Tornadoes
Eric A. PaniThe University of Louisiana at Monroe
Background
Definition: a violently rotating column of air that extends to the ground from a cumuliform cloud
Visible funnel may not be present every time
Funnel cloud if rotation does not reach ground
Most rotate cyclonically Statistics:
Average width ~ 100 m Average path length ~ 1-2
miles Average forward speed ~ 10-
20 mph Most have wind speeds < 100
mph
(Source: http://www.motorminute.com/Mixed_Nutz/Tornado.gif)
(Source:http://www.zonezero.com/exposiciones/road/images/eleventh/tornado.jpg
Life cycle
Dust-whirl stage: first sign as dust swirling upward from surface and short funnel from cloud base
Organizing stage: downward descent of funnel and increased intensed
Mature stage: funnel reaches greatest width and nearly vertical
Shrinking stage: decreasing funnel width, increasing tilt as base lags
Decay stage: vortex stretches into rope
May not go trough all stages (Source: http://wings.avkids.com/Book/Atmosphere/Images/tornado.gif)
(Source: http://www.redriver.net/tornado/tornado.jpg)
(Source: http://tornado.sfsu.edu/geosciences/StormChasing/cases/Miami/MiamiWallcloud.GIF)
Circulation and VorticityCirculation is line integral (counterclockwise) about a contour of velocity component tangent to the contour
V
ldˆ
dlVldVC cosˆ
Solid Body RotationSuppose a circular disk of radius r is rotating at an angular velocity Ω about the z axis is solid body rotation
r
z
22
0
2
2
0
2
)(
ˆ)(ˆ
and
rdrC
rdrC
ldrldVC
rV
dl
rdθ
rddl
Problem
For a large tornado, C ~ 5 104 m2s-1
If r ~ 100 m, what is the value of Ω and V?
C = 2πΩr2 and V= Ωr
Ω = C/(2πr2) Ω = (5 104)/(2π(100)2)=0.8 s-1
V = (0.8)(100) = 80 ms-1~160 kts
(Source: http://www.usatoday.com/weather/gallery/tornado/wtor4a.jpg)
Circulation and Vorticity
22
and 2 disk, For the
Thus,
ˆ
so , But,
ˆ)(ˆ
:Theorem Stokes'By
2
22
r
r
A
CrC
A
C
AdAnC
V
dAnVldVC
A
A
(Source: http://www.nws.noaa.gov/om/all-haz/Double%20Tornado%20Hi.jpg)
Circulation and Vorticity in Natural Coordinates
n
s
V
nn
VV
)( sd
n
R
V
n
V
sn
C
snR
V
n
VC
RsRs
sns
Vn
VC
nss
Vsnn
VnVsn
n
VC
nsd
snn
VsVsVdsVC
snn
VVsdsVCldVC
sn
0,lim
)( Thus,
1 Now
)(
so )(But
)(
)())((ˆ
Combined Rankin Vortex
r=a
uniform vorticity
irrotationalenvironment
arr
ara
r
π
CV
π
CK
a
Ka
πa
Car
r
KV
r
rV
rar
rπa
C
πa
CrV
πa
CrrVrdr
πa
CrVd
r
rV
rπa
C
r
V
r
Var
ar
arπa
C
rrV
,1
,
2
2)
2( ,At
)(10 , Outside
22
2)(
)(1 ,Within
, 0
,
2
2
22
2
2
02
0
2
2
V
r
Vmax
a
Pressure distribution
2
422
2
22
2
3
2
3
2
2
2
2
0
2
02
2
42
22
22
2
1
22But
)2
1(
2)|
2
1(
22
1
2
11
2
1
so ,2
, Outside
2
2let anddensity constant Assume
)2(
1
2
1
2 ,Within
gradient) pressure balances al(centripet 1
balance hiccyclostrop andon distributi pressure chydrostati Assume
0
r
appa
C
r
Cr
Cpp
r
drCdp
r
C
rr
C
r
p
r
CVar
rpprdrdp
a
Cr
a
rC
ra
Cr
r
p
a
CrVar
r
p
r
V
r
p
p r
p
p
r
(Source: http://www.ametsoc.org/AMS/image/challeng/tornado.gif)
Pressure drop
0max
2max0
max22
0
22
2
422
2
0
So,
2But . Thus,
2
1
2 and
2
,At
ppV
Vpp
VC
aapp
aa
appapp
ar
aa
ar
p0
p∞
2
2max
0
Vp
Rough estimates
Generally Vmax < 280 kts Let ρ = 1.275 kg m-3
Then p∞ – p0 = ρVmax2=(1.275)(130)2=215 mb
Generally taken to be ~ 100 mb Vertical velocities substantial (~ 80 m/s) and
not necessarily in core Inflow velocities may reach 50 m/s near
ground
Top Related