Transcript
Page 1: Eos, Vol. 69, No. 19, May 10, 1988authors.library.caltech.edu/52944/1/eost7220.pdfEos, Vol. 69, No. 19, May 10, 1988 produced interesting results. In spite of the severe problems of

Eos, Vol. 69, No. 19, May 10, 1988

p r o d u c e d in teres t ing results. I n spite o f the seve re p r o b l e m s o f d r i l l ing in v o l c a n i c mater i ­al, the results to da te s h o w that p r o g r e s s is poss ib le a n d that g rea te r d e p t h s c o u l d b e r e a c h e d . T h e p r e sen t d r i l l ho l e c a n b e d e e p ­e n e d fur ther , b u t p r o b a b l y n o t m u c h b e l o w the p re sen t d e p t h — p e r h a p s 5 0 m m o r e , at mos t . Plans a re n o w b e i n g d e v e l o p e d f o r a l a rger d r i l l ing o p e r a t i o n in the h o p e o f r e a c h i n g 5 0 0 m o r m o r e .

References

Bonat t i , E. , O . J o e n s u u , a n d H . Wan le s s , G e o l o g i c a l o b s e r v a t i o n s in the s u b m a r i n e ca lde r a o f Santor in i ( A e g e a n Sea) , Rapp. P. V. Reun. Comm. Int. Explor. Sci. Mer Medit., 20, 5 6 9 , 1 9 7 2 .

B o s t r o m , K., a n d M . N . A . Pe te r son , O r i g i n o f a l u m i n i u m - p o o r f e r r o - m a n g a n o a n sedi­m e n t s in areas o f h i g h hea t flow o n the East Pacific Rise , Mar. Geol, 7, 4 2 7 , 1969 .

B o s t r o m , K., a n d L . W i d e n f a l k , T h e o r i g i n o f i ron - r i ch m u d s at the K a m e n i islands, San­tor ini , G r e e c e , Chem. Geol, 42, 2 0 3 , 1984 .

B r u n , A . , Recherches sur VExhalaison Volcani-que, A . H e r m a n & Fils, Paris, 1 9 1 1 .

B u t u z o v a , G . Y . , I r o n o r e s ed imen t s o f the f u m a r o l e field o f the Santor in i v o l c a n o , their c o m p o s i t i o n a n d o r i g i n , Dokl. Akad. Nauk. SSSR, 168, 2 1 5 , 1 9 6 6 .

B u t u z o v a , G . Y . . R e c e n t v o l c a n o - s e d i m e n t a r y i r o n - o r e p r o c e s s in San to r in V o l c a n o Cal­d e r a a n d its e f fec t o n the g e o c h e m i s t r y o f s ed imen t s (in Russ ian) , Tr. Geol. Inst. Akad. Nauk SSSR, 114, 1 9 6 9 .

E l d e r field, H . , Effects o f v o l c a n i s m o n water chemis t ry : D e c e p t i o n is land, Anta rc t i ca , Mar. Geol, 13, M l , 1 9 7 2 .

G o r s h k o v , G . S., C a t a l o g u e o f the active v o l ­c a n o e s a n d solfatara fields o f the Kur i l e is­lands , in Catalogue of the Active Volcanoes of the World, par t 7, In te rna t iona l V o l c a n o l o g ­ical Assoc i a t i on , N a p l e s , Italy, 1958 .

H a m m e r , C . U . , H . B . C lausen , a n d W . Dans -gaa rd , G r e e n l a n d ice shee t e v i d e n c e o f post-glacial v o l c a n i s m a n d its c l imat ic i m ­pact , Nature, 288, 2 3 0 , 1 9 8 0 .

H a r d e r , H . , K o h l e n s a u e r l i n g e als e ine Eisen-que l l e d e r s e d i m e n t a r e n Eisenerze , Sedi-mentol. Ore Genesis, 2, 107 , 1964 .

H e i k e n , G , a n d F. M c C o y , Jr., C a l d e r a deve l ­o p m e n t d u r i n g the M i n o a n e r u p t i o n , T h i r a , C y c l a d e s , G r e e c e , / . Geophys. Res., 89, 8 4 4 1 , 1984 .

Pichler , H . , a n d S. Kussmau l , C o m m e n t s o n the g e o l o g i c a l m a p o f the Santor in i islands, in Thera and the Aegean World, v o l . 2 , ed i t ed b y C . D o u m a s , p p . 4 1 3 - 4 2 6 , L o n d o n , 1980 .

Puchel t , H . , R e c e n t i r o n s e d i m e n t f o r m a t i o n at the K a m e n i islands, Santor in i ( G r e e c e ) , in Ores in Sediments, e d i t e d b y G . C . A m -stutz a n d A . J. Be rna l , p p . 2 2 7 - 2 4 5 , S p r i n g e r V e r l a g , N e w Y o r k , 1 9 7 3 .

Schro l l , E., C o n t r i b u t i o n to the m i n e r a l o g y o f the i r on r i ch m u d s ed imen t s o f Santor in i , in Thera and the Aegean World, v o l . 1, ed i t ed b y C . D o u m a s , p p . 3 3 3 - 3 4 2 , L o n d o n , 1978 .

Smi th , P. A . , a n d D . S. C r o n a n , G e o c h e m i c a l d i spe r s ion o f metals in a n d a r o u n d the cal­d e r a o f T h e r a , in Thera and the Aegean World, v o l . 1, e d i t e d b y C . D o u m a s , p p . 3 4 3 - 3 6 1 , L o n d o n , 1 9 7 8 .

Smi th , P. A . a n d D . S. C r o n a n , T h e g e o ­chemis t ry o f me ta l l i f e rous s ed imen t s a n d waters associa ted with sha l low s u b m a r i n e h y d r o t h e r m a l activity (Santor in i , A e g e a n Sea) , Chem. Geol, 39, 2 4 1 , 1 9 8 3 .

N . Arvan i t ides , V . G a l a n o p o u l o s , S. K a l o -g e r o p o u l o s , G . S k a m n e l o s , C . Papavassi l iou, and S. Paritsis are with the Institute of Geology and Mineral Exploration, Athens; K. B o s t r o m is with the Department of Geology, University of Stockholm, Stockholm.

News & Announcements

Early HVO Bulletins Collected, Published

PAGE 579

T h e v o l c a n o l o g y bul le t ins p u b l i s h e d s ince the ear ly 20 th c e n t u r y b y the Hawai ian V o l ­c a n o O b s e r v a t o r y ( H V O ) are n o w available as a t h r e e - v o l u m e set. I n this co l l ec t ion , the c o n ­tents o f The Volcano Letter a n d o t h e r ear ly H V O publ ica t ions are fully i n d e x e d f o r the first t ime .

T h e H V O c o l l e c t i o n , ent i t led The Early Seri­al Publications of the Hawaiian Volcano Observa­tory, was insp i red b y the Smi thson ian Institu­t ion 's 1987 rep r in t The Volcano Letter, 1925— 1955. T h e n e w w o r k i nc ludes an i n d e x to The Volcano Letter as wel l as to the o t h e r early H V O publ ica t ions , a l o n g with a b i b l i o g r a p h y . In the p r e f a c e to the H V O v o l u m e s , c o m p i l ­ers D a r c y B e v e n s , T a e k o J ane Takahash i , a n d T h o m a s L. W r i g h t exp la in that this c o l ­lec t ion c o m p l e t e s the r ep r in t ing o f the early H V O serial pub l i ca t ions . Each v o l u m e is sep­arately pag ina t ed a n d i n d e x e d , a n d the c o l ­lec t ion spans w o r k s f r o m 1913 to 1955 .

T h e co l l ec t i on b e c a m e available M a y 1988 . Inqui r ies f o r o r d e r s c a n b e d i r ec t ed to Kathy Engl ish , Bus iness M a n a g e r , Hawai ian Natura l His to ry Assoc i a t i on , L td . , P O B o x 74 , Hawai i Nat iona l Park, H I 9 6 7 1 8 . Q u e s t i o n s r ega rd ­ing the con ten t s o f the v o l u m e s can b e d i rec t ­e d to T h o m a s L. W r i g h t , Hawai i V o l c a n o O b ­servatory , P O B o x 5 1 , Hawai i Nat iona l Park, H I 9 6 7 1 8 .

This item was contributed by T . J ane T a k a h a ­shi, Hawaii Volcano Observatory, Hawaii Nation­al Park.

Information Report Flow in an Experimental Micro—Magma Chamber

PAGES 579, 588 T h e c h e m i c a l e v o l u t i o n a n d e rup t ive b e ­

hav io r o f m a g m a s m a y b e c o n t r o l l e d largely by c o n v e c t i v e p r o c e s s e s wi thin m a g m a c h a m ­bers . A c c o r d i n g to a r e c e n t Nat iona l R e s e a r c h C o u n c i l R e p o r t [Committee on Physics and Chemistry of Earth Materials, 1 9 8 7 ] , " the style o f c o n v e c t i o n itself, w h e t h e r it is turbulent , laminar , large-scale , o f mul t ip le scales, t ie red , o r loca l i zed a n d in termit tent , is very m u c h at ques t ion . " In the U . S . Na t iona l R e p o r t to the In terna t ional U n i o n o f G e o d e s y a n d G e o ­

physics , Marsh [ 1 9 8 7 ] r e v i e w e d r ecen t theo­retical a n d e x p e r i m e n t a l d e v e l o p m e n t s relat­e d to the style o f c o n v e c t i o n in m a g m a c h a m ­bers , n o t i n g b o t h significant quanti tat ive advances a n d also the m a n y r e m a i n i n g u n c e r ­tainties. W i t h r e g a r d to doub le -d i f fus ive c o n ­vec t ion , h e stated "as ever , the critical ques ­t ion c o n c e r n s w h e t h e r o r n o t actual m a g m a c h a m b e r s c o n v e c t in this style." Similarly, Spera et al. [ 1 9 8 6 ] , in d iscuss ion o f doub le -d i f ­fusive c o n v e c t i o n , c a u t i o n e d against "app ly­ing results f r o m saltwater tanks to m a g m a c h a m b e r s . "

E x p e r i m e n t a l silicate melts are c o m m o n l y c o n s i d e r e d i n a p p r o p r i a t e f o r e x a m i n i n g c o n ­vect ive p r o c e s s e s in m a g m a c h a m b e r s b e c a u s e o f the ve ry small l eng th scales necessi ta ted by l abora to ry s tudies (mil l imeters versus mete rs to k i lomete r s in actual m a g m a c h a m b e r s ) . W e are n o t aware o f any e x p e r i m e n t a l studies o n c o n v e c t i o n in grani t ic melts with d i sso lved H 2 0 u n d e r p ressu re . Results sugges t ing the o p e r a t i o n o f c o m p o s i t i o n a l c o n v e c t i o n a c c o m ­p a n y i n g d i s so lu t ion o f a silical glass r o d in a picri t ic me l t at 1 a tm led Donaldson and Hamil­ton [ 1 9 8 7 ] to the c o n c l u s i o n "that the re is a ro l e f o r e x p e r i m e n t a l p e t r o l o g y in the investi­ga t ion o f m a g m a t i c fluid p roces ses . " H e r e w e p resen t p r e l im ina ry results sugges t ing that c o n v e c t i v e p roces se s in h y d r o u s silicate melts at e leva ted pressures may also b e a m e n a b l e to e x p e r i m e n t a l s tudy.

T h e c o v e r p h o t o g r a p h shows the result o f an e x p e r i m e n t c o n d u c t e d at 10 kbar a n d 9 5 0 ° C f o r 3 0 h o u r s . F low lines in the h y d r o u s grani te me l t (left) are revea led by trails o f g raph i te part ic les (b lack) . T h e g raph i te was p r o d u c e d b y r e d u c t i o n o f the small a m o u n t o f C 0 2 p r e s e n t in the e x p e r i m e n t ' s starting materials . T h e p h o t o g r a p h clearly s h o w s o r ­gan i zed flow: T h e m e l t e d grani te flows ho r i ­zontal ly f r o m left to r ight t h r o u g h the c e n t e r o f the c h a r g e a n d then turns to flow b o t h u p a n d d o w n t o w a r d the marg ins o f the capsu le . A d d i t i o n a l e x p e r i m e n t s have c o n f i r m e d that the flow fea tures are r e p r o d u c i b l e .

T h e r u n il lustrated is o n e o f a series o f e x ­p e r i m e n t s d e s i g n e d to e x p l o r e the effects o f c o n t a m i n a t i o n o f h y d r o u s silicic m a g m a s by mafic a n d ul t ramafic rocks near the crust-mant le b o u n d a r y . Equa l masses o f p o w d e r e d grani te a n d se rpen t i zed a m p h i b o l e pe r ido t i t e (da rk mater ia l o n far r ight o f the c o v e r p h o ­t o g r a p h ) w e r e p a c k e d in to a small g o l d c a p ­sule so that the t w o materials w e r e in con tac t at a sha rp b o u n d a r y . T h e e x p e r i m e n t was r u n with the capsu l e m o u n t e d in a p i s ton cyl ­i n d e r appara tus in the same or ien ta t ion as is s h o w n in the p h o t o g r a p h . D u r i n g the e x p e r i ­men t , b r e a k d o w n o f se rpen t ine in the p e r i d o ­tite starting mater ia l ( ~ 6 wt % H z O ) releases water a n d causes p rog re s s ive me l t i ng o f the grani te . S h o r t e r d u r a t i o n e x p e r i m e n t s s h o w that me l t i ng b e g i n s nex t to the grani te-per i -dot i te b o u n d a r y ; increas ing run d u r a t i o n re­sults in f o r m a t i o n o f a p rogress ive ly w i d e r b a n d o f grani t ic glass, g r a d i n g in to a z o n e o f crystals a n d l iqu id at the distal e n d o f the capsu le (left in the p h o t o g r a p h ) . Graph i t e trails a n d e n t r a i n e d r o u n d e d crystals de f in ing flow pat terns similar to those in the p h o t o ­g r a p h can b e seen e m a n a t i n g f r o m this crys­tal/liquid r e g i o n in s o m e e x p e r i m e n t s .

W i d e b e a m e l e c t r o n m i c r o b e analyses o f the grani t ic me l t p o r t i o n o f the c h a r g e s h o w that it is h o m o g e n e o u s a n d a lmos t ident ical in c o m p o s i t i o n to the starting grani te . T h e m o s t no tab le c h a n g e is a slight e n r i c h m e n t in M g O

Page 2: Eos, Vol. 69, No. 19, May 10, 1988authors.library.caltech.edu/52944/1/eost7220.pdfEos, Vol. 69, No. 19, May 10, 1988 produced interesting results. In spite of the severe problems of

Eos, Vol. 69, No. 19, May 10, 1988

Fig. 1. O b l i q u e aerial v i ew o f the c ra te r lake o f Poas V o l c a n o , Cos ta Rica . T h e v i ew is f r o m the sou th ; the lake is a b o u t 3 0 0 m in d i ame te r . P h o t o g r a p h cou r t e sy o f the U . S . Na ­t ional A e r o n a u t i c s a n d S p a c e A d m i n i s t r a t i o n / W a l l o p s Fl ight C e n t e r , W a l l o p s Is land, V a .

f r o m 0.1 to 0 .5 wt % . T h e d i f f e r ences b e ­tween analytical totals a n d 1 0 0 % indica te that the granit ic mel t con ta ins 7 ± 1 wt % dis­so lved H 2 0 . Calcu la t ions o f the dens i ty o f the granit ic glass [Kushiro, 1 9 8 4 ] , ba sed o n m i c r o -p r o b e analyses s p r e a d t h r o u g h o u t the c h a r g e , s h o w n o significant spatial variat ions, a n d the total r a n g e is wi thin 0 .05 g / c m 3 . T h e viscosity o f the mel t with 6 - 8 wt % H 2 0 at 9 5 0 ° C is ca lcula ted to b e 3 0 0 0 - 4 0 0 0 po i se [Shaw, 1 9 7 2 ] . T e m p e r a t u r e g rad ien ts in the c h a r g e are small a n d less than ~ 5 ° C .

T h e results s h o w n in the c o v e r p h o t o g r a p h c o n f i r m that flow in h y d r o u s grani t ic mel t is a m e n a b l e to e x p e r i m e n t a l invest igat ion. T h e r e are m a n y m e c h a n i s m s that m a y have b e e n i n v o l v e d in d r i v i n g the flow, i n c l u d i n g sur face tens ion effects , small variat ions in the melt 's water c o n t e n t , a small ho r i zon ta l t em­pe ra tu re g rad ien t , a n d v o l u m e c h a n g e s caused by crystall ization o r d i sso lu t ion . T h e o ­retical c o n s i d e r a t i o n s sugges t that a small hor i zon ta l t e m p e r a t u r e g rad ien t c o u l d p r o ­d u c e the o b s e r v e d flow. Small var iat ions in water c o n t e n t c o u l d also b e i m p o r t a n t in p r o ­m o t i n g flow o f grani t ic mel t by caus ing sig­nificant var iat ions in me l t densi ty . W e are present ly e x p l o r i n g these a n d o t h e r poss ib le d r iv ing f o r c e s .

T h i s r esea rch was s u p p o r t e d b y Nat iona l Sc ience F o u n d a t i o n gran t N S F E A R 8 5 -0 6 8 5 7 .

References

C o m m i t t e e o n Physics a n d C h e m i s t r y o f Earth Materials , Earth Materials Research, Nat ional A c a d e m y Press, W a s h i n g t o n , D . C , 1987 .

D o n a l d s o n , C . H . , a n d D . L. H a m i l t o n , C o m ­pos i t ional c o n v e c t i o n a n d l ayer ing in a r o c k mel t , Nature, 327, 4 1 3 , 1987 .

K u s h i r o , I., S t ructures a n d s o m e physical p rope r t i e s o f silicate melts o f g e o l o g i c a l in­terest, in Materials Science of the Earth's Inte­rior, ed i t ed by I. Sunagawa , p . 3 9 , T e r r a Scientific, T o k y o , 1984 .

Marsh , B . , M a g m a t i c p rocesses , Rev. Geophys., 25, 1043 , 1987 .

Shaw, H . R. , Viscosi t ies o f m a g m a t i c silicate l iquids: A n empi r i ca l m e t h o d o f p red ic t ion , Am. J. Sci., 272, 8 7 0 , 1 9 7 2 .

Spera , F. J., D . A . Y u e n , S. Clark , a n d H.-J. H o n g , Doub le -d i f fu s ive c o n v e c t i o n in m a g ­m a c h a m b e r s : S ing le o r mul t ip le layers?, Geophys. Res. Lett., 13, 153 , 1986 .

This report was contributed by M i c h a e l R. Car­rol l and Pe ter J. Wyl l i e , both of the California Institute of Technology, Pasdena, Calif.

Temperature Profiles From Poas Crater Lake PAGE 588

In 1984 , w e t o o k par t in an e x p e d i t i o n to m e a s u r e the t e m p e r a t u r e field a n d b a t h y m e ­try o f the ac id lake (F igu re 1) that has f o r m e d in the c ra te r o f Poas v o l c a n o , Costa Rica , s ince its last e r u p t i o n in 1 9 5 3 . Obta in ­ing these data was the first s tep in a l o n g -r a n g e s tudy p l a n n e d b y researchers at the C e n t e r f o r G e o p h y s i c a l Resea rch , Univers i ty o f Cos ta Rica (San J o s e , Cos ta R ica ) , a n d the C o l l e g e o f O c e a n o g r a p h y , O r e g o n State Uni ­

versity (Corval l i s ) . T h e s tudy will eventual ly c o n s i d e r all aspects o f fluid b e h a v i o r in a v o l ­can ic lake that is hea t ed o r o the rwise c o n v e c -tively d r i v e n b y e n e r g y in jec ted at the lake b o t t o m .

E v i d e n c e o f c o n v e c t i o n is clearly visible o n the sur face o f the Poas lake m o s t o f the t ime. F u m a r o l e activity has b e e n c o n t i n u o u s s ince 1953 . Phreat ic e x p l o s i o n s are qui te f r equen t , va ry ing f r o m w e a k to s t rong , a n d the he igh t o f the e j ec ted c o l u m n varies f r o m 1 to m o r e than 5 0 0 m . O n e i m m e d i a t e l y useful result o f the resea rch w o u l d b e an est imate o f the hea t t ransfer f r o m sou rce s wi th in the c o n d u i t to the o v e r l y i n g wate r c o l u m n . A s far as g e o ­physical fluid b e h a v i o r g o e s , w e are interest­e d in the tu rbu len t a n d diffusive p roces se s b y w h i c h heat a n d c h e m i c a l species are trans­f e r r ed . W e are especia l ly in teres ted in the i m ­pac t o n the dens i ty stratification o f the dens i ­ty c h a n g e s that o c c u r as part iculates settle d o w n w a r d t h r o u g h the fluid c o l u m n . T h e stratification w o u l d o the rwi se b e c o n t r o l l e d b y the tu rbu len t a n d dif fusive p roces se s dr iv­e n b y t h e r m o c h e m i c a l fac tors .

T h e lake is ~ 3 0 0 m in d i ame te r , wi th shee r vert ical walls in m o s t p laces . P e r i o d i c samples o f its sur face waters d u r i n g 1 9 8 0 - 1 9 8 1 s h o w e d t e m p e r a t u r e s r a n g i n g f r o m 2 2 to 4 9 ° C , a s t r o n g acidi ty wi th pH < 1, a n d aver­a g e sulfate c o n c e n t r a t i o n s o f ~ 5 3 , 0 0 0 p p m [Casertano et al., 1 9 8 5 ] . T h e water is ma in ly a h o t sulfur ic ac id so lu t ion , d i lu ted f r o m t ime to t ime by ra inwater a n d e n r i c h e d b y p e r i o d i c phrea t i c e x p l o s i o n s . A d e n s e v a p o r r ich in sulfur ic ac id usually c o v e r s the water 's sur­face . T h i s gas c l o u d f o r c e d us to w e a r gas masks cons tant ly w h e n w e w e r e w o r k i n g nea r the water 's e d g e .

T h e r e are n o p r i o r m e a s u r e m e n t s o f the lake's in te r io r t he rma l s t ruc ture o r its ba­thymet ry . W e u s e d a r a d i o - c o n t r o l l e d ca tama­

ran hul l with a i r sc rew p r o p u l s i o n t o ca r ry a n d l a u n c h d i s p o s a b l e t he rma l p r o b e s ( m o d i ­fied T - 7 S ipp i can C o r p o r a t i o n p r o b e s that are u s e d in o c e a n s a m p l i n g ) . T h e p r o b e pays o u t a pa i r o f wi res f r o m an internal b o b b i n , so the resis tance o f its t he rmis to r s e n s o r was r e c o r d e d a s h o r e o n a p a p e r char t a n d later c o n v e r t e d in to equ iva len t t empe ra tu r e s . P r io r l abo ra to ry c h e c k s in hea t ed s ample s o f Poas wate r s h o w e d n o m e a s u r a b l e c h a n g e in sen­sor o r w i r e p e r f o r m a n c e o v e r 10 minu tes , w h i c h is a m u c h l o n g e r t ime than that taken u p b y de l ive ry , l a u n c h , a n d fall o f p r o b e s in the field.

W e first s a m p l e d the ho r i zon t a l t e m p e r a ­ture field o v e r the n o r t h e r n ha l f o f the lake sur face , in a t r iangular t rack that was tra­v e r s e d in 8 minu te s . F r o m l a u n c h to r e tu rn the p r o b e m e a s u r e d a c h a n g e o f o n l y 1.4°C, m o s t o f w h i c h r e p r e s e n t e d a g r a d u a l w a r m ­i n g o f su r face wa te r t o w a r d the lake cen te r . T h i s was e x p e c t e d b e c a u s e act ive f u m a r o l e s o c c u r in the sou th o f the crater , a n d the lake sur face nea r the sou th s h o r e s h o w e d pa tches o f t u r b u l e n c e that w e r e e v i d e n c e o f vert ical c o n v e c t i o n . S u r f a c e t e m p e r a t u r e taken with a glass t h e r m o m e t e r at the l a u n c h site o n the n o r t h e r n s h o r e was 4 8 ° C .

F o u r vert ical d r o p s w e r e m a d e b e f o r e w e lost c o n t r o l o f the b o a t in the v a p o r c l o u d a n d a l ight ac id ra in s h o w e r . Pos i t ion c o n t r o l was p o o r b e c a u s e o f the visibility fac to r ; w e "nav iga t ed" b y the s o u n d o f the a i r sc rew m o ­tor . T h r e e o f the vert ical prof i les p r o d u c e d useful data, wi th results d e s c r i b e d b e l o w .

Bathymetry and Bottom Temperatures

W a t e r d e p t h in the cent ra l p o r t i o n o f the lake was a b o u t 5 0 m . T h i s result is b a s e d o n fall t imes o f 2 3 , 2 5 , a n d 2 6 . 5 s to a c lear ly d is t inguishable s tep f u n c t i o n in e a c h r e c o r d ­e d t race, w h i c h w e in te rpre t as the p o i n t o f


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