44
STEPHEN GRIFFIES
The top and bottom of the cell are defined by surfaces of constant
generalized vertical coordinate s = s(x, y, x , t). The generalized
surfaces do not overturn, which means that s, , is single signed
throughout the ocean.
These assumptions lead to the following results for the sides of the grid
cell
TRACER MASS ENTERING CELL WEST FACE =
dy dz (U p C + p Fx) (59)
SS
where XI 5 x 5 x2 defines the domain boundaries for the east-west
coordinate^.^ Similar results hold for the tracer mass crossing the cell
in the north-south directions. At the top and bottom of the grid cell9
To reach this result, we used a result from Section 2.5 to write the volume
flux passing through the top face of the grid cell
dA(,) ii . (v - vref) = w(') dx dy,
(63)
with w(') = z,, dsldt the dbsurface velocity component. A similar
relation holds for the bottom face of the cell. The form of the SGS flux
passing across the top and bottom is correspondingly given by
Since the model is using the generalized coordinate s for the vertical,
it is convenient to do the vertical integrals over s instead of z. For this
8We use generalized horizontal coordinates, such as those discussed in Griffies, 2004. Hence,
the directions east, west, north, and south may not correspond to the usual geographic
directions. Nonetheless, this terminology is useful for establishing the budgets, whose validity
is general.
9As seen in Section 6, for pressure-like vertical coordinates, s increases with depth. For
depth-like vertical coordinates, s decreases with depth. It is important to keep this sign
difference in mind when formulating the budgets in the various coordinates. Notably, the
specific thickness z,, carries the sign.
STEPHEN GRIFFIES
The top and bottom of the cell are defined by surfaces of constant
generalized vertical coordinate s = s(x, y, x , t). The generalized
surfaces do not overturn, which means that s, , is single signed
throughout the ocean.
These assumptions lead to the following results for the sides of the grid
cell
TRACER MASS ENTERING CELL WEST FACE =
dy dz (U p C + p Fx) (59)
SS
where XI 5 x 5 x2 defines the domain boundaries for the east-west
coordinate^.^ Similar results hold for the tracer mass crossing the cell
in the north-south directions. At the top and bottom of the grid cell9
To reach this result, we used a result from Section 2.5 to write the volume
flux passing through the top face of the grid cell
dA(,) ii . (v - vref) = w(') dx dy,
(63)
with w(') = z,, dsldt the dbsurface velocity component. A similar
relation holds for the bottom face of the cell. The form of the SGS flux
passing across the top and bottom is correspondingly given by
Since the model is using the generalized coordinate s for the vertical,
it is convenient to do the vertical integrals over s instead of z. For this
8We use generalized horizontal coordinates, such as those discussed in Griffies, 2004. Hence,
the directions east, west, north, and south may not correspond to the usual geographic
directions. Nonetheless, this terminology is useful for establishing the budgets, whose validity
is general.
9As seen in Section 6, for pressure-like vertical coordinates, s increases with depth. For
depth-like vertical coordinates, s decreases with depth. It is important to keep this sign
difference in mind when formulating the budgets in the various coordinates. Notably, the
specific thickness z,, carries the sign.