Geoscience Reference
In-Depth Information
11.4.2 The entrainment velocity
The mixed layer over land typically deepens at a few tens of meters to perhaps
100 m per hour
(
1-3 cm s
−
1
)
. But the convective velocity scale
w
∗
∼
is typically
at least 1 m s
−
1
and often larger. Why is
w
e
w
∗
?
The answer lies in the effects of stability on turbulence. In the i
nter
facial layer
the buoyant production term represents a rate of TKE loss because
wθ
is negative
there. Thus the turbulent energy budget under quasi-steady conditions says:
rate of TKE gain from shear production
,
turbulent and pressure transport
rate of TKE loss to viscous dissipation and buoyancy
.
(11.44)
If the underlying boundary layer is in free convection the shear production term
vanishes, and it can also be quite small in a barotropic CBL. In such cases the only
source term in the TKE budget is turbulent and pressure transport from below.
In the interfacial layer capping a convective ABL we can estimate this transport
term as of order
w
3
=
/h
, with
h
the thickness of the interfacial layer
(Figure 11.1)
.
If
h
scales with
z
i
,thisisoforder
w
3
∗
/z
i
. Therefore a flux Richardson number
characteristic of interfacial-layer turbulence is
∗
g
θ
0
w
e
w
3
rate of energy loss to buoyancy
rate of energy gain by transport
∼
R
f
=
.
(11.45)
/z
i
∗
Observations suggest that
R
f
in steady turbulence cannot exceed a relatively small
value, say 0.2-0.3; at larger values the turbulence is extinguished.
We can look at this another way. If in the interfacial layer
R
f
→
constant
=
a
,
then a crude statement of the TKE budget there is
a
w
3
g
θ
0
wθ
1
=
∗
z
i
−
.
(11.46)
It then follows from the definition of
w
∗
a.
This is a common
closure in CBL modeling - that the entrainment flux of temperature is a constant
negative fraction of the surface flux.
Another Richardson number, one characteristic of CBL structure
(
Prob-
lem 11.21
)
,is(
Deardorff and Willis
,
1985
)
that
−
wθ
1
/Q
0
=
g
θ
0
z
i
w
2
R
∗
=
.
(11.47)
∗
R
∗
is typically large; for example, if
z
i
=1km,
=1K,
w
∗
=1ms
−
1
,then
R
∗
=
30. The definitions of
R
f
and
R
∗
,
Eqs. (11.45)
and
(11.47)
, imply that they
are related by
