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I think there would be precipitation but it would be generated by the high pressure, like a heavy drizzle/mist.
If temperatures were freezing then I guess it would take the form of ice fog and very fine ice crystals.
Sounds like conditions to develop an inversion - the valley would be shrouded in fog so the sun wouldn't be able to reach the surface there and warm the air so temperature would decrease when going down. Does this happen in the real places?
There can be fog in Death Valley, although it looks like it's a rare event, not the type of weather that's normal there in winter.
Both at about 43N, Turpan (elevation 30 m; that is 180 m above the bottom of the Turpan Depression) is some 800 m lower than Urumqi. As you can see from the tables, Turpan's January mean is about 5C warmer and its July mean 8C warmer than Urumqi's. The diurnal range is about 2C greater in Turpan than in Urumqi, year round. Turpan is also sunnier than Urumqi, especially in winter it has over a half more sunshine hours more. This doesn't look like Turpan gets inversions in winter. Average humidity in Turpan's winter is not really low (66%) but that's not very high either.
Maybe my idea that these places would develop an inversion with the sun angle being low enough is wrong, I don't know. Maybe 43 is still a latitude that's not high enough. From what I've found it looks like it's indeed true that fall/winter inversions become more common with increasing latitude and that they're common in places like Fairbanks in Alaska, or Oymyakon in Siberia.
Remember what was the weather in Central Europe the November a year ago? A high was sitting over Central Europe the whole month so it didn't rain at all in many places. This led to inversions and foggy weather (and severe smog situation in cities) - the low elevations were cold and shrouded in fog under the inversion while in high elevations it was sunny. This is a satellite image from November 18, 2011:
Note how the Po valley (latitude 45N) is under the clouds too. I wonder what were the conditions that caused these inversions - I assume it must have been because the higher layers of the air were too warm. It's definitely possible to have clear winter days even in low elevations when a high comes so inversion is only possible in these situations, not inevitable, there must be other factors driving it than just the combination of high pressure + low sun angle.
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There are "sinks" in the intermountain west that are polar in winter compared to places outside of them that are actually higher. One of the sinks in Utah, forget the name of it, is actually the coldest place in the United States in the winter, just nobody lives in most of them. Probably the most apt comparison is Park City and Salt Lake City in Utah, very close as crow flies and about 3,000ft elevation difference. In winter, they have very similar temps, the difference comes in the high sun months when the winter inversion is broken
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