05 August 2008

The Simplest Meaningful Climate Model

It's a surprise to me that it is possible to construct a climate model that is meaningful in the space of a blog post. At least you can if you're interested in atmospheric temperatures. This isn't the case for the oceans, which is interesting for different reasons.

Here it is: T = (S*(1-a)/r^2/4/s)^(1/4)

1 line, easily dumped into your spreadsheet program of choice or calculator. I'll give some sample spreadsheet code at the bottom.

T is the temperature the earth radiates to space with.
S is the solar constant, which is about 1367 Watts per square meter (on a plane face-on to the sun, which our spheroidal earth isn't -- that's why the division by 4)
a is the albedo of the earth -- how much of the sun's energy is bounced straight back to space (now an observable quantity, it's about 0.30)
r is the average earth-sun distance, 1.00 Astronomical Unit
s is the Stefan-Boltzman constant, 5.67*10^(-8)
and ^ means 'to the power of' (x^y key on calculators, this symbol in spreadsheets).

If we plug in this values, we see that T is 255 K (-18 C, 0 F). ... and this matches the observations fairly well. (I've rounded several of the numbers above, and this one -- research the figures yourself and plug them in, and then go looking on your own for reference figures on how much energy the earth radiates to space and its equivalent temperature. This is one of the 'projects' I mentioned)

What happened? Certainly where I live (Washington DC area) seldom gets that cold, and I did say this was a meaningful model. The thing is, the model is meaningful for two reasons, first, that it does match up with the observations fairly well (temperature as radiated to space -- we have satellites circling the earth to check this figure now). Second, the model does not match up to the figure we really want -- the temperatures at the surface.

This tells us that the model is missing something important that affects the surface temperature, but not the temperature the earth shows to space. Whatever it is, its average effect is about 33 K (60 F).

But first, play around with those numbers and see what happens to temperature. In the next part, I'll go in to how to derive this model. Third part will be some looking at analyzing it. There's more here than may meet the eye.

Spreadsheeting:
If you're comfortable with naming variables, just paste this formula in to a1 and name the variables S, r, a, s elsewhere. The figures in a1 will be the temperature.
= (S*(1-a)/r^2/4/s)^(1/4)

S: 1367
a: 0.30
r: 1.0000
s: 5.67e-8 (cannot change)

if you're less comfortable with your spreadsheet, use this formula in a1:
= (b1*(1-b2)/b3^2/4/b4)^(1/4)

b1: 1367
b2: 0.30
b3: 1.0000
b4: 5.67e-8 (cannot change)

12 comments:

Tracy P. Hamilton said...

Robert, Keep up the good writing, hopefully you will get the readership you deserve.

tamino said...

Equations written in the form you show are sometimes confusing, especially to those without a lot of math (who need the help the most).

Wordpress (the blog site I use) allows LaTeX in posts, but that may not be an option for your site. One possibility is to do them in LaTex, produce a pdf (or dvi), then capture an image to insert in the post in order to display equations.

Just a thought.

P. Lewis said...

If you type in the search strings "LaTeX for Blogger" and then "jsTeXrender for Blogger" you will get an idea of what is available.

G Ernest D said...

This is a very good idea, in my opinion. Could you please do as a couple of other people have suggested, and type the equations on LaTeX?

In the formula

T = (S*(1-a)/r^2/4/s)^(1/4)

there are, apparently, 3 divisions. Since division is not associative this needs some parentheses to make sense. This is why LaTeX would help.

Can you say something about where you got this simple model?

Penguindreams said...

Thanks for the leads folks, I'll check them out and render a more readable version. The equation can be cut and pasted into a spreadsheet, which is easier for that use than LaTeX (which I use for my papers). But for reading, it's pretty bad, I agree.

Ernest, the reduced parentheses version is:
4 s r^2 T^4 = S (1-a)

And yes, I'm going to post about just where this all comes from. In brief for now: conserve energy with Stefan-Boltzmann earth output balancing solar input.

Ed D said...

What's the point of dividing by r² (which is one, anyway) when the power at the top of the atmosphere has been used? I could understand the use of r² if there was a term for the actual solar output at the surface of the sun.

Ed D said...

What's the point of the r² term when the basic energy input term S is already taken at the top of the atmosphere? I could understand if there was a term for the actual output from the Sun but, as far as I can see, r is irrelevant. r² is one, anyway, of course.

John Mashey said...

Actually, given the laudable goal of appealing to a broader audience than usual, maybe this is a good place to debate:

What's the best Web pedagogy, if a goal includes having a broad audience try things?

a) use regular math notation only via LaTex.

b) use Excel

c) show both, inline [more work]

d) use one, include the others at the end.

Of course, that in some cases, the normal math notation and the natural Excel don't translate so obviously.

Penguindreams said...

Ed D: Good point. There are two sides to the answer. For the near side, it's that I'd like to take a look down the road at what happens when we let the earth-sun distance vary through the year. We'll need a more complex model for that. But the variation is up to plus or minus 0.0167 AU. (Nearest the sun about January 3rd.)

The other side is that on long enough time scales, the annual average earth-sun distance does change (by about that same amount). This is the Milankovitch variation in eccentricity, which has a period around 100,000 years. Not an issue for us thinking about the next few decades or centuries, but important when we think about ice ages.

John: I definitely think this is a good time and place for that debate. I think I favor the 'do both' option, on grounds of covering all bases. What's your preference and why? Other folks, good time to contribute your cents.

Didjeridust said...

Love this stuff!
(this post specially, your blog generally...just got tipped by Tamino...thanx heaps...)

When do we get next part?

tamino said...

You say, "... on long enough time scales, the annual average earth-sun distance does change (by about that same amount). This is the Milankovitch variation in eccentricity, which has a period around 100,000 years."

This isn't quite correct. Changes of the semi-major axis of earth's orbit are negligible, so the "annual average" distance is remarkably constant. But the variation throughout the year changes strongly with eccentricity changes. I posted about Milankovitch cycles here and here.

Penguindreams said...

tamino, how dare you mention, much less document that I made an error.
Why, I'll have to ... er, thank you!

Thanks for the correction and link to a good description of the relevant processes. The variation isn't zero, but it's a good 100 times smaller than I'd had it. I'll correct my post later.

General:
I know I'll make mistakes, here even more than in my professional writing. So corrections are welcome. The idea is to learn about the science, which does include pointing out, or discussing, errors (and whether it really is one, which this was).