**Update**: See Part II.

I thought it would be fun to create a calculator to compute Global Warming. You can find it here: Global Warming Calculator. The method uses conservation of energy and the climate dynamics shown in the figure here, and in the Wikipedia article on the Greenhouse Effect.

The figure itself (by Robert A. Rohde, Creative Commons Attribution NonCommercial License applies) is based on the work of Kiehl & Trenberth, in their well-known article "Earth's Annual Global Mean Energy Budget" Bull AMS 78(2) Feb 1997, 197-207. The baseline data for the calculator was also taken from Kiehl and Trenberth's publication.

The method works by finding the steady state corresponding to the percentages you enter into the calculator. The default values are the ones from Kiehl and Trenberth shown in the figure. The only "difficult" part is figuring out the total flux from the ground (Eg) as a function of incoming solar flux (Es). The formula is

Eg=Es*(1-Pas+Psg*Pas)/(Pgs-Pgs*Pas+Pas)

where Pas is the portion of atmospheric radiation sent into space, Psg is the portion of solar radiation reaching the ground, and Pgs is the portion of ground flux radiated into space. Here "ground flux" includes radiation into space and the atmosphere, as well as energy transfer via evaporation, thermals, etc.

Although quite simple compared to other climate models, that doesn't mean it's not sound as far as the physics goes. Of course, it's only as good as the information you put into it.

It is interesting to see how sensitive the degree of warming is to changes in the percentages. Remember that the IPCC prediction of warming over the next 100 years is anywhere from 1 to 5 degrees.

## 9 comments:

Steve,

My calculator does get 288K. Here are my calculator's numbers for Trenberth's model:

Stefan-Boltzmann constant = 5.6704E-08 W/m^2-K^4

Fixed Fluxes= TRUE

Earth's Albedo= 0.312865497

Solar Atm Absorbed= 0.195906433

Earth's Emissivity= 1

Atm. Emissivity= 1

Latent Heat Flux= 0.158536585

Sensible Heat Flux= 0.048780488

Back Radiation= 0.624277457

Direct Radiated= 0.102564103

Solar Constant= 1368 W/m^2

Avg Solar Constant= 342 W/m^2

Avg Atm Solar= 67 W/m^2

Avg Solar Reflected= 107 W/m^2

Avg Surface Solar= 168 W/m^2

Avg Latent Flux= 78 W/m^2

Avg Sensible Flux= 24 W/m^2

Avg Surface= 492 W/m^2

Surface Temp= 287.9802655 K

Surface Radiated= 390 W/m^2

Direct Radiated= 40 W/m^2

Atm Absorbed= 519 W/m^2

Atm Temp Down = 274.9367715 K

Atm Temp = 309.3060505 K

Atm Temp Up = 242.1615729 K

Atm To Space= 195 W/m^2

Atm To Earth= 324 W/m^2

Up vs. Down Atm.= 66 W/m^2

Total Earth Radiated= 235 W/m^2

Total Radiated= 342 W/m^2

Jim-

Thanks a lot, that's great. I will check against my figures and see where the differences lie. I'll post back here what I find out. Sorry for not replying sooner, I have been running around today.

-Steve

Jim-

I agree with your figures exactly. I get that the average surface temperature is 14.83, and that this is 34.255 degrees warmer than it would be without a greenhouse effect. What I see stated, for example here: http://en.wikipedia.org/wiki/Greenhouse_effect, is that the earth is warmed 14 degrees, and this is 33 degrees warmer than it would be without the greenhouse effect. So it's the (14,33) vs (14.83,34.255)

difference that I'm wondering about. It doesn't seem to be just

rounding.

-Steve

Steve,

How are you handling the feedback?

Jim-

I'm not handling feedback. For example, a change in CO2 might change several of these percentages, including changes through feedback mechanisms. The calculator simply assumes you know the new percentages that would finally result from the CO2 change. It's pretty crude!

-Steve

Steve,

I think this model must deal with the feedback, or the results won't reflect reality. My model has to loop from 50 to 60 times before it settles down from a change.

For this model, there are only three ways GHGs can affect it: 1) direct absorption of short wave solar radiation by the atmosphere, 2) decreasing the atmosphere window opening, and 3) adjusting the atmosphere's up-down radiation ratio.

For small adjustments, only the atmosphere window opening need be changed. The other two factors won't change by much.

When adjusting the atmospheric window, I find that for a given surface temperature increase the atmosphere must warm by at least 130%. However, if you adjust the albedo down by a small fraction, then the atmosphere warms much less--in the 60% to 90% range.

The satellite data indicates that our current warming is probably due to albedo changes.

Jim- Your model is more sophisticated than mine. Mine barely qualifies as a model; it just solves a balance of energy equation based on percentages you put in. I'm sure that to be realistic you would have to put in percentages that incorporate feedback. I have very little idea how much variation in percentages is "a lot." I assume a point or two is not so much, but I could very well be wrong. For what it's worth, my calculations (the linear model at the bottom of the calculator page) suggest that the albedo, radiation from the atmosphere to the ground, and radiation from the ground to the atmosphere are most important. Radiation from ground to space (Window) is considerably less important according to my calculation. Best Regards, Steve

Steve,

You can download my model from //members.aol.com/jamesbat. The Excel file is AtmosphereFlux7.xls. I use macros to get by the self-referencing problem with Excel. The other buttons just copy one or more cells. I don't have any evil hidden operations in my macros, so you can allow them.

Hi Jim-

Thanks, I downloaded your spreadsheet and took a test drive. I think I understand your question about feedback in my calculator now. As I understand it, your spreadsheet finds the steady state equilibrium by iterating. When the various energies don't change much from one iteration to the next, the steady state has been found. I took a different approach, using algebra to find a closed-form solution for the steady state. That way I didn't have to code iteration, which I have no idea how to do in JavaScript. The key formula is Eg=Es*(1-Pas+Psg*Pas)/(Pgs-Pgs*Pas+Pas) which I mentioned in the blog. This is the formula for the steady-state energy flux from the ground, including latent and salient heat, which are not assumed fixed. The radiated portion of that is what goes into the S-B law to get temperature. Does this seem reasonable?

-Steve

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