Geoscience Reference
In-Depth Information
Since
φ
m
=
1
5,
R
f
is
z/L
R
f
=
βz/L
.
(12.38)
1
+
Thus in the very stable surface layer
R
f
→
1
/β
0
.
20.
12.3.2 Entrainment-induced stratification
As shown in
Figure 9.7
,
an ABL can be made stably stratified by the entrainment
of warmer air aloft. This is sometimes called the “inversion-capped neutral” ABL.
Here we can revisit the “eddy Richardson number”
Ri
e
for an eddy of size
r
,
introduced in
Chapter 10
:
gθr
1
/
3
θ
0
2
/
3
.
gθr
θ
0
[
Ri
e
(r)
=
=
(
10
.
38
)
2
u(r)
]
Ri
e
(r)
is the ratio of fluctuating buoyancy and inertia forces on an energy-
containing-range eddy of size
r
, with
θ
the amplitude of the fluctuations in potential
temperature. We interpreted
Eq. (10.38)
as saying that the largest eddies feel the
strongest buoyancy effects. If we write
θ
∼
∂/∂z
,then
g
θ
0
r
1
/
3
2
/
3
N
2
r
1
/
3
∂
∂z
Ri
e
(r)
=
=
2
/
3
,
(12.39)
with
N
the
Brunt-Vaisala frequency
. If we also write
∼
σ
w
/
,where
σ
w
is the
rms vertical velocity fluctuation, and take
r
=
then
Eq. (12.39)
is
N
2
2
σ
w
Ri
e
()
=
.
(12.40)
We might expect
Ri
e
()
, like the flux Richardson number, to approach a constant
under very stable stratification so that the turbulence scale approaches the
buoyancy
length scale
b
,
σ
w
u
N
.
b
∼
N
∼
(12.41)
As we shall discuss,
u/N
has been used as the length scale of the energy-containing
eddies in the stably stratified, equilibrium limit.
12.3.3 Equilibrium height of the nocturnal SBL
Derbyshire
(
1990
) determined analytically some of the properties of the equilibrium
nocturnal SBL implied by Nieuwstadt's (
1984
) local-scaling model. An underlying
