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I'm just trying to think where in the world it would be. Are there any plateaus just over 10000' in the Himalayas, preferably with 20000'+ mountains at either end?
There are to look into: the Tibet plateau and the Andean plateau.
No idea what the record is, but these places can give an idea:
Lhasa, China, recorded 30.4ºC at 3490 metres.
Uyuni, Bolivia, recorded 31.5ºC at 3680 metres.
Sucre, Bolivia, recorded 34.7ºC at 2910 metres. It falls short of meeting the criteria, but I guess the highest temperature at over 3000 m/10000 ft may be around this value.
Telescope Peak in Death Valley National Park in California is a little more than 11,000f/3300m. It sits directly above Badwater Basin (at -282 feet/-85 meters ). Badwater Basin gets temps above 130 degrees F/ 55C. The highest recorded temperature on earth occurred there back in 1913, at 134F/56.6C.
If you figure the typical lapse rate when looking at temperature decrease as elevation gains is 3.5 degrees F for every 1000 feet of altitude, then you lose 35 degrees F by going up 10,000 feet (or 19 degrees C, by going up 3048 meters).
Using this rudimentary math, on the day Badwater Basin was 134F/56.6C, it would have been 99F/37C up above, at 10000 feet, near Telescope Peak.
Of course, the climate of Death Valley is very unique, and the special factors that cause Badwater Basin to be so hot, might make the lapse rate atypical. But, this still gives us a ballpark idea of what it might have been.
I'm having trouble finding California data, but it appears it won't be as hot. Gem Lake in the Sierra recorded 96°F at 8970 feet, but the reading appears suspicious as all the highest temperatures in July were from 1933:
Station might not be trustworthy, though. 10000 feet was probably in mid 80s if correct. Note in LA, hottest temperatures are often at 5000 feet not sea level. Also caps the smog layer. Silverton, CO at 9200 recorded 90s several times:
Much of the high sun season coincides with the monsoon, depressing temperatures. And it appears less variable than the US West. Mexico City has a record of 93°F at 7500 feet.
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