Indeed this is the definition of incompressible flow: it is nondivergent—no bubbles allowed! Note that in any real fluid, Eq. 6-11 is never exactly obeyed. Moreover, despite Eq. 6-10, use of the incompressibility condition should not be understood as implying that D = 0. On the contrary, the density of a parcel of water can be changed by internal heating and/or conduction (see, for example, Section 11.1). Although these density changes may be large enough to affect the buoyancy of the fluid parcel, they are too small to affect the mass budget. For example, the thermal expansion coefficient of water is typically 2 x 10-4 K-1, and so the volume of a parcel of water changes by only 0.02% per degree of temperature change.

6.3.2. Compressible flow

A compressible fluid, such as air, is nowhere close to being nondivergent—p changes markedly as fluid parcels expand and contract. This is inconvenient in the analysis of atmospheric dynamics. However it turns out that, provided the hydrostatic assumption is valid (as it nearly always is), one can get around this inconvenience by adopting pressure coordinates. In pressure coordinates, x, y, p , the element(al fixeod "volume" is Sx Sy Sp. Since z = z (x, y, p), the vertical dimension of the elemental volume (in geometric coordinates) is Sz = dz/dp Sp, and so its mass is SM given by

Sx Sy Sp

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