Yes, although doing this would in practice be prohibitively difficult with current technology, and the effects wouldn’t be quite so neatly predictable because climate involves complicated feedback effects. (As an example, the existence of the Sahara suppresses strong hurricanes in North America because dust from the Sahara interferes with the formation of hurricanes over warm tropical waters.)
The Sonoran Desert specifically is a mix of a rain shadow from the southern edge of the Sierra Nevada to its west, and the fact that it sits in the subtropical “horse” latitudes where deserts tend to form because those latitudes mostly see sinking, rather than rising, air (they’re in the sinking part of the [Hadley cells](https://en.wikipedia.org/wiki/Hadley_cell)).
But mountains are really, really, REALLY fucking huge. To put some scale to this, I decided to cut out a roughly 45-mile-square segment of the southern Sierras and estimate how much rock you’d have to move. The average elevation on either side of the area I cut out is close to sea level, and the section I cut out averages well over a mile high. Let’s use 2000 m as a rough estimate for the average depth, and some quick drawing gets me ~3,500 km^2 of total area.
That’s 2 km * 3,500 km^2 = 7,000 km^3 of rock. Rock is pretty dense, so let’s estimate roughly 3000 kg/m^3 of rock; that would be [about 20 trillion metric tons](https://www.wolframalpha.com/input?i=7000+km%5E3+*+%283000+kg%2Fm%5E%283%29%29+in+metric+tons) of rock. That’s comparable to the scale of all man-made materials, total, everywhere on Earth, just to cut a small gap in a relatively narrow, not especially high, mountain range.
Yes. Let’s say you made the mountains higher. As the mountains rise, the air flow is forced to rise, leading to cooling of the air and condensation of water droplets, resulting in more rain. The taller mountains would also act as a barrier to air flow, creating localized pockets of air, so the desert wouldn’t be as dry.
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