I would like a bit of a review of my ideas and would appreciate it if anyone points out a serious fault with them - see ideas below:
Transporting moist air by means of natural convection in a pipe: Run a huge black pipe, that will get hot in the sun, from the sea to a few hundred metres above the area needing rain. Moist air from the sea will rise in the pipe by means of natural convection and cause convectional rain. This idea could bring rain to many areas. It would be similar to a solar updraft tower, which can deliver huge volumes of air per second to the atmosphere. Heat the seawater by concentrated solar power (or other means) near the inlet of the pipe to increase relative humidity. This system will be cheaper than solar updraft towers. Some calculations: For a 20 m diameter vertical pipe that is 500 m high with air temperature of 25 deg C outside and 30 deg C inside, a flow of about 3340 cubic metres per second can be expected. Eventually you will have a few cubic kilometres of moist air in the region if wind is weak. To do your own calculations search for "stack effect draft."
One could have a few such pipes into a region to spread humid air. One or two cubic kilometres of moist air per day can be delivered like this. Pipes could be heated more by reflecting sunlight onto them with mirrors. Rocks that the pipe rests on could be heated by solar energy so that the pipe stays warm at night and can keep on delivering moist air. It is quite likely that at night the air from just above the sea will be warmer than land air, which will cause it to rise in the pipe. Moist air is less dense than drier air, which will help it to rise in the pipe.
But here is another idea. In desert regions with hot air one can significantly change the density of the air by increasing relative humidity, because hot air holds so much water vapour and water vapour is less dense than air. At a temperature of 40 deg C with RH of 30% and P=101.325 kPa, air has a density of about 1.118 kg/cubic metre. If you raise the RH of this air to 90% it has a density of about 1.099 kg/cubic metre. This is the same as air with an RH of 30% and T=45 deg C. By increasing the RH of the air with RH = 30% to one with RH = 90% (all at T=40 deg C) you have about the same effect on density as raising the temperature of the air by 5 deg C ( from 40 to 45 deg C). In hot deserts It seems you do not have to heat the air to cause natural convection - you can just increase RH and the air will rise by natural convection in the pipe. The RH can be increased by heating seawater at the inlet of the pipe. At T=40 deg C with RH=90%, there are about 46 grams of water vapour in every cubic metre of air transported in the pipe.
What happens when the air comes out the pipe? Well, say the air with RH=90% and T=40 deg C comes out in air with temperature of 35 deg. Clouds will form with bases at about 245 metres above the outlet of the pipe (very low clouds). The clouds could display huge vertical ascent from their bases because of high RH, high dew point and so on (tall clouds with low bases and towering high tops will result). If a rain cycle results maximum, temperatures will be reduced by evaporation and minimum temperatures will increase because of increasing effective sky temperatures.
This depends on strength of sunlight, temperature of water coming into the greenhouse, heat losses and so on, but it seems that to form 1 cubic metre of 90% RH air at 40 deg C starting with 30% RH air at 25 deg C, every second, will take very roughly 200 square metres of surface irradiated by the sun. A massive greenhouse with water in could suffice to provide all the humid air needed. Similar greenhouses have been proposed for solar updraft towers. A greenhouse 1 km by 1 km could provide 5000 cubic metres of RH=90% with T=40 deg C air every second.