Something that struck me quite some time ago as an eyeball-quality correlation was that population density was higher in areas with more rainfall. A proper study of this should look through history and be rigorous about both quantities. As long distance large-scale trade of food became possible, it'll also be necessary to look at water requirement for food and for people separately.
But, as a start and something of a Fermi estimate, let's go with what my water company tells me is typical per-person water usage at home -- 70 gallons per day. That's approximately 250 liters. Typical annual rainfall around here is about 1 meter per year. So, if I captured all the rain that fell on an area, how large would that area have to be for me? If everyone else did the same, how many of us could live in 1 square km?
The 250 liters are what I'd use in 1 day if I were approximately 'average'. For a year, I need 365 times that much, for about 100,000 liters. 1 liter is a cube 0.1 m on a side, so for a year's water I need about 100 m^3. That'd fill 1 meter deep across a square 10 meters by 10 meters. Since I get about 1 meter rain per year, this says I need about 100 square meters. If we all caught absolutely all rain, and didn't lose any to trees, farming, industry, grass, ... this would let us go up to a population density of 10,000 per square km (about 25,000 per square mile). This is actually something like the density of the highest density large cities -- check out Chicago, New York, London, Paris, for instance.
Los Angeles gets about 0.4 m/year rain, and Phoenix about 0.2. Their densities are about 3200 and 1200, respectively (Wikipedia for both numbers) to the 4000 and 2000 we'd guess from the above. On the other hand, both illustrate the fact that cities don't rely on only the rain that falls on them. Both have extensive systems to bring water to them. This is not new; Rome, to support its population in the days of Empire, when they were also using 10s of gallons of water per day, built a tremendous system of aqueducts to bring water to the city. New York and Chicago, I know, also bring water from outside the city (again water systems, rivers, lakes).
If we figure on about 10% of the rainfall being captured for residential use, then we're down to about 1000 per square km, or about 2500 per square mile in my part of the country. These are densities comparable to what is seen over moderately large areas (entire metropolitan regions, small countries in northern and western Europe, ...) with meter per year rainfall.
There's a certain reasonability, then, to there being a relation between rainfall and population density. You can exceed that relation, but only at the expense of building a large scale water system -- and not letting people in the areas you bring water from have acess to it. Water rights have a lengthy and often not peaceful history. In any case, even if all the people are in one place, their requirement extends over a larger area, something more in accord with the 1000 per square km (for 1 m/year rainfall). For, say, the 20 million or so people in the Los Angeles area, with 0.4 m/year of rain, it says their footprint is more like 50,000 km^2, to the 4000 or so the metro area actually occupies. These are all still Fermi estimates, of course. It does point out to us, however, that urban areas likely have a footprint rather larger than the official area. Conversely, it means that they need to be concerned about weather and climate over a larger area than just their own borders.
The climate concern is ... what do you do if you get less rain (or less snow)? What do you do if the rain comes more in situations (thunderstorms) that are harder for you to capture the rainwater from? Either of these changes drives you to a lower population in your urban (including suburbs) area, or pins a new expense on you -- to build more extensive water systems, and systems able to handle greater rainfall rates. While we mostly expect rainfall to increase, we do expect that there will be areas which will see falls. Which ones, not so sure, but some. It's expected and observed that there'll be an increase in rain falling in heavy doses even where there's little change in total rainfall.
A friend mentioned a TV person-in-the-street interview during a local drought. The person wasn't concerned about the drought and low river levels because "I don't get my water from the river, I get it from the tap." I trust you all know that it went from the river through a processing system to her tap even if she didn't. (Or, in the case of Chicago, from Lake Michigan -- whose level seems to be falling more than the usual cycle.)
It seems more people don't realize that wells have the same problem. Across much of the Great Plains US (take Kansas for a central example state), farming and residences take advantage of the Ogallala Aquifer. The problem is, aquifers need recharging. That is, the water in the aquifer comes from rainfall elsewhere. With the Ogallala today, the usage exceeds the recharge rate. This is not very surprising, as the aquifer's recharge area is itself in fairly dry areas. For more, see the USGS web site, search on Ogallala Aquifer and recharge. Similar problems exist for many wells. The main difference between wells and lakes or rivers is that you can't see the levels dropping as easily.
Music of the Spheres
2 hours ago