Geoscience Reference
In-Depth Information
whose solution is
(Problem 12.13)
W (t
0
)e
−
if (t
−
t
0
)
t
0
)
]
,
(12.15)
with
t
0
the time at which the stress-divergence terms in
Eqs. 12.12
have decayed
to zero.
Equation (12.15)
can have important consequences for the clear-weather, night-
time behavior of winds at heights that on the previous afternoon were in the mixed
layer of the CBL. These follow from multiplying
Eq. (12.15)
by its complex
conjugate (indicated by an asterisk):
W (t)
=
=
W (t
0
)
[cos
f(t
−
t
0
)
−
i
sin
f(t
−
W (t)
2
W (t) W
∗
(t)
W (t
0
)W
∗
(t
0
)
=
=
=
constant
.
(12.16)
For the physical components of velocity this implies
(Problem 12.14)
(U)
2
(V )
2
2
2
+
≡[
U(z,t)
−
U
g
(z)
]
+[
V(z,t)
−
V
g
(z)
]
2
2
=[
U(z,t
0
)
−
U
g
(z)
]
+[
V(z,t
0
)
−
V
g
(z)
]
=
constant
.
(12.17)
Equation (12.17)
is a statement of kinetic energy conservation for a time-dependent,
mean horizontal flow having only pressure-gradient and Coriolis forces - a flow
with neither turbulent friction nor mean advection. When the flow has decayed to
a nonturbulent state we can drop the modifier
mean
and interpret
Eq. (12.17)
as
saying that the squared magnitude of the difference between the velocity and the
geostrophic velocity is constant in time. The direction of the horizontal velocity
changes with time, but its energy constraint
(12.17)
causes the tip of its vector to
trace out a circular path in the
U, V
plane. As sketched in
Figure 12.5
,
the center of
the circle is
(U
g
,V
g
)
; its squared radius is
R
2
+
V(t
0
)
−
V
g
2
,
and its angular frequency around this circle is
f
. This phenomenon, discussed by
Blackadar
(
1957
), is one cause of the
nocturnal
or
low-level jet.
We'll consider the cases where the initial state is a barotropic CBL
(
Figure
11.6)
and a baroclinic CBL
(Figure 11.8)
.
In the barotropic case from
Eqs. (11.6
),
Figure 11.5
,
and
Figure 11.6
we estimate
=
U(t
0
)
−
U
g
2
u
2
(U)
2
(V )
2
25
u
2
/f z
i
+
barotropic:
U
V
5
u
∗
,
.
(12.18)
∗
∗
In the baroclinic case, from
Figure 11.8
we take
(U)
2
(V )
2
125
u
2
baroclinic:
U
5
u
∗
,
V
10
u
∗
,
+
.
(12.19)
∗
These show that the intensity of this inertial oscillation can be strongly affected by
baroclinity. In combination with the direction change due to the mean wind spiral
