Geography Reference
In-Depth Information
We now consider a flow consisting of a single convective updraft embedded in
a basic state westerly flow that depends on z alone. Linearizing about this basic
state by letting
j du dz
+ ω (x, y, z, t) ,
U (x, y, z, t)
ω =
U
=
i u
+
and noting that the linearized form of the right-hand side in (9.55) becomes
·∇ × i w du dz
j
k
·∇ ×
( U
× ω
)
=−
k
+
we find that the linearized vorticity tendency is
∂ζ
∂t =−
u ∂ζ
∂w
∂y
du
dz
∂x +
(9.56)
The first term on the right in (9.56) is just the advection by the basic state flow.
The second term represents tilting of horizontal shear vorticity into the vertical by
differential v er tical motion.
Because du/dz is positive, the vorticity tendency due to this tilting will be
positive to the south of the updraft core and negative to the north of the updraft
core. As a result, a counterrotating vortex pair is established with cyclonic rotation
to the south and anticyclonic rotation to the north of the initial updraft, as shown in
Fig. 9.12a. Eventually, the development of negative buoyancy due to precipitation
loading generates an upper level downdraft at the position of the initial updraft and
the storm splits as shown in Fig. 9.12b. New updraft cores are established centered
on the counterrotating vortex pair.
In order to understand the generation of updrafts in the vortices on the flanks of
the storm, we examined the perturbation pressure field. A diagnostic equation for
the disturbance pressure is obtained by taking
∇·
(9.53) to yield
2 p
ρ 0
2 U
·
U
∂b
∂z
=−∇
+ ∇·
( U
× ω
)
+
(9.57)
2
The first two terms on the right in (9.57) represent dynamical forcing, whereas
the last term represents buoyancy forcing. Observations and numerical models
suggest that the buoyancy forcing in (9.57) produces pressure perturbations that
tend partly to compensate the buoyancy force in the vertical momentum equation.
Dynamically forced pressure perturbations, however, may generate substantial
vertical accelerations.
In order to compute the dynamical contribution to the disturbance pressure
gradient force in either the right or left side vortex, we use cylindrical coordinates
(r, λ, z) centered on the axis of rotation of either vortex and assume that to a first
 
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