Desert southwest Yes. By that I mean (1) it's in the Southwest (2) the climate at that place is desert climate or desert-like climate.
On a [...] lake? Yes. Although the lake doesn't always have water in it, best I know.
Two of your three previous proposals contain all the right letters. You just need to arrange them slightly differently. Think about the syllables of "Natron", maybe.
Trona is an unincorporated community in San Bernardino County, California. In 2015 it had a population of approximately 1,900. Trona is at the western edge of Searles Lake, a dry lake bed in Searles Valley, southwest of Death Valley. The town takes its name from the mineral trona, abundant in the lakebed. It is about 170 miles (274 km) northeast of Los Angeles, on State Route 178.
How do I know about Trona? California has one lonely coal-fired power plant left, and I wanted to find out where that is, because I have never seen any such power plant. Somewhere on the website of the State of California it tells me it is a "co-generation facility" in Trona. Where the heck is Trona?! Google Maps to the rescue.