Finite difference forms of the curvature terms

For the latitude-longitude projection, the curvature terms are differenced following Jacobson (2005) as

$$\begin{align*}\begin{split} F_{uc\, i,j} \,\ =& \,\, \frac{1}{2} \Bigg\lbrace \m... ...t( u_{i+1,j}+u_{i,j}\right)}{2} g_j \right] \Bigg\rbrace \end{split}\end{align*}$$ (308)

$$\begin{align*}\begin{split} F_{vc\, i,j} \,\ =& \,\, \frac{1}{2} \Bigg\lbrace \m... ...j-1}+u_{i+1,j-1}\right)}{2} g_{j-1} \right] \Bigg\rbrace \end{split}\end{align*}$$ (309)

where ${m^v}_x$ is the map factor in the $x$-direction located at a $v$ velocity point and

$$g_j = \frac{{\rm tan}\left( \phi_{r,j}\right)}{R_e}$$ (310)