The simplest climate model is the 0 dimensional model. We average over all of latitude, longitude, elevation, and time (or at least enough time). Those are the 4 dimensions we could have studied, or could get our answer in terms of. The 0 dimensional model gives us just a number -- a single temperature to describe everything in the climate system. We could expand, perhaps, to also getting a single wind, humidity, and a few other things. But it's distinctly lacking in terms of telling us everything we'd like to know. It fails to tell us why the surface averages 288 K, instead of the 255 K we see as the blackbody temperature. But it does get the blackbody temperature a start.

There is also only one 4 dimensional model -- where you include all 4 dimensions: latitude, longitude, elevation, and time. These are the full climate models, also called general circulation models (GCMs), atmosphere-ocean general circulation models (AOGCMs -- the original GCMs only let the atmosphere circulate), and a few other things. These are the most complex of the models.

But there are 14 more climate models possible: 4 one dimensional, 6 two dimensional, and 4 three dimensional.

In one dimension, we have the four which let 1 dimension vary, only:

- time
- elevation
- latitude
- longitude

Something quite close to the simplest model can be used for the time-only climate model. We would then let the earth-sun distance vary through the year, solar constant vary with the solar cycle, and albedo ... well, that would be a bit of a problem. As we've still averaged over all latitudes and longitudes, however, this model wouldn't tell us about why high latitudes are colder than low latitudes, or why land on the eastern side of oceans is warmer than land on the western side, or ... a lot. Still, it would take us another step of complexity down the road to understanding the climate system on global scale. This sort of model isn't used much professionally, but it can be a help

In elevation only, we'd (we hope) be able to look in to why the temperatures in the atmosphere do what they do -- falling as you rise through the troposphere and mesosphere, even or rising in the stratosphere. This class of models is known as the Radiative-Convective models (RCM). Namely, they include radiation and convection. The most famous early model of this sort is by Weatherald and Manabe, (1967?). We'll be coming back here.

In latitude only, we'll start being able to see why the poles are colder than the equator. Budyko and Sellers, separately but both in 1969, developed models like this. They're called energy balance models (EBM). They start with our simplest climate model, but applied to latitude belts on the earth. First you pretend that no energy enters or leaves the latitude belt except through the top of the atmosphere. Same thing as we said for the simplest model, except we applied it to the whole earth. You then compute the latitude belt's temperature, and discover that the tropics would be much warmer than they are, and the polar regions would be much colder. We're not surprised that we get the wrong answer here, but the degree of error then tells us by how much and where this 'no latitudinal energy transport' approximation is worst. You can then add the physics of 'heat flows from hot to cold' and get to work on how the climate in your model changes due to this fact.

The 4th one dimensional model, I've never seen anyone use -- a model in longitude only. This dimension is much quieter than the other two spatial dimensions. In the vertical, global average temperatures vary by something like 100 C in something like 10 km. 10 C/km; we'll get to exactly how much, where, and why, later. In latitude, temperatures vary from 30-40 C in low latitudes to -40 to -80 C in high latitudes (poles), so rounding again, about 100 C, but now across 10,000 km. About 0.01 C/km. In longitude, after we average over all year and all latitudes, ... there isn't much variation. As an eyeball matter, I'd be surprised if it were more than 10 C. (Project: Compute it. Let me know your result and sources. I may eventually do it myself.) This would be not more than 10 C, but still across 10,000 km or so, so something like 0.001 C/km at most (average absolute magnitude).

So our 4 models can be sequenced in terms of how much variation they get involved with, and, not coincidentally, it's something like the order of frequency I've seen the models in the literature:

- Elevation -- Radiative-Convective Models (RCM) -- 10 C/km, 100+ C range
- Latitude -- Energy Balance Models (EBM) -- 0.01 C/km, about 100 C range
- Time -- (not common enough to have a name I know of) -- a few C range, seasonally
- Longitude -- (never used that I know of) -- 0.001 C/km or less, a few C range

The 6 two-dimensional models are:

- time-elevation (an expanded Radiative-Convective Model)
- time-latitude (an expanded Energy Balance Model)
- time-longitude (I've never seen done as a model, but Hovmo"ller diagrams do this in data analysis)

- elevation-latitude (a cross between Radiative-Convective and Energy Balance)
- elevation-longitude (I've never seen as a model, but it's not unheard of for data analysis)
- latitude-longitude (I've never seen as a model, but common for data analysis)

In 3 dimensional modelling, we are back down to 4 models, as for 1 dimensional. This time, though, it's a matter of what we leave out:

- time (keep latitude, longitude, elevation; not common for models)
- longitude (keep time, latitude, elevation -> the straight combination of RCM and EBM; most common of the 3D models)
- latitude (keep time, longitude, elevation)
- elevation (keep time, latitude, longitude)

And then we have kitchen sink, er, 4 dimensional, modelling.

A question I'll take up later is why we would run a simpler model (1d instead of 2d, 3d instead of 4d) if we

*could*run the more complex model. Part of the answer will be that there's more than one way to be complex.

## 3 comments:

This question is probably not directly relevant, but since it's a more complex question than a zero-dimensional model, I'm curious as to how you would go about working the albedo change for a given change in ice or cloud fraction on Earth (or seasonal variations). I'm not even too sure what the estimates are for albedo during the LGM, or Cretaceous hothouse, etc.

I'll be getting more complex than the 0D model, have no fear. (Or have lots of fear, depending on your taste.)

Always a good idea to go back to definitions when trying to think about how something would change for LGM (aside: which means Last Glacial Maximum, except that since I started in Astronomy, and read a lot of science fiction, keep thinking is supposed to be 'Little Green Men') or the Cretaceous hothouse.

With sea ice (ice sheets, ...), we have a pretty fair idea of their albedo -- what fraction of the incident solar energy gets bounced out. For some surfaces, like ocean, it turns out that albedo depends significantly on the angle the sun comes in at. For a sun high in the sky, the albedo is relatively low for the ocean (0.06 or so), but for a sun near the horizon (as happens for the Arctic) it can be more like 0.2 or even higher at grazing. Ice seems to be less angle-dependent.

For plant-bearing land surfaces, you have to do a seasonal calculation. When the mid- and high- latitude plants start growing in spring, you go from black/brown backgrounds (low albedo) to green (higher albedo, but how much depends on the plant types, what soil they're over, and ...).

With clouds, as usual, things get even messier. Albedo is the fraction of energy that gets bounced back out of the atmosphere. Now, when sunlight hits a cloud drop, what happens depends on the size of the cloud drop. It also matters whether the cloud particle is liquid (a drop) or ice (particle). Of course some of the light that hits a cloud particle doesn't bounce back out, but maybe it does on the next step of its path. Maybe it bounces forward several times ('forward scattering albedo' is the term to look for), but then gets bounced backwards and not intercepted again on its way out. What will happen, in net, depends not only on how big the particles are, and whether they're frozen or not, and the type of frozen shape they have, ... but also on how thick the cloud is.

That's the long way 'round for saying "I don't know" the answer to what albedo was for the LGM or Cretaceous. Any reasonably serious attempt to derive planetary albedo for those times requires a GCM (full 4d model, preferably with elaborate clouds).

As a simple experiment, though, which we

canapproach from the simplest model, consider albedo to be a function of planetary temperature. As the planet gets seriously cold, the albedo is due to only snow and ice, about 0.8. For seriously hot Earths, we're looking at the albedo, planet-wide, of towering cumulus, also (iirc) about 0.8. In between the two, as we are now, it's something else (about 0.3; the lowest plausible is maybe 0.2). So guess an albedo function that goes through 0.8 as you get very cold or warm, and 0.3 for a planetary blackbody temperature of 255 K. One such is (sorry for the ugly typography):a = 0.8 - 0.5*exp( (T-T0)^2 / l^2)

where a is albedo, T0 is 255 K, and l is a measure of how fast you go from very cold to very hot. Say 10 K for starters.

Now go back to the simplest model, start with a temperature guess of, say, 288 K. Compute the albedo that corresponds to that. Then plug in this albedo to derive your next guess of temperature. With that temperature, make your next guess of albedo. Repeat until the numbers come back about the same between successive cycles.

Then, try again with a first temperature guess of 245 K.

If you're numerically intuitive, you'll see what's going to happen. If not, the exercise will help develop your intuition. It's an interesting result, even if not a full climate model sort of result.

arrgh. Make that:

a = 0.8 - 0.5*exp( - (T-T0)^2 / l^2)

The original version would be interesting, I suppose, but negative albedoes don't make much sense. Much less to have albedoes heading off to negative infinity.

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