or 

Using the hydrostatic relation for the basic state, this may be written as:

If the depth scale of the pressure perturbation is of the same order of magnitude as the scale height of 

pressure () then:

and so, after some further manipulation:

or

Therefore, omitting terms of second order (and higher) the vertical momentum equation can be written as:

By similar reasoning, and using the fact that the basic state terms are dependent on height alone:

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