This question on condensation and wingtip vortices, What is the mechanism by which condensation forms above wings?, attributes the reduction in temperature found in wingtip vortices to Adiabatic cooling.
The reasoning for this, as quoted from Wikipedia, is given as:
Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. When the pressure applied on a parcel of air is reduced, the air in the parcel is allowed to expand; as the volume increases, the temperature falls as internal energy decreases
My question is: why are we assuming that a parcel of air—which, according to Bernoulli's principle has a lower internal pressure—would be allowed to expand against the external pressure which would be higher?
This reasoning to me would suggest that the parcel would compress (if it was compressible) and that its temperature would increase.
Also, why are we assuming that air is now compressible below Mach?
Wouldn't it make more sense to attribute the reduction in temperature to an isochoric cooling process? This would only require the internal pressure to decrease, and would not require a change in density.