Geoscience Reference
In-Depth Information
where
G 1 is the angular momentum in the outer, potential flow region; and v c is
the azimuthal velocity at the core radius. It is thought that the smooth transition
in the data is not an artifact and that diffusion at the interface between the core
and the outer flow region must be responsible for the smooth transition. The
velocity field in the presence of diffusion, such that the radial and azimuthal wind
components are functions of radius only and the vertical wind component is a
function of height only, is called a ''Burgers-Rott vortex'' ( Figure 6.47b ). In a
Burgers-Rott vortex, the steady-state solutions are
u ð r Þ¼ ar
ð 6
:
34 Þ
r ½ 1 exp ð ar 2
r Þ¼½G=
=
Þ
ð 6
:
35 Þ
2
2
w ð z Þ¼ 2az
ð 6
:
36 Þ
where a is a positive constant;
is
the angular momentum as in (6.26). For the Burgers-Rott solutions, the azimuthal
momentum is diffused in the radial direction only, and radial and vertical momen-
tum are not diffused at all.
Both the Rankine combined vortex and the Burgers-Rott vortex are limited
because they assume a steady state and are axisymmetric: real tornadoes change
intensity quite rapidly and are not necessarily axisymmetric. A more serious
problem with the Rankine combined vortex is that it ignores radial and vertical
motions, which are responsible for vortex intensification and are necessary by-
products of surface friction. Vertical velocity in the Burgers-Rott model increases
with height and is unbounded, which is not realistic: the Burgers-Rott vortex does
not take into account boundary conditions. Furthermore, neither the Rankine
combined vortex nor the Burgers-Rott vortex allow for any sinking motion.
The ''Sullivan'' vortex, which allows for both rising and sinking motion, is also
a solution to the axisymmetric, steady-state equations of motion with diffusion.
One of its solutions is ( Figure 6.48 )
is the kinematic coecient of viscosity; and
G
r ½ 1 exp ð ar 2
u ð r Þ¼ ar þ 6
=
=
2
Þ
ð 6
:
37 Þ
rH ð ar 2
r Þ¼ A
=
=
2
Þ=
H ð1Þ
ð 6
:
38 Þ
where
ð x
e f ð t Þ dt
H ð x Þ¼
ð 6
:
39 Þ
0
and where
f ð t Þ¼ t þ 3 ð t
0 ð 1 e y
Þ=
ydy
ð 6
:
40 Þ
As x !1 , H ð x Þ=
H ð1Þ! 1, so that the azimuthal wind component varies as in a
potential vortex (6.27). The Sullivan vortex is defined in terms of integrals of
functions, which makes it quite complicated, but is easily used nowadays with
sophisticated mathematical software that is readily available. The vertical and
Search WWH ::




Custom Search