Scalar horizontal advection

The horizontal flux-form scalar advection (of $\varphi$) is expressed as

$$\nabla\cdot{\left( {\bf V}\mu\varphi\right)}_{i,j} = \Big\lbrack ... ...\overline \varphi^y}_{i,j+1} -V_{i,j}{\overline \varphi^y}_{i,j} \Big\rbrack /d$$ (126)

where $\varphi$ represents some arbitrary scalar, and the advection uses a second order approximation where the scalars are interpolated to the velocity points using

$${\overline\varphi^x}_{i,j} = \left( {\varphi}_{i-1,j} + {\varphi}_{i,j}\right)/2$$ (127)

$${\overline\varphi^y}_{i,j} = \left( {\varphi}_{i,j-1} + {\varphi}_{i,j}\right)/2$$ (128)