Vertical discretization of the pressure gradient terms

The vertical pressure gradient (Eq. 187) is descretized as

$$PG_{z,k} \,=\, {\hat{\mu}}_{d,k}\epsilon_k = {\hat{\mu}}_{d,k} \l... ..._k-p_{k-1}}{{\overline{\pi}}_{d,k}-{\overline{\pi}}_{d,k-1}}\right) - 1 \right]$$ (201)

where the bracketed term on the RHS of Eq. 203 defines the discretized form of $\epsilon_k$, and the standard vertical interpolation operation from Eq. 177 is used for $\mu_d$ and $\varphi_q$. The horizontal pressure gradient is expressed as

$$PG_{xy,k} = \mu_{d,k} \left( \nabla\Phi_k + C_{pd}\theta_{\rho,k} \nabla\Pi_k + {\overline{\epsilon}}_k \nabla\Phi_k \right)$$ (202)

where ${\overline{\epsilon}}$ is computed (interpolated to layer centers) using Eq. 172.