Geoscience Reference
In-Depth Information
In the reference frame of the density current (which in Figure 3.37 is moving from
left to right) it follows from (3.12) that
cH ¼ u ð H h Þ
ð 3
:
13 Þ
where c is the density current-relative air speed ahead of it (and the speed of the
density current); and u is the density current-relative air speed above it. In other
words, the air must speed up as it ascends over the density current and is forced
into a narrower channel in the vertical. Or the flux of air entering the domain
from the right is the same as the flux of air exiting the domain on the left, above
the dense, cold air. In nature, the air behind the gust front is not necessarily
''resting'' and there could be (in most instances there in fact is) vertical shear, so u
is a function of z.
For simplicity, it is assumed that the atmosphere is hydrostatic far ahead of
the density current and surface friction is ignored (turbulent friction is ignored
also above the ground), even though we know that there is surface drag. In terms
of specific volume, where
0 ¼ 1
= 0 , the vertical equation of motion in the resting
u ¼ v ¼
(i.e.,
w ¼ 0) density current, for z
<
h,is
p 0
0 @
=@
z ¼ gð 0 1 Þ= 0
ð 3
:
14 Þ
Above the density current, for z h,
p 0
15 Þ
So, inside the density current we find p 0 by integrating (3.14) from height z to
height h that
0 @
=@
z ¼ 0
ð 3
:
0 p 0 ¼ 0 p t þg½ð 1 0 Þ= 0 ð h z Þ
ð 3
:
16 Þ
where p t ¼ p 0 ð h Þ . Above the density current, we find from (3.15) that
0 p 0 ¼ 0 p t
17 Þ
At z ¼ 0 (where the kinematic lower boundary condition is w ¼ 0) in the density
current the steady-state, two-dimensional (
ð 3
:
@=@
y ¼ 0), inviscid, horizontal equation
of motion is
p 0
2 u 2
þ 0 p 0 Þ
1
u
@
u
=@
x þ 0 @
=@
x ¼ 0 ¼ @=@
x ð
ð 3
:
18 Þ
þ 0 p 0 is
constant, anywhere along the x-coordinate: In the reference frame of the density
current, air approaches along the ground (along the x-axis) from the right and
must come to a grinding halt as it encounters the wall of cold air, the ''nose'' of
the advancing cold air (x decreases, so p 0 must increase); therefore the air parcel
encounters an adverse pressure gradient force and it decelerates ( Figure 3.38 ). The
reader is cautioned that the relationship between p 0 and v is not causal: one does
not cause the other; each must accompany the other, as a consistency argument.
The pressure field is disturbed not just at the leading edge, but also ahead of the
leading edge of the density current in an ''action at a distance''-like manner (the
air ahead of the leading edge of the advancing density ''knows'' about the exist-
ence of the density current before encountering it as a result of sound waves
2 u 2
1
Equation (3.18) is a Bernoulli-like equation,
in that the quantity
Search WWH ::




Custom Search