Buoyancy of a Rising or Falling Air Parcel in the Atmosphere

An adiabatic air parcel moving in such an atmosphere may become more or less buoyant than the surrounding air. To study this motion, let us assume that an adiabatic air parcel is rising in such a nonadiabatic atmosphere with a lapse rate A. Let us assume that 

The change of pressure with altitude in this environment is then given by (1.3), or using this notation, dp/dz = -Maitgp/RTa(z). 

Now consider an adiabatic air parcel at z = 0 with initial temperature Tq. Its adiabatic motion as it is rising will still satisfy (16.6). Combining this with the pressure versus altitude of the atmosphere that we have just seen, we find that 

Integrating this equation from the initial position of the air parcel (z = 0 and 7o) to the final position, we obtain 

If the density of the air parcel is p and that of the atmosphere around it pa, the acceleration experienced by the air parcel a will be proportional to the density difference between the atmosphere and the air parcel  

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