AIDJEX BULLETIN No. 30

November 1975

THE EFFECT OF ICE MOTION ON THE MIXED LAYER UNDER ARCTIC ICE

--Miles G. McPhee . . . . . . . . . . . . . . . . . . . . . 1

FLUCTUATIONS AND STRUCTURE WITHIN THE OCEANIC BOUNDARY LAYER BELOW THE ARCTIC ICE COVER

--William 0. Criminale, Jr., and G. F. Spooner . . . . . . 29

INTEGRATION OF ELASTIC-PLASTIC CONSTITUTIVE LAWS --R. Colony and R. S. Pritchard . . . . . . . . . . . . . . 55

A DIFFERENCE APPROXIMATION TO THE MOMENTUM EQUATION --Robert S. Pritchard . . . . . . . . . . . . . . . . . . . 81

I I ON THE FRACTURE OF ICE SHEETS WITH PART-THROUGH CRACKS

--R. Reid Parmerter . . . . . . . . . . . . . . . . . . . . 94

THE NCAR ELECTRA FLIGHTS, A REPORT . . . . . . . . . . . . . 119

* * * * * * * * * Front cover: Atmospheric instrumentation a t the arti f ic4al

lead apegment on Elson Lagoon i n March 1974.

Back cover: We ran out of captions. Next issue, new covers.

AIDJEX BULLETIN No. 30

November 1975

* * * * * * * * * *

Financial support for AIDJEX is provided by the National Science Foundation,

the Office of Navai! Resemeh, and other U.S. and Canadian agencies.

* * * * * * * * * *

Arct ic Ice Dynamics Joint Experiment D i v i s i on o f Marine Resources

University of Washington Seatt le , Washington 98105

Division of M d n e Resources UNIVERSITY OF WASHINGTON

The AIDJEX BuZZetin aims t o provide both a forum for discussing AIDJEX probzems and a s w c e of information pertinent t o aZZ AILVEX p a r t i d - pan t s . Issues--numbered, dated, and sometimes subt i t Zed--contain technicaZ materia2 cZose Zy reZated t o AILVEX, infoma2 reports on theoretieaZ and f i e ld work, transZations of re Zevant s c i en t i f i c reports, and discussions of interim AIDJEX resutts.

BuZZetin 30 contains f i v e reports from members of the modeling group. Only the Zast art ic te , about the NCAR EZectra fZights i n JuZy, indicates that the main experiment i s going on. Bear we Zeave t o the Communique' and the newspapers for the time being, but the next BuZZetin W i l Z bring you photographs.

The breakup of B i g

Any correspondence concerning the BulZetin shouZd be addressed t o

Alma Johnson, Editor AIDJEX BuZ Zetin 4059 RooseveZt Way N . E. Seattle, Washington 98105

Miles McPhee has discovered two errors i n his paper i n BuZZetin 29 ( p . 93-111).

On page 96, i n the definit ion for e, the :mer l ine i n brackets should be

-sin €3 -cos B Onpage 103, the equation a t page bottom shouZd read

pyc, U ( 2 ) G sin B

and the definit ion for f i n the l ine .* % imediately af ter the equation shouZd

. .< be deteted. 1 &*..iib

THE EFFECT OF I C E MOTION ON THE MIXED LAYER UNDER ARCTIC PACK I C E

bY Miles G. McPhee

AIDJEX

ABSTRACT

This paper documents t he response of t he Arc t ic mixed l aye r t o synoptic weather events during March and Apr i l 1972 and ind ica t e s how measure- ments now i n progress a t fou r manned d r i f t s t a t i o n s w i l l be used f o r s i m i l a r s t u d i e s .

T i m e h i s t o r i e s of ice motion, rate of working, mixed l a y e r depths, and dens i ty , along wi th t h e dens i ty s t r u c t u r e of t h e upper pycnocline, are presented f o r t h e period 24 March - 28 Apr i l 1972. Twice during these f i v e weeks t h e mixed l a y e r r a t h e r abrupt ly shallowed t o 30 m and then deepened t o 45 m, events which were preceded by ice d r i f t speeds of 15-20 km p e r day. speed of t h e i c e relative t o the ocean w a s c a r e f u l l y observed, i t w a s

tu rbulen t energy a v a i l a b l e f o r entrainment during t h e deepening phase of a simple one-dimensional mixed l aye r model. With t h i s i n mind w e examined one event c lose ly by comparing t h e work done by t h e ice, as in fe r r ed from d i r e c t Reynolds stress measurements i n t h e oceanic bound- a ry , with changes i n the p o t e n t i a l energy of t he top 60 m of t h e ocean.

It could be argued from the r e s u l t s t h a t t he p o t e n t i a l energy increase , which w a s approximately 10% of t h e work done, w a s due t o entrainment by mechanical s t i r r i n g . However, o the r evidence, including turbulen t energy measurements, i m p l i e s t h a t convergence of m a s s along the upper l aye r s of t h e pycnocline i n response t o Ekman divergence (ice stress c u r l ) near t h e sur face w a s respons ib le f o r t h e changes, and t h i s hypothesis i n con- junc t ion wi th atmospheric pressure maps i s shown t o expla in q u a l i t a t i v e l y t h e subsequent rap id shallowing. This phenomenon, during which t h e mixed l aye r th ickness decreased by about 10 m i n 10 hours, gave r i s e t o apprec iab le b a r o c l i n i c cu r ren t s a t depths of 30-60 m. i ng occurred two days later, causing cu r ren t s i n the opposite sense, and t h e whole sequence seems t o be c lose ly connected t o a r e l a t i v e l y in t ense "eddy," documented by Hunkins, l oca l i zed a t about 100 m depth.

F ina l ly , t h e not ion is advanced t h a t synoptic-scale changes i n a w e l l - developed mixed l a y e r are dominated by sur face-s t ress c u r l r a t h e r than by mechanical s t i r r i n g , and a rudimentary o u t l i n e is given f o r using t h e AIDJEX ice model t o p r e d i c t mixed l a y e r c h a r a c t e r i s t i c s .

Since ice growth w a s s m a l l during t h e period and the

.-hoped t h a t t h e measurements might provide use fu l da t a on the amount of

A rap id deepen-

1

1. INTRODUCTION

Recent works on mixed l a y e r processes i n the open ocean have s t r e s sed

the importance of wind-induced mixing, e spec ia l ly i n descr ib ing t h e short-

term responses of mixed l aye r depth and density.

been formulated f o r descr ibing the entrainment process t h a t mixes denser

material from below up i n t o the mixed l aye r as stress is appl ied a t t h e

Various approaches have

surface. Denman and Miyake [1973] have compared summertime weathership d a t a

with a model i n which t h e rate of production of mechanical energy i s propor-

t i o n a l t o t h e cube of the 10 m wind speed, as suggested by Kraus and Turner

[1967]. G i l l and Turner [1975] have compared weathership d a t a f o r longer

per iods wi th var ious models; they found t h a t a modified Kraus-Turner model

gives the bes t f i t when t h e mechanical energy input due t o wind is constant

over t he year , r a t h e r than a func t ion of wind speed. Po l l a rd , Rhines, and

Thompson [1973] have a s se r t ed t h a t entrainment is due t o shear across t h e

i n t e r f a c e between wind-induced i n e r t i a l cur ren ts i n t h e mixed l a y e r and t h e

s t a b l e regime below.

stress reaches a l imi t ing depth a f t e r ha l f an i n e r t i a l period.

According t o t h e i r theory, t he mixed l a y e r under constant

The ice-covered Arctic Ocean af fords a unique opportunity t o i n v e s t i g a t e

t h i s aspect of t h e air-sea in t e rac t ion . F i r s t , t h e ice motion is t h e impor-

t a n t f a c t o r i n mechanical energy inpu t , no t t h e su r face wind i t s e l f . Thus

with c a r e f u l measurements of ice ve loc i ty w e can e l imina te the need f o r

considering t h e atmospheric boundary layer . Second, t h i ck ice serves as

an exce l l en t i n s u l a t o r and provides a good b a r r i e r t o s ens ib l e and l a t e n t

hea t f lux. There can exist, of course, an appreciable buoyancy f l u x under

ice t h a t is f reez ing rap id ly [Solomon, 19731, but mult iyear ice t h a t is

already th i ck grows q u i t e slowly i n t h e winter [Welch e t a l . , 19731.

2

A v a l i d objec t ion t o comparing the upper ocean under pack ice with the

open ocean is t h a t the ice e f f e c t i v e l y suppresses su r face wave ac t ion , which

must be important i n the mixing process. It should be noted, however, t h a t

the underside of t he ice is not smooth, bu t is punctmated by pressure r idge

kee l s t ha t may a f f e c t the mixed l aye r as much as s i z a b l e sur face waves.

Also, t he momentum t r ans fe r r ed t o waves i n the ocean may be comparable t o t h a t

t ransmit ted t o t h e ice.

From these considerat ions, i t w a s hoped t h a t extensive measurements of

upper ocean c h a r a c t e r i s t i c s made during the 1972 AIDJEX experiment could be

used t o i n v e s t i g a t e t h e e f f e c t and process of mechanical s t i r r i n g .

r e s u l t s suggest t h a t large-scale nonhomogeneous e f f e c t s may be more important

than is general ly thought.

The

2. DATA

The da ta described here were co l lec ted during the 1972 AIDJEX p i l o t

study by inves t iga to r s a t the main camp, s i t u a t e d i n multiyear pack i c e

400 km north-northeast of Barrow, Alaska. A l l the da t a r e s ide i n some form

i n the AIDJEX da t a bank and are ava i l ab le upon request .

The pos i t i on and ve loc i ty of t h e camp w e r e determined by Thorndike [1974]

a f t e r smoothing Navy S a t e l l i t e Navigation da ta with a sophis t ica ted f i l t e r i n g

technique.

who combined pressure measurements from AIDJEX manned camps and unmanned d a t a

Atmospheric pressure maps w e r e compiled by Brown e t al. [1974],

buoys with National Weather Service m a p s by using an objec t ive ana lys i s scheme

ca l l ed Cressman smoothing. I n view of t he sparse NWS coverage i n the Arc t ic ,

the Brown e t al. maps represent a l a r g e improvement over the s tandard maps

and c l ea r ly show d e t a i l s of weather systems t h a t passed through.

3

Two systems measured upper-level cur ren ts at the main camp. One, which

employed s tandard Savonius-type cur ren t meters mounted on a f ixed mast at t en

levels t o 100 m [Hunkins and F l i e g e l , 19741, w a s used t o ge t t h e absolu te

cur ren ts discussed i n sec t ion 4; however, t h e d i r e c t i o n measurements a t 30 m

and 40 m were ad jus ted by a constant amount, as suggested by Hunkins (personal

communication). The o the r system w a s designed t o i n v e s t i g a t e the turbulen t

s t r u c t u r e of t h e ice-ocean boundary l aye r [McPhee and Smith, 19751. I n t h e

context of t h i s paper, those measurements were used t o ca l cu la t e t h e ice-

ocean drag coe f f i c i en t and t o cor rec t t h e Lamont (Hunkins) 30 m and 40 m

d i rec t ions .

The primary d a t a set comprise CTD (conductivity-temperature-depth)

measurements made with a Guildl ine instrument t o furn ish dens i ty d a t a f o r t h e

measurements (Smith and McPhee, i n preparat ion) . A heavy schedule of shallow

casts, from t h e su r face t o about 65 m, w a s maintained f o r about seven weeks

s t a r t i n g 24 March 1972 (day 84 of t h e 1972 calendar , which is used he rea f t e r ) .

On the average, 10 c a s t s pe r day w e r e made, but t h i s number w a s increased

during t i m e s of increased ice motion.

Ind iv idua l casts showed regions of marked s a l i n i t y v a r i a t i o n i n t h e

upper p a r t of t h e pycnocline and near t h e surface. The v a r i a t i o n i n t h e

pycnocline w a s expected, s i n c e the re w a s evidence of considerable i n t e r n a l

wave a c t i v i t y ; however, no p l aus ib l e physical explanation could be found f o r

densi ty va r i a t ions of t he s i z e shown near t he surface. Possibly, as suggested

by E. L. L e w i s (personal communication), t h e conduct ivi ty response of t h e

instruments changed as they came i n t o contact wi th ice c r y s t a l s . A t any rate,

the f luc tua t ions w e r e removed by rep lac ing measurements i n t h e upper 5 m with

the average of measurements from t h e 6-15 m levels. I n t e r n a l wave e f f e c t s

w e r e removed by averaging a l l t h e casts t o 62 m depth (60 m from the ice) i n

4

each day.

teristics are taken from the averages of each day's casts.

i n sec t ion 4 t h a t demonstrate the v a r i a b i l i t y found from cast t o cas t .

Thus, unless otherwise noted, t i m e series of densi ty f i e l d charac-

Examples are given

3. GENERAL FEATURES

Figure 1 shows t h e average of CTD casts made from day 99.5 t o day 100.5.

A near ly homogeneous l a y e r extended t o about 35 m bezow the 2 m ice, bounded

below by a pycnocline with a typ ica l densi ty gradient i n its upper p a r t of

about 4 X lo-' gm cm-4 and a corresponding Brunt-V&zllg frequency of about

0.02 sec- l ,

I n wintertime t h e Arc t i c Ocean is near its freezing point throughout

the mixed layer . A t low temperatures, densi ty is almost exclusively a func-

t i o n of s a l i n i t y ; thus temperature can increase with increasing densi ty i n

the pycnocline and serves mainly as a t r a c e r f o r s a l i n i t y d i f fe rences . I n

p r a c t i c e i t : w a s found t h a t the top of the thermocline w a s o f ten easier t o

d i s t ingu i sh than the ha loc l ine , and f o r t h i s reason the -1.55'C isotherm

w a s chosen t o s i g n i f y the depth of t he mixed layer . Figure 2 shows t h i s

depth as measured from the ice as a funct ion of t i m e . Also drawn is t h e

m e a n densi ty ( i n 'Jt u n i t s ) of t h e w a t e r column above t h i s l eve l .

Figure 3 shows the ice speed as a funct ion of t i m e during the 35-day

period.

very quie t .

of 1-3 cm sec- l , which w e r e confirmed by a c t u a l measurements of t he baro t ropic

flow f i e l d [Newton, 19741. Thus i t is a f a i r l y good approximation t o consider

the i c e as being pushed by the wind across a st i l l ocean, and i n t h a t sense

w e can ca l cu la t e t h e stress exerted by the i c e and a l so t h e work done as a

I n t h e absence of wind d r i f t cur ren ts , t h e ocean i n t h i s region is

A dynamic topography compiled by Newton ind ica t e s mean cur ren ts

5

funct ion of ice speed.

d i r ec t ed toward expressing the drag exerted on t h e ice as a funct ion of t he

Much of t h e AIDJEX oceanographic e f f o r t has been

speed of t h e ice relative t o t h e undisturbed ocean. Using turbulence measure-

ments i n conjunction wi th p lane tary boundary l a y e r theory, w e found t h a t t h e

drag w a s f a i r l y constant over a speed range of 15-25 cm sec-' and could be

expressed f o r t he AIDJEX s i t e as IT^ = pcW18l2, where 181 is the speed of

t h e ice relative t o t h e ocean below t h e f r i c t i o n a l l aye r and t h e drag coef f ic i -

e n t , q,,, w a s about 0.0034 [McPhee and Smith, 19751.

G is t h e ice ve loc i ty , and t h e rate of working is &T = pr+lzl

f3 is t h e f r i c t i o n a l tu rn ing angle , found t o be about 24". Figure 4 , t h e

ca lcu la ted rate of working, demonstrates t h a t t h e energy input came mainly

during two r e l a t i v e l y short- l ived events.

I f the ocean is s t i l l , -+ -+

cos B , where

Figure 5 shows l i n e s of constant dens i ty (isopycnals) p l o t t e d from

day 84 through day 118, as compiled from the d a i l y average p r o f i l e s . From a

comparison of Figures 4 and 5 t h e r e emerges a p a t t e r n of upper pycnocline

response t o t h e energy input by i c e motion. F i r s t , t h e r e is a downwarping

of t h e upper isopycnals (and a deepening of t he mixed l aye r ) followed by an

abrupt upwarping (shallowing of t h e mixed l a y e r ) , which is followed i n tu rn

by another deepening. This second deepening is gradual a f t e r t h e f i r s t storm,

but i t occurs abrupt ly a f t e r t h e second event , on day 108. I n t h e next sec-

t i o n , t he response t o the second storm w i l l be examined i n d e t a i l .

4 . RESPONSE TO A STORM

The storm t h a t passed t h e AIDJEX camp from day 102 t o day 105 gave us

a good opportunity t o measure t h e under-ice boundary l aye r i n d e t a i l . The

6

s t r u c t u r e of turbulence and mean flow is discussed i n McPhee and Smith [1975];

those r e s u l t s pe r t inen t t o t h i s study can be summarized as follows.

F i r s t , t h e ice moved s t e a d i l y westward following i soba r i c contours.

Figure 6 shows a westward displacement of about 80 km over the ten-day per iod

with l i t t l e n e t northward motion.

the ice t h a t i n many respec ts w a s a good s c a l e model of a t y p i c a l n e u t r a l

atmospheric boundary layer ; t h a t is, t h e r e w a s a t h i n (approximately 1 m)

su r face l aye r with l a r g e shear near t h e ice below which the stress f e l l off

quickly and t h e shear came mostly i n t h e north-south (V) component, providing

a n e t northward t ranspor t .

constant during t h i s t i m e , as is i l l u s t r a t e d by Figure 7 , which shows nondimen-

s i o n a l mean p r o f i l e s f o r d i f f e r e n t i c e speeds. Although the re may have been

s m a l l i n e r t i a l f l uc tua t ions superimposed on t h i s s t r u c t u r e , they were ce r t a in ly

not the dominant mode.

A well-defined boundary l aye r developed under

The shape of t he ve loc i ty p r o f i l e s s tayed q u i t e

Second, t h e mean and . turbulen t p r o f i l e s w e r e scaled n i ce ly by u,/f, where

u, is the f r i c t i o n ve loc i ty and f t h e Cor io l i s parameter.

stress, turbulen t energy, and f r i c t i o n a l turning f e l l off at about z = -0.5u,/f.

This depth approached t h e 'mixed layer depth only during peak i c e v e l o c i t i e s ,

s o t h a t f o r t he most p a r t the f r i c i t o n a l boundary l aye r w a s shallower than

the mixed layer .

P r o f i l e s of shear

Under these condi t ions, i t w a s c l e a r t h a t t r e a t i n g the mixed l aye r as

a s l a b character ized by i n e r t i a l cur ren ts with s t rong shear a t both the sur face

and the bottom dens i ty i n t e r f a c e as suggested by Pol la rd , Rhines, and Thompson

[1973] w a s not at a l l a good approximation. S imi la r ly , i f turbulence did

no t f i l l the mixed l a y e r except at times of peak ice ve loc i ty , i t did no t s e e m

reasonable t o treat the entrainment due t o mechanical mixing a t the base of

7

the mixed l a y e r as propor t iona l t o the cube of t h e f r i c t i o n v e l o c i t y , as

suggested by s e v e r a l authors [e.g., K r a u s and Turner, 1967; Denman, 19731.

The f a c t remains, however, t h a t t h e mixed l a y e r both deepened and

became more s a l i n e during the e a r l y s t ages of t he storm.

t h e depth of t h e -1.55"C isotherm f o r each c a s t during the ten-day period.

There is considerable v a r i a t i o n from cast t o cast, probably because of i n t e r n a l

wave a c t i v i t y , but a t rend s tands out which ind ica t e s a

day 101 to day 104. The p r o f i l e f o r day 100, shown i n Figure 1, can be idea l i zed

as a mixed l aye r 35 m t h i c k ( t o 37 m depth) with at equal t o 24,034, over a

pycnocline with a l i n e a r density grad ien t of s lope 4 x lo-' gm

Figure 8 shows

5-10 m deepening from

From

t h i s i t is a simple matter t o ca l cu la t e t h e change of mixed l aye r dens i ty and

water column p o t e n t i a l energy i n terms of the change i n mixed l aye r depth,

if i t i s assumed t h a t m a s s is conserved and t h e p r o f i l e below the maximum

mixed l aye r depth is undisturbed. I f t h e mixed l a y e r thickness i n t h i s simple

model changes by 10 m from day 100 t o day 104, w e would expect i ts Ut va lue t o

change t o 24.09, which agrees w e l l wi th t h e average value observed f o r day 104

( see Figure 2).

cgs u n i t s .

f o r t he water column above 60 m.

c lo se ly with the value ca lcu la ted from t h e simple model.

W e would a l s o expect a p o t e n t i a l energy change of 4.4 x l o 5

Figure 9 shows t h e range of average p o t e n t i a l energy ca l cu la t ed

The change from day 101 t o day 104 agrees c

Given t h i s information, i t should be a simple matter t o calculate what

f r a c t i o n of t h e t o t a l work done by the ice is converted t o p o t e n t i a l energy

by entrainment.

f o r t h e ten-day period.

Figure 9 , so t h a t t h e work done between day 101 and day 104 is roughly t en t i m e s

Figure 10 shows t h e i n t e g r a l of t h e work rate curve i n Figure 4

Note t h a t t h e scale is ten t i m e s as g r e a t as t h a t i n

t h e p o t e n t i a l energy increase . Thus i t would appear t h a t t he o r i g i n a l i n t e n t

8

of t h i s paper--to i n v e s t i g a t e the p a r t i t i o n of mechanical energy i n t o energy

ava i l ab le f o r entrainment--is accomplished. However, c lose r examination of

t he da t a reveals some important inconsietencies .

F i r s t , a comparison of Figures 9 and 10 shows t h a t the p o t e n t i a l energy

increase precedes t h e energy input by about two days, i .e . , t he mixed l aye r

changes before t h e i c e moves much. Clearly i n t h i s ins tance any attempt t o

express the entrainment rate as some funct ion of t h e ice ve loc i ty would be

f r u i t l e s s .

Second, t h e average dens i ty p r o f i l e f o r day 104, shown i n Figure 11,

exh ib i t s a much smoother t r a n s i t i o n from t h e mixed l aye r i n t o the pycnocline

than the p r o f i l e s before the storm. I n a simple model l i k e the one described

above, w e would expect t h e i n t e r f a c e t o sharpen as t h e l aye r deepened.

changes occurred i n t h e upper pycnocline t h a t appear t o be as s i g n i f i c a n t as

changes i n the mixed l aye r i t s e l f .

Ins tead ,

F ina l ly , t h e event t h a t occurred soon a f t e r t he storm, when the mixed

l aye r became more shallow (day 105) and then deepened abrupt ly (day 108),

cannot be explained by a simple model. The question is t o what extent these

events are r e l a t e d t o events observed earlier when t h e i c e w a s moving f a s t e r .

Figure 1 2 presents a more complete view of the densi ty s t r u c t u r e i n

the upper pycnocline f o r days 105-110. Here the isopycnal value f o r each

cas t is ca lcu la ted and p l o t t e d i n order t o give some ind ica t ion of short-

term var ia t ion . Figure 13 demonstrates t he d i f fe rence between average p r o f i l e s

when compared with Figure 11. Together, Figures 11, 12, and 13 give the

impression t h a t w a t e r t y p i c a l of t he upper pycnocline (0, between 24.2 and

24.6) has co l lec ted , forc ing the mixed l aye r up and depressing deeper isopycnals.

Figure 9 supports t h i s i n t e r p r e t a t i o n , s ince i t shows a marked increase i n the

9

p o t e n t i a l energy of t h e w a t e r column above 60 m from day 105 t o day 107 i n

s p i t e of t h e decrease i n mixed l a y e r thickness. I n o ther words, i f no mass

had converged, t h e shallower mixed l a y e r would imply a r e d i s t r i b u t i o n down-

ward t h a t would decrease p o t e n t i a l energy.

I n an earlier work, convergence of mass i n t h e upper pycnocline w a s

discussed i n connection with ba roc l in i c cur ren ts observed e a r l y on day 106

[McPhee, 19751. The h e u r i s t i c explanation of fe red then w a s t h a t divergence

of Ekman t ranspor t i n t he boundary l aye r caused convergence J u s t below t h e

mixed layer . This argument gains considerable support when atmospheric

pressure maps i n t h e Beaufort Sea quadrant are examined.

The map quadrant extends from the North Pole i n t h e upper r i g h t corner

south along t h e 90°W meridian on the r i g h t and along the 1 8 O O W meridian at

the top , as shown by Figure 14, which a l s o shows t h e pos i t i ons of t he manned

camps. Camp Jumpsuit (JS) , whose pos i t i on is ind ica ted in Figures 15, 16,

and 17 by the numeral 0 , w a s the main camp a t which measurements i n t h i s paper

w e r e made. Two four-hour maps f o r each of days 102 through 109 are shown

because they give an o v e r a l l synopt ic view not ava i l ab le elsewhere and, some-

what su rp r i s ing ly , shed a grea t dea l of l i g h t on events i n t h e mixed layer .

S t a r t i n g with day 102, t he camp w a s loca ted j u s t south of a high

centered about 80°N, 15OoW, with the main pressure grad ien t t o the south.

As t i m e progresses, t h e main storm region, with s t rong e a s t e r l y geostrophic

winds, is advected i n t o the camp area and t h e ice d r i f t s westward, causing

northward Ekman t r anspor t i n t h e oceanic boundary layer . This pumps water

away from the low toward t h e high near t h e su r face ( the opposi te e f f e c t occurs

i n the atmospheric boundary l aye r ) .

t o the westward ice stress; and s i n c e the wind varies s p a t i a l l y , t he stress

and t r anspor t m u s t a l s o vary. Simple cont inui ty considerat ions thus r equ i r e

The northward t ranspor t is proportional.

10

a vertical ve loc i ty i n t h e boundary l a y e r and a l s o i n the mixed l aye r , since

p rope r t i e s t he re are uniform. Consider a case i n which t h e n e t t r anspor t is

a l l northward (near t h e a c t u a l case); then

0 0

o r

0

where My = I vdz is the t r anspor t , H is the depth of f r i c t i o n a l in f luence ,

WH i s t h e vertical ve loc i ty a t H, and w 0 is t h e vertical ve loc i ty a t the

Z = -H

surface.

and s i n c e t h e t ranspor t decreases t o t h e nor th on day 102, w e can expect an

Since W , - wH is t h e time rate of change of t h e mixed l aye r thickness ,

increase i n mixed l a y e r thickness which e f f e c t i v e l y en t r a ins material i n the

manner suggested by Denman [1973], who included an advective ve loc i ty i n t h e

formulation of h i s model. During days 103 and 104 the ice camp i s i n the main

stream, and although the speed i s higher and t r anspor t g rea t e r , t he north-south

gradient is s m a l l and the mixed l aye r changes much less than i t did ear l ier .

Toward t h e end of day 105 and on day 106 t h e ice lies j u s t nor th of t he low and

a gradien t i n t h e opposi te sense exists, so t h a t an upward ve loc i ty decreases

the mixed l aye r depth (but does no t g rea t ly a l ter its o ther p rope r t i e s ) .

From another viewpoint, t h i s is t h e region where water t y p i c a l of t h e upper

pycnocline has converged i n response t o Ekman divergence i n t h e boundary

layer .

On days 107 and 108, t he ice sits i n a dynamically f l a t region u n t i l ,

on day 109, another system is advected i n t o t h e region south of t he camp and

a negat ive t r anspor t g rad ien t causes t h e mixed layer t o deepen i n t h e same

manner as on day 102. 11

Thus w e are l e d t o the first main r e s u l t of t h i s paper: t h a t t he

t r anspor t divergence, which is propor t iona l t o the ice stress c u r l , is much

more important than d i r e c t tu rbulen t energy input i n determining t h e prop-

erties of sur face waters i n the wintertime Arctic. This mechanism, which has

a pronounced elffect on t h e upper pycnocline as w e l l as on the mixed l aye r ,

has o the r i n t e r e s t i n g consequences, some of which are described i n t h e

next sect ion.

5. BAROCLINIC CURRENTS I N THE UPPER PYCNOCLINE

The pecu l i a r "hump" i n the isopycnal record shown i n Figure 12 w a s

accompanied by an i n t e r e s t i n g sequence of ba roc l in i c cur ren ts below the

mixed layer . As mentioned previously, a s m a l l segment of these cur ren ts ,

from e a r l y on day 106, has been inves t iga ted [McPhee, 19751. It w a s found

t h a t t he observed north-south cur ren ts roughly f i t the geostrophic shear

equation

i f t he length scale i n the z-direct ion w a s i d e n t i f i e d with t h e d is tance t h e

ice moved. As Figure 6 demonstrates, however, ice motion v i r t u a l l y stopped

soon afterward, making it impossible t o estimate densi ty grad ien ts .

ing, i t should be noted t h a t observations from a d r i f t i n g s t a t i o n make i t q u i t e

d i f f i c u l t t o d i f f e r e n t i a t e between temporal and s p a t i a l changes.

I n pass-

Figures 18 and 19 show the mean V (north-south) and U (east-west) cur ren t

records a t s ix l e v e l s as measured by t h e Lamont group and modified as descr ibed

i n sec t ion 2. The records have a l s o been smoothed with a 13-hour running mean

12

t o remove i n e r t i a l s i g n a l s t h a t became q u i t e s t rong toward the end of t he

period.

ve loc i ty from cu r ren t s measured relative t o the ice.

The cu r ren t s shown are absolu te , determined by subt rac t ing t h e ice

The 2 m trace responds d i r e c t l y t o ice motion, showing a northwestward

mean t ranspor t . A t 20 and 30 m, cur ren ts are s m a l l almost a l l of t h e t i m e ,

wi th perhaps a small n e t westward displacement a t 30 m, r e f l e c t i n g the mean

' g e o s t r q h i c flow. A more complete p i c t u r e of t he ice-driven f l o w i s given

i n Figure 7, where i n each case t h e uppermost value is a t 2 m and t h e lowest

is a t 32 m.

The traces at 40 and 50 m can be compared with changes i n the dens i ty

f i e l d shown by Figure 12. For example, the V traces show southward cu r ren t s

on 106 as the hump bu i lds and corresponding northward flow on the o the r s i d e

(day 108).

The trace a t 100 m is p a r t i c u l a r l y i n t e r e s t i n g beclause i t appears t o

be associated with cu r ren t s a t 40 and 50 m, bu t does not follow e x p l i c i t l y

from t h e densi ty hump shown i n t h e top 50 m. I f one thinks i n terms of va r i -

a t i o n s i n t i m e only, the co l lapse of t h e hump on day 108, which is accompanied

by a sharp decrease i n p o t e n t i a l energy (see Figure 9), seems t o spawn ene rge t i c

cur ren ts a t g rea t e r depths. It should be kept i n mind tha t the k i n e t i c energy

of these cur ren ts i s l a rge r than tha t of the i ce d r i f t cu r ren t s during t h e

storm. The cur ren ts from day 108 t o day 110 give the d i s t i n c t impression of

a s loping f r o n t analogous t o those observed i n the atmosphere.

The cur ren t a t 100 m has been in t e rp re t ed by Hunkins [1974] as evidence

of an eddy, which he suggests o r ig ina t ed from ba roc l in i c i n s t a b i l i t y some

d is tance away. The i n t e r p r e t a t i o n here is t h a t t he cur ren ts a rose from a

complicated set of events following the storm and is better described as a

"front" r a t h e r than an "eddy." It is i n t e r e s t i n g t o no te t h a t a s i m i l a r

13

storm response occurring on days 85-95 (see Figure 5) w a s a l s o accompanied by

deep cur ren ts some t i m e a f t e r t he storm had passed a t t h e su r face [Hunkins,

19741.

6 . CONCLUSIONS AND FURTHER WORK

The i n t e r p r e t a t i o n of mixed layer da t a made here has suggested some

unexpected r e su l t s . F i r s t , f o r conditions encountered--the f r i c t i o n ve loc i ty

about 1 cm sec- l , the i n i t i a l mixed l aye r depth about 35 m, and very l i t t l e

sur face buoyancy flux--the growth of t h e mixed l aye r is much more dependent

on large-scale convergences and divergences than on l o c a l turbulent mixing.

Second, the mechanism t h a t cont ro ls the response of t he mixed l aye r may a l s o

be an important source f o r t ransmi t t ing k i n e t i c energy t o lower l e v e l s i n

the ocean. How t h i s ac tua l ly occurs seems t o be a complicated phenomenon,

and a more de t a i l ed discussion is not appropriate here . The oceanographic

da t a from the 1975-16-AIDJEX experiment should help i d e n t i f y the processes

invo lve d . One of the output f i e l d s from the AIDJEX dynamical ice model is t h e

c u r l of t h e ice-water stress f i e l d . It seems reasonable i n l i g h t of t h e

r e s u l t s shown here t o test a mixed l a y e r model i n which a vertical ve loc i ty

is computed from t h i s f i e l d aod i n which su r face buoyancy f l u x is ca lcu la ted

from ice growth, which is a l s o a model output. The model would be i n i t i a l i z e d

and checked with d a t a now being col lected. Careful considerat ion would have

t o be given t o the s t r u c t u r e of t h e upper pycnocline, but such a model, i f

successfu l , could prove t o be use fu l i n p red ic t ing longer-term ice dynamics

and thermodynamics and may be appl icable t o more temperate zones.

14

7. REFERENCES

Brown, R. A., P. Maier, and T. Fox. 1974. Surface atmospheric pressure f i e l d s and derived geostrophic winds, AIDJEX 1972. HDJEX BuZZetin NO. 26, 173-203.

Denman, K. L. 1973. A time-dependent model of the upper ocean. JownaZ of PhpsicaZ Oceanography, 3, 173-184.

Denman, K. L., and M. Miyake. 1973. Upper layer modification at Ocean S ta t ion Papa: observations and simulation. JownaZ of PhysicaZ Oceanography, 3, 185-196.

G i l l , A. E., and J. S. Turner. 1975. A comparison of seasonal thermocline models wi th observation. To appear i n Deep-sea Research.

Hunkins, K. 1974. Subsurface eddies i n t h e Arctic Ocean. AIDJEX BuZZetin NO. 23, 9-36.

Hunkins, K., and M. F l i e g e l , 1974. Ocean current observations a t t h e 1972 AIDJEX main camp. A I N E X BuZZetin No. 26, 75-108.

Kraus, E. B., and J. S. Turner. 1967. A one-dimensional model of t h e seasonal thermocline. 11. The general theory and i ts consequences. TeZZus, 29, 98-106.

McPhee, M. G. 1975. An experimental inves t iga t ion of t h e boundary layer under pack ice. Technical Report M75-14, Dept; of Oceanography, - University of Washington, S e a t t l e , 164 pp.

McPhee, M. G . , and J. D. Smith. 1975. Measurements of t h e turbulen t boundary l a y e r under pack ice. A I M E X BuZZetin No. 29, 49-92.

Newton, J. L. 1973. The Canada Basin: mean c i r cu la t ion and intermediate-scale flow fea tures . Ph.D. d i s s e r t a t i o n , University of Washington, Seattle.

Pol la rd , R. T., P. B. Rhines, and R. 0. R. Y. Thompson. 1973. The deepening of t h e wind-mixed layer . GeophysicaZ Flu id Dynamics, 3, 381-404.

Solomon, H. 1973. Wintertime sur face layer convection i n t h e Arctic Ocean. Deep-sea Reseaxeh, 20, 269-283.

Thomdike, A. 1974. S t r a i n ca lcu la t ions using AIDJEX 1972 pos i t i on data. AIDJEX B u Z l e t h No. 24, 107-129.

Welch, M., E. Partch, H. Lee, and J. D. Smith. 1973. Diving repor t , 1972 AIDJEX p i l o t study. AIMEX Bulletin No. 18, 31-44.

15

8. FIGURES

1

2

3

4

5

6

7

8

9

10

11

' 12

1 3

14

15

16

17

18

19

Average

Mixed 1

of CTD casts f o r day 100, 9 Apr i l 1972.

y e r depth ( s - l i d l i n e ) and sigma-t (dashed l i n e ) f o r d

Ice speed at main AIDJEX camp, days 84-117.

R a t e of working (i = p c, G3 cos B ) , days 84-117.

YS 84-117.

Isopycnal sur faces vs. t i m e , days 84-117. Contour i n t e r v a l is 0 .1 sigma-t un i t s .

E a s t (pos.) - w e s t (neg.) displacement ( s o l i d l i n e ) and nor th (pos.) - south (neg. ) displacement (dashed l i n e ) , days 101-111.

Nondimensional p r o f i l e s of boundary l a y e r cur ren ts [from McPhee and Smith, 19751.

Depth of -1.55OC isotherm (measured from the i c e ) f o r a l l c a s t s , days 101-111.

Change i n p o t e n t i a l energy of upper 60 m, days 101-111.

Work done by i c e , days 101-111.

Average of CTD casts f o r day 104.

Isopycnal contours f o r each cast , days 105-110.

Average of CTD casts f o r day 107.

Map of Alaska quadrant, showing approximate manned s t a t i o n pos i t i ons during 1972 AIDJEX p i l o t study [from Brown et al. , 19741.

Surf ace barometric pressure maps, days 102-104 [adapted from Brown e t a l . , 19741. Numeral 0 ind ica t e s pos i t i on of main camp.

Surface barometric pressure maps, days 105-107.

Surface barometric pressure maps, days 108-109.

North (pos.) - south (neg.) cur ren ts at s i x levels .measured from ice, days 101-111.

E a s t (pos.) - w e s t (neg.) cur ren ts a t s i x levels, days 101-111.

16

Fig. 1. Average of casts for day 100, 9 April 1972.

a?

I c Y CTD

e n _ _

70 I I 29.4 24.8 30.2 36.6 31.0 31.4 3i.8

SALINITY

23,. 8 2f. 2 21.6 25.0 25.4 25.8 26.2

SIGMA T

Fig. 2. Mixed layer depth ( s o l i d l i n e ) and sigma-t (dashed l i n e ) for days 84-117.

1 7

H1,m

2 5 0 0 0

a.00

15.00

1o.m

5.00

0

Fig . 3. Ice speed at main AIDJEX camp, days 84-117.

MY

Fig. 4 . Rate of working (k = pcWc3 cos 8) ,. days 84-117.

18

20

0 h

8

Fig. 5. Isopycnal surfaces vs. time, days 84-117. Contour interval is 0.1 sigma-t units.

- 4 -20

DAY

Fig. 6. East (pos.) - west (neg.) displacement (solid line) and north (pos.) - south (neg.) displacement (dashed line), days 101-111.

19

NON- DIMENSIONAL CROSS- STREAM DEFECT

( U 3 2 - U ) /u *

-2

A * a, s 3

NON-DIMENSIONAL DOWNSTREAM DEFECT

Fig. 7. Nondimensional p r o f i l e s and Smith, 19751.

- LABELS DESCRlPnON A 8 HOUR COMPOSITE OF 20-MINUTE AVERAGES TAKEN 4/11/72 PY

AVERAGE SPEED AT 32m.: 16.32 em *a<' AVERAGE BEAR~NG s6*

5 HOUR COMPOSITE OF PQMINUTE AVERAGES TAKEN 4/12/72 PY AVERAGE SPEED AT 32m : 23.67 cm ws-' AVERAGE BEARINO 97.

B

of boundary layer cu r ren t s [from McPhee

Fig. 8. Depth of -1.55OC isotherm (measured from the ice) for a l l casts, days 101-111.

20

411 1 4/13 4/15 4/17 4/19 1 a

1'32 106

MY

108 UO

Fig. 9 . Change i n potent ia l energy of upper 60 m , days 101-111.

h

10 I 0 4

X v)

2 3 VI

t

8 Y

10

8

6

4

2

0

4/11 4/15 4/19 I I I I I 1

I I I I I I I

102 104 106 110

Fig. 10. Work done by ice, days 101-111.

21

I I I I

I I I I I I I \ \

I

TEMPERATURE (OC)

-1 .6 -1 .4 -1 .2 -1 .o '-0.8

10-

20-

30-

I - E 40-

n W

I

p ig . 11. Average of CTD cas t s for day 104. t

'. ----

salinity

70 I 1 I I I 2 9 . 4 2 9 . 8 30 .2 30.6 31 .0 31.4 31.8

SALINITY

2 3 . 8 24 .2 2 4 . 6 2 5 . 0 25.4 2 5 . 8 26.2

S I G M A T

I I

I I I I

4/14 4/15 4/16 4/17 4/10 4/19

Pig. 12. Isopycnal con- tours f o r each c a s t , days 105-110.

t -* 1

t

22

-2.0 0

10-

20-

30- L:

r - Z S O - 0

50-

60-

10 29.4

Fig. 13. Average of CTD casts for day 107.

-1.8 ' -1-6 -1.4 -1.2 -1.0 -0.8 I I 1 I I __

i

DAY 107 I' 1 i- - t I -

-

-

-

- sal i n i ty -

I I I I I

31.4 31.8 29.8 30.2 30.6 31.0

Fig. 14. Alaska quadrant, with approximate pos i t ions of manned s ta t ions during 1972 AIDJEX p i l o t study. JS designates main camp. [From Brown et a l . , 1974.1

23

1 M . O d'

1 Q 3 m 0

lO'+mO

Fig. 15. Surface barometric

Wm5

m,'7

pressure maps, days 102- . 104. Adapted f r o m Brown et a l . [1974].

Numeral 0 indicates position of main camp.

24

105.0 I

106.0

lo710

105.5

F i g . 16. Surface barometric pressure maps, days 105-107.

25

I i

109.0 109.5

F i g . 17. Surface barometric pressure maps, days 108-109.

26

u! Ou! vc;J 111

u

0

I

- 0

- 0 - 0

lil

0

27

FLUCTUATIONS AND STRUCTURE W I T H I N THE OCEANIC BOUNDARY LAYER BELOW THE ARCTIC I C E COVER

by W i l l i a m 0. Criminale, Jr.

University of Washington and

G. F. Spooner Dept. of Oceanography

University of Washington

Dept. of Oceanography and Geophysics Program

ABSTRACT

Progress has been made i n developing a t h e o r e t i c a l model by l i n e a r ana lys i s t o descr ibe f luc tua t ions wi th in and beyond t h e su r face boundary l a y e r below the Arctic i ce cover. The governing system is der ived under t h e assumptions of (1) f l a t ice topography, (2) no d i f fus ion of m a s s bu t with a known source of m a s s f l u x a t t h e i c e cover, (3) a turbulen t Ekman boundary l aye r with weak s t r a t i f i c a t i o n bounded below by a s t rong pycnocline, (4) turbulence parameterized by spec i fy ing a v a r i a b l e eddy v i s c o s i t y , and (5) ve loc i ty f luc tua t ions f u l l y three-dimensional. It is shown t h a t t he general prob- l e m is s i x t h order and completely coupled, even i n the region below t h e va r i ab le shear zone. The fundamental physics under- l y i n g any o s c i l l a t i o n s are discussed and compared with a con- vent iona l tu rbulen t boundary layer .

INTRODUCTION

The i n s i t u measurements made by Smith during t h e 1970, 1971, and

1972 AIDJEX p i l o t s t u d i e s [Smith, 19741 are remarkable i n oceanic boundary

l a y e r research programs. They are complete enough t o provide a knowledge

of both t h e mean and t h e f l u c t u a t i n g f i e l d s throughout t h e su r face bound-

a ry l a y e r beneath t h e ice cover of t h e A r c t i c Ocean during t h e spr ing;

and t h e i r q u a l i t y is such t h a t they form a s o l i d base from which t o assess

t h e physics of boundary l a y e r motion as it occurs i n nature . I n s h o r t , any

29

explanations should account i n some way f o r Smith's observations.

I n terms of the mean ve loc i ty , Smith has confirmed t h e presence of a

moderately well-developed Ekman s p i r a l , a t least during periods when t h e

wind above the ice has been active long enough o r hard enough t o transmit

shear stress t o t h e water below. Unlike t h e classical p i c tu re , however, t h e

s p i r a l found by Smith w a s d i s t o r t e d and spanned t h e region between depths of

4 m and 54 m.

showed a mixed l a y e r extending from t h e ice down t o a depth of about 40 m

and, below t h a t , a well-developed pycnocline.

The flow w a s decidedly turbulen t . The mean dens i ty p r o f i l e

The o ther s a l i e n t f e a t u r e of t h e measurements w a s t h e d i s t r i b u t i o n of

t h e f luc tua t ion f i e l d wi th in t h e sur face boundary layer .

region--from the ice down t o approximately 4 m--the turbulence i n t e n s i t y

w a s roughly 10% of the mean cur ren t . It dropped t o 6% i n the upper 15-20 m

of t he mixed layer , decreased f u r t h e r t o 3% i n t h e lower p a r t of t he mixed

layer , and then remained constant throughout t he pycnocline. A l l s i x com-

ponents of t h e turbulen t stress w e r e determined along with t h e in t ens i ty .

The s t r u c t u r e Smith found i s d i f f e r e n t from t h a t of any turbulence described

previously, and not even i n i t s gross behavior can i t be explained wi th in

the present s ta te of knowledge.

I n the near-ice

Smith's measurements do have l imi t a t ions . For example, i t w a s impos-

s i b l e t o obta in absolu te ly t r u e s teady-state values f o r t he mean ve loc i ty

and dens i ty because, although several weeks were spent i n the f i e l d , un-

s teadiness w a s s t i l l prevalent .

i t y is present i n the Arctic and is a cause of some of the v a r i a t i o n s t h a t

are seen i n t h e dens i ty p r o f i l e . Then, too, t h e s i g n i f i c a n t topography on

t h e underside of t he ice influenced t h e behavior of a l l q u a n t i t i e s t h a t

w e r e measured wi th in the near-ice region, and possibly f a r t h e r down as w e l l .

On t he o ther hand, t h e measurements w e r e made a t approximately t h e same

t i m e of year f o r t h ree successive years and s u b s t a n t i a l l y t h e same r e s u l t s

w e r e obtained, s o t h a t i t would s e e m t h a t t h e observations do represent

what a c t u a l l y occurs.

tics of t h e f luc tua t ions are s h o r t e r than those f o r t h e evolut ion of the

mean f i e l d , allowing them t o b e inves t iga ted separa te ly . F ina l ly , i t is

important t o note t h a t t he measurements are complete and lead n a t u r a l l y t o

Large- and moderate-scale convective activ-

Also, t h e time scales f o r the dynamic charac te r i s -

30

a framework f o r formulating a model t h a t dea l s with t h e f l u c t u a t i o n s , and i t

is our purpose h e r e t o attempt such a formulation.

A cr i t ical parameter f o r t h e s t a b i l i t y of a s t r a t i f i e d shear flow (or

t h e maintenance of turbulence i n a s t r a t i f i e d f l u i d ) is t h e Richardson num-

ber . E s s e n t i a l l y , t h i s number is a measure of t h e relative s t r e n g t h s of

the buoyancy e f f e c t s due t o t h e s t r a t i f i c a t i o n t h a t i n h i b i t s d e s t a b i l i z a t i o n

and t h e shear e f f e c t s t h a t promote i n s t a b i l i t y . Miles [1961] and Howard

[1961] have shown t h a t a s u f f i c i e n t condition f o r s t a b i l i t y i n a s t r a t i f i e d

shear flow is t h a t t h e Richarson number be everywhere g r e a t e r than 1 / 4 . Although t h h c r i t e r i o n w a s determined by considering a l i n e a r system, i t

has been both adequate and u s e f u l when dea l ing wi th t h e f u l l problem. An

inspec t ion of t he p r o f i l e s from Smith's work shows t h a t , wi th in the upper

p a r t of t h e su r face Ekman l a y e r , t h e Richardson number is less than 1/4,

i nd ica t ing t h a t t h e turbulence i s induced by shear moderated by ice to-

pography i n the very near-ice p a r t of t h e flow.

turbulence found are comparable t o standard f l a t p l a t e measurements.

t h e deeper p a r t of t h e mixed l a y e r , however, t h e Richardson number is much

g r e a t e r than t h e c r i t i ca l va lue , while t h e tu rbu len t stresses are s t i l l

q u i t e reasonable: 6%, as compared wi th 10% with no r ap id decease. Such

values are not compatible with t h e simple i n s t a b i l i t y c r i t e r i o n and r equ i r e

another explanation f o r t h e source of t h e f luc tua t ions .

Indeed, t h e va lues of t h e

I n

A c lose r inspec t ion of t h e Smith [1974] d a t a helps t o cons t ruc t a

hypothesis f o r turbulence generation under the conditions t h a t are d i c t a t e d

by t h e s i t u a t i o n . The conjec ture is t h a t t h e turbulence is generated by

brine-driven convection, which is a c t i v e i n t h e sp r ing when the measurements

w e r e made. Such a proposal w a s put f o r t h by Smith [1974] as w e l l . It is

based on (1) t h e behavior of t h e mean dens i ty p r o f i l e s where v a r i a t i o n s

p e r s i s t t o g r e a t e r depths during meteorological ly q u i e t periods than when

t h e r e is increased atmospheric a c t i v i t y ; and (2). t h e j e t l i k e s t r u c t u r e of

t h e mean v e l o c i t y p r o f i l e t h a t corresponds t o a replenishment of f r e s h e r

w a t e r from t h e s i d e s at c e r t a i n depths a f t e r t h e b r i n e has f a l l e n v e r t i c a l l y

t o a d i f f e r e n t level. This a c t i o n is caused by leads. I n terms of t h e

f luc tua t ions , t h e convective a c t i v i t y creates a downward f l u x of ho r i zon ta l

momentum t h a t is dr iven by b r i n e plumes. The plumes are generated a t random

31

l oca t ions and t i m e s by t h e f reez ing of sea water a t t h e ice cover. From

this po in t of view, t h e b r i n e acts as a source of mass a t t h e ice cover,

u l t ima te ly being analogous t o a s i n k as it fal ls ; at mfddepths t h e r e is a

b r i n e packet t h a t is more l i k e a doublet.

un l ike t h a t of t h e c l a s s i c a l s t r a t i f i e d (or n o n s t r a t i f i e d ) shear flow, where

The n e t r e s u l t is a p i c t u r e

t h e boundaries are only pass ive and play no r o l e i n t h e dynamics of t h e

f luc tua t ions .

The general problem has two major non l inea r i t i e s : t h e governing

equations and, i f t h e topography is t o be included, t h e boundary conditions.

Solution of t h e combination is beyond t h e a b i l i t y of cur ren t techniques and

is not necessa r i ly t h e most d e s i r a b l e rou te t o follow even i f t h i s w e r e not

t h e case. Instead, a model is synthesized t h a t is t r a c t a b l e but s t i l l

r e t a i n s t h e f e a t u r e s t h a t are believed t o be of prime importance. For t h i s

purpose, t h e specula t ion t h a t c r i t i c a l physics is connected wi th t h e b r i n e

source a t the i c e cover is c e n t r a l t o t h e i s s u e and t h e construction.

The immediate, and convenient, consequence of t h i s imput is t h a t t h e

i c e topography can be ignored. Smith has reported t h a t t he turbulence f i e l d

is less s e n s i t i v e than t h e mean flow t o topographic, disturbances. However, removing topography from consideration means t h a t t h e mean v e l o c i t y used i n

the ana lys i s w i l l have t o be more i n keeping with a f l a t p l a t e boundary

l a y e r than with t h e l a y e r measured by Smith i n t h e f i e l d , s i n c e many of t h e

v a r i a t i o n s found i n h i s measurements are topographically induced. Good

p r o f i l e s f o r t h i s purpose can be found, f o r example, i n t h e work of .

Deardorff [1972].

argument [McPhee and Smith, 19751 t h a t h i s measurements are influenced by

topography much more--that is, topographic e f f e c t s pene t r a t e much deeper--

than had been h i t h e r t o believed .)

(More recent eva lua t ion of t h e Smith d a t a has l e d t o an

Using model p r o f i l e s from Deardorff is permiss ib le because t h e f i e l d

d a t a have shown t h a t t he mean zone of shear is w e l l wi th in t h e mixed l a y e r

where t h e f l u i d is only weakly s t r a t i f i e d . The pycnocline begins at a

po in t where t h e mean ve loc i ty is approximately constant. Since t h e flow is

tu rbu len t , t h e s p e c i f i c a t i o n

but ion f o r t h e turbulen t , o r

purpose can a l s o be obtained

of t h e mean f i e l d is complete when a d i s t r i -

eddy, v i s c o s i t y is given; information f o r t h i a

from Deardorff [1972]. It shourd be pointed

32

ou t , though, t h a t t h i s is tantamount t o paying only token acknowledgment t o

turbulence and is, i n t h e strictest sense, cons i s t en t only with the assump-

t i o n of f l a t ice topography. It can b e seen from t h e Smith measurements of

t h e f l u c t u a t i n g stresses t h a t a mean t r anspor t hypothesis, i.e., me defin-

ing an eddy v i s c o s i t y , does not have any real meaning when t h e corresponding

stress components move from p o s i t i v e t o nega t ive a t p laces d i f f e r e n t from

those where t h e zeroes of t h e mean v e l o c i t y grad ien t occur. This remarkable

behavior is displayed c l e a r l y . A t o the r p laces i n t h e l a y e r , t h e eddy v i s -

c o s i t y would be negative, but t h i s r e s u l t is not objectionable, s ince the re

is no reason (as t h e r e would be i n t h e case of molecular v i s c o s i t y , which is

based on thermodynamics) t h a t i t cannot be negative. It is a l s o i n t e r e s t i n g

t o note t h a t t h i s change i n s i g n has been predicted by Deardorff from numer-

i c a l modeling and by higher c losu re schemes f o r atmospheric boundary l a y e r

models i n t h e case of an uns tab le boundary l aye r . A l l func t ions are taken

as t r u e mean va lues and considered as func t ions of t h e depth only. N o real

requirement is made, however, t h a t t he func t ions be so lu t ions of a par t icu-

l a r set of equations, bu t more t h a t toge ther they are rep resen ta t ive of a

turbulen t Ekman boundary l a y e r on a f l a t surface. In t h i s sense, t h e com-

b ina t ion of t h e Deardorff work and f i e l d da t a lead t o a respec tab le evalua-

t i on .

The second non l inea r i ty is more fundamental and is impossible to c i r -

cumvent un less (a) some s o r t of c losu re scheme is put f o r t h o r (b) t h e sys-

t e m is l inea r i zed . S t r i c t l y speaking, both assumptions a r e incorporated

i n t o t h e plan. Closure is involved i n t h e assumption of a turbulen t v i s -

cos i ty , e f f e c t i v e l y parameterizing t h i s complication when determining the

mean flow with no add i t iona l equations.

l a t i n g the equations f o r t h e f l u c t u a t i n g components. This s t e p is made by

per turb ing t h e mean f i e l d as given by t h e above-mentioned turbulen t d i s t r i -

bu t ions , l i n e a r i z i n g i n terms of t h e f l u c t u a t i o n s , and then so lv ing t h e

system numerically.

There remains t h e problem of formu-

Clearly, no asymptotic state can be predic ted by t h i s

method, bu t an understanding of t h e dynamics of such a turbulen t flow can

be approached by studying t h e flow aspec t s by the methods derived from

l i n e a r s t a b i l i t y theory.

wi th in t h e realm of t h e model.

An assessment of brine-driven convection is c l e a r l y

33

c

Classical l i n e a r s t a b i l i t y theory w a s developed o r i g i n a l l y t o expla in

t h e t r a n s i t i o n t o turbulence and the re fo re was concerned with i n s t a b i l i t i e s

i n laminar flows [see Betchov and Criminale, 1967, f o r a review of t h e sub-

ject].

important t o understanding why a laminar flow cannot be maintained as a

func t ion of c e r t a i n parameters.

turbulence i n c e r t a i n kinds of flows, f o r t h e l i n e a r and t h e nonl inear

problems have much i n common when they are examined i n terms of energy argu-

ments. Extending t h e s t r a t egy t o turbulen t flows is no t a t a l l unnatural .

A well-defined turbulen t flow can be s tudied i n much the same way as l i m i t

cyc les (reached by nonl inear balances) are perturbed and inves t iga ted f o r

s t a b i l i t y o r maintenance when c e r t a i n parameters are changed. For s tandard

turbulen t f l a t p l a t e boundary l aye r s , Betchov and Criminale [1964; see a l s o

Landahl, 19671 have already made t h e extension and shown t h a t t h e ana lys i s

can be v a l i d when the proper quest ions are asked. It i s i n t h i s s p i r i t t h a t

t h e turbulen t Ekman problem is being inves t iga ted .

Although t h i s theory could not be rea l ized , i t is never the less

It a l s o he lps i n t r ac ing t h e o r i g i n s of

L i l l y [1966] has made t h e necessary ca l cu la t ions f o r t h e s t a b i l i t y of

t h e n o n s t r a t i f i e d laminar Ekman l aye r f o r two-dimensional dis turbances and

found t h a t , i n con t r a s t with a Blasius l aye r , t h e r e are two poss ib l e modes

of i n s t a b i l i t y . Brown [1970] has extended t h e ana lys i s t o t h e turbulen t

Ekman problem (with a later considerat ion of t h e ramif ica t ions of nonlinear-

i t y as w e l l ) under t h e same condi t ions.

t hese works only from t h e b a s i s of l i n e a r i t y .

source of b r ine a t t h e ice cover, t h e problem is no longer one of t he eigen-

va lue va r i e ty . With inhomogeneous boundary condi t ions, an important pa r t i c -

u l a r so lu t ion is introduced and is caused (driven) by t h e source; t h e source

is prescr ibed a pz-Jiori by specifying t h e mass f l u x as a known funct ion of

t h e va r i ab le s i n t h e plane of the ice cover and t i m e . The manner i n which

t h e governing equations are derived r e t a i n s t h e complete three-dimensionality,

making t h i s p a r t of t h e ana lys i s novel t o t h e problem.

as w e l l as nonl inear i ty , has always been a c h a r a c t e r i s t i c of turbulence t h a t

is essential t o t h e mechanics of t h e flow.

The present study i s s i m i l a r t o

Because t h e r e must be a

Three-dimensionality,

I n s t a b i l i t y i n t h e s t r a t i f i e d Ekman boundary l a y e r has been s tudied i n

atmospheric models, with t h e papers of Brown [1972] and Kaylor and F a l l e r

34

[1972] being representa t ive . However, t h e su r face boundary l aye r of t h e

ocean under t h e ice cover is not s t rong ly s t r a t i f i e d , so t h a t t h e atmos-

pher ic s t u d i e s are not very u s e f u l t o u s here. On t h e o the r hand, t he re is

an i n t e r f a c e i n t h e mean dens i ty f i e l d between t h e mixed l a y e r and t h e pyc-

noc l ine t h a t causes a l a r g e dens i ty grad ien t .

loca ted below t h e region of shear i n t h e turbulen t boundary l aye r , i.e.,

where t h e mean v e l o c i t y is approximately constant except poss ib ly f o r some

remnant of t h e s p i r a l ; t h e Ekman depth is more o r less t h e same as t h a t of

t h e mixed layer . )

f o r c e and subsequent i n t e r n a l wave generation can be predicted.

bases put f o r t h f o r so lv ing t h e l i n e a r f l u c t u a t i o n problem, the mean dens i ty

change is accounted f o r , and t h e s ign i f i cance w i l l be explo i ted thoroughly.

The Smith measurements of t h e f l u c t u a t i n g stresses extend w e l l beyond t h e

depth a t which t h e mean dens i ty changes and no doubt already bear some

imprint r e l a t i n g t o t h i s phenomenon.

(This v a r i a t i o n is usua l ly

The physics connected with t h e add i t ion of t h e g rav i ty

With the

Solutions f o r t h e l i n e a r equations t h a t govern t h e out l ined system

can be obtained f o r a l l t h e dependent v a r i a b l e s (veloci ty , densi ty , pres-

sure) by Fourier-decomposing i n terms of t h e space va r i ab le s of t he plane

of t h e i c e and t i m e , and then i n t e g r a t i n g t h e set of ordinary d i f f e r e n t i a l

equations t h a t determine t h e amplitudes as a func t ion of depth. Correla:

t i o n s (energy, stress, buoyancy) of t hese q u a n t i t i e s can be ca lcu la ted by

averaging t h e products over t h e Fourier v a r i a b l e s and t h e t i m e .

opera t ions determine t h e s t r u c t u r e as a func t ion of t h e parameters t h a t

occur n a t u r a l l y (or as inputs) wi th in the su r face boundary layer .

t h e p o s s i b i l i t i e s , t h e p r i n c i p a l parameters d e a l t wi th are t h e Reynolds and

Richardson numbers toge ther with an angle t h a t descr ibes t h e three-

dimensionality. The Rossby number p lays no r o l e ( i . e . , does not vary) s i n c e

t h e value is f ixed by t h e sca l ing t o ensure t h a t r o t a t i o n is always of t h e

required value i n t h e dynamics.

These

Among

Thermodynamics is not e x p l i c i t l y included i n t h e model. It is thermo-

dynamics a t t h e ice cover t h a t expla ins t h e source of t h e br ine, bu t t h e

details are not r e l evan t and a m a s s f l u x is taken as known; i.e., t h e prob-

lems decouple.

Boussinesq sense, removing any f u r t h e r need f o r considering temperature.

I n addi t ion , t h e f l u i d is assumed incompressible i n t h e

35

For c l a r i f i c a t i o n , t h e inhomogeneous boundary condition can be equated with

an inhomogeneous term i n t h e equations of motion. Spec i f i ca l ly , a known

func t ion of t h e required v a r i a b l e s would appear i n t h e dens i ty equation

and, with t h a t inc lus ion , t h e boundary conditions would become homogeneous.

The two representa t ions are f u l l y equivalent. When viewed from t h i s per-

spec t ive , however, i t can be seen t h a t d i f fus ion of t h e dens i ty by e i t h e r

molecular o r tu rbulen t (eddy d i f f u s i v i t y ) means is not e s s e n t i a l and can be

neglected. It is t h e

ward d i f fus ion of t he

esis of the model.

s i m p l i f i e s t h e study.

A f i n a l comment

Betchov and Criminale

advective t r anspor t of t h e mass t h a t l eads t o a down-

momentum, making t h i s s t e p cons i s t en t with t h e hypoth-

This operation has a f u r t h e r advantage i n t h a t i t

is i n order. As has already been recognized by

[1964] i n t h e computations f o r t h e turbulen t boundary

l a y e r on a f l a t p l a t e , t h e design of t he problem has a drawback t h a t stems

from spec i fy ing t h e eddy v i s c o s i t y as a function of depth alone.

p a r t i c l e t h a t moves from one region t o t h e o ther i n the v e r t i c a l must ad jus t

i ts v i s c o s i t y t o t h a t value given by the eddy v i s c o s i t y d i s t r i b u t i o n . This

seems p laus ib l e i n t h e very-near boundary region inasmuch as t h e adjustment

must be rapid. (This is t r u e f o r f l a t p U t e boundary l aye r s . ) Once ou t s ide

t h e shear zone, t h e v i s c o s i t y is constant and t h e r e is no problem. Neither

is the re a problem throughout most of t h e shear l a y e r , s ince t h e v i s c o s i t y

v a r i e s very slowly. The concern, then, is movement from ou t s ide i n t o t h e

sur face boundary l a y e r (from deeper t o shallower loca t ions ) . This simpli-

f i c a t i o n is j u s t i f i e d because (1) t h e bas i c ve loc i ty is taken t o be steady

and (2) i n a l i nea r i zed theory t h e p a r t i c l e s t h a t do make t h e c ross ing

during a cyc le occupy only a small por t ion of t h e t o t a l o s c i l l a t i n g f l u i d .

The only o the r r e s o r t would be t o r equ i r e , as Drazin [1962] did, t h a t each

f l u i d p a r t i c l e r e t a i n i t s v i scos i ty . Using t h i s cons t r a in t makes f o r more

complicated equations (higher order i n t h e de r iva t ives ) and introduces a

s i n g u l a r i t y at t h e c r i t i c a l l a y e r un less t h e r e is a d i f f u s i n g c o e f f i c i e n t

f o r t h e eddy v i scos i ty . Modeling i n t h i s fash ion does not s e e m warranted

a t t h i s t i m e .

A f l u i d

36

BASIC EQUATIONS

Define a set of axes such that the z is positive upwards and parallel

to the force of gravity; x and y then define the plane of the ice that is perpendicular to z . The mean velocity, pressure, and densify are assumed given by the functions u(z)i + v ( z ) j , F,' and c ( z ) , respectively. The

% % assumption of a turbulent viscosity in addition to the molecular value allows for T(z) = VM + V T ( z ) to be written with vT(z ) considered known, and the kinematic notation is permissible by the Boussinesq approximation.

The governing equations for the fluctuations are obtained by superimposing small perturbations onto the mean field, linearizing, and then subtracting

the mean field from the instantaneous equations.

Let us adopt the following notation: velocity vector components, - - (c + u', V + v ' , w ' ) ; pressure, P + p ' ; and density, quantities are functions of (x, y, z , t). The derivation of the equations is now straightforward, except possibly for the stress term in the momentum balance inasmuch as it does not commonly appear. culty if it is recalled that the shear comes from the divergence of the

tensor

+ p'. All primed

This step presents no diffi-

and that the fluid particle is not constrained to preserve its viscosity, i.e., there are no viscosity fluctuations. Thus, expanded, we will have

'Since will not appear in the equations for the linear perturbations, its functional dependence is immaterial. It would be expected, however, to be consistent with the definition of the mean field.

37

w h e n z 3 i s taken as z and U; = W ' .

w i l l read

I n these terms t h e governing equations

au' au ' awl

ax ay az + - + - = o -

a w l aw aw - +v-- +Ti-

a t ax aY

ae ae ae ' dz

a t ax ay dz - +v- +T-+-w' = o

- - where 8 ' = p'/po, 0 = p/p with p 0 defined as a re ference dens i ty and f t h e 0

r o t a t i o n f a c t o r t h a t includes the f a c t o r of 2.

The set of equations (1)-(5) can be transformed i n t o a set of ordinary

d i f f e r e n t i a l equations as a r e s u l t of t he l i n e a r i t y and the f a c t t h a t the

c o e f f i c i e n t s depend upon z only.

t h e v a r i a b l e s x, y , and t by assuming an expansion of t h e form

Accordingly, w e can Fourier decompose i n

e t c . , f o r v ' , w' , p ' , 8 ' . Making t h e s u b s t i t u t i o n , w e now have

38

= o

where the pressure amplitude, p^, is in dynamic units, i.e., per unit density. The primes denote derivatives with respect to z and A = d2/dz2 - (a2 + y2), the Laplace operator.

Solving the above equations in the full form has not been done. Instead,

additional simplifications have been made and then solutions are sought to the reduced system. For example, for the nonstratified problem, Lilly [1966] per-

formed rotations (originally suggested by a Faller experiment) of the

coordinates and then neglected the dependence of the dependent variables on

one of the new variables. This is tantamount to doing the problem for two- dimensional disturbances at the outset. Normally this step is not critical in stability analysis because of the well-known Squire theorem that demon- strates that it is a two-dimensional disturbance that has maximum amplification, but the theorem is not valid for the Ekman problem. Under

the current scheme, it will not be necessary to reduce the problem in any way. This is done by changing the system to an alternate problem which,

except for the stratification is no more complex than the Lilly computations.

Define the transformations

and (6) - (10) can now be rewritten as

Eliminating the pressure between (13) and (15) and invoking (12) produce

&e set of three equations for u, G, and 0 . h

The transformations, as given by (ll), can be interpreted as a projection of both the mean and the fluctuation velocity onto the z axis, i.e., and 6 are parallel. As a result, equation (12) for the incompressibility condition is analogous to a two-dimensional problem and it is by this means that deriva-

tion of equations (17)-(19) is eased considerably. The component 3 is more analogous to a vorticity and is.thus the reason that the pressure is absent

from equation (14). to assuming that the velocity can be decomposed into the sum of an irrotational and a solenoidal part. known, the velocity proper can be determined. new system is not as yet useful because of the mixed appearance of both Cartesian (a) and polar ( G ) coordinates in the equations.

In certain respects, the defining of and 5 is analogous

Once these solutions for the decomposed portions are As shown in (17)-(19), the

Introducing the

40

additional definitions

I A

W l J

a a - = - I

removes -this obstacle and the desired result becomes

..,.., ..,....., = fv' - iage

- .., .., 0' .., (U - c)0 = i - lJ. ... a

This set of equations is now expressed solely in terms of (%,$), the polar

wave coordinates. velocity, 3, transforms by the cosine rather than being unchanged, as would be the situation when Squire transformations are normally used. effects and viscosity also change by the cosine, whereas gravity scales as the square of the cosine.

It should be noted that the vertical component of the

The rotation

41

A comparison of the Lilly equations will show a formal correspondence

for (21), (22) when stratification is removed. To find answers numerically, Lilly chose the scale so that the ratio of the Reynolds to the Rossby numbers was unity, selected values for the rotation angle (E in the Lilly notation) and the Reynolds number, and then computed e as a function of 01.

present case, an angle is still required (+) as well as a Reynolds number

(together with a Richardson number when the stratified problem is treated) to

find c as a function of 5. Such computations, however, are necessary only when the eigenvalue problem is to be solved. Otherwise, judicious choices of

frequency (w) and scales (6 ) must be selected for physically meaningful results for the noneigenvalue problem. the inhomogeneous system can, in principle, be solved for any frequency or scale. The eignevalue problem, by contrast (and definition), can only admit

particular values and still satisfy the boundary conditions.

In the

This remark is to be taken with caution since

There are several ways to solve (21)-(23) for 2, 3 , and g. Either the

system of three equations--or two, since 6 is readily eliminated between (21) and (23)--is programmed for numerical integration or a single equation for one unknown--pref erably $--is obtained and then the integration is performed. The best choice is not always apparent and depends on the adroitness of the programmer and the values of the parameters used. For the moment, let us consider the task done and $ and are known. The value of is available from (23). An inversion of the transformations (11) and (20) provides the original variable amplitudes. Having done this we have

a - a

.., 72 = , w = cos @A),

and 6 = i still follows.

42

A nondimensional form of the equations can be established by choosing

a length scale, 2, and a time scale based on velocity q, = F(0)2+ v(0)2)y2.

Let z = 20 be the new coordinate in the vertical, K = GZ the new wave

number radius, and w = q 0 U , 3 = q0v, e = qoC, L? = q U, P = q V the velocity

components,respectively; 8 = 6 and 0 = 0 are already nondimensional.

parameters can be defined as

0 0 - The

... q,z Reynolds number E Re = 7

ve

- st O q;

Richardson number E J = -

40 " 3 ! Rossby number 5 = -

where we have written

; = GM + Cr = GM + GesCz,

and, in terms of the nondimensional coordinates

Substitution gives

43

I - .

Now, if the ratio Re/Ro is set equal to unity in (27) and (28) we will have

... ... .., V Ro 'e 40 e

V e % Thus,

in terms of the turbulent viscosity.

= (ve/f) in accordance with the definition of the Ekman depth but

For the Smith data, Z = 8 meters.

The important topic of boundary conditions must be discussed. An

inspection of the governing equatiom reveals that the system is sixth-

order in terms of the derivatives with respect to z . Hence, we are free

to designate six conditions in terms of the dependent variables. It

should be remarked in passing that the system would have been eighth-order

if the diffusion of mass had not been neglected. And, of course,

diffusion would have resulted in the Schmidt number to enter as another

parameter. Normally, for a boundary layer with homogeneous boundary

conditions, the requirements are at the boundary (2 = 0 here),

ut = 2)' = W ' = 0. Far away, i.e., z = - 0 3 , U' = 2) ' = W ' = 0 . The pressure

p ' and the density p ' are automatically determined throughthe velocity

constraints.

Changing to the Ekman layer with homogeneous conditions, the

determination can still be made in terms of nonrotating boundary layer

physics. It is interesting, though, that Lilly based his work on an

alternative form of the z =-- condition (the z = 0 remained the same).

Specifically, he decided that there should be neither any flow in the z

44

direction nor any torque at infinity.

and aui/az = 0 at z = 00.

and is impossible to fulfill, along with the z = 0 values, with only a

sixth-order problem. It will be seen, however, that the problem does

satisfy all of the conditions but only in a fortuitous fashion.

or not the no-torque conditions are interchangeable with the vanishing of the

velocity is an unresolved question. There is reason for doubt, judging

from initial numerical solutions of the equations made by Criminale and

Gustavsson (in preparation), where it was impossible to duplicate the Lilly

eigenvalues at low Reynolds numbers.

In his terms this means W ' = 0

Strictly speaking, this constitutes four conditions

Whether

Putting the homogeneous boundary conditions in the transformed

variables of equations (21)-(23) (or 27-29) leads to = 3 = 0 and 3' = 0

by the fact that u must vanish at both locations in z .

that 6 = 0 will also come about from the w = 0 condition.

It can be seen

The inhomogeneous problem is arrived at by recognizing that there is

Taking the flux to be a known function of a mass flux at the ice cover.

( z , y , t ) we can write

(28) ae - = F ( x , y , t ) az

at Z = 0 and F is known. In Fourier space the equivalent of (28) is

ij '(0) = f(a,y,w) (29)

where

45

It is assumed that a form f = fo(G,4,u) can be determined.

prevents the full determination of the problem in terms of the polar

coordinates i, 4.

interpreted as a new condition on w at z = 0 since and u, i.e., c' must still vanish at the ice cover, Operating on (23) enables us to

write

Any other case

Using equation (23) it can be seen that ( 2 9 ) can be

ox, in a slightly different form, (31) is

The infinity conditions are unaltered, demonstrating that all perturbation

dependent variables must be proportional to f This, then, defines the

particular solution with f 0 '

assuming the role of the driver. 0

46

TOWARDS SOLUTIONS: FUNDM4EXTAL PHYSICS

Progranpning f o r t h e numerical i n t e g r a t i o n of t h e complete system

given by equations (21)-(23) (or 27-29) toge ther wi th t h e determination

of necessary c o e f f i c i e n t s is nearing completion and w i l l be reported as

soon as enough is known. I n t h e meantime a g r e a t dea l can be learned by

examining t h e equations under various r e s t r i c t i o n s o r i n c e r t a i n regions

of t h e flow. This is poss ib l e because t h e system i s l i n e a r and has t h e

support of a g r e a t wealth of known p rope r t i e s f o r s p e c i a l types of

func t ions and opera t ions from mathematics.

L e t us adapt t h e no ta t ion t h a t t h e Vor t ic i ty t h a t is due t o t he

f l u c t u a t i n g f i e l d is defined as

I n terms of each component of t h e v o r t i c i t y t h i s means t h a t

+' 3 = % - lA' Y '

where 1 is i n t h e X d i r e c t i o n , 2 i n t he y, and 3 i n t he z .

s o l u t i o n f o r IT i s as t h e o the r va r i ab le s and the re fo re t h e Fourier i r ep resen ta t ions become

The form of t h e

47

A b i t of manipulation and t h e use of t h e d e f i n i t i o n s of t h e transformed

v a r i a b l e s leads t o

= -ia; . ( 3 6 ) 3

The physics of equations (21) and (22) is now c l e a r e r , f o r ( 3 4 ) - ( 3 6 )

show t h a t t h e equations govern the v o r t i c i t y , where t h e components

V% atid are of t h e parabolic type. The operator i n bracke ts contains

t h e d i f fus ion , convection,and unsteady terms f o r each component. Other

terms can be in t e rp re t ed as follows:

(a) 2,;; ?I;. The mean v o r t i c i t y components come from V I , ? I , making

these terms of t h e per turba t ion v o r t i c i t y t r anspor t by t h e ve loc i ty 3.

(b) ? G I ; 5 ' . These are couplings of V% t o t he o the r component of

the v o r t i c i t y 3 through t h e Cor io l i s e f f e c t .

(c) <%6. I n general t h i s is a production due t o t h e Bjerkness e f f e c t .

(d) Products wi th grad ien ts of <. An analogue he re i s harder t o

r e a l i z e , b u t must be r e l a t e d t o t r anspor t due t o v a r i a t i o n i n u. For example,

t h e 7' products c l e a r l y represent vertical movement by t h e f a c t t h a t t h e r e

is a gradien t of V% and 3 i n t h i s d i r ec t ion .

The equation f o r t h e dens i ty , (23) , merely states t h a t t h e change of

dens i ty of a f l u i d p a r t i c l e is due t o t r anspor t from t h e mean gradien t and 3.

Returning t o t h e set of equations (12)-(15),operations can be made t h a t

are equivalent t o taking t h e divergence of t h e momentum equations. Having

done t h i s w e have

48

where p/G2 = @/a2 transforms the pressure.

(Poisson) and each term in (37) influences pressure fluctuations by:

This equation is Laplace-like

-3.. - (a) 2iaU'w. Local curvature of streamlines in the direction of the

mean flow interacting with the variation of the mean speed (shear) in the

vertical . (b) &?;. Again curvature effects, but interaction with rotation

and curvature caused by vortices.

(c) GG1. Local density change and gravity force coupling in the

vertical.

(d) Eddy viscosity terms. No analogue presently available, but seems

to be "source like. It

The only analytic solutions that are readily available come from

additional simplifications for the coefficients in the equations. Specif-

ically,

that part of the flow below the zone of shear and the break in density

distribution.

the numerical integration; cf. Betchov and Criminale [1967]. Accepting

this basis we immediately see that all derivatives of all mean variables

must be very small save one, g ' , and it should be approximately

constant.

we can define a region as the free stream and it is meant to be

Practically speaking this is necessary in order to begin

Call 5' = l' and the equations read

49

r - (Go - e ) 6 = i z w a

where 6 = go and < = < i n t h e f r e e stream. The pressure equation becomes 0

The equations now have constant c o e f f i c i e n t s and are coupled through

per turbat ion-perturbat ion in t e rac t ion only. Moreover, i n t h i s form it

is i r r e v e l a n t whether o r not t h e system of t h r e e equations or one equation

f o r one unknown i s solved. For t h i s purpose, def ine t h e opera tors

... i V o

L = - ... A + (Uo - e ) a

and the system becomes

f" - "2 D ( 2 ) + L(U) = 0 a

(43) r - -i - ... w + (go - c,e = 0 . a

The grouping i n (43) is exac t ly t h e same as i t would be if we had a

matrix mult iplying a vec tor wi th components ($,5,6). Thus, t h e system is

50

-jD r- w

... V

G

c - ‘ L D L a2

0

Expanding ( 4 4 ) yields a complex operator

= 0.

Eor any one of the three

(44 1

XZ dependent var+ables.

root equation for h given either by (a) the determinant of the coefficients

of the above matrix or (b) a direct substitution in the expanded operator.

Even though the order of the system is sixth, it turns out that the

polynomial for h is cubic in h2.

readily met for any growing exponential solution, determined by any

negative real part of A , are suppressed by setting the corresponding

coefficient to zero. It is interesting to note that this requirement

also insures that all derivatives of the solutions in the free stream will

vanish as well.

All solutions will be proportional to e with the

The infinity boundary conditions which are

It is convenient to think of the operator, L , as the vorticity operator.

In other words, if there were no rotation or stratification, the vorticity

components would be uniquely determined from ( 4 3 1 , indicating that vorticity

is convected and diffused from the parabolic nature of L.

be thoughtof as the pressure operator from (41).

is seen that the pressure is determined everywhere within the flow--once

the value at the boundary is given--by the properties of the Laplace equation.

In this case, the master operator factors as L { L ( A ) ) for the determination of

the velocity fluctuations showing that they stem from both a viscous part

Similarly, A can

With this reduction, it

51

o r v o r t i c i t y and an i n v i s c i d o s c i l l a t i o n due t o pressure.

r ep resen t s t h e normal state of a f f a i r s f o r t h e free stream of standard

boundary l a y e r s where n e i t h e r r o t a t i o n nor s t r a t i f i c a t i o n e f f e c t s are

present. Moreover, i n t h e case where t h e r e is no pycnocline below t h e

shearzone (and f a r away from t h e ice boundary), t h i s s impl ica t ion would

a l s o hold.

Such a case.

I n the inv i sc id l i m i t v o r t i c i t y vanishes.

When s t r a t i f i c a t i o n alone is returned t o t h e constant c o e f f i c i e n t

problem, t h e opera tors can s t i l l be factored, bu t i n a d i f f e r e n t form,

namely,L{(co - c)L(A) - i r } . i n e r t i a by the coupling of two f i r s t - o r d e r ( i n time) equations. The

o r i g i n a l p a r t of t h e viscous so lu t ion remains unchanged but governs only

As might be expected, t h e system acqui res

t h e component of the v o r t i c i t y . The remaining f a c t o r is involved and

i s due t o t h e nonbarotropic p rope r t i e s of t h e f l u i d . I n a l i k e fashion,,

t h e pressure i s no longer determined by t h e Laplace opera tor , showing t h a t

t h e modif icat ion of t h e f a c t o r s e x i s t s even f o r an inv i sc id f l u i d . Indeed,

i n t e r n a l waves possess v o r t i c i t y .

With no s t r a t i f i c a t i o n but r o t a t i o n , t h e system a l s o has i n e r t i a by

v i r t u e of i n e r t i a l o s c i l l a t i o n s . Fac tor iza t ion of t he opera tors is no

longer poss ib l e s ince t h e p a i r of equations f o r t h e v o r t i c i t y components

are coupled by t h e r o t a t i o n f ac to r . Here

is t h e opera tor and aga in v o r t i c i t y is poss ib le even i n t h e i n v i s c i d l i m i t .

52

I n addi t ion , t h e v o r t i c i t y must possess two components (planar vec to r ) ,

i n d i r e c t c o n t r a s t t o t h e previous cases where one component i s s u f f i c i e n t

t o desc r ibe t h e motion i n these t e r m s .

Any o the r combination involves t h e f u l l opera tor

i n t h e f r e e s t r e a m .

f a c e t s of t h e physics w e l l beyond t h e su r face boundary l aye r . This state

A s a r e s u l c , t h e r e must be an i n t e r p l a y among a l l

i s assured by having only t h e mean dens i ty vary as i s t h e case wi th a

pycnocline. When viewed from wi th in , i .e., looking down from the i c e

cover, any r e s u l t a n t motion must respond t o t h e f r e e s t r e a m and a d j u s t

t o t h e su r face boundary condition, should one e x i s t . This i s t h e subjec t

of work t h a t is now i n progress.

ACKNOWLDGMENT

The au thors are g r a t e f u l t o J. Dungan Smith and R. A. Brown f o r

t h e i r review of t h i s aspec t of t h e work.

t o acknowledge the support from t h e National Science Foundation, Grant

In addi t ion , w e would l i k e

OPP 71-04031 and from t h e Air Force Off ice of S c i e n t i f i c Research, Grant

AE'OSR 74-2579.

53

REFERF3lCES

Betchov, R., and W. 0. C r i m i n a l e . 1964. Osci l la t ions of a turbulent flow. Physics of Fluids, 7 , 1920-1926.

Petchov, R., and W. 0. Criminale. 1967. Stabizcty of ParaZZeZ F l o w s . New York: Academic Press.

Brown, R. A. 1970. A secondary flow model f o r t h e planetary boundary layer . J. Atmos. Sei., 27, 742-757.

Brown, R. A, 1972. On t h e i n f l e c t i o n point i n s t a b i l i t y of a s t r a t i f i e d Ekman boundary l aye r , J. Atmos. Sei . , 29, 850-859.

Deardorff, J. W. 1972. Numerical invest igat ion of n e u t r a l and unstable planetary boundary layers . J. Atmos. Sei. , 29, 91-115.

Drazin, P. G. 1962. Onstabi l i ty of p a r a l l e l flow of an incompressible f l u i d of var iab le densi ty and v iscos i ty . Proc. Cambridge Phil. Soc., 58, 646-661.

Howard, L. N. 1961. Note on a paper of John W. Miles. J. FZuid. Mech., IO, 509-512.

Kaylor, R., and A. J. Fa l le r . 1972. I n s t a b i l i t y of t h e s t r a t i f i e d Ekman boundary layer and the generation of internal waves. J. Atmos. SI&., 29, 497-509.

Landahl, M. T. 1967. A wave guide model f o r turbulent shear flow. J . Fluid Mech., 29, 441-459.

L i l l y , D. K. 1966. On the i n s t a b i l i t y of Ekman boundary flow. J . Atmos. Sei., 23, 481-493.

McPhee, M. G., and J. D. Smith. 1975. Measurements of t h e turbulent boundary layer under pack ice. AIDJEX Bulletin, 29, pp. 49-92.

Miles, J. W. 1961. On t h e s t a b i l i t y of heterogeneous shear flows. J . Fluid Mech, , 10, 496-508.

Smi th , J, D. 1974. an ice-covered ocean. PhysicaZ Processes Responsible for the DispersaZ of PoZZutants i n the Sea, with SpeciaZ Reference t o the Near Shore Zone. Rapports e t Proces-verbaux Series (ed. J. W. Talbot and G. Kullenberg), 67,

Turbulent s t r u c t u r e of the surface boundary l a y e r i n Proceedings of the 1972 ICES Symposium on the

53-65.

54

INTEGRATION OF ELASTIC-PLASTIC CONSTITUTIVE LAWS

by R, Colony and R, S. Pritchard

Arctic Ice Dynamics Joint Experhent University of Washington, SeattZe, Washington 98105

INTRODUCTION

This work began as a descr ip t ion of the difference scheme used t o

in t eg ra t e t h e A I D J E X sea ice cons t i t u t ive model.

apparent that the present so lu t ion technique can be applied t o models other

than the A I D J M sea ice model. The in tegra t ion scheme may be used i n other

f i n i t e d i f fe rence codes, i n f i n i t e element codes tha t allow general e l a s t i c -

p las t ic material behavior, and i n three-dimensional models as w e l l as i n the

two-dimensional model appropriate f o r pack ice . It is not pa r t i cu la r ly

usefu l f o r impl ic i t schemes i n which the solut ions must be found i n a l l

c e l l s simultaneously, but i t may be used as pa r t of many difference schemes

t h a t e x p l i c i t l y advance the so lu t ion a t one c e l l independently of surrounding

c e l l s .

A s we progressed it became

The AIDJEX sea ice model assumes elastic-plastic behavior [Coon e t a l . ,

19741.

problems [Pritchard and Colony, 19741.

has been modified t o include t h e elastic strain during p l a s t i c flow [Pritchard,

1974, 19751, and i n t h i s paper we present a d i f fe rence scheme that allows us

t o in t eg ra t e t h i s modified cons t i t u t ive law.

Last year w e presented a difference scheme fo r solving one-dimensional

Since t h a t t i m e t he kinematic r e l a t i o n

A common technique f o r in tegra t ing e l a s t i c -p l a s t i c cons t i t u t ive l a w s

is t o der ive an incremental l a w between stress rate and s t re tch ing .

modulus is found by d i f f e ren t i a t ing the y ie ld function, thereby f inding a

re la t ionship between stress-rate components. This approach is used i n many

f i n i t e element programs f o r f i n i t e deformation of e l a s t i c -p l a s t i c materials

[e.g., Marcal, 1967; Hibbl t t et al., 1970; and O s i a s and Swedlow,

However, this method cannot be used f o r t he AIDJEX ice model because i t does

The

19741.

55

not always harden [Marcal and Kfng, 19671. Our model may s o f t e n o r may be

i d e a l during c e r t a i n deformatlon h i s t o r f e s .

programs a l s o use t h e d i f f e r e n t i a l form of t h e y i e l d cons t r a in t [Trul io e t

al., 19691. It has been observed Pn f i n i t e d i f f e rence programs that t h e

stress state may d r i f t away from t h e y i e l d cons t r a in t when t h e material is

p l a s t i c f o r a long time.

s a t i s f i e s t h e y i e l d cons t r a in t d i r e c t l y r a t h e r than i n a d i f f e r e n t i a l form.

W e have found it convenient t o use an i m p l i c i t d i f f e rence scheme developed

by Gear [1971] t o i n t e g r a t e systems of nonlinear ordinary d i f f e r e n t i a l

equations subjec t t o a lgeb ra i c cons t r a in t s ,

Several f i n i t e d i f f e r e n c e

These two problems lead us t o an approach' t h a t

The development and use of t h i s d i f f e rence scheme f o r t h e AIDJEX sea

i c e model is one of two main cont r ibu t ions of t h i s work. The o the r involves

t h e treatment of r o t a t i o n i n general deformation f i e l d s .

a s p e c i a l s o l u t i o n technique t h a t is both more e f f i c i e n t and more under-

s tandable because t h e r o t a t i o n has been eliminated by transformation and

t r e a t e d sepa ra t e ly from t h e a c t u a l i n t e g r a t i o n of t he e l a s t i c - p l a s t i c

response.

W e have included

\

. ELASTIC-PLASTIC CONSTITUTIVE LAW

I n t h e e l a s t i c - p l a s t i c c o n s t i t u t i v e l a w now used i n t h e AIDJEX sea

ice model, e l a s t i c response is i s o t r o p i c [Coon e t a l . , 19741:

where e ,.. is t h e e l a s t i c s t r a i n ( t o be defined by kinematics); 2 is t h e Cauchy

stress r e s u l t a n t i n excess of hydros t a t i c equilibruim; 9' is set equal t o

9 - 7 - 1 tr g, t he dev ia to r i c stress; and M,, M, are t h e bulk and shear moduli,

v a r i a b l e s which depend on t h e instantaneous thickness d i s t r i b u t i o n . The y i e l d

constragnt is a l s o i s o t r o p i c and contains one parameter [Coon e t a l . , 19741:

where I = tr (5 is t h e f i r s t moment of stress; II' = tr g'g' is t h e second

moment of t h e stress devia tor ; and p* is t h e y i e l d s t r eng th , a scalar ...

56

parameter depending on the deformation history through the thickness distribution,

The plastic flow rule is obtained by maximizing plastic working and is the associated flow rule [Coon et al,, 1974; Pritchard, 19751.

where D is the plastic stretching and X is a positive scalar. -P The linearized kinematic relation is given in the form [Pritchard, 19751

W e + e W = D - D (4) c - ...- -- - -P where

the kinematic relation assumes that elastic rotation is zero, a notational

convenience which does not affect the stress response.

and ,W are the stretching and the spin. We note that this form of

A SMPLIFYING TRANSFORMATION

Rather than introduce a difference approximation to the entire set of

governing equations, we make a transformation of variables that both simpli-

fies the equations and provides an interpretation of rotation due to spin. We use only the small elastic strain, linear kinematic approximation, but

the transformation is applicable in the more general case [Pritchard, 19751.

We introduce the rotation operator Q = &(t) which is to be determined ... as the solution of

- 6 = - - W Q (5)

subject to appropriate initial conditions.

shown to be

The components of Q may be -

~ = (cos f3 - sin 6

s i n cos f3

where f3 = -w if

and f3 is p o s i t i v e f o r a clockwise r o t a t i o n ,

then used t o eva lua te t h e r o t a t e d s t r a i n tensor given by

This orthogonal opera tor is

(7 1 t E .., = ~9

W e should po in t ou t that although ,W is the s p i n of t h e body, f is not

t h e r o t a t i o n i n t h e sense of Truesdell and No11 [1965],

defined r e l a t i v e t o a re ference configuration and 4 is defined independently

of a re ference conf igura t ion , but even apa r t from that t h e r o t a t i o n

i t s rate of change are given by Truesde l l [1966] as

The r o t a t i o n is

and

Rt = W - - 1 R(fi U-l - u-' fi)Rt _.., - 2 . . , - - 1 w . . ,

where Y is t h e r i g h t s t r e t c h tensor. It s u f f i c e s t o say t h a t g is not r i g i d

r o t a t i o n unless

constant. Therefore, we do not attempt t o i n t e r p r e t r o t a t i o n

and ,V are commutative, which is t r u e i f eigenvectors are

i n terms of g. Under t h e change of s t r a i n we f i nd t h e co ro ta t iona l e l a s t i c s t r a i n

rate t o be

- t e - g g + g y = g2g

with the governing equation f o r E taking t h e form

( 8 ) ' t _

We now introduce a similar r o t a t i o n operation f o r t h e stress tensor

8 3 8 - D - P p

by de f in ing

. (9) t p = 8 ,og

Our i n t e n t is t o e l imina te t h e sp in from t h e remaining equations used t o

eva lua te stress. The remaining elements of t h e e l a s t i c - p l a s t i c model are

transformed by t h e change of va r i ab le s . W e study each ind iv idua l ly .

Elastic response takes the ro t a t ed form

E = - ' 1 t r ~ + L ~ r .., 4M1 - - 2M, -

1 where tr 5 = tr g, 2' = S - i n general on t h e i n v a r i a n t s of r o t a t e d stress 8.

,1 tr g, and t h e moduli MI and M, may depend

The y i e l d c o n s t r a i n t i n eq. (2) is unchanged because i n v a r l a n t s of

9 and 2 are i d e n t i c a l . That is,

58

I = t r S

II' = tr ??Sr - The p l a s t i c flow r u l e is ro t a t ed t o r e f l e c t t h e change of stress

-

v a r i a b l e . I f w e in t roduce t h e r o t a t e d p l a s t i c s t r e t c h i n g L L -

Dp - 4 !?p 9 eq. (3) becomes

But i n terms of t h e r o t a t e d stress S, t h e flow r u l e is ..#

(11 1

s o t h a t i t remains i d e n t i c a l i n form t o (3).

I n summary, t h e elements of t h e transformed e l a s t i c - p l a s t i c c o n s t i t u t i v e

l a w are given by t h e transformation r e l a t i o n (5), t h e elastic response (lo), t h e y i e l d c o n s t r a i n t (2), t he flow r u l e ( l l ) , and the kinematic r e l a t i o n

i ..# = Q - 0 , (13 1 t where t h e ro t a t ed s t r e t c h i n g D L = f

no ta t iona l s impl i c i ty .

J? Q is introduced f o r consistency and

DECOMPOSITION INTO ISOTROPIC AND DEVIATORIC PARTS

I n elastic material models, and i n p l a s t i c i t y models t h a t do not allow

p l a s t i c d i l a t a t i o n , t h e response is usua l ly decomposed i n t o i s o t r o p i c and

d e v i a t o r i c components. This decomposition is p a r t i c u l a r l y s i m p l e because

t h e e l a s t i c bulk and shear moduli a f f e c t only one term and t h e p l a s t i c flow

s i m i l a r l y can affect only t h e shear response. I n our more complicated

flow r u l e t h e equations do not uncouple, but some s i m p l i f i c a t i o n does occur.

Perhaps t h e s t ronges t argument i n favor of decomposition is seen i n Appendix

4, where w e f i r s t w r i t e t h e component equations without decomposing only t o

f i n d later i n that development t h a t t h e most e f f i c i e n t va r i ab le s t o use from

a computational s tandpoin t are t h e stress trace and t h e d e v i a t o r i c components.

59

The rotation due to spin is unaffected by the decomposition.

response due to stretching that is improved by rewriting in this form.

It is only the

The elastic material response becomes

t r g = - l I 2Ml

where E' is the rotated elastic strain deviator ..,

The plastic flow rule takes the form

tr QP = -2XOI

gpr =

where we have used the fact that

and the subscript notation is used to denote partial differentiation. The yield constraint does not change.

Thus

This = a$/aI and 0111 = a@/aII'. reflects the fact that we have already expressed the loading function in terms of the appropriate stress invariants.

It is instructive to combine all elements of the material model into a set of coupled nonlinear ordinary differential equations in which stress

(I, 8') and the multiplier (A) are the only unknowns and D .., is the assumed

forcing function. These equations take the form

d 1 - -I + 2AOI = tr Q dt m.1 - - G I + 2X@,,, g' = Q' d 1 dt 2M,

subject to the algebraic constraint given by eq. (2). rotation transformation must also be considered in order to also determine

the actual stress 2.

Of course, the

60

THE APPROXXMATION OF THE DTFPERENTTAL-ALGEBRAIC EQUATIONS

The kinematic relation E = D - D and the yield constraint -P $(I, 11', p*) 5 O constitute a system of simultaneous differential-algebraic

equations. Introducing the stress-elastic strain relationship (1) and the

flow rule (3) reveals the dependent variables of the coupled system to be

S .., and A . In principle this differential-algebraic system could be trans-

formed into a system of ordinary differential equations by suitable differ-

entiation OK by the elimination of certain of the variables. However,

Gear [1971] has shown that certain implicit integration formulas naturally accommodate the additional algebraic equations. In this formulation there

is no need t o distinguish between the differential equations and the algebraic

constraints, because the difference approximations of all the equations

render them algebraic.

- -

The kinematic relation

must be replaced by a difference approximation.

equation by integrating eq. (12) over the time step (t"-l, t") :

We derive this difference

t" t" t" - dt = e(t)dt - /,,_, gp(s,A,t)dt where we have been careful to indicate the dependence of D on the solution -P

(and A ) .

The integral of the plastic stretching I '" D(t)dt is estimated from tn- 1 the midpoint rule for quadrature,

D(t)dt A A t D(tn - A t / 2 )

This rule is employed because the leap-frog scheme gives the total stretching

at the midpoint time, t =

ing in the interval (&"-I, t") is approximated by

tn-' + A t / 2 . The integral of the plastic stretch-

61

where 0 is a center ing parameter 0 L 8 5 1 t h a t allows us to use e i t h e r an

e x p l i c i t i n t e g r a t i o n method (6 = 0) o r an i m p l i c i t method (8 > 0). I f

8 = 1/2, t h e scheme is cons i s t en t t o second order r a t h e r than f i r s t order

f o r o the r values. I f 8 = 1, fewer func t ion eva lua t ions are required. The .

m u l t i p l i e r x appearing i n D may be evaluated a t e i t h e r t o r t . For

t h e case 8 = 1 (which w e have used) t h e choice is immaterial. When

0 5 8 < 1, then x must be provlded as an i n i t i a l condition; and i f 8 = 0,

then may not be independently provided as an i n i t i a l condition.

r e s u l t i n g d i f f e rence equation is

n- 1 n- 1 n -P

The

n n n where w e have shortened t h e no ta t ion f o r ep($,h ,t ) t o ,Dp.

The elastic response allows us t o e l imina te s t r a i n E" from t h e set of

v a r i a b l e s by using

1 = 1 1 tr S" + - (grin 4 M , - - 2 M ,

n The y i e l d cons t r a in t must be s a t i s f i e d a t t i n t he form

where p* is allowed t o change i n t i m e a r b i t r a r i l y ( e i t h e r i nc rease o r

decrease). The p l a s t i c s t r e t c h i n g i s evaluated i n terms of

where ac$/aI and ac$/aIIf are known funct ions of t h e i n v a r i a n t s (I, II') and

p* f o r any given y i e l d function c$ = $(I, 11', p * ) .

Equations (151, (161, (17.1, and (18) are seen t o be wholly a lgeb ra i c

i n t h e unknowns The s o l u t i o n of these a lgeb ra i c equations is

d e t a i l e d f o r a teardrop y i e l d curve i n appendixes 3, 4, and 5 .

and An.

The center ing parameter 8 is chosen i n p r a c t i c e t o g ive accura te ,

s t a b l e , and e f f i c i e n t algorithms.

schemes (e = 0) r e q u l r e t i m e s t e p l i m i t a t i o n s t o ensure s t a b i l i t y , Since

we must s a t i s f y t h e y i e l d cons t r a in t a t t i m e tn t h e scheme must be i m p l i c i t ;

there is nothing t o ga in by using an e x p l i c i t assumption t o eva lua te t h e

Gear 119711 has shown that e x p l i c i t

62

p l a s t i c s t r e t ch ing . The choice between a centered scheme (0 = 1 /21 and a

backward scheme (0 = 1) is not clear, The so lu t ion methods described i n

appendixes 3 , 4 , and 5 may be appl ied i n e i t h e r case.

i n p r a c t i c e because fewer func t ion evaluat ions are required.

W e have used 8 = 1

ALGORITHM

I n t h i s s e c t i o n we

FOR INTEGRATING DIFFERENCE EQUATIONS

o u t l i n e t h e set of operat ions required t o increment

t he stress t o t h e next t i m e s tep .

showing t h e sequence of these s t eps . I n t h e remainder of t h i s s ec t ion w e

descr ibe what is done i n each s t e p i n more d e t a i l .

I n Figure 1 we present a flow char t

We assume t h a t t h e o ld value of stress cf-’, t he ve loc i ty grad ien t n-% given by

known.

and p-’ and a l s o t h e material p rope r t i e s p * , M,, and M, are ,v

The r o t a t i o n is evaluated by assuming a reference configurat ion aligned n-% with t h e material a t t i m e t , t h e midpoint of t h e cycle. The constant of

i n t e g r a t i o n def in ing B (and f) has been shown i n Appendix 1 t o be immaterial

t o determining t h e stress tensor which means we show t h a t the d i f f e rence

equat ions are independent of a re ference configurat ion (or o r i en ta t ion ) .

For our choice of re ference configurat ion we write t h e r o t a t i o n from t t o tn as

n -4

cos AB - s i n AB

s i n AB cos AB - . = (

where AB = - A t Q / 2 may be in fe r r ed by consider ing components of Q =

in tegra ted through t h e i n t e r v a l ( t n 4 , tn).

hal f of t h e cycle , ( t n - l , tn-’), t h e r o t a t i o n is seen t o be

9 ... Simi lar ly , during t h e f i r s t

cos AB s i n AB

- s i n AB cos AB

I n t h e s o l u t i o n algorithm, only one r o t a t i o n operator is needed because

63

-I

Compute the trmefamation mstrix 8 from g-4 and determine the roteted stresstat the old t i n

5 h - v e , s - q .." z q

I

Mg s psn: ' 0 The response is

Set k - , e: - ,O The rotated stress at to bas

L

I

9 - 0 ?he a?reas is at the vertex of the yield

Rotate new strees tensor and plastic stretchins

m 9 ,s"gt, !$ Q Ql Qt I

Newton's Method 1

l

Fig. 1. Flow chart of computational algorithm.

64

The f i r s t s t e p of the so lu t ion algorithm i s the r o t a t i o n of the s t r e s s

i n t o 9-': a t t n - l Y

S"" Y = Cf $"' (Qn)t -c

n- 1 n where eq. (19) has been used t o rep lace Q by 9 . - An e l a s t i c estimate is then made by forc ing zero p l a s t i c flow. The

e l a s t i c - p l a s t i c c o n s t i t u t i v e l a w requi res the mater ia l t o be e l a s t i c unless

t h a t stress v i o l a t e s t he y i e ld cons t ra in t .

n+ n-g . The ro ta ted s t r e t ch ing D a t the midpoint is the same as D . - Y

on-$ = Dn-+ Y "d

because Qn4 Y = w 1 f o r our choice of reference configurat ion.

The e l a s t i c stress estimate is computed from h

I

S' h

-..

= In-' + 2Ml A t tr Dn-' -

and h h A

II' = tr S'S' --,

and t h e y i e ld cons t r a in t is

o r p l a s t i c . We assume t h a t

t e s t ed t o see whether t he material is e l a s t i c

M, and M, are constant i n t h i s

i n the e l a s t i c moduli and p* are a l l an order of magnitude

changes i n the stress. I n general t h e y i e ld cons t r a in t is

Y2 = @(?, f i r , p")

I f Y2 0 w e have no p l a s t i c flow. The material is ca l led

work. Changes '

smaller than

t e s t ed as

e l a s t i c and the

new ro ta t ed stress tensor i s set equal t o t h e elastic estimate. n

I f t h e material response a t t i m e t is e l a s t i c - p l a s t i c , then, as w e

show i n Appendix 2 , the e l a s t i c - p l a s t i c c o n s t i t u t i v e l a w is appropriate f o r

t h e complete t i m e i n t e r v a l ( t n - l , tn). s t r e t ch ing when t h e material changes from e l a s t i c t o e l a s t i c - p l a s t i c provides

a bonus i n computational ease,

The r e i n t e r p r e t a t i o n of p l a s t i c

When the behavior is e l a s t i c - p l a s t i c and t h e complete kinematic

r e l a t i o n and the y i e ld curve must be s a t i s f i e d simultaneously, t he f i r s t

s t e p is t o determine whether a ve r t ex (corner o r por t ion of t he curve having

no unique tangent d i r ec t ion ) is the appropriate stress state. I n general ,

65

t h i s is done by computing a provis ional p l a s t i c s t r e t ch ing tensor

the stress s ta te a t the ver tex.

t o see i f a normality r e l a t i o n fo r p l a s t i c flow can e x i s t a t the corner.

Appendix 3 gives an example f o r t he cubic y ie ld curve.

based on

Then the pr inc ipa l values of g: can be examined I

I

I f none of the corners of the y ie ld curve s a t i s f i e s the kinematics and the

y ie ld curve, then the stress s ta te must r e s ide on a smooth port ion of the y i e ld

curve. Under these circumstances the r e l a t i o n between stress and p l a s t i c s t r e t ch ing

is inve r t ib l e .

i n many ways.

[Blum, 19721. Newton's method has excel lent convergence proper t ies when the

The algebraic equation shown i n the previous sec t ion may be solved

Appendix 4 descr ibes Newton's method f o r vector-valued functions

stress state is w e l l away from the vertex.

ver tex of t he y ie ld curve, Newton's method cannot be employed and the binary search

method (Appendix 5) is used. Newton's method occasionally converges t o a wrong

When the stress s ta te is near the

root such as X < 0 o r I > 0. In t h i s case the binary search method is used.

The binary search method [Hamming, 19621 is an extremely r e l i a b l e method.

I n f a c t , exis tence and uniqueness of the so lu t ion of t he d i f fe rence equations can

be most r ead i ly seen from the formulation i n Appendix 5.

search method is not pa r t i cu la r ly e f f i c i e n t f o r computation.

depends on a mix of Newton's method and a r e l i a b l e back-up method--vfz., the

binary search--that i s guaranteed t o converge t o the cor rec t roo t even though i t

is less e f f i c i e n t .

However, t he binary

The t o t a l algorithm

The f i n a l s t e p i n the so lu t ion technique is t o r o t a t e the stress tensor and

the p l a s t i c s t r e t ch ing back t o the proper or ien ta t ion i n space:

66

APPENDIX 1

HOW THE SOLUTION I S INDEPENDENT O F REFERENCE CONFIGURATION

The rotation operator Q with components -

depends on, the constant of integration 8, from

t B - 6, = -[ w dT

6 0

If we define the reference rotation operator

then the rotation from Bo to f3 is given by

The matrix representation is

( A l . 1)

( A l . 2)

( A l . 3 )

( A l . 4 )

This relationship may be interpreted as a tensor expression and then also

holds in the more general three-dimensional case where cannot be defined in terms of rotation through an angle B - B o . the difference equations that approximate the ice model response provide the

same stress 9 independently of the choice of 8, or of T. rotations 9 and depend on time, but is a constant.

In this appendix we prove that

In eq. ( A 1 . 5 ) the -4

To show that the difference scheme is independent of choice of reference configuration, we replace 9 by task is then to show that the new computed stress f is independent of the rotation z. The modified difference equations are given by

in all difference equations. Our *

67

n- 1 n+ ( i ) r o t a t i o n of t h e o ld stress and t h e s t r e t c h i n g Q P

( i i ) eva lua t ion of old ro t a t ed s t r a i n

( i i i ) s o l u t i o n of t h e ro t a t ed d i f f e rence equations

-n n Dn = x [QT ,1 + 2Q$, (Ty] -P

( i v ) r o t a t i o n of stress back t o i ts proper o r i e n t a t i o n

n Solu t ion of t h i s system of equations provides t h e same r e s u l t i n g CT n , but

in te rmedia te s o l u t i o n s are ro t a t ed by the r e l a t i o n s

68

The s c a l a r i nva r i an t s , however, s a t i s f y

- 1 = 1

h = X

n-1 n-+ n These r e l a t i o n s hold a t each appropr ia te t i m e t , t , o r t . Since the re ference o r i en ta t ion may be chosen a r b i t r a r i l y without

a f f e c t i n g the stress, we use t h i s f l e x i b i l i t y t o s impl i fy the algorithm.

During each cycle of ca l cu la t ion the reference o r i en ta t ion is changed s o

t h a t .., $-' = .., 1 during t h a t t i m e s t ep . By thus def in ing 9 w e are a b l e t o

reduce the number of r o t a t i o n s t h a t must be performed i n the so lu t ion method.

W e are a l s o ab le t o evaluate 9 of r o t a t i o n operators that must be computed and s tored .

n- 1 n i n terms of , thereby halving the number

APPENDIX 2

TRANSITION PROM ELASTIC TO ELASTIC-PLASTIC FLOW

In a t i m e s t e p the stress s ta te may move from some poin t i n t e r i o r t o

the y i e ld curve t o a s ta te on the y i e ld curve having a f i n i t e p l a s t i c s t r a i n

rate.

p l a s t i c flow, i t is necessary t o examine the v a l i d i t y of using the e l a s t i c -

Since the c o n s t i t u t i v e l a w changes abrupt ly with the occurrence of

p l a s t i c flow r u l e f o r t h e complete t i m e s t e p A t . t o use t h e ro ta ted stress tensor g.

Again, i t w i l l be convenient

L e t t he material behavior be e l a s t i c i n the t i m e i n t e r v a l (t,t*), where

t* = t + aAt , 0 1. a 2 1. Then

k .., = [E(+*) - g(t)]/aAt = e (A2.1)

The t i m e t* is associated with a stress t h a t is on the y i e ld curve but

has j u s t reached t h a t state. Now i n the i n t e r v a l (t*, t + - A t ) p l a s t i c flow

occurs and

By adding the two equations, we obtain

(A2.3)

where (1 - a) 2 0, but a is not yet determined.

The normal flow rule relating 2, gp, and the yield curve 4 is

(A2.4)

The plastic multiplier A plays no physical role in the model. equations can be written as

Now the two

D - (1-a) X a 32 .., (A2.5) .

If w/aS .., is only a function of E(t + At)--that is, the centering

parameter for plastic stretching is unity,8 = 1--then the stress 2 is the same whether or not the cycle is split into an elastic and an elastic-

plastic part.

Although the stress at t + At is not altered, the plastic strain rate 4 deals in time scales of the order of hours, so that a small error in eP over a time scale of At will not be detected. direct physical interpretation of the plastic stretching Qp. plastic stretching is not affected by the one-step or the two-step method:

is reduced by an amount (1 - a), where a is still unknown. The model P

Furthermore, we do not have a The total

t+At - iyAt h a d t = gP dt - (1 - a) X &k dt (A2.6) aE t ai! t

There is no change when the trapezoidal rule for integration is used. In

conclusion, there is no reason to determine the transition time t* or use

a special integration scheme during transition cycles.

70

Th

APPENDIX 3

VERTEX OF THE Y I E L D CURVE

ubic y i e l d curve i s given as

I 5 0 1 $@, p") = II' - 1 2 ( 1 + 1/2p*),

f o r i nva r i an t s I and II'. The curve can be p lo t t ed (and is , below) a s a

funct ion of t h e stress i n v a r i a n t s

1 - - I *I - 2

such t h a t

where

= *I1

4 ( * Y Y p") = F b I , OI1, p * )

The p r inc ipa l stress components are ro t a t ed 7T/2 from t h e stress invar-ants:

01 = *I + *I19 a2 = UI - aII

The normal flow r u l e is

h > O P aF DII = h - aF D I p = h - acr; 9 a%

71

-t and is a geometric statement that the vector Dp=(Dp, D p ) is orthogonal t o

and d i rec ted outward from F(a (T p * ) , The vec tor &' is assumed t o have

i ts o r i g i n a t (aT, oTT) on the yjield curve,

I TI I' 11'

The p l a s t i c s t r a i n rate invar i - A L A

P P a n t s , DI , DII, must be al igned with t h e stress inva r i an t s ,

A t t h e ve r t ex of t h e y i e ld curve, oI - - aII = 0 and the re is no unique

perpendicular d i rec t ion .

+ T / 4 and - n/4 of t h e vec tor ((TI, 0). The following algorithm is used t o

determine whether g = ,O is an appropriate stress state.

a provis ionary stress rate 6 based on ) / A & , where gn-' is the

previous stress state. Then a provisionary 6 and D may be evaluated. If the vec tor of t he s t r a i n rate invariance of the provisionary o'p is perpen-

d i cu la r t o t h e ve r t ex , then g = 0 is t h e co r rec t stress state.

checked e a s i l y by examining t h e inequal i ty DIP >

zfp is not perpendicular a t the ver tex , then ,O # ,O.

A t t h i s po in t the vec tor can be d i r ec t ed between

F i r s t ca l cu la t e n-i

= (9 - g

-P

This is .u

I f the provisionary

APPENDIX 4 NEWTON'S METHOD FOR THE COMPUTATION OF ELASTIC-PLASTIC STRESS

The ro t a t ed stress tensor 8 satisfies the d i f f e r e n t i a l equations

1 1 tr s + - s ' , 4M,- - 2M2 - I+ Y = p g p , g =

S' = s - - ' 1 t r ~ - - 2 - ,.A tr S = sll + ..,

where - .?kt x > o . - X a s s -

The so lu t ion of t h e d i f f e r e n t i a l equations is subjec t t o the a lgebra ic

cons t r a in t on the y i e ld curve:

$ ( I , 11, p") = 11' - F [ I2 1 + I / 2 p * ] = 0

2

where I = tr S, - 11' = 2[ ( '11 2 s 2 2 ) + S122]. The equations f o r constant

elastic moduli are w r i t t e n i n component form as

(A4.1)

72

as12 '3'12 = dl2

[ 1 + 1/2p*I I2 @(I, II, p * ) = 11 - - 2

(A4.2)

(A4.3)

(A4.4)

The constants e,, e2, cg, are r e l a t e d to t h e e l a s t i c moduli:

The p a r t i a l de r iva t ives of t he y i e ld funct ion with respec t t o the ro ta ted

stress have been expressed as

(A4.5)

For y ie ld curve (A4.4)

&L = + (SI, - s22) as1 1 i k#L = - (Sll - s 2 2 ) as2 2 a i

2-512 &L = 2S12

3 - 3 I2 3 aI 4P * - - [I + 7;p*] = - [ (Sll + SZ2) + - (S l1 + S Z 2 ) 2 ]

The equations (A4.1-A4.5) are t o be in t eg ra t ed from t = t o t o t = t o + A t .

I f w e denote t h e stress state a t t o as 3 and t h e stress s ta te a t t o + A t

as 2, t h e d i f f e rence approximations are appropriate:

With t h i s approximation, equations (A4.1-A4.4) are nonl inear , i m p l i c i t ,

coupled equations i n t h e unknown S l 1 , s22, s12, and h , f o r A > 0.

unknowns can be thought of as t h e roo t s (or zeros) of c e r t a i n funct ions

having continuous v a r i a b l e s x l , x 2 , x3, and xq. Sui t ab le func t ions are

The

73

F2(x l ,x2 ,x3 ,x4 ) = e2(xl - s ^ , , ) / A t + c1(x2 - s ^ , , > / A t

- x - (2, + x2I2 4- hl] - d,, 4 1 3 4p* (A4. 'I)

(A4.8)

( A 4 . 9 ) - 2 (xl + 2, )2[1 + (q + z 2 ) / 2 p * ]

When x1 = s i l y x 2 = s 2 , , x, - - s12 , and 2, = 1, then Fl = F , = F, = F, = 0.

A recognized way of determining t h e r o o t s of vector-valued functions

is given by Newton's Method [Isaacson and Keller, 19661, an i t e r a t i v e

procedure i n which t h e sequence of iterates is given by

J ( $ ) = - F ( z k ) (A4.10)

where hz: - '+' = 5 k+l - zk and t h e Jacobian matr ix evaluated a t s = gk is

denoted by ,I(&). The Jacobian matr ix has < , j elements aF</aXj. Once an k k+l

I I < E l and

i n i t i a l guess sQ is prescr ibed , t h e sequence g1 , g2, . . . , g Y Q Y 0 . 0

can be computed.

I whose elements w e g ive here f o r completeness and f o r t h e in t roduct ion of

some s impl i fy ing nota t ion .

The i t e r a t i o n is terminated when I k ) I I < E,. The two norms are r e l a t ed through t h e Jacobian matrix,

- aF1 = - [ (xl + x,) + - 3 + x*)2] + (XI - x2) = y + 6 3x4 4P *

74

- - - h3 E B 3%

8x4 aF,

Y - 6, aF4

8x2 - = - - - Y + 6 ,

The l i n e a r equat ion (A4.10) now becomes

a B 0

B a 0 0 A B

0 y + 6 y - 6 2B

- - - 0 aF4 a F 4

3x3 3x4 - - - 2B,

j & 4 E-J k+i k

- 3 2 . - where a , f3, y, 6 , A , B y and F are evaluated a t

The Jacobian matr ix < can be made symmetric by mult iplying t h e t h i r d row

by 2 .

19661 can be made.

i d e n t i c a l l y zero elements) l eads u s t o perform some elementary row

column transformations which allow t h e 4 t o be computed more e x p l i c i t l y .

For example, A > e3 /A t > 0 s o Ax, is e a s i l y eliminated from t h e system,

r e s u l t i n g i n

= sk and a t L!g = *

Then t h e very e f f i c i e n t Cholesky decomposition [Isaacson and Keller,

However, t he sparseness of < ( i .e . , t he number of

and

where e = - 2 B 2 / A 5 0.

t h e first two rows, adding and subt rac t ing t h e f i r s t two columns, and

mult iplying t h e l a s t column by 1/2.

W e can s impl i fy even f u r t h e r by adding and sub t r ac t ing

where

75

AZ, = b,+4x2, AZ, = Ax1 - Ax2, a x , = 2 b b

fl = -F, - F, , 2

2BF, - - f E -F, + F, , f 3 = -F, +- A

The matrix can now be w r i t t e n as

The f i r s t two v a r i a b l e s may be eliminated by s u b s t i t u t i o n

azl = (-yhz3 + f,)/a"

k2 = wax, + f,)/B*

and t h e remaining r e l a t i o n s h i p allows us t o f i n d Az,

Y 6 ($-$.+;).., = f, 'Sf, -B"f2

The asymptotic convergence of Newton's Method is

This means that when t h e i n i t i a l guess is i n t h e neighborhood of t h e r o o t ,

t h e r a t i o of t h e consecutive I&) is r2 f o r Irl < 1.

that t h e stress state a t t is c l o s e t o t h e stress state a t to + At. a t t o i t is an t i c ipa t ed t h a t Newton's Method w i l l be very e f f e c t i v e .

event t h a t t h e guessed stress state is very small, hz, cannot be r e l i a b l y

computed.

curve is no t defined f o r equation (A4.4).

t h e s i n g u l a r i t y of t h e Jacobian matrix.

well, t h e b inary search method of Appendix 5 w i l l be employed.

It i s usua l ly the case

Hence,

In t h e

Note t h a t i n t h e s ingu la r case of S, = 0 t h e normal t o t h e y i e l d

This s i n g u l a r i t y is r e f l e c t e d i n

When Newton's Method does not work

76

APPENDIX 5

THE BINARY SEARCH METHOD FOR COMPUTING ELASTIC-PLASTIC STRESS

The binary search method, often referred to as the bisection method,

is a reliable and moderately economical algorithm for determining the real

root of a single algebraic equation when the root is known to have finite upper and lower bounds.

but the cost soon becomes prohibitive. integration of the constitutive law can be accomplished by the binary search

method.

and X are thought to be the roots of certain functions having continuous variables z 1 , z2, z 3 , and A .

This method can be generalized to more dimensions, This appendix will show how the

As in Appendix 4 the rotated stress ( s l l + s,,), (Sl1 - s,,), Sl2,

Suitable functions are

cl*(zl - 2 ,̂) = d,, + d,, + 2h1(1 + 3 z,/P*) (A5.2)

h

C2*(a2 - 2,) = d,, - d,, - 2 h 2 (A5.3)

A

e2* (z3 - z 3 ) = d,, - 2Az, (A5.4)

where el* = 1/2M,At, e,* = 1/2M2At and 4 = (z, - 3)/At for integrating from old stress state $ to the new stress state in a time step At. The conditions

< 0 and A > 0 are invoked. When equation (A5.1) is equal to zero and z1 - equations (A5.2-A5.4) are satisfied, then z1 = (s l l + s2,), z 2 = (SI, - s,,),

z 3 = s I2 , and A = A.

From the implicit function theorem it is sufficient to consider a

single function F = P ( z , ) = f [ z , , ~ , [ A ( ~ ~ ) l , Z 3 [ A ( Z 1 ) ] ] . The functions z,[A(z,)] and z,[A(z,)] are easily constructed from equations (A5.2-A5.4). Evaluation of composite functions like F ( z , ) is particularly suitable for computers. From equation (A5.1) it is clear that if the root ( s , , + s,,)

exists, it must be in the interval [-2p*, 01, Therefore, we shall restrict

z1 to the same interval,

It is convenient to define the function D(z , ) as

77

3 Z l ( l + z1/p*)

and i ts nonzero r o o t 0 as -4/3 p*. D(zl) roo t but equations (A5.2), (A5.3), and (A5.4) are s a t i s f i e d , I n eq, (A5.2) the produce m(y) must be bounded so t h a t A i s i n f i n i t e .

h

(A5.3) and (A5.4) we r equ i r e z2 and z 3 to be zero.

= 0. By considering equation (A5.1) we now see t h a t

Then D(zl) 0 f o r z 1 5 0; and

0 f o r -2p* 5 z1 i q. Consider t he s i g n of F(y) when y is not a

Simi la r ly , s i n c e

is i n f i n i t e then f o r t h e products Az2 and Az3 t o be bounded i n equations

Thus z2[A(y)l = z,[A(y)l

F ( y ) < 0

f o r t h e above-described conditions. Eq. (A5.1) a l s o shows F ( 0 ) > 0 and

F(-2p*) > 0.

[ -2p* , 01.

This suggests that the re are two r o o t s i n the i n t e r v a l

The condi t ion A > 0 w i l l determine which roo t t o select.

The e l a s t i c estimate t o t h e ro t a t ed stress sll + s z 2 i s denoted by e e

and z 1 z 1 = z 1 + (dll $. d 2 2 ) / c 1 * . Then equation (A5.2) can be w r i t t e n as

o r simply el (Z

seen t h a t

- z l e ) = 2fW(z,). The condition A > 0 is invoked and i t is

e f o r r) 5 Z 1 5 0, then Z1 5 Z 1

and f o r -2pQ 5 z 1 5 q, then z1 e z 1 .

e Furthermore, z1 e

z1 z l e =

> q => sI1 + s22 > q (case 1 )

> n =;> sll + s 2 2 < 17 (case 2)

=> sll + s22 = q (case 3)

Cases 1 and 2 are sketched below.

CASE 1 CASE Z ?

78

The i n t e r v a l containing t h e roo t may be s l i g h t l y improved by not ing that

F(zle) > 0 by v f r t u e of A(zle) = 0 , and t h a t t h e stress s ta te is not elastic.

Case 3 is such t h a t F ( q ) = 0 but A = h cannot be determined from

equation (A5.2). I n t h i s event A is eliminated from equations (A5.2, A5.31,

giving t h e l i n e a r r e s u l t

- Subs t i t u t ion i n t o f (q , , z 2 , z 3 ) then determines z 2 = sll - s 2 2 , z 3 - s12.

The binary search method is geometrically i n t u i t i v e . The roo t of a

Then f(xk) func t ion f is known t o have upper and lower bounds xk and xu. *f(xu> < 0. depending upon t h e s ign , e i t h e r t h e lower bound is replaced by a g rea t e r

lower bound o r t h e upper bound i s replaced by a smaller upper bound.

new i n t e r v a l is ha l f t h e s i z e of t h e previous i n t e r v a l . In 30 s t e p s of t h i s

a lgori thm t h e f i n a l i n t e r v a l s i z e is about lo-’ of t he o r i g i n a l i n t e r v a l

s i ze . I n addi t ion , an immediate e r r o r bound of t h e roo t is given as

The func t ion is evaluated a t t h e midpoint of t h e i n t e r v a l and,

The

x i 5 x 5 xu.

REFERENCES

Blum, E. K. 1972. Nwnerieal Analysis and Computat.iSon: Theory and Practice. Reading, Mass.: Addison-Wesley Publ, Co,

Coon, M. D . , G. A. Maykut, R. S. Pr i tchard , D. A. Rothrock, and A. S. Thorndike. 1974. Modeling the pack ice as an e l a s t i c - p l a s t i c material. AIDJEX Efulletin, 24, pp. 1-105.

Gear, C. W. 1971. Simultaneous numerical so lu t ion of d i f f e r e n t i a l - a lgebra ic equations. Transactians on C i r c u i t Theory, 18(1), 89-95.

Hamming, R. W. 1962. NwnericaZ Methods for Scient is ts and Engineers. New York: McGraw-Hi l l .

79

Hibbitt, H . D., P. V. Marcal, and J, R. Rice. 1970. A finite-element formulation for problems of large strain and large displacement, Int. J. So Zids S t m t u r e s , 6, 1069-1086.

Isaacson, E., and H . B. Keller. 1966. AnaZysis of Nwner<caZ Methods, New York: John Wiley and Sons.

Marcal, P. V. 1969. Finite-element analysis of combined problems of non- Proc. ASME Computer Confer- linear material and geometric behavior.

ence on ComputationuZ Approaches t o App Zied Mechanics, pp. 133-149.

Marcal, P, V., and I. P. King. 1967. Elastic-plastic analysis of two- dimensional stress systems by the finite element method. Int . J . Mech. Sei., 9, 143-155.

Osias, J. R., and J. L. Swedlow. 1974. Finite elastic-plastic deformation, I, theory and numerical examples. In t , J , Solids Structures, 20, 321-339.

Palmer, A. C., G, Maier, and D. C, Drucker. 1967. Normality relations and convexity of yield surfaces for unstable material or structural elements, J , App Zied Mechanics, 34, 414-470.

Pritchard, R. S. 1974. Elastic strain in the AIDJEX sea ice model. AIDJEX Bu Z Zetin, 27 , pp . 45-62.

Pritchard, R. S. 1975. An elastic-plastic constitutive law for sea ice. J . AppZied Mechanics, 42 (ser. E, no. 2), 379-384.

Pritchard, R. S., and R. Colony. 1974, One-dimensional difference scheme for an elastic-plastfc sea ice model. NonZinear Mechanics. Austin, pp. 735-744.

In ComputationaZ Methods i n Texas Institute for Computational Mechanics,

Thorndike, A. S. 1974. Strain calculations using AIDJEX 1972 position data. AIDJEX BuZZetin, 24, pp. 107-130.

Truesdell, C, A. 1966. The Elements of Continuum Mechanics. Berlin: Springer Verlag.

Truesdell, C. A., and W. Noll. 1965. The NonZinem FieZd Theories of Mechanics. Encyclopedia of Physics, vol. 3 , no. 3. Berlin: Springer Verlag.

Trulio, J. G., J. J. Germroth, W. E. Carr, and M. W. McKay. 1969. Ground Motion Studies and AFTON Code Deve Zopnent. Studies, vol. 111. Tech. Rep. No. AFWL-TR-67-27, vol. 111, Air Force Weapons Laboratory, Kirtland Air Force Base, New Mexico.

Numerical Ground Motion

80

A DIFFERENCE APPROXIMATION TO THE MOMENTUM EQUATION FOR THE A I D J E X SEA I C E MODEL

by Robert S. Pritchard

Arctic Ice Dynudcs Joint Experiment University of Washington, Seattle, Wash. 98105

ABSTRACT

A difference approximation to the momentum equations for the AIDJEX sea ice model is formulated. These equations fit into the leap-frog scheme presently used to integrate the model. The scheme considers Coriolis acceleration and allows the water stress to be specified as an arbitrary function of the ice velocity. coupled, algebraic equations that must be solved for ice velocity. The velocity is found using Newton's method. performing successfully in a two-dimensional ice dynamics code.

The resulting difference equations are nonlinear,

The scheme is

INTRODUCTION

In this paper we formulate a difference approximation to the complete

set of momentum equations used in the AIDJEX sea ice model.

the Coriolis acceleration and an arbitrary water stress contribution. difference scheme for integrating one-dimensional motions is given by

These include

A

Pritchard and Colony [1974] for the elastic-or-plastic constitutive law

described by Coon et al. [1974]. acceleration is neglected and only linear water drag laws are considered.

The treatment of elastic strain in the constitutive law [Pritchard, 1974, 19751 requires more sophisticated treatment of the numerical formulation.

Tntegration of the full elastic-plastic constitutive law has been considered by Colony and Pritchard [1975]. In this paper we present a detailed formu-

lation of the difference approximation to momentum balance and the solution technique used to integrate the equations. The equations allow quadratic water drag (or more general functions) and include the Coriolis acceleration. With this paper we now have a complete description of the difference equa- tions now used to represent the AIDJEX sea ice model.

For that difference scheme Coriolis

81

The bas i c d i f f e rence scheme used is t h e k a p - f m g scheme [Richtmyer

and Morton, 19671.

computing stress 0 Y and v e l o c i t y u_ a l t e r n a t e l y by leap-frogging over t he

o the r v a r i a b l e i n t i m e .

t i m e s a t which each v a r i a b l e is evaluated.

This scheme de r ives i t s name from t h e p r a c t i c e of

I n Figure 1 we d isp lay t h e t i m e a x i s and show t h e

tn-l tn $it1

Fig. 1. T i m e s a t which stress and v e l o c i t y are computed.

Momentum balance f o r t h e AIDJEX sea ice model [Coon et a l . , 19741 i s

w r i t t e n by accounting f o r t h e i n t e r n a l stress divergence div 9 , air stress

T -ay water stress zw, and t h e Cor io l i s fo rce mfc 8 X 2:

where m is t h e mass per u n i t area; zw = zW(u) - depends on t h e ice ve loc i ty ;

fc = 2SZsin I$ is t h e Cor io l i s parameter, varying with t h e e a r t h ' s r o t a t i o n

rate 52 and t h e l a t i t u d e 4; and ,k is a u n i t vector upward from t h e su r face

of t h e ea r th . It is t h e unique combination of forces that makes t h e

app l i ca t ion of t h e s o l u t i o n method e spec ia l ly i n t e r e s t i n g .

We use an i n t e g r a l approximation method [Pri tchard, 19703 t o de r ive

t h e appropr ia te d i f f e rence equations.

momentum equations through t h e t i m e i n t e r v a l ( t n+, tn+*) and over a

s p a t i a l region R t h a t is t o be defined later.

are then expressed i n terms of t h e average q u a n t i t i e s t h a t arise.

i n t e g r a t e over reg ion R w e use t h e Green-Gauss theorem t o eva lua te a l l g rad ien t s i n terms of l i n e i n t e g r a l s around the boundaries. This method

w a s developed by Noh [1964] t o allow accura te eva lua t ion of grad ien ts i n

nonrectangular g r ids . The a c t u a l g r id used t o compute momentum conservation

has been used by Bertholf and Benzley [1968], and Benzley et al . [1969] i n

t h e TOODY Code.

I n t h i s method we i n t e g r a t e t h e

The d i f f e rence equations

When w e

82

The grid is logically rectangular as shown in Figure 2. The integer

variables 3’ and k define position in index space. point is glven by the vector (A, yk) .

Location of each mesh Furthermore, the thickness distribu-

tion and stress are computed as averages within each rectangular mass cell.

$‘-1 ’ j X

Fig. 2. Spatial configuration of mesh,

APPROXIMATION

By integrating the momentum equations over the time and space intervals

we find

Since the grid is assumed Lagrangian, the region R follows material particles and we may interchange spatial integration and time differentiation. If we then define as the average velocity within R ,

w

, n+%

where A = /R da is the area at time tn and we assume mass is constant in the momentum cell.

83

The stress divergence is evaluated by using t h e stresses computed i n t h e

t h e surrounding mass c e l l s . For convenience w e write

n39 n-% where At = t - t is the t i m e s t e p f o r t he momentum in t eg ra t ion . The

integrand is assumed constant i n t i m e because t h e stress is evaluated a t

t h e midpoint i n t h e leap-frog d i f f e rence scheme.

of fa is important, i t is ou t s ide t h e scope of t h i s paper and we do not

pursue i t f u r t h e r .

Although t h e eva lua t ion

The a i r stress is an independent d r iv ing fo rce and t h e average may

be computed d i r e c t l y . We use a cons i s t en t approximation and l e t

-n

The water stress is assumed t o be a func t ion of the i c e ve loc i ty 2. where -ca is t h e average a i r stress over the i n t e r v a l ,

There are numerous ways t o approximate t h i s force. We l e t

tn+$ -n++

Jn-k -.w T da dt 2 A At [ew zw + ( 1 - €Iw) Z-’l t

-n+$ - where zw - zw(c . The parameter 8, is l imi ted t o t h e i n t e r v a l [0,1].

This approximation is introduced because i t allows u s t o use a second-order-

cons i s t en t approximation i f 8, = 1/2, bu t a l s o lets us use the forward

d i f f e r e n c e given by 0, = 0. Ef f ic iency of t he so lu t ion scheme is not

reduced by keeping 8, as an a r b i t r a r y parameter i n t h e program.

out t h a t water stress is expected t o be a quadra t ic func t ion of v e l o c i t y

so t h a t whenever 8, # 0 t h e equations are nonlinear and coupled and must

be solved by an appropr ia te means.

W e po in t

The Cor io l i s f o r c e is evaluated by the s a m e technique used to approxi-

m a t e t h e w a t e r stress. W e f i n d t h a t

84

where 8, is i n t h e i n t e r v a l [0,1].

i n t h e computer program.

The parameter 8, is again l e f t a r b i t r a r y

I n t h i s term, however, l i n e a r i t y of t he Cor io l i s

acce le ra t ion couples t h e two momentum equations, but does not introduce any

o the r problems.

cussed by Semtner [1974].

Several problems t h a t arise i n the choice of 8, are d i s -

A l l of t h e above approximations are introduced t o g ive t h e d i f f e rence

equation a t each momentum cell:

For n o t a t i o n a l convenience w e drop t h e overbar and no longer d i f f e r e n t i a t e

between t h e s o l u t i o n v and t h e numerically computed average v e l o c i t y F. ..,

SOLUTION METHOD

I n t h e algorithm f o r so lv ing t h e d i f f e rence equations we use the n++

momentum balance equations t o so lve f o r v e l o c i t y a t t i m e i5

eq. (1) t o so lve f o r v and write

, We rearrange n+$ -

where

and

Solu t ion of t hese equations is accomplished using Newton's method

This method r equ i r e s t h a t we eva lua te t h e [Isaacson and Keller, 19661.

Jacobian mat r ix A ( v ) as - 5

A - = a y a ? t o f i n d t h e so lu t ion .

85

Solution of equations (2) by Newton's method requires a starting estimate of the velocity that is sufflciently close to the solutions so

that the iteration does converge to the proper root.

the old velocity ZJ ..d

convergence.

step limitation imposed by the Courant number when solving the full set of

conservation and constitutive laws.

We have been using n-S as the starting value and have found satisfactory

This assumption is helped by the inertia term and by the time

n+% The vth iterate to the velocity ZJ is 2 . We compute 2 from the u -V -v+ 1

linear algebraic equation

where

and hV = 4(Zv). The starting guess zo is set equal to $-'. Iteration

continues by incrementing v = 0, 1, 2, ... until convergence occurs. Convergence of this method can be ensured if the determinant of Llv

is bounded away from zero.

imposed by the need for convergence.

In the appendix we have studied limitations

A more limited version of the method occurs when inertia is neglected.

Since driving forces of sea ice are of low frequency, many investigators have felt that inertia should be neglected. The present solution scheme

can also handle this case. Multiplying the inertia by a parameter em,

normally set to unity, gives

To neglect the inertia we choose 8, = 0, and for our normal case we set Om = 1. system of equations from hyperbolic to parabolic. The computational

properties of this difference scheme have not yet been investigated.

solution method for the generalization remains essentially unchanged.

By neglecting inertia we change the fundamental character of the

The

86

SUMMARY AND CONCLUSIONS

The d i f f e rence approximation has been used i n one- and two-dimensional

ca l cu la t ions t h a t inc lude the AIDJEX e l a s t i c - p l a s t i c c o n s t i t u t i v e l a w .

water stress and Coriolis-centered parameters have been chosen as The

I e, = - 0, =

2 1

Thus, we have a r b i t r a r i l y chosen the second-order cons i s t en t scheme. The

ana lys i s i n t h e appendix shows t h a t t h e choice of 8, = 0 a l s o has a d e s i r a b l e

property: t h e s o l u t i o n of t h e momentum equations f o r v e l o c i t y always e x i s t s .

The w a t e r stress l a w has been assumed t o be e i t h e r l i n e a r o r quadratic.

Both cases have worked s a t i s f a c t o r i l y .

a few changes t o introduce a new w a t e r stress law.

The computer program requ i r e s only

The d i f f e rence equations have been used t o c a l c u l a t e wind-driven

d r i f t by s e t t i n g parameters properly. Time-dependent c a l c u l a t i o n s t h a t

inc lude t h e e f f e c t of i n e r t i a have been performed with no coding changes

except t o set i n t e r n a l stress divergence t o zero (fa = 0).

8, = 0 i n add i t ion , quasi-steady d r i f t is ca lcu la ted .

ignored, we cannot set 0, = 0 and 8, = 0; i f w e do, t h e Jacobian mat r ix

becomes s ingular .

v e l o c i t y a t half-time s t e p s because the i c e c o n s t i t u t i v e l a w is inopera t ive .

I f we set 5

When i n e r t i a is

For wind-driven d r i f t i t is unnecessary t o compute

Therefore, w e set t h e center ing parameters t o u n i t y (ew = 1, 8, = 1) and

i n t e r p r e t t h e so lu t ion as

With t h i s choice of parameters w e a c t u a l l y so lve

n n++ , which is t h e v e l o c i t y at tn, ins tead of 2 .

n n Zw + Za - mfc $ x u n 5 = o

These are t h e standard equations f o r wind-driven d r i f t ,

arises i n t h e quasi-steady case:

i n i t i a l guess t o start Newton's method.

This has worked i n a l l cases t e s t e d ,

One complication

t h a t is, t h e r e i s no e a s i l y a v a i l a b l e

I n t h e pas t w e have used go = 0.

W e f i n d t h a t t h e d i f f e r e n c e approximation t o momentum balance equations

is a u s e f u l t o o l i n a v a r i e t y of circumstances. The method allows an

87

arbitrary choice of water stress laws and can be programmed in its most general form without having to specify parameters at the outset. solutions obtained by this technique are satisfactory and the method is efficient to compute.

Finally,

APPENDIX

CONVERGENCE OF NEWTON'S METHOP

To ensure convergence we must demand that & be nonsingular (for each iterate).

zero for all 2. This is accomplished by bounding the determinant of 4 away from

Writing components and forming the Jacobian matrix provides

where water stress and velocity components are written as

Expanding the determinant and rearranging gives

We see that the five terms that contribute to the crucial determinant include inertia, mlAt; the three independent properties of the water stress

gradient corresponding to divergence, determinant, and curl; and the

Coriolis multiplier, mfc 6,. term as

For convenience we write the divergence-like

a~~ a-ry T I = -+ - au a~

the determinant-like term as

88

and the curl-like term as

Then the determinant takes the form

where the water stress gradient, measured here by T ~ , 'r2, and T 3 , is

arbitrary . Although IAI - may be of either sign and still allow the velocity to be

determined, we require that

141 0

All of the separate reasons for introducing this limitation reduce to the fact that the determinant is positive when time steps I?* are small and water stress is zero.

the scheme must work for small water stress, so we allow only continuous variations from the limit.

We expect to limit the time step to ensure stability and

It is readily apparent that the choice 8, = 0 eliminates all singu- larities by giving

This is positive whenever 8, and 8, are not simultaneously zero, a case that gives us no "new velocity" for which to solve.

Consider the case where all multipliers are non-zero. By suitable

rearrangement of terms we may write the inequality

where l-rl 1 , IT^ I , component. Written in this form we see that a time-step limitation may be used to provide a positive determinant. Typical values of water stress

and I -c3 I are used to indicate the magnitude of each

89

may be estimated using data of McPhee [1975]. is assumed bounded by k, then

If each component of axw/ac

I.,I I 2k

1.J I 2k

IT,] L 2k2

where k is on the order of Cwv where Cw(= 0.005 gm cm-2) is the drag coeffi- cient and 2, is the ice speed. The inequality given by equation (Al) is

satisfied if upper bounds on I T , I , IT,/, and IT,] are introduced.

find a A t that guarantees 141 > 0 we solve the quadratic relation Thus to

for e@/T where T is the maximum time step allowed by Newton's method in solving momentum balance. The roots are

em m - = T ewk f J3(ewk 12+ 2(ewk ) (mfcec> - (mfcec)2

First treat the case where the discriminant is positive. algebraically larger root because this gives the smaller limitation on T .

Furthermore, for the simple case when 8, = 0 we see this to be necessary because 1141 is negative between the two roots and positive outside the

interval. Therefore, we set

We use the

Now we consider the special case when the discriminant is negative. For this case there are no real roots for which the determinant drops to

zero, and therefore no time-step limitation is imposed on the solution

scheme.

Bwk

By setting the discriminant to zero we find the relation between

and mfcec that allows us to eliminate the time step restriction. It is

1 OWk 7"fc 0,

In this case the discriminant is always negative, so that no real roots to 141 = 0 exist; this guarantees a positive determinant. k, m, fc, and 8, we can choose a 8, such that At is unrestricted.

We see that for any

90

W e summarize the time-step r e s t r i c t i o n a s follows. T i m e s t e p A t is

bounded by T :

At T where

T = a, i f 8,k < 7 m f c 1 8, ;

otherwise

1 -

8,k + [ 3 ( 8 w k ) 2 + 28,kmfcec - (mfc8c)21'

W e must consider the addi t iona l case where 8, = 0 and the i n e r t i a

t e r m is neglected.

a r e s t r i c t i o n on 8,k i n terms of mfcec. We have

There is no longer a t i m e s t e p r e s t r i c t i o n , bu t , ins tead ,

W e can ensure that 141 2 o i f

which r equ i r e s that 8, may not be zero i f i n e r t i a i s neglected.

t yp ica l values of m , f c , and 8, t o determine t h e time-step r e s t r i c t i o n s

and then i n t e r p r e t t he r e s u l t s f o r a t yp ica l value of k.

We choose

L e t

m = 300 gm cm-2

fc = 1.4 x sec-l

e, = 112

Then when i n e r t i a is considered (em = 1) the re is no time-step

r e s t r i c t i o n i f 8,k 10.007. A t t h i s l i m i t the t i m e s t e p is r e s t r i c t e d by

T = 0.433 X l o 5 sec

For l a rge r 8,k t h e t i m e s t e p is r e s t r i c t e d fu r the r .

i n e r t i a is neglected (8, = 0) , we f ind the r e s t r i c t i o n

Al te rna te ly , if

8,k I 0.00768

When i n e r t i a is included, t h e r e s t r i c t i o n is not a s s t r i n g e n t a s t h a t

imposed by the Courant condition [Pr i tchard and Colony, 19741 f o r problems

9 1

&

of interest.

on the order of 20 cm sec-’ we require that However, when inertia is neglected we see that for velocities

--a very strong limitation.

REFERENCES

Benzley, S. E., L. D. Bertholf, and G. E. Clark. 1969. TOODY 11-A: A computer program for two-dimensional wave propagation--CDC6600 version. Research Report SC-RR-69-516, Sandia Laboratories, Albuquerque.

Bertholf, L. D., and S. E. Benzley. 1968. TOODY 11: A computer program for two-dimensional wave propagation. Research Report SC-RR-68-41, Sandia Laboratories, Albuquerque.

Colony, R., and R. S. Pritchard. 1975. Integration scheme for the AIDJEX elastic-plastic constitutive law. In this Bulletin.

Coon, M. D., G. A. Maykut, R. S. Pritchard, D. A. Rothrock, and A. S. Thorndike. 1974. Modeling the pack ice as an elastic-plastic material. AIDJEX Bulletin, 24, pp. 1-105.

Isaacson, E. , and H. B.. Keller. 1966. Analysis of Numer’kal Methods. New York: John Wiley and Sons.

McPhee, M. G. 1975. Ice-ocean momentum transfer for the AIDJEX ice model. AIDJEX Bulletin, 29, pp. 93-111.

Noh, W. F. 1964. CEL: A time-dependent, two-space-dimensional coupled Eulerian-Lagrangian Code. In Methods i n Computational Physics, vol. 3 , ed. B. Alder, S. Fernbach, M. Rotenberg. New York: Academic Press, pp. 117-179.

Pritchard, R. S. 1970. Numerical approximations to the deformation of a continuum. Informal report from Eric H. Wang Civil Engineering Research Facility to Air Force Weapons Laboratory, University of New Mexico, Albuquerque.

Pritchard, R. S. 1974. Elastic strain in the AIDJEX sea ice model. AIDJEX BuZZetin, 27, pp. 45-62.

Pritchard, R. S. 1975. An elastic-plastic constitutive law for sea ice. JowvlaZ of Applied Mechanics, 42 (ser. E , no. 2) , pp. 379-384.

92

Pritchard, R. S., and R. Colony. 1974. One-dimensional difference scheme for an elastic-plastic sea ice model, Nonlinear Mechanics, Austin, pp. 735-744.

In ComputationaZ Methods i n Texas Institute for Computational Mechanics,

Richtmyer, R. D., and K. W. Morton. 1967. Difference Methods for In i t ia l - VaZue ProbZems, 2nd ed. New York: Interscience Publishers.

Semtner, A. J., Jr. 1974. An oceanic general circulation model with bottom topography, Tech. Report No. 9, Department of Meteorology, University of California, Los Angeles.

"&ai! SimuZation of Weather and Climate Series,

93

ON THE FRACTURE OF I C E SHEETS WITH PART-THROUGH CRACKS

by R. Reid Parmerter

Dept. of Aeronuut<es and Astronautics University of Washington

ABSTRACT

An e f f i c i e n t method is developed f o r ca loula t ing the stress i n t e n s i t y f a c t o r f o r ice shee t s with part-through cracks. With t h i s method crack propagation problems can be solved f o r bending, s t r e t c h i n g , o r thermal loads with var ious end conditions.

INTRODUCTION

The arct ic ice is subjected t o v e r t i c a l loads and thermal loads which

cause bending, as w e l l as t o hor izonta l loads which cause in-plane extension.

For purposes of ana lys i s , t he ice is o f t e n idea l ized as a homogeneous, i s o t r o p i c

p l a t e on an e las t ic foundation, SO t h a t stresses may be ca lcu la ted by

a f a i r l y simple theory. This ca l cu la t ion , along with an experimentally deter-

mined f a i l u r e c r i t e r i o n , may then be used t o p red ic t f a i l u r e loads f o r t h e

ice. Evans and Unters te iner [1971], f o r example, have considered t h e problem

of thermal cracking due t o cooling of t he upper su r face of t he ice. I n t h e i r

problem, crack i n i t i a t i o n a t the upper sur face can be predicted from the stress

f i e l d , bu t nothing can be s a i d from t h e i r ana lys i s about thesubsequent propaga-

t i o n of t h e crack.

Once a crack i s introduced i n t o t h e p l a t e , t he problem must be t r ea t ed

by the theory of f r a c t u r e mechanics.

of p l a t e theony no longer e x i s t , and t h e complicated three-dimensional f i e l d

around t h e crack is found t o be s ingu la r a t t he crack t i p wi th in the context

of t h e usual l i n e a r theory of e l a s t i c i t y .

f a c t do not exis t , s ince nonl inear ih ies i n the real problem modify the crack

t i p so lu t ion .

Near the crack the simple stress f i e l d s

'

Of course, s ingu la r stresses i n

Nevertheless, experience has proven [Liebowitz, 19681 t h a t f o r

94

many materials crack propagation can be predic ted from a knowledge of t he

s t r eng th K of the s i n g u l a r i t y ca lcu la ted from l i n e a r e l a s t i c theory.

The bas i c theory of f r a c t u r e mechanics may be summarized as follows.

We consider a l i n e a r i s o t r o p i c e las t ic body containing a crack. Local t o the

crack tip, t he stress f i e l d ca lcu la ted from t h e i n f i n i t e s i m a l l i n e a r theory

of e l a s t i c i t y has t h e following form i n the coordinate system of Figure 1.

[Liebowitz, 19681:

(5 YY

where V is Poisson 's r a t i o and P, 8 are polar coordinates with o r i g i n a t t h e

crack t i p and 8 measured from t h e z axis. Only t h e opening mode of deforma-

t i o n i s considered here , as i t is t h e dominant mode i n p l a t e s .

are a l s o considered i n t h e complete theory of f r a c t u r e .

stress f i e l d i s completely charac te r ized by a s i n g l e parameter K, c a l l e d t h e

stress i n t e n s i t y f a c t o r (SIF). The SIF is a func t ion of t h e geometry of t h e

Shearing modes

Notice t h a t t he l o c a l

body, including t h e length of t he crack, and i s l i n e a r l y proport ional t o t h e

appl ied loads.

I n general three-dimensional problems, K may vary along t h e crack f r o n t .

This repor t i s l imi ted to two-dimensional problems (plane s t r a in ) of long,

s t r a i g h t , part-through cracks of uniform depth.

constant along t h e f r o n t .

s t r a i n problems is thus reduced t o ca l cu la t ing K f o r var ious geometries and

load configurat ions.

For these problems, K is

The bas i c problem of f r a c t u r e mechanics f o r plane

The theory then states t h a t t h e crack w i l l propagate

95

when the SIF reaches a c r i t i c a l value K

Frac ture toughness i s a material property determined by measuring the load

and corresponding K required t o propagate a crack i n a test specimen.

then assumed t h a t i n any o ther geometry and loading condi t ion, crack propaga-

t i o n w i l l ensue i f t h e SIF exceeds Kf,

ca l l ed t h e f r a c t u r e toughness. f '

It is

Measurements of f r a c t u r e toughness f o r i c e are l imi ted , Gold 119633

has estimated the energy absorbed pe r u n i t area of new crack sur face as

the crack extends i n f r e sh i ce .

Kf. I n plane s t r a i n condi t ions,

This energy, y, can be d i r e c t l y r e l a t ed t o

E K f 2 = Y

1 - v2

where E is Young's modulus.

is 150-200 e rgs p e r em2.

expects t h a t i n sea i c e i t would be s t rongly dependent on br ine volume. How-

ever , da ta on the f r a c t u r e toughness of sea ice a r e not ava i lab le .

The value of y measured by Gold f o r f r e s h i c e

The value of Kf is temperature dependent, and one

c T--$!&->.+ / I I

c z

Fig. 1. Cross sec t ion of sea i c e shee t with part-through crack.

96

I n t h i s r epor t , t he problem of ca l cu la t ing SIF f o r long s t r a i g h t sur face

cracks of uniform depth is solved f o r t h e case i n which the loading does not

vary along the crack length.

an e las t ic foundation (eq. 2b, next sec t ion) def ines the length scale f o r t he

problem. The so lu t ions are v a l i d provided t h a t t he i c e is t h i n and the crack

is long compared with t h i s length scale,

The c h a r a c t e r i s t i c length l / h f o r a p l a t e on

FORMULATION OF THE PROBLEM

Consider an ice shee t of l a r g e hor izonta l ex ten t re la t ive t o i ts thick-

ness b , r e s t i n g on t h e sur face of t he ocean,

foundation, t h a t is, an elastic foundation which provides a r e s t o r i n g force

a t each poin t proport ional t o t h e p l a t e displacement a t t h a t po in t [Hetenyi,

19461.

The ocean acts as a Winkler

The v e r t i c a l deformation w of the p l a t e is governed by the equation

D V 4 # f k w = q (1)

where D = Eb3/12 (1 - v2) E = Young's modulus

v = Poisson's r a t i o

k = foundation s t i f f n e s s = pg

p = dens i ty of water

g = acce le ra t ion of grav i ty

q = v e r t i c a l loading per u n i t area

I n t h i s r epor t , we consider t he s p e c i a l case where the loading, geometry, and

so lu t ion depend only on the coordinate x . I n t h i s case (1) becomes the t o t a l

d i f f e r e n t i a l equation

which is i d e n t i c a l t o the equation of a beam on an elastic foundation [Hetenyi,

19461, with p l a t e s t i f f n e s s D replacing t h e beam s t i f f n e s s .

(2) are of the form

The so lu t ions of

h=c +kc s i n w * e- cos

97

The c h a r a c t e r i s t i c dampening length is l / h , where

We s h a l l a l s o consider the problem with an ex te rna l ly imposed tension

o r compression T (expressed per u n i t length) i n t h e p l a t e [Parmerter, 19741, I n t h e one-dimensional loading case, t h e problem is a l s o t r ea t ed by Hetenyi.

The d i f f e r e n t i a l equation becomes

Solu t ions of t h i s equat ion are of t he form

im sin @ w - e cos

where a2 = A 2 + T/4D

fi2 = X 2 - TI40 (4)

The inf luence of T is measured by the relative s i z e of t he terms on the right- hand s i d e of (4); i n o ther words, t h e i n t e r a c t i o n between T and t h e bending of

t h e p l a t e is neg l ig ib l e when ITl4DI << A 2 .

W e now introduce a part-through crack of depth a i n t o the ice shee t

(Figure 1).

shee t , a t a d i s t ance L / 2 from the edges.

t o reduce t h e l o c a l bending and s t r e t c h i n g s t i f f n e s s of t he p l a t e .

For s impl i c i ty , t he crack is taken t o be i n the center of t h e

The gross e f f e c t of t he crack is

The SIF i n bending and tension is known f o r t h e problem of the cracked

p l a t e without a foundation.

t h i s case would be t o consider t h e uncracked p l a t e subjected t o t h e given

loading. The so lu t ion t o t h i s problem would give the tension and bending

moment t ransmi t ted across the sec t ion where the crack is t o be introduced.

This tens ion and moment g ive rise, respec t ive ly , t o a constant and t o a

l i n e a r l y varying stress, a t t h e c ross sec t ion .

o r l i n e a r l y varying pressure appl ied t o the crack face, with a l l o the r sur-

faces free of t r a c t i o n , has been solved by several methods [Bueckner, 19711.

The superpos i t ion of t he pressure-loaded crack so lu t ion on the uncracked

The bas ic procedure f o r eva lua t ing t h e SIF i n

The SIF f o r constant pressure

98

p l a t e so lu t ion y i e l d s the so lu t ion t o the cracked p l a t e problem with the

crack faces f r e e and o ther sur faces loaded by the given t r ac t ions .

A similar procedure can be used f o r t he p l a t e on an elastic foundation.

However, i n t h i s case we r equ i r e t h e so lu t ion f o r constant o r l i n e a r l y vary-

ing pressure appl ied t o t h e crack f ace and f o r an elastic foundation boundary

condi t ion appl ied t o t h e bottom surface.

which pinned o r clamped boundary condi t ions are appl ied t o t h e p l a t e a t t he

ends x = * L / 2 .

end condi t ions is required i n t h e superposi t ion procedure. I n t h i s r epor t ,

an approximate method is described f o r ca l cu la t ing these so lu t ions from the

known so lu t ion f o r crack f ace pressures , no foundation, and f r e e ends.

W e might a l s o consider problems i n

I n t h i s case, t h e so lu t ion f o r crack f ace pressure and similar

SOLUTION METHOD

The problem of a surface-cracked p l a t e with one-dimensional loading,

f r e e edges a t x = +L/2 , and no foundation i s one of plane s t r a i n .

t i o n f o r t he x,z stresses is t h e same as t h e plane stress so lu t ion f o r a

s t r i p .

We assume t h a t t h e loadings are appl ied i n such a manner t h a t the stress

d i s t r i b u t i o n oxx a t x = 0 is l i n e a r i n Z , and that the stress components O Z z ,

C T ~ are neg l ig ib l e compared with Om. These condi t ions w i l l be m e t provided

t h a t t he s t r i p is s lender ( L >> b ) and concentrated loadings are not appl ied

c lose t o II: = 0.

t he stress a t x = 0 has t h e form

The solu-

W e consider f i r s t t h e uncracked s t r i p subjected t o t h e given loadings.

Taking z = 0 t o be i n t h e midsurface of t h e p l a t e (Figure 1) ,

*xx Z

m b / 2 u t + 0 (5)

where Cit is the d i r e c t tension stress, r e l a t e d t o t h e hor izonta l fo rce T per

u n i t l ength by ut = Tlb, and Gm is the maximum bending stress, r e l a t e d t o the

moment M per u n i t length by Qm = 6M/b2. Then from dimensional considerat ions

the SIF has the form

K

where 5 is equal t o alb.

99

For 5 5 0.5, the func t ions f and g can be derived from the r e s u l t s i n Bueckner

[ 19 711 :

f(5) = 2 , 0 0 2 r (1 + 5.089c2 + 13,50c6)

g(c) 2.002E- (1 - 1.2165 + 5.084c2 - 3.951c3

+ 13,50C6 - 8,2455’)

(7)

These expressions are accura te t o 1%.

The forms of f and g i n the l i m i t 5 -f 1 have been discussed by Bentham

The simple asymptotic form f o r g gives r e s u l t s wi th in 1%

Simi lar accuracy can be obtained f o r f w i t h an add i t iona l term

and Koi te r [1973].

f o r 5 -f 0.4.

added t o t h e form given i n Bentham and Koiter:

g(c ) = 0.663/(1 -

The accuracy of t hese expressions has been v e r i f i e d by comparing them with known

f i n i t e element so lu t ions and with (7) i n t h e i r common domain.

The large-scale e f f e c t of the crack i n the s t r i p is t o increase the s t r i p

L e t 6 be the a d d i t i o n a l x displacement of t h e poin t x = L/2, z = 0 compliance.

relative t o t h e poin t 2 = z = 0, due t o the in t roduct ion of t h e crack; and l e t 8 be t h e add i t iona l r o t a t i o n of the c ross sec t ion x = L/2 relative t o the c ross sec t ion

z = 0. Then, as R i c e and Levy [1972] have shown,

Ht 6 = -(attat E + atmom)

100

1 plane stress ( s t r i p problem)

plane strain ( p l a t e problem) where H =

1 ,- u2

Substituting equations (7) or (8) into the definitions of the compliance coefficients aij, the following expressions are obtained:

For 0 5 5 I0.5,

att = 2.00452{1 + 5.089t2 + 8.633E4 + 6.750C6 + 27.485’ + 26.04512)

amt = 2.00452{1 - 0.81075 + 5.089c2 - 4.056E3 + 8.633E4 - 5.7455’ + 6.750t6 - 5.4805’ + 27.485’ - 17.335’ + 26.04512 - 14.84513}

(10)

= 2.00452{1 - 1.6215 + 5.828E2 - 8.111t3 + 11.84E4 - 11.495’ + 10.6556 - 10.965’ + 31.495’ - 34.656’ + 10.86510 + 26.04512 - 29.68513 + 8.497514}

- --- 5*93 2.22 In (1 - 5 ) + 3.92 att - (1-512 (1-5)

0.660 0.988 + o.270 -- - 04nt - (1-5)2 (1-5)

o*220 - 0.295 %lsn = (1-<)2

101

We now consider t h e problem of a s t r i p o r a plate on an e las t ic foundation.

Solu t ions f o r t h e uncracked s t r i p under var ious loadings can be ca lcu la ted by

applying proper boundary condi t ions t o the genera l so lu t ions of (2) when

1Z?/4O1 << X2, o r t o t h e general so lu t ion of (3) when the in t e rac t ion of

bending and tens ion is s i g n i f i c a n t .

be a good approximation t o the more exact e l a s t i c i t y so lu t ions , provided t h a t

t h e s t r i p is s lender ( L >> b and l / h >> b) and provided t h a t concentrated

loads are not appl ied near the c ross sec t ion z = 0, where an accura te estimate

of t h e stresses is required. Under these assumptions, t h e stress d i s t r i b u t i o n

a t X = 0 is again l i n e a r with t h e form of (5). Comparing t h i s so lu t ion with

t h e s o l u t i o n f o r a s i m i l a r l y loaded beam without a foundation, the e f f e c t of

t h e foundation w i l l i n general be t o reduce the bending stress om. d i r e c t t e n s i l e stress ot is not a f f ec t ed by the foundation, wi th in the context

of t h e s m a l l deformation theory of equat ions (2) o r (3). Bending does, of

course, introduce s t r e t c h i n g

i n t h e x d i rec t ion ; however, t h i s s t r e t ch ing is proport ional t o t h e i n t e g r a l

of t h e square of t he s lope , (c&/&)~, an e f f e c t which is ignored i n l inear

theory and w i l l be considered neg l ig ib l e i n t h i s repor t .

The so lu t ion of these equations w i l l

The

i f t he ends x = +L/2 are re s t r a ined from motion

When a crack i s introduced i n t o the s t r i p , add i t iona l deformations occur.

These deformations create add i t iona l foundation reac t ions t h a t f u r t h e r reduce

the load t ransmit ted across the s e c t i o n x = 0. I n p a r t i c u l a r , t h e foundation

i n t e r a c t s with t h e tens ion via the compliance amt (eq. 9 ) .

po in t , consider a s t r i p on a foundation loaded only by tension T. t i o n of t h e s t r i p is e n t i r e l y i n the x d i rec t ion , and no foundation r eac t ions

occur.

so lu t ion . However, when an edge crack is introduced, t he ends of t h e s t r i p

r o t a t e through an angle 8 =

Thus foundation reac t ions must be introduced i n t o t h e problem, and these reac-

t i o n s create a moment a t x = 0 which tends t o resist the opening of the crack.

As a r e s u l t , t h e cracked s t r i p on a foundation subjected t o pure tension loading

has a lower SIF than t h e same s t r i p without t he foundation.

t h e r epor t w i l l be concerned with ca l cu la t ing t h e add i t iona l load reduct ion

t h a t occurs as a r e s u l t of t h e crack-induced deformations.

To c l a r i f y t h i s

The deforma-

The presence of t h e foundation i n no way inf luences t h e uncracked

6H crt (eq. 9) i f t h e foundation is not present .

The remainder of

The bending and s t r e t c h i n g s t r a i n s caused by t h e crack are confined t o

Locally, t he deformations are not beamlike--i.e., a small region around it.

102

plane sec t ions do not remain plane-and s t r i c t l y speaking, t he foundation

reac t ions cannot be predicted accura te ly from beam on e las t ic foundation

theory. However, these reac t ions a l s o have a small lever arm with respect

t o the sec t ion 3: = 0 and are therefore r e l a t i v e l y unimportant i n determining

the crack-induced moment a t 3: = 0. As a r e s u l t , the cracked s t r i p on a

foundation can be modeled i n the following way.

crack is removed from t h e s t r i p by c u t s a t x = +E (Figure 2 ) .

deforms under tens ion and bending loads CTt and Om according t o equation (9). Foundation r eac t ions are neglected i n t h i s s ec t ion .

of t he s t r i p , each of length L / 2 - E, are t r ea t ed as s t r i p s on an e las t ic

foundation. The nonbeamlike deformations and foundation r eac t ions near t he

ends x: = kc are ignored.

A sec t ion containing the

This s ec t ion

The remaining two p ieces

The end s t r i p s are jo ined t o the cen te r s ec t ion by

- = -

I I I I--*€

Fig. 2. A model f o r t h e cracked ice shee t .

cont inui ty r e l a t i o n s on displacement, r o t a t i o n , force , and moment. This

simple jo in ing condi t ion assumes t h a t t h e sec t ions a t 3: = +E remain plane.

For E = 0, th&s condi t ion is not m e t , bu t t he planar condi t ion is approached

rap id ly as E increases . The d is tance over which nonplanar deformations occur

is of order b - a. reac t ions i n t h i s region, t h e region is t r e a t e d as beamlike by s e t t i n g E = 0.

I n t h i s way, t h e SIF f o r a cracked p l a t e on an e las t ic foundation can be

derived from t h e known so lu t ion f o r t h e S I F of a cracked p l a t e without a

foundation, p lus so lu t ions f o r uncracked p l a t e s wi th elastic r e s t r a i n t s on a foundation.

Because of t he relative unimportance of t h e foundation

103 .

SOLUTIONS FOR T = 0

The bending problem i n the absence of in-plane forces has been considered

by Parmerter and Mukherji [1975]. The ends of t h e p l a t e may be f r e e , clamped,

o r pinned, bu t hhere must be no cons t r a in t on motion i n t h e x d i rec t ion , so

t h a t a x i a l forces are not introduced by the crack deformations,

method could be used i f t h e ends were e l a s t i c a l l y constrained aga ins t r o t a t i o n

o r z displacement.

The same

The given loads cause a moment Mc a t x = 0 i n the uncracked beam. Mc may be ca lcu la ted from (2) and boundary condi t ions and is assumed known.

s impl i c i ty , only symmetric loadings are considered.

t he moment M t h a t is t ransmit ted across x = 0 a f t e r t h e crack is introduced.

This moment is determined by the condi t ion t h a t t he r o t a t i o n of t h e cracked

sec t ion , ca lcu la ted from (9), be equal t o the r o t a t i o n a t z = 0 of t he beam

sect ion.

For

We now wish t o ca l cu la t e

The r o t a t i o n of t h e beam sec t ion is e a s i l y ca lcu la ted . As Figure 3

shows, the so lu t ion t o t h e beam problem may be cansidered as the superposi t ion

of t h e so lu t ions t o two o the r problems.

f r e e end condi t ions and depic t ing the ex te rna l loads as an end moment M bu t t h e argument is v a l i d f o r any symmetric loading t h a t produces bending and

f o r any end condi t ions t h a t do not restrict x motion of t he n e u t r a l a x i s of

t he beam.

and t h e foundation.

with the foundation and end r e s t r a i n t , The r o t a t i o n a t x = 0 f o r t h e f i r s t

problem is zero; thus the r o t a t i o n may be ca lcu la ted s o l e l y from the second

problem.

end condi t ions a t x = L / 2 has t h e simple form

The f i g u r e is s impl i f ied by showing

O ?

I n t h e f i r s t problem, Mc is i n equi l ibr ium with the appl ied loads

I n the second problem, M - Mc is i n equi l ibr ium only

The r e l a t ionsh ip between end moment Me and end r o t a t i o n f o r var ious

1 where a,, = - F(AL, end r e s t r a i n t ) T

T = Ab

104

moment Mc a t z=O i n uncracked beam /

I 4 ,moment M i n cracked beam

/ I I

The so lu t ion t o t h i s c 1

'cc[ problem 1 a l ) M O problem is t h e super-

pos i t i on of t h e so lu t ions t o problems -

1 and 2. +

1. Mc - problem 2

Fig. 3. Solut ion of t h e beam problems by superposi t ion.

Matching the r o t a t i o n of t h e beam sec t ion t o t h e r o t a t i o n of t h e cracked

sec t ion y i e l d s the equation

Solving f o r M y i e l d s

Thus t h e moment ca r r i ed a t 2 = 0 i n the cracked beam is reduced by t h e

f a c t o r J, from t h e moment M, c a r r i e d by t h e uncracked beam.

from ( 6 ) : The SIF follows

105

The beam compliance % may be found easily from solutions given by Hetenyi [1946] for various end conditions:

For a free end at z = +L/2,

sinh AL + sin AL cash XL + COS AL - 2

For a pinned end at 3c = *L/2,

cash XL + cos AL 3T sinh XL - sin XL ab = L ( '

For a clamped end at x = +L/2,

Other conditions of restraint is placed

sinh XL + sin XL cosh XL + cos XL + 2

(15)

elastic constraint may also be considered, as long as no on the motion of the ends x = kL/2 in the x direction.

Prior to the derivation of equations (13) and (14), a number of solutions for specific problems were calculated by FEM, the finite element method

[Parmerter and Mukherji, 19751.

a problem of plane stress.

sary, these solutions provide a completely independent check of the assumptions which were made in deriving (13). variety of loadings and end conditions.

In these solutions, the strip was treated as Since no assumptions of beam behavior were neces-

The FEM solutions were generated for a

Results are shown in Table 1, where the K calculated by F E M is compared with the K calculated by (13) and (14). The foundation correction factor 9 is also shown, to indicate the magnitude of the additional support generated by the foundation when the crack is introduced. results agree with the F E M results within a few percent.

In general, the analytical The magnitude and

106

TABLE 1

COMPARISON OF FINITE ELEMENT AND ANALYTICAL SOLUTIONS

Run 5

1 0.22 2 .32 3 .42 4 .52 5 .62 6 .72 7 .82 8 .22 9 .32 10 .42 11 .52 12 .62 13 .72 14 .42 15 .91

16 .22 17 .42 18 .62 19 .82 20 .91 21 .22 22 .42 23 .62 24 .82 25 .91 26 .82

27 .70

28 .70 29 0.85

KFEM dyn cmm3j2

7. 43X1O7 9.5 x107 11.8 x107 14.7 x107 18.4 x107 22.8 x107 27.3 x107 6.4 x107 8.1 x107 10.1 x107 12.5 x107 15.6 x107 19.4 x107

3.09 7.41

12.3 20.3 35.5 73.9 99.5 12.4 20.7 37.3 89.8 152.0 102.0

3.82x10

22.5 30.0

n

llJ (eq 14) K-KFEM (eq.13) dyn cm-3/2 K Conditions

0.976 .950 .906 .840 .737 ,579 .348 .978 .952 .911 .848 ,750 .595 .911 .121

.992

.967

.896

.620

.282

.996

.983

.944

.760

.433

.873

.578

.656 0.302

8. 08X107 10.3 x107 12.7 x107 15.7 x107 19.6 x107 24.3 x107 28.4 x107 6.4 x107 8.2 x107 10.2 x107 12.6 x107 15.9 x107 19.9 x107

3.20 7.02

12.6 20.8 36.6 77.6 100.0

12.7 21.1 38.6 95.2 154.0 109.0

3.90~10

21.6 28.2

8.0%...Runs 1-7: L=192 m, hL=6.20 7.4% 7.1% 6.4% 6.1% 6.2% 3.9% 0.0%. ..Runs 8-13: L-124.8 m, XL-4.03 1.2% ~=0.0969, E=iO'' dyn cm" 1.0% thermal load with free ends 0.8% 1.9% 2.5% 3.0%...Runs 14-15: L=124.8 my hL14.03

moment? with free ends 2.2%...Runs 16-20: L=64.2 my hL=2.07 2.2% ~=0.0969, E=1O1' dyn cm'2 2.9% moment? with free ends 4.7% 0.5% 2.0%...Runs 21-25: L=124.8 m, XL=2.07 2.1% ~=0.0498, Es1.44 10" dyn cm-2 3.3% moment+ with free ends 5.7% 1.3% 6.7%...Run 26: L=264 my hL=2.06

~=0.0969, E=10' ' dyn cm'2 thermal load* with free ends

-5.6% r=0.0969, E=lO1_' dyn cm'2

'rr0.0234, Es2.937 1OI2 dyn cmm2 moment? with free ends

2.0%...Run 27: L-63.3 m, XL=2.04 T-0.0969, E=10" dyn cm" uniform load of 1 dyn cm-2 with clamped ends

-4.2%...Runs 28-29: L=63.3 my AL=2.04 -6.4% ~=0.0969, E=lO1' dyn cm-2

moment? with pinned end

NOTES: For all runs, thickness b was set at 3 m and pw at 980 dyn ~ m - ~ . *Thermal load was a linear variation in temperature through the thickness of the sheet, with the top surface 2OoC colder than the bottom. ?Moment was applied as a linearly varying stress (5

The magnitude varied from z=b/2.

on the ends x=*L/2. = 1 dyn cm-2 at z=-bE to a, = -1 dyn cm'2at

-107

s i g n of t h e d i f f e rence between t h e two so lu t ions suggest t h a t most of t he

e r r o r is i n the FEM so lu t ions and t h a t t h e a n a l y t i c a l r e s u l t is wi th in one

o r two percent of t h e co r rec t e l a s t i c i t y so lu t ion .

In t h e FEM, t h e body being analyzed is broken up i n t o many elements.

The s t i f f n e s s c h a r a c t e r i s t i c s of each element are then approximated.and assem-

bled i n t o a s t i f f n e s s matr ix f o r t he e n t i r e body.

gives a s t r u c t u r e t h a t is s t i f f e r than the a c t u a l s t ruc tu re .

t he region around the crack is s t i f f e r than the ac tua l cracked region. Since

t h e craLked region becomes less s t i f f as t h e crack extends, t he FEM approxi-

mation has the e f f e c t of making the cracked region behave as though the crack

were s h o r t e r than i t a c t u a l l y is. Since the SIF normally increases with crack

length , FEM so lu t ions usua l ly underestimate t h e SIF. The magnitude of the

e r r o r depends on t h e element s i z e and shape, and on the complexihy of t he

model used t o estimate t h e element s t i f f n e s s . The g r ids t h a t were used f o r

most of t h e so lu t ions i n Table 1 were a l s o used t o ca l cu la t e t he SIF f o r t he

s t r i p without a foundation. These r e s u l t s were compared with t h e known r e s u l t s

(eq. 7). The FEM underestimated the SIF by 2%-3%.

The FEM approximation always

I n p a r t i c u l a r ,

For t h e s t r i p on a foundation, t he e f f e c t of the length and end condi-

t i o n s is shown i n Figures 4 and 5. The foundation modifies t he SIF f o r deep

cracks , s o t h a t a peak value i s reached, and then K decreases with increasing

crack length.

Looking a t Table 1 and Figure 4 , w e can now understand why t h e d i f f e rence

between t h e methods is p o s i t i v e i n most of t h e runs, but negat ive i n run 15.

Run 15 is f o r 5 = 0.91, beyond the peak of t he curve on the XL = 4.0 curve.

Thus the FEM, which under values t h e crack length , produces a SIF t h a t is

l a r g e r than t h e c o r r e c t value.

t h a t t h e FEM produces a smaller value.

5 = 0.91 is a t t h e peak of the AL = 2 , @ curve, where S I F has a s t a t i o n a r y

value. Thus the d i f f e rence caused by a s h i f t of 5 i s very small. The l a r g e r

e r r o r s ( f o r example, i n runs 19 and 24) are assoc ia ted wi th t h e s t eepes t por-

t i o n s of t he curves, where e r r o r s i n 5 produce a maximum change i n K. The

f i r s t seven runs were made with e a r l y g r ids t h a t were less ref ined , r e s u l t i n g

in somewhat l a r g e r e r r o r s .

I n most cases the SIF increases with 5 , so

I n the case of run 20, t h e value

f

108

8 -

7 -

6 -

5-

c) b

e.? \ 4 - * 3 -

2 -

I I I I 1 0.2 0.4 0.6 0.8 1.0

E +

Fig. 4 . The e f f e c t of b e a m length on normalized S I F f o r bending problems.

Runs 28 and 29 are exceptions t h a t cannot be explained i n terms of the

s t i f f n e s s e r r o r of t he FEM. These runs are t h e two examples which were solved

wi th pinned ends.

beam approximation,

while i n the FEM a r e l a t i v e l y coarse g r i d w a s used t o approximate the t r u e

e l a s t i c i t y so lu t ion i n the region.

complicated deformation i n t h e neighborhood of t he p in , with an o v e r a l l e f f e c t

equivalent t o a nonzero compliance i n t h e v e r t i c a l d i r ec t ion . This moves the

so lu t ion away from the so lu t ion f o r a r i g i d p in , toward a f r e e end condi t ion:

A vert ical r eac t ion fo rce is suppl ied by the pin. I n the

the d e t a i l s of deformation around the p in are not modeled,

The t r u e e l a s t i c i t y so lu t ion w i l l have a

109

8

7

6

5

$4 's, Y

3

2

1

free end8 without a foundation (AL = T = 0)

free ends . . (AL = 2.0, T = 0.1)

I I I 8 1 0.2 0.4 0.6 0.8 1.0

E+

Fig. 5. The effect of end conditions on normalized SIF for bending problems.

-in other words, K will be increased (Figure 5). that the FEM solutions for runs 28 and 29 yielded values of K greater than equations (13) and (14).

This is probably the reason

From this detailed discussion of the F'EM solutions and the analytical approximation, we can see that the analytical approximation is probably accurate to within 2%, except in the case of pinned ends or some other concentrated constraint where the beam model does not account for the local deformations.

The SIF plotted in Figures 4 and 5 has been normalized by the maximum stress ( J ~ in the uncracked beam at the cross section where the crack is t o

110

be introduced. This normalization separates out t he i n t e r a c t i o n between the

crack-induced deformation and t h e foundatfon o r end res t ra ints , The founda-

t i o n a l s o has a major e f f e c t i n determining 0,.

normalized curves f o r hL = 4.0 and AL = 6.0 are near ly the same (Figure 4 )

Thus the f a c t t h a t t h e

should not be in t e rp re t ed as meaning t h a t t he SIF under i d e n t i c a l loadings

is near ly the same, I n Table 1, f o r example, no te t h a t t h e SIF f o r XL = 6

and a 20°C thermal load (runs 1-7) is s i g n i f i c a n t l y g rea t e r than t h e SIF f o r

XL = 4 under t h e same loading, The d i f fe rence is due almost e n t i r e l y t o the

g r e a t e r moment a t x = 0 i n the longer uncracked beam.

SOLUTIONS FOR T # 0

The s implest in-plane problem is one i n which t h e only loading is a

tension T appl ied a t the midplane of t he p l a t e a t x: = L L / 2 * I n the uncracked

p l a t e , t he deformation i s a pure extension i n the x d i rec t ion . No motion

occurs i n t h e z d i r e c t i o n , s o t h a t no foundation r eac t ions are generated.

The stress a t x = 0 is ot = T / b independent of t h e foundation parameters.

Once a crack i s introduced i n t o t h e su r face of t h e p l a t e , bending is coupled

t o t h e problem via t h e compliance coe f f i c i en t atm (9) . As a r e s u l t , founda-

t i o n r eac t ions are produced which generate a bending moment i n t h e p l a t e

tending t o c l o s e t h e crack. Thus the foundation a c t s t o reduce the SIF.

The amount of t h i s reduct ion can be ca lcu la ted using the model

t h a t w a s introduced i n t h e previous sec t ion . The foundation reac t ions cause

a moment of unknown magnitude t o be t ransmit ted between the c e n t r a l cracked

s e c t i o n and the end sec t ions . The r o t a t i o n of t h e c e n t r a l and end sec t ions

must be matched:

Solving f o r M one obta ins

111

Subs t i t u t ing t h e moment from (18) i n t o (6) t h e SIF f o r a cracked beam on a

foundation is

The beam compliance ab i s ca lcu la ted from (15), (16), o r (171, provided t h a t

lT/4DI << A2. beam sec t ion , and ab must be ca lcu la ted taking t h i s i n t o account.

t i o n f o r f r e e ends may be derived from r e s u l t s given by Betenyi [1946,

chap. 61:

For l a r g e T, the tens ion has t h e e f f e c t of s t i f f e n i n g t h e

The solu-

(20) 2aB [f3(3a2-B2) s inh aL + a(3B2-a2) s i n BLI 3.r [B2(3a2-B2)2 cosh CLL + a2(3B2-a2)2 COS BL - (a2+B2>31 ab =

where a, B have been defined i n (4).

a and 13 tend t o A and (20) reduces t o (15).

I n the l i m i t as T approaches zero,

Although similar expressions can be derived f o r beam s t i f f n e s s with

pinned o r clamped ends, w e w i l l not de r ive them i n t h i s repor t s i n c e t h e

s t i f f e n i n g e f f e c t of tens ion seldom matters i n sea i c e problems.

strate t h i s , w e can recast t h e condi t ion 1~/401 << h2 i n another form.

Subs t i t u t ing f o r A and D , an equivalent expression is

To demon-

For sea ice,

k a 980 dyn ~ m ’ - ~

E t 10” dyn cm-2

v z 0.3

The inequa l i ty becomes

(5 << 2 x l o 6 K

where 0 is i n dynes pe r square cent imeter and b is i n centimeters.

f r a c t u r e s t r eng th of uncracked ice is between l o 6 and l o 7 dyn

The

so

112

stresses g r e a t e r than l o 6 are not of i n t e r e s t i n cracked ice.

t h i s as an upper l i m i t on 0 gives the inequa l i ty

Subs t i t u t ing

b >> 0.25 cm

which i s s a t i s f i e d f o r any problem of i n t e r e s t i n sea ice.

The accuracy of (19) was checked by running two exaq le s with t h e FEM.

I n both cases, a u n i t stress was appl ied a t t h e ends,

(XL = 2.04,

t h e FEM gave 136.

equat ion (1.9) g ives K = 431, while t h e FEM gave K = 429, The foundation

r eac t ion i s s i g n i f i c a n t here .

(6) would have been 162 i n t h e f i r s t example and 1008 i n t h e second.

I n one problem

T = 0.0969, and 5 = 0,70), equation (19) gives K = 140, while

I n t h e o the r example (hL = 1.89, T = 0.0944, and 5 = 0.901,

Without a foundation the SIF ca lcu la ted from

Problems of combined tens ion and bending can be solved by superposi t ion

of so lu t ions (19) and (14). I n general , t h e bending problems should be solved

taking i n t o account t h e s t i f f e n i n g e f f e c t of t h e tension; however, s ince w e

have seen t h a t t h i s is not s i g n i f i c a n t f o r sea ice , t h e po in t w i l l no t b e

pursued i n t h i s repor t .

SOLUTIONS WITH LATERAL END CONSTRAINTS

This technique can a l s o be used t o so lve t h e problem i n which t h e ends

I n of t he p l a t e a t x = kL/2 are constrained from motion i n the x d i rec t ion .

t he example shown i n Figure 6, a uniformly d i s t r i b u t e d load is appl ied t o ice

which i s frozen i n a channel between two r i g i d w a l l s . Bending of t h e p l a t e

w i l l produce a ho r i zon ta l as w e l l as a v e r t i c a l r eac t ion f o r c e a t t h e w a l l .

I f t h e p l a t e i s uncracked t h e only ho r i zon ta l fo rce produced w i l l be a tension

a r i s i n g from t h e - s e c o n d a r d e r e f f e c t of t h e lengthening of t he p l a t e due t o

i t s curvature .

t i ons , where only f i r s t -o rde r terms are re ta ined i n t h e s t r a i n s , and it is

negligibly. s m a l l for sea ice problems.

This e f f e c t does not show up i n normal l i n e a r e l a s t i c i t y solu-

However, once a crack i s introduced i n t o t h e p l a t e , a compressive reac- t i o n f o r c e appears a t t h e w a l l .

i n so lu t ions of t h e l i n e a r theory of e l a s t i c i t y ,

This i s a f i r s t - o r d e r e f f e c t and shows up

The o r l g i n of t h i s fo rce

113

Fig. 6. A problem with lateral end constraints.

is in the coupling term s t , which produces an extension 6 (eq. 9) of the region around the crack, when the crack is subjected to a moment. Because of the constraint on end motion at x = +L/2, this extension must be absorbed in compression of the beam sections, which results in a compressive reaction force at the wall. The compression reduces the SIF and the extension around the crack and modifies the moment transmitted across the cracked section.

The unknown values of ut and Om can be found from the equations of con- tinuity of 6 and 8. that the extension in the cracked section be equal to the compression in the beam section. Using (9) we obtain the following:

The requirement of zero displacement at x = +L/2 requires

Continuity of rotations between the sections requires '

where the right-hand side is from (12), M = b2Um/6, and Mc is the moment at x = 0 in the uncracked beam.

L

These equations simplify to

attot + atmum = - Ut

(22) %tot + h u m = ab("c - om)

114

where Oc = 6Mc/b2.

Equations (22) are two equations f o r t h e unknowns at and Om. Solving

we f ind

Subs t i tu t ing i n t o (6) y i e lds

To v e r i f y equations (23) and (24), t h e problem shown i n Figure 6 w a s

solved by t h e FEM with 5 = 0.7, T = 0,0969, and XL = 2.044, f o r a uniform

load q of u n i t magnitude appl ied on t h e sur face . From Hetenyi [19461, t h e

maximum bending stress at x = 0 i n the uncracked beam i s

sin XL/2 cosh AL/2 - cos AL/2 s i n h A1;/2) (25) s inh XL + s i n XL -

OC -

Subs t i t u t ing i n t o (23) and (24) w e ob ta in

om = 68.6

U t = -15.3

K = 2311

The plane stress so lu t ion generated by t h e FEM gave at = -14.8 and K = 2280.

I f t h e ends of t h e p l a t e had been clamped on r o l l e r s so t h a t in-plane compres-

s i o n could not be generated, t h e SIF from equation (14) would have been 3890. Thus, cons t ra in ing t h e motion of t h e ends reduces t h e SIF by about 40%.

As a p r a c t i c a l example of t h e loading shown i n Figure 6, w e consider t h e

Once the ice problem of i c e t h a t is i n t he process of f r eez ing i n a channel,

shee t becomes at tached t o t h e channel walls, t he buoyancy of t he ice as i t

accretes on t h e bottom su r face begins t o exe r t a unlform load, causing deforma-

t i o n s and stresses i n t h e uncracked sheet 'which can be ca lcu la ted from t h e

115

s o l u t i o n f o r uniform loading,

II: = 0 is given by (25).

En pa r t ecu la r , t h e maximum bending stress a t

The normalized SIF f o r t h i s problem is shown i n Figure 7, Note t h a t

This is due with increas ing AL t he SIF f i r s t increases and then decreases.

t o a maximum i n CJc a t XL = IT.

negat ive a t hL

t h e uniform loading tends t o c lose a crack on t h e top surface.

Tc changes again a t XL 2 14.1 and continues t o o s c i l l a t e wi th increas ing AL; however, t h e magnitude rap id ly tends t o zero, so t h a t only neg l ig ib l e stresses

The bending stress than decreases and becomes

7.85. I n o the r words, t h e r e is a range of lengths f o r which

The s ign of

220 !l

6q

1.0 0.0413 2.0 0.1462 3.0 0.2161

IT 0.2173 4.0 0.1858 7.0 0.0176 3.0 -0.0019

12.0 -0.0031 15.0 0.0003

5"

Fig. 7. Normalized SIF f o r t h e problem i n Figure 6.

,

116

are generated fox l a r g e AL, long enough t h a t the bend2ng is confined t o the edges and t h e cone t r a l por t ion

of t h e shee t f l o a t s i n near ly i s o s t a t i c euqi l ibr ium with minimal bending.

Cracks near the edge of t h e shee t would be heavi ly loaded i n t h i s case.

problems could a l s o be s tudied using the method described i n t h i s r epor t ,

r e s u l t s shown i n 'Figure 7 are only a simple app l i ca t ion of t he model t o a prob-

l e m wi th lateral end cons t r a in t .

TQ expla in i t i n phys ica l terns, t h e shee t becomes

Such

The

CONCLUSIONS

With t h e method developed i n t h i s r epor t , t he SIF can be ca lcu la ted f o r

cracked p l a t e s on an e las t ic foundation with var ious end condi t ions.

approximations involved i n t h e method are v a l i d provided t h a t 'I >> 1 and t h e

crack length Z is long ( 2 >> l / A ) , condi t ions t h a t are m e t i n a v a r i e t y of

p r a c t i c a l problems. The accuracy of t h e method has been checked by comparison

with plane stress so lu t ions f o r a s t r i p , ca lcu la ted by t h e f i n i t e element

method.

and minimal e f f o r t .

The

Accuracies of a few percent are obtained with closed form expressions

ACKNOWLEDGMENT

This work w a s supported by Nat ional Science Foundation Grant OPP71-04031

t o t h e Universi ty of Washington f o r t h e Arctic Sea Ice Study.

REFERENCES

Bentham, J. P., and W. T. Koiter. 1973. Asymptotic approximations t o crack problems. Mechanics of Frachre, ed, G. C. Sih. Groningen, Netherlands: P. Noordhoff, p. 131-178.

Bueckner, H, F. 1971. Weight func t ions f o r t he notched bar. ZAiWl, 51, 97-109.

Evans, R. J . , and N. Unters te iner . 1971. Thermal cracks i n f l o a t i n g ice shee ts . Jowlnal of Geophysical Research, 76, 694-703.

L. W. 1963. Crack formations i n ice p l a t e s by thermal shock. Canadian JowlnuZ of Physics, 41, 1712-1728.

117

Hetenyi, M, 1946, Beams on EZastZc Foundut$on, Ann Arbor; Unhexsfty o f

Liebowitz, H., ed, 1965. @actuz-&, vol. 2, New York: Academic Press,

Michigan Press,

Parmerter, R. R, 1974. Dimensionless strength parameters for floating ice sheets. AIDJEX Bulletin No. 23, p, 83-95.

Rice. J. R., and N. Levy. 1972, The part-through surface crack in an elastic plate. JoumaZ of AppZied Mechanics, 39, 185-194.

. .

118

THE NCAR ELECTM FLIGHTS, A REPORT

INTRODUCTION

I n Ju ly 1975 t h e NCAR Electra completed t h e f i r s t of two o r t h ree

missions t h a t have been scheduled f o r t h e a i r c r a f t during t h e AIDJEX main

experiment. t o make measurements i n the

p lane tary boundary l a y e r i n support of ATDJEX; and t o inves t iga t e the rad i -

a t i v e and macrophysical p rope r t i e s of arctic s t r a t u s clouds.

The mission had two objec t ives :

The National Science Foundation through t h e National Center f o r Atmos-

pher ic Research funded t h e program, and the Naval Arc t i c Research Laboratory

a t Barrow, Alaska, suppl ied f u e l and l o g i s t i c support . The research program

w a s organized and car r fed out under the d i r e c t i o n of R. A. Brown and F.

Carsey of AIDJEX and G. Herman of Harvard University.

THE MEASUREMENT PROGRAM

The Electra mission comprised four f l i g h t s , t h r e e of them devoted p r i -

The measure- marily t o s t r a t u s cloud s t u d i e s and one t o t h e boundary layer .

ments made from t h e Electra are l i s t e d i n Table 1. The da ta were recorded

i n d i g i t a l form on magnetic tape by an on-board computer system.

f l i g h t s t h i s system provided real-time disp lays of t he ambient wind speed

and d i r ec t ion , temperature, dew poin t , a l t i t u d e , atmospheric pressure , and

ozone concentrat ion versus t i m e .

from a forward-looking camera and the time-lapse movies from t h e down- and

side-looking cameras are now on f i l e i n t h e AIDJEX da t a bank.

recorded da ta w i l l be a v a i l a b l e i n la te October after processing by NCAR t o

remove a i r c r a f t motions.

During the

These records p lus a video tape recording

The d i g i t a l l y

CompiZed from reports by 2. Brom (AILUEX), G. Her" (Harvard), and E. Leavitt (AIDJEX) .

119

TABLE 1

MEASURE24ENTS MADE ON NCAR ELECTRA F‘LIGHTS, 16-27 JULY 1975

Measurement Spatial

Range Accuracy Resolution Resolution

Stagnation air temp. Fast-response air temp.

Humidity Latent heat flux Mean winds Radiation surface temp.

Up/down IR radiation Visible radiation Radiometric altitude

Gusts Cloud liquid water

Hydrometeor liquid water Cloud droplet spectrum

Hydrometeor spectrum Pressure altitude

-6OOC t o f50°C

-6O*C to t50’C

-5OOC to +5O0C

0-40 g kgr’ 0-127 m s-’

-4O’C to +7SoC 0-2.5 ly min”’ 0-2,5 ly min-’

0-700 m 0-40 m s-’ 0-6 g m-3 0-8 g m-3 20-280 Mm

300-4500 pm

300-1035 mb

+0.5OC - $1, o*c 21 , OOC

-- +I m sml

9.5OC

+5 m +0,1 m s-’

fl mb

0.05OC

0.05OC

0, lo‘c 1.0 g kg-’ 0.01 m s-’ 0. lac 0.1 ly min-’ 0.1 ly min-’ l m 10 cm s-l 0.6 g m-3 0.8 g mW3 -I

1 mb

100 m l m

1 km 10 m

10 m

30 m 100 m 100 m 100 m 10 m 200 m

100 m 100 m 10 km

Time-lapse (1 frame per 4 sec) 16 mm photography, down- and side-looking, was made continuously of the ice surface. and gases (ozone, fluorochlorocarbons, halogens).

Air was sampled continuously for trace particulates

Simultaneously with t h e a i rcraf t measurements, observat ions were made

i n t h e atmospheric boundary l a y e r by personnel a t Big Bear, the AIDJEX main

camp on t h e ice, These meteorological measurements, described i n d e t a i l by

Paulson and B e l l [19751, include mean p r o f i l e s of wind speed, wind d i r e c t i o n ,

and a i r temperature from a 20 m tower, p i b a l bal loon a scen t s t o measure wind

speed and d i r e c t i o n , and acous t ic soundings of the inversion height .

su r f ace a i r pressure , recorded a t da t a buoys r ing ing the main camp a t a

r a d i a l d i s tance of approximately 300 km, Is used by AIDJEX t o produce a geo-

s t roph ic wind f i e l d f o r t h i s region.

The

On t h e f i r s t E lec t r a f l i g h t , 19 Ju ly , boundary l aye r da ta were col-

l ec t ed between a l t i t u d e s of 400 m and 30 m i n two perpendicular hor izonta l

d i r ec t ions (one of them p a r a l l e l t o t he mean wind) passing over Big Bear.

Figure 1 shows t h e f l i g h t plan. The momentum f l u x w a s measured d i r e c t l y

Leg Plan A (w i th reverse check) mean

Fig. 1. Boundary l a y e r f l i g h t , 19 Ju ly 1975. Legs were flown i n the following sequence: A a t 400 m a l t i t u d e ; B a t 300 m; B, reversed (---), a t 200 m; B at 100 m; A a t 60 m; B a t 30 m. Return t o Barrow w a s flown a t 100 m a l t i t u d e ,

121

with a gust probe.

t he p r o f i l e tower and acous t i c sounder recorded continuously during the

f l i g h t . Unfortunately, t h e wind was blowing d i r e c t l y over t h e camp tower,

so t h a t sur face l a y e r wind p r o f i l e s were contaminated and are of question-

a b l e value. Instruments aboard the a i r c r a f t recorded a mean wind speed of

9 m s-l a t 400 m a l t i t u d e and a s t rong temperature inversion with aqcompany-

ing wind shear a t approximately 100 m a l t i t u d e ; t h i s la t ter measurement cor-

responds f a i r l y w e l l w i th the s i g n a l received by the acous t i c sounder.

A t Big Bear, p i b a l observat ions were taken hourly and

l h e second, t h i r d , and four th f l i g h t s of the Electra mission were

made t o i n v e s t i g a t e the s t r a t u s cloud cover i n t h e Arctic,

conaissance on 23 J u l y w a s d i r ec t ed toward determining t h e hor izonta l ex ten t

and v a r i a b i l i t y of arct ic s t r a t u s .

second f l i g h t cons is ted of 1 2 l egs a l t e r n a t e l y ascending and descending i n

a double sawtooth p a t t e r n from Barrow out t o approximately 77'30'N, 138*W.

A l l measurements l i s t e d i n Table 1 were recorded while s c i e n t i s t s on board

made v i s u a l cloud observat ions. The a i r c r a f t t raversed seve ra l cloud types,

among them su r face fogs , boundary l aye r s t r a t u s and stratocumulus, and a l to -

s t r a t u s . I

A mesoscale re-

The f l i g h t plan (Figure 2) f o r t h i s

During t h e last two f l i g h t s , on 24 and 27 Ju ly , v e r t i c a l p r o f i l e s were

made of s i g n i f i c a n t meterological va r i ab le s i n seve ra l boundary l aye r s t ra-

t u s clouds and i n one a l t o s t r a t u s cloud. The f l i g h t p lan consis ted of 7-10

minute l e g s beneath t h e cloud, i n the cloud, a t the cloud top, and above

t h e cloud, a t a l t i t u d e pressures of 850, 700, 600, and 500 mb (Figure 3).

The s o l a r radiometr ic da t a w i l l determine t h e r e f l e c t i v i t y and abso rp t iv i ty

of t h e s t r a t u s decks, and t h e da t a from the PRT-6 radiometer w i l l determine

t h e i n f r a r e d r a d i a t i v e l o s s a t cloud top.

Several b r i e f excursions were made over t he tundra near Barrow t o

measure ho r i zon ta l g rad ien ts of wind (land-sea breeze) , sur face temperature,

and t h e concentrat ion of some minor atmospheric cons t i t u t en t s , such as

ozone and fluorocarbon, from t h e coast inland.

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160' 150' 140'

Fig. 2. Mesoscale reconnaissance f l i g h t , 23 Ju ly 1975, t o determine ho r i zon ta l ex ten t and v a r i a b i l i t y of a r c t i c s t r a t u s clouds.

PROGRAM OBJECTIVES

The da ta co l l ec t ed during t h e Electra f l i g h t s w i l l not only provide

b a s i c information about the atmospheric boundary l a y e r i n the Arctic, but

w i l l a l s o serve as an e s s e n t i a l check on the t o t a l sur face a i r stress and

the geostrophic wrnd derived from t h e sur face pressure f i e l d .

measure these two va r i ab le s and understand t h e r e l a t ionsh ip between them

before we can c a l c u l a t e a i r stress f o r dr iv ing t h e ice model, which is a

p r i n c i p l e product of t h e ATDJEX experiment. Brown [1975] and Leavitt [1975]

We must

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descr ibe how t h e AIDJEX da ta w i l l be used t o develop t h e r e i u i r e d boundary

l a y e r model and calculate the momentum f l u x t o t h e pack ice,

Because of l o g i s t t c and s a f e t y r e s t r i c t i o n s , t h e A I D J E X su r face measure-

ments are confined t o f a i r l y smooth t e r r a i n , where the a i r stress is l i k e l y

t o r e f l e c t only l o c a l su r f ace condi t ions. The a i r stress from such large-

scale su r face i r r e g u l a r i t i e s as r idges and rubble f i e l d s may not appear i n

the tower p r o f i l e s , and it could be as l a r g e as the stress t h a t is recorded

(see Arya, 1973).

horizonLal and ver t ical stress d a t a from t h e Electra f l i g h t s w i l l provide a

much-needed check on how w e l l t h e sur face l a y e r da t a represent t he large-scale

stress.

Since we want t he large-scale (100 km) stress, t h e extended

The d a t a from t h e s t r a t u s cloud program w i l l be used i n two ways:

t o cons t ruc t an i n t e r n a l l y cons is ten t p i c t u r e of t he macrophysical p rope r t i e s

of t h e cloud, including hor izonta l and ver t ical s t r u c t u r e , occurrence and

(1)

Fig, 3. Radiation f l i g h t s , 24 and 27 Ju ly 1975, t o c o l l e c t vertical p r o f i l e s below, i n , and above arctic s t r a t u s clouds.

124

intensity of convective activity, fluxes and flux divergences of solar and terrestrial radiation, and liquid water content and wet- and dry-bulb temper- atures throughout the boundary layer; and (2) to test the parameterizations

in the theoretical model discussed by Herman [1975] and compare the model predictions with the actual cloud conditions observed in flight.

A second NCAR Electra mission, scheduled for later in the main experi-

ments, will essentially repeat theJuly series, except that three of the four

flights will be aimed primarily at gathering data in the boundary layer and

only one flight will concentrate on the stratus cloud study. cal measurements made at the surface simultaneously with the flights will be

supplemented by profiles of wind speed, wind direction, and temperature to 500 m altitude as measured by the NCAR Boundary Layer Instrumented Package

(BLIP).

The meteorologi-

ACKNOWLEDGEMENT

We wish to thank the pilots and technical crew of the Electra, whose efforts ensured the success of our measurement program.

REFERENCES

Arya, S. P. S. 1973. Contribution of form drag on pressure ridges to the air stress on arctic ice, J . Geogphys. Res., 78, 7092-7099.

Brown, R. A. 1975. Planetary boundary layer models and parameters for AIDJEX. AIDJEX BuZZetin No. 29 (July), 113-130.

Herman, G. 1975. Radiative-diffusive models of the arctic boundary layer. Ph.D. dissertation, MIT Meteorology Department, Cambridge, Mass., 171 pp.

Leavit, E. 1975. Determination of air stress from AIDJEX surface layer data, AIDJEX BuZZetin No. 28 (March), 11-19.

Paulson, C. A. 1975. Meteorological observations during AIDJEX main experi- ment, AIDJEX l3uZZetCn No. 28 (March), 1-10.

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