Finite volume method on a staggered grid

The Oceananigans.jl staggered grid is defined by a rectilinear array of cuboids of horizontal dimensions and vertical dimension , where index the location of each cell in the staggered grid. Note that the indices increase with increasing coordinate .

A schematic of Oceananigans.jl finite volumes for a two-dimensional staggered grid in . Tracers and pressure are defined at the center of the control volume. The control volumes are centered on the left and right edges of the pressure control volume while the control volumes are centered on the top and bottom edges of the pressure control volumes. The indexing convention places the -node on cell -faces to the left of the tracer point at cell centers.

Dropping explicit indexing, the areas of cell faces are given by

so that each cell encloses a volume  .

A finite volume method discretizes a continuous quantity by considering its average over a finite volume:

The finite volumes that discretize each of , , and are located on a grid which is "staggered" with respect to the grid that defines tracer finite volumes. The nodes, or central points of the velocity finite volumes are co-located with the faces of the tracer finite volume. In particular, the -nodes are located in the center of the "-face" (east of the tracer point), -nodes are located on -faces south of the tracer point, and -nodes are located on -faces downwards from the tracer point.