Geoscience Reference
In-Depth Information
=
√
3
/
2sgn
(
f
)
,
left side of the equati
on
be imaginary, as the right side is, gives
α
i
+
(
√
3
/
2
)i
sgn
(
f
)
.
With
α
determined
Eq. (12.47)
then yields an expression for boundary-layer
depth
h
:
so that
α
=
3
/
2
√
3
kR
f
u
∗
L
|
.
√
3
R
f
u
4
∗
h
2
=−
|
=
(12.49)
g
θ
0
Q
0
|
f
f
|
For
R
f
0
.
20-0
.
25 this yields
0
.
4
u
∗
L
|
1
/
2
0
.
4
u
∗
|
−
1
/
2
h
|
f
|
h
,
.
(12.50)
f
|
u
∗
f
|
L
This was perhaps the first analytical derivation of the power law that was originally
derived by
Zilitinkevich
(
1972
) on dimensional grounds.
Model calculations (
Businger and Arya
,
1974
;
Brost and Wyngaard
,
1978
)and
numerical simulations (
Zilitinkevich
et al
.
,
2007
) support
Eq. (12.50)
, showing
that if the SBL reaches a quasi-steady state its depth
h
follows the predicted
(u
∗
L/
)
1
/
2
dependence closely. The implied value of
h
can be quite small; for
example, if
u
∗
=0.1ms
−
1
,
L
= 100 m,
f
=10
−
4
s
−
1
, and the proportionality factor
= 0.4, then from
Eq. (12.50)
h
|
f
|
∼
100 m.
12.3.4 A constraint on the maintainence of turbulence in the SBL
Derbyshire
(
1990
) used
Eq. (12.48)
for the
T
profile to write
Eq. (12.45)
as
dW
dz
g
θ
0
Q
0
u
2
z/h)
1
−
α
∗
.
=−
R
f
(
1
−
(12.51)
∗
Integrating this from the surface to height
z
and evaluating the result at
z
=
h
gives
the geostrophic drag law
(Problem 12.22)
G
u
∗
1
kR
f
h
L
.
=
(12.52)
u
3
This drag law, with
Eq. (12.49)
and the identity
gQ
0
/θ
0
=−
/(kL)
, implies that
∗
the surface buoyancy flux is
g
θ
0
Q
0
=−
R
f
√
3
G
2
|
f
|
.
(12.53)
In Derbyshire's analytical model the parameters on the right side are all constant,
so this equation says the surface buoyancy flux is also constant. This clearly needs
interpretation.
