I spent a number of years developing what is called a “three-dimensional Eulerian photochemical grid model,” aka the “Urban Airshed Model.” I was one among many, of course, but I did make some significant contributions to the effort.
The “three-dimensional” part of the name says that a volume was divided up into a lot of compartments, “grid cells” in the jargon, and the “Eulerian” part says that the grid didn’t move around, although there was a bit of cheating on that one in that the top of the modeling region rose with the “mixing layer” in some versions of the model. The alternative to “Eulerian” is “Lagrangian” where the model volume itself moves around, usually with the fluid flow, which is to say, the wind. That’s a “trajectory model” and it usually had only a single box, although we developed some multi-box trajectory models to handle plumes like those from power plants. A single line of boxes is “one-dimensional;” a “moving wall” of boxes is “two-dimensional.” A single box, therefore, is “zero-dimensional.”
So-called “box models” are common in air pollution, and other areas of environmental modeling. They can be really simple, especially if you are dealing with pollutants that don’t react. Then all you have to do is have a source input for emissions, a “ventilation rate” for the combination of wind and diffusion that’s removing material from your box, and boundary conditions for what kind of air is replacing what’s in the box. This is the sort of model that you get on first year chemistry or physics courses; it can be expressed in a single differential equation.
You can make the box pretty big, too, provided you’re willing to take these big honking averages of everything. For either non-reactive or “first-order” (those that just decay all by themselves, without reacting with other things) pollutants, your average result for the single box calculation is the same as if you’d done the multi-box calculation and then averaged all the boxes. That’s what’s called “linear” in the biz.
I did a lot of work with box models, partly because it was easy to test chemical mechanisms with them, and the results are easy to understand also. And I got to thinking about that “ventilation rate.” And wind power.
See, if you extract energy from the wind, it slows down, and that will have an impact on the ventilation rate of any area whose air is passing by the windmills. So I did some box model calculations on the amount of energy that was being extracted from the wind at Altamont Pass near San Francisco, plus the degree of pollution that was in the air that went through the Pass. That allowed an estimate of the increase in air pollutants that would occur in San Francisco due to the decrease in ventilation.
Okay, it was a weird calculation to make in the first place, but the results weren’t that deranged. There was an effect, the largest of which was equivalent to the amount of nitrogen oxides that would have had to be emitted in order to generate the excess of ozone seen at the pass. On a per kilowatt basis, it turned out to be a little less than the amount of nitrogen oxides that would be emitted by a natural gas-fired plant, such plants being the cleanest of all fossil fueled power plants. Of course the result depended on the amount of pollution already in San Francisco; a totally clean area would see no pollution equivalent at all, and since I made those calculations, SF has reduced pollutant levels.
I wrote up my results, sent the paper off to a journal, and then received some of the most flagrantly wrong referee comments I’ve ever received on a paper. One of them showed that I was “wrong” with a calculation that was itself off by five orders of magnitude, assuming, among other things, that wind speeds are constant all the way up to the stratosphere. I think he managed to calculate the wind kinetic energy over the entire Bay Area also, rather than just through the Pass.
Well, I know when I’m licked, and it was obvious that I wasn’t going to get anyone to pay attention to that wacky idea. Even in science, sometimes I’m too clever by half, and that’s a rueful comment, not a brag.