Houze Robert A., Jr. Cloud Dynamics

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(8.10)

8.8 The Tornado

299

ized by a fragmented, contorted, but still destructive funnel. Air motions within

and near the tornado (Fig. 8.30) were determined by tracing debris particles and

identifiable cloud elements in motion pictures and by surveying surface damage

patterns. In the mature stage, tangential velocities at a radius of 200 m and heights

60-120 m above ground were estimated to be

-50-80

S-I,

in general agreement

with deductions from other documented cases.

222

At the wall cloud level, there

was strong downward motion on the southwest side of the funnel, with upward

motion on the northeast side (Fig. 8.30b). An empirical model of the life cycle

ofa

waterspout vortex is shown in Fig. 8.31. Comparison with Fig. 8.29 shows that the

waterspout life cycle resembles qualitatively that of the supercell tornado. Figure

8.31a corresponds to the organizing stage, Fig. 8.31b to the mature stage, and Fig.

8.31c to the decaying stage of the tornado.

As will be discussed further in Sec. 8.8.2 below, the tornado vortex is in

cyclostrophic balance along most of its length. The condensation marking the

funnel cloud is produced by the low pressure at the center of the vortex. Thus, the

overall shape of the cloud is a tracer of the vortex, not fluid parcels. Recalling the

component vorticity equations (2.57)-(2.59), noting that Coriolis force is negligi-

ble on the scale of the tornado, and assuming that baroclinic generation is also

negligible in this situation, we may combine (2.57)-(2.59) to obtain the three-

dimensional vorticity equation

Dro

- =

(er

V)v

where the expression on the right is the vector sum of all the stretching and tilting

terms on the right-hand sides of (2.57)-(2.59). The change of the shape of the

tornado vortex indicated by the life cycle of its cloud form seen in Fig. 8.29 is a

222 Measurements made with a small, portable Doppler radar carried to within a few kilometers of a

mature tornado have confirmed estimates such as these by showing tangential velocities as large as 60

S-I

(Bluestein and Unruh, 1989).

Figure 8.26 A postulated waterspout development process. (a) Early prefunnel stage. (b)

Intermediate stage. The different horizontal dimensions of the parent vortices (solid circles in cloud,

dashed when shown below, as on left), dark spot on ocean surface, and condensation funnel are

indicated. The heavy dashed curves in (a) denote cold downdraft air which terminates at the surface as

a gust front. The downdraft, the anticyclonic member of the parent vortex pair, and the subcloud

extension of the parent vortices are omitted in (b). In (b), the vortex center is made visible by a

condensation funnel, produced by lowering

ofthe

pressure in the central core. The dark sea-spray ring

suggests tangential winds >22 m

S-I.

(From Simpson et al., 1986.Reprinted with permission from the

American Meteorological Society.)

Figure 8.27 Empirical model of a developing waterspout. Cloud base height

H varies from 550 to

670 m and width

d, from 100 to 920 m. Dot over triangle is the standard rain shower symbol. (From

Golden, 1974a. Reprinted with permission from the American Meteorological Society.)

Figure 8.28 Horizontal pattern of wind and spray in a waterspout. Streamlines (solid) and isotachs

(dashed, m

S-I)

of boundary-layer flow are shown. (From Golden, 1974b. Reprinted with permission

from the American Meteorological Society.)

Figure 8.29

FRAME NO.

J.J

,r-!fJJJl>!jf>'

:;;;;;~'

WAll

CLOUD

300

~£-r-~

( I ' '

, " I

VORTEX

\,_\

\UP

... _

~W~

~·OTION

....

' \

_~~_--~

~MAX

DOWN

- -

DOWN

--.::::

- - -

...::::--

-"'

~ ~ ~

--'" X

Figure 8.30

8.8 The Tornado

301

dramatic visual illustration of the effects included in (8.10). As the life cycle

progresses, the initially wide, vertically oriented vortex stretches into a narrower

vortex and tilts over so that the vorticity vector along portions of the vortex is

highly canted, tending toward horizontal in places. The fact that the tornado

remains damaging in the decaying stage is clearly the effect of stretching. Al-

though the tornado is becoming narrower and affecting a smaller area, the rota-

tional wind in the affected area remains strong as convergent flow at the base of

the tornado continues to squeeze the vortex so that the vorticity in its interior

remains strong.

The tornado funnel is sometimes observed to contain one to six smaller subvor-

tices,

-0.5-50

m in diameter. These suction vortices may be stationary or orbit

around the tornado center (Fig. 8.32). They contain some of the strongest winds of

the tornado and leave a complex pattern of narrow trails of debris and extreme

damage within the tornado's general path. As will be discussed below, the suction

vortices are a result of waves superimposed on the primary tornado vortex be-

coming unstable and dominating the pattern. The suction vortices are observed to

be particularly dangerous and damaging. Again, the effect of stretching on the

intensity of vorticity is evident, as the locally intensified convergence in the small-

scale perturbations concentrates the vorticity in the suction vortices.

A schematic of the flow regions in a tornado is shown in Fig. 8.33. All of region

I is in cyclostrophic balance. Region

is called the outerflow and consists of air

that is converging and conserving its angular momentum. Region Ib is the core,

which immediately surrounds the axis and extends out to the radius of maximum

tangential wind. The core appears to be in a state of approximate solid-body

rotation (defined below). This means that, in the absence of vertical motions, the

core is stable against radial displacements and supports waves. These waves are

seen moving up or down tornado funnels. There is almost no entrainment into the

core, and flow along the axis can be either up or downward, as we will see. Region

II is the turbulent boundary layer, where friction upsets cyclostrophic balance.

The net inward force drives strong radial inflow toward region III, called the

corner region, where there is strong upflow into the core. Lying over the top of the

vortex is region IV, the buoyant updraft. The tornado must end in an updraft,

since a downdraft would destroy the vortex.

Figure 8.29 Structure of a tornado observed at several stages of its life cycle. Sketches show

funnel cloud and associated debris. Letters

A-H

indicate damaged farms. (From Golden and Purcell,

1978a. Reprinted with permission from the American Meteorological Society.)

Figure 8.30 Tornado structure from photogrammetry and damage surveys of the Union City,

Oklahoma, tornado of 24 May 1973. (a) Outline of tornado funnel and upper wall cloud showing

features tracked on movie loops. Sequential outline of cloud tags and streamers in wall cloud and

typical composite trajectory and displacement of debris aggregate are superimposed. (b) Schematic

plan view of horizontal streamlines (solid lines) and low-level vertical motion patterns (dashed lines)

around decaying tornado. (From Golden and Purcell, 1978b. Reprinted with permission from the

American Meteorological Society.)

302 8 Thunderstorms

COMPOSITE

MODEL

MS~)lt;?

CONDENSATE

SHELL

_--"'''-'-7

-'TURBULENT

WALL

VERT. EDDIES

~--

- --

--~-

----,

f-------_

-~,

I ...-

...

\'~

~~~HOWER

.,,

-';

-------

TRANSLUCENT

SMOKE

- SOMETIMES A

I FUNNEL

APPENDAGE

HERE

1llllillllliI"IllllllIlllddllll

r-Dfi

-.!..

NVISIBLE VORTEX

" CIRCULATION

(a)

(b)

, , \

, \ ,

\ , \ \

\ ' , \

1\~

\ I I

RW+,

I I

\ I

I I

\ " \

WHITE

- CAPS

.J.J

.J~

J~.>

.J.J.>

~..

.. rtf

_;.~:'_G.:~AK~

-------

....

- - -

.....

- -

....

(c)

Figure 8.31 Empirical model of a waterspout. (a) Early stage. Cloud base height H varies from 550

to 670 m, maximum funnel diameter

Djfrom

3 to 150 rn, d = 3-45 m,

= 15-760 m. (b) Mature stage.

Maximum funnel diameter ranges from 3 to 140m. (c) Decay stage, during which the funnel cloud often

undergoes rapid changes in shape and may become greatly contorted. Maximum funnel diameters

range from 3 to 105 m. (From Golden, I974a. Reprinted with permission from the American

Meteorological Society.)

8.8 The Tornado 303

Figure 8.32 Conceptual model of the subdivision of a tornado funnel into smaller suction vortices.

(From Fujita, 1981. Reprinted with permission from the American Meteorological Society.)

REGION

:m::

Figure 8.33 Schematic of flow regions in a tornado. (Adapted from Morton, 1970 and from

Davies-Jones, 1986.)

8.8.2 Tornado Vortex Dynamics-"

The dynamics of the core region of the tornado can be examined theoretically by

considering the dry Boussinesq equations written in cylindrical polar coordinates

(r,e,z).

The z-axis is vertical and centered on the vortex, which is assumed to be

axially symmetric

(a/ae

= 0). Molecular friction is ignored. Under these condi-

223 This discussion is based largely on a review article by Davies-Jones (1986) and papers by

Davies-Jones and Kessler (1974) and Rotunno (1977, 1979, 1986).

(8.11)

(8.12)

(8.13)

304 8 Thunderstorms

tions, the components of the Boussinesq version of the mean-variable equation of

motion

(2.83) are

-2

1:\*

- J

_ _ _ U

+ U u,

+wu

----+K

+-u

+ u

dr r r

- - - -

Liv

(- 1 - - vJ

vt + U Vr +

z + - = Km v

+ - Vr + Vzz - 2

r r r

1d*

- ( 1 )

+ uW

+ wW

= - -

+ B + K m w

+ - w

+ w

The mean-variable thermodynamic equation (2.78) and continuity equation (2.75)

in their Boussinesq versions (i.e., with the density factor

omitted) are

Lie

= K

(

err +

~er

+ e

)

(8.14)

and

~(rLi)

+ W

= 0

r r

(8.15)

In these expressions, U =

Dr/Dt,

v = rD@/Dt, and w =

Dz/Dt,

and the eddy-flux

terms in

(8.11)-(8.14) are parameterized in terms of a simplified K-theory (Sec.

2.10.1), in which the values of the mixing coefficients

and K

are constant. The

expressions multiplying

in (8.11)-(8.13) are the axisymmetric cylindrical-coor-

dinate components of the vector

while the factor multiplying K

in (8.14) is

the axisymmetric cylindrical-coordinate form of

ii.

The key dynamical characteristic of the tornado vortex is its extremely large

tangential wind component, which far exceeds its radial or vertical component

(i.e.,

[v]

» [u

1wi). In this case, (8.11) suggests that the flow is to a first approxi-

mation in cyclostrophic balance (Sec.

2.2.6). Studies of the dynamics of the tor-

nado vortex have therefore concentrated on simple cyclostrophic solutions to

(8.11)-(8.15). We proceed below by examining the two simplest known solutions.

The very simplest solution is the

Rankine

vortex,

which consists

oftwo

regions

separated by radius

r.: There is assumed to be no radial or vertical motion whatso-

ever (0

= w= 0); the flow is assumed to be frictionless

(K;

= K

= 0); and the

inner region is characterized by solid-body rotation, which means the angular

velocity

Vir is constant. Thus, within the inner core, the tangential velocity is

directly proportional to radial distance from the center of the vortex:

v = ar, a = constant, r

::;

(8.16)

A way of expressing the constant is obtained by first noting that, in axisymmetric

cylindrical coordinates, the vertical

vorticity'

(Sec. 2.4) is given by

1 d(rv)

,==

k·

V x v =

---

(8.17)

r dr

By applying (8.17) to (8.16), we find

, = 2a,

r::;

(8.18)

(8.19)

(8.22)

8.8 The Tornado

305

The integrated amount of vertical vorticity contained in some horizontal area A is

given by

'dA

k·

V x v dA =fV

A A

where the closed line integral, called the circulation and indicated by

I',

follows

from Stokes's theorem.

is taken around the boundary of

with VI being the

horizontal velocity component tangent to the boundary. To obtain the circulation

I', around the circle covered by the inner region of the Rankine vortex, we substi-

tute (8.18) into the left-hand expression of (8.19) and integrate over the area

surrounded by a circle of radius

r.: The result is

2anr

= r

(8.20)

From this expression, it is evident that specification of r

and I', determine the

constant in (8.16) and hence the flow across the whole inner core of the vortex.

The outer region of the Rankine vortex is characterized by

potential vortex

flow,

which is a flow whose tangential velocity is inversely proportional to r. Thus,

we have

v= blr, b = constant, r > r

(8.21)

This flow is irrotational, which

means'

0, a fact easily verified by applying

(8.17) to (8.2l). Since there is no vorticity outside

the circulation at any larger

radius remains

c; that is,

r = f

v(r)dl

v(r)2nr

= r,

which implies that the constant in (8.21) is

b=-

(8.23)

(8.24)

Thus, the constants in both (8.16) and (8.2l) may be expressed in terms of

The basic characteristics of the pressure distribution in the Rankine vortex are

easily diagnosed since under the assumed velocity conditions (0

= w= 0), the

equation for radial momentum (8.1l) reduces exactly to the cyclostrophic relation

---

The perturbation pressure p* has been replaced with the total pressure p in (8.24),

which is allowable since the base-state pressure

is a function of z only. The

assumed condition

w= 0 further implies that there is no vertical acceleration, and

the total pressure is therefore governed by the hydrostatic relation (2.38), which

may be written in the present case in mean-variable form as

- =

-Po

dz 0

(8.25)

(8.29)

(8.30)

306

8 Thunderstorms

From (8.21) and (8.23), the maximum tangential velocity is

- r

v =

(8.26)

max

2TCr

The larger this velocity is, the greater the pressure deficit

ofthe

vortex. This result

follows from integration of (8.24) from the center of the vortex to

r =

r.,

Making

use of (8.16), (8.20), and (8.26), we obtain

) _

f'c

1( rer

v~ax

P r

- Po =

--;.

2TCre2

dr = 2 (8.27)

This relation implies that if the density of the air were 1 kg m-

and the maximum

wind in the tornado 50 m

S-I,

then the pressure deficit across the vortex would be

12.5hPa.

is also evident from (8.24), that the largest pressure gradient is located

r = r

and has the value po'j2/

rc'

has been determined from photogrammetric observations that tornadoes of-

ten appear to behave to a first approximation as a Rankine vortex. This fact has

been determined by comparing the shapes of observed tornadoes to the shape of

the Rankine vortex. The outline of the funnel cloud is taken to be an isobaric

surface and is compared to the shape of an isobaric surface in the Rankine vortex.

The slope of the isobaric surface is implied by (8.24) and (8.25):

~:Ip

= -

~Iz/~Ir

(8.28)

Integration from a point along the funnel's edge

(r.z)

to the top

ofthe

funnel (oo,Zj)

yields

-2

maxre

2gr

Zf -

v;;x

:e:)'

r sr

the shape of the funnel is observed, and the funnel is assumed to be a Rankine

vortex, (8.29) provides a way of estimating the maximum wind in the vortex. If the

tip of the funnel is observed to be at height

z; and is assumed to have radius r

then (8.29) implies that v

rnax

V2g(zj

- zo).

The simplest vortex solution that allows a secondary circulation (i.e., motions

in the

rand

Z directions) is the one-cell, or Burgers-Rott vortex.P' for which the

horizontal convergence (negative divergence) is assumed to be a positive con-

stant. In this case, the axisymmetric divergence is given by

1 a (

ru =

-2c

= constant < 0

r ar

which has the solution

U =

-Cr

224 The one-cell vortex was described in early papers by Burgers (1948) and Rott (1958).

(8.31)

The continuity equation (8.15) then becomes

-2c+-

= 0

which, if

= 0 at z = 0, implies that

w= 2cz

8.8 The Tornado 307

(8.32)

(8.33)

With

0 and

given by (8.31) and (8.33), it can be verified by substitution into

(8.12) that, in the absence

eddy mixing,

d/r,

d = constant, r > e (8.34)

where e is an arbitrarily small positive number. This expression is of the same

form as (8.21). As in that case, the flow is irrotational, and the circulation

r is a

constant, with all of the vorticity concentrated within

r $ e. If eddy mixing is

included, (8.34) is replaced by

(8.35)

Thus, neither

nor

vis a function of z: the horizontal circulation is independent of

height. The three-dimensional circulation in the one-cell vortex is illustrated in

Fig. 8.34a.

The

horizontal flow spirals inward, while upwelling in the center

ofthe

vortex. In the case with eddy mixing, the viscosity associated with turbulence

establishes a core that rotates approximately as a solid body, as in the case of the

Rankine vortex.

The

maximum tangential velocity occurs at a radius r. =

12V2K

/c.

Another solution, referred to as the two-cell or Sullivan-" vortex, is illustrated

in Fig. 8.34b.

is similar to the

Burgers-Rott

single vortex, except that the inner

225 Discovered by Sullivan (1959).

;g~

!(i)

J:~U:~

Figure 8.34 Projections of vortex streamlines onto a vertical plane through the axis of the vortex

and onto a horizontal plane. (a) One-cell

Burgers-Rott

vortex. (b) Two-cell Sullivan vortex. (Adapted

from Sullivan, 1959.)

308 8 Thunderstorms

core has a secondary circulation and the boundary between the inner and outer

cells is at radius

2.38v'2K

/c. The fluid spirals inward and upward in the

outer cell, downward near the axis, and outward and upward near the outer edge

of the inner cell. Vorticity is concentrated (by stretching) in the annular zone of

convergence straddling the boundary between the two cells. This ring of concen-

trated vorticity is where multiple (suction) vortices (discussed below) tend to form

(Fig. 8.32).

Most of the above discussion of tornado vortex dynamics applies to regions Ia

and b in Fig. 8.33. The boundary layer and corner regions (II and III) entail

theoretical difficulties beyond the scope of this text. We can, however, examine

qualitatively the conceptual model of tornado vortex structure in Fig. 8.35, which

takes these additional regions into account. The model has been developed on the

basis of the one- and two-cell theories described above, laboratory simulations,

photogrammetry of real tornadoes, and numerical simulations using models like

those discussed in Sec. 7.5.3.

The flow in the conceptual model is depicted as a function of a parameter called

the swirl ratio, which is an overall measure of the ratio of tangential to vertical

flow in the vortex. As the swirl ratio increases, the flow changes as depicted in

tttttt

ttt\\\

(e)

Figure 8.35 Conceptual model of tornado vortex structure. Panels (a)-(e) show structure for

successively higher swirl ratio. (a) Weak swirl case: flow in boundary layer separates and passes

around corner region. (b) One-cell vortex. (c) Vortex breakdown. (d) Two-cell vortex with downdraft

impinging on ground. (e) Multiple vortices. The connected CL indicates the center line. (From

Davies-Jones, 1986 and Lewellen, 1976.)