Geoscience Reference
In-Depth Information
7.3 Steady-State Momentum Equation
The Ekman equation admits a steady-state solution of the momentum conserva-
tion equation, often a valuable approximation for interpreting isolated measure-
ments(Chapter9)orevencalculatingtimeevolutionofthescalarvariables(McPhee
1999).Thesteadyversionof(7.9)maybewritten
if
u
=(
K
u
z
)
z
(7.26)
Usingthesamestaggeredgridasbefore,the differenceequationis
A
i
u
i
+
1
+
B
i
u
i
+
C
i
u
i
−
1
=
0
(7.27)
where
K
i
+
1
K
i
A
i
=
C
i
=
A
i
+
C
i
)
B
i
=
−
(
if
+
(7.28)
∆
z
i
∆
zz
i
∆
z
i
∆
zz
i
−
1
D
i
E
i
u
i
+
1
, therecursionrelationsare
with
u
i
=
+
C
i
D
i
−
1
B
i
+
−
D
i
=
(7.29)
C
i
E
i
−
1
A
i
−
E
i
=
B
i
+
C
i
E
i
−
1
Themomentumequationatgridpoint
zz
1
is
u
1
)+
−
τ
0
∆
A
1
(
if
u
1
=
u
2
−
(7.30)
z
1
u
w
0
+
u
v
0
)
where
.
Using flux(stress) boundaryconditions,therecursioncalculationisstartedwith
τ
0
=
−
(
i
−
τ
0
D
1
=
z
1
A
1
+
if
(7.31)
∆
A
1
E
1
=
A
1
+
if
7.4 Distributed Sources
Solarheatingisanexampleofascalar sourcetermforwhichthefluxatthesurface
is not necessarily the properboundarycondition for the conservationequation.Let
Q
θ
(
)
z
be the source function of an arbitrary variable
θ
, for which the total flux
throughthesurfaceis
F
0
.To conserve
θ
0
F
0
θ
=
−
Q
θ
(
z
)
dz
(7.32)
−
∞
