Vertical integration of the hydrostatic equation

The hydrostatic pressures correspond to the coordinate surfaces, and are used to interpolate the non-hydrostatic pressure and Exner functions, and the geopotential in a consistent manner. The non-hydrostatic Exner function at layer centers is computed as

$$\Pi_{k} = {\left({\frac{p_k}{p_0}}\right)}^\varsigma \hskip1.in (k=1,N)$$ (195)

The hypsometric equation (the form defined by Eq. 20) is used to obtain the geopotentials on levels from

$${\hat\Phi}_k = {\hat\Phi}_{k+1} \,+\, \frac{C_{pd}\,\alpha_k\,{\overline\pi}_k}{R_d\,{\overline\Pi}_{\pi,k}} \left({\Pi_\pi}_{k+1}-{\Pi_\pi}_{k}\right)$$ (196)

$$= {\hat\Phi}_{k+1} \,+\, \vartheta_{\pi,k}\,\alpha_k\, \left({\Pi}_{\pi,k+1}-{\Pi}_{\pi,k}\right) \hskip.5in (k=N,1)$$ (197)

where ${\hat\Phi}_{N+1}=\Phi_s$ at the lower boundary. Note that one can define the following consistent relationship for the layer center geopotential as

$${\Phi}_k = {\hat\Phi}_{k+1} \,+\, \psi_{\pi,k} \left({\Pi_\pi}_{k+1}-{\overline\Pi}_{k}\right)$$ (198)

$${\Phi}_k = {\hat\Phi}_{k} \,-\, \psi_{\pi,k} \left( {\overline\Pi}_{k} - {\Pi_\pi}_{k} \right)$$ (199)

where $\psi$ can be equal to either $\vartheta_{\pi}\,\alpha$ or $C_{pd} \, \theta_{\rho}$. Subtracting Eq. 199 from Eq. 200 and solving for ${\hat\Phi}_k$ yields Eq. 197. Finally, by eliminating $\psi_{k}$ between Eq.s 199-200, an equation which depends only on ${\hat\Phi}$ and ${\Pi_\pi}$ can be defined as

$${\Phi}_k = \frac{ {\hat\Phi}_{k} \left({\Pi_\pi}_{k+1}-{\overline... ..._{k} - {\Pi_\pi}_{k} \right) }{ \left({\Pi_\pi}_{k+1} - {\Pi_\pi}_{k} \right) }$$ (200)

This equation can be useful in particular during pre or post processing steps when, for example, only ${\hat\Phi}$ and ${\Pi_\pi}$ are known. In the model, Eq. 199 is used to diagnose the geopotential at the layer centers after ${\hat\Phi}_k$ has been determined.