52
STEPHEN GRIFFIES
advection metric frequency arises since linear momentum is not conserved on the sphere.'' Hence, the linear momentum budget picks up
this extra term that is a function of the chosen lateral coordinates. The
advection metric frequency is analogous to, but far smaller than, the
Coriolis frequency.
Unlike the case of the tracer and mass balances considered in Section
3, we do not consider momentum sources interior to the fluid domain.
Such may be of interest and can be introduced without difficulty. The
goal of the remainder of this section is to consider the linear momentum
balance for finite grid cells in an ocean model.
4.2
An interior grid cell
At the west side of a grid cell, n = -2 whereas n = 2 on the east side.
Hence, the advective transport of linear momentum entering through the
west side of the grid cell and that which is leaving through the east side
are given by
TRANSPORT ENTERING FROM WEST =
dy ds z,, u (p V )
TRANSPORT LEAVING THROUGH EAST = -
dy ds z,, u (p v).
(93)
SS
Similar results hold for momentum crossing the cell boundaries in the
north and south directions. Momentum crossing the top and bottom
surfaces of an interior cell is given by
TRANSPORT ENTERING FROM THE BOTTOM =
dx dy w(') (P V )
TRANSPORT LEAVING FROM THE TOP = -
dx dy w(') (P v).
(95)
SS
llAngular momentum is conserved for frictionless flow on the sphere in the absence of horizontal boundaries (see Section 4.11.2 of Griffies, 2004).