Geoscience Reference
In-Depth Information
7.2.1 Flux of Variable
θ
Specified at Upper Surface
If the flux of
θ
is specified at the upper boundary, the difference equation for the
gridpoint
zz
1
is
t
F
1
θ
∆
Q
1
θ
(7.10)
tK
2
θ
2
∆
t
2
∆
t
2
∆
F
2
θ
F
1
θ
tQ
1
θ
=
θ
1
−
θ
1
,
j
−
1
=
−
z
1
+
z
1
+
2
∆
zz
1
(
θ
2
−
θ
1
)+
2
∆
z
1
+
∆
∆
∆
z
1
∆
orgroupingterms
t
F
1
θ
∆
Q
1
θ
(
1
−
A
1
)
θ
1
=
−
A
1
θ
2
+
2
∆
z
1
+
+
θ
1
,
j
−
1
(7.11)
so
A
1
A
1
−
E
1
=
1
t
F
1
θ
∆
Q
1
θ
−
2
∆
z
1
+
−
θ
1
,
j
−
1
=
D
1
(7.12)
A
i
−
1
7.2.2 Variable
θ
Specified at Upper Surface
A second class of boundary condition addresses the case where the value of
is
specified at the interface instead of its flux. In this case, an estimate of the surface
fluxismadefromthesurfacevalue,
θ
θ
s
,andthevalueatthefirstgridpointusingthe
time-centeredestimateoffrictionvelocity
u
∗
0
(
θ
1
−
θ
s
)
Φ
θ
F
s
=
(7.13)
Φ
θ
is the dimensionless change in the mean quantity across the distance
separatingthesurfaceandthefirstgridpoint
where
|
zz
1
|
1
ℜ
Φ
θ
=
u
∗
0
dz
(7.14)
0
where
istheeffectiveviscosityordiffusivity(notnecessarilytheeddydiffusivity).
Thedifferenceequationforthefirstgridpointis
ℜ
2
∆
t
2
∆
t
F
2
θ
F
1
θ
tQ
1
θ
θ
1
,
j
+
1
−
θ
1
,
j
−
1
=
−
z
1
+
z
1
+
2
∆
(7.15)
∆
∆
tQ
1
θ
=
−
A
1
(
θ
2
−
θ
1
)+
s
l
(
θ
2
−
θ
1
)+
2
∆
