click > click > whir, indeed!
Happy to find this thread. My wife's 2013 SE makes this cycling noise, too. It may be a small thing, but it has to create a drain on the system over time. I even thought at one point it was because we hang our keys near the cars in the garage, and perhaps the proximity caused the car to "get ready." I hope to read an answer here soon.
My S sedan (manual) makes no such noises.