Adiabatic heating and cooling of an incompressible fluid element due to vertical displacement

This note analyzes the process of displacing a fluid element vertically in a layer of fluid. It is shown that, when the thickness of the fluid layer is very small compared with the height over which the density of the fluid element changes significantly, the fluid element may be regarded as incompressible. However, it also turns out that the temperature of the element changes, no matter how thin the fluid layer may be, at the adiabatic lapse rate; it becomes apparent, consequently, that an incompressible fluid, as well as a compressible fluid, undergoes adiabatic heating and cooling.