LMAO
Gavin wrote this just for you; the equations won't copy, so go to the link. They won't bite you over there.
Learning from a simple model
gavin @ 7:59 am
A lot of what gets discussed here in relation to the greenhouse effect is relatively simple, and yet can be confusing to the lay reader. A useful way of demonstrating that simplicity is to use a stripped down mathematical model that is complex enough to include some interesting physics, but simple enough so that you can just write down the answer. This is the staple of most textbooks on the subject, but there are questions that arise in discussions here that don't ever get addressed in most textbooks. Yet simple models can be useful there too.
I'll try and cover a few 'greenhouse' issues that come up in multiple contexts in the climate debate. Why does 'radiative forcing' work as method for comparing different physical impacts on the climate, and why you can't calculate climate sensitivity just by looking at the surface energy budget. There will be mathematics, but hopefully it won't be too painful.
So how simple can you make a model that contains the basic greenhouse physics? Pretty simple actually. You need to account for the solar radiation coming in (including the impact of albedo), the longwave radiation coming from the surface (which depends on the temperature) and some absorption/radiation (the 'emissivity') of longwave radiation in the atmosphere (the basic greenhouse effect). Optionally, you can increase the realism by adding feedbacks (allowing the absorption or albedo to depend on temperature), and other processes - like convection - that link the surface and atmosphere more closely than radiation does. You can skip directly to the bottom-line points if you don't want to see the gory details.
The Greenhouse Effect
The basic case is set up like so: Solar radiation coming in is , where is the albedo, TSI the solar 'constant' and the factor 4 deals with the geometry (the ratio of the area of the disk to the area of the sphere). The surface emission is where is the Stefan-Boltzmann constant, and is the surface temperature and the atmospheric radiative flux is written , where is the emissivity - effectively the strength of the greenhouse effect. Note that this is just going to be a qualitative description and can't be used to quantitatively estimate the real world values.
There are three equations that define this system - the energy balance at the surface, in the atmosphere and for the planet as a whole (only two of which are independent). We can write the equations in terms of the energy fluxes (instead of the temperatures) since it makes the algebra a little clearer.
Surface:
Atmosphere:
Planet:
The factor of two for A (the radiation emitted from the atmosphere) comes in because the atmosphere radiates both up and down. From those equations you can derive the surface temperature as a function of the incoming solar and the atmospheric emissivity as:
If you want to put some vaguely realistic numbers to it, then with S=240 W/m2 and =0.769, you get a ground temperature of 288 K - roughly corresponding to Earth. So far, so good.
Point 1: It's easy to see that the G (and hence ) increases from S to 2S as the emissivity goes from 0 (no greenhouse effect) to 1 (maximum greenhouse effect) i.e. increasing the greenhouse effect warms the surface.
This is an extremely robust result, and indeed has been known for over a century. One little subtlety, note that the atmospheric temperature is cooler than the surface - this is fundamental to there being a greenhouse effect at all. In this example it's cooler because of the radiative balance, while in the real world it's cooler because of adiabatic expansion (air cools as it expands under lower pressure) modified by convection.
Radiative Forcing
Now what happens if something changes - say the solar input increases, or the emissivity changes? It's easy enough to put in the new values and see what happens - and this will define the sensitivity of system. We can also calculate the instantaneous change in the energy balance at the top of the atmosphere as or changes while keeping the temperatures the same. This is the famed 'radiative forcing' you've heard so much about. That change (+ve going down) is:
where are the small changes in solar and change in emissivity respectively. The subscripts indicate the previous equilibrium values We can calculate the resulting change in G as:
so there is a direct linear connection between the radiative forcing and the resulting temperature change. In more complex systems the radiative forcing is a more tightly defined concept (the stratosphere or presence of convection make it a little more complex), but the principle remains the same:
Point 2: Radiative forcing - whether from the sun or from greenhouse gases - has pretty much the same effect regardless of how it comes about.
Climate Sensitivity
The ratio of is the sensitivity of to the forcing for this (simplified) system. To get the sensitivity of the temperature (which is the more usual definition of climate sensitivity, ), you need to multiply by i.e. . For the numbers given above, it would be about 0.3 C/(W/m2). Again, I should stress that this is not an estimate for the real Earth!
As an aside, there have been a few claims (notably from Steve Milloy or Sherwood Idso) that you can estimate climate sensitivity by dividing the change in temperature due to the greenhouse effect by the downwelling longwave radiation. This is not even close, as you can see by working it through here. The effect on due to the greenhouse effect (i.e. the difference between having and its actual value) is , and the downward longwave radiation is just , and dividing one by the other simply gives - which is not the same as the correct expression above - in this case implying around 0.2 C/(W/m2) - and indeed is always smaller. That might explain it's appeal of course (and we haven't even thought about feedbacks yet...).
Point 3: Climate sensitivity is a precisely defined quantity - you can't get it just by dividing an energy flux by any old temperature.
Feedbacks
Now we can make the model a little more realistic by adding in 'feedbacks' or amplifying factors. In this simple system, there are two possible mechanism - a feedback on the emissivity or on the albedo. For instance, making the emissivity a function of temperature is analogous to the water vapour feedback in the real world and making the albedo a function of temperature could be analogous to the ice-albedo or cloud-cover feedbacks. We can incorporate the first kind of physics by making dependent on the temperature (or for arithmetical convenience). Indeed, if we take a special linear form for the temperature dependence and write:
then the result we had before is still a solution (i.e. ). However, the sensitivity to changes (whether in the greenhouse effect or solar input) will be different and will depend on . The new sensitivity will be given by
So if is positive, there will be an amplification of any particular change, if it's negative, a dampening i.e. if water vapour increases with temperature that that will increase the greenhouse effect and cause additional warming. For instance, , then the sensitivity increases to 0.33 C/(W/m2). We could do a similar analysis with a feedback on albedo and get larger sensitivities if we wanted. However, regardless of the value of the feedbacks, the fluxes before any change will be the same and that leads to another important point:
Point 4: Climate sensitivity can only be determined from changes to the system, not from the climatological fluxes.
Summary
While this is just a simple model that is not really very Earth-like (no convection, no clouds, only a single layer etc.), it does illustrate some relevant points which are just as qualitatively true for GCMs and the real world. You should think of these kinds of exercises as simple flim-flam detectors - if someone tries to convince you that they can do a simple calculation and prove everyone else wrong, think about what the same calculation would be in this more straightforward system and see whether the idea holds up. If it does, it might work in the real world (no guarantee though) - but if it doesn't, then it's most probably garbage.
N.B. This is a more pedagogical and math-heavy article than most of the ones we post, and we aren't likely to switch over exclusively to this sort of thing. But let us know if you like it (or not) and we'll think about doing similar pieces on other key topics.
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