Basic Radiative models/Earth’s climate system analysis Pt. 2

From Part 1 we should be able to calculate the energy balance of a planet, and should be able to calculate the equilibrium blackbody temperature of an isothermal spherical zero-albedo planet, as a function of distance from a sun having a given photospheric temperature (the outer layer of the sun).

The concluding calculation for Earth’s temperature in part 1 was off by 33 K. Recall we did this by using 4 pi R^2 εσT^4 for energy emission and 0.25 S (1-A) to find the equilibrium OLR. This 33 K gap is made up for the fact that terrestrial infrared radiation is delayed to space by the presence of an atmosphere opaque to infrared wavelengths…i.e., it has a greenhouse effect. There is an upward surface flux of infrared radiation at ~390 W/m-2, while only ~240 W/m-2 at the TOA, so around 150 W/m-2 is absorbed by the atmosphere. The two major contributors to the greenhouse effect are water vapor and carbon dioxide. Other greenhouse gases are included, but the actual surface temperature of Earth can be nearly accounted for on the basis of the CO2 greenhouse effect with water vapor feedback. Removing either CO2 or Water Vapor from the atmosphere produces a surface temperature too cold for habitability, but taking out just CO2 also makes the atmosphere too cold to hold much water vapor. At least 30 W/m-2 more longwave radiation would escape to space by removing the CO2 from the atmosphere.

By now we have a basic radiative model of S/4(1-a)+ G = σT^4

This can be further written by a time function of,

c (dT/dt) = S/4(1-a) + G – σT^4. Here, c is the specific heat capacity of the ocean ( 1.7 × 10^10 J/m^2/K) so values can now vary in time

It is strongly noted here, that water vapor acts as a feedback to climate change since its concentration is not set by sources and sinks, but by temperature and circulation. Although a powerful greenhouse gas, it can produce just about no effect in forcing a climate change, although once a climate change has been initiated it can further amplify the initial forcing and make the climate more sensitive to changes. Since a warmer climate can hold more water vapor and a cooler climate can hold less, if conditions are pushed by some climate forcing (ex. solar increase, a change in CO2, increase in anthropogenic aerosol concentration, volcanic eruption), the water vapor concentration will adjust accordingly to the climate change (on timescales of days) and make it either warmer or colder (in the first two examples, the effect is warmer temperatures, so a positive feedback from water vapor making it still further warmer; in the latter two examples with all other things equal the net effect is cooling, and so a positive feedback from water vapor gives still further cooling). Although the feedback is positive, it will not produce a “runaway” effect except for the most extreme of cases such as what happened on Venus.

Vapor pressure in equilibrium with a water surface increases quasi-exponentially with temperature at a rate in accord with Clausius-Clapeyron. If the relative humidity remains about constant as temperature and specific humidity increase, then water vapor greenhouse feedback roughly doubles the sensitivity of climate. The changes in specific humidity, with little change in relative humidity, have been documented recently and in accordance with our understanding of modern global climate change.

All gases are condensible at low enough temperatures and/or high enough pressures. CO2, for example, is condensible on Mars though not in present-day Earth climate. This happens when the partial pressure of a gas is equal to the saturation vapor pressure (P_sat). P_sat increases with temperature, since molecules move faster and it becomes more difficult for condensation. This temperature dependence of P_sat comes from the Clausius-Clapeyron equation

P_sat(T) = P_sat(To) exp − L/Rv (1/T – 1/To)

where Rv is the Gas Constant for the gas doing the condensing and To is a reference temperature


(es refers to P_sat here)

So what happens when you add greenhouse gases?

The most simplistic explanation you will often hear, and the best dinner table explanation, is that you delay more of the outgoing infrared radiation from escaping to space due to higher concentrations of gases which are preferentially absorbing radiation in the infrared region.

From Beer’s law, the concentration of gases in a medium effects the amount of absorption of photons; specifially, for some specific wavelength, the logarithm of the transmission through a mixture is equal to the negative of the sum of the absorbance (Ai) of each IR-absorbing species:
ln(1 – εA) = − ∑ Ai (for each wavelength)

Ai depends on the concentration, molar absorptivity, and thickness of the atmosphere. εA is the emissivity of the atmosphere which is a function of GHG concentration.

Let’s get more specific. It is helpful here not to think of the atmosphere as a single slab where this “infrared absorption” is taking place, but a layered one. Think of infrared as moving up into the atmosphere layer-by-layer until it finally reaches an altitude so cold and thin that it can escape to space. Some IR is absorbed in each layer, where it can re-radiate up and may interact with higher layers, or it may heat the surrounding air by interacting with nearby molecules, or it may go downward to the surface.

By adding more greenhouse gases, you increase the IR absorbed in high areas that would otherwise let the radiation through. To escape the atmosphere with increased GHG concentration, infrared must be radiated from higher altitudes which is colder and thinner. It is the top layer which determines the heat balance (that determines the radiation loss to space), and call the pressure at this altitude Px. At its temperature, pressures equal to or greater than Px (below in altitude) radiate strongly. In reality, gases absorb IR selectively, and most of the atmosphere doesn’t interact with IR at all. So the OLR looks like σT(Px)^4. So if you add more greenhouse gases, the regions here become optically thicker for those wavelengths of infrared radiation (meaning you need to go to higher altitudes to find the effective radiating level {ERL}, or that region where radiation is lost to space). So, you decrease Px. Therefore, the temperature of the ERL will not change much as you increase greenhouse gas concentration, but all layers below will warm. This is because the colder, higher layers do not radiate as well and you create a situation where more solar radiation comes in, than infrared goes out. The planet is now taking in about 1 W/m-2 more than it releases to space. Eventually, the lower areas will warm, as this is necessary to achieve radiative equilibrium. Remember from part 1 that an atmosphere in equilibrium has net shortwave radiation at the TOA equal to the OLR at the TOA. This only happens as the lower levels (of altitude) get hotter until the upper levels get hot enough to radiate as much energy back to space as the planet receives, so you are extrapolating along the adiabat a greater distance to reach the ground and hence get a higher temperature.

A planet not in radiative equilibrium either has more solar radiation coming in that infrared going out (warming), or more infrared going out than solar is coming in (cooling). If you add more CO2 and delay more IR back to space, then you will knock off the equilibrium with more solar coming in, so the planet must warm until equilibrium is established. Currently, man is adding CO2 faster than the oceans can absorb, and so the concentration is rising rather fast. Remember that for the process to work there needs to be colder layers above you to work with, since you replace surface radiation with higher, colder radiation to inhibit loss. In fact, the greenhouse effect works because it allows a planet to radiate at a temperature colder than the surface, so the lapse rate is a necessary condition for the operation of the greenhouse effect. As long as there is colder air aloft, adding more GHG’s will make the surface warmer.

The absorption of energy at some frequency is proportional to the number of molecules of absorber encountered (assuming the mixing ratio of the absorber to be constant within a layer of some thickness). Due to Kirchoff’s law, a good absorber is a good emitter of radiation. So the specific absorption of some layer depends on the number of molecules of each greenhouse gas encountered by the beam, and the absorption characteristics of the greenhouse gas. The CO2 band at 15 microns is where the gas absorbs and emits strongly, and CO2 absorbs strong near the peak of the Planck function for Earth-like temperature. In the atmospheric window regions (where the atmosphere is transparent on the sides of the CO2 band), radiation comes from the warmer, lower regions of the atmosphere.

The expressions for radiative balance at the surface, atmosphere, and TOA will be given respectively by σT(s)^4, 2 * σT(a)^4 (due to radiation in both directions), σT(top)^4. The atmosphere, however, is not totally transparent to incoming shortwave radiation, nor completely opaque to outgoing longwave radiation, so let’s assume the atmosphere absorbs a fraction ε of the longwave. Emissivity arises because the atmosphere is not radiaitng as a perfect blackbody, and is letting some longwave through. Let us represent the rate of surface emission by U (which equals σT(s)^4) and 1 – ε escapes to space. The atmosphere will then emit radiation at εσT(a)^4 (or B), so that S/4 (1-a) + B = U.

Use T(a) = 1/(2^1/4)*T(s) and T(s) = T(earth) * (2/2-ε)^1/4. A value of ε = 0.769 approximates real values. However, this is for simple purposes, and when dealing with more complex radiaitve models where we treat the atmosphere as multiple layers, you may need to deal with more numbers (e.g. ε=0.95) when you take into account clouds, and other things, and using one value for one layer can give you a good estimate for the surface, but not for the atmosphere, and vice versa. The above does a good job at the surface of T = 288 K.

The back radiation depends on both the greenhouse gas content of the atmosphere (which determines its emissivity) and the temperature profile. A layer that has low emissivity, and so low absorptivity, in some given wavelength band is referred to as being optically thin; in contrast, an optically thick atmosphere which is opaque to infrared effects temperature accordingly as the ERL approaches a black-body like condition. To be simple, adding greenhouse gases increases both the absorption and emission of the atmospere.
How much warming?

This page gives the formulas for forcing-responses for various GHG’s. ΔT of ~1.2 K from just 2x CO2 is very easy and based on simple physics (hint, apply the new OLR of ~3.7 W/m-2 forcing to Stefan-Boltzmann in part 1 and extrapolate ΔT). However, the literature (see IPCC, 2007) now suggests ΔT (2x CO2) is 2-4.5 K. This comes from adding positive and negative feedbacks (water vapor, ice-albedo, lapse rate, clouds, etc) which are net positive, and so responsible for the higher climate sensitivity. The equation for the change of CO2 in terms of radiative forcing comes from (see chart in linked)

ΔF = 5.35 ln (Cf/Ci)

where Cf and Ci are final and initial CO2 levels, respectively. “5.35” comes from line-by-line radiative transfer codes, and is a constant (Myhre et al., 1998 .) This would be similar to a change in solar output of about 1.6% which can be calculated from:

0.0156 * 1370 * 0.69/4 = 3.7

Here, .0156 was the unknown, and I found that by finding the necessary percent of solar output change (with albedo and geometry of earth taken into account) to produce a 3.7 w/m-2 forcing which is similar to that of 2x CO2. The doubling of CO2 can be from any value (i.e. 200 to 400 ppmv gives the same radiative forcing as 400 to 800 ppmv, or 150 to 300 ppmv). This is due to the fact energy intensity that is output in W/m-2 goes up proportionally to the log of the CO2 concentration, rather than proportionally to the CO2 concentration itself (as is shown two formulas up). This is because a greenhouse gas at relatively high concentrations will be less effective than a dilute gas on a molecule-by-molecule basis. This is very good for us since if it worked without the logarithmic relationship we’d be burning up! Still, the forcing change at even higher concentrations is significant enough to affect living things on the planet, or trigger climate changes like glacial-interglacial cycles.

The current radiative forcings from today, relative to pre-industrial conditions are shown in Figure 10 of my post, here

Part 3 still to come, have to go over the surface budget a bit, and atmospheric/ocean circulation

References for pt 1 and 2

Kiehl, J. T. and Trenberth, K. E. (1997). “Earth’s Annual Global Mean Energy Budget”. Bulletin of the American Meteorological Association 78: 197-208.

Myhre et al. (1998). “New estimates of radiative forcing due to well mixed greenhouse gases”. Geophysical Research Letters Vol. 25, No. 14, 2715–2718



2 responses to “Basic Radiative models/Earth’s climate system analysis Pt. 2

  1. Pingback: Basic Radiative models/Earth’s climate system analysis Pt. 3 « Climate Change

  2. J. R. Leicester

    Fully 45 % of the sun’s radiation is infrared. It exists in the spectrum wavelengths between 760 nm and 3E-4 cm. Where is this accounted for? CO2 does not absorb infrared in these wavelengths..

    How is CO2 “layered” in the atmosphere? By temperature? Mass? What is the absorptivity of CO2?

    Noting that, with a doubling of CO2 in the atmosphere, there are virtually no changes in the capability of the gas to either transfer heat or diffuse it, how does it become such a ‘superinsulator’?

    The temperature differential between earth’s surface and the average atmospheric temperature indicates infrared radiation from earth to be, at best, insignificant. Most of the heat carried to the atmosphere is by conduction and convection. It is only at the atmospheric/space boundary where the radiant heat transfer becomes significant because of the lack of atmosphere. How can earth’s radiative heat loss be measured accurately with such a wide variance of surface temperatures throughout the globe?

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