Radiative Forcing and Climate Change

The IPCC (as well as other research groups) frequently use the term radiative forcing to describe the effects of certain anthropogenic and natural processes on the global temperature. Radiative forcing is a neat way of bringing all these different processes to a common denominator and it even allows a pretty painless estimation of how the global temperature can be expected to develop in the long run. Though some physics is required to fully appreciate it.

Every object emits thermal radiation. The higher its temperature T, the higher the intensity I at which it emits thermal radiation. Intensity here refers to the power emitted in watt per square meter m² of surface area, so its unit is W/m². The Stefan-Boltzman law tells us the specific value for I:

I_thermal = k*T^4

So the intensity of the thermal radiation is equal to some constant times the temperature (in Kelvin) to the power of four. Doubling the temperature thus means that the intensity of the thermal radiation goes up by a factor of 2^4 = 16. Since the intensity is measured in watt per square meter, the total power emitted by the object is just the above expression times the total surface area S:

P_thermal = k*S*T^4

Say you are given an object of very low temperature and put this object under a beam of intense radiation. The radiation reaches the object with an intensity I_incoming. Given that the cross-sectional area of the object relative to the source of radiation is C (its shadow area), the object will absorb energy with the rate:

P_incoming = I_incoming*C

Since we agreed to start with a low-temperature object, we can expect that initially P_incoming > P_thermal. The object absorbs more energy than it emits thermally, so it’s temperature will rise. For how long? The temperature will continue to increase until a balance is established and the absorbed power is equal to the emitted power:

P_incoming = P_thermal

I_incoming*C = k*S*T^4

Solving for T, this allows us to calculate the equilibrium temperature from the intensity of the incoming radiation and the ratio of cross-sectional area to surface area:

T = ( (1/k)*(C/S)*I_incoming ) ^ (1/4)

Granted, it’s not a beautiful formula, but all the more useful. If the intensity changes only by very little in relation to its base value, we can simplify this formula for maximum convenience. The derivation uses a bit of calculus, which I’ll ban to the appendix, but it’s good enough to know that using the above formula, we can arrive at the following rule. Changing the intensity of the incoming radiation by an amount dI changes the temperature by the amount dT:

dT = 0.8*dI

A very simple relationship. Change the intensity by dI = +5 W/m² and the temperature will change by dT = +4 °C. Actually, the value 0.8 is not exact science. Inputting pre-industrial values (temperature 287 K and 340 W/m² at the surface on average) gives you a constant K = T_initial/(4*I_initial) = 0.2 rather than 0.8. Why not use this? The above model does not include any effects of weather and climate. 0.2 corresponds to a body with no surface features. Taking into account an atmosphere which can delay the release of heat, trap it someplace, exhibit complex feedback loops, and so on, changes this constant to 0.8 with a fair amount of uncertainty (0.6-1.0). So it may be more accurate to say that a change in intensity by dI = +5 W/m² will, to the best of our knowledge, lead to a change in global temperature of anywhere between dT = +3 °C and dT = +5 °C, with the best guess around dT = +4 °C.

Now we come to the term radiative forcing. Looking at the IPCC documentation, you will find that anthropogenic land use (for example turning grass into wheat fields) is associated with a radiative forcing of dI = -0.15 W/m². It means that the cumulative effect of all these changes is equivalent to turning the sun down by -0.15 W/m². It takes all the complex changes in environment and condenses them down to a single number. A number that tells us how strong the effect is when converted into an equivalent change in incoming solar radiation. And by extension, using the simplified Stefan-Boltzman law, what its individual effect on global temperature in equilibrium is. The radiative forcing of anthropogenic land use dI = -0.15 W/m² is thus expected to decrease global temperature by anywhere between dT = -0.09 °C to dT = -0.15 °C, with the best guess being around dT = -0.12 °C.

The effect of the ozone layer in its current state corresponds to a radiative forcing of dI = +0.35 W/m², equivalent to turning the sun up by +0.35 W/m² (by the way just a change of 0.09 % of the total incoming solar radiation). From it, we can expect the global temperature to go up by dT = +0.28 °C compared to the pre-industrial global temperature. You can see that the term radiative forcing is pretty useful since it takes two very different effects and brings them to a common basis. It allows us to compare climate effect of processes, that are otherwise very difficult to compare.

For the climate effect of CO2 accumulated in the atmosphere we can use the formula:

dI = 5.4*ln(c/280)

Where c is the current concentration of CO2 in the atmosphere in ppm (280 ppm is the pre-industrial value). For the effect of all greenhouses gases combined, with the concentration of CO2 as a proxy for all others, you can use the constant 8.1 instead of 5.3. ln is the natural logarithm. Currently the atmosphere contains CO2 at a concentration of 410 ppm. Inserting this gives us a radiative forcing of dI = +2.1 W/m² for CO2 alone and dI = +3.1 W/m² for all greenhouse gases combined. This corresponds to a change in global temperature of dT = +1.7 °C from CO2 alone and dT = +2.5 °C when taking all greenhouse gases into account.

You might be wondering: Why is this so high? Currently we are at +1.0 °C above pre-industrial, which is far from the +2.5 °C calculated here. Is the calculation wrong? The calculation is correct, but remember that what we are calculating here is the equilibrium temperature we would eventually reach if maintaining this state. We did not look at the changes over time when deriving the formula, we only considered the final steady-state P_incoming = P_thermal. So the temperature calculated via radiative forcing should always be interpreted as the temperature that is “baked in” given that no changes will occur. Staying at 410 ppm CO2 and maintaining the concentrations of the other greenhouse gases as well leads to a temperature change dT = +2.5 °C in the long run. That’s when P_incoming = P_thermal.

Of course, aside from being great for comparison, we can also combine radiative forcings of different effects to calculate a combined effect. We have dI = +3.1 from greenhouse gases, dI = +0.35 from ozone, dI = -0.9 from the build-up of aerosols (high uncertainty) and dI = -0.15 from land use. Adding all leads to a rounded combined effect of dI_total = +2.4 W/m² and a corresponding change in equilibrium temperature from anywhere between dT = +1.4 °C to dT = +2.4 °C compared to pre-industrial times, with the best guess being dT = +1.9 °C.

Note that since each of the radiative forcings has its own uncertainty, the possible range for changes in global temperature is actually larger. Best case we have dI = +2.6 from greenhouse gases, +0.1 from ozone, -1.9 from aerosols and -0.3 from land use, which leads to dI _total = +0.5 W/m² and the range dT = +0.3 °C to dT = +0.5 °C. Worst case we have dI = +3.6 from greenhouse gases, +0.5 from ozone, -0.1 from aerosols and -0.1 from land use, then we get dI _total = +3.9 W/m² and the range dT = +2.3 °C to dT = +3.9 °C. So taking into account all these individual uncertainties, the range that covers all conceivable outcomes goes from dT = +0.3 °C to dT = +3.9 °C. Though thankfully uncertainties usually cancel out to a large degree when adding many variables, the more variables included the better, so our initial dT = +1.4 °C to dT = +2.4 °C is highly likely to include the correct equilibrium temperature.

The loss of arctic sea-ice, which has been accelerating in recent decades and is now happening at a rate faster than predicted by almost all climate models, can also be incorporated with a proper radiative forcing. A recent calculation puts it at around dI = +0.7 W/m², thus adding another dT = +0.6 °C on top if the trend continues.

With these basics, you should be able to find this document by the IPCC a much more enjoyable read. An updated and in-depth look at radiative forcing of aerosols can be found here. Remember, radiative forcing just tells you how strong an effect is in terms of turning the intensity of the sun up or down by a bit. A useful basis for comparison and rule-of-thumb prediction of global temperature.

An added technical afterthought you may or may not find interesting: Radiative forcing does not just inform us about the value of the equilibrium temperature, but also about the rate of change in temperature (in °C per year or decade). Adding the heat dE to a body changes its temperature by c*m*dT with c being the specific heat capacity and m the mass. Using the power P = dE/dt with which heat is added we get: P = c*m*dT/dt or dT/dt = P/(c*m). If not in equilibrium, the power is P = P_incoming-P_thermal, so:

dT/dt = (I_incoming*C – k*S*T_current^4)/(c*m)

In equilibrium we have: I_incoming*C = k*S*T_equilibrium^4. This leads to:

dT/dt = n*(T_equilibrium^4 – T_current^4)

With the constant n = k*S/(c*m). Rewriting this as:

dT/dt = n*((T_pre+dT)^4 – (T_pre+dT_current)^4)

And using a binomial expansion assuming dT << T_pre and dT_current << T_pre:

dT/dt = n*(T_pre^4 + 4*T_pre^3*dT – T_pre^4 – 4*T_pre^3*dT_current)

dT/dt = m*(dT-dT_current)

With the constant m = 4*T_pre^3*n. Using reference data dT_current = 1 °C, dT = 2 °C and dT/dt = 0.2 °C per decade in recent times, we get the rough estimate m = (0.2 °C per decade) per (°C gap to equilibrium temperature). The brackets are only here to aid understanding. So given a radiative forcing dI, we can estimate the corresponding change in equilibrium global temperature dT = 0.8*dI and combine this with the current excess in global temperature dT_current to estimate the rate of change in temperature dT/dt = 0.2*(dT-dT_current) in °C per decade. Naturally, once dT_current = dT, the equilibrium state is reached and the rate of change dT/dt becomes zero. This idea also works works backwards in the sense of measuring the current rate of change as well as the current excess temperature and deducing from this the equilibrium temperature dT and the corresponding total radiative forcing dI.

Here’s as promised the derivation of dT = constant*dI

P_incoming = P_thermal

I*C = k*S*T^4

I = k*(S/C)*T^4

dI/dT = 4* k*(S/C)*T_0^3 = 4*k*(S/C)*T_0^4/T_0 = 4*I_0/T_0

dT/dI = T_0/(4*I_0)

dT = constant*dI with constant = T_0/(4*I_0)

Using T_0 = 13.6 °C = 287 K and I_0 = 340 W/m² (pre-industrial):

constant = 0.2 °C/(W/m²)

(Note that for changes in temperature, K is the same as °C)

As mentioned, this only holds true for a body with no surface features. The presence of an atmosphere with climate dynamics leads to constant = 0.8 °C/(W/m²)

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