Non-Hydrostatic Surface Pressure

The non-hydrostatic surface pressure is obtained by integrating $\epsilon+1$ from the top of the model atmosphere ($\eta=0$) to the surface ($\eta=1$)

$$p_s \,=\, p_t \,+\, \int_1^0 \left(\epsilon+1\right) \mu \,d\eta$$ (55)

where $p_t=p_1=\pi_1$ and as $\epsilon \rightarrow 0$, $p_s \rightarrow \pi_s$. Note that $p_s$ is merely a diagnostic and is not used in the solution of the dynamic equations.