SOME OCEAN MODEL FUNDAMENTALS
39
2.5
Dia-surface transport
We seek an expression for the flux of fluid passing through a surface of
constant generalized vertical coordinate. The result will be an expression
for the dia-surface transport. It plays a fundamental role in generalized
vertical coordinate modelling. Our derivation here follows that given in
Section 6.7 of Griffies, 2004.
At an arbitrary point on a surface of constant generalized vertical
coordinate (see Figure 5), the flux of fluid in the direction normal to the
surface is given by
seawater flux in direction ˆ
n = v · ˆ
n,
(41)
with
ˆ
n = ∇s |∇s|
−1
(42)
the surface unit normal direction. Introducing the material time derivative ds/dt = s ,t + v · ∇s leads to the equivalent expression
v · ˆ
n = |∇s|
−1 (d/dt − ∂ t ) s.
(43)
That is, the normal component to a fluid parcel’s velocity is proportional
to the difference between the material time derivative of the surface and
its partial time derivative.
Since the surface is generally moving, the net flux of seawater penetrating the surface is obtained by subtracting the velocity of the surface
v (ref) in the ˆ
n direction from the velocity component v · ˆ
n of the fluid
parcels
flux of seawater through surface = ˆ
n · (v − v
(ref) ).
(44)
The velocity v (ref) is the velocity of a reference point fixed on the surface,
and it is written
v
(ref) = u
(ref) + w
(ref) ˆ
z.
(45)
Since the reference point remains on the same s = const surface, ds/dt = 0
for the reference point. Consequently, we can write the vertical velocity
component w (ref) as
w
(ref) = −z ,s (∂ t + u
(ref)
· ∇ z ) s,
(46)
where equation (16) was used with ds/dt = 0. This result then leads to
ˆ
n · v
(ref) = ˆ
n · u
(ref) + ˆ
n · ˆ z w
(ref)
= −s ,t |∇s|
−1 ,
(47)
39
2.5
Dia-surface transport
We seek an expression for the flux of fluid passing through a surface of
constant generalized vertical coordinate. The result will be an expression
for the dia-surface transport. It plays a fundamental role in generalized
vertical coordinate modelling. Our derivation here follows that given in
Section 6.7 of Griffies, 2004.
At an arbitrary point on a surface of constant generalized vertical
coordinate (see Figure 5), the flux of fluid in the direction normal to the
surface is given by
seawater flux in direction ˆ
n = v · ˆ
n,
(41)
with
ˆ
n = ∇s |∇s|
−1
(42)
the surface unit normal direction. Introducing the material time derivative ds/dt = s ,t + v · ∇s leads to the equivalent expression
v · ˆ
n = |∇s|
−1 (d/dt − ∂ t ) s.
(43)
That is, the normal component to a fluid parcel’s velocity is proportional
to the difference between the material time derivative of the surface and
its partial time derivative.
Since the surface is generally moving, the net flux of seawater penetrating the surface is obtained by subtracting the velocity of the surface
v (ref) in the ˆ
n direction from the velocity component v · ˆ
n of the fluid
parcels
flux of seawater through surface = ˆ
n · (v − v
(ref) ).
(44)
The velocity v (ref) is the velocity of a reference point fixed on the surface,
and it is written
v
(ref) = u
(ref) + w
(ref) ˆ
z.
(45)
Since the reference point remains on the same s = const surface, ds/dt = 0
for the reference point. Consequently, we can write the vertical velocity
component w (ref) as
w
(ref) = −z ,s (∂ t + u
(ref)
· ∇ z ) s,
(46)
where equation (16) was used with ds/dt = 0. This result then leads to
ˆ
n · v
(ref) = ˆ
n · u
(ref) + ˆ
n · ˆ z w
(ref)
= −s ,t |∇s|
−1 ,
(47)