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The dynamics of iceberg drift Napoleoni, Jean-Gerard Pascal 1979

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c THE DYNAMICS OF ICEBEEG DRIFT by JEAN-GERARD PASCAL NAPOLEONI Ingenieur, Ecole C e n t r a l e des A r t s e t Manufactures, P a r i s , 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August 1979 (c) Jean-Gerard P a s c a l Napoleoni, 1979 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my written permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 6 i i ABSTRACT T h i s t h e s i s p r e s e n t s numerical models c o n s t r u c t e d f o r the p r e d i c t i o n of i c e b e r g d r i f t , . The need f o r such models a r i s e s p r i m a r i l y from the need t o p r o t e c t d r i l l i n g v e s s e l s o f f s h o r e Labrador from the r i s k of c o l l i s i o n with d r i f t i n g i c e b e r g s . A d i s c u s s i o n of the d i f f e r e n t p o s s i b l e d r i f t models i s presented. A f t e r commenting on the numerical r e s u l t s o b t a i n e d with these models, a method i s proposed f o r a n a l y z i n g the past t r a j e c t o r y of an i c e b e r g i n order to determine c o e f f i c i e n t s necessary f o r p r e d i c t i n g i t s d r i f t . i i i TABLE OF CONTENTS ABSTRACT .. i i TABLE OF CONTENTS i i i TABLE OF FIGURES vi. ACKNOWLEDGEMENTS v i i 1- HISTORICAL BACKGROUND, ,, , ................... ,, 1 1.1, Greek merchants and V i k i n g emigrants,. ............ 1 1.2. Twentieth century. . 2 2. CURRENT KNOWLEDGE OF AND INTEREST-IN ICEBERGS, . . . 5 2,1,. C a l v i n g and general d r i f t p a t terns...,,,.,.,,,.,,., 5 2.2. Shapes, s t a b i l i t y and draught. 7 2.3. Present t e c h n o l o g i c a l problems. .................. 9 2.3.1. Valdez. 9 2.3.2. Labrador S h e l f , 10 2.3.3. Towing and d e s t r u c t i o n ; the need f o r d r i f t p r e d i c t i o n . . ,.„•., «•«.. ,,,, ...... 1 2 2.4. Fresh water, oceanography and i c e b e r g s . ...,..,.,.14 3« ICEBERG DYNAMICS,. .,...,.,,,..,.,....,.,,.,,.,.-,,.,.,,.16 3.1. L i s t of symbols. 16 3.2. I n t r o d u c t i o n . 17 3.3. Forces a c t i n g on i c e b e r g s . ... 18 3.4. D i f f i c u l t measurement of i c e b e r g parameters. ...... 20 3.5. T h e o r e t i c a l models, .............................. 21 3.5.1. P r e l i m i n a r y d i s c u s s i o n . ,.,...,.......,....21 iv 3 . 5 . 2 . Nonaccelerated flow. . . . 22 3 , 5 . 3. Acc e l e r at ed flow. ....... . . , , . . . . . . . . , .» . . . . 2 3 3 . 5 . 4 . The case of a m u l t i - l a y e r ocean,. ....... 25 1 I 4 , NUMERICAL MODELLING., , , . , , . , , , • , „ , , „ , , , , 27 4.1, . T h e o r e t i c a l r e p r e s e n t a t i o n of the i c e b e r g , , , , . . . , . 2 7 4 . 1 . 1 . D e f i n i t i o n o f a dynamic surface,. . . . . . . . . . . 2 7 4 . 1 . 2 . V a r i a t i o n s o f the dynamic s u r f a c e . ........28 4 . 1 . 3 . I n f l u e n c e of fieynolds number. . . . . - . . . , - . . . . . 3 0 4 . 1 . 4 . Need f o r the modelling of i c e b e r g r o t a t i o n . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 4 . 1 . 5 . S o t a t i o n equation. ........................ 32 4 . 2 . M o d e l l i n g of oceanographic, atmospheric and as t r o n o m i c a l parameters. ............... . . . . . . . . . . 35 4. 2. 1- I n t r o d u c t i o n ................ . , . , . . . . . . . . . . . 35 4 , 2 , 2 , Wind and c u r r e n t s , ....... . , . . . . . . . . . . . . . . . 35 4 . 2 , 3 - P r e s s u r e f i e l d s and towing. ...............36 5. NUMERICAL RESULTS, 37 5 , 1 , Order of magnitude of i c e b e r g a c c e l e r a t i o n s , d r i f t v e l o c i t i e s and Reynolds number. ,,,,,,.,,.,,,,..,.,..,.,»,»,, . 37 5 .2 - Frame of r e f e r e n c e , 38 5 . 3 , D i s c u s s i o n of t h e o r e t i c a l t r a j e c t o r i e s . , . . , , . . . . , 3 9 5 , 3 . 1. Foreword. ........................... .39 5 . 3 . 2 , Importance o f C o r i o l i s terms. ............. 39 5 . 3 . 3 , I n f l u e n c e of i c e b e r g shape i r r e g u l a r i t i e s . 42 5 . 3 . 4 , Importance o f i c e b e r g r o t a t i o n . ...........44 5 . 3 . 5 , On the i n f l u e n c e of pressure f i e l d s - , , , . . . 4 7 5 . 3 . 6 , Towing e f f i c i e n c y , . . . 4 9 V 6. INVERSION ., ........ . ...... 52 6.1. Concept. ........... . ,, ... ............ 52 6.2. Theory. . 52 6.3. D i s c u s s i o n , ........................ ........,,,,,.55 6.3.1. L i m i t t o the number of i c e b e r g unknowns. ,,55 6.3.2. Unfavourable c o n d i t i o n s . ..................55 6.4. I n f l u e n c e of the q u a l i t y of the data on the i n v e r s i o n r e s u l t s , . . , . . . , , , • , 5 6 7, CONCLUSION, .............,,, , ..........., .,, .,, ..,,58 REFERENCES ,,,, .,. , , ,. .,, , ,,, . , . ., , ......... . 59 APPENDIX : HOT WATER DRILLING ON A COLD GLACIER (MANUSCRIPT) , .,, ,, , , i , ,,,, 65 A b s t r a c t . ,..,.,,,,,,..,.,,»,,, ... . , . , 6 6 1. I n t r o d u c t i o n . ., .,. . ........ .............. 67 2. Theory, , ..,, ........ ..,.. .,67 3. Design. .., .,.......................................76 4. F i e l d r e s u l t s , . . . . . . . . . . . . . . . . . . . . . . . . .............. 80 5. Acknowledgements, . . . . , . . , , , . , , , , , » , , , , 8 7 References, , . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7 Appendix, --...........................................88 v i T A B L E OF F I G U R E S Figure no. Page 1. Theoretical representation of the iceberg, f ..,...,29 2, Lever arm functions,. 34 3. Influence of C o r i o l i s terms on tr a j e c t o r i e s . , , . , , 4 0 4, Wind parameters relevant to Figures 3 , 6 . . . . . . 4 1 5,i Influence of shape i r r e g u l a r i t i e s on tra j e c t o r i e s . . . . . . . . . . . , . . . . . . . . . . . . . , . . . . . . , , . . , 4 3 6, Influence of iceberg rotation on t r a j e c t o r i e s . ,,.,, . .,,,,.,,.4.••••..,.45 7, Influence of ocean pressure f i e l d s on t r a j e c t o r i e s , , , . . . . . . . . , , , . . . . . . . . . . . . , , . . . . . . . . . . . . 4 8 8, Towing e f f i c i e n c y curves, ,.,.,,,.,.„.,.,.,.,,,,.,50 v i i ACKNOWLEDGEMENTS I am extremely g r a t e f u l t o Dr,- G, K, C, .Clarke f o r h i s e n t h u s i a s t i c s u p e r v i s i o n and f o r h i s constant and always h e l p f u l guidance, I wish t o thank Eastcan E x p l o r a t i o n Canada f o r making t h e i r data and p r a c t i c a l knowledge. a v a i l a b l e , and Dr. R. Goodman f o r h i s v a l u a b l e advice,. The t h o u g h t f u l comments of a l l those who read the manuscript have been h i g h l y a p p r e c i a t e d . I am very t h a n k f u l t o N„ F o r s t e r and E. . D. Waddington f o r t h e i r h e l p i n w r i t i n g and e d i t i n g t h i s t h e s i s . Without Mr. Waddington's s k i l f u l and superb p o l i s h i n g of the manuscript, and h i s power f u l l y e f f i c i e n t completion of a l l b u r e a u c r a t i c requirements a f t e r my departure from Vancouver, t h i s t h e s i s would not have been s u c c e s s f u l l y f i n i s h e d - . T h i s work was supported by the O.B,C. Committee on A r c t i c and A l p i n e Research (Grant 65-0446), the N a t i o n a l Research C o u n c i l of Canada (Grant 67-4327) and Environment Canada (Grant 65-1654) through grants to G, K. C, C l a r k e , I f i n a l l y wish t o thank the Canada C o u n c i l who made my stu d y i n g i n Canada p o s s i b l e through i t s generous f i n a n c i a l support. 1 V HISTORIC AL BACKGROUND 1,1. Greek merchants and V i k i n g emigrants. Around 300 B.C, Pytheas set s a i l from M a r s e i l l e on a journey of no r t h e r n e x p l o r a t i o n . His e x p e d i t i o n , commissioned by the merchants of the Greek colony, was aimed a t opening new t r a d i n g r o u t e s i n t o unexplored northern Europe.. He landed i n England, and f o l l o w i n g i n d i c a t i o n s r e c e i v e d there, ventured f u r t h e r n o r t h i n search of Thule, a my t h i c a l hyperborean l a n d . I t i s unfortunate t h a t a l l we know about Pytheas was w r i t t e n by the Greek t r a v e l l e r and geographer Strabo, who d i d not t r u s t the accounts of the c a p t a i n from M a r s e i l l e . Indeed Strabo t r i e d t o d i s c r e d i t Pytheas by q u e s t i o n i n g the c r e d i b i l i t y of h i s d e s c r i p t i o n of northern seas where " a f t e r one day's journey to the n o r t h o f Thule, men come to a s l u g g i s h s ea, where there i s no s e p a r a t i o n of sea, land and a i r , but a mixture of these elements l i k e the substance o f j e l l y f i s h through which one can n e i t h e r walk nor s a i l " (Synge, 1912), T h i s account g i v e s Pytheas the best c l a i m among a n c i e n t s a i l o r s as the f i r s t t o d i s c o v e r i c e i n f e s t e d , fog-bound, A r c t i c waters, = Although the Spitsbergen g l a c i e r s c a l v e i c e b e r g s t h a t o f t e n d r i f t as f a r south as the range of Pytheas' p o s s i b l e journey, Strabo's t e x t does not i n d i c a t e t h a t the Greek saw any of them. Near the end of the tenth century, I c e l a n d was reached by waves o f V i k i n g s e t t l e r s . C u r i o u s by nature, and pushed by 2 l o c a l d i s p u t e s and r a p i d overcrowding of t h i s n o r t h e r n i s l a n d , I c e l a n d e r s s a i l e d f u r t h e r west to Greenland and the n o r t h e a s t e r n American c o a s t . . For them, the g l a c i e r s and i c e b e r g s o f these A r c t i c areas, were such a common s i g h t t h a t the sagas only mention them i n pas s i n g (Magnusson and P o l s s o n , 1 966),. 1.2. Twentieth century. By the end of the n i n e t e e n t h c e n t u r y , i c e b e r g s i n both hemispheres had been s i g h t e d , measured and photographed innumerable times by whalers, s e a l e r s , e x p l o r e r s , and oceanographers, and had a l s o been the cause . of many sea d i s a s t e r s . Indeed, u n t i l the end of World War I I and the i n v e n t i o n of r a d a r , there was l i t t l e a c a p t a i n could do to avoid c o l l i d i n g with an i c e b e r g while s a i l i n g on a fog-bound ocean. V i s u a l o b s e r v a t i o n which o f t e n gave too l a t e a warning, fog horns which sometimes provided a r e t u r n echo s i g n a l l i n g the p r o x i m i t y of a berg, low s a i l i n g speeds, and water t i g h t compartments i n s h i p s , were the only defence ag a i n s t the danger. The inadequacy of these .methods was made only too obvious when, on A p r i l 12, 1912 the T i t a n i c sank a f t e r having h i t one of the numerous i c e b e r g s c a l v e d d u r i n g an e x c e p t i o n a l l y bad year: t h a t same year a bergy b i t was s i g h t e d at 42° 30 N, 300 miles west of the Spanish c o a s t . T h i s stunning d i s a s t e r l e d to the c r e a t i o n of the I n t e r n a t i o n a l Ice P a t r o l whose job has been t o gather 3 i n f o r m a t i o n on f l o a t i n g i c e t h a t c o u l d be hazardous to North A t l a n t i c s h i p p i n g l a n e s . New techniques f o r the e a r l y d e t e c t i o n of i c e b e r g s were i n v e s t i g a t e d with mixed success, Waidner e t a l , (1914) attempted to use continuous measurement of sea s u r f a c e temperature. T h i s method was found i n e f f e c t i v e as the r e g i o n s of c o o l e r waters are o f t e n not centered around the i c e b e r g and d e t e c t i o n depended upon the d i r e c t i o n from which the s h i p reached the berg. During the.same p e r i o d , the f i r s t sonars were being developed and the echoes r e f l e c t e d back t o the e m i t t i n g s h i p by the underwater p o r t i o n of the i c e b e r g , c o u l d , i n f a v o u r a b l e c o n d i t i o n s , warn of the p r o x i m i t y of the i c e . General use o f on-board radar and o b s e r v a t i o n s gathered by the Ice P a t r o l from i t s own v e s s e l s as w e l l as from commercial s h i p s , reconnaissance f l i g h t s and s a t e l l i t e imagery, have improved North A t l a n t i c s a i l i n g s a f e t y c o n s i d e r a b l y . In some i n s t a n c e s , however, methods used to l o c a t e i c e b e r g s may remain i n e f f e c t i v e : heavy p r e c i p i t a t i o n and rough seas make i t impossible f o r radar to d e t e c t s m a l l e r i c e b e r g s which are s t i l l p o t e n t i a l l y dangerous t o modern v e s s e l s , j u s t as extended p e r i o d s of fog or low cloud cover may p a r t i a l l y hinder a i r p l a n e o b s e r v a t i o n and prevent s a t e l l i t e use. Furthermore, g l a c i e r i c e sometimes c o n t a i n s l a r g e amounts of rock because of i t s past flow along g l a c i e r beds or moraines. Icebergs made of such i c e have a higher d e n s i t y and may f l o a t so low above water t h a t they may become v i r t u a l l y i m p o s s i b l e to s i g h t even when atmospheric c o n d i t i o n s are f a v o u r a b l e . 4 When o b s e r v a t i o n methods are i n e f f e c t i v e one must r e l y on a p r e d i c t i o n of the i c e b e r g ' s d r i f t as c a l c u l a t e d from i t s l a s t s i g h t e d p o s i t i o n . 5 2, CURRENT KNOWLEDGE OF AND INTEREST IN ICEBERGS, 2.1. C a l v i n g and ge n e r a l d r i f t p a t t e r n s , . The great m a j o r i t y of A n t a r c t i c i c e b e r g s are produced by i c e s h e l v e s , These f l o a t i n g sheets of i c e , produced by l a r g e tidewater g l a c i e r s , have s i z e s ranging from l e s s than 100 km2 to 2300 km2 f o r the Ross i c e - s h e l f , with a t y p i c a l t h i c k n e s s of 500 m, F l o a t i n g i c e sh e l v e s , because of t h e i r l a r g e s i z e and the extremely low i n t e r n a l s t r a i n r a t e , are i n v i r t u a l l y p e r f e c t i s o s t a t i c e q u i l i b r i u m and the s t r e s s tensor w i t h i n the i c e i s h y d r o s t a t i c . T h i s e x p l a i n s why, when l a r g e i c e b e r g s c a l v e from the s h e l f under the i n f l u e n c e of wind, t i d e or ocean c u r r e n t s , they do not d i s i n t e g r a t e . Since i c e shelves have f a i r l y l e v e l upper and lower s u r f a c e s , these i c e b e r g s have roughly t a b u l a r shapes.. These.tabular i c e b e r g s can be as long as 170 k i l o m e t r e s and l e n g t h s of a few k i l o m e t r e s are extremely common. Due to t h e i r s i z e and to the e x i s t e n c e of the A n t a r c t i c o c e a n i c convergences, such i c e b e r g s , while f l o a t i n g i n c o l d water, can s u r v i v e f o r many years. Icebergs o f the Northern Hemisphere are produced by numerous tidewater g l a c i e r s of Greenland, the n o r t h e a s t e r n Canadian A r c t i c , S p i t s b e r g e n , the S i b e r i a n i s l a n d s and south e a s t e r n Alaska, The area where bergs are most f r e q u e n t l y encountered and where they i n t e r f e r e most with man's a c t i v i t i e s i s i n B a f f i n Bay, the Labrador Sea and the Grand 6 Banks of Newfoundland. Since the t e r m i n i of t i d e w a t e r g l a c i e r s are h i g h l y c r e v a s s e d , i c e b e r g s produced by these g l a c i e r s are on the average much s m a l l e r than t h e i r A n t a r c t i c c o u n t e r p a r t s and o f t e n e x h i b i t very e r r a t i c shapes. Icebergs seldom exceed a few k i l o m e t r e s i n l e n g t h and by the time they reach t h e southern Labrador Sea they are r a r e l y longer than one k i l o m e t r e . Murray (11969) compiled o b s e r v a t i o n s made by the I n t e r n a t i o n a l Ice P a t r o l and concluded t h a t of about 40,000 i c e b e r g s annually produced by Greenland g l a c i e r s , o n l y an average of 380 c r o s s the 48th p a r a l l e l , Most i c e b e r g s are c a r r i e d south by the main Labrador c u r r e n t while the r e s t remain i n i s o l a t e d f j o r d s , d r i f t i n a n o r t h e a s t e r n flow along the western Greenland c o a s t , or e l s e become imprisoned i n A r c t i c pack i c e . Iceberg d e t e r i o r a t i o n i s q u i t e r a p i d i n warm waters and very few North A t l a n t i c i c e b e r g s s u r v i v e more than 8 months (Hobe, 1977), Murray estimated t h a t a berg 30 m high and 90 m long would disappear i n about 24 days i n water at 3°C, The r a t e of d e t e r i o r a t i o n i s i n c r e a s e d by the f o r m a t i o n of growlers c a l v e d by sea waves along p r e - e x i s t i n g c r e v a s s e s . While e r o s i o n i s at work, s l o w l y reducing the s i z e of the d r i f t i n g b e rg, the s t a b i l i t y of t h i s f l o a t i n g body may be a f f e c t e d . Indeed, i c e b e r g s r o l l over numerous times i n the course of t h e i r d e t e r i o r a t i o n . T h i s u n p r e d i c t a b l e behaviour renders work on, or even near i c e b e r g s , h i g h l y hazardous. 7 2,2, Shapes, s t a b i l i t y and draught, To f a c i l i t a t e the task of r e c o g n i z i n g i c e b e r g s from one reconnaissance f l i g h t t o the next, the I n t e r n a t i o n a l I c e P a t r o l produced a l i s t of most t y p i c a l shapes e x h i b i t e d by the emerged p o r t i o n s of i c e b e r g s . A l l a i r e (1973), i n order to e v a l u a t e the s t a b i l i t y of i c e b e r g s as a f u n c t i o n of the shape of t h e i r above-water p o r t i o n , n u m e r i c a l l y t r e a t e d some simple cases which had g e o m e t r i c a l shapes corresponding to the Ice P a t r o l c l a s s i f i c a t i o n . By assuming a r a t i o of 1:0,88 between water and i c e d e n s i t i e s , he obtained two s e t s of r e s u l t s : minimum s t a b l e width to abovewater height r a t i o s and draught to abovewater h e i g h t r a t i o s . I t i s i n t e r e s t i n g t o note t h a t the draught t o he i g h t r a t i o ranges from 19 :1 to 1:1 with t y p i c a l values between 3:1 and 7 : 1 , i n good agreement with f i e l d o b s e r v a t i o n s conducted by Smith (1931), S h i l ' n i k o v (1960) and Lebedev (1965). A l l a i r e concluded t h a t "the s t a b i l i t y depends l a r g e l y on the abovewater c h a r a c t e r i s t i c s of the berg and i s r e l a t i v e l y independent of below water shape." T h i s statement can be c r i t i c i z e d s i n c e A l l a i r e . a s s u m e d r a t h e r bulky shapes f o r the submerged p a r t o f h i s t h e o r e t i c a l bergs, when i t i s c l e a r t h a t an unstable p i n n a c l e d i c e b e r g w i l l have, a f t e r r o l l i n g over, a p i n n a c l e d underwater shape, T h e r e f o r e , while these r e s u l t s may be used t o diagnose p o t e n t i a l i n s t a b i l i t y , they should not be r e l i e d upon to assume i c e b e r g s a f e t y and s t a b i l i t y . I t i s unf o r t u n a t e t h a t i n f o r m a t i o n about the underwater shape of i c e b e r g s i s d i f f i c u l t to o b t a i n , 8 s i n c e the draught, the p o s i t i o n of the c e n t r e of buoyancy and the form drag f a c t o r of the underwater p o r t i o n are a l l very important parameters necessary i n an e v a l u a t i o n of the r i s k i n v o l v e d i n using bottom i n s t a l l a t i o n s , and i n s t u d i e s of i c e b e r g d r i f t and s t a b i l i t y . Very o f t e n , as they r o l l over, i c e b e r g s expose p a r t of t h e i r p r e v i o u s l y hidden underwater p o r t i o n but the i n f o r m a t i o n i s by then i r r e l e v e n t although i t was n o t i c e d on such o c c a s i o n s t h a t underwater s u r f a c e s are smoother than abovewater p a r t s . T h i s i s due to the f a s t e r melting by sea water o f sharp edges around which water c i r c u l a t i o n i s more t u r b u l e n t , Most data on underwater i c e b e r g shapes were gathered from grounded i c e b e r g s (Lebedev, 1965 and B r e s l a u , 1970) and on these o c c a s i o n s , s i d e - s c a n n i n g sonars, scuba d i v e r s , underwater photography and bathymetry were used to o b t a i n i n f o r m a t i o n . From the p o i n t of view of the o i l companies, the i n f o r m a t i o n most sought a f t e r i s t h a t p e r t a i n i n g to draught s i n c e , i f underwater i n s t a l l a t i o n s are to be p r o t e c t e d , i t i s c r u c i a l to know very soon a f t e r an i c e b e r g has been s i g h t e d , whether or not i t has deep enough draught t o endanger w e l l heads and p i p e l i n e s , . At present, the i only promising method i s r a d a r . Severalj d i f f e r e n t radar sounders were t e s t e d i n the beginning of 1977 (personal communication from B, Narod and B. . Goodman) and i n some cases the draught was obtained with an estimated a b s o l u t e e r r o r of 20 metres. One of the d i f f i c u l t i e s h i n d e r i n g the success of radar measurements comes from the f a c t t h a t i n the case of sharp i c e b e r g k e e l s with s t e e p l y i n c l i n e d s i d e s . 9 echograms e x h i b i t a l a r g e number of m u l t i p l e r e f l e c t i o n s , are i n f l u e n c e d by d i f f r a c t i o n , and thus become. d i f f i c u l t to i n t e r p r e t . I t i s hoped t h a t computer treatment of echogram records may g r e a t l y improve the r e s o l u t i o n of t h i s method. 2.3. Present t e c h n o l o g i c a l problems. 2,3. 1. Valdez, . The present i n t e r e s t i n i c e b e r g s i s not l i m i t e d t o North A t l a n t i c s h i p p i n g l a n e s , . Valdez, at the south end of the Alyeska o i l p i p e l i n e i s l o c a t e d i n an i n l e t of P r i n c e W i l l i a m Sound (Alaska) i n t o which the t i d e w a t e r Columbia G l a c i e r c a l v e s i c e b e r g s , . Although the g l a c i e r ' s terminus i s p r e s e n t l y thought t o be s t a b l e and grounded on a s h o a l , i t s r e t r e a t c o u l d l e a d to a c a t a s t r o p h i c break-up accompanied by a dramatic i n c r e a s e i n i c e b e r g p r o d u c t i o n t h a t would s e r i o u s l y hinder the o i l tanker t r a f f i c (Mark Meier, p e r s o n a l communication). T h i s p a r t i c u l a r example i l l u s t r a t e s the t h r e e main problems a s s o c i a t e d with i c e b e r g s : the mechanics of i c e b e r g c a l v i n g from a grounded or f l o a t i n g g l a c i e r terminus or i c e s h e l f , the p h y s i c s of berg d r i f t under the i n f l u e n c e s of sea c u r r e n t s and wind and, f i n a l l y the. techniques t h a t could be used to d e s t r o y or tow the i c e b e r g s . 10 2. 3. 2. . Labrador Shelf. The Labrador Sea, where d r i f t i n g ice i s not a potential r i s k but a daily r e a l i t y , has been nicknamed "Iceberg a l l e y " by oilmen. Offshore o i l and gas exploration started there i n the late 1960's and icebergs as well as sea ice were immediately found to be the greatest environmental obstacles (Yungblut, 1975), Furthermore, i t was the f i r s t time that such a problem was tackled and new ideas as well as innovative technology had to be introduced, ; The present need for iceberg d r i f t prediction models originates primarily from o i l exploration a c t i v i t y on the Labrador Shelf, I t i s therefore useful to examine in d e t a i l the problems that the o i l industry encounters i n t h i s area. During the exploration phase, d r i l l i n g ships or platforms must be able to remain at the same location for many months. In iceberg infested waters, t h i s may be impossible i f a d r i f t i n g iceberg threatens to c o l l i d e with the d r i l l i n g i n s t a l l a t i o n . The requirement of being able to leave the d r i l l i n g s i t e on short notice, along with environmental factors such as rough seas, have given to the - method of dynamic ship positioning a t o t a l supremacy i n the f i e l d of offshore d r i l l i n g i n i c e infested waters (Verlet,1975). By t h i s system, d r i l l i n g vessels are not anchored but instead are kept in position by a set of continually operating thrusters. Absolute positioning with respect to the well head i s provided by an array of acoustic beacons set on the ocean f l o o r . . The 11 marine r i s e r i s t h e r e f o r e the only p h y s i c a l l i n k between the s h i p and the sea bottom and a s p e c i a l l y designed blow-out pre v e n t e r t h a t a l l o w s r a p i d d i s c o n n e c t i o n of t h e r i s e r from the well-head makes i t p o s s i b l e f o r the d r i l l i n g ship to l e a v e i n a matter of minutes, In the Labrador area sea i c e seldom disapp e a r s f o r more than f i v e months thus r e s t r i c t i n g the d r i l l i n g season. Safety r u l e s aimed a t p r o v i d i n g enough time t o d r i l l a r e l i e f hole before freeze-up, should a blow-out occur l a t e i n the summer, f u r t h e r reduce the working season. I t appears t h a t although the technology i s a v a i l a b l e to a v o i d i c e b e r g damage while d r i l l i n g , the time l o s t d u r i n g d i s c o n n e c t i o n , waiting p e r i o d s and r e c o n n e c t i o n i s so v a l u a b l e t h a t i t i s worthwhile t o spend c o n s i d e r a b l e e f f o r t t r y i n g t o reduce down-time as much as p o s s i b l e , _ O i l and gas production r e q u i r e s permanent i n s t a l l a t i o n s . Some of these, such as l o a d i n g t e r m i n a l s , are abovewater while others, such as w e l l heads or c o l l e c t i n g p i p e l i n e s , must be submerged. D i f f i c u l t i e s a s s o c i a t e d with s u r f a c e s t r u c t u r e s could t h e o r e t i c a l l y be solved using dynamically p o s i t i o n e d , r a p i d l y , d i s c o n n e c t a b l e f l o a t i n g s t r u c t u r e s , although such i n s t a l l a t i o n s may prove uneconomical to b u i l d and operate. Bottom i n s t a l l a t i o n s o f f e r a more formidable c h a l l e n g e : the depth a t which they would be l o c a t e d would i n many i n s t a n c e s be l e s s than 200 metres and i c e b e r g s d r i f t i n g i n the Labrador Sea o f t e n have draught exceeding t h i s value,. , Indeed Lewis e t a l . . (1977) have found scours caused by the dragging of i c e b e r g k e e l s down to depths of 275 metres on the B a f f i n 12 Shelf. I t i s debatable whether t h i s extremely deep scouring occurred recently or during times when icebergs were larger or the sea shallower, although i t i s c e r t a i n that bergs with a draught of 200 metres exist at present i n the;Labrador Sea. Unless pipelines and well heads are buried, they w i l l be threatened by destruction by iceberg keels ploughing the sea f l o o r , Needless to say, entrenching such equipment on a rough sea bed would be extremely d i f f i c u l t and c o s t l y . 2,3,3. Towing and destruction; the need for d r i f t prediction. k possible alternative to the defensive method of b u r i a l , i s active influence on icebergs. Eastcan Exploration, who have conducted most of the exploration work i n the Labrador area, have since 1970 run a program of iceberg towing i n the v i c i n i t y of t h e i r d r i l l i n g s i t e s (Hydrospace, 1970), Supply boats are usually required around offshore i n s t a l l a t i o n s and were readily available; these were s l i g h t l y modified for the task of iceberg towing.. The program was considered successful (Duval B., 1975), since many icebergs were deflected from c o l l i s i o n t r a j e c t o r i e s , but due to the limited p u l l that can be applied to the iceberg the method was found to be i n e f f e c t i v e for bergs having a mass larger than a m i l l i o n tons. D i f f i c u l t i e s l i m i t i n g the towing p u l l are not so much due to the design of the towing boat, winch or cable strength, but rather are the r e s u l t of the manner i n which the 13 cable i s attached to the iceberg, J, Duval (1971) reported that a single looped cable, or a group of such loop cables t i e d i n the shape of a hammock, were used to tow the berg. These towing cables hang underwater from f l o a t s at the supposed depth of the iceberg* s centre of gravity, i n order to reduce the berg*s tendency to r o l l over, I t so happens that as soon as the towing speed reaches a c e r t a i n value, the iceberg has a tendency either to r o l l over uncontrollably, or simply to break up, I f either of these two events occur while the iceberg i s under tow at close proximity to the d r i l l i n g ship, the additional time that i s l o s t or the lack of equipment to tow more than one iceberg, may force the d r i l l i n g ship to leave l o c a t i o n . The reported ineffectiveness of towing large icebergs i s only due to the lower possible towing speed caused by a greater iceberg drag.. I f the amount of time available between the beginning of towing operations and the predicted impact with the d r i l l i n g vessel could be increased, towing would remain e f f e c t i v e even f o r larger bergs. Thus, a l l other variables being equal, any method, such as d r i f t prediction, that w i l l allow early recognition of p o t e n t i a l l y dangerous bergs w i l l decrease the r i s k of having to displace the d r i l l i n g ship because of towing f a i l u r e , Destruction by fragmentation i s another means of disposing of dangerous icebergs. Barnes (1927 a and b) while observing the erosion of d r i f t i n g Labrador icebergs, noticed that the largest growlers were breaking o f f on sunny mornings and concluded that d i f f e r e n t i a l heating, and the stress i t 14 i n d u c e s , were the main f a c t o r s i n i c e b e r g e r o s i o n . Due to the slow r a t e at which heat flows through i c e , i t seems t h a t the phenomenon t h a t Barnes observed must have been f a i r l y s u p e r f i c i a l . However, by using charges of t h e r m i t he obt a i n e d slow fragmentation of l a r g e i c e b e r g s where high e x p l o s i v e s remained i n e f f e c t i v e . I t seems t h a t c o n s t r u c t i n g i n s t a l l a t i o n s t h a t would be completely i c e b e r g r e s i s t a n t , although f e a s i b l e from a purely t e c h n i c a l p o i n t o f view, would be unacceptably expensive. Instead, a c a l c u l a t e d r i s k c o u l d be made accept a b l e i f kept at a low l e v e l by long range i c e b e r g d r i f t p r e d i c t i o n t h a t c o u l d provide s u f f i c i e n t warning time t o permit towing or d e s t r u c t i o n of the menacing i c e b e r g s , The concept of lo w e r i n g the r i s k t o an acceptable l e v e l i m p l i e s t h a t the chances of d e s t r u c t i o n , although s m a l l , would s t i l l e x i s t . . 2.4. F r e s h water, oceanography and i c e b e r g s . T h i s survey o f the i c e b e r g q u e s t i o n would not be complete without mentioning dream-like schemes t h a t have s p o r a d i c a l l y s u r f a c e d f o r the l a s t twenty years concerning the towing of huge A n t a r c t i c i c e b e r g s t o r e g i o n s l a c k i n g f r e s h water. The i d e a was f i r s t s c i e n t i f i c a l l y e v a l u a t e d by Weeks and Campbell (1974) who concluded t h a t the concept was econ o m i c a l l y and t e c h n o l o g i c a l l y f e a s i b l e with regards t o p r o v i d i n g water to A u s t r a l i a and the Atacama d e s e r t ( C h i l e ) . More r e c e n t l y ( A s s o c i a t e d P r e s s , June 1977) i t was re p o r t e d 15 that a French group had started work on behalf of the government of Saudi Arabia to provide water to that dry country. Towing icebergs from Antarctica to Saudi Arabia would involve a much greater order of magnitude in terms of iceberg s i z e , tow l i n e tension and t r a v e l time when compared to those encountered i n Labrador by the o i l industry; but, undoubtedly, experience acquired i n the northern hemisphere w i l l prove very h e l p f u l to a c t i v i t i e s conducted i n the southern hemisphere. Even i f icebergs were of no i n d u s t r i a l importance they would remain of great i n t e r e s t to oceanographers, f o r the study of these huge natural d r i f t i n g buoys can y i e l d large quantities of oceanographical data.. From t h i s point of view, icebergs o f f e r the advantage of being large enough so that they can be seen using s a t e l l i t e imagery,. . Cloud cover permitting, sequences of daily s a t e l l i t e pictures can provide iceberg t r a j e c t o r i e s which, i f analyzed using d r i f t models, may y i e l d valuable r e s u l t s on ocean dynamics,. 16 ICEBERG DYNAMICS 3 , 1 . L i s t of symbols. C^ form drag c o e f f i c i e n t f o r portion of iceberg exposed to wind Cd^j form drag c o e f f i c i e n t for portion of iceberg below sea l e v e l C^ skin f r i c t i o n drag c o e f f i c i e n t C R e Reynolds c o e f f i c i e n t E rotation damping term (m2 kg s - 2 ) F w t o t a l water pressure force (N) buoyancy force (N) Fda: dynamic a i r drag (N) F ^ dynamic water drag (N) F T towing force (N) g acceleration of gravity (m s - 2 ) 3+ moment of i n e r t i a of iceberg (m2 kg) K water f r i c t i o n c o e f f i c i e n t (m2 kg s -*) l a lever arm for a i r drag force (m) l w lever arm for water drag force (m) M<; iceberg mass (kg) Re Reynolds number S a area of the abovewater portion of iceberg perpendicular to wind d i r e c t i o n (m2) S w area of the underwater portion of iceberg perpendicular to r e l a t i v e water flow d i r e c t i o n (m2) 17 dynamic drag area of above water portion (m2) s c l w dynamic drag area of underwater portion (m2) s ocean surface slope (rad) T a torque due to a i r dynamic drag (m N) T w torque due to water dynamic drag (m N) v^ wind velocity (m s - 1 ) v ( iceberg v e l o c i t y (m s _ 1 ) v w water velocity (m s _ 1 ) v w^ water v e l o c i t y with respect to the iceberg (m s - i ) v ^ iceberg velocity with respect to the water (m s - i ) tf^ water acceleration (m s - 2 ) ^ iceberg acceleration (m s - 2 ) wind d i r e c t i o n i n iceberg reference frame (rad) p. d i r e c t i o n , in iceberg reference frame, of r e l a t i v e flow of water with respect to berg (rad) )) water absolute v i s c o s i t y (m2 srl) water density (kg m-3) a i r density (kg m-3) (p l a t i t u d e (rad) C$ angular v e l o c i t y of the earth (rad s -*) 3.2. Introduction. At the beginning of the twentieth century, the explorer Nansen on board the Fram i n the Greenland Sea, noticed that f l o a t i n g ice did not d r i f t in the di r e c t i o n of 18 the wind but 20° to 40° to the r i g h t of that d i r e c t i o n . Working on an explanation of t h i s fact, Ekman produced a theory f o r wind driven c i r c u l a t i o n and probably became the f i r s t s c i e n t i s t to analyze the movements of d r i f t i n g i c e . In the l a s t ten years considerable i n t e r e s t has been shown in the study of a r c t i c sea ice and to a lesser extent i n icebergs of the Davis S t r a i t and Labrador Sea, Unfortunately, although sea ice research has been very intense, sometimes carried out by whole teams such as the a r c t i c Ice Dynamics Joint Experiment (AIDJEX), only a small percentage of t h i s work has any relevance to iceberg d r i f t . There are two main reasons for t h i s f a c t : unlike icebergs, s o l i d pack ice can be physically treated as a continuous, two-dimensional medium under s t r a i n while ocean c i r c u l a t i o n i s dramatically altered by the presence of ice. 3.3. Forces acting on icebergs. Dynamic models are based on Newton's second law of motion. I f we use the earth's surface as reference frame, the dynamic equation for iceberg d r i f t i s where F^ and F d a are the forces applied to the berg by sea water and wind. . F^ can be expressed as the sum of F b v 0 (buoyancy forces r e s u l t i n g from the action of s t a t i c pressure forces) and which i s the water dynamic drag force. I f i t 19 i s assumed t h a t buoyancy f o r c e s and weight are i n d i r e c t o p p o s i t i o n and i f we express F ^ and F ^ as - ' 4 P w S w - v r w i i >\rW(t-z] Eguation (1) can be r e w r i t t e n as [ 3 1 M<- % - Fdw + F«,a + 2 t f « - (ox v;-Cockanoff et a l , (1971) used an analog computer to p l o t i c e b e r g t r a j e c t o r i e s i n i n e r t i a l c o n d i t i o n s and under the i n f l u e n c e of a v a r i a b l e c u r r e n t . In t h e i r computation based on Equation (3) , r e p r e s e n t s the mass of the berg to which i s added "a v i r t u a l mass which i s dependent upon the shape c h a r a c t e r i s t i c s of the body and the flow c h a r a c t e r i s t i c s around the body, while v^- i s r e p l a c e d by v^ u . The pros and cons of t h i s l a s t c h o i c e of v a r i a b l e w i l l be d i s c u s s e d i n S e c t i o n [3.5.3], . Sodhi and Dempster (1975) s t u d i e d the response of i c e b e r g s t o r o t a r y t i d a l c u r r e n t s and sudden changes i n t r a n s l a t o r y c u r r e n t v e l o c i t y . They obtained exact s o l u t i o n s f o r t r a j e c t o r i e s and response times f o r these w e l l d e f i n e d s i t u a t i o n s . I t i s c l e a r t h a t i n t h i s type of m o d e l l i n g , the major unknowns are the form drag c o e f f i c i e n t s and p r o j e c t e d s u r f a c e s . 20 3.4. D i f f i c u l t measurement of i c e b e r g parameters,. A l l authors, i n agreement with Hoerner (1965), suggest form drag c o e f f i c i e n t s r a nging from 0,6 t o 1,2 • Banke and Smith (1974) towed s m a l l i c e b e r g s on the Labrador coast t o o b t a i n values of . The underwater shapes of the bergs being towed were obtained from phototographs and d i r e c t measurements were taken by scuba d i v e r s . The l a r g e s t of the towed i c e b e r g s d i s p l a c e d 1030 metric tons and the maximum towing speed reached with a s m a l l e r berg was 1 m/s. They obtained 1.2 as a mean value f o r with a standard d e v i a t i o n o f 0,2 . . As d u r i n g these experiments the Reynolds number remained between 7 x U0 3 and 8 x 10*, w e l l w i t h i n the range of the s u b c r i t i c a l zone,,the values are i n agreement with those p u b l i s h e d by Hoerner,. E t t l e (1974) analyzed i c e b e r g d r i f t data gathered from e i g h t bergs observed from U.S. Coast Guard v e s s e l s . H i s r e s u l t s confirmed the importance of strong winds on both ocean and i c e b e r g movement.„ Assuming t h a t steady s t a t e c o n d i t i o n s p r e v a i l e d and using a /S a r a t i o equal to 3.5 as proposed by Budinger i n an unpublished work, E t t l e o b t a i n e d v a l u e s f o r C ^ / C ^ r a t i o t h a t range between 1.5 and 7,1 . As he mentioned, "the o b s e r v a t i o n a l e r r o r s 1 are magnified by squaring the r e l a t i v e r a t i o " , Aware of t h i s f a c t and i n view of the d i f f i c u l t i e s encountered when t r y i n g to o b t a i n r e l i a b l e values f o r S a , S^, C ^ , C ^ , Dempster (1974 and p e r s o n a l communication) proposed a kinematic model where the v e l o c i t y of the i c e b e r g i s a l i n e a r combination of the 21 d i f f e r e n t v e l o c i t i e s of ocean l a y e r s and wind. The c o e f f i c i e n t s o f t h i s c o m b i n a t i o n would have t o be d e t e r m i n e d h a l f t h e o r e t i c a l l y , h a l f e m p i r i c a l l y . From the p u b l i s h e d l i t e r a t u r e on d r i f t m o d e l l i n g , i t appears t h a t most e f f o r t s have been d i r e c t e d e i t h e r towards d a t a a c q u i s i t i o n i n the.hope o f d e t e r m i n i n g and Q w and o t h e r shape f a c t o r s , or towards t h e o r e t i c a l n u m e r i c a l o r a n a l y t i c a l t r e a t m e n t o f s i m p l i f i e d d r i f t s i t u a t i o n s . . A d e t a i l e d d i s c u s s i o n of the d i f f e r e n t p o s s i b l e ways i n which models c o u l d be c o n s t r u c t e d , and of what terms s h o u l d be i n c l u d e d or n e g l e c t e d , seems very a p p r o p r i a t e , 3,5. T h e o r e t i c a l models, 3.5 , 1 , P r e l i m i n a r y d i s c u s s i o n . . The main c r i t i c i s m o f dynamic models i s t h a t many of t h e parameters used a r e d i f f i c u l t t o e v a l u a t e and t h a t s q u a r i n g the r e l a t i v e v e l o c i t y o f t h e b e r g w i t h r e s p e c t t o t h e water i s an a d d i t i o n a l s o u r c e o f i n a c c u r a c y s i n c e both water and i c e b e r g speeds are of t h e same o r d e r o f magnitude and d i f f i c u l t t o measure. However, models d i s c u s s e d h e r e , a r e d e r i v e d from the dynamic e q u a t i o n s i n c e i t was f e l t t h a t a model based on t h e t r u e r e p r e s e n t a t i o n of t h e p h y s i c a l phenomena can a l w a y s be improved by more d e t a i l e d s t u d y o f t h e s t a t e o f the ocean a l o n g w i t h t h e geometry and dynamics of i c e b e r g s . F u r t h e r m o r e , i t w i l l be shown t h a t , ocean p o t e n t i a l 22 f i e l d s and not water dynamic drag forces, are the d r i v i n g influence of iceberg dynamics. Thus, water drag forces have a damping r o l e , and errors made on t h e i r evaluation are not as c r u c i a l . 3,5.2. Nonaccelerated flow. At f i r s t , the case of an i n f i n i t e , nonviscous, homogeneous ocean on which no wind exists i s considered. It i s also assumed that under the influence of a uniform slope f i e l d , a perfect geostrophic c i r c u l a t i o n i s taking place. Water motion i s described by Equation (4) As these conditions define one of the simplest cases of oceanic c i r c u l a t i o n , the equations used i n the models should at l e a s t in t h i s case, provide exact solutions. It i s obvious that an iceberg, f l o a t i n g on such an ocean and o r i g i n a l l y d r i f t i n g at the same speed as the water (vw^=0) , w i l l not see i t s motion altered,. Indeed, according to Archimedes' p r i n c i p l e , and since the r e l a t i v e speed of the berg with respect to the water i s zero, the sum of forces applied by the water to the berg i s exactly i d e n t i c a l to the sum of forces that would be applied to an equal volume of water. If s* i s the slope of the ocean surface, buoyancy forces and weight of the iceberg are linked by 23 while the presence of geostrophic equilibrium y i e l d s Equation (1) now becomes 3. 5- 3, accelerated flow. Equation (6) i s v a l i d whenever the assumption of a geostrophic nonaccelerated flow i s acceptable, Unfortunately such a s i t u a t i o n almost never occurs, In the case of a flow accelerated under the influence of a pressure f i e l d . Equation (5) holds true, while Equation (6) no longer does. The iceberg must then obey [si « F ^ ^ i l i O K ^ + 9 M i s Equation (8) i s i d e n t i c a l to Equation (4) with the exception of the term F^uo* The iceberg w i l l therefore move i n exactly the same manner as does the water; the potential f i e l d influences both iceberg and water in an egual manner. In t h i s case i t i s inadequate to speak of response time of the berg under a change in current velocity since both current and berg accelerate i n phase. In agreement with t h i s conclusion Bayly (1971), commenting on Cockanoff's model, i n s i s t e d on the necessity of taking into account the effect of the pressure 24 f i e l d on the i c e b e r g : " p a s s i n g through a p o t e n t i a l f i e l d a f l o a t i n g body would be expected t o experience the same a c c e l e r a t i o n s as the water flow, i n phase with the water. . Thus i t would be a mistake t o assume t h a t an i c e b e r g would be a c c e l e r a t e d by the. flow only a f t e r the flow had i t s e l f a c c e l e r a t e d , as a time l a g i n the v e l o c i t y p r o f i l e would be i n t r o d u c e d , T i d a l c u r r e n t s , f o r example, are d r i v e n by changes i n the t i d e r a i s i n g p o t e n t i a l and any study of the e f f e c t s of t i d a l c u r r e n t s t h a t does not take i n t o c o n s i d e r a t i o n the e f f e c t upon the sea slope of such p o t e n t i a l changes would be erroneous. Equation (8) r e q u i r e s knowledge of the sea s l o p e . U n f o r t u n a t e l y , as has already been mentioned, such i n f o r m a t i o n cannot a c c u r a t e l y or r a p i d l y be obtained by d i r e c t measurement i n order t o be u s e f u l i n r e a l time modelling of i c e b e r g d r i f t . T h i s d i f f i c u l t y may be avoided by using the water dynamics Equation (4) from which i t i s p o s s i b l e to d e r i v e s as a f u n c t i o n of both the speed and the a c c e l e r a t i o n of the water. I n t r o d u c i n g s obtained i n Equation (4) i n t o Equation (8) y i e l d s [ < f ] £ « t , + ^ / k - - 2 " « ^< Equation (8) p r e d i c t s a steady motion of the i c e b e r g with a zero r e l a t i v e d r i f t with r e s p e c t to the water. The C o r i o l i s term of t h i s equation i s i d e n t i c a l t o the one used by Cockanoff (1971). However Cockanoff d i d not i n c l u d e a water a c c e l e r a t i o n term. Supposing t h a t there e x i s t s a wind which 25 a c t s upon the emerged p o r t i o n of the i c e b e r g but does not p e r t u r b the water .flow, the movement of the berg i s d e s c r i b e d by [ 1 0 ] K = 1 + p u , * 5 a ] / M , . - Z 3.5.4. The case of a m u l t i - l a y e r ocean. Equation (10) p r o v i d e s a v a l i d d e s c r i p t i o n of the d r i f t ( as l o n g as the berg i s t o t a l l y immersed i n a l a y e r of water which has a v e l o c i t y constant everywhere, and which i s a c c e l e r a t e d by the presence of a p o t e n t i a l f i e l d . As e x p l a i n e d before, the i n t e r n a l pressure f i e l d , along with s t r e s s a p p l i e d to the upper and lower boundaries of the ocean, causes water v e l o c i t y to vary with depth. In the case of a two-layer ocean, the v e l o c i t y of the water i s supposed to be c o n s t a n t w i t h i n each of the two l a y e r s , M, and r e p r e s e n t the masses of water d i s p l a c e d by the i c e b e r g w i t h i n the f i r s t and second l a y e r . Using Equation (10) to express the f o r c e a p p l i e d on the p o r t i o n of the i c e b e r g immersed i n l a y e r 2 one o b t a i n s [ 1 1 ] Ft « M t JL. + fiU + 2M> " " & -^"0 * F « where F 2 1 i s the f o r c e a p p l i e d by p o r t i o n 1 of the i c e b e r g to p o r t i o n 2. I f the f o r c e s a p p l i e d on p o r t i o n 1 are expressed i n a s i m i l a r manner, the t o t a l a c c e l e r a t i o n of the i c e b e r g i s 26 obtained [12] C M , +^V ; . - r t .vL, - f i z < * An equation of the same type as Equation (6) could be written to describe the movement of an iceberg f l o a t i n g i n an n-layer ocean. . Although the use of such an equation would represent a t h e o r e t i c a l improvement over the use of Equation (12), since the physical phenomena are better described, the introduction of new c o e f f i c i e n t s related to the iceberg shape may, i n most cases, more than o f f s e t the t h e o r e t i c a l improvements. For example, the new c o e f f i c i e n t s introduced by Equation (12) are * i * sw, * S w z , c ^ , , Cdu)z , which replace , , used i n the one layer case. Thus, attemptinq to describe more than three d i f f e r e n t layers i s surely u n r e a l i s t i c and i n most sit u a t i o n s i t should be possible to obtain useful r e s u l t s by only representing one or two layers, 27 4. NUMERICAL MODELLING A numerical model based on the equations discussed in Chapter 3 was created to test the influence of various meteorological and oceanographic conditions on iceberg t r a j e c t o r i e s . 4,1. Theoretical representation of the iceberg. 4.1.1. D e f i n i t i o n of a dynamic surface. The only iceberg related parameters that are needed to study berg d r i f t according to Equation (10), are i t s mass and the shape c o e f f i c i e n t s Sa, , C^a a n d cd<*j* Actually, the two c o e f f i c i e n t s S and C appear only once, as a. product, i n the expression of Equation (10) through the evaluation of F^ (Equation (2)). I t i s therefore j u s t i f i a b l e to define a dynamic drag surface S^a as - Cja . Similar dynamic surface c o e f f i c i e n t s may be defined f o r the submerged portion of the iceberg. In order to allow for the modelling of any shape of iceberg, S^a was expressed as a function of 0a<; , where 6kt- i s defined as the di r e c t i o n toward which the wind blows, measured i n a frame of reference t i e d to the iceberg. According to i t s d e f i n i t i o n , %«.(/fl!a<) i s a continuous, s t r i c t l y p ositive function with a period of 2TT , 28 4,1 .2 . V a r i a t i o n s o f the dynamic s u r f a c e . A v a r i a t i o n r a t i o f o r S^ f ^ - ) f o r a given i c e b e r g i s d e f i n e d as the r a t i o of the maximum t o the minimum value reached by S^tOa*) when 0^ v a r i e s from 0 t o 2 i f . In order to ev a l u a t e t h i s v a r i a t i o n r a t i o , i t i s assumed t h a t the l a r g e s t l e n g t h t o width r a t i o f o r the emerged p o r t i o n of any i c e b e r g i s 5 : 1 , I t i s a l s o supposed that Cj^ s t a y s w i t h i n the range 0,.6 - 1 ,2 f o r a l l i c e b e r g s . Therefore, according t o these hypotheses, the. v a r i a t i o n r a t i o f o r S^fQi,,-) may be as high as 10. I s t h i s value overestimated? I f the abovewater p o r t i o n of an i c e b e r g i s elongated i n the d i r e c t i o n 0a<-, i t i s probable t h a t i t s shape i s somewhat s t r e a m l i n e d along t h i s d i r e c t i o n ( c f . F i g u r e 1). Thus, f o r t h i s p a r t i c u l a r i c e b e r g , both cAa(6ai) a n d sda.(&a<) r e a c n minimum v a l u e s f o r the same angle &a<-. S i m i l a r i l y i t c o u l d be argued, as shown i n Fig u r e 1 t h a t l a r g e values f o r and S«_ may o f t e n be reached s i m u l t a n e o u s l y . T h i s i n t u i t i v e argument supports the f a c t t h a t the v a r i a t i o n may indeed be as l a r g e as a whole order of magnitude. Underwater shapes are assumed, f o r modelling purposes, t o have a l s o a maximum v a r i a t i o n r a t i o of 10 f o r sduo t although i t c o u l d be argued t h a t underwater shapes should be more r e g u l a r and thus have s m a l l e r v a r i a t i o n r a t i o s f o r c o e f f i c i e n t s , T r y i n g to r e l a t e the v a r i a t i o n s of , to those o f Sji^j d i d not appear f r u i t f u l and was not attempted. I n order to provide a model of i c e b e r g d r i f t which would be f l e x i b l e enough t o adapt to a l l p o s s i b l e cases, T h e o r e t i c a l representation of the iceberg. 30 s<fcx. a n d sdoj(&) were rep r e s e n t e d by F o u r i e r s e r i e s . 4.1.3. I n f l u e n c e of Reynolds number, I t has been shown by Hoerner (1965) t h a t and Cj^ depend on the Reynolds number. The c h a r a c t e r i s t i c l e n g t h used t o compute Re i s the square r o o t of the dynamic drag s u r f a c e . T h i s c h o i c e e m p i r i c a l l y takes i n t o account the f a c t t h a t bodies with h i g h form drag c o e f f i c i e n t s induce t u r b u l e n t flow at lower values of the Reynolds number: i t a l s o o f f e r s the advantage of i n t r o d u c i n g no new parameter. The flow of a i r around the emerged p o r t i o n of the i c e b e r g i s t u r b u l e n t as soon as the l i g h t e s t winds blow. Indeed, a i r v i s c o s i t y i s c l o s e to 1.4 x 1 0 - s m2 s _ 1 , and assuming a c h a r a c t e r i s t i c l e n g t h of 100 metres, Reynolds numbers l a r g e r than 10 6 are obtained as soon as the a i r v e l o c i t y i s l a r g e r than 0.14 m s _ 1 . For t h i s reason, i s assumed to be constant s i n c e the data p u b l i s h e d by Hoerner show t h a t form drag c o e f f i c i e n t s do not vary much w i t h i n the t u r b u l e n t zone. M o d e l l i n g r e s u l t s (Chapter 5) show t h a t the Reynolds number o f the water flow around the i c e b e r g i s g e n e r a l l y l a r g e r than 10 s, w e l l w i t h i n the t u r b u l e n t zone f o r r e g u l a r smooth bodies such as spheres or c y l i n d e r s . The onset of t u r b u l e n c e may, however, happen a t much lower Reynolds numbers f o r the f o l l o w i n g reasons: i ) Rough s u r f a c e d c y l i n d e r s with a r a t i o g r a i n s i z e diameter t o c y l i n d e r diameter as s m a l l as 2 x 10~ 2 have a 31 c r i t i c a l Reynolds number close to 2 x 10* . Icebergs surfaces are much rougher than these c y l i n d e r s so a lower c r i t i c a l Reynolds number would be expected. i i ) Since the v e l o c i t y p r o f i l e of the ocean i s not constant, the Reynolds number of the flow w i l l vary with depth. Although the flow at some depth may have a small Reynolds number, turbulence onset may s t i l l occur caused by turbulent flow i n adjacent water layers. i i i ) Scuba divers have noticed the constant production of a i r bubbles released from the melting of the i c e (Dempster, personal communication). The upward flow of water around the ice created by these hubbies, along with the response of the berg to swell, prove that, i n the v i c i n i t y of the berg's surface, r e l a t i v e water motions may have s i g n i f i c a n t v e r t i c a l components. Thus the evaluation of Reynolds number based uniquely on the r e l a t i v e d r i f t speed i s probably an underestimation, the f l u c t u a t i o n s of the v e r t i c a l water v e l o c i t y may favor turbulence onset. The model assumes that C^^j i s constant as long as the flow i s f u l l y turbulent and S ^ represents the dynamic surface for turbulent flow. i n order to allow f o r changes i n the value C j ^ which occur when the flow becomes c r i t i c a l or s u b c r i t i c a l laminar, a Reynolds c o e f f i c i e n t C p e i s introduced, which i s equal to unity f o r turbulent flow and varies with respect to Re i n c r i t i c a l and s u b c r i t i c a l flow conditions. The curve representing the v a r i a t i o n s of t h i s c o e f f i c i e n t i s 32 one of the i n p u t data o f the model and i s continuous and piecewise l i n e a r as a f u n c t i o n of the Reynolds number.. The dynamic water drag i s expressed as where S^(©) = g c ^ T T © ) + tw S(»(nTr9) 4.1,4. . Need f o r the modelling of i c e b e r g r o t a t i o n , The model t a k e s i n t o account the f a c t t h a t the coupled a c t i o n s of the sea and the wind cause the i c e b e r g to r o t a t e . T h i s modelling o p t i o n was i n c l u d e d t o i n v e s t i g a t e the e f f e c t of such r o t a t i o n because the l a r g e v a r i a t i o n s t h a t occur i n the value of and suggest t h a t i c e b e r g r o t a t i o n s can d r a m a t i c a l l y i n f l u e n c e the d r i f t . The p h y s i c a l d e s c r i p t i o n of the phenomenon however can e a s i l y be c r i t i c i z e d f o r being too s i m p l i s t i c i n i t s approach. R e s u l t s o b t a i n e d when t h i s o p t i o n i s i n e f f e c t are merely q u a l i t a t i v e , 4. 1.5. R o t a t i o n equation. The movement of an i c e b e r q t h a t does not r o l l over but i n s t e a d keeps a constant f l o t a t i o n l i n e while d r i f t i n g can be d e s c r i b e d as the sum of a t r a n s l a t i o n and a 33 r o t a t i o n around the v e r t i c a l a x i s p a s s i n g through i t s c e n t r e of g r a v i t y . Newton's second law of motion a p p l i e d to r o t a t i n g systems s t a t e s t h a t the angular a c c e l e r a t i o n of a s o l i d allowed to r o t a t e around an a x i s p a s s i n g through i t s c e n t e r o f g r a v i t y i s e q u a l to the sum of a l l torques a p p l i e d to the s o l i d d i v i d e d by i t s moment of i n e r t i a with r e s p e c t to t h i s a x i s . Let Jt* be the moment of i n e r t i a of the . i c e b e r g with r e s p e c t to the v e r t i c a l a x i s passing through i t s c e n t r e of g r a v i t y . Supposing t h a t the berg i s under the e f f e c t of a wind f o r c e F^Q(Qai) a n d supposing t h a t the d i s t a n c e between the l i n e of a c t i o n of t h i s f o r c e and the r o t a t i o n a x i s i s la(6>ai-)* the torque due to t h i s f o r c e i s [14] I(e^) = Fja(A0 1(6..) The l e v e r arm f u n c t i o n l a(&<) i s a continuous f u n c t i o n with a p e r i o d o f 2 TT and i s represented i n the model by a F o u r i e r s e r i e s . k s i m i l a r l e v e r f u n c t i o n e x i s t s i n the.model f o r the underwater p o r t i o n of the i c e b e r g ( c f , Figu r e 2) , The r o t a t i o n equation of the berg i s then Ll*] 0 = UteO + Tu(euJ - E where Qi i s the angle measuring the r o t a t i o n of the frame of re f e r e n c e f i x e d t o the i c e b e r g with r e s p e c t t o a frame f i x e d on the ea r t h ' s s u r f a c e . E i s a damping term and was made J Q e g u a l t o K- ^  where K i s a p o s i t i v e water f r i c t i o n c o e f f i c i e n t with the dimension of [ Mjji L ij2L T j} - 1 . F u r t h e r refinement i n the c h o i c e of the damping term i s not j u s t i f i e d FIGURE 2 Lever arm functions. 35 because E q u a t i o n (15) a l r e a d y c o n s t i t u t e s a q u a n t i t a t i v e l y q u e s t i o n a b l e p a r t of t h e model. I n f a c t , E would be b e t t e r approximated by a f u n c t i o n such as E = K Cd?) fa*^^ where c< and ^ depend on t h e R e y n o l d s number, a c c o r d i n g t o e x p e r i m e n t a l r e s u l t s r e l a t i n g t o s k i n f r i c t i o n d r a g . However i t was f e l t t h a t t h e use of such a f u n c t i o n would s i m p l y c o m p l i c a t e t h e model w i t h o u t making i t more r e a l i s t i c , 4,2. M o d e l l i n g o f o c e a n o g r a p h i c , a t m o s p h e r i c and a s t r o n o m i c a l parameters, 4.2.1. I n t r o d u c t i o n , No attempt was made t o c o r r e l a t e t h e v a r i o u s parameters r e l a t i n g t o ocean c u r r e n t s , and a t m o s p h e r i c c i r c u l a t i o n or p r e s s u r e g r a d i e n t s . A l l such parameters were t r e a t e d as independent i n p u t v a r i a b l e s , as t h e scope o f t h i s work was n o t t o fcuild a model f o r ocean-atmosphere i n t e r a c t i o n s . C o r i o l i s a c c e l e r a t i o n i s always computed f o r 500N, 4.2.2. Wind and c u r r e n t s , i ) Wind and ocean c u r r e n t s a r e a l l o w e d t o v a r y i n d e p e n d e n t l y i n d i r e c t i o n and i n t e n s i t y as c o n t i n u o u s p i e c e w i s e l i n e a r f u n c t i o n s o f t i m e . i i ) A t w o - l a y e r ocean i s r e p r e s e n t e d by l a y e r s h a v i n g 36 independent v e l o c i t i e s v a rying as d e s c r i b e d above. I f a two l a y e r ocean i s modelled a c c o r d i n g to Equation (6), extraneous i c e b e r g c o e f f i c i e n t s must be introduced as mentioned i n Chapter 3. These new c o e f f i c i e n t s are defined according t o the same methods used i n the one l a y e r s i t u a t i o n , i i i ) T i d a l c u r r e n t s are represented by a c u r r e n t v e c t o r t h a t d e s c r i b e s an e l l i p s e . The input parameters are the maximum and minimum v e l o c i t i e s of the c u r r e n t , the d i r e c t i o n of the c u r r e n t as i t reaches one of i t s two maximum v e l o c i t i e s (azimuth of the g r e a t a x i s o f the e l l i p s e ) , and the t i d a l p e r i o d and phase. T h i s t i d a l c u r r e n t may be superimposed on the other p r e v i o u s l y mentioned ocean c u r r e n t s . 4.2.3, Pres s u r e f i e l d s and towing, Sea topography i s represented by sea l e v e l v a l u e s a t each p o i n t of a square g r i d , . The slope i s computed from the s l o p e of the s m a l l e s t t r i a n g l e of the g r i d t h a t i n c l u d e s the l o c a t i o n the berg, A second o p t i o n allows the c a l c u l a t e d time d e r i v a t i v e of water v e l o c i t y t o be i n c l u d e d i n Equation (12) to take i n t o account the e f f e c t of pressure f i e l d s . Towing i s represented by a f o r c e t h a t v a r i e s c o n t i n u o u s l y piecewise l i n e a r l y i n both d i r e c t i o n and i n t e n s i t y . 37 5, NUMERICAL RESULTS, ( 5.1. Order of magnitude of i c e b e r g a c c e l e r a t i o n s , d r i f t v e l o c i t i e s and Reynolds number. T h i s s e c t i o n provides rough es t i m a t e s of the magnitude of a c c e l e r a t i o n s experienced by an i c e b e r g , The berg under c o n s i d e r a t i o n i s a c y l i n d e r , 200 m i n diameter, f l o a t i n g with i t s a x i s v e r t i c a l , 100 m i n t o t a l h e i g h t . Assuming = 1030 kg m - 3 and = 850 kg m - 3 the r a t i o s v o / s a i s equal to /(p^-(3r ) = 5,7 , Because of the c y l i n d r i c a l shape, dynamic s u r f a c e s do not depend on c W or Q^i and are set equal to = 3,400 m2 and S ^ = 16,600 m2 . With a mass o f 2.7 x 10» kg, such an i c e b e r g i s medium-sized by Labrador standards. . With the n o t a t i o n s i n t r o d u c e d e a r l i e r , the a c c e l e r a t i o n s experienced by the i c e b e r g a re ( i n m s ~ 2 ) i ) C o r i o l i s 9.0 x 10- s v ^ i i ) Water drag 3.2 x 10- 3 v w i 2 i i i ) A i r drag 8.2 x 10~ 7 v a 2 iv) Sea slope 9,8 s (s i n rad) v) Towing 3.7 x 10-io F T For r e a s o n a b l e values o f the v a r i a b l e s such as v^ = 0.2 m s - i , v ^ = 0.1m s - i , v^ = 10 m s ~ i , s = 10~« i t i s c l e a r t h a t a l l a c c e l e r a t i o n s are of comparable order of 38 magnitude, around 1 0 - s m s - 2 , T h e r e f o r e , i t i s not j u s t i f i e d , a p r i o r i , to n e g l e c t any of the terms of Equations (10) or (12). For steady s t a t e c o n d i t i o n s , the d r i f t r a t i o v,^ / v a i s equal to 1.6 x 1 0 - 2 , i n good agreement with Murray (1969) who g i v e s e m p i r i c a l values of the r a t i o vi /v^ ranging from 2.5 x 10- 2 to 3,4 x 1 0 - 2 . Towing t e n s i o n s can be expected t o range from 10 s N to 5 x 10 s N , which, f o r t h i s berg, would r e s u l t i n a steady s t a t e towing speed of 0.1m s~* to 0.24 m s _ 1 with r e s p e c t t o the water, Since the dynamic v i s c o s i t y of the water near 0°C i s = 1.8 x 10 -* m2 s - i , the Reynolds number of the flow around t h i s i c e b e r g i s 7 x 10s vW(.- . Although v W t- i s o f t e n s m a l l enough f o r Re to be w e l l w i t h i n the c r i t i c a l zone, CR& i s always s e t e q u a l t o u n i t y f o r the numerical s t u d i e s s i n c e changes i n t r a j e c t o r y due t o v a r i a t i o n s of C^ e are comparable t o the v a r i a t i o n s observed when the dynamic s u r f a c e s are a l t e r e d as i n example [5.3.3.].. 5,2. Frame of r e f e r e n c e . A l l t r a j e c t o r i e s are p l o t t e d using an orthogonal r e f e r e n c e frame with the x - a x i s p o i n t i n g east and the y - a x i s p o i n t i n g n o r t h . The v e l o c i t y v e c t o r s f o r wind and ocean c u r r e n t are d e f i n e d i n c o u n t e r c l o c k w i s e p o l a r c o o r d i n a t e s u s i n g the x ^ a x i s as o r i g i n . 39 5,3, D i s c u s s i o n of t h e o r e t i c a l t r a j e c t o r i e s , 5. 3, 1. Foreword, The best use t h a t can be made of a numerical model such as the one presented i n Chapters 5 and 6, i s t o compare observed t r a j e c t o r i e s t o computed t r a j e c t o r i e s , when the o c e a n i c and atmospheric input parameters of the model are those measured i n s i t u while the i c e b e r g i s t r a c k e d . U n f o r t u n a t e l y , no such data are a v a i l a b l e i n the p u b l i c domain, and the model co u l d not be used at i t s b e s t p o t e n t i a l . As a consequence, v a r i o u s t r a j e c t o r i e s analyzed i n the next s e c t i o n s r e p r e s e n t simple s i t u a t i o n s t h a t best i l l u s t r a t e a p a r t i c u l a r p o i n t , while i n t r i c a t e t r a j e c t o r i e s r e s u l t i n g from h i g h l y varying c u r r e n t s and winds were avoided s i n c e , i n the absence o f data, they would r e l a t e more, t o g r a p h i c a r t than d r i f t m o delling. 5,3.2. Importance o f C o r i o l i s terms ( c f . F i g u r e 3), The three t r a j e c t o r i e s o f F i g u r e 3 are obt a i n e d with the i c e b e r g d e s c r i b e d i n s e c t i o n 1.5.1. j] and i d e n t i c a l wind and c u r r e n t c o n d i t i o n s . Hater v e l o c i t y i s co n s t a n t , eastward at 0,25 m s - * . Hind v e l o c i t y and d i r e c t i o n are shown as f u n c t i o n s of time i n F i g u r e 4. Equation J.4JJ shows t h a t the sea slope s i s down to the n o r t h . The d r i f t was computed a c c o r d i n g t o 40 NORTH (km) 0.0 0.7 1.4 2.1 2.8 3.5 P I 1 1 1 « — 1 to . . . - • • - . oo FIGURE 3.. Influence of C o r i o l i s term on t r a j e c t o r i e s . The water velocity i s constant eastward at 0,25' m s - 1 . The wind velocity i s given in Figure 4, The t o t a l d r i f t time i s 7 hours. I Trajectory 1: no C o r i o l i s term. Trajectory 2: C o r i o l i s term uses vt- , iceberg ve l o c i t y r e l a t i v e to earth's surface. In e f f e c t , t h i s amounts to ignoring the sea slope. Trajectory 3: C o r i o l i s term uses vt'w , iceberg velocity r e l a t i v e to the water. This i s the best treatment of the three, 41 2TT (a) W i n d d i r e c t i o n 8 ( r a d i a n s ) 377/2 77 - . — _ -TT/2 > 0 (b) W i n d s p e e d (m s 1) 30 20 10 \ \ \ 0 i \ 1 2 3 4 5 Time ( h o u r s ) FIGURE 4. Time dependent wind vector used f o r d r i f t s i m u l a t i o n s i n F i g u r e s 3, 5> and 6, _ (a) d i r e c t i o n 6 to which the wind blows, 9 i s zero to the e a s t , and i n c r e a s e s c o u n t e r c l o c k w i s e . (b) magnitude of wind v e c t o r i n m s _ 1 , 42 AUk = Fia ^ where v i s kept equal to zero f o r the f i r s t t r a j e c t o r y , and r e p l a c e d by v. and v*^ f o r r e s p e c t i v e l y the second and t h i r d t r a j e c t o r i e s . Since the water v e l o c i t y i s c o n s t a n t , the water a c c e l e r a t i o n term MIXJ i s zero. The s i g n i f i c a n t d i f f e r e n c e s observed i n these three t r a j e c t o r i e s prove the importance o f the C o r i o l i s a c c e l e r a t i o n and the n e c e s s i t y to i n v e s t i g a t e c a r e f u l l y , as i n s e c t i o n s [ 3 . 3 . U and [ 3 . 5 . | how i t should be computed. 5.3 .3 . I n f l u e n c e of i c e b e r g ' s shape i r r e g u l a r i t i e s ( c f . F i g u r e 5 ) , Wind and c u r r e n t c o n d i t i o n s are i d e n t i c a l t o those s p e c i f i e d i n s e c t i o n [ 5 , 3 . 2 , ] and the d r i f t equation i s — — _ » . hi 1- = FH* + F^ + 2 y\i w x v , - w The mass o f the i c e b e r g i s 2,7 x 10 9 kg and i t s shape f u n c t i o n s are s d c ( 6 ) = 1 0 3 £ 3 - 4 + 1 - ° cos.(2 0 ) 3 . ScLu ( 0 > = 1 0 4 I 1 - 7 + ° ' 7 c o s < 3 © ) .1 The i c e b e r g i s not allowed to r o t a t e and the f o u r t r a j e c t o r i e s are obtained f o r i n i t i a l o r i e n t a t i o n of the i c e b e r g , ( i . e . the c o u n t e r c l o c k w i s e angle from east to the i c e b e r g 0=0 axis) egual t o 0, TT/2, TT and 3TT/2 . 43 FIGURE 5, Influence of i r r e g u l a r shape on t r a j e c t o r i e s , Water velocity i s constant eastward at 0.25 m s _ l , giving sea slope s downward to the north i n the geostrophic approximation, The wind velocity i s given i n Figure .4,. The t o t a l d r i f t time i s 7 hours. The shape factors S (j( w and Sot a for the iceberg are given i n Section 5,3.3, The t r a j e c t o r i e s shown re s u l t from the following i n i t i a l values of 9), the counterclockwise angle from east to the 0=0 axis of the iceberg. Trajectory 1: <p=0 Trajectory 2: <p= TT/2 Trajectory 3: <p= Tt Trajectory 4: (j>=3 TT/2 44 Although the v a r i a t i o n r a t i o s f o r and are only 1.8 and 1.4 , the i c e b e r g d r i f t s along very d i f f e r e n t t r a j e c t o r i e s depending on i t s i n i t i a l o r i e n t a t i o n . S i m i l a r l y , i c e b e r g s with d i f f e r e n t shapes, even i f they have i d e n t i c a l masses, may e x h i b i t s t r i k i n g l y d i f f e r e n t t r a j e c t o r i e s , . 5.3.4. Importance o f i c e b e r g r o t a t i o n ( c f . F i g u r e 6) , These t r a j e c t o r i e s are obtained f o r d r i f t d u r a t i o n , wind and c u r r e n t c o n d i t i o n s and i c e b e r g parameters i d e n t i c a l t o those d e s c r i b e d i n £5.3,3.], F o r t h i s example, however, the i c e b e r g i s allowed t o r o t a t e a c c o r d i n g to T <£± - F, $ + FJ 9 - K d e yJc - -fa. + rciw ^ 5~t where K = 1.5 x 10 - 3 k g m 2 s - i and J= 1.3 x 10 * 3 kg m* . In a l l three c a s e s , i n i t i a l o r i e n t a t i o n and abovewater l e v e r arm f u n c t i o n s are i d e n t i c a l and equal t o 8<-0 = 0 and l a ( 0 ) = 0 The underwater l e v e r arm f u n c t i o n s are i ) l w = 30 s i n 9 i i ) 1^= 10 s i n 0 i i i ) 1VJ= -20 s i n 0 r e s p e c t i v e l y f o r i c e b e r g s used i n t r a j e c t o r i e s 1, 2 and 3. For c l a r i t y , the o r i g i n s of the t r a j e c t o r i e s have been s h i f t e d and the arrows i n d i c a t e the o r i e n t a t i o n of the i c e b e r g s every hour. These t r a j e c t o r i e s show t h a t i c e b e r g r o t a t i o n can d r a m a t i c a l l y a l t e r the d r i f t . Thus, f u r t h e r 45 FIGURE 6, I n f l u e n c e of i c e b e r g r o t a t i o n on t r a j e c t o r i e s . Note t h a t , f o r c l a r i t y , the o r i g i n i s s h i f t e d eastward at each subsequent s i m u l a t i o n . The i c e b e r g shape f a c t o r s are given i n Secti o n 5.3,3, In a l l three c a s e s , the 0=0 a x i s of the i c e b e r g i s i n i t i a l l y a l i g n e d east, The below-water moment arm f u n c t i o n s l w ( 0 ) f o r the t r a j e c t o r i e s shown are T r a j e c t o r y 1: l w ( 9 ) = 3 0 s i n (9) T r a j e c t o r y 2: l w ( 0 ) = 1 0 s i n ( 9 ) T r a j e c t o r y 3: l w ( 0 ) =-20sin (0) 47 f i e l d i n v e s t i g a t i o n on the occurrence of such r o t a t i o n s i s needed, 5,3,5, On the i n f l u e n c e of pressure f i e l d s ( c f . Fi g u r e 7 ) . T r a j e c t o r y no. 1 r e p r e s e n t s the movement of a p a r t i c l e of water d u r i n g 24 hours. Such displacement i s meant t o simulate c u r r e n t v a r i a t i o n s due to changes i n atmospheric g r a d i e n t s . T i d a l c u r r e n t s are then superimposed, The a c t i o n of wind i s not taken i n t o account and i c e b e r g t r a j e c t o r i e s 2, 3 and 4 are obtained according to T r a j e c t o r y no. 3 i s obtained using the i c e b e r g d e s c r i b e d i n s e c t i o n £ 5 . 1 . J, while t r a j e c t o r y no, 2 i s obtained using an i c e b e r g of the same shape but with the l i n e a r dimensions s c a l e d down by a f a c t o r of two. For no, 4, the l i n e a r dimensions of the i c e b e r g are doubled. The d i f f e r e n c e s i n d r i f t t r a j e c t o r i e s i l l u s t r a t e the d i s c u s s i o n i n [3,5,3,]. Indeed, i f the changes i n c u r r e n t v e l o c i t y are due to v a r y i n g pressure f i e l d s , a l l i c e b e r g s should d r i f t along t r a j e c t o r y no, . 1 a c c o r d i n g to Equation (10) I f t he term X w i s n e g l e c t e d , t h e o r e t i c a l t r a j e c t o r i e s 48 NORTH (km) -7.0 -4.0 -1.0 2.0 5.0 8.0 FIGURE 7. I n f l u e n c e o f ocean p r e s s u r e f i e l d on t r a j e c t o r i e s . 1: motion of a p a r c e l o f water due t o changes i n a t m o s p h e r i c g r a d i e n t s , w i t h t i d a l c u r r e n t s superimposed. A l l i c e b e r g s s h o u l d f o l l o w t h i s t r a j e c t o r y . 2 t h r o u g h 4 i l l u s t r a t e t h e ^  e r r o r i n t r o d u c e d by n e g l e c t i n g water a c c e l e r a t i o n %w i n e q u a t i o n £10], The d e v i a t i o n from 1 i n c r e a s e s w i t h i c e b e r g mass., 2: c y l i n d r i c a l i c e b e r g i n S e c t i o n £5.1] s c a l e d down by f a c t o r of 2 i n a l l d i m e n s i o n s . 3: c y l i n d r i c a l i c e b e r g i n S e c t i o n £5.1], 4: i c e b e r g i n S e c t i o n £5,1] w i t h a l l d i m e n s i o n s d o u b l e d . 49 w i l l depart from no. 1, and the l a r g e r the i c e b e r g s , the l a r g e r the e r r o r , s i n c e more massive i c e b e r g s have l o n g e r response times. 5.3.6. Towing e f f i c i e n c y ( c f . F i g u r e 8 ) , For t h i s study, the c u r r e n t i s assumed c o n s t a n t , f l o w i n g eastward at 0.25 m s - 1 and the wind c o n s t a n t l y blows northward a t 10m s " 1 . D r i f t t r a j e c t o r i e s r e p r e s e n t 4 hours of d r i f t a c c o r d i n g t o H i t = + ^ + ^ a 5 . The i c e b e r g parameters are those of s e c t i o n £ 5 . 1 , ] . T r a j e c t o r y no. 1 i s obtained i n the absence of towing. T r a j e c t o r i e s 2, 3, 4 and 5 are obtained when a towing f o r c e of 5 x 10 s N i s a p p l i e d from t = 1 hr to t = 4 h r s , The towing d i r e c t i o n i s c o n s t a n t , and i s r e s p e c t i v e l y s e t to 0, TT/2, TT and 3 TT/2 f o r t r a j e c t o r i e s 2 r 3, 4 and 5, Other constant towing d i r e c t i o n s p r o v i d e d r i f t t r a j e c t o r i e s ending on the curve ( I ) . T r a j e c t o r i e s i n s i m i l a r c o n d i t i o n s with towing p u l l 2.5 x 10 s N end on curve ( I I ) . These curves show t h a t towing e f f i c i e n c y depends on ocean and wind c o n d i t i o n s . T h i s mainly occurs because of the dependence o f Fj on the square of v W c- ,. When some d r i f t p r e d i c t i o n i s a v a i l a b l e , such curves w i l l h e l p decide towing s t r a t e g i e s . 50 FIGURE 8, Towing e f f i c i e n c y . E f f e c t of constant towing f o r c e F a p p l i e d i n d i r e c t i o n 9 (counterclockwise from east) from t=1 hour to t=4 hours on the c y l i n d r i c a l i c e b e r g of S e c t i o n J. 5. 1 j j . Wind i s northward (9= TT/2) at 10 m s ' 1 anc c u r r e n t i s eastward at 0.25 m s - i . . Curve (I) i s l o c u s of endpoints f o r F T =5 x 10 s N and a l l 9. T r a j e c t o r y 1: no towing T r a j e c t o r y 2: 9=0 (east) T r a j e c t o r y 3: 9= TT/2 (north) T r a j e c t o r y 4: 9= TT (west) T r a j e c t o r y 5: 9=3 TT/2 (south) Curve (II) i s l o c u s of endpoints f o r F r =2,5 x 1 0 s N and a l l 9, 51 NORTH (km) -1.0 0.0 1.0 2.0 3.0 4.0 52 6.. INVERSION 1 6.1. Concept. Numerical models such as those d i s c u s s e d i n pre v i o u s c h a p t e r s may be used f o r the p r e d i c t i o n of i c e b e r g d r i f t as long as ocean and atmosphere c o n d i t i o n s can be f o r e c a s t and i f i c e b e r g parameters are known. T h i s chapter i n v e s t i g a t e s how i c e b e r g parameters may be obtained from an a n a l y s i s of past t r a j e c t o r i e s of the bergs. 6.2,. Theory. For t h i s d i s c u s s i o n , i t i s assumed t h a t past, present and f u t u r e c u r r e n t s and wind data are a v a i l a b l e . In order to s i m p l i f y the d i s c u s s i o n , i t i s supposed t h a t water v e l o c i t y does not vary with depth. Equation (10), d e s c r i b i n g i c e b e r g dynamics i n the case of a one-layer ocean, may t h e r e f o r e be used. I t i s a l s o assumed t h a t a p a r t i c u l a r i c e b e r g has been t r a c k e d f o r some time i n the past, and t h a t i t s s u c c e s s i v e p o s i t i o n s and o r i e n t a t i o n s 6^ are known as a f u n c t i o n of time. L e t P t and P 2 be two t r a j e c t o r y l o c a t i o n s , reached at time t t and t 2 , and l e t v t i be the i c e b e r g v e l o c i t y at time t x . The displacement P.P^  may be expressed as [16] P^2 - ( t , - t O £ * C(%(t'Ut'dt t t which, a c c o r d i n g t o Equation (10), y i e l d s 53 _^ ± z t Y [17] R Pz - + [ \ t(i']di'di ' k \ \ ( ^ ¥ ^ ut *> t< t,t, *Z.Z*$[(^-V»)cH'clt ±,i, The f i r s t term i n the r i g h t hand s i d e of Equation (17) r e p r e s e n t s the water displacement and may be computed from water v e l o c i t y measurements as \ dt . The values v"tl and v-fc) necessary to e v a l u a t e both the f i r s t and the f o u r t h term may be d e r i v e d from t r a j e c t o r y data. The t h i r d term may be expressed as the sum of • n i {[ FA ^')<kt'dt ix t and of a s i m i l a r e x p r e s s i o n f or Mt- ^dt'dtt , P r o j e c t e d on the axes of the r e f e r e n c e frame ( i , j ), Equation (18) y i e l d s 00 Qui = ^ ^ - n ^ ^ k ^ X ^ ri - O where [20 and Q w , Rtov\' are s i m i l a r q u a n t i t i e s obtained by r e p l a c i n g i by j i n (20),. In the same manner Q a and 54 may be obtained as 0^ = \ \ Fdc_(t')dt'dt e t c , According to e a r l i e r assumptions, a l l q u a n t i t i e s Q W , Kn' T w „ # Q! , B l n i T ^ , can be computed from a v a i l a b l e data. L e t P be d e f i n e d as t - r Y [21] p = f^pz- c t . - t i )^ - u ^ j t ^ w x ^ y ^ - ^ d t dt *, t, t, and U and U be U- p-i u'~ p-j With t h i s n o t a t i o n Equation (16) becomes 22 which may be w r i t t e n as 23 For each segment of t r a j e c t o r y , two l i n e a r equations such as (23) may be w r i t t e n , whose unknowns are the c o e f f i c i e n t s of the F o u r i e r s e r i e s d e s c r i b i n g the i c e b e r g shape f u n c t i o n s . P r o v i d e d the p r e l i m i n a r y assumptions hold t r u e , the i n v e r s i o n problem i s t h e o r e t i c a l l y s o l v a b l e , 55 6.3, D i s c u s s i o n . T h i s s e c t i o n d i s c u s s e s the p o t e n t i a l accuracy o f the i n v e r s i o n method depending on atmospheric and oceanographic c o n d i t i o n s and on the q u a l i t y of the a v a i l a b l e data. 6,3 ,1 , L i m i t t o the number of unknowns. Attempting to re p r e s e n t the i c e b e r g shape f u n c t i o n with F o u r i e r s e r i e s of more than three f r e q u e n c i e s i s u n r e a l i s t i c c o n s i d e r i n g t h a t nothing but an i l l u s i o n o f accuracy would be.gained. Whenever the upper ocean must be represented by a m u l t i - l a y e r model, more c o e f f i c i e n t s are needed t o re p r e s e n t the i c e b e r g . In such cases the i n v e r s i o n may not provide any r e l i a b l e r e s u l t s but f o r the f i r s t and perhaps second terms. 6, 3, 2, , Unfavourable c o n d i t i o n s . In some circumstances, i t may become i m p o s s i b l e t o o b t a i n any i n f o r m a t i o n out of the t r a j e c t o r y data. For example i f no wind were to blow, the i c e b e r g would d r i f t at e x a c t l y the same speed as the water and a l l c o e f f i c i e n t s of the system of equations would vanish. The abovewater p o r t i o n of the i c e b e r q may be such t h a t the berg w i l l always expose the same s i d e to the wind, In t h a t case i t i s i m p o s s i b l e to determine any of the and b ^ 56 c o e f f i c i e n t s , although (0 a t * ) m a Y be computed, where 0 a <.- i s the constant o r i e n t a t i o n of t h i s berg with r e s p e c t to wind d i r e c t i o n . I f the behaviour of the i c e b e r g does not change S d a (0Qt) ^ s a H t h a t i s needed to p r e d i c t the t r a j e c t o r y , 6,4. I n f l u e n c e of the q u a l i t y of the data on the r e s u l t s . A computer proqram was used to t e s t the s e n s i t i v i t y of the i n v e r s i o n method t o e r r o r s i n the data. U n f o r t u n a t e l y , no r e a l data were a v a i l a b l e and a r t i f i c i a l t r a j e c t o r i e s such as those presented i n Chapter 5 were used as i n p u t f o r the i n v e r s i o n , In order to s i m u l a t e i n a c c u r a c i e s found i n a c t u a l d a t a , a random n o i s e was added to c u r r e n t , wind, i c e b e r g l o c a t i o n s and o r i e n t a t i o n s values. The amplitude of t h i s noise.was based on the accuracy of p r e s e n t l y a v a i l a b l e measuring d e v i c e s (R. Goodman, p e r s o n a l communication). Current speed ± 0,01 m s _ 1 Current d i r e c t i o n ± 5<> Wind speed ± 0.15 m s _ 1 Wind d i r e c t i o n ± 5° Iceberg p o s i t i o n t 30 i Iceberg o r i e n t a t i o n ± 1 0 ° The e r r o r on the l o c a t i o n o f the i c e b e r g corresponds to what may be expected from a radar echogram obtained from a d i s t a n c e 57 of a few k i l o m e t r e s . The.error i n o r i e n t a t i o n i s an estimate s i n c e d i r e c t o b s e r v a t i o n may be the only method unl e s s some s p e c i a l i n s t r u m e n t a t i o n i s developed. The r e s u l t s i n d i c a t e t h a t the i n v e r s i o n provides poor r e s u l t s , the f i r s t c o e f f i c i e n t s being determined with a 30% e r r o r , i f such a s i g n i f i c a n t e r r o r on the l o c a t i o n of the berg i s made, R e l i a b l e values of (Q .•) or S^{Q^) are o b t a i n e d f o r i s o l a t e d values of Qai or 6W<: i f f a v o u r a b l e c o n d i t i o n s are met: i ) The v e l o c i t y c o n t r a s t between the ocean l a y e r s i s l a r g e , and the wind i s strong, i i ) Wind and c u r r e n t d i r e c t i o n s do not vary f o r at l e a s t a few hours, i i i ) The i c e b e r g does not r o t a t e w i t h i n a few hours. P r e s e n t l y , due to the poor accuracy of d i s t a n c e measurements obtained from r a d a r data, the i n v e r s i o n method may only provide rough estimates of the average shape f u n c t i o n s of the bergs. T h i s q u e s t i o n should be examined i n g r e a t e r d e t a i l when r e a l data become a v a i l a b l e . I t i s hoped t h a t i t w i l l then become c l e a r t h a t i f adequate i n s t r u m e n t a t i o n i s developed to track i c e b e r g s , more i n f o r m a t i o n on the bergs c o u l d be gathered from t r a j e c t o r y a n a l y s i s . 58 7. CONCLUSION S u c c e s s f u l p r e d i c t i o n of i c e b e r g d r i f t r e q u i r e s the knowledge of some i c e b e r g parameters, which c o u l d be obtained from an a n a l y s i s of past t r a j e c t o r i e s , and the a b i l i t y to f o r e c a s t the v a r i a t i o n s of oceanic c u r r e n t s and wind. P r e s e n t l y , the : g r e a t e s t need i s f o r s m a l l s c a l e , d e t a i l e d study o f ocean behaviour i n order to develop numerical models f o r s h o r t term p r e d i c t i o n of oceanic movements. 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Proceedings of the Ninth world petroleum Congress, V o l . 4, p. 47-50 Main L i b r a r y TN863 W6 1975:4 64 APPENDIX 65 APPENDIX 1 HOT WATER DRILLING ON A COLD GLACIER by Jean-Gerard P. Napoleoni and Garry K. C, C l a r k e Department of Geophysics and Astronomy U n i v e r s i t y of B r i t i s h Columbia Vancouver V6T 1W5, Canada 1 T h i s manuscript has been p u b l i s h e d i n Canadian J o u r n a l of Earth S c i e n c e s , V o l , 15, No, 2, p, 316-321, 1978, 66 ABSTRACT An open c i r c u i t hot water d r i l l u sing a propane water heater and a g a s o l i n e - d r i v e n pump i s d e s c r i b e d . The d r i l l i s designed t o reach depths of 300 m i n c o l d i c e f o r h o l e s o f 3 cm i n diameter. A maximum d r i l l i n g r a t e of 120 m/hr was obtained d u r i n g f i e l d t e s t s , and a 220 m h o l e was d r i l l e d i n f o u r hours,. RESUME La sonde a eau chaude d e c r i t e i c i e s t a c i r c u i t ouvert. Concue pour f o r e r en g l a c e f r o i d e , des t r o u s ayant 300 m de profondeur et 3 cm de diametre, e l l e comprend un chauffe-eau au propane et une pompe mue par un moteur a essence. Sur l e t e r r a i n , l a v i t e s s e de forage a t t e i g n i t 120 m/hr e t un t r o u de 220 m de profondeur f u t fore.en guatre heures. 1. INTRODUCTION The hot water d r i l l d e s c r i b e d i n t h i s paper was b u i l t f o r the purpose of p l a c i n g t h e r m i s t o r c a b l e s i n c o l d g l a c i e r s . I t was designed to d r i l l h o l e s 300 m deep and 3 cm wide. The p r i n c i p l e of an open water c i r c u i t was adopted because i t i s f a s t e r and l e s s impaired by d i r t y i c e than the e l e c t r i c a l hot po i n t or the c l o s e d hot water c i r c u i t : t h e j e t of water c a r r i e s heat e f f i c i e n t l y and, i n d i r t y i c e , c o n s t a n t l y c l e a n s the h o l e . A s t e a m - d r i l l c o u l d not have been used, s i n c e i t i s only p r a c t i c a l f o r shallow h o l e s . Hot water d r i l l s have been s u c c e s s f u l l y used on temperate g l a c i e r s by F. G i l l e t ( 1971, and p e r s o n a l communication ), P, T a y l o r ( pers o n a l communication ) and H, . R o t h l i s b e r g e r (personal communication t o M. Meier ). F. M u l l e r a l s o used t h i s system i n c o l d i c e ( p e r s o n a l communication ) . Our purpose, i n w r i t i n g t h i s r e p o r t , i s t o provide a d e t a i l e d d e s c r i p t i o n of the necessary equipment and i t s use, 2, THEORY i Symbols c heat c a p a c i t y of water (J k g - 1 ° C _ 1 ) e e f f i c i e n c y of the d r i l l i n g t i p E l i n e h e a t i n g source s t r e n g t h (W m _ 1) 68 H depth of the hole (m) thermal conductivity of hose walls (W m_1 °C - 1) k^  thermal conductivity of i c e (W m_1 °C-i) L latent heat of fusion of ice (J kg - 1) g flow rate of water (m3 s _ l ) E D i n i t i a l radius of d r i l l hole (m) Rt inner radius of the hose (m) R2 outer radius of the hose (m) S maximum cross-sectional area of the d r i l l i n g t i p (m2) T 0 temperature of the hot water at the surface (°C) T t(z) temperature of the hot water at depth z inside the hose (oc) (z) temperature of the water at depth z outside the hose (OC) T 5 temperature of the water outside the nozzle (°C) i n i t i a l temperature of gla c i e r ice (°C) v d r i l l i n g speed (m s - 1 ) p density of water (kg m-3) density of i c e (kg m-3) | heat flux (W m-2) i i Examination of heat loss in the hose Figure 1 shows a v e r t i c a l section of the.hose and hole at depth z. The following assumptions are made : (a) The flow inside and outside the hose i s perfectly COLD WARM WATER WATER N \ HOSE WALLS t '2(Z)I •++•4 Ri R 2 F I G U R E 1 V e r t i c a l s e c t i o n o f t h e h o l e a n d h o s e a t d e p t h 70 t u r b u l e n t and the water i s t h e r e f o r e t h e r m a l l y homogeneous,. (b) The temperature of the upcoming water i s T = 0 <>C a t any depth. These two assumptions are p e s s i m i s t i c and l e a d to o v e r e s t i m a t i o n of the heat l o s s . The v e r t i c a l temperature g r a d i e n t of the water f l o w i n g i n s i d e the hose may then be expressed as as T 2 = 0°C , T, i s then obtained as a f u n c t i o n of depth T h i s r e s u l t l e a d s to the f o l l o w i n g o b s e r v a t i o n : the percentage of heat l o s t d u r i n g the passage of water i n the hose does not depend on the i n i t i a l temperature T 0 of the water. However, t h i s percentage l o s s i s diminished by i n c r e a s i n g the flow and the r a t i o R 2 / R 1 • Thus, f o r a g i v e n amount o f energy a v a i l a b l e a t the s u r f a c e , minimum l o s s i s achieved with high flow a t low temperature, i i i M e l t i n g processes at the d r i l l i n g t i p d_T = 2Tr/rt, (Tz -Ti) Let us d e f i n e the c o n s t a n t As shown i n F i g u r e 2 , two l e v e l s are d e f i n e d ; above l e v e l A, the d r i l l i n g t i p and the hose have a c o n s t a n t s e c t i o n F I G U R E 2 V e r t i c a l s e c t i o n o f t h e n o z z l e a n d h o l e a t t h e b o t t o m o f t h e h o l e . 72 S ; from l e v e l A down to l e v e l B the s e c t i o n of the t i p decreases. The advance of the n o z z l e depends on the r a t e of melting of the i c e w a l l s between l e v e l s A and B . Any me l t i n g which occurs above l e v e l A only c r e a t e s a u s e l e s s widening of the.hole so t h a t a l l the heat t h a t the water c a r r i e s past A i s l o s t . I t i s t h e r e f o r e u s e f u l to d e f i n e e as the e f f i c i e n c y of the t i p where T^ i s the temperature of the water when i t flows upward past A , An i d e a l l y shaped nozzle w i l l use a l l the g i v i n g an e f f i c i e n c y equal t o u n i t y . i y D r i l l i n g speed I t i s assumed t h a t i c e i s a t melting p o i n t s i n c e m e l t i n g o f i c e o r i g i n a l l y a t -10 <>C r e q u i r e s only 7 % more energy than melting of the same mass at melting p o i n t . T h i s makes the assumption a c c e p t a b l e as a b a s i s f o r e s t i m a t i n g d r i l l i n g r a t e i n c o l d i c e . The d r i l l i n g speed can then be w r i t t e n TiCH) - Tz a v a i l a b l e heat between A and B , so t h a t T equals 73 i v E e f r e e z i n g hole In c o l d i c e , there i s a p o s s i b i l i t y t h a t water w i l l s t a r t t o r e f r e e z e i n the hole while d r i l l i n g i s s t i l l i n p r o g r e s s . In the worst case the hose and n o z z l e may become locked i n the i c e and i m p o s s i b l e to recover. I t i s t h e r e f o r e worthwhile t o examine the c o n d i t i o n s which lead to premature c l o s u r e of the h o l e . Let us c o n s i d e r a hole i n c o l d i c e , f i l l e d with water. I f E i s the r a d i u s of the h o l e , the ice-water boundary w i l l move a c c o r d i n g to the equation [6] =*•-<!>-where >^ i s the heat f l u x through the water and ^ the heat f l u x through the i c e . I f the heat l o s s e s through the hose w a l l s are represented by a h e a t i n g l i n e E » ^ 1 can be approximated by p-(5 =r r i ZTX R while w i t h i n the i c e _ A f i n i t e d i f f e r e n c e scheme was used to s o l v e Equation (6) ( J a r v i s and C l a r k e , 1974) and an example of computed r e s u l t s i s shown i n F i g u r e 3 . I n i t i a l c o n d i t i o n s are f o r a hole r a d i u s B 0 i n i s o t h e r m a l i c e at temperature . I f d r i l l i n g progresses at 100 m / hr , w i t h i n one.minute a f t e r a given depth i n the i c e has been reached by the t i p of the n o z z l e , the hot water j e t i s already f a r enough below t h i s 7 4 FIGQBE 3 S e f r e e z i n g of a water f i l l e d hole i n c o l d i c e . The i n i t i a l r a d i u s of the hole i s R = 2 c m and the l i n e h e a t i n g s t r e n g t h i s E = 72 H a—* . The r a d i u s of the hole i s given as a f u n c t i o n of time f o r d i f f e r e n t v a l u e s o f the i n i t i a l i c e temperature T i -T I M E (min) 7 5 depth to a l l o w the assumpt ion t h a t a l l the heat t h a t r e a c h e s the i c e w a l l s i s p r o v i d e d by l o s s t h r o u g h the hose . As one minute i s s m a l l compared to the t y p i c a l time c o n s t a n t o f such a sys tem, the proposed i n i t i a l c o n d i t i o n s are r e a s o n a b l e , Because they n e g l e c t the amount o f heat r e c e i v e d by the i c e w i t h i n the f i r s t minute , they l e a d to o v e r e s t i m a t i o n o f the r a t e of r e f r e e z i n g . A l i n e source h e a t i n g r a t e o f 72 W / m , as used f o r the example shown i n F i g u r e 3 , may be o b t a i n e d with water a t 20 °C f l o w i n g through a hose o f the type used i n the f i e l d t e s t s ( c f . 3 - i i ) . In t h a t c a s e , wi th a maximum d iameter o f 3 cm f o r the hose and the n o z z l e , i t i s found t h a t d r i l l i n g w i l l be safe i n i c e as c o l d as -12 °C . T y p i c a l l y , the t e m p e r a t u r e of a c o l d g l a c i e r i s minimum at a few t e n s of metres below the s u r f a c e , and from t h e r e , i n c r e a s e s s l o w l y wi th d e p t h . As the t e m p e r a t u r e o f the downward- f lowing d r i l l i n g water d e c r e a s e s wi th depth i t i s not o b v i o u s , a p r i o r i . where t h e g r e a t e s t r e f r e e z i n g r i s k w i l l be e n c o u n t e r e d . I t i s a l s o worth m e n t i o n i n g t h a t , because o f the moving boundary , r e s u l t s as shown i n F i g u r e 3 are n o n l i n e a r wi th r e s p e c t to T^ , H 0 and E . T h e r e f o r e , the r e s u l t s do not s c a l e i n a s imple way wi th r e s p e c t t o these v a r i a b l e s . 76 3, DESIGN i Heater A commercially a v a i l a b l e propane heater was used. T h i s heater c o n s i s t s of a 3 m copper pipe c o i l e d i n the shape of a long t r u n c a t e d cone , A l a r g e burner blows a flame i n s i d e the cone, thus heating the water f l o w i n g through the pipe. At sea l e v e l , water f l o w i n g at il2 1/mn can be heated from 0 °C to 45 oc the propane consumption being 5 kg per hour. i i Hose Equation (1) i n d i c a t e s t h a t high flow r a t e reduces heat l o s s . On the other hand, a narrow hose r e q u i r e s a s m a l l c r o s s ^ s e c t i o n S f o r the hole, thus i n c r e a s i n g the d r i l l i n g r a t e . A compromise between these two f a c t o r s l e d to the c h o i c e of a hose with i n n e r r a d i u s 1, 2 cm and outer r a d i u s 2.5 cm ,. For t h i s case B 2 /B^ = 2 ; higher values f o r t h i s r a t i o cannot be obtained without u s i n g expensive s p e c i a l i z e d hoses,. No data were a v a i l a b l e on the value of thermal c o n d u c t i v i t y f o r the w a l l m a t e r i a l of the hose. An experiment shown i n F i g u r e 4 was conducted to determine kj^ . The l o n g i t u d i n a l temperature g r a d i e n t of a flow of c o l d water c i r c u l a t i n g through a sample of hose immersed i n warm water was measured with t h e r m i s t o r s . The r e s u l t i n g curve was f i t t e d 77 FIGORE 4 Experimental setup to determine thermal c o n d u c t i v i t y of hose w a l l s . 1 c o l d water ( near 0 °C ) s u p p l i e d by the r e g u l a t e d bath; 2 e l e c t r i c a l wires l i n k i n g the t h e r m i s t o r s to the ohmmeter; 3 t h e r m i s t o r s ; twelve of them were used over a l e n g t h of 3 m ; The i n c r e a s e i n temperature between the f i r s t one and the l a s t was around 0,45 °C with a water flow of 2.5 1/mn ; 4 warm water ( near 15 °C ); 5 sample of hose; 6 water r e t u r n i n g t o the c o l d bath; t CD 7 9 t o Equation (3) ; k^ was found t o be 0,4 W m _ 1 <>K-» . Surface t r a n s p o r t a t i o n o f a 300 m length of hose i s i m p r a c t i c a l under f i e l d c o n d i t i o n s . . Three 100 m long s e c t i o n s were s t o r e d on l i g h t wooden r e e l s and c a r r i e d s e p a r a t e l y . Before d r i l l i n g they were connected and r o l l e d on a l a r g e m e t a l l i c r e e l f i t t e d with a feed-through s w i v e l j o i n t . D r i l l i n g of an e n t i r e hole c o u l d then proceed without i n t e r r u p t i o n . The r e e l can be used as a winch s i n c e i t i s equipped with a geared crank,. i i i Pumps Pumping 12 1/mn r e q u i r e s a head of about 20 atm to overcome f r i c t i o n through the conn e c t i o n s , t h e no z z l e and 300 m of hose. k s m a l l high pressure g a s o l i n e - d r i v e n pump was used and as such pumps have v i r t u a l l y no s u c t i o n power, and the water supply may be some d i s t a n c e from the d r i l l i n g s i t e , a s e l f - p r i m i n g c e n t r i f u g a l pump was used a t the head of the l i n e . i v D r i l l i n g t i p Equation (5) shows t h a t d r i l l i n g r a t e i s p r o p o r t i o n a l t o t i p e f f i c i e n c y so i t i s p r o f i t a b l e to use a well-designed n o z z l e . For t h i s reason d i f f e r e n t shapes of nozzle were t e s t e d by d r i l l i n g through l a r g e b l o c k s of c l e a r i c e , , Some 80 had grooved s u r f a c e s made to c r e a t e more t u r b u l e n t flow i n the hope of i n c r e a s i n g the speed at which i c e melts,. However, the t e s t s showed t h a t these a s p e r i t i e s are u s e l e s s because they c a t c h a g a i n s t the w a l l s of the hole and slow the advance. Best r e s u l t s were obtained with a long smoothly-tapered such as the one used i n the f i e l d and shown i n Figure 5 , As c o u l d i n t u i t i v e l y be expected, combination of high water flow at low temperature was most e f f i c i e n t with a l o n g t i p , v F i e l d equipment F i g u r e 6 i s a diagram of the complete eguipment d u r i n g f i e l d o p e r a t i o n . Information about the v a r i o u s components i s given i n the appendix, 4. FIELD RESULTS The d r i l l was used i n August 1976 on the Hazard G l a c i e r i n the S t e e l e G l a c i e r area ( S a i n t E l i a s Mountains, Yukon T e r r i t o r y , Canada). The i c e was presumably c o l d because other g l a c i e r s i n the area are known to be c o l d . D r i l l i n g s i t e s were l o c a t e d near the upper l i m i t of the a b l a t i o n zone and t h e i r a l t i t u d e ranged from 1800 m t o 2100 m , Near the s i t e s , the g l a c i e r was f a i r l y c l e a n , with rocks c o v e r i n g l e s s than 5 % of the i c e s u r f a c e . D r i l l i n g went as f a s t as 120 m/hr near the s u r f a c e . . The deepest hole (220 m) took 4 hours to d r i l l and, near i t s 81 F I G U R E 5 L o n g i t u d i n a l s e c t i o n o f t h e d r i l l i n g t i p u s e d i n t h e f i e l d . A l l d i m e n s i o n s a r e i n m i l l i m e t r e s . 82 FIGUBE 6 D r i l l i n g equipment d u r i n g f i e l d use, 1 f i n e s t r a i n e r ; s i l t i s h a r m f u l t o the p i s t o n pump; 2 s u c t i o n hose; d i a m e t e r s 2.5 - 3,1 cm ; 3 low p r e s s u r e c e n t r i f u g a l pump w i t h manometer; 4 o r d i n a r y garden hose ( maximum l e n g t h : 100 m , d i a m e t e r s 1.25 cm - 2 cm ) ; 5 f l o w s w i t c h ; , 6 war n i n g b e l l ; t h e s w i t c h t r i g g e r s i t when the water f l o w drops because t h e burner must th e n be shut o f f ; 7 water h e a t e r ; 8 b u r n e r ; 9 copper c o i l e d p i p e ; 10 propane tank w i t h manometer; 11 h i g h p r e s s u r e p i s t o n pump w i t h surge t a n k , manometer and r e l i e f v a l v e ; 12 hose r e e l ; 13 geared hand c r a n k ; 14 h i g h p r e s s u r e hose; 15 p u l l e y ; 16 bronze c y l i n d e r s ; they add weight t o t h e n o z z l e ; 17 b r a s s n o z z l e ; 83 84 completion, the d r i l l i n g speed was 50 m/hr , The flow was maintained at 12 1/mn and the water temperature at the s u r f a c e was around 30 °C , Water pressure o s c i l l a t e d between 14 and 19 atmospheres with o c c a s i o n a l peaks to 23 atmospheres. The c i r c u i t was p r o t e c t e d by a r e l i e f valve s e t at 26 atmospheres. As the burner was not a u t o m a t i c a l l y r e g u l a t e d , t h e r e was a p o s s i b i l i t y of overheating the c o i l , should the water flow have stopped. This r i s k was avoided by using a f l o w - s w i t c h connected t o a warning b e l l . One 45 kg propane tank was ab l e t o provide 5 kg of f u e l per hour even when a i r temperature dropped to -5 °C , A nearby water source i s a p r e r e g u i s i t e f o r t h i s system. However, once d r i l l i n g has s t a r t e d , i f the water source dwindles, i t i s p o s s i b l e to r e c y c l e the water which flows out of the hole. The d r i l l i n g s i t e must be c l e a r e d t o prevent s m a l l r o c k s from f a l l i n g i n the hole and i m p a i r i n g the progress of the n o z z l e . At the s u r f a c e , the diameter of the hole reached 6 cm i n about 15 mn and then g r a d u a l l y expanded up t o 8 cm . B e f r e e z i n g of the hole never c r e a t e d any d i f f i c u l t y . A f t e r the f i r s t 20 m had been d r i l l e d , the hose u n r o l l e d from the r e e l under i t s own weight. Using a p u l l e y to guide the hose i n t o the hole was not e s s e n t i a l s i n c e , even when l e f t on the g l a c i e r s u r f a c e , the hose s l i d smoothly i n t o the h o l e . No de v i c e was used t o c o n t r o l v e r t i c a l i t y . Badar soundings made a t the d r i l l s i t e s gave i c e t h i c k n e s s e s 10-15 % s m a l l e r than the a c t u a l l e n g t h of the 85 h o l e s , which tends t o i n d i c a t e t h a t a s i g n i f i c a n t d e f l e c t i o n had o c c u r r e d . I t was u s u a l l y impossible t o be sure t h a t the g l a c i e r bed had a c t u a l l y been reached and t h a t the n o z z l e had not i n s t e a d encountered a l a r g e rock. I n one i n s t a n c e , d r i l l i n g came t o a h a l t at 110 m depth,. Water c i r c u l a t i o n was continued f o r about 20 mn at which time s i l t y water appeared at the s u r f a c e . . Since s i l t i s most l i k e l y to be found on the bed rock as a product of g l a c i e r s l i d i n g , the appearance of such s i l t y water i s , i n the absence of other i n f o r m a t i o n , a good h i n t i n a s s e s s i n g whether or not the bed was a c t u a l l y reached,, I t i s t h e r e f o r e u s e f u l t o estimate how long one should wait i n order to take f u l l b e n e f i t of such an i n d i c a t i o n ; assuming t h a t the average diameter of the h o l e i s 6.5 cm , the annular space between the hose and the i c e has a volume of 2,8 1/m which, f o r t h a t p a r t i c u l a r h o l e , g i v e s a t r a v e l time of 25 mn i n reasonable agreement with the observed v a l u e . P r o v i d e d that the, hole was kept f i l l e d with water, p u l l i n g up the hose was f a i r l y easy and i t was not necessary t o use the r e e l crank. The hose had to be drained t o prevent water from f r e e z i n g i n s i d e during the n i g h t . This was e a s i l y done by blowing through the hose compressed propane from the f u e l b o t t l e . Propane was a l s o c o n v e n i e n t l y used f o r camp stoves and, as both g a s o l i n e engines c o u l d be adapted t o run on t h i s f u e l , propane i s probably the bes t c h o i c e f o r t h i s type o f f i e l d work. Three h o l e s were d r i l l e d i n t h r e e days by t h r e e 86 people. The d i s t a n c e between s i t e s was approximately one k i l o m e t e r of f a i r l y smooth i c e and s l e d s were used to c a r r y the equipment which, a l l i n c l u d e d , weighed 400 kg , The h e a v i e s t piece was the propane b o t t l e ; when f u l l i t weighed 80 kg . The f i e l d t e s t s showed reasonable agreement with the the o r y : a c c o r d i n g t o s u r f a c e measurements and using Eguation (5) , the e f f i c i e n c y of the d r i l l i n g t i p i s 0.6 . D r i l l i n g r a t e decay with depth i n d i c a t e s t h a t 60 % of the heat i s l o s t by conduction through the hose when the hole i s 2 20 m deep. I t i s probable t h a t an extremely long tapered t i p would have an e f f i c i e n c y as high as 0.8 >• The burner should be mo d i f i e d : f o r a constant flow of water and propane, the temperature of the heated water i s 45 °C a t sea l e v e l where at an a l t i t u d e of 1800 m i t drops from 32 °C t o 28 °C f o r a 300 m ga i n i n e l e v a t i o n , T h i s i s most l i k e l y due to incomplete combustion caused by r a r e f a c t i o n of the a i r . These m o d i f i c a t i o n s could i n c r e a s e the speed by as much as 30 % , Holes as deep as 500 m could then be d r i l l e d i n l e s s than 8 hours, because the pump can pro v i d e enough head and the hose can withstand strong t e n s i o n s . 87 5,. ACKNOWLEDGEMENTS We wish t o thank E.B, LaChapelle and P h i l T a y l o r of the U n i v e r s i t y of Washington f o r t h e i r h e l p f u l a d v i c e and the use of t h e i r t e s t equipment. We are a l s o very g r a t e f u l to B. B. . Narod f o r h i s a s s i s t a n c e i n the design and t o S.G. C o l l i n s f o r h i s v a l u a b l e help during f i e l d use. The development of the d r i l l was supported by the N a t i o n a l Eesearch C o u n c i l ( Canada), Environment Canada and the U n i v e r s i t y of B r i t i s h Columbia Commitee of A r c t i c , and Al p i n e Research. The coo p e r a t i o n o f Parks Canada i s g r a t e f u l l y acknowledged. J.-G. P, Napoleoni wishes to thank The Canada C o u n c i l f o r i t s generous f i n a n c i a l support, EEFEEENCES Carslaw, H.S., and Jaeger, J - C , 1959, Conduction of heat i n s o l i d s , second e d i t i o n - Oxford U n i v e r s i t y Press, London. G i l l e t , F. 1975- Steam, hot-water and e l e c t r i c a l thermal d r i l l s f o r temperate g l a c i e r s . J o u r n a l of G l a c i o l o g y , V o l , 14 , No. 70 , p., 170 -76 . J a r v i s , G, T. , and C l a r k e , G, K, C, , 1974, Thermal e f f e c t s of c r e v a s s i n g on St e e l e G l a c i e r , Yukon T e r r i t o r y , Canada. J o u r n a l of G l a c i o l o g y . V o l . 13, No. 68, p 243-54. 88 Yin-Chao Yen, and C h i T i e n , 1976, Heat t r a n s f e r c h a r a c t e r i s t i c s o f melting and r e f r e e z i n g a d r i l l h o l e through an i c e s h e l f i n A n t a r t i c a . Hanover, New-Hampshire, Cold Regions Research and E n g i n e e r i n g Laboratory, Report 76-12, APPENDIX Information about f i e l d equipment. Numbers r e f e r to F i g u r e 6 . Low pressure pump (3), C e n t r i f u g a l s e l f - p r i m i n g pump; Monarch I n d u s t r i e s L i m i t e d , Model BVG4 d r i v e n by a 2HP Bri g g s and S t r a t t o n g a s o l i n e engine,. With the engine running at 2700 rev/mn i t provides 15 1/mn under 2 atm . Weight: 30 kg , ($220), Flow s w i t c h (5),. Mc Donnell and M i l l e r ITT, Model PS4-3T , I I t f i t s p i p e s with a diameter of 19 mm and can withstand 10 atm o f s t a t i c p r e s s u r e . The 'fl o w 1 s e t t i n g i s a d j u s t a b l e from 7 to 16 1/mn and the 'no-flow' from 3.5 to 4,2 1/mn , ($6 0) Water-heater (7). Edvan Agencies, New-Westminster, B r i t i s h Columbia, Model C u l v e r t King* At sea l e v e l i t heats 12 1/mn of water from 0 °C to 45 °C while burning 89 5 kg of propane per hour. I t s dimensions are 1,20 x 0,40 x 0,20 m and i t weighs 17 kg . ($500). High pressure p i s t o n pump (9),. Monarch I n d u s t r i e s L i m i t e d , Model C5330 HB b e l t d r i v e n by a 5HP Briggs and S t r a t t o n g a s o l i n e engine. I t i s equipped with a surge tank, manometer and r e l i e f v a l v e and can provide 12 1/mn under 30 atm , I t weighs 32 kg . ($400).. Hose r e e l (10), BNG Equipment L t d , Burnaby, B r i t i s h Columbia, Custom made. . I t can s t o r e 300 m of 2,5 cm hose, has a geared hand crank and a feed-through s w i v e l j o i n t . Dimensions are 1.00 x 0,90 x 0.90 m . Weight i s 60 kg . ($300),. Hose (11). BTB I n d u s t r i e s Canada L t d , Model Conquest  A g r i c u l t u r a l . Working pressure i s 53 atm and diameters are 1.3 cm and 2.6 cm , I t weighs 47 kg per 100 m,. ($200 per 100 m) Hose c o u p l i n g s . Long shank ( 5 c m ) with swivel b o l t s . I n s i d e diameter i s 0.95 cm and l a r g e s t diameter i s 3 cm , Attached t o the hose with Band I t s t e e l clamps. A s m a l l t o o l a l l o w s one to r e p l a c e clamps e a s i l y and r a p i d l y . 90 Hose and c o u p l i n g s were t e s t e d f o r t e n s i l e : s t r e n g t h and they withstood 6 metric tons f o r 2 min with no apparent damage. 

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