Geoscience Reference
In-Depth Information
We'll illustrate with the entrainment flux of potential temperature, relaxing
slightly our usual assumption of horizontal homogeneity to allow a nonzero mean
vertical velocity
W
. This implies a nonzero divergence of the horizontal wind
field, since to a good approximation
∂U/∂x
+
∂V/∂y
+
∂W/∂z
=
0
.
The mean
potential temperature equation is then
∂
∂t
+
W
∂
∂wθ
∂z
=
∂z
+
0
.
(11.39)
We integrate
Eq. (11.39)
from
z
i
, the height of maximum negative temperature flux
Leibnitz' rule in the form
h
2
(t)
h
2
∂
∂t
(h
2
)
∂h
2
(z
i
)
∂z
i
∂
∂t
dz.
=
−
+
dz
(11.40)
∂t
∂t
z
i
(t)
z
i
For the second term we use the mean-value theorem in the form
h
2
W
∂
=
−
i
≤
h
m
≤
∂z
dz
W(h
m
)
[
(h
2
)
(z
i
)
]
,
h
2
.
(11.41)
z
i
Since
wθ
vanishes at
h
2
, the third term integrates to
wθ(z
i
)
.
The interfacial layer thickness typically is in the range 0.2-0.6
z
i
, but nonetheless
it is conventional here to take the limit as
h
2
−
−
→
0, giving a
jump model
of
the interfacial layer. In this zero-thickness limit several terms in the integral of
Eq. (11.39)
vanish
(Problem 11.21)
. The resulting expression is
z
i
∂z
i
∂t
−
W(z
i
)
(h
2
)
(z
i
)
wθ(z
i
)
≡
wθ
1
=−
−
=−
w
e
.
(11.42)
w
e
,the
entrainment velocity
, is the mean rate of erosion of the nonturbulent fluid
by the turbulent fluid. If
w
e
=−
W
the rate of entrainment is balanced by the mean
subsidence, so
∂z
i
/∂t
0. This happens often in the marine boundary layer under
steady synoptic conditions. Over land,
w
e
usually exceeds
=
in the morning and
early afternoon hours, allowing
z
i
to grow. Under a high-pressure system Ekman
pumping
(Chapter 9)
causes
W
to be negative, suppressing the growth of
z
i
and
thereby limiting the formation of the small cumulus at the top of the mixed layer.
We often have clear skies in high-pressure areas.
The jump equations for the fluxes of momentum and conserved scalar
c
are
|
W
|
uw
1
=−
w
e
U,
vw
1
=−
w
e
V ,
wc
1
=−
w
e
C.
(11.43)
Clearly, the entrainment velocity
w
e
is an important parameter.
