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