and likewise for the y-component of the pressure gradient force. Therefore, ignoring terms of second order or higher, the vector momentum equation using this linearized pressure gradient force becomes:
If is the density scale height for an isothermal atmosphere defined by:
then the vertical momentum equation may be written as:
For liquids, where very much exceeds the depth of fluid motions, the compressibility of the fluid (and therefore the term involving ) can be neglected. As it stands, eq.(6) is not ideal for the representation of deep motions in a compressible atmosphere. An improved form can be obtained by expressing the buoyancy force in terms of the entropy perturbation . Using the perfect gas equation , (to within an arbitrary constant) and the definition of potential temperature:
where then it is easy to show that:
where is the ratio of specific heats and the arbitrary constant in the definition of has been chosen to remove residual constant terms.
Expanding , and about their basic state values then leads to: