40
STEPHEN GR IFFIES
which says that the normal component of the surface's velocity vanishes
when the surface is static, as may be expected. It then leads to the
following expression for the net flux of seawater crossing the surface
Hence, the material time derivative of the generalized surface vanishes
if and only
throughout
if no water parcels cross it. his important result is used
ocean theory and modelling.
Figure 5. Surfaces of constant generalized vertical coordinate living interior to the
ocean. An upward normal direction n is indicated on one of the surfaces. Also shown
is the orientation of a fluid parcel's velocity v and the velocity dref) of a reference
point living on the surface.
Expression (48) gives the volume of seawater crossing a generalized
surface, per time, per area. The area normalizing the volume flux is that
area dA(fi) of an infinitesimal patch on the surface of constant generalized
vertical coordinate with outward unit normal a. This area can be written
(see equation (6.58) of Griffies, 2004)
dA(fi) = lz,, Vsl dx dy.
Hence, the volume per time of fluid passing through the generalized
surface is
and the magnitude of this flux is