global warming potential attempts to make a reasonable comparison between the long-term climate effects of these disparate greenhouse gases.

Let's look at IPCC's definition of global warming potential: "The GWP has been defined as the ratio of the time-integrated radiative forcing from the instantaneous release of 1 kg of a trace substance relative to that of 1 kg of a reference gas":

GWP \left(x\right) = \frac{\int_0^{TH} a_x \cdot \left[x(t)\right] dt} {\int_0^{TH} a_r \cdot \left[r(t)\right] dt}

where TH is the time horizon over which the calculation is considered; ax is the radiative efficiency (same, as far as I can tell, as radiative forcing) due to a unit increase in atmospheric abundance of the substance (so its units are Wm−2 kg−1) and [x(t)] is the time-dependent decay in abundance of the substance following an instantaneous release of it at time t=0. The denominator contains the corresponding quantities for the reference gas CO2.

If I read the description of GWP in AR3 correctly, this definition uses a perturbative approach: GWP is the global warming response to the emission of a small additional amount of greenhouse gas x , in the presence of the other greenhouse gases that are already around. It's analogous to the marginal cost in economics. It depends, among many other things, on the time scale over which you look (TH) and the concentration of all the greenhouse gases already present.

The reference gas, CO2, is an unfortunate choise, because it is such an unruly beast (non-linear density dependence, rather intractable lifetime). So GWP may not be the greatest choice of measures by which to compare the badassedness of a particular greenhouse gas.

But this is the measure we've got, and as long as you remember all the caveats, which none of the journalists ever mention, GWP does give you the relative impact of a small additional amount of a given greenhouse gas, at least as compared to the major bad actor, CO2.

A list of GWP for the most common greenhouse gases shows the values for TH=20 years and TH=100 years. Another caveat: water is too slick to handle because its GWP depends on a huge number of parameters, not the least fluctuating of which is the temperature. So that's left out of the discussion altogether, even though we know from daily experience that a cloudy sky or even high humidity makes for warmer nights than a clear one.

It was pointed out quite early on (and repeated in AR5) that there is no scientific case for choosing to report GWP at 100 years rather than over some other time horizon. The choice of time horizon is a "value judgement". Meaning, there is space here for manipulation and misinterpretation. Also, if we ever get in the situation of runaway global warming, the relevant time horizon becomes the lifetime of the main cause of the positive feedback, very probably methane.


If you are interested in the global warming caused by the emissions of a unit amount of gas, you don't normalise to CO2, you would simply retain the numerator of the equation defining GWP, and integrate out to infinite time, to get the absolute global warming potential, AGWP, which has units Jm-2kg-1.

AGWP = ∫0∞ ax·[x(t)] dt

If you want to know how much heat the earth retains as the result of greenhouse gas emission, the AGWP needs to be multiplied by the surface of the earth, and the total mass M of the emitted greenhouse gas:

ΔE = 4πR2 M ∫0∞ ax·[x(t)] dt

The units for ΔE is Joules. As an example, the oceans are where 93% of the planet excess heat is absorbed. The change in heat content of the oceans since we started burning fossil fuels is now close to 2x1023 Joules.

I repeat the caveat: The above formula only holds for incremental amounts of GHG emission (small M), for those GHGs which don't have a linear concentration dependence. See the discussion of nonlinearities under radiative forcing.


If you're interested in what kind of temperature rise a given amount of GHG emissions would cause, you need the Global Temperature change Potential, or GTP. Here, "temperature" refers to the global mean surface temperature, so GTP depends also on how the retained heat is distributed among the atmosphere, the land and the oceans. So its estimate encompasses the entire geological complexity that makes this planet such a wonderful place to live on.

It is a testament to climate scientists' prowess and tenacity that global warming predictions have been so accurate.



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