From AMS Glossary
A fluid in which the density remains constant for isothermal pressure changes, that is, for which the coefficient of compressibility is zero.
Expansion and contraction of an incompressible fluid under diabatic heating or cooling is thus allowed for. In the more usual problem of isothermal processes, the fluid may or may not be stratified (have density differences within it), but motion of a parcel from higher to lower pressure or vice versa will not change the density of that parcel. Stated mathematically, the density gradient ∇ρ and the local derivative ∂ρ/∂t may not be zero, but the individual derivative Dρ/Dt vanishes. By the equation of continuity, it follows that the total divergence vanishes:
where u is the velocity with components u, v, and w. For many purposes in meteorology, the atmosphere is treated as a heterogeneous fluid in which only vertical motions show compressibility. Together with the assumption of hydrostatic equilibrium, this has the effect of eliminating compression waves (including sound waves).