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If it were possible to build a giant geodesic dome tens of miles in diameter, what effect would it have on nearby weather?

If built in an arid climate like the Sahara or Australia, would it be likely to cause rainfall under or nearby it (the way a mountain range causes preciptiation)?

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Arid climates like Sahara and Australia you name are caused by low humidity air circulating those region because of the global atmospheric circulation.

Obviously adding a mountain or a mountain range in those region won't increase the amount of humidity in the air, and likewise won't increase the amount of rain. It would probably increase the amount of mist. (I am aware of the 2016 news about a middle east desertic country looking to build an artificial mountain to increase rain, but since then nothing has happened, so I take it as a P.R. stunt)

But if you have sufficient humidity in the air and what you lack is just the cooling to have that humidity condense into rain than yes, an artificial elevation can increase the amount of rain.

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I don't know about Australia, but I do know that in the Sahara there are quite a few spectacular mountain ranges; the Ahaggar Mountains come to mind, as well as the Tibesti, the Adrar des Ifoghas and the Aïr. They are not known for their lush rainy climate...

Hoggar National Park, Assekrem, Tamanrasset, Algeria

Hoggar National Park, Assekrem, Tamanrasset, Algeria. Photograph by Mohammed Amri, available on Wikimedia under the Creative Commons Attribution-Share Alike 4.0 International license.

Satellite image of Tarso Toon (top right) and the Ehi Yéy (bottom left)

Satellite image of Tarso Toon (top right) and the Ehi Yéy (bottom left) extinct volcanoes in the Tibesti mountain range. Image available on Wikimedia, public domain.

Aïr Mountains in the Sahara

Aïr Mountains in the Sahara. Landsat 8 satellite image, May 26, 2013. Photograph available on Wikimedia, public domain.

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Mountains receive more rainfall than lower lands.because the temperature on top of mountains is lower than the temperature at sea level or low lands.

Winds carry moist air over the land. When air reaches the mountain, it rises because the mountains are in the way. As the air rises, it cools, and because cool air can carry less moisture than warm air, there is usually precipitation (rain).

So artificial big structures as mountain can attract more rain. But building a huge structure like that is much hard so we should be attentive to attract rains by planting more trees and stopping cutting trees.

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