Tornadoes Eric A. Pani The University of Louisiana at Monroe.
Transcript of Tornadoes Eric A. Pani The University of Louisiana at Monroe.
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