A MODEL OF A TROPICAL CYCLONE IN THE STEADY STATE

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A MODEL OF A TROPICAL CYCLONE IN THE STEADY STATE

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Xerox University Microfilms 300 North Z eeb Road Ann Arbor, M ichigan 48106

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10 where th e u n i t s o f th e c o o rd in a te s r and z a re i n c e n tim e te rs and th e v e l­ o c i t i e s i n c e n tim e te rs p e r second.

The c o n s ta n t C v as d eterm in ed t o e q u al

7000 cm s e c ~ l i n o r d e r to c o rresp o n d t o a s u rfa c e p r e s s u r e d i f f e r e n c e , be­ tw een th e c e n te r and 5 0 0 -k ilo m e te r r a d i u s , o f a p p ro x im a te ly 70 m i l l i b a r s , th e p r e s s u r e d if f e r e n c e i n th e New E ngland H u rric a n e .

These fu n c tio n s

summarize th e p r i n c i p a l known f e a t u r e s o f th e o b serv ed wind d i s t r i b u t i o n from anemometer l e v e l to th e to p o f th e l a y e r o f f r i c t i o n a l in f lu e n c e , a s w i l l be d is c u s s e d below .

A f u n c tio n which would a ls o re p ro d u c e th e wind

d i s t r i b u t i o n i n th e lo w e st te n m e te rs o f th e atm osphere would be c o n s id e r­ a b ly more complex b ecau se th e wind s h e a r i n t h i s lo w e s t la y e r i s s e v e r a l m ag n itu d es g r e a te r th a n i n th e re m a in in g 1 .5 k ilo m e te rs o f th e f r i c t i o n a l la y e r.

The c o n tr ib u tio n to th e l a r g e - s c a l e flo w p ro c e s s o f th e c y clo n e

by t h i s lo w e st 1 0 -m e te r la y e r m ust be sm a ll and th e u se o f a more com pli­ c a te d f u n c tio n d id n o t seem p r o f i t a b l e . The d ep th o f th e la y e r o f f r i c t i o n a l in flu e n c e h a s im p o rta n t im p li­ c a tio n s f o r th e whole p ro c e s s , s in c e m ost o f th e in flo w m ust o c cu r w ith in th is la y e r.

The d e p th o f th e la y e r o f a i r flo w in g i n t o th e cyclone and

th e v e r t i c a l la p s e r a t e o f e q u iv a l e n t - p o te n ti a l te m p e ra tu re 0j; i n t h i s l a y e r l a r g e l y d e term in e th e r e s u l t i n g r a d i a l d if f e r e n c e i n Qj; i n th e cy­ c lo n e and hence th e a v a ila b le r a d i a l d if f e r e n c e s i n th e h e a tin g by th e c o n d e n sa tio n p r o c e s s .

F o r th e p u rp o se o f t e s t i n g th e th e o ry a low e s t i ­

m ate o f th e d ep th o f th e f r i c t i o n a l la y e r and a v a i l a b l e h e a tin g i s d e s i r a b l e . A t th e to p o f th e f r i c t i o n a l la y e r th e r a d i a l component o f t h e v e l ­ o c i t y v a n ish e s} t h i s h e ig h t i s u s u a l l y i d e n t i f i e d a s th e g ra d ie n t-w in d le v e l.

Only i n d i r e c t o b s e rv a tio n s o f th e h e ig h t o f th e g ra d ie n t-w in d

l e v e l i n t r o p i c a l c y c lo n e s a re a v a i l a b l e .

C lin e (1926) h a s summarized

11 o b s e r v a tio n s o f cloud movements I n t r o p i c a l cy clo n es a c c o rd in g to clo u d ty p e .

A lthough th e h e ig h t o f th e g ra d ie n t-w in d l e v e l u n d o u b ted ly i s d i s ­

t r i b u t e d a sy m m e tric a lly a b o u t th e c e n te r and v a r ie s w ith r a d iu s , th e s e o b s e r v a tio n s in d ic a te t h a t th e g ra d ie n t-w in d l e v e l l i e s betw een th e lo w e r-c lo u d and m id d le -c lo u d l e v e l s .

R o ss ty and Montgomery (1935; P* 21)

have giv en a t h e o r e t i c a l e x p re s s io n f o r th e h e ig h t o f th e g ra d ie n t-w in d l e v e l a s a f u n c tio n o f th e g ra d ie n t-w in d speed and th e a n g le made by th e is o b a r s w ith th e d i r e c t i o n o f th e s u rfa c e w ind.

T h is e x p re ssio n was de­

r iv e d on th e b a s is o f s e v e r a l assu m p tio n s in c lu d in g th e s te a d y s t a t e and s tr a ig h t is o b a rs .

H aurw itz (1936) has d is c u s s e d th e e f f e c t o f c e n t r i f u ­

g a l fo r c e on th e h e ig h t o f th e g ra d ie n t-w in d l e v e l ; t h i s e f f e c t m ig h t be a p p ro x im a te ly e x p re ss e d as th e c o r r e c tio n f a c t o r

w here v

g i s th e g ra d ie n t-w in d speed, t o be a p p lie d to th e h e ig h t o f th e g r a d i e n t wind l e v e l computed f o r s t r a i g h t flo w .

To i l l u s t r a t e th e m agnitude o f

th e h e ig h t o f th e g ra d ie n t-w in d l e v e l g iv en by th e th e o ry , v a lu e s a t two p o in ts on th e r a d iu s a re g iv e n .

The assumed v a lu e s o f Vg and th e a n g le

made by t h e s u r f a c e wind w ith th e is o b a r s were s u b s t i t u t e d i n th e formu­ l a f o r s t r a i g h t flo w and th e n m u lti p l ie d by th e above c o r r e c tio n f o r cen­ trifu g a l fo rce .

The fo llo w in g v a lu e s f o r th e h e ig h t o f th e g ra d ie n t-w in d

l e v e l were o b ta in e d :

1 .6 km a t 100 km r a d iu s ; and 2 .1 km a t 500 km r a d i u s .

F or th e model th e h e ig h t o f t h e g ra d ie n t-w in d l e v e l was assumed t o be th e h o r iz o n ta l p la n e , 2 - 1 .5 k®.

T h is assum ption does n o t s e r io u s ly c o n f l i c t

w ith th e o ry o r o b s e rv a tio n and d o es ap p ear t o r e p r e s e n t a low e s tim a te o f th e h e ig h t a s d e s ir e d . The r a d i a l d i s t r i b u t i o n o f th e t a n g e n t i a l component o f v e lo c i ty a t anemometer l e v e l i s th e b e s t d e fin e d and m ost c o n s is te n t f e a tu r e o f th e

12 wind f i e l d i n th e t r o p i e a l c y c lo n e .

T here i s g e n e r a lly a c e n t r a l “eye" o f

l i g h t winds whose r a d iu s i s v a r i a b l e and may be as g r e a t as 25 k ilo m e te r s . O u tsid e t h i s calm eye th e wind sp eed I n c re a s e s r a p id l y w ith in c r e a s in g r a d iu s to a v e ry s h a rp maximum a f t e r which th e wind speed d e c re a se s a s r “^ o r

a c c o rd in g to v a r io u s a u th o r s .

O b se rv a tio n a l ev id en ce s u p p o rts

th e r “^ d i s t r i b u t i o n assum ed f o r th e m odel.

The fu n c tio n which b e s t f i t s

th e average d i s t r i b u t i o n o f v e l o c i t y i n th e t r o p i c a l cy clo n es f o r which d a ta has been ta b u la te d by C lin e , i s V*t = c o n s ta n t (U. S . W eather B ureau, H u rrica n e N o te s, 194-8) • The assumed r a d i a l d i s t r i b u t i o n o f v a t anemometer l e v e l i s shown i n F ig u re 1 .

The d i s t r i b u t i o n o f v computed from o b s e rv a tio n s made a t Tampa

i n th e F lo r id a H u rric a n e o f 18-19 O cto b er 1944- have a ls o been e n te r e d on t h i s diagram .

A lthough o f s m a lle r dim ension and i n t e n s i t y th a n th e m odel,

t h i s storm was chosen f o r com parison b ecau se an u n u s u a lly com plete re c o rd o f th e r a d i a l d i s t r i b u t i o n o f v e l o c i t y was a v a i l a b l e .

The c e n te r o f t h i s

c y clo n e , i n s o f a r a s i t may be d e te rm in e d , p a sse d d i r e c t l y o v er Tampa, where o b s e rv a tio n s w ere made a t 1 5-m inute i n t e r v a l s n e a r th e c e n te r o f th e storm.-*-

A graph o f th e d i s t r i b u t i o n o f th e wind v e lo c i ty w ith tim e a s ob­

se rv e d a t Tampa was c o n v e rte d to t h e d i s t r i b u t i o n w ith r a d iu s d e p ic te d i n F ig u re 1 by m u ltip ly in g th e tim e c o o rd in a te by th e observ ed speed o f th e cyclone c e n te r .

The components o f v e l o c i t y were determ ined from th e Tampa

d a ta by assum ing c i r c u l a r I s o b a r s and th e p l o t t e d v a lu e s a re th e mean o f th e v a lu e s on o p p o s ite s id e s o f th e c y c lo n e . The v e r t i c a l d i s t r i b u t i o n o f th e t a n g e n t i e l component o f v e l o c i t y i s f a i r l y w e ll known from th e o ry and o b s e r v a tio n .

N ear th e s u rfa c e th e wind

speed in c r e a s e s r a p i d l y , v e ry n e a r ly a s th e lo g a rith m o f th e e le v a tio n U n ite d S ta te s W eather B ureau Form 1130, Airway w eather r e p o r t s , Tampa, F lo r id a , 18-19 O cto b er 1944*

i 35

30

v msec*

25

20

100

200

300

400

r km j F ig » 1 « The d i s t r i b u t i o n o f th e t a n g e n t ia l component o f v e l o c i t y a t anemom eter l e v e l ; th e s o l i d l i n e i s th e assum ed d i s t r i b u t i o n ; th e c i r c l e s r e p r e s e n t th e v a lu e s o b serv ed a t Tampa i n th e h u r r ic a n e 18-19 O cto b er 19Ulw The d ashed l i n e i s an a r b i t r a r i l y smoothed r e p r e s e n t a tio n o f t h i s o b serv ed d is trib u tio n *

j j 1 j j '

r

z km

1.0

30

20

35

v msec"'

F i g , 2 , The v e r t i c a l d i s t r i b u t i o n o f th e t a n g e n t ia l component” o f v e l o c i t y i n th e f r i c t i o n a l l a y e r j th e a s­ sumed d i s t r i b u t i o n a t U00 km r a d iu s i s g iv en by th e so ],id l i n e 5 th e c i r c l e s r e p r e s e n t a lo g a rith m ic d i s t r i b u t i o n w hich y i e l d s th e assumed v a lu e s a t 10 m e te rs and 15>0 m e te rs above th e anemometer l e v e l , ■i

•J

33 a c c o rd in g to s e v e ra l i n v e s t i g a t o r s .

The d e p th o f t h i s l a y e r o f lo g a rith m ic

in c r e a s e i s about one te n th th e t o t a l d e p th o f th e f r i c t i o n a l la y e r acco rd ­ in g to R ossby and Montgomery (1935)*

Above t h i s s u rfa c e la y e r th e wind

speed g ra d u a lly in c r e a s e s t o a v a lu e a t th e g r a d ie n t l e v e l o f approxim ate­ l y tw ic e t h a t a t anemometer l e v e l .

The r a t i o o f th e s u rfa c e wihd speed to

th e g r a d ie n t wind speed was fo u n d to be 0 .6 1 f o r s tro n g w inds a cco rd in g to T a y lo r (1 9 1 5 ), w hile F a ir g r ie v e (1913) g iv e s 0 .4 5 a s th e av erag e v a lu e o f th e r a t i o f o r a la r g e number o f o b s e r v a tio n s .

The f u n c tio n assumed f o r

th e v e r t i c a l v a r i a t i o n o f v ,

£- £ . ^ , r e p r e s e n ts some compromise z + .2 x 10* betw een th e s e two f e a tu r e s b u t a p p ea rs t o y i e l d a r e a l i s t i c v e lo c i ty p ro ­ file .

The v e r t i c a l p r o f i l e o f th e assumed ta n g e n t i a l v e l o c i t y i s shown i n

F ig u re 2 .

D ots have been e n te r e d i l l u s t r a t i n g a t r u e lo g a rith m ic v e lo c i ty

p r o f i l e i n th e lo w e st 150 m e te r s .

T h is lo g a r ith m ic p r o f i l e f i t s th e

assumed wind d i s t r i b u t i o n a t t h e two p o i n t s , 10 m e te rs and 150 m eters above anemometer l e v e l .

T hese two p o in ts on t h e lo g a rith m ic p r o f i l e a re

s u f f i c i e n t t o determ in e ( a f t e r Rossby and Montgomery) th e ro u g h n ess p a ra ­ m eter z 0 , Z2 * zo lo g -= - * —— S O Wind speed a t e le v a tio n z . z. +. z lo g - 2 ^

Wind speed a t e le v a tio n z2 —

I

■ —

■■

I I

■ «■

I

^

■■

W

where th e e le v a tio n i n t h i s e x p re s s io n i s th e d is ta n c e above th e e a r t h 's s u r f a c e (u n lik e z in th e f u n c tio n f o r th e assumed wind d i s t r i b u t i o n where z i s th e h e ig h t above th e anem om eter.) n e s s p a ra m ete r i s 6 cm.

F o r th e case i l l u s t r a t e d th e rough­

T h is v a lu e a p p e a rs to be i n th e c o r r e c t ran g e

o f m agnitude a c c o rd in g t o th e d e te r m in a tio n s o f zQ from W tlst's d a ta (R ossby and Montgomery, 1936) o f from 4 cm f o r m oderate ocean s w e ll to

u th e maximum v a lu e o f 20 cm.

However, even th e c o r r e c t o rd e r o f m agnitude

o f th e roughness p a ra m e te r m ust rem ain i n do u b t b ecau se o f th e la c k o f ob­ s e r v a tio n a t th e h ig h wind v e l o c i t i e s i n a t r o p i c a l h u r r ic a n e .

One m ust

co n clu d e t h a t , a lth o u g h th e o rd e r o f m agnitude o f th e ta n g e n t ia l compon­ e n t o f v e lo c i ty i s i n agreem ent w ith o b s e r v a tio n , th e o rd e r o f m agnitude o f th e f i r s t d e r iv a tiv e I * . i s i n d o u b t i n th e lo w e st few d ek am eters. second d e r i v a t i v e ,

The

3 2I ^ z

may be i n e r r o r by a s much aB s e v e r a l o rd e rs o f

m agnitude b u t t h i s q u a n tity does n o t e n t e r d i r e c t l y i n th e su b seq u en t c a l­ c u la tio n s . The d i s t r i b u t i o n o f th e r a d i a l component o f v e l o c i t y i n th e f r i c t i o n a l l a y e r i s n o t w e ll e s ta b lis h e d e i t h e r by th e o r y o r by o b s e r v a tio n .

The

q u a n tity d i r e c t l y o b serv ed i s th e a n g le made by th e i s o b a r s w ith th e d i ­ r e c t i o n o f th e w ind.

F o r th e wind a t th e s u rfa c e o v e r th e o cean s, th e

a n g le i s known to l i e c o n s i s t e n t l y i n th e ran g e o f from 15 to 30 d e g re e s . The mean o f th e s e extrem e v a lu e s a g re e s w ith F ie rc e * s e s tim a te f o r th e New E ngland H u rric a n e .

T h ere b e in g no e v id en c e f o r any o th e r r e p r e s e n t a ti v e

v a lu e f o r c y c lo n ic c o n d itio n s , a f u n c tio n was assumed t o p ro v id e t h i s mean v a lu e o f th e an g le a t th e s u rfa c e th ro u g h o u t m ost o f t h e c y c lo n e , b u t w ith th e a d d itio n a l c o n d itio n t h a t th e a n g le ap p ro ach es z ero a t th e c e n te r .

As

one p ro g r e s s e s from th e o u te r edge o f t h e cy clo n e model tow ard th e c e n te r th e m ajor f e a tu r e s o f t h e v a r i a t i o n o f t h e a n g le made by th e is o b a r s and th e assumed s u rfa c e wind a r e :

a slow d e c re a s e from a v a lu e o f 2 2 .3 ° a t

500 km r a d iu s t o a minimum o f 1 6 .7 ° a t 100 km r a d iu s ; an in c r e a s e to a sec o n d a ry maximum o f 1 7 .6 ° a t 50 km r a d i u s ; fo llo w e d by a d e c re a se to zero a t th e c e n te r .

The s l i g h t maximum i n th e a n g le i n a p p ro x im a te ly th e same

p o s i t i o n on th e r a d iu s was found by K o e h le r b u t was deduced u n d er somewhat

15 a r b i t r a r y assum ptions f o r th e form o f th e f r i c t i o n a l s t r e s s and th e d i s t r i ­ b u tio n o f v .

The l a t e r com putation o f th e s u r f a c e s t r e s s f o r th e assumed

wind d i s t r i b u t i o n in d ic a te d a p ro b a b le in c o n s is te n c y i n t h e m agnitude assumed f o r u i n th e g e n e ra l re g io n w ith in 100 km o f th e c e n te r o f th e cy­ c lo n e (s e e th e l a t t e r p a r t o f t h i s s e c tio n ) so t h a t o n ly th e d e c re a se o f th e a n g le tow ard th e c e n te r can be d efen d ed a s r e a l i s t i c .

A graph d e s c r ib ­

in g th e assumed d i s t r i b u t i o n o f th e r a d i a l component o f v e l o c i t y a t ane­ mometer l e v e l i s given i n F ig u re 3 .

V a lu es o f th e r a d i a l component o f v e l ­

o c i t y computed from th e Tampa o b s e r v a tio n s i n th e h u r r ic a n e o f 18-19 O cto­ b e r 1944 a re in d ic a te d by th e c i r c l e s .

The l a t t e r v a lu e s a r e so i r r e g u l a r

t h a t no a tte m p t was made to draw an a v erag e c u rv e . O nly a few m ajor f e a t u r e s o f th e v e r t i c a l v a r i a t i o n o f u a r e known. The r a d i a l component would be ex p ected t o in c r e a s e a t l e a s t w ith in th e la y e r i n which th e lo g a rith m ic in c r e a s e h o ld s and th e n to d e c re a s e t o zero a t th e g ra d ie n t-w in d l e v e l .

Some guide to t h e r e l a t i v e m agnitude o f u t o v

i s p ro v id e d by K oehler (1 9 4 7 ), who d e term in e d u f o r a g iv e n v and an assumed d i s t r i b u t i o n o f th e c o e f f i c i e n t o f eddy v i s c o s i t y .

The f u n c tio n

/g-f./X fQ g

was assumed to d e s c rib e th e v e r t i c a l v a r i a t i o n o f th e r a d i a l component o f th e v e lo c ity .

T h is fu n c tio n h a s a s l i g h t maximum a t 125 m e te rs e le v a tio n *

and o f c o u rse becomes z ero a t an e le v a tio n o f 1 .5 k ilo m e te r s .

The assumed

v e r t i c a l d i s t r i b u t i o n o f u and t h e r e l a t i v e m ag n itu d es o f u to v a r e i l l u s ­ t r a t e d by th e hodograph o f th e v e l o c i t y d i s t r i b u t i o n a t 400 km r a d iu s (F ig u re 4 ) .

15

10 o

CD

£

5

0 L-o* 0

200

100

300

400

r km F i g , 3* The d i s t r i b u t i o n o f th e r a d i a l component o f v e l o c i t y a t anemometer l e v e l ; th e assumed d i s t r i b u t i o n i s g iv e n b y th e s o l i d l i n e ; th e d o ts r e p r e s e n t th e d i s t r i b u t i o n i n t h e h u r r ic a n e 18-19 O cto b er 19i*2i* d e term in e d from th e Tampa o b s e rv a tio n s *

10 i

o

surface

a> c/)

- " 'z = ..|5 k m

_zs..5km

E 3

r » l.0km —

0

r»l.5hm 30

v m sec F i g , U« hodograph o f th e assumed wind d i s t r i b u t i o n a t U00 km r a d i u s .

40

16 The re m a in in g v a r i a b l e s - te m p e ra tu re , p r e s s u r e , th e v e r t i c a l compon­ e n t o f v e l o c i t y and th e f r i c t i o n a l s t r e s s e s - were th en made c o n s is te n t w ith th e assum ed h o r iz o n ta l wind f i e l d by a p ro c e s s o f s u c c e s s iv e a p p ro x i­ m a tio n .

The a c c u ra c y o r th e deg ree o f c o n s is te n c y re q u ire d o f th e rem ain­

in g v a r i a b l e s i s governed l a r g e l y by th e re q u ire m e n ts o f th e s o lu tio n above th e f r i c t i o n a l l a y e r ; f o r th e v a lu e s a t th e to p o f th e f r i c t i o n a l l a y e r , i n e f f e c t , become th e boundary c o n d itio n s f o r th e s o lu tio n i n th e u p p er l a y e r . The s o l u t i o n above t h e f r i c t i o n a l la y e r i s o b ta in e d by th e e x tr a p o la tio n o f th e v e r t i c a l d e r i v a t i v e s .

I n m ost c a se s th e v e r t i c a l d e r iv a tiv e o c c u rs i n

th e e q u a tio n s a s a d if f e r e n c e o f s e v e r a l l a r g e r te rm s, and t h e r e f o r e a h ig h d e g re e o f c o n s is te n c y i n th e v a rio u s term s o f th e e q u a tio n s i s r e q u ir e d a t *

th e to p o f t h e f r i c t i o n a l l a y e r . i s th e v e r t i c a l d e r i v a t i v e

The m ost s t r i k i n g example o f such a term

i n e q u a tio n ( 2 ) . d 2

If

i s to be d e t e r d z

mined t o t h r e e f i g u r e s , th e n v and lo g p (when v i s la r g e ) m ust be known to a p p ro x im a te ly e ig h t s i g n i f i c a n t f i g u r e s .

However, a t t h i s p o in t i n th e

com putation i t was n o t o b v io u s what amount o f v a r i a t i o n i n th e boundary c o n d itio n would s i g n i f i c a n t l y a l t e r th e s o lu tio n f o r th e u p p er l a y e r .

Nor

was i t known to what e x te n t th e s e u p p er boundary c o n d itio n s were d e t e r ­ mined by t h e known f e a t u r e s o f th e h o r iz o n ta l wind d i s t r i b u t i o n i n th e f r i c t i o n a l l a y e r o r t o w hat e x te n t th e y a r e a r b i t r a r y .

In re tro s p e c t i t

a p p e a rs t h a t th e deg ree o f c o n s is te n c y o f th e term s a t th e to p o f th e f r i c t i o n a l l a y e r i s s u f f i c i e n t f o r th e i n t e g r a t i o n b u t th e accu racy w ith which th e in d iv id u a l te rm s were o b ta in e d i s g r e a te r th a n t h a t w a rra n te d by th e assu m p tio n s and by some o f th e c a l c u l a t i o n s . The te m p e ra tu re and p r e s s u r e f i e l d which would be c o n s is te n t w ith th e d i f f e r e n t i a l e q u a tio n s and th e assumed h o r iz o n ta l wind f i e l d w i l l

depend l a r g e l y upon th e c h o ic e o f boundary c o n d itio n s f o r th e te m p e ra tu re and th e p r e s s u r e .

I f s a t u r a t i o n o c c u r s , th e m o istu re d i s t r i b u t i o n i s a ls o

im p o rta n t i n d e te rm in in g th e te m p e ra tu re .

From th e su b seq u en t com putations,

i t w i l l a p p e a r t h a t , u n d e r t h e a ssu m p tio n s f o r th e flo w , th e p re s s u re and te m p e ra tu re f i e l d s w i l l be c o m p le tely d e fin e d by th e c h o ice o f th e v e r t i c a l d i s t r i b u t i o n o f p r e s s u r e and te m p e ra tu re and m o istu re - th o s e p r o p e r tie s which would be d e term in e d by an a e r o l o g i c a l sounding - a t th e p o in t where th e a i r e n te r s th e f r i c t i o n a l l a y e r o f th e m odel.

I t i s n e c e ssa ry t h a t

t h i s so u n d in g , which w i l l p ro v id e a boundary c o n d itio n , be r e p r e s e n ta tiv e o f th e c o n d itio n s i n a t r o p i c a l c y c lo n e .

The case b e s t known from o b se rv a ­

ti o n i s t h e mean s t a t e o f th e atm osphere i n th e t r o p i c s , which sh o u ld c o r r e s ­ pond c lo s e ly to th e c o n d itio n s a t th e p o i n t where th e a i r e n te r s th e c y c lo n e . The mean sounding a t Swan I s la n d f o r th e month o f Septem ber was chosen f o r t h i s p u rp o s e .

A f te r t h i s c h o ice o f soun d in g th e a u th o r became aware o f th e

more r e p r e s e n t a t i v e mean sounding f o r t h e h u rr ic a n e seaso n g iv en by S c h a c h t (1 9 4 6 ).

However, th e assumed sounding and th e mean t r o p i c a l n ig h t sounding

a re q u i t e s im ila r (T ab le I ) . T able I .

The assumed sounding and t h e mean t r o p i c a l sounding. Sounding assum ed f o r Mean t r o p i c a l sounding a i r e n te r in g t h e model ( a f t e r Schacht)

S u rfa c e p re s s u re

1000 mb

1013.6 mb

S u rfa c e te m p e ra tu re

300° K

298 ° K

S u rfa c e m ixing r a t i o

17 % o

16*9 ° / 00

T em perature la p s e r a te *

5 .0 °

km” 1

M ixing r a t i o la p s e r a te *

3 .5 % o to*"1

4 .0 % )0 km 1

E q u iv a le n t p o t e n t i a l te m p e ra tu re la p s e r a te *

5 .3 °

6.0 ° km”1

km” 1

5 .5 °

*The la p s e r a t e s a re a v erag e s f o r th e lo w e st 2 k ilo m e te rs .

km”1

18 The v e r t i c a l component o f v e l o c i t y was computed from th e e q u a tio n o f c o n tin u ity (4-).

E q u a tio n (4-) g iv e s f o r

, th e v e r t i c a l d iv e rg e n c e ,

= - J - J J u r)

and w was o b ta in e d by i n t e g r a t i o n w ith r e s p e c t to h e ig h t.

The f i r s t term

on th e r i g h t , th e h o r iz o n ta l d iv e rg e n c e , depends o n ly upon th e assumed wind f i e l d and i s th e r e f o r e known to any d e s ir e d a c c u ra c y .

The av erag e m agnitude

o f th e h o r iz o n ta l d iv e rg e n c e i n th e f r i c t i o n ^ l a y e r i s te n tim e s th e average m agnitude o f th e d e n s ity change te rm , — iL £ . S in c e th e v a r i a t i o n o f ternS dt p e r a tu r e and p re s s u re o f t h e a i r i s in c lu d e d i n t h i s l a t t e r te rm , th e com­ p u ta tio n o f w i s a f f e c t e d o n ly i n th e second s i g n i f i c a n t f ig u r e by jg change i n th e p r e s s u r e and te m p e ra tu re f i e l d s .

T hus, c o n s id e ra b le a ccu racy i n th e

d e te rm in a tio n o f w i s p o s s ib le w ith o n ly a ro u g h ap p ro x im atio n to th e tem­ p e r a tu r e and p re s s u re f i e l d s . The d e n s ity change term i s r e a d i l y ap p ro x im ated t o s e v e r a l s i g n i f i c a n t f ig u r e s w ith th e in fo rm a tio n now a v a i l a b l e .

I f th e term — f o r th e / dt steady state is expanded by substitution from the equation of state, one

obtains:

On th e average i n th e f r i c t i o n l a y e r , th e te rm u (— hP-E - A i n eq u aP a r " a* t i o n (8 ) i s a p p ro x im a te ly one o r d e r o f m agnitude l e s s th a n th e term w (A ^ P- - 1 ^ T ). p Jz T gradient term, A

of

th e r a d i a l d e r i v a t i v e s , th e r a d i a l te m p e ra tu re

is about an order of magnitude less than the radial

19 p r e s s u r e g ra d ie n t te rm , A A-E . The r a d i a l te m p e ra tu re g r a d ie n t was t h e r e p 2 -r f o r e assumed to be z e ro . T h is assu m p tio n a f f e c t s th e a c c u ra c y o f th e com­ p u ta tio n o f th e v e r t i c a l component o f v e l o c i t y , on th e a v e ra g e , o n ly i n th e f o u r th s i g n i f i c a n t f i g u r e . To an acc u ra cy o f two s i g n i f i c a n t f i g u r e s th e h o r iz o n t a l lo g a rith m ic p re s s u re g ra d ie n t does n o t depend pn h e i g h t and may be computed from th e assumed wind d i s t r i b u t i o n a t th e to p o f th e f r ic tio f P ^ la y e r by means o f th e g ra d ie n t wind e q u a tio n . T h is f i r s t a p p ro x im a tio n

t o th e

h o r iz o n ta l p r e s s u r e

g ra d ie n t w i l l be denoted a s f ^ ( r ) , F i r s t appro x im atio n to £ ((o ^

)

\ U (7-sjXj

(9a)

where T^ i s th e e s tim a te d mean te m p e ra tu re f o r th e to p o f th e f r i c t i o n a l l a y e r ; and v (1 .5 ) i s th e ta n g e n t i a l component o f v e l o c i t y a t 1 .5 Ion e le v a ­ tio n . The l a r g e r te rm , w (A ? P- - A -^_A) , wass i m i l a r l y ap p ro x im ated . The p d z T d z temperature term, A -gJL is less than one tenth of the magnitude of the T dz p r e s s u r e te rm , A ? p and may be ap proxim ated to two s i g n i f i c a n t f i g u r e s by p oz assum ing a c o n s ta n t la p s e r a t e th ro u g h o u t th e f r i c t i o n a l l a y e r . Upon s e ttin g

= cyi.

, where oi. i s a c o n s ta n t, one o b ta in s f o r th e v e r t i c a l

lo g a rith m ic te m p e ra tu re g r a d ie n t,

~

/

of

—--------

where T0 i s th e mean te m p e ra tu re a t t h e s u r f a c e .

The v e r t i c a l p r e s s u r e

g r a d ie n t term i s th e n known to an ap p ro x im ate a c c u ra c y o f th r e e s i g n i f i ­ c a n t f ig u r e s upon th e s u b s t i t u t i o n o f t h i s same v a lu e o f th e te m p e ratu re i n th e h y d r o s ta tic e q u a tio n , L.

=. — —2 _

20 S u b s titu ti o n o f th e above ap p ro x im a tio n s f o r th e g r a d ie n ts o f tem pera­ t u r e and p re s s u re in e q u a tio n ( 8 ) le a d s to th e fo llo w in g e x p re s s io n f o r th e term -i />



dt

X

i f .

s

u

4-

V J -f___$

{ ,( * ) -

S dt

V‘RIT0+ ^ ? )

_____ - ________ ^ 7 TZ W i /

I f th e d i s t r i b u t i o n assumed f o r u i n e q u a tio n ( 6 ) i s a b b re v ia te d u =

a s fo llo w s:

gjz)

w here fgC r) sud gg(z) a re th e fu n c tio n s f o r t h e r a d i a l

and v e r t i c a l

v a r ia ­

t i o n o f th e r a d i a l component o f v e l o c i t y , r e s p e c t i v e l y , th e e q u a tio n o f con­ t i n u i t y (A) w ith t h i s a b b re v ia tio n f o r u and th e above ap p ro x im a tio n f o r i J> d t

i s a d i f f e r e n t i a l e q u a tio n o f th e form

- J bl

t

cl*

v

i

$t(T0-*c&)

s

[

O

i

z th e re g io n i n which th e s o lu tio n i s t o be d e term in e d by th e method o f f i n i t e

d if f e r e n c e s .

O b se rv a tio n s o f th e w ind f i e l d a r e n o t s u f f i c i e n t t o v e r i f y

th e s e o r any assumed v a lu e s o f th e a c c e le r a ti o n a t t h i s l e v e l .

A ll t h a t can

be s a id i s t h a t th e term

i s a t l e a s t o f th e same o rd e r o f m agnitude as >z found by K o eh ler (194.7), and t h a t th e v e r t i c a l g r a d ie n t o f th e t a n g e n t ia l component o f v e l o c i t y iB v e ry s m a ll (s e e F ig u re 2 ) .

The i n t e g r a t i o n th e n

p ro ceed ed by th e m ethod o u tlin e d u s in g a v e r t i c a l i n t e r v a l o f 0 ,0 5 km, which e x p e rie n c e in d ic a te d t o be a maximum p e r m is s ib le i n t e r v a l f o r th e e x tr a p o la tio n •

34 C om putation o f th e r a d i a l d e r i v a tiv e s p re s e n te d some d i f f i c u l t i e s .

A

l i n e a r approxim ation t o th e r a d i a l d e r i v a t i v e s o f p r e s s u r e and v e lo c i ty , a c c u ra te to th r e e o r f o u r s i g n i f i c a n t f i g u r e s , would r e q u ir e a h o r iz o n ta l i n t e r v a l f o r th e com putation no l a r g e r th a n one k ilo m e te r i n th e re g io n o f r a p id ch an g es.

A co m p u tatio n n e t o f such dim en sio n s would in c re a s e th e

la b o r beyond re a s o n , u n le s s h ig h -s p e e d a u to m a tic com puters were a v a il a b le . As a s u b s t i t u t e m easure th e r a d i a l d e r i v a t i v e s were approxim ated by f i t t i n g a fo u rth -d e g re e p o ly n o m ial to th e v a lu e s o f th e v a r i a b le s a t f iv e p o in ts i n th e n e t and ta k in g t h e d e r iv a tiv e o f t h i s p o ly n o m ial a t th e p o in t a t which th e d e r iv a tiv e i s d e s i r e d .

T h is i s a s ta n d a rd method o f n u m e ric al

a p p ro x im atio n and i s p a r t i c u l a r l y w e ll s u i t e d f o r com putations w ith c a lc u ­ l a t i n g m achines p o s s e s s in g th e f e a t u r e s o f a u to m a tic m u lt ip l ic a tio n and d iv is io n .

I n th e h o r i z o n t a l a g rad ed n e t i n t e r v a l was u s e d :

i n th e re g io n

from 0 t o 120 km r a d iu s an i n t e r v a l o f 10 km was u se d ; in th e re g io n 150 to 500 km, where th e r a d i a l v a r i a t i o n i s more r e g u la r , an i n t e r v a l o f 50 km was u s e d . The i n t e g r a t i o n was h a l t e d a t an e le v a t io n o f 2 .5 km a f t e r computing 20 l e v e l s .

C e r ta in f e a t u r e s o f th e s o lu ti o n were a lre a d y a p p a re n t b u t

th e reasons f o r h a l t i n g th e work a t t h i s s ta g e were th e m ech an ical d i f f i ­ c u l t i e s e n c o u n te re d .

U nder th e m ost f a v o r a b le c o n d itio n s th e r a d i a l d e r­

iv a t i v e s a re d i f f i c u l t to compute a t th e boundary.

I n th e in te g r a t io n

perform ed h e re th e s o l u tio n e x h ib ite d r a t h e r v i o l e n t o s c i l l a t i o n s , and u n d e r such c o n d itio n s th e s o lu tio n i s w o rth le s s a t th e boundary.

The

r a d iu s o f th e re g io n i n which th e s o l u t io n i s v a l i d , t h e r e f o r e , d e c re a se s w ith a l t i t u d e .

35 DISCUSSION OF THE CICLONE MODEL The s o lu tio n o f th e e q u a tio n s d e te rm in e d by th e n u m e ric a l i n t e g r a t i o n i s p re s e n te d g r a p h ic a lly i n F ig u re s 7 th ro u g h 1 1 .

F ig u re 7 i s a v e r t i c a l c ro s s

s e c tio n o f th e flo w showing th e r a d i a l and v e r t i c a l components o f th e v e l ­ o c ity .

For th e sake o f c l a r i t y , o n ly h a l f th e computed v e c to r s a re shown.

The d i s t r i b u t i o n o f th e rem ain in g v a r i a b l e s i s shown i n F ig u re s 8 th ro u g h 11 by a com parison o f t h e i r r a d i a l p r o f i l e s a t 1 .5 km and a t 2 .5 km.

The

computed v a lu e s o f th e v a r ia b le s a t e ac h l e v e l a re g iv en i n Appendix I I I . 6.

O s c i l l a t i o n o f th e flow The m ost pronounced f e a tu r e o f t h e s o l u t io n in th e re g io n above 1 .5 km

e v id e n t in F ig u re 7 , i s th e o s c i l l a t i o n o f th e flo w ; e x c e p t n e a r th e c e n te r , th e r i s i n g a i r i s a l t e r n a t e l y d i r e c t e d outw ard from th e cy clo n e and th e n in w a rd .

The am plitude o f th e s e o s c i l l a t i o n s i s so g r e a t a s to o b scu re th e

g e n e ra l tr e n d o f th e s o lu tio n everyw here e x c e p t n e a r th e c e n te r o f th e cy­ c lo n e . O s c ill a tio n i s a c h a r a c t e r i s t i c e f f e c t o f th e f i n i t e - d i f f e r e n c e method o f s o lu tio n ; and i t i s c e r ta in t h a t a l a r g e amount o f th e o s c i l l a t i o n ob­ se rv e d i n F ig u re 7 would be e lim in a te d by u s in g a s m a lle r i n t e r v a l i n th e com putational net. However, th e v e r t i c a l wave le n g th , i f i t may be so term ed , o f some o f th e o s c i l l a t i o n s i s q u ite l a r g e .

F o r exam ple, a t a r a d iu s o f

80 km, l e s s th a n one h a l f o f a com plete o s c i l l a t i o n i s o b serv ed in a v e r ­

t i c a l i n t e r v a l o f one k ilo m e te r .

S in c e t h e i n t e g r a t i o n was perform ed i n

r e l a t i v e l y pmal1 f i n i t e s te p s o f 0 .0 5 km, th e o s c i l l a t i o n a t 80 km radiuB does n o t appear to be th e r e s u l t o f th e f i n i t e d if f e r e n c e m ethod. W hile i t i s im p o ssib le to s e p a r a te th e r e a l o s c i l l a t i o n from t h a t i n ­ tro d u c e d by th e f i n i t e d if f e r e n c e m ethod, u n le s s th e s o lu tio n i s recom puted w ith a n e t o f f i n e r s p a c in g , i t seems l i k e l y t h a t o s c i l l a t i o n w i l l r e s u l t

UJ>) U 0 U D A 9 I 3

*Z

3

£e> a 6

• H -

t«6

* •4

too

soo

400

r km

F ig , 8 . The d istr ib u tio n of temperature a t 1 ,5 km e le v a tio n (broken lin e ) and a t 2*5 Ion e le v a tio n ( s o lid l i n e ) .

1 T .0

o

r th a n th e te rm , R

, a t th e c e n te r o f th e boundary i s ab o u t 100 tim e s l a r g e r

and th e d e c re a se o f th e in w a rd ly d i r e c t e d a z r p r e s s u r e - g r a d ie n t f o r c e w ith h e ig h t i s t h e r e f o r e d eterm in ed by th e r e l a t i v e ­ l y h ig h e r te m p e ra tu re s a t th e c e n t e r .

A lthough th e d if f e r e n c e i s so sm all

t h a t i t i s n o t a p p a re n t in th e r a d i a l p re s s u re p r o f i l e a t 2 .5 km (F ig u re 1 1 ), th e p r e s s u r e a t 30 km r a d iu s i s g r e a t e r th a n a t AO km r a d i u s .

(F o r th e nu­

m e ric a l v a lu e s , see Appendix I I I . ) A t th e to p o f th e f r i c t i o n a l l a y e r w ith in a r a d iu s o f 100 km o f th e c e n te r , a re g io n o f h ig h e r te m p e ra tu re c o in c id e s w ith th e minimum i n th e p r e s s u r e p r o f i l e (compare F ig u re s 8 and 1 1 ) .

I f th e r a d i a l te m p e ra tu re

g r a d ie n t re m a in s unchanged th e h o r iz o n ta l p r e s s u r e - g r a d ie n t f o r c e w i l l r e ­ v e r s e a t lo w e r e le v a tio n n e a r th e c e n te r and r e v e rs e a t g r e a t e r e le v a tio n w ith in c r e a s in g r a d iu s from th e c e n te r .

Once th e p re s s u re g r a d ie n t i s

d ir e c te d outw ard th e o u tflo w in c r e a s e s r a p i d l y w ith e l e v a tio n .

A s ig n ifi­

c a n t c h a r a c t e r i s t i c o f t h i s p ro c e s s i s t h a t th e m agnitude o f th e n e g a tiv e r a d i a l te m p e ra tu re g r a d ie n t i s n o t c o n s ta n t b u t in c r e a s e s w ith e le v a tio n i n th e upward and outw ard flo w .

Thus th e v e r t i c a l d e c re a se o f th e inw ard­

l y - d i r e c t e d p r e s s u r e - g r a d ie n t f o r c e which p e rm itte d th e o u tflo w , i s i n ­ t e n s i f i e d i n th e o u tflo w .

As a r e s u l t th e o u tflo w in c r e a s e s r a p id ly w ith

e le v a tio n so t h a t a l l th e a i r a sc e n d in g w ith in th e re g io n e n c lo s e d by t h i s r a d iu s (w here th e p r e s s u r e g r a d ie n t h a s re v e rs e d ) i s e v ic te d w ith in a l a y e r o f s m a ll v e r t i c a l e x te n t . The s tre n g th e n in g o f th e r a d i a l te m p e ra tu re g ra d ie n t i n th e o u tflo w accom panies th e a d v e c tio n j t h i s e f f e c t w i l l be more e a s i l y re c o g n iz e d a f t e r th e d is c u s s io n i n th e fo llo w in g s e c tio n .

A nother s l i g h t c o n tr ib u tin g f a c ­

t o r i s t h e s te a d y in c r e a s e i n t h e r a d i a l te m p e ra tu re d if f e r e n c e w ith de­ c re a s in g p r e s s u r e due to a c h a r a c t e r i s t i c o f th e m o is t- a d ia b a tic p r o c e s s .

38 T h is e f f e c t , i l l u s t r a t e d by th e s p re a d in g a p a r t , a t lo w er p r e s s u r e , o f th e l i n e s o f c o n s ta n t e q u iv a l e n t - p o t e n t i a l te m p e ra tu re on a te p h ig ra m , w i l l be r e l a t i v e l y u n im p o rta n t i n th e lo w er la y e r s b u t w i l l become v e ry im p o rta n t a l o f t .

The 2 -d e g re e warming by f r i c t i o n a l d i s s i p a t i o n , f o r ex­

am ple, produces a d if f e r e n c e o f a p p ro x im a te ly 10 d e g re e s i n 0g.

The m o ist

a d ia b a ts r e p r e s e n tin g t h i s 10 -d e g re e d if f e r e n c e i n fig a re s e p a ra te d by a lm o st 10 d e g re e s o f te m p e ra tu re a t a p r e s s u r e o f 100 mb.

An in c re a s e

w ith e le v a tio n , i n th e r a d i a l d if f e r e n c e o f th e h e a tin g i s a p p aren t n e a r th e c e n te r o f th e c y clo n e m odel.

W ith in 100 km o f th e c e n te r , th e slo p e

o f th e cu rv e o f th e f u n c tio n p

i s g r e a t e r a t 2 .5 km e le v a tio n th a n a t

dp

1 .5 km (F ig u re 9 ) . Both th e a d v e c tio n and th e momentum a s s o c ia te d w ith th e s tro n g o u t­ flow te n d t o ex ten d th e re g io n o f o u tflo w w ith in c r e a s in g e le v a tio n , f o r from e q u a tio n ( 2 )

The term RT

becomes in c r e a s in g ly n e g a tiv e w ith h e ig h t because o f a- r th e n e g a tiv e te m p e ra tu re g ra d ie n t which in c r e a s e s w ith th e o u tflo w . I n th e zone o f convergence a t th e o u te r r a d iu s o f s tro n g o u tflo w , and n e g a tiv e and th e a c c e le r a ti o n a l term u

av

i s la r g e

i s t h e r e f o r e la r g e and n eg -

a tiv e . No co m putations w ere made i n th e downward flo w becau se th e method o f com putation f a i l s a s th e v e r t i c a l v e l o c i t y ap p ro ach es z e ro j th e r a t i o o f th e v e r t i c a l to th e h o r i z o n t a l i n t e r v a l o f com p u tatio n m ust d e c re a se w ith th e r a t i o o f th e v e r t i c a l component o f v e lo c i ty t o th e h o r iz o n ta l component. Even i f one were t o e x tr a p o la te th e d if f e r e n c e s from o u ts id e th e troublesom e p o in t a t which w v a n is h e s , th e d e sc e n d in g a i r would be u n s a tu r a te d , p r e s e n t-

39 in g a new problem o f d i s c o n t i n u i t i e s in t h e a i r p r o p e r t i e s . U n fo rtu n a te ly th e accuracy o f th e s o lu tio n n e a r th e c e n te r i s n o t g re a t.

The downward m otion a t th e c e n te r rem oves p o in ts from th e computa­

t i o n , and th e r a d i a l d e r iv a tiv e s become a lm o s t a s d i f f i c u l t t o d eterm in e a t th e c e n te r a s a t th e o u te r ed g e.

The m agnitude o f th e e r r o r i n th e

com putation o f th e ta n g e n t ia l component o f v e l o c i t y a t th e end p o in t i s a p p a re n t, because th e computed v a lu e exceeds th e l i m i t im posed by th e boundary c o n d itio n s f o r v .

For c y c lo n ic flo w a t th e boundary, th e tangen­

t i a l component o f v e lo c i ty w i l l alw ays be g r e a t e r th a n ( 1 9 ) , s e c tio n 8 ) .

At 30 km r a d iu s t h i s l i m i t w i l l be:

( see e q u a tio n v - - .75 m s e c ~ \

The computed t a n g e n t ia l v e lo c i t y a t 2 .5 km e le v a tio n and 30 km r a d iu s i s - 3 .0 0 m sec”'®’ (F ig u re 10) and th e e r r o r i n th e computed v a lu e must be a t l e a s t 2 .2 5 m sec“ ^ .

However, even a llo w in g a l l p o s s ib le l a t i t u d e o f th e

r a d i a l d e r iv a tiv e s , th e outflow a t 10 and 20 km r a d iu s i s so g r e a t t h a t c o n tin u ity r e q u ir e s downward m otion o v e r th e c e n te r , e x te n d in g below 2 .5 km e l e v a tio n . 8.

On th e form o f th e s o lu tio n above th e re g io n o f com putation The com putation o f th e s o lu tio n was o n ly c a r r ie d to an e le v a tio n o f

2 .5 km, and n e a r th e c e n te r , o n ly to th e re g io n o f downward m o tio n .

I n th e

re g io n t o which th e com putation does n o t e x te n d , a few f e a tu r e s o f th e s o lu tio n may be deduced by e x ten d in g some o f th e r e l a t i o n s f o r th e flo w o b se rv e d i n th e computed s o lu tio n and by th e c o n s id e ra tio n t h a t , u n d er th e o r i g i n a l assum ptions f o r th e flo w , s e v e r a l p r o p e r t ie s a r e conserved i n th e flo w above th e f r i c t i o n a l l a y e r .

The o r i g i n a l assu m p tio n s on which th e

co m putations a re b ased a re e q u iv a le n t to s p e c ify in g t h a t th e s tre a m lin e s a re l i n e s o f c o n s ta n t p o t e n t i a l te m p e ra tu re 9 in th e u n s a tu r a te d a i r and l i n e s o f e o n sta n t e q u iv a l e n t- p o te n ti a l te m p e ra tu re ©e i n th e s a tu r a te d a i r .

AO The a b s o lu te v a lu e o f 6 g f o r a s tre a m lin e i n th e s a tu r a te d a i r a sc e n d in g from th e f r i c t i o n a l la y e r i s d eterm in ed by th e v a lu e a t th e to p o f th e f r i c t i o n 3^l a y e r .

In th e d e sc e n d in g a i r th e s tre a m lin e s a r e l i n e s o f con­

s t a n t 6 , b u t th e a b s o lu te v a lu e o f © i s n o t d eterm in ed u n le s s f u r t h e r boundary c o n d itio n s a re a p p lie d a t th e p o in t where t h i s a i r e n te r s th e model o r u n le s s te m p e ra tu re c o n tin u ity i s demanded a t some a r b i t r a r y p o in t where th e u n s a tu r a te d and th e s a tu r a te d a i r m eet. The c o n s e rv a tio n o f © o r % im p lie s a c e r t a i n th e rm a l s t r u c t u r e f o r any giv en flo w . been n o te d :

C e rta in c h a r a c t e r i s t i c s o f th e flow i n th e model have

a re g io n o f s tro n g o u tflo w s t a r t s a t th e c e n te r j u s t above

th e f r i c t i o n a l la y e r and sp re a d s outw ard from th e c e n te r w ith in c r e a s in g e le v a tio n ; th e o u te r s u rfa c e o f t h i s re g io n o f o u tflo w ro u g h ly co rresp o n d s to th e s u rfa c e o f zero h o r iz o n ta l p re s s u re g r a d ie n t and i s a re g io n o f s tro n g h o r iz o n ta l convergence; th e o u tflo w i s s u f f i c i e n t l y g r e a t to r e q u ir e a c e n t r a l c o re o f descen d in g a i r . = O d t

U

I n th e d escen d in g a i r 4

?f

Upon r e a r r a n g in g term s one o b ta in s , Pz

^

= - ^

.

S in c e f o r s t a b l e a i r ,

0 and f o r th e c ase c o n sid e re d h e re w 0 , i t fo llo w s t h a t

^ ^ 0 . I n th e descen d in g a i r , p o t e n t i a l l y c o ld e r a i r l i e s tow ard th e Pr c e n te r ; w h ile i n th e a sc e n d in g a i r , th e p o t e n t i a l l y warmer a i r l i e s tow ard 1m th e c e n t e r . T hus, e x c e p t f o r th e u n lik e ly c ase t h a t ~ — m ig h t s t i l l be i7 r

n e g a tiv e , th e o u tw a rd ly d ir e c te d p re s s u re g r a d ie n t w i l l d e c re a s e w ith e le ­ v a tio n in th e d escen d in g a i r ; and, by anology w ith th e s o lu tio n i n th e re g io n below , th e o u tflo w would a ls o be e x p ec te d to d e c re a se w ith e le v a tio n . T h is re g io n o f d e c re a sin g o u tflo w w ith e le v a tio n i s p ro b a b ly a re g io n o f h o r iz o n ta l d iv e rg e n c e s in c e th e o u tflo w a t g r e a t e r r a d iu s s t i l l in c r e a s e s

a w ith e le v a tio n . O v e rly in g th e a sc e n d in g s a tu r a te d a i r th e r e w i l l be a la y e r o f ascen d ­ in g u n s a tu r a te d a i r w hich p re v io u s ly had descended o v e r th e c e n te r o f th e c y clo n e m odel.

T h is c ase has been found i n th e com putations a t 20 km

r a d iu s ( th e o u tflo w from th e s id e s o f th e c y lin d e r o f 20 km r a d iu s i s g r e a t­ e r th a n th e in flo w a t th e b o tto m ).

I f t h e r e i s to be re g io n o f downward

m otion above, t h i s la y e r m ust be one o f s tr o n g d iv e rg e n c e .

The th e rm a l

s tr u c tu r e i n t h i s la y e r i s n o t d e fin e d p a r t l y because th e n e c e s s a ry bound­ a ry c o n d itio n s have n o t been in tro d u c e d , b u t i t may be o b serv ed t h a t an abnorm ally l a r g e te m p e ra tu re g r a d ie n t would be re q u ire d to m a in ta in conver­ gence.

F o r convergence,

( ^ J . < 0 ; much s tr o n g e r o u tflo w i s re q u ire d a t ? r th e c e n te r th a n a t th e o u te r r a d iu s and th e d e c re a se o f th e in w a r d ly - d ir e c t­ ed p r e s s u r e g ra d ie n t w ith e le v a tio n m ust be m ost r a p id a t th e c e n te r . Hence t h e r e m ust be a r a p id in c r e a s e i n te m p e ra tu re tow ard th e c e n te r , which i s p ro v id e d i n th e s a tu r a te d a i r by th e d i f f e r e n t i a l warming due to f r i c t i o n a l d i s s i p a t i o n and th e o r i g i n a l ©g d i s t r i b u t i o n .

T h is in c r e a s e i n

te m p e ra tu re tow ard th e c e n te r m ust be m a in ta in e d w ith in th e d ry a i r i n o r­ d e r to s u s ta in convergence. The f e a t u r e s o f th e s o lu tio n d is c u s s e d above have been in c o rp o ra te d i n a sch em atic model (F ig u re 1 2 ) , showing a c ro s s s e c tio n o f t h e flow and th e accompanying th e rm a l s t r u c t u r e .

S in c e th e boundary c o n d itio n s f o r p , T and

v a re s im ila r w ith in 80 km o f th e c e n t e r , i t would seem l i k e l y t h a t th e c h a r a c t e r i s t i c s o f th e s o lu tio n in th e a sc e n d in g a i r m ight be extended from th e c e n te r o f th e cy clo n e t o a t l e a s t t h i s r a d i u s .

The d ia m e te r o f th e

re g io n o f downward m otion c an n o t be e s tim a te d from th e p r e s e n t d a ta .

The

v e r t i c a l te m p e ra tu re s t r u c t u r e i n F ig u re 12 h a s been i n f e r r e d from th e ©g and © d i s t r i b u t i o n .

The p ro m in en t c h a r a c t e r i s t i c o f th e v e r t i c a l te m p e ra-

0,-1 0,-4 0-3 ,8'

N

NJ

,

0-1 -H

•e*3

,0E“l B

F ric tio n a l L ayer.

A

VERTICAL TEMPERATURE DISTRIBUTION ABOVE THE FRICTIONAL LAYER

r—► VERTICAL CROSS SECTION OF THE FLOW

r HORIZONTAL "PRESSURE AND TEMPERATURE DISTRIBUTION

F ig* 1 2 * S chem atic diagram o f th e flo w n e a r the' c e n te r o f th e c y clo n e and th e c o rre sp o n d in g p re s s u re and te m p e ra tu re d is t r i b u t i o n *

42 t u r e curve w i l l be th e in v e r s io n i n th e s tro n g o u tflo w , a c c e n tu a te d by a te m p e ra tu re d is c o n t in u ity a t th e ju n c tu r e o f th e s a tu r a te d and u n s a tu r a te d flo w ;

th e a i r o v e rly in g th e in v e rs io n i s r e l a t i v e l y u n s ta b l e .

These cu rv es

were c o n s tr u c te d w ith r e f e r e n c e t o th e a c tu a l o b s e r v a tio n s j an d , ex cep t f o r th e d is c o n t i n u i t y i n te m p e ra tu re i n th e m odel, th e g e n e ra l f e a t u r e s o f th e s e cu rv es a g re e w ith th e a v a ila b le d a ta .

The p u rp o se o f th e s e c u rv e s i s mere­

l y to show t h a t th e o bserved th e rm a l s tr u c t u r e may be r e c o n c ile d w ith th e deduced c h a r a c t e r i s t i c s o f th e flo w . For com pleteness o f th e model i t i s d e s ir a b le t o d is c u s s s e v e r a l p r i n ­ c i p l e ^ a c c e p te d by some i n v e s t i g a t o r s , to in d i c a t e t h a t t h e assu m p tio n s employed i n t h i s stu d y impose so g r e a t a r e s t r i c t i o n on th e flo w as to make a com plete s o lu tio n im p o s s ib le .

Sawyer (1947) h a s n o te d t h a t th e assum ptions

o f a s te a d y s t a t e , c i r c u l a r symmetry and no f r i c t i o n , im ply some l i m i t a t i o n on th e d i s t r i b u t i o n o f th e v o r t i c i t y .

S tu d ie s o f th e D u r s t - S u t c l i f f e e f f e c t

( f o r exam ple, K o e h ler, 1947) m ight be i n t e r p r e t e d as i n d i c a t i n g t h a t th e o u tflo w produced by th e p ro c e s s e s c o n sid e re d i n t h i s stu d y c o u ld n o t be a s g r e a t as th e in flo w i n th e f r i c t i o n la y e r . The problem a r i s i n g from th e v o r t i c i t y d i s t r i b u t i o n may be seen from th e fo llo w in g :

Under th e o r i g i n a l assu m p tio n s f o r th e flo w above th e f r i c ­

t i o n l a y e r , e q u a tio n ( 1 ) may be w r itte n U = ~W - 2 * s : where jr r S u tc liffe . EL

“ 0.

(18)

, th e v e r t i c a l component o f th e a b s o lu te v o r t i c i t y i s eq u al to X )•

T h is i s th e form o f th e e q u a tio n employed by D u rst and

From (18) i t ap p ears t h a t u becomes zero a s w v a n is h e s , u n le s s I n th e flo w d e p ic te d i n F ig u re 12, a s tr e a m lin e , such as t h a t

43 la b e l e d 0 = 1 , p a s s e s th ro u g h a p o in t where w = 0 , b u t u> 0 , so t h a t

^

m ust a ls o be zero a t t h i s p o in t. To i l l u s t r a t e t h a t such a v o r t i c i t y d i s t r i b u t i o n i s p o s s ib le , con­ s i d e r e q u a tio n ( 1 ) a s i t a p p lie s above th e f r i c t i o n l a y e r

Uj£lf + utctof

+

jr

u jt

4. U >

=

r

o

and s in c e u =* — th e e q u a tio n may be w r itt e n dt

-

and i n i n te g r a te d form =

'f'V +

c o n s ta n t

(19)

Z.

where th e c o n s ta n t f o r a s tre a m lin e i n t h e a scen d in g a i r i s d eterm in ed by th e v a lu e o f v a t th e to p o f th e f r i c t i o n a l l a y e r .

I n t h i s form th e equa­

t i o n i s more e a s i l y re c o g n iz e d a s a s ta te m e n t o f th e c o n s e rv a tio n o f angu­ l a r momentum.

From (19) and. from th e c o n s id e r a tio n o f th e boundary condi­

t io n s f o r v , i t w i l l be o b serv ed t h a t ih - + -

w ith in th e s a tu r a te d a i r , ? as th e ju n c tu r e o f th e s a tu r a te d and u n s a tu r a te d a i r i s a p p ro a ch e d . ■■■

Hence

one needs o n ly to impose th e boundary c o n d itio n v = - A s f o r th e d ry a i r descen d in g a t th e c e n te r i n o r d e r to s a t i s f y th e c o n d itio n s o f c o n tin u ity f o r v and th e v a n is h in g o f £ &.

The re q u ire m e n t on th e v o r t i c i t y m erely

r e p r e s e n ts a l i m i t a t i o n on th e ad m issab le boundary c o n d itio n s f o r th e de­ scen d in g a i r . The fo re g o in g d is c u s s io n in d ic a t e s some o f th e d i f f i c u l t i e s en co u n te re d

i n computing th e amount o f o u tflo w from th e f l u r s t - S u t c l i f f e e q u a tio n ( 1 8 ) . W hile under o u r a ssu m p tio n s, e q u a tio n ( 18 ) m ust h o ld , i t does n o t o f f e r a u s e f u l method f o r com puting u , s in c e none o f th e term s on th e r i g h t i s known w ith s u f f i c i e n t p r e c is io n f o r th e c o m p u ta tio n .

F o r exam ple, i n th e

model

where V r e p r e s e n ts th e t o t a l wind speed and en erg y p e r u n i t m ass.

t u 3 i

+

“'

W

(2 0 )

, t h e r e f o r e , th e k i n e t i c

By th e h y d r o s ta tic e q u a tio n ,

s

*

i t

th e in c r e a s e in p o t e n t i a l energy i s v e ry n e a r ly b a la n c e d by th e work done by th e v e r t i c a l p r e s s u r e - g r a d ie n t f o r c e .

The work done by th e f r i c t i o n a l

f o r c e s , re p re s e n te d by th e term J i ( u. ^ 'tc + yj- ^ ) i s n e g a tiv e i n th e /> ' Hfe f f r i c t i o n a l la y e r o f th e model and zero i n th e re g io n ab o v e. I t i s a p p a re n t t h a t i n a c o o rd in a te system moving w ith th e a i r p a r c e l , a g a in in k i n e t i c energy i s b ro u g h t ab o u t by th e work done on th e p a r c e l by th e h o r iz o n ta l p r e s s u r e - g r a d ie n t f o r c e . - -fa

I n th e f r i c t i o n a l la y e r th e work term i s p o s i t i v e ,

> Oj w h ile above, i n g e n e r a l, work i s done by th e a i r p a r c e l

a g a in s t th e p r e s s u r e f o r c e s , - ^

4. 0 .

I n th e s te a d y s t a t e th e problem

o f m aintenance o f th e k i n e t i c energy i s re s o lv e d i n t o th e problem o f main­ te n a n c e o f th e h o r iz o n ta l p r e s s u r e d i s t r i b u t i o n w ith in flo w .

The h o riz o n -

46 t a l p r e s s u r e d i s t r i b u t i o n i s b u t i n d i r e c t l y r e l a t e d t o th e d i s t r i b u t i o n o f p o t e n t i a l e n e rg y .

The r e l a t i o n s h i p i s co m p lic ate d by th e e f f e c t s o f com press­

i b i l i t y and n o n a d ia b a tic h e a tin g and w i l l o n ly be d eterm in ed by th e d e s c r ip ­ t i o n o f th e flow p ro c e s s e s from th e s u rfa c e to th e to p o f th e atm osphere. However, th e e s s e n t i a l f e a t u r e o f th e p ro c e s s f o r m a in ta in in g th e h o riz o n ­ t a l p r e s s u r e d i s t r i b u t i o n h a s been in d i c a te d .

I t i s th e r a d i a l d if f e r e n c e

o f warming, w ith g r e a te r warming and h ig h e r te m p e ra tu re s a t th e c e n te r .

In ­

flow i n th e f r i c t i o n a l la y e r i s accompanied ty h e a tin g due b o th t o conden­ s a tio n and t o f r i c t i o n a l d i s s i p a t i o n , b o th te n d in g to produce g r e a t e s t warm­ in g a t th e c e n t e r .

The accompanying r e v e r s a l o f th e h o r i z o n ta l p re s s u re

g r a d ie n t a t th e c e n te r i s s u f f i c i e n t l y g r e a t to r e q u ir e s tr o n g h o r iz o n ta l o u tflo w and downward m otion o v e r th e c e n te r o f th e c y c lo n e , so t h a t a t h i r d h e a tin g e f f e c t w i l l o ccu r a t th e

c e n te r due to th e com pression o f d escen d in g

a ir. The c o n d itio n o f energy c o n se rv a tio n h a s a lre a d y been imposed upon th e s o lu tio n , and th e en erg y tra n s fo rm a tio n s a re i m p l i c i t i n th e d e s c r ip tio n o f th e s o lu tio n ; so t h a t no new in fo rm a tio n i s g ain ed by th e d is c u s s io n o f en erg y tr a n s f o r m a tio n s .

The m agnitudes o f th e energy changes were computed

m erely t o a id i n th e com parison o f th e p r e s e n t model w ith o th e r sy ste m s. An i n c i d e n t a l purpose i s t o make c le a r th e danger o f em ploying term s such a s " th e e n erg y o f c y clo n e s" when r e f e r r i n g s o le l y to th e k i n e t i c en erg y . I n th e s te a d y s t a t e th e r e l a t i o n s h i p o f th e changes o f th e rm a l en erg y and m ech a n ic al energy i n a flow p ro c e ss a r e m ost e a s i l y e x p re sse d by th e u se o f th e e n th alp y -ch a n g e e q u a tio n commonly employed i n e n g in e e rin g p ro b ­ le m s.

Addding th e f i r s t law o f therm odynam ics (5) and th e e q u a tio n o f

k i n e t i c en erg y ( 20 ) one o b ta in s :

47

*?

■ 4

e lt

* + £ + C v T) =

^

4 $ - + - ^ - ( U 'TV+ * r * J

c ii

f

(a)

32

s in c e by th e a ssu m p tio n s o f a s te a d y s t a t e and c i r c u l a r symmetry, d£ = u

* w ^_E

a t

a t

.

o z

W ith th e a id o f th e id e a l- g a s r e l a t i o n s h i p (21) becomes

* C? J )

where C

r

=

+ 1 - ‘L - { U T r + V ' T f )

^

(22)

i s th e s p e c i f i c h e a t o f a i r a t c o n s ta n t p re s s u re and C T i s c a l l e d ir

th e e n th a lp y . The n e t change i n any o f th e form s o f en erg y w ith in th e f r i c t i o n a l la y e r i s found by i n t e g r a t i n g o v e r th e volume o f th e f r i c t i o n a l l a y e r .

F o r exam ple,

th e n e t change i n e n th a lp y i s g iv en by th e fo llo w in g e x p re ss io n :

£

4 1 ^ 6

r

d t

f r i c t i o n a l la y e r where d 6

i s th e volume e le m e n t.

F o r th e s te a d y s t a t e and w ith th e a id o f

th e e q u a tio n o f c o n tin u ity , t h i s volume i n t e g r a l may be tra n sfo rm e d in to a s u rfa c e i n t e g r a l by G a u ss 's theorem ,

where VQ i s th e norm al component o f v e l o c i t y d ir e d te d outw ard; and dA th e elem ent o f a r e a .

I n th e s te a d y s t a t e th e n e t r a t e o f change i n th e e n tro p y

o f th e flo w i s th e n e t f l u x o f e n tro p y th ro u g h th e boundary o f th e system . F or th e f r i c t i o n a l la y e r o f th e m odel, th e i n t e g r a l becomes - 5 0 0 km

j J C p jO Vn T d A

= J C e, j > u / T ( i n < ') d r + J C ?/> u .T [ z n x i-* ii> ',) d i

f r i c t i o n a l la y e r

( £ s / .y / f m )

( T =■

km)

The changes o f k i n e t i c e n e rg y , p o t e n t i a l e n e rg y and l a t e n t h e a t added i n th e f r i c t i o n a l la y e r a re computed i n s im ila r f a s h io n .

However, th e rem ain in g

term s i n e q u a tio n ( 22 ) , r e p r e s e n tin g th e r a t e o f work a g a in s t f r i c t i o n a l f o r c e s , p ro v id e no c o n tr ib u tio n when i n t e g r a t e d o v er th e volume o f th e f r ic ti o n a l la y e r.

F or upon tra n s fo rm in g by G auss’ s theorem , one o b ta in s :

f r i c t i o n a l la y e r s in c e

to p and bottom o f f r i c t i o n a l la y e r

a 0 a t th e to p o f th e f r i c t i o n a l l a y e r and u = v - 0 a t

th e s u r f a c e . E s tim a te s o f th e m agnitude o f th e en erg y changes i n th e re g io n above th e f r i c t i o n a l la y e r a re a ls o p o s s i b l e .

The te m p e ra tu re o f th e s t r a t o ­

sp h ere su rro u n d in g a c t u a l c y clo n e s does n o t v a ry g r e a t l y from i t s v a lu e , in th e u n d is tu rb e d s t a t e , o f 200° K e lv in .

A re a s o n a b le e s tim a te o f th e

a v erag e te m p e ratu re o f th e a i r a t o u tflo w would l i e i n th e ran g e o f from 250° to 200°.

The te m p e ra tu re a t in flo w i n t h i s la y e r i s ap p ro x im ately

290 ° so t h a t th e c o rre sp o n d in g a v e ra g e e n th a lp y change would be from

ft 8 A x 10° to 9 x 10 e rg s p e r gram o f a i r .

The av erag e m o istu re l o s s p e r

gram o f a i r must be l e s s th a n th e m o is tu re c o n te n t a t in flo w o f 0 .015 grams and i s p ro b a b ly g r e a te r th a n 0 .0 0 5 gram s, c o rre sp o n d in g t o a ran g e o f added l a t e n t h e a t p e r gram o f a i r o f from 1 x 10

to 3 x 10

e rg s.

The k i n e t i c

energy w i l l d e c re a se i n t h i s re g io n and s in c e th e v e lo c i ty can n o t v a n is h th e l i m i t to th e av erag e change w i l l be t h e av erag e v a lu e o f th e k i n e t i c energy o f th e flow le a v in g t h e f r i c t i o n a l l a y e r (10^ e rg s gm- ^ ) .

These

e s tim a te s o f average changes o f en erg y may be c o n v erted i n t o r a t e s o f change by m u ltip ly in g by th e r a t e o f mass flo w i n th e model ( 2 .1 x 10 ^

49 grams sec"'*').

The rem ain in g te rm o f e q u a tio n ( 2 2 ) , th e r a t e o f change o f

p o t e n t i a l en erg y , i s th e r e f o r e o f th e same o r d e r o f m agnitude a s th e sum o f th e r a t e o f e n th a lp y change and th e r a t e a t which h e a t i s added. The v a lu e s o f th e r a t e o f change o f t h e v a r io u s form s o f en erg y o f e q u a tio n (22) a re g iv en in T able IV .

V alu es f o r th e f r i c t i o n a l l a y e r o f th e

model were computed as d e s c rib e d above; th e re m a in in g v a lu e s a r e e s tim a te d r a t e s o f change.

Average r a t e s o f change o f e n erg y p e r gram o f a i r i n th e

volume a re given a s w e ll a s th e n e t change o r en erg y f l u x . T able IV .

R ate o f change o f en erg y o f th e flo w (e n e rg y flu x ) i n th e cyclone m odel.

Above O v e ra ll F r i c t i o n a l L ayer F lu x Avg. F lu x Avg. F lu x Avg. Form o f Energy e rg s sec ”1 e rg s sec x e rg s sec ”1 e rg s se c ”1 e rg s s e c ”1 e rg s ”1 gm"1 gm- 1 gm"1 F r i c t i o n a l L ayer

L a te n t H eat

8 10 *420

6 .8 x 102

50

* * - 2 .1 x 1 0 20

K in e tic

1 .4 x 1 0 20

1 .1 x ' 102

d is s ip a tio n

.84

.6 8 x 102

X

1 020

K

-1 1 .3 x ic £

1

- H .2 x 10^0

O

E n th a lp y

190 x 1020 29 . x 102

H8

1 7 .3 x 102

Q

2 1 .5 x 1020

r

P o te n tia l

X

10 20

8x10s

210 x 10 20 27. x 102 -1 5 0 x i o 2o 60 x 1020

- 19Lx 102 8 x 102

The d i s s i p a t i o n does n o t a p p e a r i n t h i s en erg y e q u a tio n ; th e tra n s fo rm a tio n , by f r i c t i o n a l d is s ip a tio n , o f k i n e t i c e n erg y i n t o h e a t i s in c lu d e d in th e above f i g u r e s . However, th e v a lu e o f th e d i s s i p a t i o n i s g iv en f o r com parison. *^E)im iting v a lu e o f f lu x o f k i n e t i c e n e rg y .

50 From T able IV i t may be seen t h a t th e r a t e o f k i n e t i c en erg y change i s one o rd e r o f m agnitude l e s s th a n th e m agnitude o f th e r a t e o f change o f th e o t h e r form s o f e n e rg y .

I f th e volume o ccu p ied by th e cy clo n e i s c o n sid e re d

a s a thermodynamic engine tra n s fo rm in g th e l a t e n t h e a t i n t o k i n e t i c energy o f th e flo w , t h i s tra n s fo rm a tio n i s v e ry i n e f f i c i e n t .

O nly th e r a d i a l d i f ­

f e re n c e in h e a tin g d e riv e d from th e l a t e n t h e a t a id s th e m aintenance o f th e h o r iz o n ta l p re s s u re g ra d ie n t and i s r e l a t e d to th e p ro d u c tio n o f k i n e t i c energy o f th e flo w .

A lthough th e k i n e t i c en erg y produced i n th e flo w in th e

f r i c t i o n a l la y e r i s r e l a t i v e l y l a r g e f o r a tm o sp h e ric p r o c e s s e s , i t i s n o t a p p a re n t t h a t , f o r th e cyclone a s a w hole, th e n e t change o r th e n e t f lu x o f k i n e t i c energy w i l l be p o s i t i v e . I t i s i n t e r e s t i n g to n o t e . t h a t , by in tr o d u c in g th e o b serv ed l i m i t to th e e n th a lp y change, a l i m i t to th e d ep th ( s u r f a c e p r e s s u r e d if f e r e n c e ) o f th e cyclone f o r a given r a d i a l d if f e r e n c e o f ©g i s a ls o im p led .

As p o in te d

o u t i n th e p re c e d in g s e c tio n , a v e ry s l i g h t r a d i a l ©g d i f f e r e n c e , i f main­ ta in e d i n a deep l a y e r , would be s u f f i c i e n t t o e q u a liz e th e s u r f a c e p re s s u re d if f e r e n c e o f th e c y c lo n e .

O b se rv a tio n l i m i t s t h e th ic k n e s s o f t h i s la y e r

s in c e th e o u tflo w does n o t o c c u r a t any a r b i t r a r y te m p e ra tu re down to zero d e g re e s K e lv in , b u t i s lim ite d a t a te m p e ra tu re n e a r 2 0 0 °.

T h is i s n o t to

s u g g e s t a p h y s ic a l li m i t a t i o n t o th e p ro c e s s b u t a l i m i t a t i o n t h a t would be r e q u ir e d by a boundary c o n d itio n in tro d u c e d t o d u p lic a te th e ob serv ed co n d i­ t i o n s in th e atm ospheric p r o c e s s . i n o th e r w ays.

T h is l i m i t i n g c o n d itio n m ig h t be s t a t e d

For example i f i t were r e q u i r e d t h a t th e o u tflo w ta k e p la c e

p r i n c i p a l l y w ith in th e tro p o s p h e re le a v in g th e s tr a to s p h e r e u n d is tu rb e d , th e same e f f e c t i s p ro d u ced .

51 10.

Comparison w ith o b s e rv a tio n s The m ajor f e a tu r e s o f th e flow n e a r th e c e n te r o f th e cyclone model

have a lre a d y been review ed i n s e c tio n 8 .

They a r e :

s tr o n g o u tflo w i n a

n arrow re g io n which i s i n th e shape o f a bowl w ith f l a r e d s id e s j th e o u te r p o r ti o n o f th e re g io n o f s tro n g o u tflo w i s a re g io n o f h o r iz o n ta l co n v er­ gence and upward motion} th e in n e r re g io n i s one o f h o r iz o n ta l divergence} and n e a r th e c e n te r , th e d iv e rg e n ce i s accom panied by downward m otio n . f e a tu r e s a re c o n s is te n t w ith th e a v a il a b le o b s e r v a tio n s .

These

The c lo u d le s s s k ie s

and o c c a s io n a l h ig h e r te m p e ra tu re s o b serv ed i n th e c e n t r a l "ey e 11 have lo n g l e d m e te o ro lo g is ts to s p e c u la te on th e p o s s i b i l i t y o f downward m o tio n .

R ie h l

( 1948 a) has a n aly z e d an a e r o lo g ic a l sounding ta k e n i n th e c e n te r o f a h u r r i ­ cane to d em onstrate th e p r o b a b il ity o f d e sc e n d in g a i r .

The cloud p a t t e r n

y i e l d s f u r t h e r i n d i r e c t evidence o f t h i s flo w p a t t e r n .

I n some in s ta n c e s th e

c e n t r a l c le a r a re a in c r e a s e s in d ia m e te r w ith e le v a tio n (W exler, 1945 and U n ite d S ta te s W eather B ureau, 1 9 4 8 ).

The edge o f th e c lo u d s su rro u n d in g th e

c e n t r a l c le a r re g io n i s u s u a lly v e ry s h a rp , which would seem to i n d i c a t e a v e ry a b ru p t t r a n s i t i o n from downward to upward m o tio n . The o n ly e v id e n t f e a tu r e o f th e s o lu tio n a t th e o u te r r a d iu s o f th e cy­ c lo n e model i s th e o s c i l l a t i o n o f th e flo w .

However, t h i s r e s u l t c an n o t be

d e f i n i t e l y e s ta b lis h e d , f o r o s c i l l a t i o n i s a c c e n tu a te d by th e method o f com­ p u t a t i o n , and w ith v i o l e n t o s c i l l a t i o n th e m ethod o f com putation f a i l s .

In

th e a c tu a l cyclone some evidence o f o s c i l l a t i o n may be deduced from th e cloud p a tte rn .

I n th e o u te r re g io n o f th e c y c lo n e , a lth o u g h th e r e seems t o be a

g e n e r a l convergence o f th e s u rfa c e w ind, more o r l e s s i s o l a t e d to w e rin g cumulus a p p e a r.

N earer th e c e n te r th e clo u d s o f te n o c c u r in bands s p i r a l i n g

in tow ard th e c e n te r o f cyclone (M aynard, 1945)•

These clo u d p a t t e r n s prob­

a b ly r e p r e s e n t two d i s t i n c t c la s s e s o f o s c i l l a t i o n s n e i t h e r o f which c o rre s ­

52 ponds e x a c tly t o th e o s c i l l a t i o n s observ ed i n th e s o lu ti o n . One f u r t h e r p o in t i n re g a rd to th e o s c i l l a t i o n seems p e r t i n e n t .

When­

e v e r in flo w o c c u rs i n th e s e o s c i l l a t i o n s i t r e p r e s e n ts th e a d v e c tio n o f po­ t e n t i a l l y c o ld e r o v e r p o t e n t i a l l y warmer a i r . o v e rtu rn in g would be e x p e c te d .

I f t h i s a d v e c tio n i s v e ry l a r g e ,

The s c a le o f m otion re p re s e n te d by t h i s o v e r­

tu r n in g would p ro b a b ly be o f th e m agnitude o f th e m otions which were tre a te d a s f r i c t i o n a l e d d ie s o r tu rb u le n c e r a t h e r th a n a s l a r g e - s c a l e mean m otion. I t i s u n l i k e l y , t h e r e f o r e , t h a t th e flow o u ts id e th e re g io n o f o v e rc a s t s k ie s can be s a t i s f a c t o r i l y t r e a t e d by assum ing e i t h e r s a t u r a t i o n o r n e g lig ib le fric tio n .

M oreover, i t i s p o s s ib le t h a t th e d i s s i p a t i o n o f v e r t i c a l m otion

by h y d r o s t a t i c a l l y u n s ta b le e d d ie s may be an im p o rta n t p a r t o f th e atmos­ p h e r ic c y c lo n ic p r o c e s s .

I n th e case c o n sid e re d i n th e m odel, th e tem p era­

t u r e o f th e a sc e n d in g a i r w i l l be d eterm in ed by th e m o is t- a d ia b a tic p ro c e s s , and th e te m p e ra tu re a l o f t a t th e o u te r r a d iu s o f th e c y clo n e w i l l have a d e f­ i n i t e minimum v a lu e depending upon th e boundary c o n d itio n s f o r 0 jj,

I n th e

u n d is tu rb e d atm osphere o f th e t r o p i c s , th e v e r t i c a l la p s e r a t e o f te m p e ra tu re i s g r e a t e r th a n th e m o is t- a d ia b a ti c j and i f th e upward m otion c e a s e s , th e a i r te m p e ra tu re above t h i s p o in t w i l l be d eterm in ed by th e u n d is tu rb e d a i r and a g r e a t e r r a d i a l te m p e ra tu re g ra d ie n t would be p o s s ib le th a n co u ld be a t t a i n e d u n d e r t h e s im p lify in g assum ptions made i n th e c o n s tr u c tio n o f th e m odel. As d is c u s s e d i n s e c tio n 8 th e q u a l i t a t i v e f e a tu r e s o f th e ob serv ed th e rm a l s t r u c t u r e a r e c o n s is te n t w ith th e flo w p a t t e r n o f th e m odel.

The

v e r t i c a l te m p e ra tu re s tr u c t u r e o f th e model (F ig u re 13) may be compared w ith a c t u a l a e r o lo g ic a l so undings made in th e h u rric a n e o f 18 -1 9 O ctober 1944» w hich have been p l o t t e d on a tephigram (F ig u re 1 4 ).

The sounding a t Miami

was ta k e n a t ap p ro x im a te ly 300 km from th e c e n te r o f th e sto rm and th e sound­ in g a t Tampa in th e calm c e n t r a l e y e .

The m o is t- a d ia b a tic curve drawn on

Jti

fO

3

53 t h i s diagram ( la b e le d a s 355 ° p s e u d o - e q u iv a le n t- p o te n tia l te m p e ra tu re ) re p ­ r e s e n ts th e curve o f a d ia b a tic a s c e n t o f th e s u rfa c e a i r a t th e c e n te r o f th e c y c lo n e , a s d eterm in ed from th e s u rfa c e o b s e rv a tio n s a t Tampaj tem pera­ t u r e , 74° F a h re n h e it; dew p o i n t , 74° F a h r e n h e it; p r e s s u r e , 967 mb.

The th r e e

cu rv es o f F ig u re 14 p ro v id e a com parison o f th e h e a tin g e f f e c t s o f condensa­ ti o n and o f downward m o tio n .

I f t h i s a n a ly s is i s c o r r e c t , th e two e f f e c t s

a re a p p ro x im a te ly o f th e same m agnitude, f o r th e m o is t- a d ia b a tic curve l i e s midway betw een th e u p p er p o r tio n o f th e Tampa and th e Miami so u n d in g s. The te m p e ra tu re d if f e r e n c e in th e s a tu r a te d a sc e n d in g a i r in th e model i s alm o st o f th e same m agnitude a s th e v a lu e i n f e r r e d f o r th e F lo r id a h u r r i ­ cane (com pare F ig u re s 13 and 1 4 ) .

I f we make th e f u r t h e r assum ption t h a t

th e te m p e ra tu re d if f e r e n c e produced in th e u n s a tu r a te d a i r w i l l be o f th e same m agnitude a s i n th e F lo r id a h u r r ic a n e , i t i s p o s s ib le t o e s tim a te th e a l t i t u d e a t which th e s u rfa c e p r e s s u r e d if f e r e n c e w i l l be e q u a liz e d .

As

H aurw itz (1935) h a s n o te d t h i s a l t i t u d e w i l l be g iv en by th e e x p re s s io n ,

t2 3 )

where th e s u b s c r ip t 1 r e f e r s t o th e o u te r r a d iu s and 2 r e f e r s to th e c e n te r o f c y c lo n e ; T i s th e mean te m p e ra tu re o f th e a i r column; P i s th e s u rfa c e p r e s s u r e ; H th e a l t i t u d e a t which th e s u rfa c e p re s s u re d if f e r e n c e i s e q u al­ iz e d .^ - The approxim ate v a lu e s o f th e term s a re T„ - T- = 10° C .; lo g ~ • * p2 0 .0 7 ; and ( o b ta in e d from th e therm odynamic diagram a f t e r some t r i a l and From th e c o n s id e r a tio n o f th e r a d i a l te m p e ra tu re and p r e s s u r e d i s t r i b u t i o n a t th e to p o f th e f r i c t i o n a l l a y e r , th e h o r iz o n ta l p r e s s u r e g ra d ie n t i s b e lie v e d to r e v e r s e a t lo w er l e v e l s n e a r th e c e n te r o f th e c y c lo n e . The h e ig h t a t which th e o v e r a l l p re s s u re d if f e r e n c e i s e q u a liz e d co rresp o n d s to th e h e ig h t a t which th e p r e s s u r e g r a d ie n t r e v e r s e s a t th e o u te r r a d iu s o f th e c y c lo n e .

54e rro r)

(260 )^j

hence H i s a p p ro x im a te ly 14 km.

The h e ig h t o f e q u a li z a tio n o f th e h o r i z o n t a l p re s s u re d if f e r e n c e i n a cyclone h a s n o t been e x a c tly d eterm in ed by o b s e r v a tio n .

O nly two soundings

have been made a t th e c e n te r o f a h u r r ic a n e (Sim pson, 1 9 4 7 ), and th e h e ig h t o f e q u a liz a tio n o f th e p re s s u re g r a d ie n t i n th e s e two c a se s i s o b scu red by th e asymmetry o f th e p re s s u re f i e l d a t h ig h l e v e l s .

However, i t seems s a fe

to p la c e th e h e ig h t o f e q u a liz a tio n o f th e p r e s s u r e d if f e r e n c e below 14 km. The u n d is tu rb e d s tr a to s p h e r e a t th e o u te r r a d iu s o f th e cyclone h a s been more f r e q u e n tly o b s e rv e d , and t h i s f a c t would a ls o seem to im ply a l i m i t to th e v e r t i c a l e x te n t o f th e c y c lo n ic p ro c e s s i n th e v i c i n i t y o f th e tr o p o p a u se . C e rta in d e t a i l s o f th e s o lu t io n depend q u ite c r i t i c a l l y upon th e assump­ tio n s .

The assu m p tio n s o f a s te a d y s t a t e , c i r c u l a r symmetry and n e g l i g i b l e

f r i c t i o n p e rm it o n ly a v e ry r e s t r i c t e d ty p e o f flo w . i s t i c s o f t h i s flow a r e :

Some o f th e c h a r a c te r ­

th e t a n g e n t i a l component v e l o c i t y must become

a n tic y c lo n ic b e fo re th e o u tflo w from th e cy clo n e i s com plete; a r e s t r i c t e d d i s t r i b u t i o n o f v o r t i c i t y i n th e d e sc e n d in g a i r ; th e o n ly mechanism which would p e rm it la r g e s c a le o u tflo w from th e cy clo n e i s th e d e c re a se and r e ­ v e r s a l o f th e h o r iz o n ta l p re s s u re g r a d ie n t f o r c e w ith e le v a tio n .

I n each

c a se good re a s o n s can be advanced f o r b e lie v in g t h a t th e s e c h a r a c t e r i s t i c s a re o b ta in e d a s a r e s u l t o f th e r e s t r i c t i v e n a tu r e o f th e a ssu m p tio n s.

The

a n tic y c lo n ic v e l o c i t y i n th e m odel o c c u rs i n a re g io n o f s tro n g r a d i a l v e l­ o c i t y g r a d ie n t (F ig u re 9) and. th e f r i c t i o n a l s t r e s s e s u n d er th e s e c o n d itio n s , would be e x p ec te d t o modify t h i s v e l o c i t y d i s t r i b u t i o n .

I t seems u n lik e ly

t h a t th e atm osphere i n th e u n d is tu r b e d s t a t e would p o s se ss any p a r t i c u l a r d is trib u tio n o f v o r tic ity .

The g r e a te r freedom o f th e e q u a tio n s i n th e

g e n e ra l case c o n c e iv a b ly p e rm it o th e r d i s t r i b u t i o n s o f v o r t i c i t y and o th e r

55 mechanisms o f o u tflo w (Sawyer 1 9 4 7 ). The r e s t r i c t i o n s do em phasize th e o b s e rv a b le f a c t t h a t th e c i r c u l a r l y sym m etric cyclone i n th e s te a d y s t a t e must be a r a r e a tm o sp h eric phenomenon, b u t th e r e s t r i c t i o n s do n o t d e m o n strate a l i m i t to th e g e n e ra l a p p l i c a b i l i t y o f th e p h y s ic a l p ro c e s s e s c o n s id e re d .

In p a r t i c u l a r , th e s e r e s t r i c t i o n s do

n o t g iv e any re a so n to doubt th e im p o rtan ce o f th e o u tw a rd ly -d ir e c te d p r e s ­ su re g r a d ie n t fo rc e in th e mechanism o f o u tflo w .

U n fo rtu n a te ly , i f th e r e ­

gion o f o u tw a rd ly -d ir e c te d p r e s s u r e g r a d ie n t i s narrow , a s in d ic a te d by th e s o lu tio n , th e a b s o lu te p re s s u re d if f e r e n c e i s s l i g h t and i t i s u n lik e ly to be v e r i f i e d by o b s e r v a tio n . 11.

C onclusions The a tte m p te d c o n s tr u c tio n o f a cyclone model by i n t e g r a t i n g th e equa­

t i o n s f o r th e flow h a s proved to be p a r t l y s u c c e s s f u l.

Some c h a r a c t e r i s t i c s

o f th e s o lu tio n were o b ta in e d and compared w ith o b s e r v a tio n s , le a d in g to th e c o n c lu s io n s summarized below: 1.

The flo w i n th e f r i c t i o n a l la y e r d e m o n stra te s th e p ro c e s s by which

a warmer c e n te r o f th e cyclone may be produced by a mechanism a c tin g i n th e lo w e s t la y e r s o f th e atm osphere o f th e t r o p i c s .

The r e l e a s e o f l a t e n t h e a t

by th e c o n d en sa tio n o f w a ter v a p o r and th e h e a t added by f r i c t i o n a l d i s s i ­ p a tio n a re b o th im p o rta n t in p ro d u c in g t h i s r a d i a l te m p e ra tu re d if f e r e n c e a t th e to p o f th e f r i c t i o n a l l a y e r .

A lthough c o n s e rv a tiv e e s tim a te s o f th e

r a d i a l d if f e r e n c e i n h e a tin g were made, th e r a d i a l te m p e ra tu re d if f e r e n c e in th e ascen d in g a i r o f th e model i s o f th e same m agnitude a s t h a t in f e r r e d f o r th e a sc e n d in g a i r i n an a c t u a l c y c lo n e .

I n th e a c tu a l c y c lo n e , wanning by

com pression o f d escen d in g a i r a p p ea rs to c r e a te an a d d it io n a l r a d i a l tem per­ a tu r e d if f e r e n c e o f a p p ro x im a te ly th e same m agnitude.

56 2.

The r a d i a l d if f e r e n c e i n te m p e ra tu re produced by th e a d d itio n o f

l a t e n t h e a t to th e a sc e n d in g a i r i s s u f f i c i e n t i n i t s e l f to re v e r s e th e h o r i­ z o n ta l p re s s u re g r a d ie n t o f th e model i f a s c e n t i s m a in ta in e d f o r a d e p th o f 20 km (computed from e q u a tio n (23)].

T h is h e ig h t would seem to exceed th e

l i m i t i n g h e ig h t in d ic a te d by o b s e r v a tio n .

I f t h i s l i m i t to th e h e ig h t o f th e

c y clo n e i s a c c e p te d , th e n th e a d d i t i o n a l h e a tin g by d e sc e n d in g a i r i s r e q u ire d to m a in ta in a s u r f a c e p re s s u re d if f e r e n c e o f th e m agnitude o f th e m odel. 3.

The boundary c o n d itio n s d e riv e d f o r th e re g io n above th e f r i c t i o n a l

l a y e r when imposed upon th e s im p lif ie d e q u a tio n s o f flo w , r e q u ir e downward m otion o v e r th e c e n te r o f th e cy clo n e as a s o lu ti o n .

An im p o rta n t f e a t u r e o f

th e boundary c o n d itio n s i n d e te rm in in g t h i s s o lu tio n i s th e com b in atio n o f h ig h e r te m p e ra tu re s and f l a t p r e s s u r e p r o f i l e a t th e c e n te r which would r e ­ q u ir e an o u tw a rd ly -d ir e c te d p r e s s u r e g ra d ie n t a s h o r t d is ta n c e above th e s u rfa c e . A.

1

The p ro d u c tio n o f s u f f i c i e n t o u tflo w does n o t ap p ear to be a p rob ­

lem s e p a r a te from th e r e v e r s a l o f th e h o r iz o n ta l p r e s s u r e - g r a d ie n t f o r c e . I f th e p r e s s u r e - g r a d ie n t f o r c e i s d ir e c te d outw ard and o f s u f f i c i e n t magni­ tu d e , th e r e does n o t a p p ea r to be a l i m i t to th e amount o f o u tflo w . 5.

The s o lu tio n a t la r g e d is ta n c e from th e c e n te r s u g g e s ts th e o s­

c i l l a t i o n o f th e flo w .

However, i f any la r g e amount o f o s c i l l a t i o n i s ad­

m itte d (and th e cloud p a t t e r n s su g g est t h a t r e l a t i v e l y l a r g e v e r t i c a l e d d ie s a re common) th e n i t seems n e c e s s a ry to d is p e n s e w ith th e assu m p tio n o f neg­ l i g i b l e f r i c t i o n i f t h i s phenomenon i s to be t r e a t e d s a t i s f a c t o r i l y .

I t is

p o s s ib le t h a t th e s e o s c i l l a t i o n s o r e d d ie s a t t h e o u te r r a d iu s o f th e cy­ clo n e a r e an im p o rta n t p a r t o f th e c y c lo n ic p ro c e s s . ^The c h a r a c t e r i s t i c s o f th e s o lu tio n s t a t e d i n th e o rd e r o b se rv e d su g g e st an o rd e r o f cau se and e f f e c t . However th e c o r r e c t i n t e r p r e t a t i o n i s t h a t th e s e c h a r a c t e r i s t i c s a re c o n s is te n t w ith th e e q u a tio n s and th e im posed boundary c o n d itio n s .

57 6.

I f th e c y clo n e i s c o n sid e re d a therm odynam ic en g in e f o r tra n s fo rm ­

in g h e a t energy i n t o k i n e t i c energy i t s thermodynamic e f f i c i e n c y i s sm all be ca u se o n ly th e r a d i a l d if f e r e n c e i n h e a tin g a id s th e p ro c e s s and th e to ­ t a l amount o f th e h e a t added i s n o t im p o rta n t. 7.

The assum ptions made when u s in g th e s im p lif ie d e q u a tio n s o f flo w

a p p e a r to y i e l d a s a t i s f a c t o r y ap p ro x im atio n to th e c y c lo n ic p ro c e s s in th e c e n t r a l re g io n o f th e c y c lo n e .

The d i f f i c u l t i e s o f th e f i n i t e method

o f s o lu tio n p r e s e n te d by o s c i l l a t i o n s and d is c o n tin u o u s a i r p r o p e r tie s and th e d i f f i c u l t y i n fo rm u la tin g boundary c o n d itio n s f o r th e d e sc e n d in g a i r have n o t been re s o lv e d , so t h a t i t i s n o t e v id e n t t h a t th e method would y i e l d a com plete s o lu tio n even i f s m a lle r i n t e r v a l s were u sed f o r th e co m p u tatio n .

58 APPENDIX I LIST OF SYMBOLS AMD CONSTANTS Components o f th e c y l i n d r i c a l c o o rd in a te system employed r

I

h o r i z o n t a l d is ta n c e from th e c e n te r , p o s itiv e d i r e c t i o n outw ard

i ;;

sj/

h o r i z o n t a l a n g le , p o s i t i v e d ir e c tio n m easured c o u n te r-c lo c k w is e from e a s t

z

e le v a tio n above th e anemometer, p o s i t i v e d i r e c t i o n upward

.j

Components o f v e lo c i ty i n c y lin d r ic a l c o o rd in a te s , p o s i t i v e v e lo c i ty c o rre sp o n d s t o m otion i n th e p o s itiv e c o o rd in a te d i r e c t i o n

-j

u

r a d i a l component o f v e lo c i ty

v

t a n g e n t i a l component o f v e lo c ity

w

v e r t i c a l component o f v e lo c i ty

yO

d e n s it y

p

p re ssu re

T

te m p e ra tu re

T’

f i r s t appro x im atio n to th e mean te m p e ra tu re f o r th e l e v e l z = 1 .5 km

T"

second appro x im atio n to th e mean te m p e ra tu re f o r th e l e v e l z = 1 .5 km

T*

mean te m p e ra tu re o f sm a ll r a d i a l i n t e r v a l a t 1 .5 kme le v a tio n

R

gas c o n s ta n t f o r a i r = 2 .8 7 x 10^ e rg s gm~^ deg”^

Cp

s p e c i f i c h e a t f o r a i r a t c o n sta n t p r e s s u r e = 1 .0 0 2 x 10 ^

Cv

s p e c i f i c h e a t f o r a i r a t c o n sta n t

h

v e r t i c a l i n t e r v a l o f e x tr a p o la tio n * 0 .0 5 x 1 0 ^ cm

y

C o r io lis p a ra m ete r (assum ed c o n sta n t) » 0 .5 x 10“ ^ s e c ~ l o a c c e le r a ti o n o f g r a v ity * 9 8 0 .6 cm s e c ” **

(p

volume

la titu d e

q)

a n g u la r v e lo c i ty o f th e e a r t h f s r o t a t i o n

q

h e a t added to th e system (o n ly l a t e n t h e a t c o n sid e re d )

e rg s gm“ l d eg~ l

r a d i a l component o f th e eddy s t r e s s t a n g e n t i a l component o f th e eddy s t r e s s t h e s t r e s s a t th e s u rfa c e computed from S v e rd ru p 's fo rm u la s a t u r a t i o n v ap o r p re s s u re o f w a te r vapor l a t e n t h e a t o f co n d en sa tio n o f w a te r vapor t o t a l wind speed norm al component o f v e lo c i t y d ir e c te d outw ard from th e system h e ig h t a t which th e s u rfa c e p re s s u re d if f e r e n c e i s e q u a liz e d p r e s s u r e a t th e s u rfa c e v e r t i c a l component o f th e a b s o lu te v o r t i c i t y * (j y *■ v A) 3r r f i r s t a p p ro x im atio n t o th e r a d i a l p re s s u re g r a d ie n t i n th e f r i c t i o n ­ a l la y e r given by e q u a tio n (7a) th e fu n c tio n d e s c r ib in g th e r a d i a l v a r i a t i o n o f u i n th e f r i c t i o n a l la y e r

70OOX —- T Z ~ .

iT t- .S ’x /o '*

*

J

to1 ± r ± / 0 1C-W

'■7' 107v

and f-^ ( r ) i s d e fin e d by e q u a tio n ( 9 a ) . 2.

Second ap p ro x im atio n to th e p re s s u re g r a d ie n t and p r e s s u r e d i s t r i b u t i o n

a t th e to p o f th e f r i c t i o n a l l a y e r . As d e s c rib e d i n th e t e x t , th e n e x t ap p ro x im a tio n to th e h o r iz o n ta l p r e s s u r e g r a d ie n t i s given by e q u a tio n ( 9 b ) ,

A second e s tim a te , T% o f th e mean te m p e ra tu re f o r th e u p p e r boundary o f t h e f r i c t i o n a l l a y e r , i s p o s s ib le s in c e th e approxim ate p r e s s u r e a t th e c e n te r i s known from e q u a tio n (9a) •

The p r e s s u r e a t 500 km r a d iu s and

1 .5 km e l e v a tio n i s 8^0 mb a s d eterm in ed by th e mean sounding assumed a t

th is ra d iu s .

From e q u a tio n (9a) i t may b e found t h a t th e p r e s s u r e a t th e

c e n te r o f th e cyclone a t 1 .5 km e le v a tio n i s a p p ro x im ately 775 mb.

The

a i r a r r i v i n g a t th e c e n te r m ust have o r i g i n a te d n e a r th e s u r f a c e a t th e o u te r r a d iu s o f th e cyclone where p « 1000 mb} T » 30 0 °, and th e s p e c if ic h u m id ity = 17 °/oo .

The ap p ro x im atio n to th e te m p e ra tu re a t th e c e n te r ,

a s th e r e s u l t o f a d ia b a tic l i f t i n g o f th e s u rfa c e a i r from 1000 mb to 775 mb, was d eterm in ed from th e thermodynamic diagram to be 2 8 8 °.

The

mean o f th e te m p e ra tu re a t th e c e n te r and a t 500 km r a d iu s ( 290 ° ) i s th e v a lu e f o r T" s u b s t i t u t e d i n e q u a tio n ( 9 b ) . The d i s t r i b u t i o n o f v i s known and th e o n ly rem ain in g q u a n tity to be e v a lu a te d i s th e term w -2-= . 2 Z t i o n (3 1 ) , and from ( 7 a ) ,

P2

A t 1 .5 km:

* 10“* x F ( r ) , e q u aw 3** .5639 .

r + .f x /o '’

I n th e e x p re s s io n f o r w, o n ly th e d iv e rg e n c e term was c o n sid e re d s in c e computed v a lu e o f lo g p o n ly i n th e f i f t h s i g n i f i c a n t fig u re .

Hence

and th e lo g a rith m ic p r e s s u r e g r a d ie n t, e q u a tio n ( 9 b ) , becomes:

65 Upon i n t e g r a t i n g w ith r e s p e c t to r , one o b ta in s ,

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5 T here­

f o r e , v a lu e s o f i -^-E. were ta b u la te d a t i n t e r v a l s o f 10 km r a d iu s in th e p av ran g e 0 - r - 1 0 0 km and th e i n t e g r a l was approxim ated by S im pson's r u l e , 3.

The t r a j e c t o r i e s The t r a j e c t o r y i n th e v e r t i c a l c ro s s s e c tio n was o b ta in e d from th e

r e l a t i o n — = S. . dr u I n th e fo llo w in g d is c u s s io n th e e x p re s s io n f o r w, e q u a tio n (3 0 ) , i s a b b re ­ v ia te d u s in g th e n o ta tio n **= g 3

F ( r ')

where g^ (z ) i s th e f u n c tio n f o r th e v e r t i c a l v a r i a t i o n o f wj and F ( r ) was d e fin e d i n e q u a tio n ( 2 8 ) ,

S in c e u = f ^ ( r ) g^ (z) from e q u a tio n s ( 7 ) ,

( 24 ) and ( 2 5 ) , i n t h i s n o ta tio n

- C M

f- ' ’

(33)

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99

R eferen c es

C lin e , I . M ., 1926: 301 pp.

T r o p ic a l C y clo n es.

M acm illan C o ., New York.

D u rs t, C. S . , and S u t c l i f f e , R. C ., 193$: The im p o rtan ce o f v e r t i c a l m otion i n th e developm ent o f t r o p i c a l re v o lv in g sto rm s. Q u a rte rly J o u r n a l o f th e R oyal M e te o ro lo g ic a l S o c ie ty , v . 6 4 , 7 5 -8 4 . F a ir g r ie v e , J . , 1913: On th e r e l a t i o n betw een th e v e l o c i t y o f th e g ra d ie n t wind and t h a t o f th e o b serv ed w ind. G re at B r ita i n M e te o ro lo g ic a l O ffic e G eo p h y sical M emoirs, n o . 9> 189-207. K ie f e r , P . J . , 1941: The thermodynamic p r o p e r tie s o f w ater and w ater v a p o r. M onthly W eather Review, v . 69, 329-331* K o e h ler, E ., 1947: C ir c u la tio n model o f t r o p i c a l c y c lo n e s. S.M. t h e s i s , D epartm ent o f M eteorology. New York U n iv e rs ity ( u n p u b lis h e d .) 58 p p . B a u rw itz , B ., 1935: The h e ig h t o f t r o p i c a l c y c lo n e s and th e eye o f th e sto rm . M onthly W eather Review, v . 6 3 , 45-49. 1936: On th e v e r t i c a l wind d i s t r i b u t i o n in a n tic y c lo n e s , e x t r a t r o p i e a l and t r o p i c a l c y c lo n e s u n d e r t h e in f lu e n c e o f e d d y v i s c o s i t y . G e r l a n d s B e rtr & g e z u r G e o p h y s ik . v . 47, 206-214. 1941: D ynam ic M e te o r o l o g y . York and London, 365 p p .

McGraw H i l l B ook C o ., New

1948: The energy o f c y c lo n e s. P a p e r p re s e n te d a t th e s p r in g m eetin g o f th e American M e te o ro lo g ic a l S o c ie ty . A b s tra c t p u b lis h e d 1949: R ep o rt on s tu d ie s o f atm o sp h eric e n e rg y . D epartm ent o f M eteo ro lo g y . New York U n iv e rs ity (m im eographed), 38 p p . Maynafd, R. H ., 1945: R adar and w e a th e r. v . 2 , 214-226.

J o u r n a l o f M eteo ro lo g y ,

M ille r , J . E . , 1945: C y clo g en e sis i n th e A t la n ti c C o a s ta l R egion o f th e U n ite d S t a t e s . D epartm ent o f M eteo ro lo g y . New York U n iv e rs ity (m im eographed.) 77pp. 1949: On en erg y e q u a tio n s f o r th e atm osphere. o f M e te o r o l o g y . New Y o rk U n i v e r s i t y . 61 pp.

D epartm ent

M itc h e l, C. L . , 1924: West In d ia n and o th e r t r o p i c a l c y clo n e s o f th e N orth A tla n tic O cean. M onthly W eather Review. Supplem ent 24, 47 pp.

100

Normand, C. W. B ., 1931: R ecent i n v e s t i g a t i o n s on s t r u c t u r e and movement o f t r o p i c a l storm s in In d ia n S e a s . G erlan d s B eitr& ge zu r G eophysik. v . 23. P ie r c e , C. H ., 1939: The m e te o ro lo g ic a l h i s t o r y o f th e New England h u rric a n e o f Septem ber 2 1 , 1938. M onthly W eather Review. v . 67, 237-285. R e f s d a l, A ., 1930: Der F e u c h tla b ile N ie d e rs c h la g . G eo fy sisk e P u b lik a s .io n e r. O slo , v . 5, n o . 12, 69 p p . R ie h l, H ., 194-8 a: A ra d io so n d e o b s e r v a tio n i n th e eye o f a h u r r ic a n e . Q u a rte rly J o u r n a l o f th e R oyal M e te o ro lo g ic a l S o c ie ty , v . 74, 194. _____________ 1948: On th e fo rm a tio n o f ty p h o o n s. v . 5, 247-264.

J o u r n a l o f M eteo ro lo g y .

F.ossby, CtG ., and Montgomery, R. B ., 1935: The ]a y e r o f f r i c t i o n a l in flu e n c e in wind and ocean c u r r e n t s . P a p e rs i n P h y s ic a l O ceanography and M ete o ro lo g y . M a ssa c h u se tts I n s t i t u t e o f Technology and Woods H ole O ceanographic I n s t i t u t i o n , v . 3 , n o . 3 , 101 pp. _____________ 1936: On th e momentum t r a n s f e r a t t h e . s e a s u r f a c e . P a p e rs i n P h y s ic a l O ceanography and M eteo ro lo g y . M assa c h u se tts I n s t i t u t e o f Technology and Woods H ole O ceanographic I n s t i t u t i o n , v . 4 , n o . 3 , 30 p p. Saw yer, J . S ., 1947: N otes on th e th e o ry o f t r o p i c a l c y c lo n e s . Q u a rte rly J o u r n a l o f th e R oyal M e te o ro lo g ic a l S o c ie ty , v . 73, 101-126. S c h a c h t, E ., 1946: A mean h u r r ic a n e sounding f o r th e C aribbean a r e a . B u lle tin o f th e American M e te o ro lo g ic a l S o c ie ty , v . 27, 324-327. Shaw, S i r N a p ie r, 1922: On th e b i r t h and d e a th o f c y c lo n e s . An in tr o d u c ­ ti o n t o : Newnham, W. V ., H u rric a n e s and t r o p i c a l re v o lv in g s to r m s ., G re a t Br i t a i n M e te o ro lo g ic a l O f f ic e G eo p h y sical Memoirs. n o . 19, 213-225. Sim pson, R. H ., 1947: A n o te on th e movement and s t r u c t u r e o f t h e F lo r id a h u rr ic a n e o f O ctober 1946. M onthly W eather Review, v . 75, 53-58. S o u th w e ll, R. V ., 1946: R e la x a tio n Methods i n T h e o r e tic a l P h y s ic s . O xford U n iv e r s ity P r e s s , 248 p p . S te w a rt, H. J . , 1942: The energ y e q u a tio n f o r a v is c o u s co m p re ssib le f l u i d . . P ro ce e d in g s o f th e N a tio n a l Academy o f S c ie n c e s . v . 28, 161-164.

101

S v e rd ru p , H. U ., Jo h n so n , M. W., and F lem ing, R. H ., 194-2: P r e n tic e H a ll I n c . , New Y ork, 1087 pp.

The O ceans.

T a y lo r, G. I . , 1915: Eddy m otion i n th e atm o sp h ere. P h ilo s o p h ic a l Tran s a c tio n s o f th e R oyal S o c ie ty o f London. S e r ie s A, v . 215, 1- 26 . 1935: S t a t i s t i c a l th e o ry o f tu rb u le n c e . P ro ce e d in g s o f th e Royal S o c ie ty o f London. S e r ie s A, v . 151, 421-478. U n i t e d S ta te s W eather B ureau, 1948: H u rric a n e N o te s. n o . 1 (m im eographed.) 210 p p .

T r a in in g p a p er

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