which, when expressed in terms of reduced gravity, g' = g(p1 - p2)/Pi, leads to Eq. 8-9.

A.1.3. Internal energy for a compressible atmosphere

Internal energy for a perfect gas is defined as in Eq. 8-12 of Section 8.3.4:

p dz, where the ideal gas law has been used, dA is an area element such that dV = dA dz, and where zs is the height of the Earth's surface. If we neglect surface topography, so that zs = 0, then, integrating by parts,

Since we saw in Chapter 3 that pressure decays approximately exponentially with height, (zp) ^ 0 as z ^x. Therefore, using hydrostatic balance, we have p dz = g pz dz, Jo and so the internal energy is zp dV.

This is the form given in Eq. 8-i2.

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