Geoscience Reference
In-Depth Information
From (2.71) we see that to the left and right of a buoyant air parcel vorticity is
generated in the y and þ y-directions, respectively ( Figure 2.7d-f ). For a ( Figure
2.7b ) fixed value of B, more air must be displaced laterally outward above a flat-
tened buoyant air parcel and more air must be displaced laterally inward below a
flattened buoyant air parcel than that for a more spherical air parcel ( Figure 2.7a ).
It follows then that the adverse pressure gradient force is more intense for the
flattened air parcel. The smaller the aspect ratio (D
L, where D is the depth and
L is the width) of the air parcel, the more the downward-directed pressure gradi-
ent force opposes upward Archimedean buoyancy. As D
=
L approaches zero, the
atmosphere becomes hydrostatic. Also, when the aspect ratio is small, the effects
of the gradient of B in generating vertical circulations are concentrated only along
the far edges of the air parcel. When the aspect ratio approaches infinity very little
air is displaced directly above and below the air parcel, very high up and very low
below ( Figure 2.7c ). Figures 2.7d, e, and f repeat a similar analysis of the effect of
the aspect ratio on vertical accelerations, but from the perspective of horizontal
vorticity produced by the vertical variation of the horizontal hydrostatic pressure
gradient force.
The qualitative analysis in the last paragraph is made more quantitative by
considering a region of buoyancy having the following distribution in space:
B ð x
=
;
z Þ¼ B 0 sin ð
z
=
H Þ cos ð
x
=
L Þ
ð 2
:
72 Þ
This idealized two-dimensional distribution of buoyancy represents a maximum in
the middle of the troposphere (z ¼ H
2) and at x ¼ 0; it is periodic in x. It is like
a quasi-spherical slab of a bubble placed in the middle of the troposphere ( Figure
2.8, left panel). (Not including variations in the y-direction does not change the
results of our analysis qualitatively.)
The divergence equation (2.62) for Archimedean buoyancy only is
=
2 p 0 b ¼ @
1
=
J
B
=@
z
ð 2
:
73 Þ
It follows from (2.72) and (2.73) that the following
p 0 b =
2
2
¼ð B 0 =
H Þ=½ð=
L Þ
þð=
H Þ
cos ð
z
=
H Þ cos ð
x
=
L Þ
ð 2
:
74 Þ
is a solution to the divergence equation (2.73) subject to boundary conditions of
vanishing p 0 b =
at z ¼ 0
;
H (the top and bottom of the model troposphere) and at
x ¼ L
2, etc.
Using (2.74), we can now calculate the acceleration induced by the vertical
perturbation pressure gradient force (inferred from Figure 2.8, right panel)
=
p 0 b =@
2
1
=
@
z ¼½ B 0 sin ð
z
=
H Þ cos ð
x
=
L Þf 1
=½ð H
=
L Þ
þ 1 g
ð 2
:
75 Þ
p 0 b =@
As
the
aspect
ratio H
=
L
approaches
zero,
1
=
@
z
becomes
B 0 sin ð
z
=
H Þ cos ð
x
=
L Þ , which is
equal and opposite
to B (2.73), and
=
Dt ¼ 0, which is the hydrostatic case. On the other hand, when H
=
Dw
L
p 0 b =@
approaches infinity, 1
=
@
z vanishes, so that Archimedean buoyancy B con-
tributes solely to vertical accelerations in the frictionless equation of motion (2.7).
This analysis suggests that infinitely narrow convective bubbles are most ecient
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