Using the hydrostatic equation for the basic state pressure eq.(4), it can be shown that:

or

Adding this equation to eq.(43) gives:

and the global integral is:

As in the total energy equation of the primitive equation set, this quasi-Boussinesq form involves the sum of kinetic energy, gravitational potential energy and enthalpy and is always positive. 

For steady flows eq.(47) can be written as:

and since    to the accuracy of the Boussinesq approximation

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