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Velocity microstructure measurements in the western and central equatorial Pacific Moum, James Norman 1984

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VELOCITY MICROSTRUCTURE MEASUREMENTS IN THE WESTERN AND CENTRAL EQUATORIAL PACIFIC by JAMES NORMAN MOUM B . A . S c . , U n i v e r s i t y Of T o r o n t o , 1978 M . A . S c . , U n i v e r s i t y Of T o r o n t o , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Phys i c s And Oceanography Departments We accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard THE UNIVERSITY OF BRITISH COLUMBIA May 1984 © James Norman Mourn, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ^fuL) ? /ftf DE-6 (3/81) i i A b s t r a c t Measurements of v e l o c i t y m i c r o s t r u c t u r e were made i n two q u i t e d i f f e r e n t ocean i c regimes us ing the f r e e - f a l l i n g p r o f i l e r , Camel I I I . In c o n j u n c t i o n wi th the P a c i f i c E q u a t o r i a l Ocean Dynamics (PEQUOD) e x p e d i t i o n , p r o f i l e s were made at or near the equator between 138°W and 153°W. E s t ima tes of the r a t e of d i s s i p a t i o n of t u r b u l e n t k i n e t i c ene rgy , e, made from the v e l o c i t y m i c r o s t r u c t u r e measurements are s u r p r i s i n g l y sma l l i n magni tude . Averaged va lues at 70, 90 and 110 meters ( i n the r eg i on of l a rge mean shear j u s t above the co re of the e q u a t o r i a l unde rcu r ren t ) are more than ten t imes sma l l e r than those p r e v i o u s l y r e p o r t e d . The d i s s i p a t i o n i n t e g r a t e d from the l e v e l of no zona l v e l o c i t y (=* 70 meters) to the undercu r ren t co re i s l e s s than 10% of an es t imate made of the work done by the zona l p r e s su re g r a d i e n t . I t i s p o s s i b l e tha t the proposed ba lance between the work done by the zona l p r e s s u r e g r ad i en t and the t u r b u l e n t f r i c t i o n does not ho ld at a l l p l a c e s at a l l t imes f o r the e q u a t o r i a l u n d e r c u r r e n t . A second set of measurements was made a long 152°E between 27°N and 4 2 ° N , south of the Ku rosh io Ex tens ion c u r r e n t . A s t rong main the rmoc l i ne between 500 and 800 meters depth and south of 34°N man i f e s t ed i t s e l f as a secondary subsu r f a ce maximum in buoyancy f r equency , N, which concu r red w i th a subsu r f a ce maximum in averaged d i s s i p a t i o n . A p l o t of e vs N i n d i c a t e s that e s c a l e s wi th N r a the r than dep th . A s imple model was deve loped to e x p l a i n the r e l a t i v e l y g rea te r occu r rence of t u r b u l e n t pa tches in the main t he rmoc l i ne which assumes tha t the t u rbu l ence i s genera ted by i n t e r n a l waves. The p r e d i c t i o n of the p r o b a b i l i t y of occu r rence of sma l l R i cha rdson number i s - 1 / N p r o p o r t i o n a l to e which p r e d i c t s the shape of the d i s t r i b u t i o n of the t u rbu l ence r e l a t i v e l y s u c c e s s f u l l y . i v Tab le of Contents A b s t r a c t i i L i s t of Tab l e s v i L i s t of F i g u r e s v i i i Acknowledgement . . . . x i i Chapter I " INTRODUCTION 1 Chapter II BACKGROUND THEORY 7 2.1 Mean K i n e t i c Energy Equa t ion 7 2.2 Tu rbu l en t K i n e t i c Energy Equa t ion 9 2.3 The Es t imate Of e 11 2.4 S t r a t i f i e d Flow Parameters 13 Chapter III EXPERIMENTAL CONSIDERATIONS 15 3.1 A B r i e f D e s c r i p t i o n Of Camel III 17 3.2 S i g n a l P ro ces s i ng 18 3.3 The Study Areas 21 3.3.1 PEQUOD 21 3.3.2 WESPAC 25 Chapter IV RESULTS FROM WESPAC 28 4.1 The Drops South Of The Ring 35 4.2 e And Eddy K i n e t i c Energy 39 4.3 e And N 42 Chapter V A MODEL OF TURBULENCE IN AN INTERNAL WAVE FIELD 51 5.1 I n t e r n a l Wave Energy P r o f i l e s 52 5.2 I n t e r n a l Wave Shears 55 5.3 The D i s t r i b u t i o n Of Shear And R i cha rdson Number . . . 5 8 5.4 Comparison With The Data 61 5.5 D i s c u s s i o n 68 Chapter VI RESULTS FROM PEQUOD . .72 6.1 C u r r e n t s And Hydrography 72 6.2 P r e v i ous E q u a t o r i a l M i c r o s t r u c t u r e Measurements . . . 7 5 6.3 PEQUOD M i c r o s t r u c t u r e 76 6.4 The On-equator P r o f i l e s . . . 8 2 6.5 e And The Zonal P r essu re G rad i en t 90 6.6 e And N, S, R i • 95 6.7 S t a t i s t i c s Of Ri And e 101 Chapter VII ESTIMATES OF EDDY COEFFICIENTS 107 7.1 V a r i o u s E s t ima to r s 108 7.2 Comparison Of E s t ima tes 111 V 7.3 Deep Ocean E s t ima tes 117 7.4 Comparison With E q u a t o r i a l Model Va lues 118 Chapter VI I I COMPARISON OF DATA SETS AND PATCH SIZE STATISTICS 123 8.1 Lognormal P r o p e r t i e s Of e 126 8.2 Patch S i z e S t a t i s t i c s 133 Chapter IX DISCUSSION AND CONCLUSIONS 148 BIBLIOGRAPHY 152 APPENDIX A - HYDRODYNAMICS 157 APPENDIX B - PRESSURE 160 APPENDIX C - FALL RATE 163 APPENDIX D - TEMPERATURE 166 APPENDIX E - VELOCITY SHEAR 169 APPENDIX F - CALCULATION OF SPECTRA 184 APPENDIX G - RESOLUTION OF THE DISSIPATION MEASUREMENT . .196 APPENDIX H - ERRORS IN THE DISSIPATION CALCULATION 203 APPENDIX I - UNITS OF e 214 APPENDIX J - TREATMENT OF WHITE HORSE VELOCITY AND CTD DATA 215 APPENDIX K - PEQUOD DROPS 219 APPENDIX L - WESPAC DROPS 248 v i L i s t of T a b l e s Tab le 1 - S c a l i n g f a c t o r s fo r n o r m a l i z a t i o n and t i t l e s g iven to the data se t s 65 Tab le 2 - C o r r e l a t i o n c o e f f i c i e n t s fo r the da ta s e t s d e f i n e d in Tab le 1 . 66 Tab le 3 - Depth , t h i c k n e s s and patch-averaged d i s s i p a t i o n s fo r the patches l o c a t e d near 500 meters depth fo r the drops w i t h i n 1/2° of the equator at 145°W 79 Tab le 4 -Tab le 5 -Tab le 6 -Tab le 7 -Tab le 8 -Tab le 9 -Tab le 10 -Tab le 11 -Comparison of e averaged over 20-140 meters from 1982 and 1979 drops 89 Comparison of v e r t i c a l eddy c o e f f i c i e n t s f o r f i v e d i f f e r e n t e q u a t o r i a l da ta se t s 112 Average va lues of e from PEQUOD and WESPAC data se t s compared to Vancouver I s l a n d s lope va lues from Lueck, Crawford and Osborn ( l983) 124 Patch s i z e s t a t i s t i c s f o r the PEQUOD data set over the depth range 20-300 meters 135 Patch s i z e s t a t i s t i c s f o r the PEQUOD data set over the depth range 300-1000 meters 136 Patch s i z e s t a t i s t i c s f o r the WESPAC data set over the depth range 20-300 meters . . . . . . 1 3 7 Patch s i z e s t a t i s t i c s fo r the WESPAC data set over the depth range 300-1000 meters 138 Patch s i z e s t a t i s t i c s f o r the WESPAC da ta set f o r depths > 1 000 meters 139 Tab le B.1 - Camel III p r e s su re c a l i b r a t i o n da ta 162 Tab le D.1 - Camel III temperature c a l i b r a t i o n data and r e s u l t of cub i c po l ynomia l f i t 168 Tab le F.1 - The s p e c t r a l r o u t i n e used to es t imate the v e r t i c a l shear spectrum and sample output 186 Tab le F.2 - Comparison of cumu la t i v e v a r i a n c e s c a l c u l a t e d us i ng the s p e c t r a l r o u t i n e w i th the expected va lues ( un i t s are b i t s 2 ) 193 v i i Tab le F.3 -Tab le G.1 -Tab le G.2 -Tab le K.1 -Tab le L.1 -Comparison of cumu la t i v e , v a r i a n c e s c a l c u l a t e d us ing the s p e c t r a l r o u t i n e w i th the expected va lues ( u n i t s are s e c " 2 ) ' 195 I n t e g r a t i o n of the no i se s p e c t r a l d e n s i t y f u n c t i o n and c a l c u l a t i o n of e q u i v a l e n t d i s s i p a t i o n due to the inheren t no i se of the shear probe 199 Comparison of no i se l e v e l s due to FM and tape r e c o r d i n g systems on a l l of the shear channe ls 202 PEQUOD drop l og 220 WESPAC drop l og 249 v i i i F i g u r e 1 -F i g u r e 2 -F i g u r e 3 -F i g u r e 4 -F i g u r e 5 -F i g u r e 6 -F i g u r e 7 -F i g u r e 8 -F i g u r e 9 -F i g u r e 10 -F i g u r e 11 -F i g u r e 12 -F i g u r e 13 -F i g u r e 14 -F i g u r e 15 ^ F i g u r e 16 -F i g u r e 17 -L i s t of F i g u r e s Schematic of Camel III 16 Schematic of s i g n a l p r o c e s s i n g 19 Map of P a c i f i c Ocean showing the WESPAC and PEQUOD study r eg ions . . . . 2 2 PEQUOD c r u i s e t r a ck ( February , 1982) 23 WESPAC c r u i s e t r a ck (May/June, 1982) 26 T o t a l l eng ths of data r e co rd f o r the t o t a l PEQUOD and t o t a l WESPAC data s e t s 27 Temperature s e c t i o n a long 152°E in May/June 1982 30 S a l i n i t y s e c t i o n a long 152°E in May/June 1982 32 Turbu len t k i n e t i c energy d i s s i p a t i o n averaged v e r t i c a l l y over 100 meter i n t e r v a l s f o r 10 drops made a long 152°E i n May/June, 1982 33 V e r t i c a l p r o f i l e of averages of 25 meter es t imates of buoyancy f requency over WESPAC drops 2 , 4 , 5 , 6 , 8 . . . 3 6 V e r t i c a l p r o f i l e of averages of 25 meter e s t ima tes of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n over WESPAC drops 2 , 4 , 5 , 6 , 8 38 a) Eddy k i n e t i c energy . b) t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over depth 41 S c a t t e r p l o t s of buoyancy f requency and e 43 a) 100 meter v e r t i c a l averages of buoyancy f requency averaged over a l l of the WESPAC d r o p s . b) l00 meter v e r t i c a l averages of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over a l l of the WESPAC drops 45 Log- log p l o t s of e vs N from F i g u r e s 14a,b . . . . 4 6 Log- log p l o t s of 7 vs N 49 P l o t s of i n t e r n a l wave p o t e n t i a l energy (PE ) , k i n e t i c energy(KE) and t o t a l energy(TE) 54 ix F i g u r e 18 -F i g u r e 19 -F i g u r e 20 -F i g u r e 21 -F i g u r e 22 -F i g u r e 23 -F i g u r e 24 -F i g u r e 25 -F i g u r e 26 -F i g u r e 27 -F i g u r e 28 -F i g u r e 29 -F i g u r e 30 -F i g u r e 31 -P l o t of the dependence of the shor t wavelength c u t o f f , X + , on the upper l i m i t , j + , of the v e r t i c a l mode number 57 V e r t i c a l p r o f i l e s of P r (R i< l ) and P r (R i< l/4 ) 62 V e r t i c a l p r o f i l e s of f r a c t i o n of t u r b u l e n t water column (PCT) 63 V e r t i c a l p r o f i l e s of no rma l i zed PCT from F i g u r e 20 and no rma l i z ed Pr(Ri<1/4) from F i g u r e 19 67 V e r t i c a l p r o f i l e s of 50 meter v e r t i c a l l y averaged v a l ues of l og e from PEQUOD 80 Averaged d i s s i p a t i o n s observed d u r i n g the P a r i z e a u c r u i s e in 1979, the A t l a n t i s II c r u i s e in 1974 and the Thomas G.Thompson c r u i s e in 1982 81 E igh t v e r t i c a l p r o f i l e s of v e r t i c a l shear as e s t ima ted from White Horse h o r i z o n t a l v e l o c i t i e s taken w i t h i n 1/2° of the equator i n F eb rua r y , 1 982 . .84 E i g h t v e r t i c a l p r o f i l e s of B r u n t - V a i s a l a f requency measured s i m u l t a n e o u s l y as the shears of F i g u r e 24 85 V e r t i c a l p r o f i l e s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over 25 meters depth and which are n e a r l y s y n o p t i c w i th the da ta of F i g u r e s 24 £ 25 86 Time v a r i a t i o n s of e averaged over 20 to 140 metersand cube of d a i l y wind speed 91 S c a t t e r p l o t s of l og e vs l og N, l og S and l og Ri from PEQUOD 96 P l o t of l og e vs l og S from PEQUOD .99 P l o t of l og e vs l og Ri from PEQUOD 100 S c a t t e r p l o t s of l og N vs l og S from PEQUOD 102 F i g u r e 32 - R e l a t i v e f requency of occur rence of l og Ri e s t ima ted from the White Horse da ta taken in F eb rua r y , 1982 104 F i g u r e 33 -F i g u r e 34 -F i g u r e 35 -F i g u r e 36 -F i gu re 37 -F i gu re 38 -F i g u r e 39 -F i g u r e 40 -F i g u r e 41 -F i g u r e 42 -F i gu re 43 -F i g u r e 44 -F i gu re 45 -Norma l ized f requency of occur rence of l og e per h a l f decade i n t e r v a l 105 V e r t i c a l p r o f i l e s of K (equat ion 7.4) and K 0 G (equat ion 7.5) f o r the PEQUOD data 115 V e r t i c a l p r o f i l e s of K and K fo r the WESPAC 0 G data 116 V e r t i c a l p r o f i l e s of 25 metre averages of K and 0 K (equat ion 7.11) from PEQUOD 120 rPP V e r t i c a l p r o f i l e s of 25 metre averages of K and V K (equat ion 7.10) from PEQUOD 121 vPP Cumulat ive d i s t r i b u t i o n of the base 10 l oga r i t hm of d i s s i p a t i o n va lues from PEQUOD 20-300m 128 Cumulat ive d i s t r i b u t i o n of the base 10 l oga r i t hm of d i s s i p a t i o n va lues from PEQUOD 300-1000m 129 Cumulat ive d i s t r i b u t i o n of the base 10 l o g a r i t h m of d i s s i p a t i o n va lues from WESPAC 20-300m 130 Cumulat ive d i s t r i b u t i o n of the base 10 l oga r i t hm of d i s s i p a t i o n va lues from WESPAC 300-1000m 131 Cumulat ive d i s t r i b u t i o n of the base 10 l oga r i t hm of d i s s i p a t i o n va lues from WESPAC >l000m 1 32 Log- log p l o t of average patch-averaged d i s s i p a t i o n s vs average pa tch t h i c k n e s s fo r the PEQUOD data below 300 meters 141 Log- log p l o t of average patch-averaged d i s s i p a t i o n s vs average pa tch t h i c k n e s s fo r the WESPAC data below 300 meters 142 Log- log p l o t of average buoyancy l eng th s c a l e vs average pa t ch t h i c k n e s s fo r the PEQUOD data below 300 meters . 1 45 x i F i g u r e 46 - Log- log p l o t of average buoyancy l eng th s c a l e vs average pa tch t h i c k n e s s fo r the WESPAC data below 300 meters .146 F i g u r e B.1 - Camel III p r e a m p l i f i e r and lowpass f i l t e r -a m p l i f i e r 161 F i g u r e C.1 - Camel III f a l l r a t e c i r c u i t (p ressure d i f f e r e n t i a t o r ) 164 F i g u r e C.2 - Camel III P r e s su re d e r i v a t i v e t r a n s f e r f u n c t i o n 165 F i g u r e D.1 - The rmis to r p r e a m p l i f i e r and temperature c i r c u i t 167 F i gu re E. 1 - The a i r f o i l probe showing flow components . . . . 170 F i g u r e E. 2 - T y p i c a l a i r f o i l probe c a l i b r a t i o n curve . . . , , 1 72 F i g u r e E. 3 - Camel III v e l o c i t y shear s i g n a l p r o c e s s i n g . . .174 F i g u r e E. 4 - Ve loc i t y . . . 175 F i g u r e E. 5 - V e l o c i t y shear p r e a m p l i f i e r t r a n s f e r f u n c t i o n 177 F i g u r e E. 6 - V e l o c i t y , , 178 F i gu re E. 7 - Complete shear c i r c u i t t r a n s f e r f u n c t i o n . . , , 179 F i gu re E. 8 - Ve loc i t y . . .181 F i gu re E. 9 - V e l o c i t y f u n r t i on shear h igh ga in a m p l i f i e r t r a n s f e r 182 F i g u r e G. 1 - Noise s p e c t r a l d e n s i t y measured w i th p o t t e d shear 1 98 F i g u r e H. 1 - Four shear s p e c t r a from Qua t s ino Sound on F i g u r e H. 2 - Percentage of the v a r i a n c e r e s o l v e d by the shear . . . 2 0 7 F i g u r e H. 3 - , . .210 F i g u r e H.4 - Spec t ra the t ime from the four ad jacent b l o c k s (57- 60) of . . .211 x i i Acknowledgement I would l i k e to express my g r a t i t u d e to P r o f e s s o r T .R .Osbo rn fo r h i s gu idance and fo r h i s support of my p r o j e c t . As w e l l , I would l i k e to thank P r o f e s s o r P .H .LeB lond fo r h i s f r e e l y - g i v e n time and though t s . I have b e n e f i t t e d from time to t ime by be ing ab le to t a l k w i th P r o f e s s o r s S .Pond, L.A.Mysak and R .W .Bu r l i ng . I am a l s o indebted to D r .R .G . Lueck fo r many v a l u a b l e d i s c u s s i o n s and much-apprec ia ted t e c h n i c a l adv i ce and to Dr .W.R .Crawford fo r h i s input to the a n a l y s i s of the e q u a t o r i a l d a t a . I would l i k e to thank Dr . J .G .R i chman f o r a l l o w i n g me to p a r t i c i p a t e i n the PEQUOD e x p e d i t i o n . D r s . P . P . N i i l e r and W. J .Schmi tz pe rm i t t ed me to take measurements d u r i n g the WESPAC c r u i s e . D r . J . R . L u y t e n s u p p l i e d the White Horse data and D r . P . N i i l e r the CTD data from WESPAC. The c a p t a i n and crew of the R/V Thomas G.Thompson p r o v i d e d exper t h e l p i n d e p l o y i n g and s u c c e s s f u l l y r e c o v e r i n g Camel I I I . V a r i ous members of the Woods Hole Buoy Group l e n t c o n s i d e r a b l e t ime and e x p e r t i s e to the seago ing o p e r a t i o n du r i ng both c r u i s e s . R.Noel a s s i s t e d wi th the o p e r a t i o n on the PEQUOD c r u i s e and R .M.Ninn is p l a yed the r o l e of t e c h n i c a l a s s i s t a n t and d e v i l ' s advocate on the WESPAC t r i p . S . M i l a i r e made the shear probes and, w i th B .Anderson, p r o v i d e d s u b s t a n t i a l t e c h n i c a l a s s i s t a n c e du r i ng the development s tage of Camel I I I . H.Heckl d i d most of the mach in ing . F i n a l l y , I would l i k e to thank the many graduate s tudents and p o s t d o c t o r a l f e l l o w s w i th whom I have been f o r t u n a t e to a s s o c i a t e w i th and l e a r n from d u r i n g my tenure as a graduate s tudent at UBC. The N a t u r a l S c i ences and E n g i n e e r i n g Research C o u n c i l of Canada suppor ted me wi th a pos tg radua te s c h o l a r s h i p du r i ng my s t u d i e s . 1 I. INTRODUCTION The study of t h r ee-d imens iona l t u rbu l ence in the ocean i s e s s e n t i a l l y a study of the sma l l s c a l e mechanisms which are r e s p o n s i b l e fo r the m i x i n g , d i s p e r s i n g , d i f f u s i n g of hea t , s a l t , momentum and other q u a n t i t i e s . To unders tand the l a r g e r s c a l e mot ions in the sea i t i s necessa ry to unders tand what i s happening at the sma l l s c a l e s . Due to the h i g h l y n o n l i n e a r na ture of these smal l s c a l e p r o c e s s e s , the ex tent of the a n a l y t i c a l t reatment has been l i m i t e d . I n s t e ad , the study of t u rbu l ence i n v o l v e s expe r imen ta t i on and e m p i r i c a l p a r a m e t e r i z a t i o n of the f low parameters in order to d e s c r i b e how they a f f e c t the l a rge s c a l e dynamics . The f i r s t s t u d i e s of ocean ic t u rbu l ence by G ran t , Stewart and M o i l l i e t ( 1 9 6 2 ) at the Canadian Defence Research E s t a b l i s h m e n t , P a c i f i c (DREP) were mot i va ted by a fundamental i n t e r e s t in the nature of the t u rbu l ence i t s e l f . From exper iments conducted under c o n d i t i o n s of s u f f i c i e n t l y h igh Reynolds number (10 8 in Seymour Nar rows ) , they were ab le to p rov i de the f i r s t s o l i d ev idence of the e x i s t e n c e of the i n e r t i a l subrange of tu rbu lence proposed by Ko lmogoro f f . For a number of y e a r s , these were the on l y workers measur ing ocean t u rbu l ence u n t i l , in the l a t e 1960's a group headed by C .S .Cox at S c r i p p s I n s t i t u t e of Oceanography deve loped i n s t rumen ta t i on capab le of r e s o l v i n g the sma l l s c a l e s of the temperature f i e l d . Around t h i s t ime , the term m i c r o s t r u c t u r e came i n t o use to d e s c r i b e the s c a l e s rang ing from ^ 1 meter down to the sma l l e s t s c a l e s which e x i s t in the ocean (a coup le of c en t ime te r s fo r 2 v e l o c i t y , one cen t imete r fo r temperature and a few m i l l i m e t e r s fo r s a l t ) The techn ique for measur ing v e l o c i t y m i c r o s t r u c t u r e in the ocean was deve loped by T .R .Osborn at the U n i v e r s i t y of B r i t i s h Columbia and d e s c r i b e d in Osborn (1974) . The measurement was made u s i n g an a i r f o i l (or shear) probe capab le of r e s o l v i n g the s m a l l e s t s c a l e s of the c ross-s t r eam v e l o c i t y f l u c t u a t i o n s (except i n r eg ions of very i n tense t u r b u l e n c e ) . The probe was mounted on a v e r t i c a l p r o f i l e r and a measure was ob t a i ned of the v e r t i c a l shear of the h o r i z o n t a l v e l o c i t y f l u c t u a t i o n s from which an es t imate was made of the t u r b u l e n t k i n e t i c energy d i s s i p a t i o n . The probe proved u s e f u l in o b t a i n i n g a d e s c r i p t i o n of the v e r t i c a l v a r i a b i l i t y of v e l o c i t y m i c r o s t r u c t u r e in the ocean . New r a p i d p r o f i l i n g t echn iques are p r o v i d i n g v a l u a b l e i n f o r m a t i o n on the tempora l and h o r i z o n t a l v a r i a t i o n . The s i g n i f i c a n c e of the measurements made w i th the shear probes was demonstrated by Crawford and Osborn (1979b) who were ab le to ba lance the work done by the zona l p r e s su re g r ad i en t ( in the e q u a t o r i a l A t l a n t i c ) which d r i v e s the e q u a t o r i a l unde r cu r r en t wi th the t u r b u l e n t k i n e t i c energy d i s s i p a t i o n , thereby l i n k i n g the tu rbu l ence w i th the l a rge s c a l e dynamics . Added c o n f i d e n c e in the es t imate made of the d i s s i p a t i o n w i th the shear probe measurements was r e c e n t l y g i ven by Oakey( l982) who showed the agreement of d i s s i p a t i o n e s t ima tes u s i ng two d i f f e r e n t t e c h n i q u e s . From the h igh wavenumber c u t o f f of the temperature g r ad i en t spec t rum, the d i s s i p a t i o n was es t ima ted to agree w i t h i n a f a c t o r of two to the shear probe e s t ima tes when 3 the temperature g r a d i e n t spectrum was w e l l - r e s o l v e d . A l though c o n s i d e r e d to be a t e s t of the method of computing d i s s i p a t i o n from temperature d a t a , an independent es t imate c o n s i s t e n t wi th tha t made by the shear probe was p r o v i d e d . The sources of t u rbu l ence are most l y c o n c e n t r a t e d near the boundar ies of the ocean . Su r f ace p rocesses (wind, wave, d i f f e r e n t i a l hea t ing and c o o l i n g ) o f t e n c r e a t e a r e l a t i v e l y wel l-mixed upper l a y e r . S t rong c u r r e n t s capab le of p roduc ing l a r g e mean shears which may generate t u rbu l ence (as in e q u a t o r i a l r eg ions ) a re g e n e r a l l y c o n f i n e d to the upper l a y e r s . Other boundary l a ye r e f f e c t s at the ocean bottom and s i d e s have been l e s s w e l l s t u d i e d but appear to show g r e a t e r l e v e l s of t u r b u l e n c e . In the i n t e r i o r of the ocean , the pr imary source fo r the t u rbu l ence i s thought to be the l o s s of energy from the i n t e r n a l wave f i e l d , a l though the e f f e c t s of double d i f f u s i v e mix ing may a l s o p lay a s t r ong r o l e in some p l a c e s . It i s d i f f i c u l t to a ssess the r o l e of double d i f f u s i v e p rocesses on the measured d i s s i p a t i o n s d i s c u s s e d h e r e , due to the l a ck of s imul taneous temperature and s a l i n i t y measurements. I t i s not g e n e r a l l y p o s s i b l e to d e t e c t s a l t f i n g e r s w i th a v e r t i c a l p r o f i l e r s i n c e the s i g n a t u r e of s a l t f i n g e r s i s e s s e n t i a l l y h o r i z o n t a l (Schmit t and Evans , 1979). However, i f the f i n g e r s are t i l t e d from the h o r i z o n t a l i n some way they may be d e t e c t e d . R e c e n t l y , La rson and Gregg( l983) were a b l e to es t imate the r e l a t i v e importance of t u rbu l ence produced by the buoyancy f l u x of double d i f f u s i o n u s i n g t h e i r measurements of temperature and s a l i n i t y f i n e s t r u c t u r e c o i n c i d e n t w i th v e l o c i t y 4 m i c r o s t r u c t u r e measurements. Regions above l o c a l T-s maxima were found which e x h i b i t e d the d i s t i n c t i v e s teps in temperature and s a l i n i t y c h a r a c t e r i s t i c of the d i f f u s i v e regime of double d i f f u s i o n . Here , the buoyancy f l u x was s u f f i c i e n t l y l a rge to be r e s p o n s i b l e fo r the measured d i s s i p a t i o n . Where the T-s r e l a t i o n showed the necessa ry form fo r s a l t f i n g e r i n g , the measured d i s s i p a t i o n s were g rea te r by a f a c t o r of 5 to 10 than i n the d i f f u s i v e regime case and were much g r ea t e r than the e s t ima ted buoyancy f l u x , as w e l l . It was suggested tha t the p r o d u c t i o n of t u r b u l e n t energy by the mean shear working a g a i n s t the Reynolds s t r e s s was at l e a s t p a r t l y r e s p o n s i b l e . S t ud i e s of i n t e r n a l waves in the l a s t 20 yea rs have shown the spectrum of i n t e r n a l wave energy in the deep ocean to be remarkably cons tan t (except in some q u i t e s p e c i a l cases ) in shape and l e v e l throughout the w o r l d ' s oceans . G a r r e t t and Munk(1979) d i s c u s s a concept which may e x p l a i n the u n i v e r s a l i t y of the i n t e r n a l wave spect rum. The s p a t i a l cons tancy may be a r e s u l t of l o n g - l a s t i n g waves, which, i f they do not d i s s i p a t e q u i c k l y , w i l l propagate long d i s t a n c e s thereby d i f f u s i n g energy from a l o c a l source over a l a rge r e g i o n . The tempora l cons tancy may be due to l o c a l l y enhanced s p e c t r a l l e v e l s s a t u r a t i n g the i n t e r n a l wave spec t rum, r e s u l t i n g in l o c a l l y i n c r e a s e d d i s s i p a t i o n of i n t e r n a l wave energy i n t o the t u r b u l e n c e . S i n c e , as we s h a l l see , the energy in the i n t e r n a l wave f i e l d s c a l e s as the l o c a l va lue of the buoyancy f r equency , one suspec t s a s i m i l a r r e l a t i o n fo r the t u r b u l e n t k i n e t i c energy d i s s i p a t i o n . In t h i s t h e s i s , I w i l l d e s c r i b e a set of measurements of 5 v e l o c i t y m i c r o s t r u c t u r e (made u s i n g the shear probes a l r e a d y d i s c u s s e d ) which I conducted in two reg ions of the P a c i f i c Ocean and I w i l l attempt to put these measurements i n t o t h e i r proper c o n t e x t . In February of 1982 a set of v e l o c i t y m i c r o s t r u c t u r e p r o f i l e s were made in the c e n t r a l e q u a t o r i a l P a c i f i c . For t h i s r e g i o n the re a l r eady e x i s t s a r e l a t i v e l y l a r g e data bank of m i c r o s t r u c t u r e measurements and i t i s i n t e r e s t i n g to compare t h e s e . In May and June of 1982, measurements were made in the Western P a c i f i c Ocean and a s i g n i f i c a n t amount of data was c o l l e c t e d at depths g rea te r than 1000 meters ( r e p r e s e n t i n g v i r t u a l l y a l l of the v e l o c i t y m i c r o s t r u c t u r e da ta anywhere at these d e p t h s ) . In Chapter 2 the equa t ions which govern the t u rbu l ence are deve loped and the a p p r o p r i a t e ocean ic ba lance d i s c u s s e d . In Chapter 3 there i s a d i s c u s s i o n of seago ing expe r imen ta l p r o c e d u r e s , the on-deck s i g n a l p r o c e s s i n g , the two c r u i s e t r a c k s and o ther measurements made d u r i n g the c r u i s e s . Chapter 3 i s supplemented by the Appendices in which more d e t a i l e d d e s c r i p t i o n s of the i n s t r u m e n t a t i o n , l ab c a l i b r a t i o n s and data h a n d l i n g are found . As w e l l , a l l of the v e r t i c a l p r o f i l e s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n , CTD data and h o r i z o n t a l v e l o c i t i e s are p resen ted in Append ices K and L. The r e s u l t s from the western P a c i f i c (WESPAC) c r u i s e are d i s c u s s e d in Chapter 4. The no tab le r e s u l t a r i s i n g from the a n a l y s i s of t h i s da ta se t i s the s t rong r e l a t i o n between the t u r b u l e n t k i n e t i c energy d i s s i p a t i o n and the buoyancy f requency in a r eg ion wi th a w e l l - d e f i n e d subsur face maximum in buoyancy f r equency . An 6 attempt to e x p l a i n t h i s wi th a .model based on i n t e r n a l wave shear i n s t a b i l i t y i s deve loped in Chapter 5. In Chapter 6 the r e s u l t s from the e q u a t o r i a l P a c i f i c (PEQUOD) are p r e s e n t e d . The PEQUOD data set i s enhanced by the v e r t i c a l p r o f i l e s of mean h o r i z o n t a l v e l o c i t y as we l l as by CTD data made w i th independent i n s t r u m e n t a t i o n , and nea r l y s y n o p t i c e s t ima tes of shear and R i cha rdson number were o b t a i n e d . V a r i o u s e s t ima te s of t u r b u l e n t eddy c o e f f i c i e n t s are d i s c u s s e d and compared in Chapter 7. A compar ison of mean va lues of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n , lognormal s t a t i s t i c s of d i s s i p a t i o n and es t ima tes of t u r b u l e n t pa t ch p r o p e r t i e s and d i s t r i b u t i o n s are g i ven in Chapter 8. I t i s hoped that t h i s may p rov ide a format fo r comparing o the r data s e t s . Chapter 9 c o n t a i n s a shor t d i s c u s s i o n and the c o n c l u s i o n s . 7 I I . BACKGROUND THEORY 2.1 Mean K i n e t i c Energy Equa t ion In the f o l l o w i n g d i s c u s s i o n of the govern ing e q u a t i o n s , C a r t e s i a n t ensor n o t a t i o n i s used . S p a t i a l c o o r d i n a t e s x, = x, x 2 = y are h o r i z o n t a l , and x 3 = z i s p o s i t i v e upwards. The r e s p e c t i v e v e l o c i t i e s are U, = U, U 2 = V and U 3 = W. The d e n s i t y of the f l u i d i s p, M i s the c o e f f i c i e n t of v i s c o s i t y , v = M/P the k inemat ic v i s c o s i t y , g the a c c e l e r a t i o n due to g r a v i t y , K the thermal d i f f u s i v i t y and 6 the Kronecker d e l t a . i j The c o n s e r v a t i o n of momentum i s expressed by the Nav ie r-S tokes equa t ions f o r an i n c o m p r e s s i b l e , - n o n - r o t a t i n g , v i s c o u s , s t r a t i f i e d f l u i d ( P h i l l i p s , 1 9 7 7 ) p(9U / d t + U 9U /dx ) = - 9p/9x - pg6 + M 9 2 U /dx dx ( 2 . 1 ) i j i j i i3 i j j The c o n s e r v a t i o n of mass i s expressed by d p / d t + 9(pU ) / d x = n . d 2 p / d x dx i i i i wh ich , i f the f l u i d i s i n c o m p r e s s i b l e and the d i f f u s i o n terms are s m a l l , reduces to dp/dt + U 9p/9x = 0 i i and 9U /9x = 0 i i ( 2 . 2 ) 8 It i s u s u a l l y common to separa te the f low f i e l d i n t o mean p l u s f l u c t u a t i n g components ( r e f e r r e d to as Reyno lds ' expans ion a f t e r Osborne Reynolds who f i r s t suggested the s e p a r a t i o n ) . The measurements made us ing the shear probe are s e n s i t i v e to s c a l e s of about 1-50 cm and i t i s b e l i e v e d tha t at these s c a l e s the motion i s random and i s r e s p o n s i b l e fo r mix ing ( i . e . , the motion i s t u r b u l e n t ) . At s c a l e s g r ea t e r than 50 cm (or a t l e a s t g r ea t e r than 1 meter) in the ocean , the spectrum of v e l o c i t y shear does not s c a l e w i th t u rbu l ence parameters (Garget t et a l . ( l 9 8 l ) , Ga rge t t et a l . ( 1 9 8 4 ) ) and other p rocesses dominate the f low f i e l d . For the purposes of t h i s t h e s i s , the s c a l e s of the f l u c t u a t i n g components w i l l be taken as the s c a l e s of the t u rbu l ence ( those s c a l e s measured by the shear probe) and a l l l a r g e r s c a l e s w i l l be r e f e r r e d to as the mean motion.. T h i s i m p l i e s tha t the mean motion i n c l u d e s c u r r e n t s , edd ies and a l l s c a l e s of waves down to the s m a l l e s t i n t e r n a l waves. Reyno lds ' expans ion of the v a r i a b l e s i n t o mean p lus f l u c t u a t i n g components i s expressed b y U = u + u ' , p = p + p ' , p = p + p ' . Equa t i ons i i i d e s c r i b i n g the mean and f l u c t u a t i n g components of the f low are s e p a r a t e l y d e r i v e d . The overbar denotes an average and w i l l h e n c e f o r t h be dropped from the mean q u a n t i t i e s u, p and p. The e rgod i c h ypo thes i s i s assumed, whereby s p a t i a l averages = t ime averages = ensemble a ve rages . Averages of i n d i v i d u a l f l u c t u a t i n g components are equa l to z e r o . The Bouss inesq approx imat ion i s used so tha t the r a t i o p '/p i s on l y of importance to the g r a v i t y term in ( 2 . 1 ) . 9 To o b t a i n the equa t i on d e s c r i b i n g the k i n e t i c energy of the mean f i e l d , (2.1) i s m u l t i p l i e d by u and averaged so tha t i J_p{3(u u )/9t + u 3(u u )/9x } = - u 9p/9x - pgu 6 2 i i j i i j i i i i 3 + nu 9 2 u /9x 9x - p9(u u'u*)/9x + pu'u'9u /9x (2.3) i i j j i i j j i j i j The mean k i n e t i c energy i s J_(u u ) and, from ( 2 . 3 ) , depends 2 i i on the work done by the h o r i z o n t a l p r e s su re g r ad i en t s i n ce the v e r t i c a l h y d r o s t a t i c p r e s su re g r a d i e n t accounts f o r the g r a v i t y term (under the Bouss inesq a p p r o x i m a t i o n ) , v i s c o u s d i f f u s i o n (which i s g e n e r a l l y c o n s i d e r e d to be sma l l f o r the mean m o t i o n ) , the d i ve rgence of the mean advec t i on of the Reynolds s t r e s s e s u ' u ' , and the Reynolds s t r e s s e s working aga in s t the mean shea r , i j T h i s l a s t term p r o v i d e s the means by which the energy i s t r a n s f e r r e d from the mean to the t u r b u l e n t f i e l d . 2.2 Tu rbu l en t K i n e t i c Energy Equa t ion The t u r b u l e n t k i n e t i c energy equa t ion i s d e r i v e d by m u l t i p l y i n g (2.1) by u ' , a ve rag ing and d i v i d i n g by p to g i ve i J_{9/9t(u'u') + u 9/9x ( u 'u ' ) } = - J_9/9x {u 1 (p f + u ' u ' / 2 ) } 2 i i j j i i p i i j j (I) ( I I ) - gp 'w ' - u'u'9u /9x + *>9/9x {u'(9u'/9x +3u'/9x )} p i j i j j i i j j i ( I I I ) (IV) (V) 10 - J ^ O u ' / B x + 3 u ' / 3 x ) ( 3 u ' / 9 x + 9 u ' / 9 x ) . ( 2 . 4 ) 2 i j j i i j j i (VI) I r e p r e s e n t s the t o t a l d e r i v a t i v e of the t u r b u l e n t k i n e t i c energy J _ ( u ' u ' ) . II i s the work done by the t o t a l dynamic 2 i i p r e s s u r e of the t u r b u l e n c e . A l t e r n a t i v e l y , t h i s i s the d i ve rgence of the t r a n s p o r t of the p r e s s u r e - v e l o c i t y c o r r e l a t i o n p l u s tha t of the t r a n s p o r t of t u r b u l e n t energy by the t u r b u l e n c e . The p-u c o r r e l a t i o n i s e s s e n t i a l in r e d i s t r i b u t i n g the t u r b u l e n t energy among the flow components. I l l i s the work done by the buoyancy f l u x . IV i s the work done by the Reynolds s t r e s s e s working a g a i n s t the mean shea r . T h i s term i s of o p p o s i t e s i g n to the i d e n t i c a l term in ( 2 . 3 ) and i s r e s p o n s i b l e fo r energy exchange between the two f i e l d s . V i s the work done by the v i s c o u s shear s t r e s s e s of the t u r b u l e n t mot ion . VI i s the r a t e of d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy , e. Terms II and V r ep resen t a l o c a l s p a t i a l r e d i s t r i b u t i o n of the t u rbu l ence energy and do not change the t o t a l energy . I f the t u rbu l ence i s c o n s i d e r e d to be homogeneous, then s p a t i a l d e r i v a t i v e s of mean t u r b u l e n c e q u a n t i t i e s are equa l to zero and the a d v e c t i v e pa r t of term I i s z e r o . F u r t h e r , i f the f low i s s t eady , then the t ime r a t e s of change are z e r o , as w e l l . T h i s r e s u l t s in a ba lance between the mechan ica l p r o d u c t i o n of t u rbu l ence by the Reynolds s t r e s s e s and the mean shear , the p r o d u c t i o n by the buoyancy f l u x and the d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy by v i s c o s i t y , or - £ p ' w ' - u ' u ' 9 u / 3 x - e = 0 ( 2 . 5 ) P i j i j 11 Osborn ( l980 ) d i s c u s s e s the r a t i o of the f i r s t two terms of (2.5) in l i g h t of l a b o r a t o r y measurements, p o i n t i n g out tha t the buoyancy f l u x i s the l e a s t s i g n i f i c a n t of the terms in ( 2 . 5 ) , r e p r e s e n t i n g at most 20% of each of the o ther two te rms. In s t r a t i f i e d t u r b u l e n t f lows wi th t h i s expected r a t i o , i t i s u s u a l l y conven ien t to r e t a i n t h i s term in the form of the f l u x R i cha rdson number, R = aJT^/(u 7 "u r 3u /3x ). f p i j i . j 2.3 The Es t ima te Of e With the shear probe sensors used to de t e c t the v e l o c i t y m i c r o s t r u c t u r e , an es t imate i s o b t a i n e d of e, p r o v i d e d a major assumpt ion i s made p e r t a i n i n g to the form of the t u r b u l e n c e . In f u l l component form, e i s w r i t t e n as e = _ M 3 u ' / 3 x +3u'/3x ) 2 2 i j j i = f { 2 ( 3 u ' / 3 x ) 2 + 2 ( 3 v ' / 3 y ) 2 + 2 ( 3 w ' / 3 z ) 2 + ( 3 u ' / 3 y + 3 v ' / 3 x ) 2 + (3u ' /3z+3w '/3x ) 2 + (3v '/3z+3w*/3y ) 2 } . T h i s can be s i m p l i f i e d c o n s i d e r a b l y i f we make the assumpt ion that the t u rbu l ence i s i s o t r o p i c . T h i s may be a good assumpt ion a c c o r d i n g to Garge t t et a l . ( l 9 8 4 ) . From a submers ib l e p l a t f o r m mounted w i th shear probes to de t e c t the c ross-s t r eam t u r b u l e n t f l u c t u a t i o n s and a hot f i l m probe to measure the streamwise f l u c t u a t i o n s , measurements made over a wide range of e i n d i c a t e tha t i s o t r o p y at d i s s i p a t i o n s c a l e s may be a r e l a t i v e l y sa fe JL assumpt ion to make down to a lower l i m i t of e = (75)*t>N 2, where 1 1 2 N i s the l o c a l buoyancy f r equency . In the deep ocean where N i s = 0.001 rad/sec and v =* 0.01 c m 2 / s e c , t h i s r e s u l t s in e ^ 1 3 x 1 0 " 7 W/m 3, which i s p r e c i s e l y the l e v e l which has been d e f i n e d as the i n s t rumen ta l no i s e l e v e l (see Mourn and Lueck (1984 ) ) . In the western P a c i f i c t he rmoc l i ne ( d i s c u s s e d in Chapter 4) where N i s 0.005 rad/sec and e =* 8X10~ 6 W/m3 t h i s may degrade the 1 r e l i a b i l i t y of the sma l l e s t ima tes of e but shou ld not s e r i o u s l y a f f e c t the averages which are dominated by i n d i v i d u a l e s t ima tes of e near or g r ea t e r than e . 1 H i n z e ( l 9 7 5 , p219) g i v e s the f o l l o w i n g i s o t r o p i c r e l a t i o n s : ( 9 u ' / 9 x ) 2 = ( 9 v ' / 3 y ) 2 = (9w ' /9z ) 2 ( 9 u ' / 9 y ) 2 = ( 9 u ' / 9 z ) 2 = ( 9 v ' / 9 x ) 2 = 2 ( 9 u ' / 9 x ) 2 = . . . (9u '/9y) (9v '/9x) = (9u ' /9z ) (9w'/9x) = - J _ ( 9u ' /9x ) 2 = 2 The shear p r o b e s measure the c ross-s t r eam components of the t u r b u l e n t v e l o c i t y f i e l d . V e r t i c a l p r o f i l e s y i e l d v e r t i c a l d e r i v a t i v e s so that (as d i s c u s s e d in Appendix E) an es t imate i s ob ta ined of the v e r t i c a l d e r i v a t i v e s , 3u ' /9z and 9v ' /9z (hence fo r th the pr imes w i l l be d ropped ) . Rewr i t i ng e in terms of these two components y i e l d s e = V5v(9u2+9v2). (2.6) 4 3z 9z T h i s i s , of c o u r s e , the k inemat i c form fo r e wi th u n i t s of [ L 2 / T 3 ] . The u n i t s used in t h i s t h e s i s are W/m3 r e q u i r i n g e to be w r i t t e n as e = YSpv ( 9u 2 + 9v 2 ) 4 9z 9z 1 3 o r e = JJ5M(9U 2 + 9V 2 ) . 4 9z 9z 2.4 S t r a t i f i e d F l o w P a r a m e t e r s The B r u n t - V a i s a l a o r b u o y a n c y f r e q u e n c y i s d e f i n e d by N 2 = -c[9p/9z - ( g / c ) 2 where p i s t h e i n s i t u d e n s i t y a n d i s a m e a s u r e P o f t h e l o c a l s t a t i c s t a b i l i t y i n t h e f l u i d . A l t e r n a t i v e l y , N may be t h o u g h t o f a s t h e n a t u r a l f r e q u e n c y o f v e r t i c a l o s c i l l a t i o n o f a p a r c e l o f f l u i d w h i c h h a s been d i s t u r b e d f r o m i t s e q u i l i b r i u m p o s i t i o n . F r e q u e n c i e s g r e a t e r t h a n N a r e r a p i d l y a t t e n u a t e d o r do n o t p r o p a g a t e a s waves w h i l e f r e q u e n c i e s l e s s t h a n N t r a v e l a s waves a n d a r e g e n e r a t e d by a • w i d e r a n g e o f s o u r c e s i n t h e o c e a n . The p a r a m e t e r N r e p r e s e n t s t h e u p p e r f r e q u e n c y l i m i t t o t h e i n t e r n a l wave s p e c t r u m o f t h e o c e a n . On t h e o t h e r h a n d , N i s a s s o c i a t e d w i t h t h e l a r g e s t s c a l e s o f t h e t u r b u l e n c e . The l e n g t h s c a l e L = ( e / N 3 ) ^ w h i c h b i s d i s c u s s e d i n C h a p t e r 8 r e p r e s e n t s t h e s c a l e a t w h i c h b u o y a n c y e f f e c t s a r e o f t h e same o r d e r a s t h e n o n l i n e a r e f f e c t s a n d t h e t u r b u l e n c e a t l a r g e r s c a l e s i s s u p r e s s e d by t h e b u o y a n c y . V a r i o u s R i c h a r d s o n numbers w h i c h r o u g h l y d e s c r i b e i n one way o r a n o t h e r t h e r e l a t i v e e f f e c t s o f t h e l o c a l s t a t i c a n d d y n a m i c s t a b i l i t y a r e d i s c u s s e d i n t h e t e x t . The f l u x R i c h a r d s o n number, d e f i n e d a b o v e , i s t h e r a t i o o f t h e g a i n i n p o t e n t i a l e n e r g y by r a i s i n g mass ( t h e b u o y a n c y f l u x ) t o t h e k i n e t i c e n e r g y r e q u i r e d t o a c c o m p l i s h t h i s ( t h e s h e a r p r o d u c t i o n 1 4 te rm) . The g rad i en t R i cha rdson number i s R = N 2 / ( 9 u ) 2 . From g 9z the PEQUOD White Horse data e s t ima tes of N and S = Au/Az are made over 25 meter depth i n t e r v a l s and a d i f f e r e n c e R i chardson number, R i = N 2 /S 2 = g_ApAz/(Au) 2 i s c a l c u l a t e d as an es t ima te of P R . U n f o r t u n a t e l y , t h i s es t imate i s on l y as good as the 9 d i f f e r e n c e es t imates of p and u and c o n c e a l s the i n f o r m a t i o n at sma l l e r s c a l e s . 15 I I I . EXPERIMENTAL CONSIDERATIONS D e t a i l e d d i s c u s s i o n s of the i n s t r u m e n t a t i o n aboard Camel III are i n c l u d e d in the append ices to t h i s t h e s i s and w i l l on ly be r e f e r r e d to h e r e . In t h i s c h a p t e r , sh ipboa rd procedures f o r deployment and recovery of the inst rument and the s i g n a l p r o c e s s i n g are d i s c u s s e d . As w e l l , I w i l l b r i e f l y d i s c u s s c r u i s e t r a c k s and other measurements made d u r i n g the PEQUOD and WESPAC c r u i s e s . A gene ra l p o i n t to be made concerns the ' d e s i g n ' of exper iments such as t h i s . A c e r t a i n l ack of c o n t r o l over expe r imenta l c o n d i t i o n s d i s t i n g u i s h e s t h i s type of exper iment from a l a b o r a t o r y se tup . In the l a b , one has the luxury of be ing ab le to ad ju s t parameters and repeat exper iments over sma l l t ime s c a l e s u n t i l s a t i s f i e d wi th the r e s u l t s . C e r t a i n l y , ocean ic measurements may be r epea t ed , but o f t e n due to t ime and expense, an extended p e r i o d may l apse be fo re do ing so and ocean i c c o n d i t i o n s may be r a d i c a l l y changed in the meantime. With t h i s in mind, i t would be i d e a l to make coherent se t s of measurements of v a r i o u s parameters which are s ynop t i c in space and time and to ensure tha t the s c a l e s of the measurements are such as to p rov i de a b a s i s f o r compar i son . E s p e c i a l l y in measur ing t u r b u l e n t q u a n t i t i e s , which have sma l l s p a t i a l s c a l e s and shor t t ime s c a l e s , t h i s problem can be q u i t e grave and may cause the exper iment to resemble a f a c t f i n d i n g m i s s i o n ra the r than a s tudy which concen t r a t e s on r e l a t i n g p r o c e s s e s . In t h i s t h e s i s , I hope tha t I s h a l l make proper r e f e r ence to t h i s l i m i t a t i o n in d i s c u s s i n g the r e s u l t s of the expe r imen ts . 16 BALLAST RELEASE MECHANISM LAUNCHER XBT WIRE LINK LIGHT STROBE MAIN PRESSURE HOUSING AND ELECTRONICS 2 SHEAR PROBES LEAD BALLAST PRE-AMP THERMISTOR CAMEL HI F i g u r e 1 - Schematic of Camel III as i t was c o n f i g u r e d fo r the WESPAC and PEQUOD c r u i s e s . 1.7 3.1 A B r i e f D e s c r i p t i o n Of Camel III A schemat ic view of Camel III i s shown in F i g u r e 1. Two shear probes p l u s a t h e r m i s t o r are mounted on a s t r e a m l i n e d nose p i e ce which con t a i n s the p r e s su re t r ansduce r and the c i r c u i t r y fo r p r e a m p l i f y i n g the shear probe s i g n a l . Underwater m u l t i c o n d u c t o r cab l es l i n k the p r e a m p l i f i e d s i g n a l s to the e l e c t r o n i c s i n the main p res su re h o u s i n g . Lead b a l l a s t i s a f f i x e d to the p r e a m p l i f i e r - main body t r a n s i t i o n cone by wire l i n k s . These l i n k s are r e l e a s e d by p r e s s u r e a c t i v a t e d b a l l a s t r e l e a s e mechanisms. The schemat ic a l s o shows the launcher and recovery a i d s ( f l a s h e r , r a d i o , and underwater p i n g e r ) . The r i n g around the top of Camel III i s used to a t t a c h a snap hook and l i n e fo r r e c o v e r y . The method used fo r d e p l o y i n g Camel III depends to a l a r g e extent on the equipment a v a i l a b l e . A new che r r y p i c k e r c rane was f i t t e d on the R/V Thomas G. Thompson and t h i s was used on the P a c i f i c t r i p s fo r l i f t i n g the Camel out of i t s c r a d l e and over the s i d e i n t o the water . Once the weight of the ins t rument has been taken up by the water the l auncher i s a c t i v a t e d to r e l e a s e the i ns t rument . At a f a l l speed of about 80 cm/sec and a r i s i n g speed on ly s l i g h t l y f a s t e r , the r e tu rn t r i p to 1000 meters i s about 45 minutes and a l i t t l e over 90 minutes to 2000 meters s i n c e Camel III s lows wi th d e p t h . The s u r f a c i n g t ime was chosen to c o i n c i d e w i th the r e t r i e v a l of the CTD on the WESPAC t r i p wh i le we g e n e r a l l y a t tempted to squeeze a Camel drop i n s i d e of a White Horse drop (which l a s t e d approx imate l y 2-3 hours) on the PEQUOD t r i p . 18 S u r f a c i n g of the inst rument was i n d i c a t e d by the r a d i o s i g n a l p i c k e d up by the sh ipboard OAR r e c e i v e r u n i t which was p re tuned to the f requency of the Camel III t r a n s m i t t e r p r i o r to l a u n c h . At n i g h t , s i g h t i n g of the inst rument was a i ded s u b s t a n t i a l l y by two s t robe f l a s h e r s mounted on the recovery end of the i n s t rumen t . Dur ing d a y l i g h t hou r s , however, s u r f a c e g l i t t e r c o u l d make i t q u i t e d i f f i c u l t to spot the s u r f a c e d i n s t rumen t . The l onges t time r e q u i r e d to spot the inst rument was j u s t under two hou r s , w i th a dozen p a i r s of eyes s t r a i n i n g and the r a d i o d i r e c t i o n f i n d e r ze roed i n to an a rc of about 4 5 ° . We expe r i enced no f a i l u r e s of OAR f l a s h e r s or t r a n s m i t t e r s on any of the d r o p s . Once the s u r f a c e d inst rument had been l o c a t e d , the s h i p was manouvered a l o n g s i d e and downwind of the i n s t rument , so tha t the snap hook c o u l d be a t t a ched from the hydrograph i c p l a t f o r m . With the snap hook in p l a c e , the ins t rument was a l l owed to d r i f t towards the s t e r n of the s h i p , where i t was r ecove red us i ng the caps tan l i n e through a sna tch b lock i n the s h i p ' s A-frame. The ins t rument was then secured to the c r a d l e and moved i n s i d e the s h i p ' s l a b f o r d i sassemb ly in o rder to r ep l a ce the g e l - c e l l b a t t e r i e s and c a s s e t t e t a p e s . 3.2 S i g n a l P r o c e s s i n g The i n s t r u m e n t a t i o n and e l e c t r o n i c s are d i s c u s s e d in d e t a i l in the append ices to t h i s t h e s i s . In t h i s s e c t i o n I w i l l d e s c r i b e the b a s i c p r o c e s s i n g of the FM s i g n a l w i th r e f e r ence to the b lock d iagram F i g u r e 2. 19 PRESSURE TRANSDUCER THERMISTOR ACCELEROKETERS SHEAR PROBES d/dt DP I t T t d/dt .d/dt r r imp a m p amp I 2 T d/dt DT A l A2 SA1 SI SA2 S2 VCOI VC02 VC03 VC08 VC04 VC05 VC06 VC09 VCO 7 VCOI 400Hz 560Hz 730Hz 3000Hz 960Hz 1300Hz 1700Hz 3900Hz 2300Hz 5400Hz CHART RECORDER (REAL TIME) INTERNAL CASSETTE RECORDER FM DISCRIMINATOR *»CHART RECORDER DIGITIZER MAGNETIC TAPE AUTOMATIC DEGLITCHER CLEANED SHEAR SIGNALS HIGHPASS FILTERS FURTHER EDITING CALCULATE DISSIPATION igure 2 - Schematic of signal processing 20 In a l l , ten s i g n a l s are input to v o l t a g e - c o n t r o l l e d o s c i l l a t o r s (VCOs) which conver t the s i g n a l v o l t a g e s to f requency modulated (FM) s i g n a l s . These are summed to produce the FM m u l t i p l e x e d s i g n a l . T h i s s i g n a l i s r eco rded on c a s s e t t e tape i n s i d e Camel I I I . To view the measurements in r e a l t i m e , an XBT wire l i n k i s used to t r ansmi t the s i g n a l to the s h i p , where i t i s d i s c r i m i n a t e d (the f requency i s conve r t ed back to a v o l t a g e ) and viewed on a cha r t r e c o r d e r . A b i g advantage of r e a l t ime v iew ing i s the peace of mind which goes wi th a s u c c e s s f u l l y o p e r a t i n g sys tem. However, due to impedance l o s s e s when the wire i s extended and the f i n i t e l e n g t h of the w i r e , there i s a p r a c t i c a l l i m i t of about 900 meters in depth when us ing the XBT wire l i n k . S ince the m a j o r i t y of the drops in t h i s p r o j e c t were deeper , the c a s s e t t e r eco rde r system was the pr imary t o o l fo r s i g n a l r e c o r d i n g . The c a s s e t t e was immediate ly cop i ed once aboard the s h i p and viewed on a cha r t r e c o r d e r . Back in the l a b , the s i g n a l s were d i g i t i z e d us ing an LSI—11 computer and reco rded on magnet ic t ape . The h igh f requency shear s i g n a l s are sub j e c t to s e v e r a l forms of g l i t c h e s (bad data p o i n t s ) which are d i s c u s s e d in Lueck , Crawford and Osborn (1983) . To e r a d i c a t e these (which may be 1-100 data p o i n t s long) an automat ic d e g l i t c h i n g r o u t i n e was used which p r o v i d e s an o b j e c t i v e c r i t e r i o n f o r the d e t e r m i n a t i o n of bad da ta and agrees wi th ' e y e b a l l e d ' data b e t t e r than 90 percent of the t ime . Once the data were s u i t a b l y ' c l e a n ' they were p roces sed as in Appendix F. 21 3.3 The Study Areas F i g u r e 3 i s a map of the P a c i f i c Ocean w i th the PEQUOD and WESPAC c r u i s e r eg ions enc l o sed by r e c t a n g u l a r boxes . The PEQUOD r eg ion was occup i ed in F eb rua r y , 1982 where n ine teen Camel III p r o f i l e s were made. T h i r t e e n Camel III p r o f i l e s were made in the WESPAC reg ion in May and June of 1982. 3.3.1 PEQUOD The l o c a t i o n of c u r r e n t meter moor ings , White Horse nets and Camel III p r o f i l e s for the PEQUOD t r i p are shown in F i g u r e 4. Two t r a n s e c t s of the equator were made a long 138°W and 145°W. F i v e a d d i t i o n a l s t a t i o n s were occup i ed a long the equa to r . J .R ichman and C . E r i k s e n ma in ta ined the moorings wh i le J . L u y t e n was r e s p o n s i b l e fo r the White Horse p r o f i l e s . S ince I have not worked w i th the c u r r e n t meter d a t a , I w i l l not d i s c u s s these any f u r t h e r . The White Horse i s a f r e e l y - f a l l i n g , a c o u s t i c a l l y s e l f -p o s i t i o n i n g dropsonde which i s used to determine v e r t i c a l p r o f i l e s of h o r i z o n t a l ocean c u r r e n t s . I t i s p o s i t i o n e d by a set of th ree t r ansponders moored to the bottom which i n t e r r o g a t e the ins t rument as i t f a l l s at about 1 m/sec. The bottom t r ansponde rs are r e f e r r e d to as a t ransponder n e t . The White Horse a l s o has a N e i l Brown m i c r o - p r o f i l i n g CTD mounted on i t . V e l o c i t i e s are computed at 25 meter i n t e r v a l s and temperature and s a l i n i t y at 2 meter i n t e r v a l s . A d e t a i l e d d e s c r i p t i o n of the White Horse i s g i ven by Luy t en , Neede l l and Thomson (1982) . 22 120* E H0° E Figure 3 -160 E 180" E 160" W HO H LONGITUDE 120 H 100 W 80 H Map of P a c i f i c Ocean showing the WESPAC and PEQUOD study regions 23 UJ Q 3 or A l 9 ( 9 2 S a ) . 1 6 ( 9 3 0 a ) ° # A * ^ , i S ( 9 0 0 a > •ooo in o U 7 P#0- «§OOQ-J A - r t . # A l 3 ( 9 0 0 a > " O A l 2 < 9 0 0 a > « « o o o A I 0 ( 8 t 0 a A 1 « ( 9 2 0 » ) M A ! 7 < 9 3 0 « » ; in o o . 8 3 S a > ( 8 2 t o ) » # A 2 < 1 5 0 0 B ) D 0 A 3 ( 1 3 0 0 a ) K #OA 4 I 9 0 0 B ) • -r # A s < 9 2 0 a ) G # A 6 ( 9 0 0 a ) 0 WHITE HORSE O MOORIMG A CAMEL 111 157 H 152 W 147 W 142 W LONGITUDE 137° W 132° H Figure 4 - PEQUOD cruise track (February, 1982). The s o l i d dots with accompanying l e t t e r s represent White Horse nets. Open c i r c l e s are at locations of current meter moorings. Open tr i a n g l e s and associated Camel III drop numbers are followed by the depth of the drop in parentheses, concentration necessitated the The large of drops near 0°, 145°W lower l e f t hand blowup. 24 For the purpose of t h i s s tudy , the White Horse data was used to determine the s t a b i l i t y of a s t r a t i f i e d f l u i d in terms of two pa ramete rs . The buoyancy f r equency , N = (-gAp/pAz ( g / c ) 2 ) ^ (where c i s the speed of sound in seawater) , i s a measure of the s t a t i c s t a b i l i t y of the f l u i d wh i l e S = AU/Az i s a measure of the dynamic s t a b i l i t y . Smal l v a lues of N and l a r g e v a l ues of S tend to d e s t a b i l i z e the water co lumn. A d i s c u s s i o n of the t reatment of the White Horse v e l o c i t y and CTD data i s g i ven i n Appendix J . A t o t a l of n ine teen Camel III p r o f i l e s were made du r i ng the PEQUOD c r u i s e . S i x t een of these y i e l d e d u s e f u l d a t a , t o t a l l i n g 12,335 meters of d a t a . Due to a p r e s su re leak in the p r e a m p l i f i e r c a se , drops were l i m i t e d to 900 meters a f t e r drop 3. E l even drops were made which were nea r l y s ynop t i c w i th White Horse p r o f i l e s . P r i o r to drop 13 Camel III was not dep loyed u n t i l the White Horse had been brought back s a f e l y aboard s h i p . The maximum time spac ing between Camel III and White Horse p r o f i l e s was about four h o u r s . From drop 13 on , the Camel was dep loyed immediate ly a f t e r the White Horse and was brought back on board be fo re the White Horse s u r f a c e d , r e s u l t i n g i n a t ime l a g of on l y m inu tes . A drop l og f o r PEQUOD, Camel III d i s s i p a t i o n p r o f i l e s , White Horse v e l o c i t y , t empera tu re , s a l i n i t y and buoyancy f requency p r o f i l e s are i n c l u d e d in Appendix K. 25 3.3.2 WESPAC The l o c a t i o n s of ten c u r r e n t moorings and twenty-three CTD s t a t i o n s o c cup i ed a long 152°E from 27°N to 41 °N are marked in F i g u r e 5. W.Schmitz dep loyed the moorings wh i le P . N i i l e r commissioned the CTD d a t a . T h i r t e e n Camel III p r o f i l e s were made, e l even of which y i e l d e d 13,070 meters of good d a t a . Having s o l v e d the l e a k i n g p r e a m p l i f i e r p rob lem, i t was p o s s i b l e to make drops to g r ea t e r depths than were made a long the equa to r . Consequen t l y , 5805 meters of data below the depth of 1000 meters were ob ta ined and th ree drops were made to n e a r l y 2300 mete rs . F i g u r e 6 shows the depth d i s t r i b u t i o n of data from both the PEQUOD and WESPAC t r i p s . Near l y e q u i v a l e n t t o t a l amounts of data were o b t a i n e d but the PEQUOD d a t a ' are concen t r a t ed above 900 meters wh i l e WESPAC data ranges to 2300 mete rs . For the WESPAC t r i p , Camel III was dep loyed so that i t broke s u r f a c e s h o r t l y a f t e r the CTD was brought aboard s h i p . Appendix L i n c l udes a drop l og fo r WESPAC, Camel III d i s s i p a t i o n p r o f i l e s and t empera tu re , s a l i n i t y and buoyancy f requency p r o f i l e s from the CTD data (a d i s c u s s i o n of the t reatment of CTD data i s g i ven in Appendix J ) . 26 Q ZD f-> cn 45° N 40° N 35" N 30" N-2 5 ° N 2 0 ° N 142° Z 147 € 2 5 . 2 1 . 2 3 A . O ^ I 3 ( 2 J 4 0 B ) «-a> 2 1 A. O 2 0 4 ^ 1 2 ( 2 2 7 0 n ) 1 9 ^ 0 / ^ . 1 I ( 2 2 4 0 a ) 1 7 * 0 I 5 * 0 ^ l 0 ( i 5 i 0 a ) 1 4 . A . U * 0 / ^ 9 < 1 « 5 a ) 1 2 * 9 * 0 B A ^ 6 ( 9 7 5 m ) 7 * O . £ t 5 ( 6 7 0 a ) - - -£ • 5 A A 4 ( H 0 0 a ) • A O 3 ^ i a O / ^ . 2 ( M 7 0 a > A l ( B « 0 a ) .aV CTD O MOORING CAMEL I I I 152 E L O N G I T U D E 157" E 162" E F i g u r e 5 - WESPAC c r u i s e t r a ck (May/June, 1982). S o l i d t r i a n g l e s and numbers to l e f t of 152° mer id ion r ep resen t CTD c a s t s , open c i r c l e s are l o c a t i o n s of c u r r e n t meter moor ings and open t r i a n g l e s are Camel III p r o f i l e s . The drop number accompanies the Camel III p o s i t i o n and the d rop depth (dbar) i s in pa r en theses . 27 GOOD DATA dbar 500 1000 1500 F i g u r e 6 - T o t a l l eng ths of da ta r e c o r d from the s p e c i f i e d 100 dbar depth i n t e r v a l s f o r the t o t a l PEQUOD and t o t a l WESPAC data s e t s . 28 IV. RESULTS FROM WESPAC Camel III p r o f i l e s were made in May and June of 1982 in the Western Nor th P a c i f i c Ocean a long 152°E from 28°N to 42°N (and a s i n g l e p r o f i l e at 2 3 ° N , 148 °W ) . These were made in c o n j u n c t i o n w i th the f i n a l r ecovery of the WP1 a r r a y of moored ins t ruments which were o r i g i n a l l y dep loyed in mid-summer of 1980. P r e l i m i n a r y r e s u l t s are p resen ted i n Schmitz et a l . ( l 9 8 2 ) . The r eg ion rough ly bounds the most e n e r g e t i c segment of the Kurosh io E x t e n s i o n , a s i t e which has been r e l a t i v e l y s p a r s e l y sampled when compared to the ana logous r eg i on of the Western North A t l a n t i c (bounding the Gu l f S t ream) . The WP1 a r r a y was p lanned as an e x p l o r a t o r y a r r ay to c h a r a c t e r i z e the g e o g r a p h i c a l v a r i a b i l i t y of the b a s i c t ime averages and f requency d i s t r i b u t i o n of low f requency c u r r e n t s and temperatures which have p e r i o d s of two days or l o n g e r . Schmitz et a l . ( l 9 8 2 ) r e f e r to these low f requency f l u c t u a t i o n s as edd ies and p o i n t out that the a r r ay was not in tended to r e s o l v e the s p a t i a l s c a l e s . The CTD measurements made a long 152°E ( k i n d l y s u p p l i e d by P. N i i l e r ) g i ve a s p a t i a l snapshot of the temperature and s a l i n i t y f i e l d s at the t ime of the m i c r o s t r u c t u r e measurements. Because of t h e i r importance in t h i s s tudy , I have i n c l u d e d the T , S and N p r o f i l e s in Appendix L a long wi th the d i s s i p a t i o n p r o f i l e s from WESPAC. There have been no p r e v i o u s l y r epo r t ed measurements of m i c r o s t r u c t u r e in t h i s r e g i o n . As w e l l , the re have been no measurements of v e l o c i t y m i c r o s t r u c t u r e at depths g r e a t e r than 1000 meters anywhere. In these r e s p e c t s a l o n e , the ensu ing 29 pages d e s c r i b e comp le te l y new r e s u l t s . A l a r g e s c a l e d e s c r i p t i o n of the s t r u c t u r e of the water column a long 152°E from 28°N to 42°N in May/June, 1982 i s g i ven i n terms of F i g u r e s 7-9 which a r e , r e s p e c t i v e l y , s e c t i o n s of t empera ture , s a l i n i t y and e p r o f i l e s . The temperature s e c t i o n ( F igu re 7) shows a number of i n t e r e s t i n g l a r g e s c a l e f e a t u r e s at the t ime of the 1982 WESPAC c r u i s e . A l a r g e c o l d core r i n g - l i k e f e a tu r e was c e n t r e d at about 34 °N (I w i l l h ence fo r t h r e f e r to t h i s as a r i n g a l though w i th on l y a s i n g l e s e c t i o n one cannot be sure tha t t h i s i s s o ) . T h i s i s an important f e a tu r e i n s e p a r a t i n g the Camel III p r o f i l e s i n t o d i f f e r e n t r e g i o n s . Drops 1-8 were made south of # the r i n g wh i l e drop 9 was made d i r e c t l y in the middle of the r i n g and drops 10-13 were n o r t h of the r i n g (but due to inst rument problems these l a t t e r drops c o n t a i n on ly a sma l l f r a c t i o n of good data above 500 me te r s ) . The a x i s of the Kurosh io E x t e n s i o n i s d e f i n e d by Schmitz e t a l . ( l 9 8 2 ) as the l o c a t i o n of the 15°C i so therm at 200 meters dep th , and t h i s occurs at about 3 7 . 5 ° N in e a r l y summer of 1982, which r e p r e s e n t s a northward s h i f t of app rox ima te l y 2 ° from i t s p o s i t i o n in J u l y , 1980 d u r i n g the o r i g i n a l deployment of WP1. The s i g n a t u r e s of the r i n g and of the Kurosh io Ex t ens i on f r o n t are both q u i t e deep, apparent at l e a s t as deep as 1800 mete r s . The upward s l o p i n g i so therms no r th of the Ku rosh io Ex t ens i on f r o n t i n d i c a t e the f r o n t of the Oyash io , which f lows southward from the Be r i ng Sea and Sea of Okhotsk. As w e l l , no r th of the r i n g , the re i s c o n s i d e r a b l e i n t e r l e a v i n g of water masses as seen by the c l o s e d 30 F i g u r e 7 - Temperature s e c t i o n from CTD measurements taken along 152°E i n May/June 1982. Isotherms are p l o t t e d every 1°C. A c o l d core r i n g i s c e n t r e d at 34°N, the Kuroshio E x t e n s i o n f r o n t at 38°N and the Oyashio f r o n t at 42°N. 31 con tours i n both the T and S s e c t i o n s . South of the r i n g , the main the rmoc l i ne l i e s between 350 and 800 meters depth and the temperature dec reases f-rom 15°C to 6 °C a c r o s s t h i s range. The s t r a t i f i c a t i o n i s o f f s e t somewhat by the s a l i n i t y which decreases to a minimum between 600 and 800 meters , t h e r e a f t e r i n c r e a s i n g w i th depth ( F igu re 8 ) . The i sotherms of the main the rmoc l i ne are n e a r l y p a r a l l e l south of the r i n g and t i l t upwards to the s o u t h . The buoyancy f requency in .the main the rmoc l i ne has maximum va lues of 0.006-0.008 rad/sec and the maximum s h i f t s downwards to the n o r t h . F i g u r e 8 shows the s a l i n i t y s e c t i o n a long 152°E from e a r l y summer, 1982. The r i n g and f r o n t a l f e a t u r e s are r e a d i l y apparent in the s a l i n i t y s e c t i o n as are the deep s i g n a t u r e s of each of t h e s e . A coherent s a l i n i t y minimum bounded by the 34.1 ppt i s o h a l i n e s s t r e t c h e s a c ros s the s e c t i o n , g e n e r a l l y s l o p i n g downward south of the Oyash io f r o n t , c o n t o r t e d by the r i n g and then s l i g h t l y upward south of the r i n g c o r r e s p o n d i n g to the upward t i l t of the i sotherms south of the r i n g . In a l l of the p r o f i l e s south of the r i n g the maximum in N l i e s about 200 meters above the minimum in S. I n d i v i d u a l p r o f i l e s of e w i l l be d i s c u s s e d i n d e t a i l in pages to f o l l o w . Here I w i l l r e f e r to F i g u r e 9 which r ep r e sen t s averages of e over 100 meter v e r t i c a l i n t e r v a l s (or app rox ima te l y 50 independent e s t ima tes of e ) . The p r o f i l e s are numbered on F i g u r e 9 fo r compar ison w i th the d e t a i l e d p r o f i l e s in Appendix L. The s c a l e of the e bar graphs i s l i n e a r in o rder to emphasize d i f f e r e n c e s in ad jacent va lues which may be reduced 32 4 5 N F i g u r e 8 - S a l i n i t y s e c t i o n from CTD measurements taken a long 152°E in May/June 1982. I s o h a l i n e s are - p l o t t e d every 0.1 p p t . Note the shaded s a l i n i t y minimum bounded by the 34.1 ppt i s o h a l i n e s . 33 XSN SON SSN 40N 4SN F i g u r e 9 - Turbulent k i n e t i c energy d i s s i p a t i o n averaged v e r t i c a l l y over 100 meter i n t e r v a l s f o r 10 drops made al o n g 152°E i n May/June, 1982. The histogram s c a l e i s l i n e a r and p r o p o r t i o n a l t o the s i z e i n d i c a t e d i n the lower l e f t hand c o r n e r . 34 by the heavy a v e r a g i n g . Blank s e c t i o n s of i n d i v i d u a l p r o f i l e s r ep resen t bad d a t a . Except fo r drop 8, the near s u r f a c e averaged d i s s i p a t i o n s are l o c a l maxima. Oakey and E l l i o t t ( 1 9 8 2 ) showed tha t a cons tan t f r a c t i o n of the energy input by the wind i s d i s s i p a t e d by t u r b u l e n t mix ing in the su r f a ce mixed l a y e r . The sma l l near-s u r f a c e va lue in drop 8 i s l i k e l y due to reduced energy input at the s u r f a c e . U n f o r t u n a t e l y , I do not have a complete m e t e o r o l o g i c a l r e co rd but I d i d r e c o r d wind speeds at the time of each Camel III d r o p , when the s h i p was s t a t i o n a r y . These show that the wind speed r e l a t i v e to the s h i p dropped to n i l two days p r i o r to drop 8, r i s i n g s low l y to 4 knots at the t ime of the d r o p . These were the sma l l e s t wind speeds recorded on the WESPAC c r u i s e . Drop 2 i s no tab l e due to the h igh tu rbu l ence l e v e l s throughout the e n t i r e d rop , even when averaged over 100 mete rs . A l l of the p r o f i l e s south of the r i n g except drop 4 have subsur f ace secondary maxima in e. These are q u i t e d i s t i n c t in drops 2, 6 and 8. a l though l e s s apparent in drop 5. The tendency i s f o r the maximum in e to i n c r ea se in depth northwards towards the r i n g . R e c a l l that the depth range of the i sotherms i n the main t he rmoc l i ne encompasses the e maxima and tha t these s lope down towards the nor th between 2 7 . 5 ° N and 3 2 . 5 ° N , ( F igure 7 ) . Drop 9 made i n s i d e the c o l d co re r i n g i s r e l a t i v e l y q u i e t except fo r a sma l l maximum in e near 400 mete rs . The drops no r th of the r i n g are i n t e r e s t i n g in tha t they r ep resen t the 35 f i r s t measurements of v e l o c i t y m i c r o s t r u c t u r e at these dep ths . They show s i g n a l l e v e l s g r ea t e r than 10~6W/m3 o c c u r r i n g over 12% of the da ta from depths > 1000 meters . Most of the patches are not t h i c k (< 5 meters) w i th a no tab l e e x c e p t i o n be ing the 30 meter t h i c k pa tch at 2050 meters from drop 11. T h i s has a pa tch-averaged d i s s i p a t i o n of 3x10" 6 W/m 3 . 4.1 The Drops South Of The Ring Drops 1 , 2 , 4 , 5 , 6 and 8 shown in Appendix L were made south of the r i n g which i s ev iden t in the T and S s e c t i o n s a long 1 5 2 ° E . Drop 1 was a t e s t drop made we l l south of the r i n g and the CTD data were not a v a i l a b l e . CTD p r o f i l e s c o i n c i d i n g w i th these drops south of the r i n g i n d i c a t e a sha l low seasona l t he rmoc l i ne and a s t rong s a l i n i t y minimum at the bottom of the main t h e r m o c l i n e . The main t he rmoc l i ne i s d i s t i n g u i s h e d by a peak in N which i s e v iden t in F i g u r e 10. F i gu re 10 r ep re sen t s averages of N over 25 meter depths over the f i v e drops south of the r i n g tha t have a s s o c i a t e d CTD d a t a . The maximum in F i g u r e 10 at 500 meters i s more than twice the va lue of the minimum at 300 mete rs . The minimum in N i s app rox ima te l y equa l to N at 900 mete rs , below which N g r a d u a l l y d e c r e a s e s . I n d i v i d u a l p r o f i l e s e x h i b i t t u r b u l e n t patches r ang ing in s i z e from < 2 meters to n e a r l y 50 meters t h i c k and these extend over the depth range sampled (the deepest drop south of the r i n g was drop 2 which was to 1450 m e t e r s ) . Drop 2 was the most e n e r g e t i c of the p r o f i l e s at dep th , from both PEQUOD and WESPAC 36 1 4 0 0 H — r F i g u r e 10 - V e r t i c a l p r o f i l e of averages of 25 meter e s t i m a t e s of buoyancy frequency over WESPAC drops 2,4,5,6,8. 37 t r i p s , bea r i ng l i t t l e resemblance in e i t h e r magnitude or ' p a t c h i n e s s ' to any of the o ther p r o f i l e s from WESPAC. In f a c t i t most c l o s e l y resembles drop 13 from the F ine and M i c r o s t r u c t u r e Exper iment (FAME) r epo r t ed by Garge t t and O s b o r n ( l 9 8 l ) and which was made w i th in the 2000 meter ba thymet r i c contour about the i s l a n d of Bermuda. Us ing the o b j e c t i v e method d e s c r i b e d in Chapter 8 to determine pa tch t h i c k n e s s , 38% of the water column i s t u r b u l e n t in drop 2 ( i . e . , > lO" 6 W/m 3 ) compared to an average va lue of 19% over a l l of the WESPAC d r o p s . There are two d i s t i n c t f e a t u r e s of the drops made south of the r i n g . F i r s t l y , in the depth r eg ion bounded roughly by 200 and 450 mete rs , there are c o n s i s t e n t and ex t ens i v e (50-200 meters) i n t e r v a l s which are at or near the inst rument no i se l e v e l of 3x lO " 7 W/m 3 . These r eg ions concur w i th the subsur f ace minima in the p r o f i l e s of buoyancy f r equency , N. Second l y , the subsur f ace maxima of N c o n t a i n both more f r e q u e n t l y o c c u r r i n g t u r b u l e n t pa tches and h ighe r l e v e l s of e. F i g u r e 11 i s a p l o t of 25 meter v e r t i c a l averages of e which were then averaged over drops 2 , 4 , 5 , 6 , 8 south of the r i n g . The minimum in t u r b u l e n t a c t i v i t y between 200 and 400 meters s tands out d i s t i n c t l y . In f a c t , seven ad jacent p o i n t s in the range 200-400 meters are the s m a l l e s t va lues over the e n t i r e depth range of the averages to 1350 meters (averag ing was not done deeper than 1350 meters s i n c e on l y drop 2 was deeper whi le at l e a s t th ree drops were averaged i n t o the remainder of the p o i n t s ) . The maximum va lue of e i s a t 500 meters and i t i s F i g u r e 11 - V e r t i c a l p r o f i l e of averages of 25 meter e s t i m a t e s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n over WESPAC drops 2,4,5,6,8. 39 l a r g e r than the averaged va lue nea res t to the s u r f a c e . Below 500 meters' there i s c o n s i d e r a b l e v a r i a b i l i t y of e, ad jacen t 25 meter averages d i f f e r i n g by up to a f a c t o r of s i x . T h i s v a r i a b i l i t y i s in c o n t r a s t to the r e l a t i v e l y smoothly v a r y i n g averaged e p r o f i l e over 200-450 mete r s . 4.2 e And Eddy K i n e t i c Energy In the ensu ing pages and i n the next c h a p t e r , I w i l l t r y to show that the t u rbu l ence in the drops south of the r i n g i s r e l a t e d to the i n t e r n a l wave f i e l d , and p rov i de some reasons to support t h i s c o n t e n t i o n . As t h i s set of measurements was made in a r eg ion of the ocean fo r which a s t rong energy source e x i s t s in the form of the western boundary c u r r e n t whose energy c o n t r i b u t i o n to the su r round ings may be es t ima ted by de te rm in ing the l o c a l eddy k i n e t i c energy , K , r e s u l t s from Schmitz et E a l . ( 1 9 8 2 ) are p resen ted to at tempt to conv ince the reader of the l a ck of c o r r e l a t i o n between e and K . K i s c a l c u l a t e d u s i n g E E the mean va lues of the v a r i a n c e s of the h o r i z o n t a l v e l o c i t y components measured by moored c u r r e n t meters which were dep loyed in mid-summer, 1980 and r e cove red in e a r l y summer, 1981. Ave rag ing i s done over about e l e ven months of data wh ich , h o p e f u l l y , d e s c r i b e s the c l i m a t e of K in t h i s pa r t of the E ocean . Schmitz et a l . ( l 9 8 2 ) d i s c u s s the data h a n d l i n g in more d e t a i l . Schmitz et a l . ( l 9 8 2 ) d e s c r i b e s e l e c t e d r e s u l t s from the f i r s t s e t t i n g of the WP1 mooring a r r a y from mid-summer 1980 to 40 mid-summer 1981. A CTD s e c t i o n from J u l y 1980 ( F igu re 2 from Schmitz et a l . ( l 9 8 2 ) ) i n d i c a t e s a c o l d core r i n g at 3 3 . 5 ° N and the f r o n t of the Ku rosh io Ex t ens i on at 3 5 . 5 ° N , which r ep resen t s a southward s h i f t of 2 ° f o r the f r o n t and about 1° f o r the r i n g when compared to the 1982 T and S s e c t i o n s of F i g u r e s - 7 " a n d 8 ( t h i s i s l i k e l y an e n t i r e l y d i f f e r e n t r i n g from 1982). With t h i s in mind c o n s i d e r F i g u r e 12a which i s d e r i v e d from F i g u r e s 5 and 8 of Schmitz et a l . ( l 9 8 2 ) . These data are eddy k i n e t i c e n e r g i e s e s t ima ted a long 152°E fo r the i n i t i a l WP1 deployment , at th ree depth i n t e r v a l s which co r r espond to depth i n t e r v a l s cove red by the Camel d r o p s . In F i g u r e 12b are the d i s s i p a t i o n s from the Camel drops averaged over the c o r r e s p o n d i n g depth i n t e r v a l s . The two p l o t s have been s h i f t e d 1°.. w i th r espec t to each o ther to a l i g n the p o s i t i o n s of the r i n g s and Kurosh io Ex t ens i on f r o n t s from 1980 and 1982. A severe l i m i t a t i o n to t h i s compar ison i s the r e l a t i v e a ve r ag ing of the two data s e t s . The c u r r e n t meter data are h e a v i l y averaged over t ime wh i l e the d i s s i p a t i o n p r o f i l e s r ep resen t s p a t i a l snapshots through the f i e l d of t u r b u l e n c e , which does i t s e l f vary c o n s i d e r a b l y . There are two d i s t i n c t t r ends to the l a t i t u d i n a l s e c t i o n of K . . F i r s t of a l l , f o r a l l th ree depth i n t e r v a l s , K i n c r e a s e s E E northward from 28 °N by about a f a c t o r of t e n , w i th peak va lues at ( f o r the deeper meters) or j u s t no r th of ( fo r the sha l low meters) the p o s i t i o n of the r i n g . Second l y , a t each mooring p o s i t i o n , v e r t i c a l l y ad jacen t and deeper meters r e co rd s u b s t a n t i a l l y lower v a lues of K . V e r t i c a l l y ad jacen t va lues E 41 F i g u r e 12 - a )Eddy k i n e t i c energy es t imated by Schmitz et a l ( l 9 8 2 ) f o r c u r r e n t meters l o c a t e d at depth - i n t e r v a l s 250-300m(dots) , 500-700m(t r iang les ) and 1000-1500m(squares) a l ong 152 ° E . Long i tude i s marked above the p l o t s . b ) t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over depth i n t e r v a l s c o r r e s p o n d i n g to a) one year a f t e r the r e cove r y of the moorings by Schmitz e t - a l ( l 9 8 2 ) . The p l o t i s s h i f t e d 1° to the south fo r reasons d i s c u s s e d in the t e x t . Note that ( in c o n t r a s t to F i g u r e s 7-9) south i s to the r i g h t . 42 d i f f e r by up to a f a c t o r of t e n . These two t r a i t s are in c o n t r a s t to the e s e c t i o n . Indeed, the 500-700 meter (mid-depth) averaged va lues of e are the g r e a t e s t f o r every drop south of the r i n g and these reach a peak which i s l o c a t e d about 2 ° south of the r i n g where the peak in K o c c u r s . There i s no E apparent peak in the average va lue of e in e i t h e r of the 250-300 meter or 1000-1500 meter segments. Drop 9 ,which was made i n s i d e the r i n g , a c t u a l l y has some of the s m a l l e s t va lues of averaged e. From these a l b e i t rough and q u a l i t a t i v e r e s u l t s there i s no apparent reason to suggest a r e l a t i o n between tu rbu l ence l e v e l s and k i n e t i c energy in the eddy f i e l d . 4.3 e And N The other a v a i l a b l e i n f o r m a t i o n to which the t u rbu l ence may be compared i s the CTD d a t a . As d i s c u s s e d in Appendix L, v a lues of N were c a l c u l a t e d from the T , S data over 25 meter depth i n t e r v a l s . For each of the da ta s e t s , independent v a lues of e were averaged over co r r e spond ing 25 meter depth i n t e r v a l s to produce the set ( e , N ) 2 5 . Subsets of these data were chosen so that the upper(20-300m) and lower (300m-bottom) p o r t i o n s of the water column c o u l d be c o n s i d e r e d s e p a r a t e l y ( t h i s g roup ing i s more c r u c i a l f o r the PEQUOD data where the e f f e c t s of the c u r r e n t f i e l d s are g e n e r a l l y c o n f i n e d to the upper 300 m e t e r s ) . From the s c a t t e r p l o t s of F i g u r e 13 i t i s d i f f i c u l t to d i s t i n g u i s h any t r e n d s . However, i t i s noted tha t the h ighes t va lues of N are a s s o c i a t e d wi th the h ighes t v a l ues of e wh i le the lowest va lues of N c o i n c i d e w i th lower va lues of e, except 43 io-10" WESPAC 20-300m a) ,e io-»' 8 •4 10-'] 10" *"1 1 1—' I ' I M l • 1 • I I I • l| < 1—I I I I ' I io-10" " • " ' 1 • <••••' WESPAC 300m-bottom b) ^ io-s-: 10-«i io-7= io-»-I* • • • • -<—i—i 111111 i ' i ><i •—i—i i 111 LOG N (r a d / s e c ) i i S c a t t e r p l o t s of buoyancy frequency and t u r b u l e n t k i n e t i c energy d i s s i p a t i o n , each estimated f o r 25 meter depth i n t e r v a l s . Averages were made over 1/3 decade i n t e r v a l s i n N ( s o l i d d o t s ) . 44 fo r the low N, near su r f a ce va lues in 13a which a re s t r o n g l y t u r b u l e n t due to wind m i x i n g . The l a r g e b l a ck do ts represen t averages of e made over a l l the va lues l y i n g w i t h i n h a l f decade i n t e r v a l s of N. A p a u c i t y of v a l ues in the upper N b in of 13b p r e c l u d e d mean ingfu l a v e r a g i n g . In each of the th ree p l o t s , there i s a s t r ong p o s i t i v e c o r r e l a t i o n of the averaged va lues between e and N (because of wind m i x i n g , the averaged va lue in the lowest N b in in 13a i s not r e p r e s e n t a t i v e ) . The l i n e drawn in 13c has a s l ope of 1. A d i s t i n c t l y d i f f e r e n t form of a ve rag ing i s p resen ted in F i g u r e s 14 and 15. These averages of e and N were made by ave rag ing v e r t i c a l l y over 100 meters and then over a l l of the d r o p s . The r e s u l t a n t v e r t i c a l p r o f i l e s are 14a ,b . Even though the drops made in and no r th of the c o l d core r i n g have now been i n c l u d e d in the a v e r a g i n g , the maxima and minima in and above the t he rmoc l i ne are s t i l l apparent ( c f . F i g u r e s 10 and 11) . The h igh averaged e va lue at 2100 meters i s dominated by the event in d rop 11 due to the sma l l amount of data a v a i l a b l e at t h i s d e p t h . The e and N p r o f i l e s of 14a,b are combined in F i g u r e 15 to show a r e l a t i v e l y s t rong c o r r e l a t i o n of e w i th N over one f u l l decade i n each parameter . Bars were drawn about the 850 meter data p o i n t i n order to p rov i de an i n d i c a t i o n of the spread of the data about the mean. For each data p o i n t , th ree c o n s e c u t i v e 25 meter v a l u e s of e were averaged f o r each drop and then averaged over a l l of the d r o p s . For the 850 meter da ta p o i n t , 45 LOG t (W/m1) 2200 ' ' ' 1 1—' ' 1 — * * * ' — " '—•—'—• • i • i i Figure 14 - a)100 meter v e r t i c a l averages of buoyancy frequency averaged over a l l of the WESPAC drops. b)100 meter v e r t i c a l averages of turbulent I kinetic energy dissipation averaged over a l l of I ' the WESPAC drops. 46 47 there were 20 p o i n t s in the depth range 825<z<875 mete rs . The average va lue was 1.0xl0~ 6W/m 3 and the s tandard d e v i a t i o n about the mean was 1.8x10" 6 W/m 3 . 90% of the p o i n t s had v a l ues of e < 1.7xl0" 6 W/m 3 and 90% had va lues > 2x10" 7 W/m 3 . The midd le 80% of the data ,~~"tTTen, i s bounded by 2x 1 0" 7<e< 1 . 7x 1 0" 6 W/m 3. These u n c e r t a i n t y or data spread bars are drawn in F i g u r e 15. A p o i n t to note concerns the d i f f e r e n c e s i n a v e r a g i n g . A l though the ave rag ing done in F i g u r e 15 i s over depth i n t e r v a l s as compared to N - i n t e r v a l s in F i g u r e 13, both e x h i b i t a s t rong p o s i t i v e s l ope which i s not much d i f f e r e n t from 1. As noted by Lueck , Crawford and Osborn (1983) , the o b s e r v a t i o n tha t e/N may be a cons tan t fo r midocean measurements at any l o c a t i o n i s not we l l e s t a b l i s h e d . However, the re e x i s t arguments based on i n t e r n a l wave dynamics which suggest tha t the ra te of energy l o s t by the i n t e r n a l wave f i e l d to the t u rbu l ence 7 s c a l e s w i th N where y i s between 1 and 2. WKB s c a l i n g i n d i c a t e s tha t the t o t a l energy in the i n t e r n a l wave f i e l d (TE) i s l i n e a r l y p r o p o r t i o n a l to N (Munk( l98 l ) ) and , i f r i s the decay t ime s c a l e of the i n t e r n a l wave f i e l d , then e = T E / T a N 1 . As w e l l , a recent d i s c u s s i o n by Ga rge t t and Hol loway(1984) suggests tha t 7 i s 1 or 1.5 depending on the a p p r o p r i a t e s c a l i n g f o r the v e r t i c a l v e l o c i t y v a r i ance of the i n t e r n a l wave f i e l d . V a r i o u s a c c o u n t s , t hen , suggest t h a t , when e has a s t rong p o s i t i v e dependence on N, the source of the t u r b u l e n t energy i s the i n t e r n a l wave f i e l d . T h i s w i l l be pursued f u r t h e r in the f o l l o w i n g c h a p t e r . 48 To complete the p r e s e n t a t i o n of the e-N s t a t i s t i c s , a compar ison i s made wi th other data s e t s . Lueck , Crawford and Osborn ( l983 ) made a, s e r i e s of measurements over the c o n t i n e n t a l s l ope of Vancouver I s l and in May, 1980. The data were c l u s t e r -averaged over the depth i n t e r v a l s l00-200m, 200-500m, and 500m-bottom. These c l u s t e r averages are p l o t t e d as s o l i d t r i a n g l e s in F i g u r e 16. Garge t t and O s b o r n ( l 9 8 l ) averaged e fo r N 2cph and these are the open c i r c l e s in F i g u r e 16. The upper and lower c i r c l e s are j o i n e d by a bar i n d i c a t i n g the range w i t h i n which l i e s the t rue va lue of e. S o l i d c i r c l e s are WESPAC va lues reproduced from F i g u r e 15 and s o l i d squares are from PEQUOD (only the da ta below 300m i s shown from PEQUOD). A l i n e has been drawn wi th a s lope of 1 and the same i n t e r c e p t as F i g u r e 15. A compar ison of the data se t s i s best accompl i shed by c o n s i d e r i n g the cons tan t of p r o p o r t i o n a l i t y of each e s t ima ted from e = a 0 N . T h i s was done in a r e l a t i v e l y crude manner by ' e y e b a l l i n g ' a bes t f i t l i n e of s lope = 1 through each da ta set (Lueck, Crawford and Osborn ( l983) and Garge t t and O s b o r n ( l 9 8 l ) have a l r e a d y done t h i s f o r t h e i r r e s p e c t i v e data s e t s ) . These r e s u l t in the f o l l o w i n g va lues of a 0 ( i n u n i t s of m 2 / s e c 3 • s e c ) : WESPAC - 2 . 2 x 1 0 " 7 PEQUOD - 1.4x10" 7 Lueck , Crawford and Osborn ( l983) - 1 . 8X1 0 ' 7 Garge t t and O s b o r n ( l 9 8 l ) - 4 . 0 X 1 0 - 7 . 49 LOG N (rad/sec) Log-log p l o t s of e vs N f o r the four data sets i n the legend and f u r t h e r d i s c u s s e d i n the t e x t . The WESPAC and Sargasso Sea values are gr e a t e r than those from PEQUOD and the Vancouver I s l a n d s l o p e . 50 The Ga rge t t and O s b o r n ( l 9 8 l ) i n t e r c e p t was c a l c u l a t e d us ing an a d d i t i o n a l data p o i n t l y i n g o u t s i d e the range of the present p l o t . A l though the u n c e r t a i n t y i n the es t imate of a 0 i s q u i t e l a r g e , i t i s c e r t a i n l y i n d i c a t i v e of the r e l a t i v e l e v e l s of e from each r e g i o n . From F i g u r e 16, i t i s q u i t e apparent that the WESPAC l e v e l s are g r ea t e r than both of the PEQUOD and Vancouver I s l and s l ope l e v e l s and tha t the PEQUOD data set i n c l u d e s the s m a l l e s t averages of t h i s p resen t p l o t . In the s p i r i t of Lueck, Crawford and Osborn (1983) , t hen , F i g u r e 16 i s o f f e r e d as an appended g u i d e l i n e wi th which to compare other v a lues of e/N in the ocean . A l though i t i s c l e a r l y imposs ib l e to determine an i n d i s p u t a b l e p a r a m e t e r i z a t i o n of the dependence of e and N wi th the a v a i l a b l e d a t a , i t i s none the l ess encourag ing and perhaps s u r p r i s i n g that the data behave so w e l l . An omnipresent problem in r e l a t i n g the r a p i d l y changing t u rbu l ence parameters to the 'mean' s t a t e i n v o l v e s the a ve rag ing p rocess r e q u i r e d to get a s t a b l e e s t ima te of e. With a few s p a t i a l snapshots of the v e r t i c a l f i e l d of the t u rbu l ence at numerous l o c a t i o n s , t h i s c r i t e r i o n i s met on l y to a minimal deg ree . However, there do e x i s t d i s t i n c t t r ends and these shou ld not be i g n o r e d . With the r e l a t i o n s h i p i n f e r r e d from F i g u r e s 13, 15 and 16, and the apparent agreement w i th the i n t e r n a l wave s c a l i n g , one i s encouraged to examine f u r t h e r the dependence of the tu rbu lence on the i n t e r n a l wave f i e l d . 51 V. A MODEL OF TURBULENCE IN AN INTERNAL WAVE FIELD In t h i s c h a p t e r , an attempt i s made to model the d i s t r i b u t i o n of shear i n s t a b i l i t y in a f i e l d of i n t e r n a l waves. An o f t - q u o t e d c o n d i t i o n fo r s t a b i l i t y in a s t r a t i f i e d shear f low r e q u i r e s tha t the R i chardson number, Ri = N 2 / ( u ' ) 2 , > 1/4 everywhere i n the f low (Turner (1973) d i s c u s s e s expe r imenta l r e s u l t s and M i l e s ( l 9 6 1 ) makes the i n i t i a l d e r i v a t i o n fo r a v a r i a b l y s t r a t i f i e d water column which expe r i ences a mean s h e a r ) . For the purposes of t h i s c h a p t e r , u' r e f e r s to the v e r t i c a l d e r i v a t i v e of the mean f low ( that i s , the mean s h e a r ) . L o c a l l y the R i cha rdson number may be reduced to a va lue l e s s than 1/4 due to the s u p e r p o s i t i o n of i n t e r n a l wave shears and one may i n f e r tha t Ri < 1/4 i s an i n d i c a t o r of i n s t a b i l i t y in the i n t e r n a l wave f i e l d ( b r e a k i n g ) . Indeed, p r o v o c a t i v e ev idence of a R i cha rdson number c u t o f f at 1/4 in an i n t e r n a l wave f i e l d i s p r o v i d e d by E r i k s e n ( l 9 7 8 ,Tab le 1) . E r i k s e n d e s c r i b e s an a lmost f l a t d i s t r i b u t i o n of the f requency of occu r rence of R i cha rdson number ve r sus a r c t a n ( R i ) down to Ri = 1/4, below which the re i s an abrupt c u t o f f . E r i k s e n makes the f o l l o w i n g remarks : 'The measurements may be i n t e r p r e t e d as ev idence tha t b reak ing does o c c u r . The remarkable expe r imenta l r e s u l t i s tha t a c u t o f f at a R i cha rdson number of 1/4 e x i s t s over v e r t i c a l s c a l e s of the o rder of a few m e t e r s ' . To s imu l a t e the e f f e c t of a f i e l d of randomly superposed i n t e r n a l waves, an es t imate f o r the shear i s r e q u i r e d to e va lua t e the R i cha rdson number from a g iven d i s t r i b u t i o n of N (the d i s t r i b u t i o n used i s the p r o f i l e of F i g u r e 10). With the 52 a i d of the G a r r e t t and Munk e x p r e s s i o n s fo r the i n t e r n a l wave dynamics , the shea r , u ' , i s c a l c u l a t e d from the mean.square of the p r e d i c t e d shear spec t rum. A R a y l e i g h d i s t r i b u t i o n i s used to es t imate the l i k e l i h o o d of occu r rence of a shear of s u f f i c i e n t magnitude to c r e a t e a R i chardson number sma l l enough f o r i n s t a b i l i t y . T h i s i s then compared to the d a t a . 5.1 I n t e r n a l Wave Energy P r o f i l e s On the b a s i s of l i n e a r theory and the a v a i l a b l e o b s e r v a t i o n s , G a r r e t t and Munk( l972, 1975, 1979) have s y n t h e s i z e d a model of the complete frequency-wavenumber spectrum of i n t e r n a l waves. T h i s model has w i ths tood the t e s t of v i go rous exper imenta l and t h e o r e t i c a l i n v e s t i g a t i o n s i n ce i t s concep t i on and now p r o v i d e s a s t andard fo r the compar ison of new r e s u l t s . T h i s s tudy w i l l beg in wi th an es t imate of the energy in the i n t e r n a l wave f i e l d c a l c u l a t e d from the measured buoyancy f r equency , N, and the GM s p e c t r a l l e v e l s . Munk( l98 l ) g i v e s a recent d i s c u s s i o n of i n t e r n a l waves and sma l l s c a l e p r o c e s s e s . The most recent forms of the GM s p e c t r a are p resen ted as f o l l o w s , w i th F the spectrum of v e r t i c a l S d i s p l a c e m e n t s , F tha t of h o r i z o n t a l v e l o c i t i e s and F = tha t of u e the t o t a l wave energy ; F (u , j ) = b 2 N 0 (g ) 2 -f 2 )E (o ) , j ) 5 N <P~ (5 .1) F = b 2 N 0 N(gj 2 + f 2 ) E ( g j , i ) (5 .2) F •(«, j ) = M F +N 2F ) = b 2 N 0 NE(co, j ) e 2 u $ (5 .3) 53 where j i s the v e r t i c a l mode number, b i s the e - f o l d i n g s c a l e of N(z) and i s taken to be 1300 meters , N 0 = 5 .2X10~ 3 s e c " 1 (3 cphl i s the s u r f a c e - e x t r a p o l a t e d buoyancy f requency used by GM, and f = 7 . 3X1 0 " 5 s e c ' 1 i s the C o r i o l i s f r equency at 30° l a t i t u d e . E (CJ,J ) i s the d imens ion l e s s energy d e n s i t y d e f i n e d by ~~ E (CJ,J ) = B (u )H ( j ) E , B(w) = 2 f/U7T(c j 2 -f 2 )^ ) , N J" B(o))da> = 1 , f H(j) = ( j 2 + J o 2 ) - V ( 2 ( j 2 + j 0 2 ) - 1 ) , 1 00 LH(j) = 1, 1 where j 0 = 3 i s a mode s c a l e number. E i s the d i m e n s i o n l e s s i n t e r n a l wave ' energy parameter ' and i s set at 6 . 3 X 1 0 ' 5 . Ex t ens i v e measurements have proven E to be remarkably u n i v e r s a l ( to w i t h i n a f a c t o r of two) . The c o n t r i b u t i o n to F of the e v e r t i c a l v e l o c i t y spectrum i s c o n s i d e r e d to be n e g l i g i b l e by Munk(l98l) compared to those of F and F (compare F i g u r e s 7 and $ u 14 of E r i k s e n ( 1 9 7 8 ) ) . The mean-square q u a n t i t i e s are , t h e n , ( fo r f<<N) <$ 2> = JdcoIF (w,j) = J_b 2 EN 0 = 53N 0 [m 2] $ 2 N N <u 2> = /dcjZF = 3b 2 EN 0 N = 44x10" f l N [m 2 /s 2 ] U • 2 N 0 from which the p o t e n t i a l ( P E ) and k i n e t i c ( K E ) e n e r g i e s are c a l c u l a t e d as a f u n c t i o n . o f dep th , z ; 54 Figure 17 - Plots of i n t e r n a l wave p o t e n t i a l energy(PE), k i n e t i c energy(KE) and t o t a l energy(TE) calculated from the equations in section 5.1 of the text and) the buoyancy frequency p r o f i l e of Figure 10. 4 / 55 PE = J_N2 ( z )<$ 2 (z)> 2 KE = J_<u2 (z)> 2 TE = PE+KE = { 5 3 N o / 2 + 2 2X1 0 - * / N O } N [ m 2 / s e c 2 ] (5.4) Us ing the N p r o f i l e in F i g u r e 10, which r e p r e s e n t s the WESPAC drops south of the r i n g , and the c a l c u l a t i o n of <u 2> and <$ 2> g i ven above, PE, KE and TE are p l o t t e d in F i g u r e 17 and w i l l be r e f e r r e d to l a t e r . 5.2 I n t e r n a l Wave Shears The mean-square va lue of the v e r t i c a l shear due to the i n t e r n a l wave f i e l d i s : < ( u ' ) 2 > = <(3u) 2 > + <(3v) 2 > = m2<u2:j-3z 3z The d i s p e r s i o n r e l a t i o n fo r l i n e a r i n t e r n a l waves i s ( O l b e r s , 1 9 8 3 ) : m2 = k 2 (N 2 -g; 2 ) , (5.5) u 2 - f 2 where m i s the v e r t i c a l and k the h o r i z o n t a l wavenumber. The v e r t i c a l v e l o c i t i e s may be expressed as w = a s i n ( m z ) . For a cons tan t va lue of N the requirement that w = 0 at the ocean bot tom, H, y i e l d s mH = jit, j = 1 , 2 , . . . When N i s a f u n c t i o n of z , a WKB s o l u t i o n i s r e q u i r e d and the c o n d i t i o n mH = J7r becomes (Munk , l98 l ) m = J 7 r / ( N 2 - c j 2 ) . (5.6) b v/N 0 2-w 2 T h i s form of the boundary c o n d i t i o n assu res a t t e n u a t i o n o u t s i d e the waveguide. Because of the r e l a t i v e i n t r a c t a b i l i t y of t h i s form fo r m, the l i m i t i n g case of co«H i s taken so tha t 56 m = JTTN . (5.7) bN 0 T h i s s i m p l i f i c a t i o n i s made w i th some concern about the c o n t r i b u t i o n of h igh f requency waves t rapped in the t h e r m o c l i n e . However, there i s ev idence to i n d i c a t e tha t a l t hough the shear spectrum i s r e l a t i v e l y f l a t in the v e r t i c a l wavenumber domain (Garget t et al.(l98l)), i t s l opes downward from low to h igh f r e q u e n c i e s (McComas and Muller(1981)) imp l y ing tha t the i n t e r n a l wave shear i s concen t r a t ed in the low f requency modes, and tha t the s i m p l i f i c a t i o n suggested here may be j u s t i f i e d . The e x p r e s s i o n fo r the mean-square shear becomes <(u' ) 2> = Jdcj2m2F (CJ, j ) u wh ich , w i th (5.2) and (5.7) i s N < ( u ' ) 2 > = 2 7r f EN 3 Z j 2 H ( j ) J* o r 3 (co2 + f 2 ) dc j . N 0 f / ( co z -f z ) The i n t e g r a l i s 37r/4f and the summation L j 2 H ( j ) = j + J , where J = 2.13 and i t i s r e q u i r e d that j + , the upper l i m i t to the f i n i t e es t imate of 2 j 2 H ( j ) , be >> j 0 = 3. Thus , <(u' ) 2> = 3 T T 2 N 3 E J + J . (5.8) 2 N 0 The upper l i m i t to the v e r t i c a l mode number, j + , i s a c r i t i c a l parameter which i s r e l a t e d to the v e r t i c a l wavelength by m+ = 27r/X+ = j + 7TN. bN 0 A parameter study showing the dependence of X+ on j + f o r a range of r e a l i s t i c N va lues i s shown in F i g u r e 18. The cho i c e of j + must concur w i th the upper wavenumber c u t o f f of the i n t e r n a l 57 0 400 800 P l o t of the dependence of the upper wavelength c u t o f f , X +, on the upper l i m i t , of the v e r t i c a l mode number f o r four d i f f e r e n t v a l u e s of buoyancy frequency, N. A h o r i z o n t a l l i n e a t X, = 10 meters i s drawn f o r comparison t o the * 10 meter break i n the v e r t i c a l shear spectrum found "by G a r g e t t et a l ( l 9 8 1 ) . 58 wave spect rum. A composi te spectrum of v e r t i c a l shear in the ocean measured n e a r l y s imu l t aneous l y by th ree separa te v e l o c i t y p r o f i l e r s w i th d i f f e r i n g s p a t i a l bandwidths was compi l ed by Ga rge t t et a l . ( l 9 8 l ) . For the s p e c t r a p r e s e n t e d , there i s a c o n s i s t e n t break in s lope at about 10 meters s e p a r a t i n g the s p e c t r a l range which s c a l e s w i th i n t e r n a l wave parameters from tha t which s c a l e s w i th buoyancy pa ramete rs . Gregg( l977) found tha t the break in s lope of the temperature g r a d i e n t s p e c t r a fo r the i n t e r n a l wave bandwidth was a l s o near 10 mete rs . A h o r i z o n t a l r e f e r e n c e l i n e at 10 meters i s drawn in F i g u r e 18. For j + < 100, the wavelengths become many t imes g r ea t e r than 10 mete rs . For j + > 600, the wavelengths fo r N = 2-4 cph converge at < 5 mete rs . Choos ing 200^j + ^400 a l l ows v e r t i c a l wavelength c u t o f f s at 4 cph of 10(j + = 200) and 5 m e t e r s ( j + = 400) and at 2 cph of 20( j + =200) and 10 me te r s ( j + =400 ) . F u r t h e r c a l c u l a t i o n s are r e s t r i c t e d to t h i s range of j + . 5.3 The D i s t r i b u t i o n Of Shear And R i cha rdson Number In order to c a l c u l a t e the l i k e l i h o o d of sma l l R i cha rdson number in an ocean of known N, the i n t e r n a l waves must be superposed in a r e a l i s t i c f a s h i o n i n order to es t imate the t o t a l shear in the f i e l d . Desaubies and Smi th ( l982) use Gauss ian s t a t i s t i c s f o r shear and s t r a t i f i c a t i o n to deve lop a d i s t r i b u t i o n f u n c t i o n fo r R i . Indeed, Evans ( l982 ) suggests tha t shear measurements from the main the rmoc l ine in the Nor th A t l a n t i c show no depar tu re from a normal d i s t r i b u t i o n based on a X 2 t e s t . However, the data were not t e s t e d fo r any o ther • d i s t r i b u t i o n and there i s no r eason , a p r i o r i , to r e j e c t the 59 hypo thes i s tha t the data f o l l ow another d i s t r i b u t i o n ( fo r example, the Ray l e i gh d i s t r i b u t i o n ) . A p r a c t i c a l c o n s i d e r a t i o n i s t h a t , a l t hough an es t ima te fo r the mean-square va lue of the shear i s e a s i l y ob ta ined (as we s h a l l s e e ) , there i s no obv ious way to get an es t imate fo r the mean va lue of the shear from the a v a i l a b l e d a t a . Both of these va lues are r e q u i r e d f o r the normal d i s t r i b u t i o n but on l y the mean-square va lue fo r the Ray l e i gh d i s t r i b u t i o n . There are a l s o some s o l i d grounds fo r assuming the l a t t e r d i s t r i b u t i o n as a d e s c r i p t o r of the s t a t i s t i c s of a l a rge number of independent shear v e c t o r s . The u s e f u l p r o p e r t y of the Ray l e i gh d i s t r i b u t i o n (which was used by Longuet-Higg ins (1952) to d e s c r i b e the s u p e r p o s i t i o n of a f i e l d of s u r f a c e waves) i s tha t i t d e s c r i b e s the d i s t r i b u t i o n of amp l i tudes of a l a rge number of independent two-dimens iona l v e c t o r s hav ing random phase (Ma i s e l ( 1971 ) ) . The form of the p r o b a b i l i t y d e n s i t y f u n c t i o n i s p(x) = x exp [- j _x 2 /a 2 ], f o r x > 0 ~a7 2 = 0, f o r x < 0 where a 2 = x~2"/2 and x"7 i s the mean square v a l u e . The cumu la t i ve x d e n s i t y f u n c t i o n i s J* p (x )dx , or — 0 0 p ' (x ) = 1 - e x p [ - l x 2 / a 2 ] . 2 In terms of i n t e r n a l wave shea r , p ' ( u ' ) = 1 - e x p [ - ( u ' ) 2 / < ( u ' ) 2 > ] d e s c r i b e s the p r o b a b i l i t y tha t the shear i s l e s s than u ' . The upper l i m i t to the c o r r e s p o n d i n g R i chardson number, Ri = 60 N 2 / ( u ' ) 2 r e q u i r e s u' to have a maximum va lue d e s c r i b e d by 1 p ' ( u ' ) so tha t P r (R i ) = 1-p ' (u ' ) = e x p [ - ( u ' ) 2 / < ( u ' ) 2 > ] d e s c r i b e s the p r o b a b i l i t y that the R i chardson number i s l e s s than some v a l u e , R f H With ( 5 . 8 ) , P r (R i ) =. exp[ -2NQ(U' ) 2 ] (5 .9) 3 i r 2 N 3 E j t J The squared shear co r r e spond ing to an a r b i t r a r y cons tan t Ri c 0 - 1 = N 2 / ( u ' ) 2 i s ( u ' ) 2 = c 0 N 2 so tha t P r ( R i < c 0 - 1 ) = exp[ - 2 N 0 c 0 ] . (5.10) 3 i r 2 N E j + J -1/N The e x p r e s s i o n (5.10) g i v e s an e dependence which i s an i n t u i t i v e l y a c c e p t a b l e c h o i c e . T h i s i m p l i e s tha t the l i k e l i h o o d of l o c a l shear i n s t a b i l i t y in the i n t e r n a l wave reg ime, fo rmu la ted in terms of sma l l R i cha rdson number, i s g r ea t e r i n r e g i o n s of l a r g e N, such as the t h e r m o c l i n e , than i n r eg ions of sma l l N. Munk( l98 l ) t akes a d i f f e r e n t approach to determine tha t ' l a y e r s of l a r g e s t g r a v i t a t i o n a l s t a b i l i t y ( l a r g e s t N) a re a l s o l a y e r s of l a r g e s t shear i n s t a b i l i t y ( l a r g e s t u ' / N ) ' . C a r r i e d one s t ep f u r t h e r , i f the shear i n s t a b i l i t y r e s u l t s i n wave b reak ing and the b reak ing causes t u r b u l e n c e , then one expec t s to f i n d the type of e-N dependence found i n the p r e v i o u s c h a p t e r . 61 5.4 Comparison With The Data In F i g u r e 19 v e r t i c a l p r o f i l e s of P r (R i< l ) and P r (R i< l/4 ) c a l c u l a t e d from (5 .10 ) .have been p l o t t e d fo r j + = 200, 300 and 400. The va lues of N used are from F i gu re 10 to s imu la te the WESPAC reg ion south of the c o l d core r i n g . ' The c r i t i c a l dependence on j + u n f o r t u n a t e l y makes i t imposs i b l e to a c t u a l l y determine a number fo r P r (R i ) at any p o s i t i o n . On t h i s s c a l e , the cu rves of P r (R i< l/4 ) are a lmost i n d i s t i n g u i s h a b l e from the r i g h t hand a x i s except fo r h i g h N and j + . The shape of the c u r v e s , however, i n d i c a t e the s t r o n g N-dependence. To compare wi th the p r o b a b i l i t y d i s t r i b u t i o n of sma l l R i , an es t ima te was made of the f r a c t i o n of the water column that was t u r b u l e n t over 50 meter depth i n t e r v a l s ( c a l l t h i s PCT fo r pe rcen t t u r b u l e n t ) f o r the p r o f i l e s 2 , 4 , 5 , 6 , 8 south of the r i n g . The c r i t e r i o n used fo r t u rbu l ence d e t e c t i o n was a c r i t i c a l t h r e s h o l d l e v e l of d i s s i p a t i o n , e . The t h i c k n e s s of pa tches c w i th s u c c e s s i v e independent e s t ima te s of e which were > e were c added to c a l c u l a t e the t h i c k n e s s of i n d i v i d u a l p a t c h e s . Over the 50 meter i n t e r v a l the t u r b u l e n t f r a c t i o n i s Z (pa t ch t h i c k n e s s e s ) / 5 0 . The s c a l e used in F i gu re 20 fo r PCT i s the same as in F i g u r e 19. The c h o i c e of e i s s u b j e c t i v e and i s , a t c the low end, l i m i t e d by the no i se l e v e l of the i n s t r u m e n t a t i o n . A l t h o u g h i t i s not c l e a r how to i n t e r p r e t the magnitudes of F i g u r e 20 (as we l l as F i g u r e 19) , the shapes of the v e r t i c a l p r o f i l e s appear not to be a f f e c t e d by the c h o i c e of 62 P r ( s m a l l R i ) 0.15 l 0.30 l 0.45 0.60 200 300 400 200 H u (0 JD TJ « W C/J KJ (X CU 6 0 0 H I I / I 1 / M / /•/ / / ii 1/ •1.000 Hi 200 300 J.-400 Pr(Bi<1) Pr(»i<l/4) 1400 F i g u r e 19 - V e r t i c a l p r o f i l e s of Pr(Ri<1) and Pr(Ri<1/4) c a l c u l a t e d from eq u a t i o n (5.10) and the buoyancy frequency p r o f i l e of F i g u r e 10 f o r j t = 200, 300, 400. 6 3 PCT 0.60 1400 F i g u r e 20 - V e r t i c a l p r o f i l e s of f r a c t i o n of t u r b u l e n t water column (PCT) estimated f o r two d i f f e r e n t t h r e s h o l d l e v e l s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n . The h o r i z o n t a l s c a l e i s i d e n t i c a l to that i n F i g u r e 19. 64 e . Two va lues of e are used in F i g u r e 20. 1 0 " 6 W/m3 i s c c 3 x ( i n s t r u m e n t a l no i se l e v e l = 3 X1 0 " 7 W/m3) wh i l e 3 X 1 0 ' 6 W/m3 i s I 0x (no i se l e v e l ) . The d i f f e r e n c e in PCT u s i n g these two t h r e s h o l d l e v e l s v a r i e s by f a c t o r s of 2-6. But both curves i n d i c a t e that the water column i s s i g n i f i c a n t l y l e s s t u r b u l e n t (by a f a c t o r of 4-5) in the low N reg ion above the the rmoc l i ne than in the t he rmoc l i ne i t s e l f and t h i s dec reases below the t h e r m o c l i n e . Presumably , o ther c h o i c e s of e would g i ve s i m i l a r c shapes ( u n t i l such a low va lue were chosen tha t i t would be n o i s e - s a t u r a t e d or such a h igh va lue chosen tha t i t would not e x i s t in the set of measured v a l u e s ) . F i g u r e 20 may be compared to F i gu re 11(the averaged e over 25 meter depth i n t e r v a l s f o r the drops south of the r i n g ) . In g e n e r a l , l o w PCT concurs w i th low e and v i c e - v e r s a . Anomal ies to t h i s t r end are i n d i c a t i v e of e i t h e r r e l a t i v e pa t ch t h i c k n e s s or r e l a t i v e t u rbu l ence magni tude. Regions where the r a t i o e/PCT i s g r ea t e r than normal l e v e l s are i n t e r p r e t e d as hav ing r e l a t i v e l y t h i n and/or s t r o n g l y t u r b u l e n t pa tches and where the r a t i o e/PCT i s s m a l l e r , the patches are t h i c k e r and/or more weakly t u r b u l e n t . A rough t e s t was d e v i s e d to i n d i c a t e the degree to which the shape of the model (5.10) agrees wi th the measured data p l o t t e d in F i g u r e 20. S ince i t has been acknowledged tha t an a ve r ag ing problem e x i s t s w i th the l i m i t e d a v a i l a b l e data s e t , i t was dec i ded tha t i t was a c c e p t a b l e to smooth the p r o f i l e s in 65 F i g u r e 20 w i th a t h r ee-po in t runn ing ave rage . In order to compare these d i r e c t l y w i th each o ther and wi th the p r o f i l e s of P r (R i<1/4 ) , each p r o f i l e was no rma l i z ed us ing the peak va lues neares t to the t h e r m o c l i n e . The s c a l i n g f a c t o r s and the t i t l e s g i ven to each data set a re g iven in Tab l e 1 . With the s c a l i n g f a c t o r , one may recover the o r i g i n a l v a l ues of Pr (Ri<1/4) and PCT. Data Set T i t l e S c a l i n g F a c to r e = 3 X1 0 " 6 PCT1 .38 @ 700m c e = I 0 " e PCT2 . .087 @ 500m c P r (R i<!/4 ) from (5.10) j + = 200 PR200 6 . 9 x 1 0 " 7 @ 500m j + = 300 PR300 1 . 7X1 0 " 5 @ 500m j + = 400 PR400 8 . 3 X 1 0 " 5 @ 500m Tab le 1 - S c a l i n g f a c t o r s f o r n o r m a l i z a t i o n and t i t l e s g i ven to data s e t s . P r o f i l e s of no rma l i zed PCT and P r (R i< l/4 ) are shown in F i g u r e 21. Above 600 mete rs , a depth which i s s t i l l in the main t h e r m o c l i n e , the agreement wi th the da ta i s very good. However, below about 700 meters the agreement i s p o o r . Sample c o r r e l a t i o n c o e f f i c i e n t s were c a l c u l a t e d as r = Z(x -x) (y -y)/ (/ (Z(x - x ) 2 ) / ( Z ( y -y ) 2 ) ) i i i i i i i and these a re l i s t e d in Tab le 2. Over the e n t i r e depth i n t e r v a l 100-1250 me te r s , the c o r r e l a t i o n s are poo r . But over the i n t e r v a l 100-600 meters the c a l c u l a t e d c o r r e l a t i o n c o e f f i c i e n t 66 SETS PCT1,PR200 PCT1,PR200 PCT2,PR200 PCT2,PR200 PCT1,PR300 PCT1,PR300 PCT2,PR300 PCT2,PR300 PCT1,PR400 PCT1,PR400 PCT2,PR400 PCT2,PR400 DEPTH RANGES 100-1250m 100-600 100-1250 100-600 100-1250 100-600 100-1250 100-600 100-1250 100-600 100-1250 100-600 .34 .89 .94 .37 ; .90 . 12 .94 . .37; .89 .11 .94 Table 2 - Correlation c o e f f i c i e n t , r, calculated for the pairs of data sets defined in Table 1 and over the depth ranges s p e c i f i e d . 67 V e r t i c a l p r o f i l e s of n o r m a l i z e d PCT from F i g u r e 20 and n o r m a l i z e d P r ( R i < l / 4 ) from F i g u r e 19. The p r o f i l e s were n o r m a l i z e d by the maximum v a l u e s n e a r e s t the t h e r m o c l i n e . The n o r m a l i z a t i o n f a c t o r s are l i s t e d i n T a b l e 1. 6 8 i s b e t t e r than . 8 9 and i s . 9 4 between the set PCT2 ( e = 3 x l 0 ~ 6 c W/m3) and each of the f am i l y PR200, PR300 and PR40.0. 5.5 D i s c u s s i o n A number of assumpt ions which were made in o rde r to d e r i v e (5.10) shou ld be emphas ized. F i r s t of a l l , the GM model upon which t h i s model r e l i e s was deve loped w i th the use of WKB t heo r y , r e q u i r i n g a s l ow l y v a r y i n g N ( z ) . The degree to which N ( z ) in the the rmoc l i ne i s s low ly v a r y i n g i s open to q u e s t i o n . Second ly , i t was assumed tha t the h i g h f requency i n t e r n a l wave components (u> - N ) do not c o n t r i b u t e s i g n i f i c a n t l y to the shear v a r i a n c e . T h i r d l y , a major problem i s the h igh s e n s i t i v i t y of the model to j . and the i n a b i l i t y to make a c l e a r c h o i c e fo r t h i s parameter . The o ther c h o i c e to be made i s tha t of e c wh ich , a l t hough i t does not a f f e c t the model , does a f f e c t the data to which i t i s compared. F i n a l l y , i t was at tempted to c o r r e l a t e two q u i t e d i f f e r e n t pa ramete r s . PCT i s an a c t u a l measure of the f r a c t i o n of the water column which has e l e v e l s g r ea t e r than a d e t e c t o r t h r e s h o l d , e . T h i s i s compared to the c p r o b a b i l i t y tha t the water column at any p o i n t meets the c o n d i t i o n f o r shear i n s t a b i l i t y ( P r (R i< l/4 ) ) due to the l o c a l s u p e r p o s i t i o n of i n t e r n a l waves. The i m p l i c i t assumpt ion made r e q u i r e s the i n s t a b i l i t y to man i f es t i t s e l f as t u r b u l e n c e at s c a l e s which may be measured by Camel I I I . The model i s unable to p r e d i c t the magnitude of PCT. In f a c t , the u n c e r t a i n t y in the cho i c e of j + a lone r e s u l t s in a 69 s c a l i n g f a c t o r d i f f e r e n c e of >100 between j + = 200 and 400. As w e l l , below about 700 mete rs , n e i t h e r shape nor magnitudes of PCT or P r (R i< l/4 ) ag ree . As has been ment ioned , though, t h i s may be in pa r t due to the u n r e p r e s e n t a t i v e i n t ense e l e v e l s e x h i b i t e d in drop 2 which i s d i s c u s s e d in the p r e v i o u s c h a p t e r . (Perhaps a g r ea t e r number of drops in the r eg i on would have reduced the i n f l u e n c e of drop 2 on the a v e r a g i n g . C o n v e r s e l y , s i n c e drop 2 appears to be q u i t e d i f f e r e n t in c h a r a c t e r from drops 4 , 5 , 6 , 8 i t may be argued tha t i t c o u l d be dropped from the a ve rag ing scheme in order to improve the behav iour of the a ve r ages . T h i s s u b j e c t i v e tamper ing c o u l d not be accepted on e t h i c a l g rounds , however, and i t was dec ided to l i v e wi th the shor tcomings of the l i m i t e d da ta s e t , w i th the unders tand ing tha t the ave rag ing i s not l i k e l y ' c o r r e c t ' ) . With due regard to the i nadequac i es s t a t e d above , a number of p e r t i n e n t comments may be made in l i g h t of the l i m i t e d -1/N success of the model . Appa ren t l y the e behav iour of the model i s f o l l owed by PCT in at l e a s t pa r t of the water co lumn. T h i s p o s i t i v e dependence of PCT on N agrees q u a l i t a t i v e l y wi th the r e l a t i o n e a N 1 suggested in the p r e v i o u s c h a p t e r . The f a c t that the magnitudes of PCT and Pr(Ri<1/4) d i f f e r so g r e a t l y i s i n pa r t due to the c h o i c e s of j + and e . However, c i t seems c l e a r that the shear i n s t a b i l i t i e s occur on l y r a r e l y (the s c a l i n g f a c t o r f o r PR400 i s 1000 t imes sma l l e r than that of PCT2) . To attempt to match PCT and P r ( R i < l / 4 ) , one might i n f e r e i t h e r a l a r g e r j + ( f o r which we have a l r e a d y determined would 70 g i ve an u n r e a l i s t i c a l l y h igh wavenumber c u t o f f in l i g h t of Ga rge t t et a l . ( l 9 8 l ) ) or a d e t e c t o r t h r e s h o l d l e v e l , e , which c i s g r ea t e r by some f a c t o r s of 10. T h i s l eads to the q u e s t i o n of the a c t u a l amount of energy which i s r e l e a s e d by a s i n g l e b reak ing event and the mechanism by which the energy i s conve r t ed to t u r b u l e n c e . For t h i s , no answers are g i v e n . However, i t i s noted t h a t , from the e n t i r e WESPAC data s e t , which c o n s t i t u t e s 13,070 meters of d a t a , r e p r e s e n t i n g approx imate l y 7,000 independent e s t ima tes of e, there i s not a s i n g l e occu r rence of e > 10" " W/m3 and, below 100 mete rs , < 1% of the data i n d i c a t e s e > 1 0 " 5 W/m3 (which t r a n s l a t e s to a l o t of ze roes in a v e r t i c a l p r o f i l e such as F i g u r e 20 ) . Hence, the re i s no ev idence of the d i s s i p a t i o n l e v e l s r e q u i r e d to s a t i s f y the above argument. In s h o r t , i t appears that the p r o b a b i l i t y of a shear i n s t a b i l i t y l e a d i n g to a b reak ing event es t ima ted from (5.10) i s much s m a l l e r than the measured l e v e l s of PCT in the ocean . An i n t e r e s t i n g s i d e l i n e i n v o l v e s the q u e s t i o n 'how long does an i n t e r n a l wave l a s t ' . I f the waves are l o n g - l a s t i n g , then energy may propagate long d i s t a n c e s , thereby e f f e c t i v e l y d i s p e r s i n g energy in the i n t e r n a l wave f i e l d f a r from a l o c a l source and, as suggested by G a r r e t t and Munk( l979) , may h e l p to e x p l a i n why the oceans are ' f i l l e d to a lmost the same e q u i l i b r i u m s p e c t r a l l e v e l e ve r ywhe re ' . The es t imate made fo r the i n t e r n a l wave energy (5.4) and the r e l a t i o n e = a 0 N may be used to es t ima te the time s c a l e r e p r e s e n t i n g the decay of the i n t e r n a l wave spec t rum, p rov ided tha t e, which i s a parameter of 71 the t u r b u l e n c e , i s r e p r e s e n t a t i v e of the energy l o s t from the i n t e r n a l wave f i e l d . D e f i n i n g r = TE/e , T = {53N o /2+22x10- 4 /N o } /a o . Lueck , Crawford and Osborn ( l983) used a va lue of .0045 rad/sec f o r N 0 . F i g u r e 14a i n d i c a t e s that a va lue of N 0 e x t r a p o l a t e d from > 900 meters from WESPAC (to a v o i d the t he rmoc l i ne maximum) i s .0055 r a d / s e c . A s u i t a b l e va lue from PEQUOD e x t r a p o l a t e d from > 300 meters i s .005 rad/sec ( F igu re 25 ) . Ga rge t t and O s b o r n ( l 9 8 l ) show p r o f i l e s of N which i n d i c a t e tha t N 0 i s « .0055 rad/sec-. Together wi th the a p p r o p r i a t e va lues of a 0 , these y i e l d r 46 days fo r PEQUOD, 36 days fo r the Vancouver I s l and s lope d a t a , 29 days fo r the WESPAC data and 16 days f o r the Sargasso Sea d a t a . The f a c t o r of two v a r i a b i l i t y in the energy parameter , E, v i a equa t ion (5.4) l i m i t s these e s t ima te s to w i t h i n a f a c t o r of - two u n c e r t a i n t y . I f the l a r g e s c a l e , low f r equency , e n e r g y - c o n t a i n i n g i n t e r n a l waves t r a v e l at 10 cm/sec (McComas and Mu l l e r (1981 ) ) then they w i l l propagate 200-500 km, which i s c o n s i d e r a b l y sma l l e r than the s c a l e of an ocean b a s i n . G a r r e t t and Munk(1979) use a p ropaga t i on speed of 20 cm/sec, but t h i s on ly i n c r e a s e s the range to 400-1000 km, which i s s t i l l l e s s than bas in s c a l e . 72 V I . RESULTS FROM PEQUOD 6 . 1 Cu r r en t s And Hydrography The l a r g e s c a l e s t r u c t u r e of the c u r r e n t s and hydrograph i c parameters i n e q u a t o r i a l r eg i ons i s now r e l a t i v e l y we l l known. Many s tandard oceanograph ic t e x t s (Pond and P i c k a r d ( 1 9 8 3 ) , G i l l ( l 9 8 2 ) ) c o n t a i n d i s c u s s i o n s of e q u a t o r i a l o b s e r v a t i o n s and dynamics . P r e l i m i n a r y t h e o r e t i c a l and expe r imen ta l work i s r e g u l a r l y r epo r t ed i n the T r o p i c a l Ocean-Atmosphere News le t te r (D .Ha lpe rn , J ISAO, U n i v e r s i t y of Washington, e d i t o r ) . A recent review i s g i ven by Leetmaa, McCreary and Moore (198 l ) . The e q u a t o r i a l c u r r e n t regime i s b a s i c a l l y compr ised of a s e r i e s of su r f a ce and subsur f ace zona l j e t s which are of the order 10 6 meters long by 10 5 meters wide and 10 2 meters deep. P r e v a i l i n g e a s t e r l y winds in the t r o p i c s d r i v e the westward f l ow ing South E q u a t o r i a l Cur ren t (SEC) at l a t i t u d e s of 10°S to 4 ° N , and the North E q u a t o r i a l Cu r ren t (NEC) between 10°N and 2 0 ° N . In the r eg ion of the doldrums from 4 °N to 10 °N , the winds are c o n s i d e r a b l y weaker (a l though s t i l l e a s t e r l y ) and the r e s u l t i n g m e r i d i o n a l wind shear r e s u l t s in an eastward f l ow ing Nor th E q u a t o r i a l Coun te r cu r r en t (NECC). More remarkable and more p e r t i n e n t to t h i s s tudy i s the e x i s t e n c e of the E q u a t o r i a l Undercur ren t (EUC). Due to the p redominant l y westward t r a n s p o r t near the equa to r , a b u i l d u p occurs at the western boundary of the ocean r e s u l t i n g i n a west-eas t p r e s s u r e g r a d i e n t . Near the s u r f a c e , the wind s t r e s s i s s t rong enough to dominate , r e s u l t i n g in the westward f l ow ing 73 SEC, but w i th i n c r e a s i n g depth the e f f e c t of the winds i s d i m i n i s h e d and , in the the rmoc l i ne below the mixed l a y e r , the p r e s s u r e g r ad i en t dominates . The r e s u l t i s the eastward f low ing EUC. I n c luded i n Appendix F C w i t h the Camel III p r o f i l e s from PEQUOD are v e r t i c a l p r o f i l e s of h o r i z o n t a l c u r r e n t , t empera ture , s a l i n i t y and the c a l c u l a t e d buoyancy f r equency . These were measured w i th the White Ho r se . The t reatment of the White Horse data i s d i s c u s s e d in Appendix J . The b a s i c f e a t u r e s of the c u r r e n t s and hydrograph i c parameters are seen in the p r o f i l e s of net D which i s a s s o c i a t e d w i th the Camel III drop 3 at 1/2°N, 138°W (see Appendix K ) . At net D the su r f a ce SEC f lows westward at about 75 cm/sec wh i le the co re of the EUC i s s i t u a t e d at about 120 meters and has a maximum eastward v e l o c i t y of about 75 cm/sec ( u n f o r t u n a t e l y , i t i s q u i t e p o s s i b l e tha t the r e a l maximum of the EUC core i s missed due to the 25 meter depth r e s o l u t i o n of the White Horse v e l o c i t i e s ) . The r e s u l t i n g mean shear i s 150 cm/sec in 120 meters or about .01 s e c - 1 . Beneath the EUC core the v e l o c i t y dec reases to near z e r o . Between 600 and 700 meters the re i s s t r u c t u r e i n the v e l o c i t y f i e l d which can be de t e c t ed above the 4 cm/sec r e s o l u t i o n (see Appendix J) of the White Horse measurements. T h i s s t r u c t u r e i s e v i den t in some of the o ther p r o f i l e s (no tab l y net K on 20/02/82 and net Q) but shou ld not be c o n s i d e r e d t y p i c a l . The temperature data (Appendix K) g e n e r a l l y i n d i c a t e a sha l low wind mixed l a ye r (the winds in mid-winter are t y p i c a l l y 74 s l a c k e r than a ve r age ) . A low g r a d i e n t in T u n d e r l i e s t h i s and the very s t r ong the rmoc l i ne i s s i t u a t e d at about 100-140 meters d e p t h . The nea r-su r f a ce s a l i n i t y shows the e f f e c t s of wind m i x i n g . The t y p i c a l s t rong s a l i n i t y maximum l o c a t e d in the main the rmoc l i ne i s not ev iden t in a l l of the p r o f i l e s (but can be seen in nets L and F in Appendix K, f o r example ) . A l l of the s a l i n i t y p r o f i l e s e x h i b i t s t rong f i n e s t r u c t u r e f e a t u r e s in the r eg ions near and in the t h e r m o c l i n e . The maximum va lue of N in the the rmoc l i ne i s about .016-.024 r a d / s e c . A second maximum of about .005 rad/sec p e r s i s t s from p r o f i l e to p r o f i l e near 350 meters dep th , which i s j u s t below the depth of the thermostad (a r eg i on of cons tan t temperature of about 12°C which has been c o n s i s t e n t l y found in e q u a t o r i a l CTD p r o f i l e s ) . A unique and i n t e r e s t i n g f e a t u r e i s the l o c a l minimum in many of the N p r o f i l e s above the t he rmoc l i ne in the l a r g e mean shear r eg ion of the SEC-EUC i n t e r f a c e . Ev iden t in nets D and E e s p e c i a l l y and to a l e s s e r extent in o ther p r o f i l e s , the minimum appears to be due to a combina t ion of l o c a l l y cons tan t temperature and l o c a l weakening of the s a l i n i t y g r a d i e n t above the t h e r m o c l i n e . M.McPhaden (pe r sona l communicat ion) has found a co r r e spond ing minimum in the tempora l v a r i a t i o n s of p o t e n t i a l temperature and the v e r t i c a l d e n s i t y g r a d i e n t . I t w i l l be i n t e r e s t i n g to compare the m i c r o s t r u c t u r e data i n t h i s r e g i o n . 75 6.2 P rev ious E q u a t o r i a l M i c r o s t r u c t u r e Measurements The p r e v i o u s no tab le e q u a t o r i a l measurements of m i c r o s t r u c t u r e are r epo r t ed in G r e g g ( l 9 7 6 ) , Crawford and Osborn(1979a, 1979b, 1981), Osborn and B i lodeau(1980) and Crawford (1982 ) . Gregg(1976) made s i x p r o f i l e s of temperature m i c r o s t r u c t u r e at 155°W on the equator in J u l y , 1972. The drops were l i m i t e d to the upper 500 meters and the c h a r a c t e r i s t i c s emphasized were those of the l a r g e mean shear r eg ion of the SEC-EUC i n t e r f a c e , the core of the EUC and the the rmostad . The i n t e n s i t y of the t u rbu l ence was found to be e x c e p t i o n a l l y h igh in the SEC-EUC i n t e r f a c e , q u i t e weak in the EUC c o r e , moderate in the thermostad between 300 and 400 meters and very weak below the the rmostad . Osborn and B i lodeau (1980) c o l l e c t e d temperature m i c r o s t r u c t u r e da ta i n the e q u a t o r i a l A t l a n t i c between 24°W and 33°W in June and J u l y , 1974. Nea r l y c o i n c i d e n t v e l o c i t y m i c r o s t r u c t u r e da ta a l l owed them to make the f o l l o w i n g p o i n t s ; temperature m i c r o s t r u c t u r e i s g e n e r a l l y found c o n c u r r e n t l y w i th v e l o c i t y t u rbu l ence f l u c t u a t i o n s un l e s s the temperature g r ad i en t m i c r o s t r u c t u r e i s c o n s i s t e n t l y of one s i g n ; t u r b u l e n t pa tches extend h o r i z o n t a l l y over at l e a s t some tens of mete rs ; the temperature m i c r o s t r u c t u r e and f i n e s t r u c t u r e are more i n tense i n the Osborn and B i l o d e a u A t l a n t i c measurements than in the Gregg P a c i f i c measurements. V e l o c i t y m i c r o s t r u c t u r e measurements in the e q u a t o r i a l A t l a n t i c in June and J u l y , 1974 were made by Crawford and Osborn(1979a,b) and in the e q u a t o r i a l P a c i f i c in January and 76 Feb rua r y , 1979 (Crawford and Osborn(1981 a ) , C rawfo rd (1982 ) ) . A g a i n , the measurements were c o n f i n e d to the upper 500 mete rs . These r e s u l t s i n d i c a t e l a r g e t u r b u l e n t i n t e n s i t i e s i n the SEC-EUC i n t e r f a c e and low i n t e n s i t i e s in the EUC c o r e . In the A t l a n t i c , h i gh t u r b u l e n t i n t e n s i t i e s were a l s o found below the EUC core but were not found below the EUC core in the P a c i f i c . The remarkable r e s u l t of these s t u d i e s was the c o n n e c t i o n made between the t u rbu l ence measurements and the l a r g e s c a l e dynamics of e q u a t o r i a l c u r r e n t s . The A t l a n t i c data i n d i c a t e d tha t the r a te of d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy in the EUC above the core i s comparable in magnitude to the r a te at which the EUC ga ins energy from the zona l p r e s su re g r a d i e n t , thereby d e f i n i n g the r o l e of t u r b u l e n t f r i c t i o n as a s ink of k i n e t i c energy in e q u a t o r i a l c u r r e n t s . 6.3 PEQUOD M i c ros t ruc tu i r e P r o f i l e s of turbu lent- k i n e t i c energy d i s s i p a t i o n , e, c a l c u l a t e d from the v e l o c i t y m i c r o s t r u c t u r e da ta o b t a i n e d from the c e n t r a l e q u a t o r i a l P a c i f i c in F eb rua r y , 1982 are con t a i ned in Appendix K. T h i r t e e n p r o f i l e s were made at or w i t h i n of 2 the equator ( these w i l l be r e f e r r e d to as on-equator ) and the o ther th ree were made w i t h i n 2 ° of the equator (and w i l l be r e f e r r e d to as o f f - e q u a t o r ) . Without e x c e p t i o n , the on-equator p r o f i l e s i n Appendix K d i s p l a y h i g h l e v e l s of d i s s i p a t i o n in the l a r g e mean shear r eg ion of the SEC-EUC i n t e r f a c e . The es t ima ted e i s u s u a l l y much sma l l e r i n the v e l o c i t y co re of the EUC .where the mean 77 shear i s a minimum. Below the c o r e , the mean shear i s h i gh but on ly o c c a s i o n a l l y i s the e very l a r g e in t h i s r eg ion (as found by Crawford and Osborn(1981b) f o r the P a c i f i c da ta but which i s c o n t r a r y to the r e s u l t s from the A t l a n t i c ) . For example, a l t hough drop 3 has the second l a r g e s t va lue of e averaged over 20 to 140 meters d e p t h , in the l a r g e mean shear r eg ion at the base of the EUC, the i n d i v i d u a l e s t ima tes of e are q u i t e s m a l l . However, the maximum e va lues from drop 5 are at the base of the EUC. The s t rong the rmoc l i ne appears to be an e f f e c t i v e b a r r i e r to the exchange of t u r b u l e n t energy from the s u r f a c e to the deeper wate rs . In f a c t , a r e c u r r e n t f e a tu r e in many of the p r o f i l e s i s the very h igh l e v e l of t u rbu l ence w i th v a r y i n g degrees of i n t e r m i t t e n c y in the upper 150 meters or so , below which the re i s a sharp c u t o f f and i n d i v i d u a l e s t ima tes of e are at or near the no i se l e v e l . At g r ea t e r depths in the e q u a t o r i a l p r o f i l e s the occur rence of t u r b u l e n t pa tches i s much l e s s f requent and the l e v e l s of t u rbu l ence are much lower . Another i n t e r e s t i n g f e a t u r e , e s p e c i a l l y in l i g h t of McPhaden's es t imate of low f i n e s t r u c t u r e a c t i v i t y in the r eg ion of minimum s t a t i c s t a b i l i t y above the EUC core and which was b r i e f l y mentioned in the p r e v i o u s s e c t i o n , i s s t r ong ev idence in at l e a s t th ree drops of low m i c r o s t r u c t u r e a c t i v i t y i n the low N reg ion above the EUC c o r e . Drop 2 (100 m e t e r s ) , drop 3 (65 meters) and drop 6 (80 meters) a l l show reduced e in the minimum N r e g i o n ; at l e a s t one and up to two decades sma l l e r compared to t h i c k pa tches of t u rbu l ence immediate ly above and below. In 78 other drops the minimum in N i s . e i t h e r very weak or nonex i s t en t and the re i s no obv ious minimum e r e g i o n . Of e i g h t drops made w i t h i n 1/2° of the equator at 145°W, four (12 ,14 ,15 ,17 ) show s t r i k i n g l y s i m i l a r t u r b u l e n t patches at 500 meters d e p t h . Drop 10 shows no s i g n of a 500 meter t u r b u l e n t p a t c h , drops 7 and 8 have t h i nne r patches at 500 meters and drop 13 has th ree n e a r l y e q u i d i s t a n t 5 meter t h i c k patches sepa ra ted by - 25 meters and c e n t r e d at 500 meters . These are summarized in Tab le 3. F i gu re 22 compares on-equator and o f f - e q u a t o r p r o f i l e s . Independent 2 meter e s t ima tes of c were averaged v e r t i c a l l y over 50 meters and then over a l l of the drops w i t h i n the r e s p e c t i v e l a t i t u d e r ange . The two upper v a lues (50, 100 meters) i n d i c a t e averaged e g r e a t e r near the equator by a f a c t o r of 4-5 over the o f f - equa to r d r o p s . Averaged over the upper 200 meters , the d i s s i p a t i o n i s 1 .6x10" 5 W/m3 f o r the on-equator data and 3 . 3 x l 0 " 6 W/m3 f o r o f f - e q u a t o r d r o p s . At 150 mete rs , c o r r e s p o n d i n g to a depth just , below the EUC c o r e , the d i f f e r e n c e i s on ly a f a c t o r of two. Below 200 mete rs , on ly three (of four teen ) separa te on-equator averages of e are sma l l e r than o f f - equa to r a ve r ages . When averaged over the depth range 200-900 meters e i s 5 . 6x10 " 7 W/m3 f o r on-equator drops and 3 .3x10 " 7 W/m3 f o r o f f - e q u a t o r d r o p s . As a compar ison w i th the measurements of Crawford and Osborn(1979a,b) and Crawford (1982) , d i s s i p a t i o n s were averaged over the depth range 20-140 meters and p l o t t e d on F i gu re 3 of Crawford(1982) which i s shown here as F i g u r e 23. The two s i n g l e 79 DROP PATCH CENTRE THICKNESS e(W/m 3xlQ 7) 7 8 10 12 13 14 15 17 no s i g n 483 meters 35 meters 3 patches each 5 meters 510 15 540 15 492 10 518 27 492 35 weak weak 50 10-20 10 30 40 20 40 Ta b l e 3 - Depth, t h i c k n e s s and patch-averaged d i s s i p a t i o n s f o r the patches l o c a t e d near 500 meters depth f o r the drops w i t h i n 1/2° of the equator at-145°W. 80 10-200 u m x> pa oi Ul cn 04 600 LOG e (W/m3) / 10" 6 10 - 5 J - 1 . 1 -i i : — I T . I r n t i 1 — i — i i i i i F i g u r e 22 - V e r t i c a l p r o f i l e s of 50 meter v e r t i c a l l y averaged val u e s of l o g e averaged over a l l of the Camel III p r o f i l e s taken w i t h i n 1/2° of the e q u a t o r ( d o t s ) and over the p r o f i l e s o u t s i d e of 1° of the e q u a t o r ( t r i a n g l e s ) . 81 AVERAGE DISSIPATION • P a c i f i c , 1979 20 to 140 meters depth O Atlantic, 1974 20 meters to undercurrent core /V P a c i f i c , 1982 20 to 140 meters depth „ 3 3 * 2 O • N O R T H Jo* I I I I 1 I I 15 M 13 12 11 to 9 8 7 6 5 4 3 O S O U T H F i g u r e 23 - Averaged d i s s i p a t i o n s observed d u r i n g the P a r i z e a u c r u i s e i n 1979, the A t l a n t i s II c r u i s e i n 1974 and the Thomas G.Thompson c r u i s e i n 1982. The 1982 data were added t o F i g u r e 3 from Crawford(1982). 82 l a r g e s t v a l ues from 1982 are w i t h i n 1/2° of the equator and the th ree o f f - equa to r va lues are a l l l e s s than 7x10~ 6 W/m 3, in gene ra l agreement wi th Crawford (1982 ) . However, of the t h i r t e e n on-equator d r o p s , on ly the two drops ment ioned are d i s t i n g u i s h a b l e in magnitude from the o f f - e q u a t o r d r o p s . The l a r g e and r e c u r r e n t peak at 0 ° found by Crawford(1982) fo r both the A t l a n t i c , 1974 and P a c i f i c , 1979 e q u a t o r i a l data i s c l e a r l y not so dominant fo r the 1982 d a t a , which were t aken , i t shou ld be no t ed , we l l p r i o r to the onset of the 1982 E l Nino even t . 6.4 The On-equator P r o f i l e s The l o n g i t u d e of on-equator s t a t i o n s occup i ed where j o i n t Camel-White Horse p r o f i l e s were made are l i s t e d in Tab le 4 and shown in F i g u r e 26. Three of the p r o f i l e s are from 138°W, four from 145°W and one from 153°W. The o ther f i v e e q u a t o r i a l p r o f i l e s were made at or near 145°W but were not s ynop t i c w i th White Horse p r o f i l e s . Crawford made n ine teen p r o f i l e s w i t h i n 1/2° of the equator in e a r l y 1979, a l l a t 150°W, and n ine of these were accompanied by ove r- the-s ide c u r r e n t meter p r o f i l e s from which the shear over the depth range 20 to 140 meters was e s t i m a t e d . White Horse v e l o c i t y p r o f i l e s were smoothed u s i n g a 3-point runn ing mean f i l t e r and then f i r s t - d i f f e r e n c e d to es t imate the magnitude of the shear over the 25 meter range . The Brunt-V a i s a l a f requency was c a l c u l a t e d over app rox ima te l y 25 meters to co r respond to the c a l c u l a t e d shea r . The e i g h t i n d i v i d u a l p r o f i l e s are shown in F i g u r e s 24 and 25, where the t h i c k l i n e r ep r e sen t s the average va lue at each depth of the e i gh t 83 p r o f i l e s . F i g u r e 26 i n c l u d e s the symbol key f o r these d iagrams . The mean shear p r o f i l e shows a minimum near the su r f a ce in the South E q u a t o r i a l Cur rent (SEC) which i n c r e a s e s to a maximum va lue of about .014 s e c " 1 i n the r eg i on of the SEC-Equa to r i a l Undercur rent (EUC) i n t e r f a c e . Va lues span a lmost a f a c t o r of 3 above the EUC core from .008 to .022 s e c " 1 , w i th the l a r g e r shears g e n e r a l l y found in the eas t e rn p r o f i l e s . As we l l as can be de te rm ined , g i ven the 25 meter depth r e s o l u t i o n of the White Horse v e l o c i t y measurements, the EUC.core was about 130 meters deep at 138°W, on l y s l i g h t l y deeper at 145°W and about 150 meters deep at 153°W. The core i s i n d i c a t e d by the minimum in the shear p r o f i l e . Below the EUC core i s a second shear maximum of about h a l f the magnitude of the upper one. The B r u n t - V a i s a l a p r o f i l e shows a subsur f ace maximum of average va lue .018 rad/sec l o c a t e d in the EUC c o r e . T h i s maximum deepens westward. A second maximum l o c a t e d at approx imate l y 350 to 420 meters i s c o n s i d e r a b l y sma l l e r but h i s t o r i c a l l y p e r s i s t e n t . A compar ison of shear and Brunt-V a i s a l a p r o f i l e s i n d i c a t e bulk R i cha rdson numbers (Ri ) l e s s than one on l y above the c o r e , where in f a c t there are many i n d i v i d u a l o c cu r r ences of R i< l/4 when c a l c u l a t e d over 25 meter i n t e r v a l s . V e r t i c a l p r o f i l e s of e are shown in F i g u r e 26. I n d i v i d u a l e s t ima tes of e were averaged over 25 meters and p l o t t e d . I n d i v i d u a l 25 meter averages range over 4 decades from 2 x 1 0 " 4 W/m3 above the EUC core on drop 3 to 3 x 1 0 " 8 W/m3 at dep th . The t h i c k l i n e r ep r e sen t s the average over the e i g h t Camel p r o f i l e s which were a s s o c i a t e d wi th White Horse p r o f i l e s and shows a near 84 E i g h t v e r t i c a l p r o f i l e s of v e r t i c a l shear as e s t i m a t e d from White Horse h o r i z o n t a l v e l o c i t i e s taken w i t h i n 1/2° of the equator i n February, 1982. Longitudes of i n d i v i d u a l p r o f i l e s are l i s t e d i n the key to F i g u r e 26. The t h i c k l i n e i s the average of the e i g h t p r o f i l e s . 85 i I BRUNT VAISALA FREQUENCy (RAD/SEC) 0 .01 .02 0"J I I I l _ L 1000 n 1 n 1 r E i g h t v e r t i c a l p r o f i l e s of B r u n t - V a i s a l a frequency measured simultaneously as the shears of F i g u r e 24. The symbols are keyed i n F i g u r e 26. The t h i c k l i n e i s the average of the e i g h t p r o f i l e s . 86 V e r t i c a l p r o f i l e s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over 25 meters depth and which are n e a r l y s y n o p t i c with the data of F i g u r e s 24 & 25. The t h i c k l i n e i s the average of the e i g h t p r o f i l e s . Large s o l i d dots are 20 meter averages over nineteen p r o f i l e s taken at 150°W and w i t h i n 1/2° of the equator by Wm.Crawford i n January/February, 1979. The l a r g e diamonds are 20 meter averages over t h i r t e e n e q u a t o r i a l p r o f i l e s from February, 1982. 87 s u r f a c e maximum of 4 x 1 0 " 5 W/m3 and a gene ra l dec rease of e wi th dep th , w i th the excep t i on of a peak near 500 meters which i n d i c a t e s averaged d i s s i p a t i o n s th ree t imes g r e a t e r than v e r t i c a l l y ad jacen t v a l u e s . The 500 meter peak in e l i e s d i r e c t l y beneath the peak in B r u n t - V a i s a l a f r equency . With the i n c l u s i o n of the f i v e Camel p r o f i l e s at 145°W which were made independent l y of White Horse p r o f i l e s ( these a re the l a rge diamonds and were averaged over 20 meters to compare to C rawfo rd (1982 ) ) , the near su r f a ce and 500 meter peaks are enhanced wh i l e the q u a l i t a t i v e d e s c r i p t i o n remains unchanged. In marked c o n t r a s t to the 1982 data are the 1979 p r o f i l e s of C rawfo rd . The 20 meter averages of n ine teen p r o f i l e s at 150°W are denoted by l a r g e s o l i d d o t s . These d i f f e r somewhat from F i g u r e 5 of Crawford(1982) due to a s l i g h t l y d i f f e r e n t a ve rag ing scheme used . Crawford averaged over i n d i v i d u a l days ' t o reduce the b i a s caused by the tendency of m u l t i p l e p r o f i l e s to be on days of h igh t u r b u l e n t i n t e n s i t y ' . For the present pu rpose , we have averaged over i n d i v i d u a l p r o f i l e s fo r d i r e c t compar ison wi th our d a t a . The da ta are from Crawford and Osborn(1981 a ) . The 1 979 p r o f i l e s were much sha l l ower ( to 300 meters) and show q u a n t i t a t i v e agreement below 160 meters wi th the 1982 data ( t h i s g i v e s us con f i dence t h a t the two se t s of measurements are comparab le ) . However, in the l a r g e mean shear r eg ion above the EUC core the 1982 p r o f i l e s show 20 meter averaged v a l ues which are sma l l e r by more than a f a c t o r of ten at 70, 90 and 110 meters ! In f a c t , the 13 p r o f i l e s from Feb rua r y , 1982, depth-averaged from 20 to 140 meters g i ve 7 of 88 0 . 2 8 x 1 0 " ' W/m3 wh i l e ave rag ing over the 19 e q u a t o r i a l p r o f i l e s from 1979 g i v e s 1.2x10"" W/m 3, more than a f a c t o r of four d i f f e r e n c e . U n f o r t u n a t e l y , n e i t h e r the 1979 p r o f i l e s nor the 1982 p r o f i l e s have comp le te l y s ynop t i c shear measurements. Nine of the 1979 d i s s i p a t i o n p r o f i l e s were a s s o c i a t e d w i th c u r r e n t meter p r o f i l e s from which the shear was e s t ima ted wh i le e i g h t of the 1982 p r o f i l e s were s ynop t i c w i th White Horse p r o f i l e s . Averaged d i s s i p a t i o n over 20 to 140 meters fo r these da ta i n d i c a t e abso lu t e va lues which are sma l l e r by about 50 pe rcen t from those mentioned above, but e es t ima ted from the 1979 e q u a t o r i a l drops remains about four t imes g r e a t e r . T h i s i s shown in Tab le 4 a l ong wi th the shear and Ri from 20 to 140 meters es t ima ted by d i f f e r e n c i n g between those dep ths . D i s s i p a t i o n s averaged over the p r o f i l e s were four t imes g rea t e r in J anuary/February 1979 than in February 1982 wh i le the averaged shear was c o n s i d e r a b l y l e s s and Ri c o n s i d e r a b l y g r e a t e r . C r aw fo rd ' s 1979 data i n d i c a t e tha t the l a r g e r v a lues of e are accompanied by sma l l e r va lues of R i and l a r g e r v a lues of shea r , as one might r easonab l y expec t . T h i s t r end does not appear to be so c l e a r f o r the 1982 data s e t , e i t h e r on a d rop by drop or an averaged b a s i s . However, c o n s i d e r i n g on l y the average of the four e q u a t o r i a l drops made i n February 1979, th ree of which are 33, 36 and 38 in Tab le 4, 7 from 20 to 140 meters i s 0 .29x10" " W/m3 which agrees s u r p r i s i n g l y we l l w i th the f i g u r e noted in the p r e ced ing pa rag raph . FEBRUARY, 19B2 JANUARY t FEBRUARY, 1979 Drop ttW/m'xIC") AU/Az Ri'NV(AU/Az) ] Drop t(W/m'»l0«) AO/Az Ri-N'/(AU/Az) ] 3(13B0W) .66 .011 1.3 13-14 1.5 .0094 .53 4(13B°W) .16 .0090 2.2 21 .66 .0096 1.3 5( 13B°W) .071 .015 .61 22-24 .71 .0083 2.0 10(145°W) .011 1.5 33 .42 .0051. 1.2 13(14SCW) .17 .0070 2.0 36 .58 .0047 4.3 14(145°W) .082 .0048 3.9 38 .12 .0035 3.3 CO 10 17(U5°W) .16 .0089 1 .8 7 - .77x10-*H/m* 19(153DW) .036 T • . 19x10" *W/oi* AOTAZ • .0093. sec"' .0069 1.0 AU/Az - .0074 sec" 1 RT • 2.5 •'I i • - j . RT - 1.8 Table 4 - Comparison of e averaged over 20-140 meters from 1982 and 1979 drops. The ve l o c i t y and depth are differenced over 20-140 meters by Crawford(1982) for the 1979 data and over 12.5-137.5 meters for the 1982 data. At the bottom are averages over the l i s t e d drops. 90 The e f f e c t of the winds on the es t ima ted e v a lues i s shown in F i g u r e . 2 7 . Averaged over 20 to 140 mete rs , the d i s s i p a t i o n i s p l o t t e d a g a i n s t t ime . Each day , s i x measurements of wind speed and d i r e c t i o n were made. The d i r e c t i o n was r e l a t i v e l y s teady and e a s t e r l y . Wind speeds were averaged over the s i x d a i l y measurements and the cube of the wind speed p l o t t e d at 1200 hours of the p a r t i c u l a r day i n F i g u r e 27. For compar i son , the data of Crawford(1982) are a l s o p l o t t e d . The a b s c i s s a of F i g u r e 27 i s c a l enda r days and the d i s t i n c t i o n i s emphasized that C r a w f o r d ' s va lues are from 1979 and the o the r s from 1982. The winds are comparable in s t r e n g t h between the two data se t s w i th somewhat l e s s v a r i a b i l i t y f o r the 1982 data a s i d e from the l a rge th ree day drop in wind speeds from February 20 to 22. The two s m a l l e s t v a l ues of averaged e occur du r i ng and j u s t a f t e r the drop i n wind speed . However, t he re i s no other i n d i c a t i o n to g i ve one c o n f i d e n c e of any r e l a t i o n between wind speed and e, in agreement w i th the f i n d i n g of C rawford (1982 ) . 6.5 € And The Zonal P ressu re G rad i en t Crawford and Osborn(1979b) sugges ted tha t a reasonab le ba lance of the t u r b u l e n t k i n e t i c energy equa t ion (2.4) in the upper e q u a t o r i a l waters i s between the p r o d u c t i o n and the d i s s i p a t i o n te rms . I f t h i s ho lds t r u e , an es t imate may be made of the t u r b u l e n c e p r o d u c t i o n term in the mean k i n e t i c energy equa t ion (2 .3) from the d i s s i p a t i o n measurements. With t h i s b a s i c p r em i se , a ba lance of the mean k i n e t i c energy was p roposed . From the su r f a ce to the l e v e l of no zona l v e l o c i t y , 91 Time v a r i a t i o n s of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over 20 to 140 meters. Dots represent s t a t i o n s w i t h i n 1/2° of the equator and c r o s s e s are o u t s i d e of 1° of the equator. Small dots and c r o s s e s are from 1979 and l a r g e dots and c r o s s e s from 1982. The s o l i d l i n e i s the cube of the average d a i l y wind speed from 1979 and the d o t t e d l i n e i s that from 1982 ( r e v i s e d from F i g u r e 4 of Crawford(1982)). 92 i t was found tha t the energy input by the wind s t r e s s at the s u r f a c e was approx imate l y ba lanced by the sum of the work done on the zona l p r e s s u r e g r a d i e n t , udP/dx, i n p i l i n g up water towards the west p l u s the l o s s e s to t u r b u l e n t f r i c t i o n , e = u ' w ' d u / d x 3 . Below the l e v e l of no zona l v e l o c i t y , the work done by the zona l p ressu re g r a d i e n t in d r i v i n g the EUC was approx imate l y ba lanced by l o s s e s to t u r b u l e n t f r i c t i o n . At the same t ime , i t was suggested tha t the ba lance of terms was be t t e r than cou ld be expected from the accuracy of the o b s e r v a t i o n s and tha t other terms fo r which no es t imate c o u l d be made may be s i g n i f i c a n t . F i ve White Horse CTD p r o f i l e s on the equator are a v a i l a b l e f o r computing dynamic h e i g h t s . These p r o f i l e s were made at net E at 138°W on 02/09/82, net K at 145°W on 02/15/82, 02/20/82 and 02/21/82 and net Q at 153°W on 02/24/82. Dynamic he i gh t s computed from the CTD data at the su r f a ce r e l a t i v e to 1000 dbar (0/1000 dbar in the s tandard n o t a t i o n ) are 15.7 m 2 / s 2 at net E, 17.8 m 2 / s 2 a t net Q but range from 16.0 to 16.7 m 2 / s 2 a t net K. The v a r i a t i o n at net K i s much g r ea t e r than that suggested by Wy r tk i ( l 983 ) of ±0.2 m 2 / s 2 due to i n t e r n a l waves and t i d e s . A c l o s e look at the data i n d i c a t e s that a 20 to 30 meter d e p r e s s i o n of the the rmoc l i ne occu r r ed at net K on 02/20/82. The a = 24, 25 and 26 s u r f a c e s were depressed by 20 to 30 t meters and the p o s i t i o n of the measured v e l o c i t y maximum in the EUC deepened by 25 meters . C o i n c i d e n t a l l y , the nea res t su r f a ce v e l o c i t y measurement (at 12.5 meters) i n d i c a t e d a r e v e r s a l from 93 s t rong westward (-53 cm/sec) to s t r o n g eastward (+57 cm/sec) f low between 02/15/82 and 02/20/82 and back to 10 cm/sec westward f low on 02/21/82. A l s o , on 02/21/82, the depressed a t s u r f a c e s as we l l as the EUC core r e tu rned to w i t h i n 1 meter of the o r i g i n a l depths of 02/25/82. Due to t h i s l a r g e v a r i a b i l i t y , i t was i m p o s s i b l e to o b t a i n a good es t imate of the zona l p ressu re g r a d i e n t from the PEQUOD d a t a . However, there e x i s t s h i s t o r i c a l data from which e s t ima tes may be made of the zona l p r e s s u r e g r a d i e n t . Knauss ( l966) computed a g r a d i e n t of 2 . 6 x 1 0 " 7 m/s 2 at the depth of the EUC core between 140°W and 104°W in May, 1958. Lemasson and P i t on ( l 968 ) show a dynamic he i gh t s e c t i o n from which Katz et a l . d 9 7 7 ) computed a g r ad i en t of 4 . 8 x 1 0 " 7 m / s 2 at 50/700 dbar and 3 . 3 x l 0 " 7 m / s 2 a t 100/700 dbar between 160°E and 105°W. An AXBT s e c t i o n a l ong the P a c i f i c equator from 172°E to 110°W made in A p r i l and May, 1979 i s r epo r t ed in Ha lpe rn (1980 ) . A number of c o i n c i d e n t CTD p r o f i l e s p r o v i d e d Ha lpe rn wi th the i n f o r m a t i o n to compute a z o n a l p r e s su re g r a d i e n t at 0/270 dbar of 5 . 4 x 1 0 " 7 m/s 2 between 153°W and 133°W. From the AXBT s e c t i o n I have made an es t imate of the dynamic h e i g h t s at 100/270 dbar at 150°W and 130°W u s i n g i n f e r r e d s a l i n i t i e s . These i n d i c a t e a zona l p r e s su re g r a d i e n t of 2 . 8 x l 0 " 7 m / s 2 at 100/270 dba r . A r easonab le e s t i m a t e , t h e n , of the zona l p r e s s u r e g r a d i e n t at the s u r f a c e i s 5 x l 0 " 7 m / s 2 and i s 3 x l 0 " 7 m / s 2 near the depth of the EUC c o r e . F o l l o w i n g the method of Crawford and Osborn (1979b) , the work done by the zona l p r e s su re g r a d i e n t and the l o s s to the t u r b u l e n t d i s s i p a t i o n i s i n t e g r a t e d over the two 94 r e g i o n s . Between the s u r f a c e and the l e v e l of no zona l v e l o c i t y 70 meters) the average v e l o c i t y i s = 30 cm/sec westward and the average p ressu re g r a d i e n t i s 4 . 5 x 1 0 " 7 m / s 2 . The work done by the zona l p r e s su re g r a d i e n t , t hen , i s (.3 m/s ) (4 .5x10 " 7 m/s 2 ) ( 70 m)( l028 kg/m 3) = I0xl0~ 3 w/m 2 . The d i s s i p a t i o n averaged over the e q u a t o r i a l drops between 20 and 70 meters i s 6x l0 " 5 W/m 3 , r e s u l t i n g in a d e p t h - i n t e g r a t e d d i s s i p a t i o n of 4x l0~ 3 W/m 2 . I f the ba lance of Crawford and Osborn ( l979 ) h o l d s , t h i s i n d i c a t e s a net input by the wind at the su r f a ce of 14xl0~ 3 W/m 2 , which i s i d e n t i c a l to the wind s t r e s s in the e q u a t o r i a l A t l a n t i c quoted by Crawford and Osborn(1979b) at the t ime of t h e i r measurements. U n f o r t u n a t e l y , I do not know of a r e l i a b l e e s t ima te of wind s t r e s s which i s s ynop t i c w i th the PEQUOD d a t a . Below the l e v e l of no zona l v e l o c i t y and to the EUC core (which i s = 110 to 140 m e t e r s ) , the average zona l p r e s su re g r a d i e n t i s 3 . 5 x 1 0 " 7 m / s 2 , the average v e l o c i t y i s - 45 cm/sec eastward and the average d i s s i p a t i o n i s 1x l0~ 5 W/m 3 . The r e s u l t i n g work done by the zona l p r e s su re g r ad i en t i s 11x10" 3W/m 2 and the d e p t h - i n t e g r a t e d d i s s i p a t i o n i s about 1xl0~ 3 W/m 2 or about 10% of the work done by the zona l p r e s su re g r a d i e n t . R e c a l l tha t the 1982 d i s s i p a t i o n s at 70, 90 and 110 meters ( F igure 26) were 10% of the 1979 v a l u e s . Presumably , t h e n , the 1979 data y i e l d s a c l o s e r ba lance to the es t imate made of the work done by the h i s t o r i c a l zona l p r e s s u r e g r a d i e n t . S i nce the re i s no reason to doubt the r e l i a b i l i t y of the d i s s i p a t i o n e s t i m a t e s , one may suspec t tha t the zona l p r e s su re g r a d i e n t in Feb rua r y , 1982 was much s m a l l e r . A l though the 95 e s t ima te s of dynamic he igh t from PEQUOD are c e r t a i n l y not r e p r e s e n t a t i v e of the mean s t a t e , they do i n d i c a t e a much l a r g e r z o n a l p ressu re g r ad i en t r a t he r than a sma l l e r one. A p p a r e n t l y , t h e n , there i s good reason to b e l i e v e tha t there must be another s i n k fo r the energy which i s input to the EUC by the zona l p r e s s u r e g r a d i e n t . Crawford and Osborn(1979b) have suggested tha t m e r i d i o n a l d i ve rgence terms which c o u l d not be es t ima ted from the A t l a n t i c data set may p l a y a r o l e . As w e l l , there i s some i n d i c a t i o n from the PEQUOD data and other data s e t s tha t the s t r e n g t h of the undercur ren t i s not s teady due to e i t h e r meandering or p u l s i n g and hence the t ime ra te of change of the mean k i n e t i c energy may p l a y a s i g n i f i c a n t r o l e in the ba lance of terms i n ( 2 . 3 ) . 6.6 e And N, S, Ri As in Chapter 4, the data from PEQUOD are compared to the o the r a v a i l a b l e data from the c r u i s e in order to determine the e x i s t e n c e of t r e n d s . With the White Horse v e l o c i t y data i t i s p o s s i b l e to es t imate shea r , S, and the d i f f e r e n c e R i cha rdson number, Ri = N 2 / S 2 , as we l l as the buoyancy f r equency . The e r r o r s i n v o l v e d in c a l c u l a t i n g R i , S and N are d i s c u s s e d in Appendix J . F i g u r e 28 shows s c a t t e r p l o t s of the th ree parameters N, S, R i c a l c u l a t e d over 25 meter depth i n t e r v a l s p l o t t e d a g a i n s t e averaged over 25 mete rs . The l a r g e b l ack do ts represen t averages of e over r e s p e c t i v e N, S, and Ri b i n s . The data span approx ima te l y one and one h a l f decades in N, two decades in S and th ree decades in R i . In each of the th ree p l o t s the s c a t t e r 96 io-» E 3 " 1 o- »1 § j 10 io-» E 3 — 1 0 - •] 10-'i E \ 3 10 io-» io-» io-ftOUOO 20K-8OTTDH •tBUQD 2 0 H - B 0 T T 0 * PWUOD 20K-BOTTOT 10° 10' LOG RI * LOG N ( r a d / s e c ) 1 0 LOG S (1/sec) 1 0 " ' 10J F i g u r e 28 - S c a t t e r p l o t s of l o g ( t u r b u l e n t k i n e t i c energy d i s s i p a t i o n averaged over 25 meter v e r t i c a l i n t e r v a l s ) vs log(buoyancy frequency(N)), l o g ( v e r t i c a l shear(S)) and l o g ( d i f f e r e n c e Richardson number(Ri) c a l c u l a t e d from N and S ) . The data represent the e n t i r e water column sampled s y n o p t i c a l l y by both Camel I I I and by White Horse. Large black dots are averages over 1/3 decade i n t e r v a l s i n N, 1/2 decade i n t e r v a l s i n S and over the ranges <1, 1-4, and 4-40 i n R i . 97 i s c o n s i d e r a b l e . G e n e r a l l y , however, the l a r g e s t va lues of e are a s s o c i a t e d wi th l a r g e S and N and sma l l R i . S i m i l a r l y , the sma l l e s t v a l ues of e are a s s o c i a t e d w i th sma l l S and smal l N. For Ri > 2 the re i s no t r end a l t hough the sma l l e r va lues of e are a s s o c i a t e d w i th Ri > 2 but are i n d i s t i n g u i s h a b l e from those f o r Ri > 10. C e r t a i n l y , the sma l l e v a l ues are not a s s o c i a t e d w i th Ri < 1. The b in-averaged d i s s i p a t i o n s support these t r e n d s . A s i g n i f i c a n t l i m i t a t i o n i n making t h i s type of compar ison i s due to the 25 meter s p a t i a l r e s o l u t i o n of N, S and Ri compared to the much f i n e r r e s o l u t i o n of the e measurements. A c u r s o r y g l ance at the data in Appendix K i n d i c a t e s that many of the pa t ch s i z e s , e s p e c i a l l y below 200 mete rs , are much sma l l e r than 25 mete rs . But these t h i n patches dominate the 25 meter average of e. On the o ther hand, the e s t ima tes of S and N from which Ri i s c a l c u l a t e d are d i f f e r e n c e d over f u l l 25 meter i n t e r v a l s , thereby s u b s t a n t i a l l y o b s c u r i n g l o c a l l y l a r g e g r a d i e n t s . Hence, the compar ison i s best made on ly fo r the h e a v i l y averaged parameters as i s done in F i g u r e s 16, 29 and 30. In F i g u r e 16, the s o l i d squares r ep resen t 100 meter v e r t i c a l averages of e and N which have then been averaged over a l l of the d r o p s . Only the data below 300 meters have been i n c l u d e d in F i g u r e 16. As d i s c u s s e d in Chapter 4, the arguments which suppor t the type of r e l a t i o n e a N 1 are based on i n t e r n a l wave s c a l i n g . S ince the l a r g e mean shear in the su r f a ce l a y e r s of the e q u a t o r i a l data would seem to dominate the tu rbu l ence g e n e r a t i o n , i t was dec ided tha t p l o t t i n g the upper 300 meter v a l ues would be i n a p p r o p r i a t e . The p l o t t e d data do not d i f f e r 98 s u b s t a n t i a l l y from those of the o ther data s e t s . A l i n e of s l ope = 1 g i v e s a reasonab le d e s c r i p t i o n of the d a t a . As p r e v i o u s l y no t ed , the PEQUOD da ta r ep resen t the sma l l e s t averaged data measured to d a t e , i n d i c a t i n g a dea r th of t u r b u l e n t a c t i v i t y in the deeper waters of the c e n t r a l e q u a t o r i a l P a c i f i c . I d e n t i c a l ave rag ing as was done to produce F i g u r e s 15 and 16 was done to generate S and Ri vs e p l o t s . N e i t h e r of F i g u r e s 29 nor 30 o f f e r s t rong ev idence f o r t r ends at h i gh l e v e l s of Ri or low shea r , but fo r low Ri and h i g h shear ( in the upper 300 m e t e r s ) , the expected t rends are q u i t e appa ren t . The upper th ree shear v a l ues e x h i b i t s u c c e s s i v e l y and s u b s t a n t i a l l y l a r g e r v a lues of e, as do the lower th ree va lues of R i . Of c o u r s e , due to the dependence of Ri on S, i t i s not s u r p r i s i n g tha t these are the same e v a l u e s . In the upper 300 meters , the re appears to be a s t r ong dependence of e on both S and Ri c a l c u l a t e d over 25 meter depth i n t e r v a l s and averaged over 100 meters d e p t h . The v e r t i c a l s c a l e s of the shear are r e s o l v e d by the White Horse measurement in the upper 300 mete rs . But at g r e a t e r depths the s i t u a t i o n i s q u i t e d i f f e r e n t , and we b e l i e v e tha t the mechanism by which the t u rbu l ence i s generated i s d i s t i n c t from that in the upper waters due in p a r t to the i s o l a t i o n of the deep water by the s t r ong t h e r m o c l i n e . In f a c t , the s c a l i n g e a N 1 has h i n t e d tha t the t u r b u l e n c e i s due t o i n t e r n a l waves. I f the t u rbu l ence i s due to i n t e r n a l waves, there i s l i t t l e chance tha t the shear i s r e s o l v e d by the White Horse measurement. As was ment ioned in 99 LOG S (1/sec) F i g u r e 29 - P l o t of l og e vs l og S, where e and S have been v e r t i c a l l y averaged over 100 meter i n t e r v a l s and then over a l l of the PEQUOD drops w i th s ynop t i c Camel I l l -Wh i t e Horse d a t a . 100 io-10 5 -8 3 10"« 10 CD PEQUOD - " o -CD < ' T 1 1 1 1 I l • I I o • ! 1 1 1 1 l"T 10 - 1 10° LOG RI 10 F i g u r e 30 - P l o t of l o g e vs l o g R i , where e and R i have been v e r t i c a l l y averaged over TOO meter i n t e r v a l s and then over a l l of the PEQUOD drops w i t h s y n o p t i c Camel I l l - W h i t e Horse d a t a . 101 Chapter 5, Ga rge t t et a l . ( l 9 8 l ) p rov ide ev idence fo r an upper wavenumber l i m i t to the i n t e r n a l wave h o r i z o n t a l v e l o c i t y shear spectrum c o r r e s p o n d i n g to ten meters whi le the measurements of E r iksen (1978) i n d i c a t e b reak ing at v e r t i c a l s c a l e s of s e v e r a l mete rs . One may specu l a t e a l i t t l e on the r e l a t i v e dependence of e 7 a B on S and R i . If e a N, e a S and e a R i , then Ri = N 2 / S 2 a 2/7 2/a e A . T h i s r e q u i r e s \/B = 2/7 - 2/a. But i t has been proposed that 7 = 1 and a p p a r e n t l y , B < 0, thereby imp l y i ng tha t 0 < a < 1, or e has a s t ronge r dependence on N than S. Two f a c t o r s tend to obscure any t rends of both S and Ri at depths g r ea t e r than 300 mete r s . One of these i s the sma l l dynamic range in both pa ramete rs . And the o ther i s due to the huge e r r o r in e s t i m a t i n g l a r g e va lues of R i . 6.7 S t a t i s t i c s Of Ri And e In F i g u r e 31 N and S c a l c u l a t e d over 25 meter depth i n t e r v a l s are p l o t t e d a g a i n s t each o t h e r . The d i agona l s ' R i = 1/4, 1, 4 have been p l o t t e d fo r r e f e r e n c e . The da ta are grouped by depth i n t e r v a l s , 20 - 300 meters and 300 meters - bottom of d r o p . The c h a r a c t e r of each i s q u i t e d i s t i n c t i v e . In the upper wate rs , there e x i s t both h ighe r v a lues of N and S and a l s o l a r g e r dynamic ranges of at l e a s t S i f not N. S u c c e s s i v e v e r t i c a l da ta p o i n t s have been j o i n e d and these i n d i c a t e ( e s p e c i a l l y f o r R i> l ) t h a t , in the upper wate rs , Ri i s reduced more f r e q u e n t l y by i n c r e a s e d S r a the r than reduced N s i n c e the l i n e s are more n e a r l y h o r i z o n t a l than v e r t i c a l . In the deep 102 i Figure 31 - Scatter p l o t s of log N vs log S estimated over 25 meter inter v a l s for the depth ranges noted in the p l o t s . Adjacent v e r t i c a l points are joined. 103 wa te r s , no p a t t e r n i s o b v i o u s . The 20 - 300 meter p l o t i n d i c a t e s many va lues of l/4<Ri<1. In the 300 meter - bottom range most v a lues of Ri a re g rea te r than 4. These two data se t s are p l o t t e d on the same p l o t l a b e l l e d 20 meters - bottom fo r compar i son . The complete range of Ri fo r the PEQUOD data i s shown in F i g u r e 32. About 2.7% has R i<0 .3 . T h i s i s remarkably s i m i l a r to the 2.5% tha t E r i ksen (1978 ) found to be <0.25 from h i s i n t e r n a l wave a r r a y . However, due to the a l t o g e t h e r d i f f e r e n t regimes from which the da ta were t aken , these shou ld not be compared f u r t h e r . Most of the da ta (30.5%) l i e s in the b in 3<Ri<l0. Very h igh v a l ues of Ri are p a r t l y a r t i f i c i a l , s i n ce in the c a l c u l a t i o n , Ri was set to 1000 when S approached zero to a v o i d d i v i s i o n by z e r o . These Ri data were then grouped a c c o r d i n g to the a s s o c i a t e d e over the r e s p e c t i v e 25 meter depth i n t e r v a l s . In each Ri b i n , the number of va lues which had e g r e a t e r than a g i ven va lue were coun t ed . These were then no rma l i zed at 1 0 " 8 W/m3 ( aga i n , note tha t averaged va lues of e l e s s than the no i se l e v e l are ob t a i ned by s e t t i n g e = 0 i f e < no i s e l e v e l ) . The amount of data used to c r e a t e each curve in F i gu re 33 d i f f e r s . Less than 3% was a v a i l a b l e f o r 0.1<Ri<0.3 wh i le more than 30% was in the range 3<Ri<lO. T h i s l a ck of data i s l i k e l y why the upper curve i s l e s s smoothly v a r y i n g than the o t h e r s . A l s o , the r e l a t i v e e r r o r s d i f f e r from curve to curve due to both the sample s i z e and the percentage e r r o r in R i . As one might e x p e c t , r e l a t i v e l y more va lues of e above a 104 t -H i - t / 4 Ri«1 ! i Ri»4 LOG R i R e l a t i v e f requency of occu r rence of l o g Ri es t imated from the White Horse da ta taken in Feb rua ry , 1982 over the depth ranges c o i n c i d e n t w i th Camel III d a t a . 105 i i \ -7 -6 -5 -4 LOG e ( » / • » ) ; Figure 33 Normalized frequency of occurrence of log e per half decade i n t e r v a l for the ranges of Ri noted in the key. 106 g i ven va lue e x i s t fo r the lower v a l ues of R i , as i s expressed by the r e l a t i v e l e v e l s and r o l l o f f s of the cons tan t Ri b in curves of F i g u r e 33. At l a r g e r R i , the d i s t i n c t i o n i s b l u r r e d and the lower two cu rves are v i r t u a l l y i n d i s t i n g u i s h a b l e from each ~oThTe r. 1 07 V I I . ESTIMATES OF EDDY COEFFICIENTS A s e r i o u s problem encountered by mode l l e r s of f l u i d dynamics i s the p a r a m e t e r i z a t i o n of those mot ions which have tempora l and/or s p a t i a l s c a l e s sma l l e r than the time s t ep or g r i d s i z e of the model . The problems p r e sen t ed to mode l l e r s of ocean i c f lows are d i s c u s s e d by G a r r e t t ( 1 9 7 9 ) . E s s e n t i a l l y , the mode l l e r s r e q u i r e es t imates of energy s i nk s at s c a l e s sma l l e r than t h e i r models can r e s o l v e and a means by which these e f f e c t s can be i n c l u d e d in the equa t i ons of mo t i on . G e n e r a l l y , t h i s i s done by i n t r o d u c i n g an eddy v i s c o s i t y ( fo r momentum) and eddy d i f f u s i v i t y ( f o r mass) , which are used in an analogous f a s h i o n to t h e i r mo lecu l a r c o u n t e r p a r t s . In c o n t r a s t to the mo lecu l a r c o e f f i c i e n t s , v and K, which are p e c u l i a r to the f l u i d i t s e l f , the eddy c o e f f i c i e n t s are p r o p e r t i e s of the flow f i e l d . As such , the va lues used must be c a r e f u l l y c o n s i d e r e d in the con tex t of the f low to be m o d e l l e d . In a l a r g e s c a l e ocean c i r c u l a t i o n model one must c o n s i d e r the e f f e c t s of momentum and mass t r a n s p o r t due to edd i e s (most ly h o r i z o n t a l ) , waves (momentum on ly ) and t u rbu l ence (most ly v e r t i c a l or at l e a s t a c r o s s i s o p y c n a l s ) . As the g r i d s i z e i s reduced the edd ies may be we l l r e s o l v e d r e q u i r i n g reduced eddy c o e f f i c i e n t s and a g r ea t e r r e l a t i v e dependence on the t u r b u l e n c e . In models which concen t r a t e on the su r f a ce l a y e r s of the ocean and which r e q u i r e a f i n e v e r t i c a l g r i d such as the e q u a t o r i a l model of Pacanowski and P h i l a n d e r ( 1 9 8 1 ) , the v e r t i c a l mix ing i s expected to be e n t i r e l y due to the t u r b u l e n c e . In t h i s c h a p t e r , a number of eddy c o e f f i c i e n t s es t ima ted from m i c r o s t r u c t u r e data are 108 compared us i ng the r e s u l t s of Crawford (1982) , Osborn( l980) Gregg(1976) and the 1982 PEQUOD measurements. A h e a v i l y averaged p r o f i l e of eddy d i f f u s i v i t y i s p resen ted and compared to tha t of the WESPAC data s e t . 7.1 V a r i o u s E s t i m a t o r s An a c c e p t a b l e ba lance fo r the t u r b u l e n t k i n e t i c energy equa t i on (as d i s c u s s e d in Chapter 2) i n v o l v e s the t u r b u l e n t p r o d u c t i o n , d i s s i p a t i o n and the work done a g a i n s t buoyancy, and i s w r i t t e n as u ' u ' 9 u / 9 x = - e - p 'w 'g/p . (7.1) i j i J Two of these terms may be i n c o r p o r a t e d i n t o the f l u x R ichardson number, d e f i n e d as the r a t i o of the buoyancy f l u x to the t u r b u l e n t p r o d u c t i o n , R = g ' p ^ T r / ( p u T u r 9 u / 9 x ) (7.2) f i j i j S i nce p'w' = -K 9 p / 9 z = K p N 2 / g , P P K = gJrV r/(pH2), (7.3) P where K r e p r e s e n t s the c o e f f i c i e n t f o r the v e r t i c a l d i f f u s i o n P of d e n s i t y . With ( 7 . 1 ) , ( 7 . 2 ) , and (7.3) the r e l a t i o n fo r K i s P K = R / ( 1 - R ) e (7.4) p f f N 7 Osborn ( l980 ) recommends an upper bound of 0.15 fo r the va lue of R , above which the energy go ing i n t o the buoyancy f l u x i s f 109 s u f f i c i e n t to suppress the t u r b u l e n c e . T h i s l i m i t on R f r e q u i r e s K < 0 . 2 e / N 2 . Oakey( l982) makes an independent P e s t ima te of K from temperature m i c r o s t r u c t u r e measurements (K , P T to be d e f i n e d s h o r t l y ) to es t imate K / (e/N 2 ) which he e s t ima tes P to be 0 . 2 6 ± 0 . 2 1 . S ince the independent e s t ima tes of Osborn and Oakey do not d i f f e r s i g n i f i c a n t l y , g i ven the e s t ima ted f a c t o r of two e r r o r in e, the f a c t o r 0.2 w i l l be c a r r i e d th rough fo r t h i s s tudy . The es t ima ted K , w i th R = 0.15 (or R /(1-R ) = 0.2) p f f f w i l l h e n c e f o r t h be denoted as K . 0 A smoothed es t imate of K i s o b t a i n e d by i n v o k i n g e = a 0 N , 0 which was d i s c u s s e d in Chapter 4. C a l l i n g t h i s K (as i t was G o r i g i n a l l y suggested by G a r g e t t ( 1 9 8 4 ) ) , K = 0 .2a o /N (7.5) G To es t ima te the eddy v i s c o s i t y , K , (7.1) i s aga in used V w i th the t u r b u l e n t eddy v i s c o s i t y d e f i n e d as K = u ' w ' / O u / 3 z ) . (7.6) V In t h i s c a s e , u r e p r e s e n t s the magnitude of the h o r i z o n t a l v e l o c i t y and K i s then the v e r t i c a l c o e f f i c i e n t of eddy V v i s c o s i t y . With (7.1) and ( 7 . 2 ) , K = e/(.1-R )S 2 (7.7) V f 110 Us ing the upper bound of R = 0 .15 , K < 1 .2e/S 2 . f V The a v a i l a b l e measurements a l l ow es t ima tes to be made of K 0 and K from the WESPAC data and K , K and K from PEQUOD. G 0 G V With measurements of temperature m i c r o s t r u c t u r e , an es t imate can be made of the c o e f f i c i e n t of v e r t i c a l d i f f u s i o n of hea t , K . No temperature m i c r o s t r u c t u r e measurements are T r epo r t ed here but the c o e f f i c i e n t s d i s c u s s e d above are compared to es t imates of K made by Gregg ( l976 ) in the e q u a t o r i a l P a c i f i c T and Osborn and B i l odeau (1980) in the e q u a t o r i a l A t l a n t i c . O r i g i n a l l y d e r i v e d by Osborn and COX ( 1 9 7 2 ) , the model i s based on a form of the temperature v a r i a n c e equa t ion which ba lances the p roduc t i on of temperature v a r i a n c e by the buoyancy f l u x and the d e s t r u c t i o n by mo lecu la r d i f f u s i o n , w~ rT r9T/9z = - K ( 9 T ' / 9 Z ) 2 . The eddy c o e f f i c i e n t f o r v e r t i c a l d i f f u s i o n of heat i s d e f i n e d by K = -w~ rT T/(9T/9z) „ (7.8) T so tha t K = K ( 9 T ' / 9 Z ) z / ( 9 T / 9 Z ) 2 (7.9) T Having d e r i v e d the e s t ima te s K , K , K , and K the 0 G T V c o n d i t i o n s of a p p l i c a b i l i t y f o r each must be s t a t e d . The 111 ba lance equa t i ons used fo r the t u r b u l e n t k i n e t i c energy and temperature v a r i a n c e equa t ions have exc luded the e f f e c t s of a d v e c t i o n . These c o e f f i c i e n t s must on l y be a s s o c i a t e d wi th the sma l l s c a l e t u rbu l ence c a u s i n g l o c a l c r o s s - i s o p y c n a l d i f f u s i o n . F u r t h e r , the upper bound of R has been i n f e r r e d from f measurements of t u rbu l ence in shear f lows and Ke l v i n-He lmho l t z i n s t a b i l i t i e s . As Osborn ( l980) n o t e s , the es t ima te K cannot O account fo r double d i f f u s i v e m i x i n g . The same must be t rue fo r K . In e s t i m a t i n g K , an approx imat ion based on i n t e r n a l wave V G arguments has been made, r e l e g a t i n g the use of K to deep ocean G env i ronments o n l y , where the dominant t u r b u l e n t energy source i s thought to be i n t e r n a l waves, and away from r e g i o n s such as e q u a t o r i a l s u r f a c e waters . 7.2 Comparison Of E s t ima tes A compar ison of the v e r t i c a l eddy c o e f f i c i e n t s K , K , K 0 G T and K from the e q u a t o r i a l A t l a n t i c and P a c i f i c i s shown in V Tab l e 5. For the 1982 d a t a , the ranges shown are es t ima tes made from averages over those e q u a t o r i a l drops w i th concu r ren t White Horse measurements. The thermostad es t ima tes a re from drop 14 which was the on l y drop w i th a we l l -deve loped thermostad . The e s t ima tes of K made by Crawford(1982) used a p r o d u c t i o n -V d i s s i p a t i o n ba lance of the t u r b u l e n t k i n e t i c energy equa t ion and hence no f a c t o r of 1/(1 —R ) appears as i t does i n ( 7 . 7 ) . f P a c i f i c . 1 9 8 2 P a c i f i c , 1 9 7 9 C r a w f o r d * 1 9 8 2 ) K K A t l a n t i c , 1 9 7 4 A t l a n t i c . 1 9 7 4 P a c i f i c . 1 9 7 2 C r a w f o r d ( 1 9 8 2 ) O s b o r n ( 1 9 8 0 ) G r e g g ( 1 9 7 6 ) a b o v e c o r e 1 . - 5 . . 0 5 - . 0 7 1 . - 1 0 0 . . 8 - 4 0 . 1 . - 1 0 0 . 1 . - 1 0 . 8 . - 4 0 . 4 . ( 2 x 1 ) x U 9 >5. c o r e . 0 0 3 - . 0 9 . 0 0 7 - . 2 . 1 - 1 0 . . 0 3 - . 2 . 2 - 1 . . 0 2 - . 6 . 1 - 4 . . 0 1 5 ( 2 ± 1 ) x . 0 l < 5 0 x m o l e c u l a r v a l u e b e l o w c o r e . 0 0 0 7 - . 2 . 0 2 - . 0 7 . 0 2 - . 3 .8 1. . 3 - 3 . ( 2 ± 1 ) x ( . 0 4 - . 2 ) t h e r m o s t a d 1. .1 1. . 1 5 be low 300m . 0 0 2 - . 9 . 0 5 - . 5 . 0 2 - 1 0 0 . Table 5 - Comparison of v e r t i c a l eddy c o e f f i c i e n t s ( in u n i t s of cm 2 /sec ) fo r f i v e d i f f e r e n t e q u a t o r i a l data s e t s . The mo lecu la r va lue f o r the theraml d i f f u s i v i t y of water i s = 0.0015 c m 2 / s e c . 113 C r a w f o r d ' s e s t ima tes would be 20% l a r g e r wi th the buoyancy term i n c l u d e d . Osborn ( l980) f i n d s good agreement between K and K in and 0 T above the EUC core from the A t l a n t i c . Fu r the rmore , these a l s o agree w i th the e s t ima tes of K made by Gregg( l976) in the T P a c i f i c in and above the EUC c o r e . T h i s l eads one to agree w i th Munk(l966) tha t the eddy c o e f f i c i e n t s fo r d i f f e r e n t s c a l a r p r o p e r t i e s are the same. It a l s o g i ves c o n f i d e n c e in the assumpt ions made to d e r i v e (7.4) and ( 7 . 9 ) . The es t ima tes g i ven fo r the P a c i f i c , 1982 data and from Crawford(1982) r ep resen t ranges of K and K . R e f l e c t i n g the lower va lues of e found in 0 V the P a c i f i c in 1982, K i s s i g n i f i c a n t l y lower than.any of the 0 o ther e s t ima tes of K and K . In f a c t , in and below the EUC 0 T c o r e , the lower bound i s at or sma l l e r than the mo lecu la r va lue fo r the d i f f u s i o n of hea t , imp l y ing tha t there may be o c c a s i o n s when mo lecu l a r e f f e c t s may r i v a l the t u r b u l e n t f l u x e s . The d e r i v a t i o n of K r e l i e s on an assumpt ion of i n t e r n a l G wave dependence which cannot be expec ted to h o l d t rue in the upper e q u a t o r i a l waters and, as expec ted , K i s much sma l l e r G than K except at depths below 300 mete rs . The va lues of K 0 V below the co re from the P a c i f i c , 1982 are c o n s i d e r a b l y sma l l e r than those from the A t l a n t i c , 1974 and , as has been d i s c u s s e d , c o n s i d e r a b l y sma l l e r va lues of e were found below the core in 1 14 the P a c i f i c i n both 1979 and 1982 than in the A t l a n t i c in 1974. In and above the EUC c o r e , the agreement between the th ree data s e t s i s q u i t e good. F i g u r e 36 shows the 25 meter averaged e s t ima tes of K fo r 0 the e i g h t e q u a t o r i a l p r o f i l e s of PEQUOD (open c i r c l e s ) . The two upper va lues (20-70 meters) are between 1 and 2 c m 2 / s e c . The minimum va lues in the co re of the EUC are . two f a c t o r s of ten sma l l e r or about 10K. The d i s t i n c t i v e minimum at 400 meters i s a s s o c i a t e d w i th the maximum in N below the thermostad and the l o c a l minimum in e which occu r s j u s t above the l o c a l maximum at 500 meters (see F i g u r e s 25 and 26 ) . Below 500 mete rs , K ranges 0 from 0.06 to 0.1 c m 2 / s e c . In F i g u r e 34 the va lues of e and N p l o t t e d in F i g u r e 16 were used to generate more h e a v i l y averaged es t ima tes of K and 0 K from the PEQUOD d a t a . The l ack of agreement above 400 meters G i s , of c o u r s e , due to the d i f f e r i n g assumpt ions i n v o l v e d in the two e s t i m a t e s . Below 400 meters the agreement i s q u i t e good. In t h i s averaged p r o f i l e the s u r f a c e c u r r e n t s a re p o o r l y r e s o l v e d . The deep va lues of K and K are about 0.1 cm 2 /sec at 0 G 900 mete rs . F i g u r e 35 i n c l u d e s s i m i l a r l y d e r i v e d p r o f i l e s from WESPAC c a l c u l a t e d from the e and N va lues of F i g u r e s 14a ,b . At 900 mete rs , K and K are s l i g h t l y g r ea t e r than 0.1 c m 2 / s e c . 0 G The g r ea t e r va lues of K and K are expected in l i g h t of the 0 G g r e a t e r t u rbu l ence l e v e l s from the deeper waters of WESPAC and 115 V e r t i c a l p r o f i l e s of K (equation 7.4) and K 0 G (equation 7.5) for the PEQUOD data, e and N were averaged v e r t i c a l l y over 100 meters and then over a l l of the p r o f i l e s . t/1 w 117 i n c o r p o r a t e d i n t o the c o e f f i c i e n t a 0 e s t imated from F i g u r e , 16. Below 900 meters , K s t e a d i l y i n c r e a s e s and i s about 0.5 cm 2 /sec 0 below 2000 mete rs , assuming that K i s t r u l y a smoothed G r e p r e s e n t a t i o n of K . 0 7.3 Deep Ocean Es t ima tes The deep e s t ima tes of K approach tha t of Munk(l966) who P es t ima ted a cons tan t va lue of K to be about 1 cm 2 /sec in the P deep ocean from a ba lance of the d e n s i t y equa t ion between the upward advec t i on of dense water and the t u r b u l e n t d i f f u s i o n downwards of l i g h t e r water . The ba lance can be expressed as wp - (K p ) = 0, z p z z where the s u b s c r i p t z r ep re sen t s d i f f e r e n t i a t i o n w i th r e spec t to the v e r t i c a l c o o r d i n a t e z . T h i s may be r e w r i t t e n (and Garget t (1984) does t h i s ) as p (w-(K ) ) - K p =0. z p z p zz Munk(l966) assumed K to be a cons tan t va lue i n the deep ocean P between 1 and 4 k i l o m e t e r s in order to s i m p l i f y the a n a l y s i s . With the e s t ima tes of F i g u r e 35 t h i s assumpt ion may be checked . By e s t i m a t i n g the r a te of fo rmat ion of A n t a r c t i c Bottom Water and assuming a un i fo rm spread ing ( r i s i n g ) r a t e over the remainder of the w o r l d ' s oceans , Munk(1966) es t ima ted a g l o b a l l y averaged upwe l l i ng speed , w, of 1x10 " 5 cm/sec. From F rgure 35, (K ) i s e s t ima ted from the v a l ues at 500 and 2000 meters to be p z 118 0.3 cm 2 /sec in 1500 meters or 2X10~ 6 cm/sec. T h i s i s 20% of Munk's va lue f o r w and does not a f f e c t the order of magnitude es t imate made f o r K . A p o i n t to note i s that a cons tan t (K ) P P z = 2X10~ 6 cm/sec g i v e s a va lue of K = 1 cm 2 /sec at 5000 mete rs . P The i n d i c a t i o n i s tha t K i n c r e a s e s wi th d e p t h . A qu i ck P c a l c u l a t i o n , though, shows tha t t h i s t r end does not n e c e s s a r i l y mean tha t the mass f l u x i n c r e a s e s w i th dep th . The mass f l u x i s p7"*7" = -K p . P z With the es t imate K fo r K , G p p'w' = - 0 . 2 a o p /N z but N 2 = -gp /p so tha t z p T w T = 0 .2a o pN/g which, i n d i c a t e s dec reased mass f l u x w i th dep th . 7.4 Comparison Wi th E q u a t o r i a l Model Va lues In a recent paper by Pacanowski and Ph i l ande r (1981 ) the authors express the need fo r a proper p a r a m e t e r i z a t i o n of v e r t i c a l m i x i n g , e s p e c i a l l y as i t a p p l i e s to t h e i r numer i ca l model of the t r o p i c a l ocean . The forms used f o r the eddy v i s c o s i t y and d i f f u s i v i t y a re K = 1 + 5 0 / ( l + 5 R i ) 2 (7.10) vPP K = 0.1 + K /(1+5Ri) (7.11) rPP vPP 119 In F i g u r e s 36 and 37 the va lues of K and K are compared to rPP vPP K and K e s t ima ted from (7.4) and ( 7 . 7 ) . The va lue of Ri used 6 V to eva lua te K and K i s c a l c u l a t e d from the N and S rPP vPP p r o f i l e s of F i g u r e s 24 and 25. The nea res t su r f a ce va lues of K and K are each too rPP vPP low by about a f a c t o r of 4-5, l i k e l y due to the i n a b i l i t y of the Ri-dependence to account fo r the wind mixed l a y e r . The va lue of K in the l a r g e shear r eg ion above the EUC co re i s more n e a r l y rPP equa l to K b u t , beneath t h i s , and to 300 mete rs , K i s sma l l e r 0 0 than K by at l e a s t a f a c t o r of two and up to a f a c t o r of t e n . rPP In the r eg ion of the EUC core i t s e l f , the d i f f e r e n c e i s about a f a c t o r of s i x . The asympto t i c va lue of 0.1 cm 2 /sec appears to be in good agreement w i th K but as d i s c u s s e d above, we expect 0 K to i n c r ea se w i th dep th . I t i s not known whether the s t rong 0 minimum at 400 meters i s anomalous. In g e n e r a l , the shapes of K and K agree in the upper water column i n d i c a t i n g tha t the 0 rPP R i cha rdson number dependence g i v e s the r i g h t sense fo r K rPP The c o n s i s t e n t l y g r ea t e r va lues of K compared to K shou ld rPP 0 not be of g rea t concern due to the lower e s t ima te s of e from 1982 which may r ep resen t an anomalous ly low data s e t . C e r t a i n l y , the va lues of K are in b e t t e r agreement wi th the rPP 120 EDDY DIFFUSIVITY (cm 2/sec) '} 10 200-(0 TJ 400 tx D tn i n ca cu 600 800 1000--2 r IO' 1 • 10< •' ' ' I I I I 10 I I I I I I Vs K - 0.2*/** o K - 0.1*K / ( » * 5 R i ) rPP v P P . - i — i i i 11 i n i i i i i 11 II i i i i I i n F i g u r e 36 - V e r t i c a l p r o f i l e s of 25 metre averages of K and 0 K (e q u a t i o n 7.11) from PEQUOD. rPP 121 EDDY VISCOSITY (cm 2/sec) 10" 2 10" 1 10° 10 0 j . - i . i . i . . i i — i 1 1 1 1 1 i i — i — i i i . i i 2004 u to 4> 400 ca « in LO Ed a. 600 8001 1000 R - 1 . 2 f / S * v R • i+50/( 1+5RD* vPP i i 111 1 1—i i i i i 11 1 r-F i g u r e 37 - V e r t i c a l p r o f i l e s of 25 metre averages of K and V K ( e q u a t i o n 7.10) from PEQUOD. . . n n vPP 122 o ther e s t ima tes of Tab l e 5. In the l a r g e shear r eg ion of the upper e q u a t o r i a l ocean , t h e n , the Ri dependence appears to agree reasonab l y w e l l . However, we expect K to i n c r ease w i th d e p t h . 0 In F i g u r e 37, the cu r va tu r e of R' a c t u a l l y m i r r o r s tha t of V K . But , be s i des the very low va lue of K ( f a c t o r of ten l e s s vPP V than K ) a t 75 mete rs , the e s t ima tes are w i t h i n a f a c t o r of vPP. two agreement ( t h i s i s b e t t e r than the e s t ima ted e r r o r fo r the es t imate of K when one c o n s i d e r s tha t 5e/e 1 ) . In the EUC V core i t s e l f , the e s t ima tes of 1-2 cm 2 /sec agree we l l w i th a l l of the d a t a . Below the c o r e , K i s much g r ea t e r than K but i s vPP V c l o s e r to the A t l a n t i c es t imate of Crawford(1982) in Tab le 5. Perhaps , d i f f e r e n t oceans r e q u i r e d i f f e r e n t p a r a m e t e r i z a t i o n s of eddy c o e f f i c i e n t s . The asympto t i c va lue of K at depth appears V to be too low. 123 V I I I . COMPARISON OF DATA SETS AND PATCH SIZE STATISTICS Some s t a t i s t i c s of e and the d i s t r i b u t i o n of the tu rbu l ence are p resen ted in t h i s c h a p t e r . These p rov ide a common ground fo r compar ison of the two data s e t s i n v o l v e d in t h i s study as we l l as to o ther data se t s a l r e a d y in e x i s t e n c e and to those yet to be c o m p i l e d . Tab le 6 l i s t s h e a v i l y averaged va lues of e over the depth ranges i n d i c a t e d fo r PEQUOD, WESPAC, and a l s o the Vancouver I s l and s lope data of Lueck, Crawford and Osborn (1983 ) . Because of the r e l a t i v e l y sma l l amount of da ta from depths g r e a t e r than 1000 meters from PEQUOD, no r e s u l t s a re g iven fo r tha t range . For the upper range of data from PEQUOD (20-300 m e t e r s ) , 7 i s 85x10 " 7 W/m 3, which i s about e i g h t t imes l a r g e r than e from the comparable range from WESPAC and a l s o e i gh t t imes 7 from 25-500 meters from the Vancouver I s l a n d s l ope d a t a . T h i s l a r g e va lue i s , of c o u r s e , due to the enormous i n f l u e n c e of the s u r f a c e c u r r e n t s t r u c t u r e near the equa to r . Whi le a lmost 50% (% turbu lent or PCT) of the independent e s t ima tes of e a re > 1 0 " 6 W/m3 from t h i s range from PEQUOD, on l y 28% are t u r b u l e n t from WESPAC. The r e l a t i v e r a t i o s 7/PCT i n d i c a t e t h a t , as w e l l as a g r ea t e r p o r t i o n of the water column be ing t u r b u l e n t , the d i s s i p a t i o n averaged over the t u r b u l e n t p o r t i o n i s more than four t imes g r ea t e r (compare 7/PCT = 1.7 fo r PEQUOD 20-300, .47 fo r PEQUOD 300-1000, .39 fo r WESPAC 20-300, .42 f o r WESPAC 300-124 20-300m 300-1000m >1000m 20m-bottom PEQUOD q u a n t i t y o f d a t a ( d b a r ) *(W/m') xlO 7 % t u r b u l e n t (>10-*W/m') WESPAC q u a n t i t y o f d a t a ( d b a r ) e(W/m») xlO 7 % t u r b u l e n t 37B0 8 5 . 49 2180 11. 28 VANCOUVER ISLAND SLOPE e(W/m») xlO' 8015 4 . 7 10 5085 9 . 2 22 25-500m 10. 540 5805 3 .8 12 >500m 12335 3 5 . 21 13070 6 . 5 19 Tab l e 6 - Average v a l ues of e from PEQUOD and WESPAC data s e t s compared to Vancouver I s l and s lope v a l u e s from Lueck , Crawford and Osborn (1983) . These va lues may s l i g h t l y underes t imate t r ue v a l ues of e s i n ce no i s e l e v e l s were set = 0. The q u a n t i t y of data r e f e r s to the t o t a l amount of da ta taken i n each depth range . 1 25 1000, and .32 fo r WESPAC >1000 m e t e r s ) . More i n t e r e s t i n g i s the range 300-1000 mete rs , f o r which e from WESPAC i s twice tha t from PEQUOD and a l s o of the Vancouver I s l a n d s lope da ta fo r the range des i gna t ed by Lueck , Crawford and Osborn ( l983 ) as > 500 meters ( these i n c l ude th ree drops to 1100 me te r s ) . The reason fo r the l a r g e va lue of 7 here i s because twice as much of the water column i s t u r b u l e n t ( s i nce the r e l a t i v e va lues of e/PCT are about equa l from PEQUOD and WESPAC, a l though PCT was not computed fo r the Vancouver I s l a n d s lope data s e t ) . T h i s f a c t i s p a r t i c u l a r l y i n t e r e s t i n g in l i g h t of the e = a 0N— r e l a t i o n posed in Chapter 4. From the i n f o r m a t i o n in Tab le 6 and to f o l l o w , i t appears tha t the va lues of e in the range 300-1000 meters from WESPAC ( i n the range of the second maximum in N) are h i g h e r , not because i n d i v i d u a l  e s t ima tes are s u b s t a n t i a l l y h igher but because more of the water  column i s t u r b u l e n t . Below 1000 meters , thw WESPAC data show the sma l l e s t 7 es t ima te s yet made (3 .8x10" 6 W/m 3 , which i s on ly m a r g i n a l l y above the i n s t r umen ta l no i se l e v e l ) . However, a c o n s i d e r a b l e amount of t u r b u l e n c e s t i l l e x i s t s at depth (^  12% of a lmost 6000 meters of v e r t i c a l p r o f i l i n g ) , i n d i c a t i n g tha t the deep ocean i s not q u i e t . In f a c t , a s l i g h t l y g r ea t e r f r a c t i o n from WESPAC > 1000 meters i s t u r b u l e n t than in the sha l l ower depth range 300-1000 meters from the e q u a t o r i a l d a t a . 1 26 8.1 Loqnormal P r o p e r t i e s Of e I t has been proposed by Gu r v i ch and Yaglom(1967) as we l l as o the r s tha t sma l l s c a l e t u r b u l e n t p r o p e r t i e s such as e a re random v a r i a b l e s which f o l l o w a lognormal d i s t r i b u t i o n . S tewar t , W i l son and Bur l i ng (1970 ) and G i b s o n , Stegen and W i l l i ams (1970 ) s t u d i e d the t u r b u l e n t boundary l a y e r of the atmosphere over the ocean . T h e i r work i n d i c a t e s tha t both the s p a t i a l d e r i v a t i v e s of t u r b u l e n t v e l o c i t i e s and t h e i r squares (as are used to c a l c u l a t e e ) have lognormal p r o p e r t i e s over a range of t h e i r v a lues which i s s u f f i c i e n t l y above the i n s t r umen ta l no i se l e v e l but below the l a r g e s t expec ted v a l u e s . Lueck and Osborn ( l982) b r i e f l y p resen ted the lognormal p r o p e r t i e s of e in and below the wind-mixed l a y e r and t h e i r data i n d i c a t e tha t the re are ranges over which e behaves l o g n o r m a l l y . The d i s s i p a t i o n v a l ues from each subgroup in Tab le 6 were grouped in qua r t e r decade i n t e r v a l s . The cumu la t i v e f requency of occu r rence of each q u a r t e r decade was computed and p l o t t e d on normal p r o b a b i l i t y pape r . Each p o i n t i n F i g u r e s 38-42 r ep r e sen t s the cumula t i ve percentage of o b s e r v a t i o n s wi th v a l ues of e l e s s than or equa l to tha t i n d i c a t e d on the h o r i z o n t a l a x i s . A s t r a i g h t l i n e f i t of the data on normal p r o b a b i l i t y paper i n d i c a t e s that the random v a r i a b l e f o l l o w s a normal p r o b a b i l i t y d i s t r i b u t i o n . S i m i l a r l y , a s t r a i g h t l i n e f i t of the l o g a r i t h m of the random v a r i a b l e i n d i c a t e s tha t the random v a r i a b l e f o l l o w s a lognormal d i s t r i b u t i o n . The form of the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n i s (from S tewar t , W i l son and B u r l i n g d 970)) P(y) = exp[ - ( l n ( y ) - M ) 2/2a 2 ]/(2v)*oy 1 27 where u = l n ( y ) and a2 = l n ( y ) 2 - l n ( y ) 2 . The mean va lue of y i s g i ven by y = exp[u + o2/2]. S ince the va lues of e in F i g u r e s 38-42 are p l o t t e d as base 10 l o g a r i t h m s , i t i s necessary to change the parameters u and a to base 10 l oga r i t hms from n a t u r a l l oga r i t hms ( l og (y ) = 2.3031n(y) ) to c a l c u l a t e y . In F i g u r e s 38-42, u and a are e s t ima ted from the p l o t t e d s t r a i g h t l i n e s and the mean v a l u e , e 0 i s e s t ima ted from t h e s e . For compar i son , the mean va lue e e s t ima ted from the da ta and l i s t e d in Tab le 6 i s i n c l u d e d . The number of e s t ima tes of e used to c o n s t r u c t each p l o t i s n. At sma l l v a l ues of e (<3x l0 _ 7 W/m 3 which i s the no i se l e v e l ) there i s a d i s t i n c t d e v i a t i o n of the behav iour of the data from that of the data above the no i s e l e v e l . The f requency of occu r rence of sma l l v a lues of e p r e d i c t e d from the lognormal p l o t i s too s m a l l . . In f a c t , the d i s t r i b u t i o n i s t r unca t ed at the no i s e l e v e l of the i n s t r u m e n t a t i o n and sma l l v a lues are not measured a l though the re i s no reason to b e l i e v e tha t they do not e x i s t . A somewhat d i f f e r e n t problem e x i s t s at h i gh va lues of e which was d i s c u s s e d by S tewar t , W i l son and Bur l i ng (1970 ) and G i b s o n , Stegen and W i l l i ams (1970 ) and Lueck and Osborn ( l982) and which i s e s p e c i a l l y apparent in F i g u r e s 38, 40 and 41. I t was found tha t the h i g h e s t va lues of e g e n e r a l l y d e v i a t e d from the s t r a i g h t l i n e f i t and t h i s d e v i a t i o n was a t t r i b u t e d to undersampl ing of the very few expec ted l a rge v a l u e s . Each of the F i g u r e s 38-42 e x h i b i t ranges of e over which the data f o l l o w s a s t r a i g h t l i n e r easonab l y w e l l . One es t imate 99.95 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0. 05 10" 8 : 1 1 1 1 1 1 1 1 | T 1 1 1 1 1 1 I | — 1 1 1—r T T I 1 | 1 1— — i — i — t » i i -PEQUOD 20-300m U = -6.05 -• a = .90 -- e = 85x10" 7 W/m3 — • = 76x10" 7 W/m3 - n = 1855 • -• • I l 1 1 1 A J 1 1 1 1—1 1 1 1 1 . A . 1 1J.1 i i _ i i l i l t . 10 - 7 1 0 • 6 LOG e (W/m3) 10 - 5 10 - o M CO F igu re 38 - Cumulat ive d i s t r i b u t i o n of the base 10 l oga r i t hm of d i s s i p a t i o n va lues from PEQUOD 20-300m p l o t t e d a g a i n s t normal p r o b a b i l i t y c o - o r d i n a t e s . The parameters u and a a re the mean and s tandard d e v i a t i o n of the n a t u r a l l oga r i t hm of e, e s t ima ted from the s t r a i g h t l i n e . e 0 i s es t imated from n and a. 7 i s the observed mean from Tab le 6. The number of independent 2 meter e s t ima tes of e i s n. F igu re 39 - Cumulat ive d i s t r i b u t i o n of the base 10 l o g a r i t h m of d i s s i p a t i o n va lues from PEQUOD 300-1000m. F igure 40 - Cumulat ive d i s t r i b u t i o n of the base 10 l o g a r i t h m of d i s s i p a t i o n va lues from WESPAC 20-300m. 10- 8 1 0 " 7 1 0 " 6 1 0 " 5 1 0 " 4 LOG e (W/m3) F igure 41 - Cumulat ive d i s t r i b u t i o n of the base 10 l o g a r i t h m of d i s s i p a t i o n va lues from WESPAC 300-1000m. F igure 42 -133 of the degree to which the s t r a i g h t l i n e f i t t e d to the p o i n t s i s r e p r e s e n t a t i v e of the data i s the agreement of the es t imate e 0 made from the l i n e wi th e. The es t ima tes of the mean va lues of the 20-300m data se t s are the b e s t , l i k e l y due to the extended dynamic range , e s p e c i a l l y in the case of the PEQUOD d a t a . A l l of the e s t i m a t e s , however, a re w i t h i n a f a c t o r of two of the observed mean v a l u e s . 8.2 Patch S i z e S t a t i s t i c s To get an idea of the d i s t r i b u t i o n of both t u r b u l e n t pa tch s i z e s and t h e i r r e l a t i v e c o n t r i b u t i o n to the s p a t i a l l y averaged d i s s i p a t i o n va lues d i s c u s s e d above, a t u r b u l e n t pa tch was d e f i n e d . -For the purposes of t h i s s tudy , a t u r b u l e n t pa tch must have at l e a s t one independent es t imate of e > 1 0 " 6 W/m3 ( thereby l i m i t i n g the s m a l l e s t pa tch s i z e to ^ 2 m e t e r s ) . I n s ide of the pa t ch i t s e l f , no two s u c c e s s i v e independent e s t ima tes of e may be < 1 0 " 6 W/m 3. T h i s second c r i t e r i o n does not a c t u a l l y change the s t a t i s t i c s c o n s i d e r a b l y but does a l l ow f o r s i n g l e 2 meter t h i c k q u i e t s t r e t c h e s to occur w i t h i n a much l a r g e r p a t c h . The m o t i v a t i o n f o r t h i s came from l o o k i n g at l ab s t u d i e s of K e l v i n -He lmhol tz b i l l o w s (Turner(1 973) , Van D y k e d 982 ) ) , which appear to be q u i t e s t a b l e in the c en t r e d u r i n g the i n i t i a l s tages of i n s t a b i l i t y wh i l e the l a r g e shear at the edges suggests t u r b u l e n c e . Whi le t h i s d e f i n i t i o n i s c e r t a i n l y not meant to be r i g o r o u s i t does p rov ide an o b j e c t i v e c r i t e r i o n which can be used to d e r i v e some i n t e r e s t i n g compar i sons . P r i o r to d i s c u s s i n g the con ten ts of Tab l e s 7-11, the means 134 of d e r i v i n g these are shown. From the r e s p e c t i v e data s e t s , t u r b u l e n t pa tches (as d e f i n e d above) were s o r t e d a c c o r d i n g to t h e i r v e r t i c a l s c a l e (t = t h i c k n e s s or pa t ch s i z e ) . The number of pa tches which f e l l i n each Range (<3 , 3-10, 10-30, >30 meters) were counted and l i s t e d under #Patches. These ranges r ep resen t h a l f decade and t h e r e f o r e order of magnitude r anges . The pa tch s i z e s in each range were summed and l i s t e d under T o t a l T h i c k n e s s . T h e i r r e l a t i v e importance was es t ima ted by d i v i d i n g the t o t a l t h i c k n e s s of each range by the t o t a l data (H) i n the data subse t , and the average pa tch s i z e , t , was computed. W i th in each p a t c h , the patch-averaged d i s s i p a t i o n , e , was t c a l c u l a t e d , summed over a l l of the patches in the range , m u l t i p l i e d by t and d i v i d e d by He where e i s the average d i s s i p a t i o n over the e n t i r e data subset (and i s i n c l u d e d in Tab le 5 ) . T h i s s t a t i s t i c , lOOt le /(He), i n d i c a t e s the r e l a t i v e t c o n t r i b u t i o n of each range to the v e r t i c a l l y i n t e g r a t e d d i s s i p a t i o n . The d i s s i p a t i o n averaged over a l l of the pa tches in a s i n g l e range i s e and t h i s i s o l a t e s the a c t u a l magnitude t of e v a l u e s w i t h i n the range from the r e l a t i v e c o n t r i b u t i o n (which i s a l s o a f u n c t i o n of the f r a c t i o n of the water column which i s t u r b u l e n t ) . As e x p e c t e d , i n the upper 300 meters of the PEQUOD data (Table 7 ) , l a r g e patches due to the l a r g e mean shear dominate the water co lumn; 34% of the e n t i r e upper 300 meters has Range •Patches T o t a l Thickness % T o t a l t Zt t tZe t I00t£e /(Hi) t =tZe /Zt t t t <3 33 76 2.0% 2.3 5.4x10' 1 12x10" 5 0.4% 1.6x10"* 3-10 32 211 5.6 6.6 7.6 50 1.6 2.4 10-30 16 266 7.0 17. 6.4 106 3.3 4.0 >30 20 1290 34. 65. 40. 2570 80. 20. 49% 85% t o t a l data •* 3780 meters H " t o t a l data t « 85.x10- 7 W/m1 t = drop averaged d i s s i p a t i o n € = i n d i v i d u a l patch average He « 32.xIO"' W/m' t e • b i n averaged d i s s i p a t i o n PEQUOD 20-300 meters t t = patch t h i c k n e s s t <* b i n averaged patch t h i c k n e s s LO cn Table 7 -Range ^Patches . T o t a l Thickness % T o t a l Le tZt 100t£e /{Hi) t -tl« /Et t t t <3 3-10 10-30 >30 64 43 20 3 139. 218 293 134 1.7% 2.7 3.7 1.7 10% 2.2 5.1 15. 45. 1 1 . x l O " 5 10. 5.1 1.3 23x10" 5 52 75 58 6. 1% 14. 20. 15. 55% 1.6x10"' 2.4 2.6 4.3 t o t a l data =• 80)5 meters t » 4.7x10"' W/m1 He * 3.8x10'' W/mJ PEQUOD 300-1000 meters H = t o t a l d a t a c «= drop averaged d i s s i p a t i o n < = i n d i v i d u a l p a t c h average t ( » b i n averaged d i s s i p a t i o n t t " patch t h i c k n e s s t •= b i n averaged p a t c h t h i c k n e s s CT\ Table 8 - Patch s i z e s t a t i s t i c s f o r the PEQUOD data set over the depth range 300-1000 mete rs . Ranqe #Patches T o t a l Thickness % T o t a l t <3 29 64 2.9% 2.2 3-10 20 109 5.0 5.5 10-30 12 254 12. 21 . >30 4 191 8.8 48. 28% t o t a l data <* 2180 meters 7 » 11.xlO" 7 W/m' H7 * 2.4x10' 1 W/m2 WESPAC 20-300 meters Ee t i e I00t£e /(He) e =tLe / E t t t t t t 4 . 4 x l 0 " s 10x10"* 4.1% 1.5x10"' 4.2 23 9.5 2.1 4.8 100 42. 4.0 1.3 64 27^ 3.3 82% H « t o t a l data e = drop averaged d i s s i p a t i o n e = i n d i v i d u a l p a t c h average t 7 ° bin averaged d i s s i p a t i o n t t * patch t h i c k n e s s t • b i n averaged patch t h i c k n e s s Table 9 - Patch s i z e s t a t i s t i c s f o r the WESPAC data set over the depth range 20-300 meters. Ranqe ((Patches T o t a l Thickness % T o t a l t Zt t tZt t I00t£« /(Ht) 7 -tZt /Zt t t t <3 73 159 3.1% 2.2 12.X10-' 26x10 - 5 5.6% 1.7xl0-« 3-10 62 387 7.6 6.2 14. 86 18. 2.2 10-30 27 401 7.9 15. - 8.7 130 28. 3.2 >30 5 181 3.6 36. 2.7 98 21 • 5.4 22% 73% t o t a l data • 5085 meters H - t o t a l data < * 9.2x10" 7 W/m* < - drop averaged d i s s i p a t i o n c = i n d i v i d u a l patch average H7 » 4.7x10-' V/m2 t t - b i n averaged d i s s i p a t i o n WESPAC 300-1000 meters t t • patch t h i c k n e s s t - b i n averaged patch t h i c k n e s s C O C O Table 10 - Patch s i z e s t a t i s t i c s f o r the WESPAC da ta set over the depth range 300-1000 mete r s . Ranqe #Patches T o t a l Thickness % T o t a l t <3 51 107 1.8% 2.1 3-10 47 280 4.8 6.0 10-30 17 257 4.4 15. >30 1 33 0.6 33. 12% t o t a l data =• 5805 meters 7 =• 3.8x10" 7 W/m1 H7 » 2.2x10-' W/mJ WESPAC > 1000 meters 2e tZe 100tE« /(Ht) t =tZe / E t t t t • t t 8.8x10" s 1 9 x 1 0 s 8.4% 1.7x10"' 8.7 52 24. 1.9 3.8 58 26. 2.2 .33 11 4_j9 3.3 63% H = t o t a l data, L O 7 » drop averaged d i s s i p a t i o n < >• i n d i v i d u a l p a t c h average t 7 • b i n averaged d i s s i p a t i o n t t • patch t h i c k n e s s t •= b i n averaged p a t c h t h i c k n e s s Tab le 1 1 -1 40 t u r b u l e n t pa tches > 30 meters t h i c k (t = 65 me te r s ) . 80% of the v e r t i c a l l y i n t e g r a t e d d i s s i p a t i o n (He) i s concen t r a t ed in t h i s 34% of the d a t a . Below 300 meters (Table 8 ) , o n r y T " " pa tches wi th t > 30 meters were found , r e p r e s e n t i n g < 2% of the t o t a l d a t a . I n d i c a t i v e of the r e l a t i v e l y qu i e s cen t deep wa te rs , on l y 10% i s t u r b u l e n t . F u r t h e r , on l y 55% of the v e r t i c a l l y i n t e g r a t e d d i s s i p a t i o n i s con t a i ned in pa tches wi th e > 1 0 " 6 W/m 3, i n d i c a t i n g tha t a r e l a t i v e l y l a r g e c o n t r i b u t i o n must be from r eg ions where e i s j u s t above the no i se l e v e l but < 1 0 " 6 W/m3 ( s i n c e , in a v e r a g i n g , v a l ues of e < no i se were set = 0 ) . The no tab l e t r end i s f o r the g r ea t e r pa tch-averaged d i s s i p a t i o n s , e , to be c o n c e n t r a t e d in the l a r g e r p a t c h e s . In f a c t , t h i s t t r end i s e v i den t in a l l of the o ther data subse ts wi th a minor c o n t r a d i c t i o n i n Tab le 9. S u c c e s s i v e l y l a r g e r pa tch s i z e s have s u c c e s s i v e l y l a r g e r averaged d i s s i p a t i o n s . But the r e l a t i v e c o n t r i b u t i o n to the v e r t i c a l l y i n t e g r a t e d d i s s i p a t i o n i s in a l l cases (o ther than the upper e q u a t o r i a l waters) c o n c e n t r a t e d in the pa tches of range 10-30 meters s imply because a g r ea t e r p r o p o r t i o n of the t u rbu l ence in the water column e x i s t s i n t h i s range of pa t ch s i z e s . The e -t dependence i s shown in F i g u r e s 43 and 44. These t are p l o t t e d on l o g - l o g s c a l e so tha t h a l f decade ranges (or b i n s ) a re e q u a l l y spaced (a l though s u c c e s s i v e e s t ima tes of t are 141 • • I I I I 1 ] I I 1_ PEQUOD >300 METERS E 10" 10° 1 I I I M 10' -i—I'M' 10J LOG t (m e t e r s ) Log- log p l o t of average pa tch-averaged d i s s i p a t i o n s vs average pa t ch t h i c k n e s s f o r the PEQUOD data below 300 m e t e r s . 142 10" _J i i i i 1 1 - i i i • • • I 9 I* 8 •J VESPflC >300 METERS 1 0" '-) 1 ( — i — | — i | i i | 1 — | — i — | — i | i i 10° 10' LOG t (meters) 10J F i gu re 44 - Log- log p l o t of average patch-averaged d i s s i p a t i o n s vs average pa t ch t h i c k n e s s f o r the WESPAC data below 300 me te r s . 143 not n e c e s s a r i l y e q u a l l y spaced ) . The PEQUOD data below 300 meters i s shown in F i g u r e 43 (the numbers a re d i r e c t l y from Tab le 8) wh i le F i g u r e 44 combines the data of T a b l e s 10 and 11. The p l o t s show q u i t e c l e a r l y tha t s u c c e s s i v e l y l a r g e r pa tch s i z e s have s u c c e s s i v e l y l a r g e r v a l u e s of e . S i m i l a r p l o t s were t made us ing s m a l l e r , l i n e a r l y spaced b ins (0-3, 3-6, 6-9, . . . ) and the t r end was e q u a l l y ev iden t w i t h , however, somewhat more no i se due to the sma l l sample s i z e s in some b i n s . A compar ison of F i gu r e s 43 and 44 shows that n e i t h e r the e s t ima te s of e nor t the es t imates of t d i f f e r s i g n i f i c a n t l y between the data s e t s . However, h a l f of the data from WESPAC are below 1000 meters compared to none in t h i s range from PEQUOD. The a p p r o p r i a t e l eng th s c a l e to which these pa t ch s i z e s may be compared and which can be e s t ima ted from the a v a i l a b l e da ta i s the buoyancy l eng th s c a l e , d e f i n e d by L =(e/N 3 )^ b (Turner (1973 ) , p143 ) , which r e p r e s e n t s the s c a l e of motion where buoyancy f o r c e s become of the same order as the i n e r t i a l f o r c e s . L , t hen , i s an a p p r o p r i a t e s c a l e f o r the l a r g e s t edd ies which b are ab l e to o v e r t u r n , g i ven the background tu rbu l ence and s t r a t i f i c a t i o n . Ga rge t t et a l . ( 1 9 8 l ) d i s c u s s the s i g n i f i c a n c e of L and d e s c r i b e s the s u c c e s s f u l s c a l i n g of the buoyancy b subrange of h i g h wavenumber ocean i c energy s p e c t r a u s i n g buoyancy pa ramete rs , e and N. 144 L was es t ima ted by de t e rm in ing pa tch averages over a l l of b the data below 300 meters fo r which the re were a l s o CTD data to c a l c u l a t e N. For each p a t c h , e and N were e s t i m a t e d , grouped t i n t o h a l f decade b ins and f u r t h e r averaged to get L f o r each t . b These v a r i a b l e s are p l o t t e d in F i g u r e s 45 and 46. From PEQUOD, L ranges from 25 to 40 cm and i s between 30 b and 40 cm from WESPAC. The es t imate fo r the upper b i n from PEQUOD (> 30 meters) i s not p l o t t e d because the re were too few patches f o r a comparable a ve rage . A l though the 10-30 meter va lue of L from PEQUOD i s h i ghe r than the o ther two the re i s no b ev idence of a t r end from these p l o t s . In f a c t , L i s remarkably b c o n s t a n t . I n d i v i d u a l e s t ima tes a c t u a l l y range from 9 t o 113 cm but most f a l l in the range 20-40 cm, as do the a ve rages . (The va lue of 113 cm i s from 2030 meters of WESPAC drop 11 where e = t 3 x 1 0 " 6 W/m3 and N was es t ima ted from the nea res t CTD s t a t i o n at drop 12 to be ^ 0.0013 r a d / s e c . T h i s i s by f a r the l a r g e s t va lue of L ). U n f o r t u n a t e l y , these s c a l e s are sma l l e r than the b v e r t i c a l s p a t i a l r e s o l u t i o n of the e measurements and are concea l ed i n the lower b in (< 3 m e t e r s ) . 114 of 311 patches counted from both data se t s below 300 meters were < 3 meters , r e p r e s e n t i n g 46% of a l l of the p a t c h e s . However, these on l y 1 45 _i i—i i i i • I I I i i i 10c -i—i—i—r T - r -10" 1 1 — I — i — r - i - r 10J LOG t (meters) Log- log p l o t of average buoyancy l e n g t h s c a l e vs average pa t ch t h i c k n e s s f o r the PEQUOD data below 300 m e t e r s . 146 10° • • i i i i • i • i — i i i • i 8 -I r — | — I | • • 10" 10C -i r io1 10J LOG t (meters) F i g u r e 46 - Log- log p l o t of average buoyancy l eng th s c a l e vs average pa t ch t h i c k n e s s f o r the WESPAC da ta below 300 mete r s . 147 rep resen t 310 of 2079 meters of da ta wi th e > 1 0 " 6 W/m 3, or about 15%, and c o n t r i b u t e on l y 6% to the v e r t i c a l l y averaged d i s s i p a t i o n . The dominant p a t c h e s , t hen , are many t imes L in b t h i c k n e s s . 148 IX. DISCUSSION AND CONCLUSIONS The main c o n t r i b u t i o n of t h i s t h e s i s i s the a d d i t i o n to the g l o b a l data set of ocean t u rbu l ence measurements. In comparing the measurements of t h i s t h e s i s to those of o the r workers , the 7-N p l o t of F i g u r e 16 p r o v i d e d a conven ien t s t a n d a r d . In order to make b e t t e r use of the v a r i o u s da ta se t s which now e x i s t , f u r t h e r mean ing fu l formats must be e s t a b l i s h e d . C e r t a i n l y , a ve rag ing i s one . But , a l s o , the s t a t i s t i c a l compar ison of data made here in Chapter 8 p r o v i d e s a number of s tandards fo r comparing l a r g e data s e t s . Not on l y are averages made over d i s c r e t e v e r t i c a l i n t e r v a l s , but e s t ima tes are made of the f r a c t i o n of the water column which i s t u r b u l e n t ; pa tch t h i c k n e s s e s ; c o n t r i b u t i o n of ranges of pa tch s i z e s to the v e r t i c a l l y i n t e g r a t e d d i s s i p a t i o n ; and patch-averaged d i s s i p a t i o n s . The data from depths > 300 meters i n d i c a t e a s t r ong r e l a t i o n s h i p between h e a v i l y averaged va lues of e and N. F u r t h e r , i t i s seen tha t t h i s t r end i s due to more f r e q u e n t l y o c c u r r i n g t u r b u l e n c e r a the r than to much h ighe r i n d i v i d u a l e s t ima tes of e . S c a l i n g arguments f o r energy and ra te of l o s s of energy in the i n t e r n a l wave f i e l d suggest a r e l a t i o n e a N 1 . The cons tan t of p r o p o r t i o n a l i t y , a 0 , i s e s t ima ted fo r four data s e t s and ranges from (1.4 to 4 )X1 0 ~ 7 m 2 / s e c 3 « s e c . An es t imate of the t ime s c a l e fo r the decay of the i n t e r n a l wave f i e l d i s r = T E / e , and depends on the s u r f a c e - e x t r a p o l a t e d buoyancy f requency , N 0 , and a 0 . The best range of e s t ima tes from the 149 data se t s are 10-100 days , which i s i n the range of e s t ima tes made independent l y by 01be rs ( l 983 ) and G a r r e t t and Munk( l979) . A major impediment to the r e l i a b i l i t y of the es t imate i s the f a c t o r of two u n c e r t a i n t y in the GM s p e c t r a l e s t i m a t e s . A b e t t e r es t imate of the t ime s c a l e must await j o i n t measurements of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n and i n t e r n a l wave s p e c t r a . A very s imple model i s p resen ted which accounts fo r the d e s c r i b e d dependence of e and N in at l e a s t a q u a l i t a t i v e manner. The p r e d i c t i o n i s based on the assumpt ion tha t the l o c a l t u r b u l e n t d i s s i p a t i o n i s due to the b reak ing of i n t e r n a l waves and tha t the occur rence of b r eak ing events i s determined by the p r o b a b i l i t y that the R i cha rdson number i s l o c a l l y reduced to a va lue l e s s than 1/4 due to a random s u p e r p o s i t i o n of i n t e r n a l wave shea r s . A reasonab le f i t i s made to the d i s t r i b u t i o n of the t u rbu l ence i n the water column through the -1/N e dependence- of P r ( R i ) , a l though i t i s not c l e a r how to r e l a t e the magnitudes of the da ta parameter PCT (percent t u rbu l ence ) and the model parameter P r (R i<1/4 ) . Nea r l y s y n o p t i c Camel III and White Horse measurements from the e q u a t o r i a l P a c i f i c permi t a compar ison of the t u r b u l e n t d i s s i p a t i o n es t ima tes to the l o c a l shear and d i f f e r e n c e R i cha rdson numbers. Independent 25 meter e s t ima te s i n d i c a t e a poor c o r r e l a t i o n between both e-S and e - R i . On a h e a v i l y averaged b a s i s , however, a s t r ong dependence e x i s t s between e and S and e and Ri ( f o r Ri < 10) . The poor c o r r e l a t i o n s fo r i n d i v i d u a l e s t ima tes shou ld not be too s u r p r i s i n g i f one 150 c o n s i d e r s tha t l a r g e shears (and low R i ) which cause the t u rbu l ence may, in f a c t , l o c a l l y decrease ( i n c r ea se ) due to the o v e r t u r n i n g even t . In t h i s c a s e , the t u rbu l ence measured may be in a s l i g h t l y decayed s t a t e from the o r i g i n a l e ven t . C o n v e r s e l y , r eg ions of l a r g e shear (or low R i ) which have not yet o ve r tu rned may have q u i t e low va lues of e. Perhaps a s imp l e r reason i s the f a c t tha t the Camel measurements were not s ynop t i c w i th the White Horse measurements in e i t h e r t ime or space . The s u b s t a n t i a l l y sma l l e r averaged d i s s i p a t i o n s from the e q u a t o r i a l P a c i f i c in 1982 (as opposed to the 1979 r e s u l t s of Crawford(1982) ) must be c o n s i d e r e d to be i n d i c a t i v e of the s p a t i a l and tempora l v a r i a b i l i t y of the t u rbu l ence in the r e g i o n . Appa ren t l y the v a r i a t i o n in the mean shear i s not as g rea t as the v a r i a t i o n in e, e i t h e r from a compar ison of i n d i v i d u a l drops or from a compar ison of 1979 and 1.982 data s e t s . Nor i s a d i r e c t r e l a t i o n s h i p (on the b a s i s of i n d i v i d u a l drops ) found between shea r , Ri or wind speed . The f i n d i n g s may not be s u r p r i s i n g in l i g h t of the p r o d u c t i o n - d i s s i p a t i o n model of the EUC proposed by Crawford and Osborn(1979b, 1981b). In t h i s b a l a n c e , the dominant terms are e and the p r o d u c t i o n of t u r b u l e n t k i n e t i c energy by the Reynolds s t r e s s e s working a g a i n s t the mean shear and hence to r e l a t e the v a r i a b i l i t y of e r e q u i r e s a measure of the v a r i a b i l i t y of the Reynolds s t r e s s e s as w e l l as of the mean shea r . In terms of the ba lance of terms of the mean k i n e t i c energy e q u a t i o n , the much sma l l e r d i s s i p a t i o n s .measured in 1982 are not l a r g e enough to ba lance 151 the work done by the zona l p r e s su re g r a d i e n t between the l e v e l of ze ro zona l v e l o c i t y and the unde rcu r ren t c o r e . Hence, o ther terms in the mean k i n e t i c energy equa i ton must be impor t an t . E s t ima tes of eddy c o e f f i c i e n t s from a number of sources compare f a vou rab l y w i th those made here rang ing from 1-40 cm 2 /sec in the l a r g e mean shear above the EUC core and from .003-.5 cm 2 /sec in the r eg ion of the core i t s e l f . At depths g r ea t e r than 300 mete rs , the e s t ima tes made from both the e q u a t o r i a l and the western P a c i f i c data agree very c l o s e l y . The eddy c o e f f i c i e n t fo r mass, K , i n c r e a s e s wi th d e p t h , P e x t r a p o l a t i n g to Munk ' s ( l966 ) va lue of 1 cm 2 /sec at 5000 mete r s . The i n c r ea se of K w i th depth does not imply g r ea t e r t u r b u l e n t P f l u x e s wi th d e p t h . An a n a l y s i s of pa tch s t a t i s t i c s -indicates tha t i ) a v e r age va lues of d i s s i p a t i o n averaged over a s i n g l e p a t c h , e , are t l a r g e r fo r t h i c k e r patches than fo r t h i nne r patches and i i ) t h e buoyancy l e n g t h s c a l e L i s v i r t u a l l y c o n s t a n t , r e g a r d l e s s of b pa t ch s i z e , t , imp l y ing that pa tches are many t imes L in b t h i c k n e s s . The parameter L i n d i c a t e s the l a r g e s t o v e r t u r n i n g b s c a l e s , and hence , a s i n g l e t u r b u l e n t pa t ch would be compr i sed of a number of ad jacen t o v e r t u r n i n g e ven t s . 152 BIBLIOGRAPHY Crawford ,W .R . (1976 ) : Tu rbu l en t energy d i s s i p a t i o n in the A t l a n t i c e q u a t o r i a l u n d e r c u r r e n t . Ph.D. t h e s i s , I n s t i t u t e of Oceanography, U n i v e r s i t y of B r i t i s h Co lumb ia , 150pp. C rawfo rd ,W .R . 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Ph.D. t h e s i s , Department of Oceanography, U n i v e r s i t y of B r i t i s h Co lumbia . Oakey ,N . S . ( 1 982) : De te rmina t ion , of the r a t e of d i s s i p a t i o n of t u r b u l e n t energy from s imul taneous temperature and v e l o c i t y shear m i c r o s t r u c t u r e measurements. J .Phys .Oceanogr . , J_2 , 256-271. Oakey ,N .S . and J . A . E l l i o t t ( 1 9 8 2 ) : D i s s i p a t i o n w i t h i n the s u r f a c e mixed l a y e r . J . Phys .Oceanogr . , _1_4 , 171-185. 0 1 b e r s , D . J . ( 1 9 8 3 ) : Models of the ocean i c i n t e r n a l wave f i e l d . Rev.Geophys.Space P h y s . , 2_1. > 1567-1606. O s b o r n , T . R . ( 1 9 7 4 ) : V e r t i c a l p r o f i l i n g of v e l o c i t y m i c r o s t r u c t u r e . J . P h y s . O c e a n o g r . , 4 , 109-115. O s b o r n , T . R . ( 1 9 8 0 ) : E s t ima t e s of the l o c a l r a t e of v e r t i c a l d i f f u s i o n from d i s s i p a t i o n measurements. J . P h y s . O c e a n o g r . , 10 , 83-89. O s b o r n , T . R . and C . S .Cox (1972 ) : Oceanic f i n e s t r u c t u r e . G e o p h y s . F l u i d . D y n . , 3 , 321-345. O s b o r n , T . R . and L . E . B i l o d e a u ( 1 9 8 0 ) : Temperature m i c r o s t r u c t u r e measurements in the e q u a t o r i a l A t l a n t i c . J . P h y s . O c e a n o g r . , 7 , 66-82. O s b o r n , T . R . and W.R .Crawford (1980) : An a i r f o i l probe f o r measur ing t u r b u l e n t v e l o c i t y f l u c t u a t i o n s in water . in A i r - S e a I n t e r a c t i o n , Dobson, Hasse and D a v i e s , e d s . , 369-386. 156 Pacanowsk i ,R .C . and S . G . H . P h i l a n d e r ( 1 9 8 1 ) : P a r a m t e r i z a t i o n of v e r t i c a l m ix ing in numer i ca l models of t r o p i c a l oceans . J . P h y s . O c e a n o g r . , , 1443-1451. P h i l l i p s , 0 . M . ( 1 9 7 7 ) : The dynamics of the upper ocean , 2nd e d i t i o n , Cambridge U n i v e r s i t y P r e s s , Cambr idge, 336pp. Pond,S . and G . L . P i c k a r d ( 1 9 8 3 ) : I n t r o d u c t o r y Dynamic Oceanography , 2nd e d i t i o n , Pergamon P r e s s , O x f o r d , 329pp. S c h m i t t , R . W . , J r . and D . L . E vans (1979 ) : An es t ima te of the r a t e of v e r t i c a l m ix ing due to s a l t f i n g e r s based on o b s e r v a t i o n s i n the North A t l a n t i c C e n t r a l Water. J . G e o p h y s . R e s . , 83 , 2913-2919. S c h m i t z , W . J . , J r . , P . P . N i i l e r , R . L . B e r n s t e i n and W .R .Ho l l and (1982 ) : Recent long-term moored ins t rument o b s e r v a t i o n s i n the western Nor th P a c i f i c . J . G e o p h y s . R e s . , 87 , 9425-9440. Stewart ,R.W. and A .A .Townsend(1952) : S i m i l a r i t y and s e l f -p r e s e r v a t i o n i n i s o t r o p i c t u r b u l e n c e . P h i l . T r a n s . R o y . S o c . L o n d o n , 243A , 359-386. S tewar t ,R .W. , J . R . W i l s o n and R .W .Bu r l i ng (1970 ) : Some s t a t i s t i c a l p r o p e r t i e s of smal l s c a l e t u rbu l ence in an a tmospher i c boundary l a y e r . J . F l u i d . M e c h . , 4j_ , 141-152. T u r n e r , J . S . ( 1 9 7 3 ) : Buoyancy E f f e c t s in F l u i d s , Cambridge U n i v e r s i t y P r e s s , Cambridge, 368pp. Van Dyke ,M . (1982 ) : An Album of F l u i d Mot ion , P a r a b o l i c P r e s s , S t a n f o r d , 176pp. W y r t k i , K . ( 1 9 8 3 ) : An attempt to moni tor the E q u a t o r i a l U n d e r c u r r e n t . J . G e o p h y s . R e s . , 88 , 775-777. 157 APPENDIX A - HYDRODYNAMICS The i n s t rumen t , Camel I I I , i s des igned to be p o s i t i v e l y buoyant in water . B a l l a s t i s p r o v i d e d by l e a d weights f a s tened to the body by AWG20 wire which i s in s e r i e s w i th a p r e s su re s e n s i t i v e shear wire c y l i n d e r r e l e a s e sys tem. The buoyant f o r c e i s c h i e f l y due to the main p r e s su re tube , which measures 30.5 cm in d i amete r , 152 cm long and 1.9 cm t h i c k . The m a t e r i a l i s an aluminum a l l o y (6061-T6) w i th a s p e c i f i c g r a v i t y of 2 .77 . The l i f t g i ven by t h i s geometry and m a t e r i a l i s then s imply c a l c u l a t e d to be 42.0 kg over the tube l eng th (us ing a nominal seawater d e n s i t y of 1028 kg/m 3 ) . The end caps to the p res su re tube are a c a s t aluminum' a l l o y (A536-T6). The shape i s h e m i s p h e r i c a l but w i th an i r r e g u l a r su r f a ce making i t d i f f i c u l t to c a l c u l a t e the net e f f e c t on the buoyancy of the i ns t rument . I t c a n , however, be measured. The remain ing components of the i n s t rument , o ther than the p r e a m p l i f i e r c a s e , ac t s o l e l y as b a l l a s t . The ins t rument was weighed in Monterey harbour from the R/V A c a n i a , opera ted by the Un i t ed S t a t es Naval Pos tgraduate S c h o o l . It was found to be 1.2 kg heavy a f t e r 12.2 kg of expendable l e ad weights p l u s 3.1 kg of l e a d a t t a ched to the nosep iece were added. Another 2.0 kg were added to the nosep iece to ensure that the ins t rument would con t inue to s ink in a r eg ion where very l a r g e d e n s i t y change w i th depth o c c u r s . The added weight a c t s f u r t h e r to s t a b i l i z e the inst rument (by add ing mass below the c en t r e of g r a v i t y ) and i n c r ease the f a l l r a t e , thereby changing the shear s e n s i t i v i t y . An u n d e s i r a b l e s i de e f f e c t i s 158 the i n c r e a s e d energy a v a i l a b l e fo r v i b r a t i o n , which i s the l i m i t i n g f a c t o r in the r e s o l u t i o n of the d i s s i p a t i o n c a l c u l a t i o n (see Appendix G ) . Two o ther s a f e t y c r i t e r i a are met by the sys tem. F i r s t of a l l , the ins t rument i s p o s i t i v e l y buoyant shou ld on l y one weight f a l l o f f the i ns t rument , a l l o w i n g f o r some margin of s a f e t y in case of a m a l f u n c t i o n in the p r e s su re r e l e a s e mechanism. As . w e l l , the weak l i n k in the system i s thought to be the p ressu re t r ansduce r l i n k to the ambient p r e s s u r e , which i s a threaded connec t i on to the p r e a m p l i f i e r c a s e . The ins t rument w i l l be p o s i t i v e l y buoyant shou ld the p r e a m p l i f i e r case f l o o d , and on ly one weight f a l l o f f . The f a l l speed of the inst rument i s governed by the nega t i ve buoyancy of the inst rument and the drag f o r c e on . the ins t rument in mot ion . The drag f o r c e i s e m p i r i c i z e d i n the form of a squared drag law D = kW2 where k = pAC /2g (A i s the d f r o n t a l c r o s s - s e c t i o n and C i s the drag c o e f f i c i e n t ) i s a d q u a n t i t y which can be d i r e c t l y c a l c u l a t e d knowing the mass and the f a l l r a t e of the i n s t rument . The drag parameter , k, was de te rmined from drops made in Howe Sound in A p r i l , 1981 aboard the CNAV Endeavour and these measurements were con f i rmed on o ther c r u i s e s . With 8.5 kg of added l e ad weight (7.7 kg in water) the measured f a l l r a te was 1.1 m/s, r e s u l t i n g in a va lue fo r k of 27.5 kg/m. I t i s i n t e r e s t i n g to c a l c u l a t e the drag c o e f f i c i e n t , C , d f o r the ins t rument under these c i r c u m s t a n c e s . The a rea 159 p r o j e c t e d to the f low i s A = TT ( .305/2) 2 = . 0 7 m 2 . Us ing a nominal v a lue of 1028 kg/m 3 f o r seawater , C = 0 .8 . To compare, d Hoerner (1965) c i t e s C = 0.35 as a measured drag c o e f f i c i e n t f o r d a to rpedo and C = 0.8 fo r a b lun t-nosed c y l i n d e r a l i g n e d d l o n g i t u d i n a l l y to the f l ow . The shape of Camel III i s c e r t a i n l y not b l u n t but added p r o j e c t i o n s on the su r f a ce and e s p e c i a l l y a f t of the main body l i k e l y c o n t r i b u t e s u b s t a n t i a l l y to the d r a g . 160 APPENDIX B - PRESSURE The p ressu re i s sensed by a V i a t r a n model 104 s t r a i n gauge type p r e s s u r e t r a n s d u c e r . The p res su re s i g n a l i s p r e a m p l i f i e d by the c i r c u i t shown in F i gu re B .1 . The second stage a m p l i f i e r i s a l s o shown. Swi tches on the second stage a m p l i f i e r a l low d i f f e r e n t f u l l s c a l e ga ins to be s e l e c t e d . The s i n g l e po le R 1 6 C 2 i s a low pass f i l t e r w i th a measured -3db p o i n t at 1.4 Hz . C a l i b r a t i o n of the p res su re s i g n a l i s conducted in the l ab us ing an Amthor type 452 dead weight t e s t e r . The c a l i b r a t i o n curve i s l i n e a r over the f u l l t r ansduce r range . A t y p i c a l c a l i b r a t i o n data t a b l e i s shown in Tab le B .1 . The accuracy of the t r ansduce r quoted by the manufacturer i s 0.2 percent of f u l l s c a l e . For the PEQUOD and WESPAC c r u i s e s a t r ansduce r w i th 0-5000 p s i range was used , r e s u l t i n g in a worst case accuracy of 10 p s i or 6.8 dba r . The r e s o l u t i o n of the p res su re measurement i s governed by the e l e c t r o n i c no i s e added to the s i g n a l by the FM/tape r e c o r d i n g sys tem. T h i s no i se can be measured in the l a b . A tape i s r ecorded on the Camel III tape c a s s e t t e s w i th DC i npu t s to each VCO. The r eco rded s i g n a l i s p l a yed back on a JVC KDA11 tape deck and demodulated us ing a Sonex FM d i s c r i m i n a t o r system. The s p e c t r a l d e n s i t y of the no i se on a p a r t i c u l a r FM channe l i s determined wi th an HP3582A spectrum a n a l y s e r . I n t e g r a t i o n of the spectrum over the f requency range of i n t e r e s t y i e l d s the no i se expected due to the s i g n a l p r o c e s s i n g . Peaks in t h i s spectrum are a t t r i b u t e d to incomple te tape speed compensa t ion . On the 400 Hz FM channe l (which i s tha t used to c a r r y the 161 p ressu re s i g n a l ) there i s an rms n o i s e l e v e l of 1 mv over 0-0.5 Hz which i s e q u i v a l e n t to 1.0 p s i or 0.7 dbar . I f the peak to peak va lue of the no i se i s taken to be f i v e t imes the rms l e v e l the r e s o l u t i o n of the p r e s su re measurement i s about 3.5 dbar . in SMt — W W -^IvVvV* Alt -VWVvV 1 fizA V , V are Viatran Model 104 outputs Mr in *W4 -AWVW— —<vvw-1—'WW-"-—• gain select ->vwv— w Figure B.I - Camel III preamplifier and lowpass f i l t e r -a m p l i f i e r . 162 date - Oct 27, 1981 t r a n s d u c e r - V i a t r a n model 104, s e r i a l #116419, 0-5000psi Pre s s u r e V P _P f i t 25 -2.440 24 50 -2.414 49 100 -2.363 99 200 -2.260 200 300 -2.157 300 400 -2.054 401 500 -1.952 501 600 -1.850 600 800 -1 .645 801 1000 -1.440 1001 1200 -1.236 1201 1 400 -1.031 1401 1600 -0.827 1600 1800 -0.622 1801 2000 -0.418 2000 2200 -0.214 2199 2400 -0.009 2400 2600 0. 195 2599 2800 0.400 2800. 3000 0.605 3000 3200 0.809 3199 3400 1.014 3400 3600 1 .219 3600 3800 1 .423 3799 4000 1 .628 4000 4200 1 .833 4200 4400 2.038 4401 HP33E l i n e a r r e g r e s s i o n r o u t i n e g i v e s P =977.4V +2408.6. f i t P Tab le B.1 - Camel III p r e s su re c a l i b r a t i o n d a t a . 163 APPENDIX C - FALL RATE The f a l l r a t e of the ins t rument i s c a l c u l a t e d by e l e c t r o n i c d i f f e r e n t i a t i o n of the s i g n a l from the p r e s s u r e p r e a m p l i f i e r . The c i r c u i t d iagram i s shown in F i g u r e C . I . A c a l i b r a t i o n s i m i l a r to that d e s c r i b e d in Appendix B y i e l d s a l i n e a r f i t to the output v o l t a g e , V , of the p res su re p r e a m p l i f i e r of the P1 form V = a+bP P1 , where P i s the measured c a l i b r a t i o n v o l t a g e . Then, av / a t = b a p / a t . p1 The d i f f e r e n t i a t o r output i s v = K av / a t = K b a p / a t . f P p i p 3P/at i s e a s i l y conve r ted to 3z/3t = W which r ep r e sen t s the i n s t r u m e n t ' s f a l l r a t e . The d i f f e r e n t i a t o r ga in i s p l o t t e d in F i g u r e C . 2 . The ga in i s 412±5 seconds w i th the -3db p o i n t measured at 0.8 Hz . A s i m i l a r measurement to tha t d e s c r i b e d in Appendix B shows the r e s o l u t i o n to be about 1.5 cm/s. 164 "ft 4m F i g u r e C.1 - Camel III f a l l d i f f e r e n t i a t o r ) . r a t e c i r c u i t ( p ressure —i 1 1 1 1 1 — r - i 1 1 i 52 xt T J • • • • • • • tt o 50 EH >< t-H EH z w tt u Pu i / p HP3582A random noise source t o PP-1 1 o/p PP-12 uniform window used f o r t r a n s f e r f u n c t i o n 48 LOG FREQUENCY (Hz) 01 0.1 ' • • 1.0 F i gu re C.2 - Camel III P ressure d e r i v a t i v e t r a n s f e r f u n c t i o n . 166 APPENDIX D - TEMPERATURE Temperature i s measured u s i n g a Thermometr ies F a s t i p Thermoprobe model FP07. The t h e r m i s t o r p r e a m p l i f i e r and low pass f i l t e r are shown in F i g u r e D . 1 . R r ep resen t s the t h e r m i s t o r i n the b r i d g e . The t low pass f i l t e r has a measured -3db p o i n t at 7 Hz . C a l i b r a t i o n of i n d i v i d u a l t h e r m i s t o r s i s c a r r i e d out in a temperature bath in the l a b . Ice water i s a l l owed to warm to room temperature very s low ly wh i l e the bath i s tho rough l y mixed us ing a j e t s t i r r e r . The f low past the t h e r m i s t o r i s of the o rder of 1 m/s. The temperature i s measured us ing a Dymec model 2801A qua r tz thermometer which i s p o s i t i o n e d as c l o s e as p o s s i b l e to the t h e r m i s t o r . The i c e p o i n t i s r e c o r d e d , as are the v o l t a g e s V and V . V i s the input to the temperature VCO, t s t 4 t s t 5 t s t 5 whi le V i s d i f f e r e n t i a t e d in o rder to c a l c u l a t e the t s t 4 temperature g r a d i e n t . A c u b i c po l ynomia l f i t i s made to the temperature d a t a , which g i ves agreement to about . 0 1 ° C . A t y p i c a l c a l i b r a t i o n sheet and the r e s u l t i n g f i t a re shown in Tab le D . 1 . The a c cu racy i s l i m i t e d by the e l e c t r o n i c no i s e added by the FM/tape r e c o r d i n g system. No ise measurements show tha t the temperature r e s o l u t i o n of the system i s about 0 . 1 ° C . F i gu re D.1 - Thermis tor p r e a m p l i f i e r and temperature c i r c u i t . 168 date Oct 29, 1981 The r m i s t o r #B2 i c e p o i n t -0.020 eC T V T V TST5 f i t TST4 TST4 6.500 -1 .061 6.497 -0.844 0. 162 7.510 -0.856 7.510 -0.681 0.161 8.500 -0.656 8.500 -0.523 0.161 9.550 -0.443 9.558 -0.353 0.160 10.500 -0.254 10.500 -0.202 0. 159 11.540 -0.047 11.537 -0.036 0. 158 12.600 0. 164 12.601 0. 130 0. 157 13.500 0.341 13.499 0.272 0.156 14.500 0.537 14.500 0.427 0.155 15.500 0.731 15.499 0.582 0. 154 16.510 0.926 16.511 0.736 0. 153 17.500 1.114 17.497 0.887 0.151 18.510 1 .305 18.508 1.039 0. 150 19.500 1 .490 19.498 1.185 0. 148 20.500 1 .675 20.499 .1.333 0. 146 21.520 1 .862 21.522 1 .482 ' 0.145 22.500 2.041 22.513 1 .623 0.143 23.500 • 2.217 23.499 1 .764 0.141 24.510 2.394 24.503 1 .905 0. 139 T = 11.77 +6.32V • 0.12V 2 + 0.04V 1 f i t TST4 TST4 TST4 T = 11.77 +5.03V + 0.07V 2 + 0.02V 3 f i t TST5 TST5 TST5 Tab le D.1 - Camel III temperature c a l i b r a t i o n data and r e s u l t of c u b i c po l ynomia l f i t . 169 APPENDIX E - VELOCITY SHEAR The c a l i b r a t i o n and the behav iour of the a i r f o i l probes used at the Department of Oceanography, U n i v e r s i t y of B r i t i s h Columbia have been d e s c r i b e d by Osborn and Crawford (1980 ) . The same techn ique f o r probe c a l i b r a t i o n was used f o r t h i s s tudy . The e l e c t r o n i c p r o c e s s i n g of the s i g n a l , however, i s somewhat d i f f e r e n t . The c r o s s stream fo r ce per u n i t l eng th of the probe i s f = J_pU 2 (dA/dx) s i n (2a) 2 where p i s the d e n s i t y of the f l u i d , U i s the f low speed , a i s the ang le of a t t a ck and dA/dx i s the r a te of change of c r o s s -s e c t i o n p e r p e n d i c u l a r to the f low wi th d i s t a n c e a long the body. T h i s i s i l l u s t r a t e d in F i g u r e E . 1 . The net f o r c e i s ob ta ined by i n t e g r a t i n g over x from the probe t i p to the p o i n t at which dA/dx = 0 and i s F = l p U 2 A s i n ( 2 a ) 2 bu t , s i n c e the downstream and c r o s s stream components of the f low a r e , r e s p e c t i v e l y , W= Ucosa and u = U s i n a , and us i ng the t r i g o n o m e t r i c i d e n t i t y s in (2a ) = 2 s i n a c o s a , F i s g i ven by F = pAWu (E.1) The p i ezoce ram i c beam mounted i n the t i p of the probe genera tes a v o l t a g e which i s p r o p o r t i o n a l to the f o r c e a p p l i e d to i t . S ince A i s cons tan t wh i le p and W are both s low ly v a r y i n g and independen t l y measured, the probe f o r c e / v o l t a g e has a l i n e a r dependence on the c r o s s stream v e l o c i t y component of 170 i *6.4 mm i* > i i '•"Wh [T "Jl.Smm . P iezocercmic bending 0.5 mm j moment sensor W u U -V? + u F i g u r e E.1 - The a i r f o i l probe showing f low components. 171 the f l ow , u, which i s expressed in ( E . 1 ) . For c a l i b r a t i o n pu rposes , the probe i s mounted in a j e t of water (the c a l i b r a t o r i s d e s c r i b e d in d e t a i l by C rawfo rd (1976 ) ) . The ang le of a t t a c k of the f low a g a i n s t the shear p robe , a, i s v a r i e d from -22° to +22 ° , and the probe i s r o t a t e d about i t s a x i s . The s i n u s o i d a l vo l t age genera ted by the r o t a t i n g motion in the j e t i s t r a n s m i t t e d to a p r e a m p l i f i e r , bandpass f i l t e r and rms meter . The ins tan taneous v o l t a g e , E pU 2 s i n2as ina ; t , (where u> i s the r o t a t i o n a l f requency in r a d i a n s / s e c o n d ) , i s d i r e c t l y p r o p o r t i o n a l to the c r o s s stream fo r ce on the p robe . I t i s conven ien t to c o n s i d e r the output of the rms meter , E , rms *E p U 2 s i n ( 2 a ) . rms The cons tan t of p r o p o r t i o n a l i t y , termed the probe s e n s i t i v i t y , S, i s determined by p l o t t i n g E / (pU 2 ) vs s i n ( 2 a ) . The rms s e n s i t i v i t y i s the s l ope of t h i s c u r v e , S = d(E / ( p u 2 ) ) / d ( s i n ( 2 a ) ) rms A t y p i c a l c a l i b r a t i o n curve i s shown in F i gu re E .2 . S has u n i t s of v o l t s / ( d y n e / c m 2 ) , and i s approx imate l y cons tan t over ang l e s of a t t a ck of l e s s than 10° (see Osborn and C rawfo rd (1980 ) ) . At g r ea t e r ang les of a t t a ck the s e n s i t i v i t y i n c r e a s e s , i l l u s t r a t i n g the importance of m a i n t a i n i n g v e h i c l e s t a b i l i t y . Any l a r g e t i l t i n g of the inst rument w i l l cause the probe to ' s e e ' a l a r g e ang le of a t t a c k thereby changing the s e n s i t i v i t y of the p robe . Acce l e romete r s mounted in the 172 F i g u r e E.2 - T y p i c a l a i r f o i l probe c a l i b r a t i o n c u r v e . 173 inst rument body are mon i t o r ed . Expe r i ence i n d i c a t e s tha t o c c a s i o n a l t i l t i n g of 2 ° in r eg ions of l a r g e mean shear i s both a maximum va lue and r a r e . Thus , a cons t an t va lue of S may be used . E x p r e s s i n g the probe vo l t age in terms of the s e n s i t i v i t y , E = SpU 2 s i n (2a ) rms = 2SpWu and the peak v o l t a g e , E, f o r a s i n u s o i d i s »/2E , so tha t rms E = 2v/2SpWu (E.2) T h i s s i g n a l i s t r a n s m i t t e d to a p r e a m p l i f i e r , a 100 Hz low pass f i l t e r , and to a second stage a m p l i f i e r p r i o r to be ing sent to a v o l t a g e c o n t r o l l e d - o s c i l l a t o r (VCO) and added to the FM s i g n a l . T h i s p rocessed s i g n a l i s rou ted to a f u r t h e r a m p l i f i c a t i o n stage and t h i s i s a l s o added to the FM s i g n a l . The purpose of t h i s second a m p l i f i e d s i g n a l i s to improve the s i g n a l to no i se r a t i o at low va lues of v e l o c i t y shea r . However, a vo l t age l i m i t i n g c i r c u i t i s r e q u i r e d to prevent s a t u r a t i o n of the VCO at h igh va lues of v e l o c i t y shear and p o s s i b l e con tamina t ion of ad jacen t FM bands. A b lock diagram of the p r o c e s s i n g ( F igure E.3) i s i n c l u d e d . The p r e a m p l i f i e r d i f f e r e n t i a t e s the input s i g n a l from the probe ( F igu re E . 4 ) . I t s t r a n s f e r f u n c t i o n i s g i ven by V /V = 1 + ( j w R ^ z J/Cd+ j ^ R i C j d + j c o R z C z ) } o i where the c i r c u i t parameters are g i ven in F i g u r e E .4 . The po l e s are probe S i probe S 2  I 1 A B C D sx f>re**j> 1 — 1 1 — r -at 1 1 S4( CfT Ul I I VCO Ctl Mtoht iff as TV! SI ma VtolS at SZ too hi 71111 3 C 6 lower e l e c t r o n i c s package D-connector underwater connector to preamp housing underwater connector to endcap upper e l e c t r o n i c s package 25-way D-connector SZ 7SOT vteti Cfl4 SI YCO SZ VCo S41 SAl 4*f CfK Vtotl Vco F igu re E.3 - Camel III v e l o c i t y shear s i g n a l p r o c e s s i n g . 175 Szout F i g u r e E.4 - V e l o c i t y shear p r e a m p l i f i e r s . 1 76 (2JTRTC2 )" 1 = 0.2 Hz ( 27rR! C , ) - 1 = 200 Hz (2TTR 2C 2 )" 1 = 253 Hz The measured p r e a m p l i f i e r ga in i s shown in F i g u r e E . 5 . I t i s p l o t t e d as a cons tan t d i f f e r e n t i a t o r ga in in dbv (the a c t u a l ga in i s d i v i d e d by 27rf and conve r t ed to dbv = 201og(V /V ) The o i ga in i s approx imate l y cons tan t at 0.85 seconds , and the -3db p o i n t i s at 150 Hz . The 100 Hz Cauer e l l i p t i c f i l t e r i s manufactured by Frequency Dev i ces (#7438). The second stage of a m p l i f i c a t i o n i s p r o v i d e d by the bandpass a m p l i f i e r shown in F i g u r e E . 6 . T h i s was used p r i o r to the i n c l u s i o n of the 7438 f i l t e r i n t o the c i r c u i t and has been r e t a i n e d as an a m p l i f i e r a l t hough no longer r e q u i r e d as a low pass f i l t e r . The t r a n s f e r f u n c t i o n of t h i s a m p l i f i e r i s V /V = ( - jcuR 7C + jcoR 7C (, ) ( 1 + J C J R 5 C ^ + j ( jR 6 C , + jo>R5C ^ ( 1 + jcoR 6C 1 ) ) } o i the po l e s are (2?TR 7C 1 ) " 1 = .11 Hz (27TR6C, )" 1 = 0.5 Hz (27TR 7C„ ) " 1 = 706 Hz (2TTR 5C 2 )" 1 = 234 Hz (27rR5C 1 ) - 1 = 1 .6 Hz The t o t a l measured t r a n s f e r f u n c t i o n of the complete c i r c u i t which p rocesses the probe s i g n a l and sends i t to the VCO i s p l o t t e d as a d i f f e r e n t i a t o r ga in in F i g u r e E .7 . I t i s seen tha t the ga in i s approx imate l y 2.7 seconds w i th -3db p o i n t s at f 1 I I I I I I I 1 ~ r 1 — i — i l i l t — i 1 1 — i — i — i i i -~-CM \ o II ad -1.0 T l < o tt o EH < EH W -2.0 tt W |X4 En n Q * * I 4 • • • 4 • S, K,=.85±.02 sees + S 3 K 2=.85±.02 sees LOG FREQUENCY (Hz) 1.0 10. i i i i i i i i i I I l I I i 1 1 1— l — i — I — I — L F i g u r e E.5 - V e l o c i t y shear p r e a m p l i f i e r t r a n s f e r f u n c t i o n . 178 1 • VWVU-ci IHHot F i g u r e E. 6 - V e l o c i t y shear a m p l i f i e r s . T 1 1—I I I I I T I I i l i ~l 1—I—I I I •12 •« CM \ o II X h 8 k4 < « o E-< I-i E-i 2 W « W Cu Cn i—i Q • + • S, K,«8.75 db (2.74 sees) t S 2 K,=8.62 db (2.70 sees) LOG FREQUENCY (Hz) 1 . 0 1 0 , i i i i i i I i i — i — I I I l i j I i I i i F igu re E.7 - Complete shear c i r c u i t t r a n s f e r f u n c t i o n . 180 < 1 Hz and f > 100 Hz . F u r t h e r a m p l i f i c a t i o n fo r low va lues of v e l o c i t y shear i s p r o v i d e d by the a m p l i f i e r and vo l t age l i m i t i n g c i r c u i t shown in F i g u r e E .8 . The ga in i s g i ven by V /V = (R,+R 2 )/R 2 o i which i s t h e o r e t i c a l l y equa l to about 30.5 g i ven the nominal c i r c u i t pa ramete rs . A c t u a l va lues are s l i g h t l y d i f f e r e n t and the measured t r a n s f e r f u n c t i o n i s p l o t t e d in F i g u r e E .9 . P r o c e s s i n g of the v e l o c i t y shear s i g n a l can be r ep resen ted by a d i f f e r e n t i a t o r w i th a ga in of GK, as below, E 9 t > »~E vco E = G K E k = G / 2 7 r f VCO where E i s the probe v o l t a g e and E i s the VCO input v o l t a g e , vco Then, E = GKE = Kk9E/9t vco where k i s the d i f f e r e n t i a t o r g a i n . I f E i s r ep resen ted by a f u n c t i o n a l form exp( icot ) , E = wKkE = 27rf KkE. vco Us ing equa t i on ( E . 2 ) , E = Kk9E/9t = Kk2|/2SpW9u/9t vco 181 F i g u r e E.8 - V e l o c i t y shear h igh ga in a m p l i f i e r . T 1 1 1—I—I I I I 1 1 1 1—I 1 I I I 1 1 1 1—I I I I 30 ~ « 4 • 4 4 4 4 * * * f 4 4 4 4 4 4 4 4 * * 4 * * 4* + -3 db-> S 33 Hz i . ^ A. * t . ^ S 30 Hz * •O . S G=29 db A2 T3 Al 10 S A2 * S G=29 db - -10 LOG FREQUENCY (Hz) 1.0 10. I i l I i i l i | i i i i i l l i I I I 1 I—I—L_L F igu re E.9 - V e l o c i t y shear h igh ga in a m p l i f i e r t r a n s f e r funct i o n . 183 p r o v i d e d tha t u i s the on l y t ime dependent parameter . Then, i n vok ing T a y l o r ' s f rozen f low h y p o t h e s i s , which can be s t a t ed as 9u/9t = W9u/9z, one o b t a i n s 9u/9z = E /(2»/2KkSpW 2 ) . vco It i s necessa ry to i n c l ude another f a c t o r to account fo r the i nhe ren t ga in of the VCO-FM system (which i s termed a ) . Then, 9u/9z = aE /(2v/2KkSpW2) , d where E i s the vo l t age of the d e m u l t i p l e x e d s i g n a l . The ga in d f a c t o r , K, i s equa l to 1 fo r the s tandard v e l o c i t y shear s i g n a l and equa l to the ga in of the f i n a l s tage a m p l i f i e r (10 fo r PEQUOD and 30 fo r WESPAC) when the a m p l i f i e d shear s i g n a l i s used f o r low s i g n a l v a l u e s . 184 APPENDIX F - CALCULATION OF SPECTRA The method of c a l c u l a t i n g s p e c t r a i s d e s c r i b e d h e r e . The s p e c t r a l v a lues are c a l c u l a t e d in u n i t s of b i t s 2 / H z and conve r t ed , u s i n g the s u i t a b l e c a l i b r a t i o n cons t an t s (see Appendix E) to u n i t s of v e l o c i t y shear s p e c t r a l d e n s i t y ( s e c " 2 / H z ) . Due to the importance of t h i s c a l c u l a t i o n to the t h e s i s , the method of c a l c u l a t i n g the s p e c t r a and the c a l i b r a t i o n of the r o u t i n e fo r do ing t h i s c a l c u l a t i o n i s o u t l i n e d h e r e . The data were o r i g i n a l l y r eco rded onto ana logue tape u s i n g the Phideck tape r e c o r d i n g u n i t s mounted i n s i d e of Camel I I I . Cop ies of each tape were made on board sh i p and the two i d e n t i c a l s e t s of tapes were t r a n s p o r t e d to UBC by d i f f e r e n t means. At UBC the data were d i g i t i z e d at a r a t e of 400 Hz u s i n g an LS I-11 computer . The mode of d i g i t i z a t i o n was as 12-bit two 's complement numbers w i th f u l l s c a l e c o r r e s p o n d i n g to ±5 v o l t s . The c o n v e r s i o n from b i t s to v o l t s , t h e n , i s -2048 b i t s = -5 v o l t s 0 b i t s = 0 v o l t s 2047 b i t s = 4.997 v o l t s 1 b i t = (5/2048) v o l t s The s p e c t r a l r o u t i n e was o r i g i n a l l y w r i t t e n at UBC by P. Leszko f o r the Do lph in data of T . R . O s b o r n , and subsequent l y t a i l o r e d to the needs of t h i s p r o j e c t . The c a l i b r a t i o n scheme fo r the r o u t i n e used b locks of 4096 data p o i n t s and ope ra ted as f o l l o w s : 185 a) f i r s t d i f f e r e n c i n g to de t rend the d a t a , b) removal of the mean c) a c o s i n e taper i s a p p l i e d to the f i r s t and l a s t 200 p o i n t s of the b l o c k , to f o r c e the t ime s e r i e s to zero at the end p o i n t s . The form of the taper i s J_( l-cos(k-2 1)/200) , d) an FFT i s done on the t a i l o r e d time s e r i e s , e) the c a l i b r a t i o n da ta are used to conve r t the f requency domain da ta to the a p p r o p r i a t e v e l o c i t y shear u n i t s , f ) the s p e c t r a l v a lues are squared and , g) averaged over 8 ad j acen t f requency bands . The output data are p r i n t e d toge ther w i th the f requency and the cumula t i ve v a r i a n c e . The lowest f requency band, 0.98 Hz , i s not i n c l u d e d in the cumu la t i ve v a l u e . A sample program and output l i s t i n g aire shown in Tab le F . 1 . The average f a l l r a te over the 4096 p o i n t s i s used to c a l c u l a t e the v e l o c i t y shear and i s p r i n t e d on the bottom of the output l i s t i n g . F i n a l s p e c t r a l c a l c u l a t i o n s were made u s i n g 1024 p o i n t s f o r improved s p a t i a l r e s o l u t i o n and the a p p r o p r i a t e changes were made to the r o u t i n e . The c o s i n e taper reduces the v a r i a n c e of the s i g n a l . For a s i g n a l hav ing a white spec t rum, the v a r i a n c e w i th a f u l l c o s i n e taper i s 3/8 's of the t o t a l v a r i a n c e . In t h i s case on l y 400 of the p o i n t s a re t a p e r e d . The v a r i a n c e , t hen , i s 1-400/4096(1-3/8) = .939 of the t o t a l v a r i a n c e . To recover the t o t a l v a r i a n c e the output i s d i v i d e d by the f a c t o r . 939 . 1 a C CALCULATE SPECTRA FROM TAPED DATA USING 4088 PT8 * 3 c THIS PROGRAM REQUIRES THE FOLLOWING INPUTS • 4 c :TAPE ON. UNIT B • 9 c CALIBRATION OATA FILE(777) ON UNIT4 • 6 c :PREVIOUSLY CALCULATED FALL RATES ON DATA • 7 c FILE(1234) ON UNIT 7 a c UNIT 6 IS THE OUTPUT * 9 c IF1LE-FILE NO. TO BE READ FROM • • 10 c 1RECSTARTING RECORD NO. • 11 c ICHAN-CHANNEL NO. • 12 c K-NO. OF SPECTRA • 13 c L-NO. OF SPECTRA TO BE AVERAGED OVER * 14 c A-2.5/4 • 15 c RO-DENSITY OF WATER IN GMS/CM«3 (1.028) • 16 c S-PROBE SENSITIVITY IN V0LTS/(0YNE/CM»«2) • 17 c GF-DIFFERENTIATOR GAIN IN SEC/RAD • .8 c NTO-NO. OF RECORDS ON THE FILE • 19 c * 20 c • 21 c 22 23 COMMON/TAPEM/CNTR.LENGHT.MESG(100) 24 COMMON/PARAM/A.RO.USUB.S.GF.NLO 25 COMMQN/MAT/US(2500) 26 LOCICAL'1 MESQ 27 INTEGER CNTR,LENGTH 28 INTEGERS LIST(t) 29 DATA LIST/1H*/ 30 CALL PLCTRLCSCAL'. .6666) 31 CALL PALPHA('ROMAN.2 '.0.0) 32 777 REAOU. LIST. EN0-899)IFIL£. IREC. KHAN.K.L. A, R0.S.6F.NT0 33 WRITE(6.LIST)IFILE.IREC,ICHAN.K.L,A.RO,S.GF.NTO 34 REA0(7,1234)(US(IB),IS»1,NT0) 35 1234 FORMAT!10E12.4) 36 NLO.IREC 37 CALL TP0SN(IFILE,5) 39 CALL SK!P{0.IREC-1.5,8700.8800,8900) 39 CALL PLOHS. ,0. .-3) 40 00 1 11-1.K 41 1 CALL SPEC(ICHAN.L) 42 GO TO 777 43 999 CALL PLOTND 44 STOP 45 too WRITE(6,6100) 46 6100 FORMAT)' UNIT 5 NOT ATTACHED TO A LEGAL TAPE7 ') 47 STOP 48 200 WRtTE(e,6200)CNTR 49 8300 FORMAT(' ERROR FROM TAPE DSR: RETURN CODE > ',15) 50 STOP 51 300 WRITE(6.6300)CNTR. (MESGd ), I • 1. LENGTH) 52 6300 FORMAT(' ERROR FROM TAPE DSR: RETURN CODE -'.IS. 83 8 -MESSAOE RETURNED FOLLOWS BELOW 7'/1X.100*1 > 54 STOP 55 400 STOP 400 66 500 STOP SCO 57 600 STOP 600 58 700 STOP 700 59 800 STOP 800 60 800 STOP 900 61 62 63 64 65 68 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 . 84 89 86 87 88 89 80 91 92 93 94 99 96 97 98 89 100 101 102 103 104 103 106 . 107 108 108 1 10 1 I 1 113 113 114 115 116 117 118 118 120 END SUBROUTINE TPOSN POSITIONS READ HEAD TO CORRECT FILE. C C c C • • c ••••••••• • SUBROUTINE TPOSNt IPSN, IUNIT ) COMMON /TAPEM/ CNTR,LENGTH.MESG(100) INTEGER CNTR.LENGTH INTEGERS PSN(5)/'P0','SN'.'«»',2«' '/.LEN/10/ LOGICAL* 1 CHARM), STAR/'**/, BLANK/' '/.MESG EQUIVALENCE (CHAR(1),PSN(4)) CALL BT0(IPSN.CHAR(1),3,N0,' ') NI-3-N0 NJ-4-NI IFCNI .LE.OjGOTO I NK'NI+1 CALL M0VEC(ND,CHAR(NK),CHAR(1)> LJ-NU*! DO 2 L-LJ.4 CHAR(L j-BLANK 2 CONTINUE 1 CHAR(NJ)-STAR CALL CNTRL(PSN,LEN, IUNIT.CNTR,8100,»IOI,4102) GOTO 3 100 WRITE(G.tO) 10 FORMATP ILLEGAL UNIT SPECIFICATION?') CALL QUIT 101 WRITE(6.103) CNTR CALL QUIT 102 WRITEC6.103) CNTR,(MESO(I).1-1.LENGTH) 103 FORMAT(' ','RC-".13./' DSR ERR-',100A1) CALL OUIT 3 RETURN END SUBROUTINE SPEC(177,IFLA1) COMMON/PARAM/A,R0,USUB,S.GF,NLO COMMON/MAT/US(2500) INTEGERS C( 10,4097) REAL«4 P(8192).AVP(B12),FREQ(812),AINT(B12) COMPLEX'S 0ATA(8192) DATA PI/3.1415927/ DATA FREQ.AVP/1024*,0/ c N-4098 DT',002 WRITE(6,6000)177,1FLA1,N,DT 6000 FORMAT(//26X.80( 1H»)/36X,1H*,78X,1H»/26X.1H«.5X, 8 30X,'C(',I 1,')',SX,'AVERRAGEO BY ',12.28X, IH«/ 826X, 1H*,78X,1H*/26X, IH*,5X,15. 14H DATA POINTS,54X,1H*/26X,1H*t 578X. 1H«/26X, 1H«.5X.F5.4.3 IH TIME INTERVAL ( SEC). 47X. 1HV26X . 1H» 878X,1H*/26X.80(1H*)///IX.3(33HP0SN. FREQU. VAR. CUMULATIVE, 86X)/) c NDIM-N/2 RN-FLOAT(NDIM) N8-N/B IN-N/16 N3-IN/3*1 DF-4./RN/0T CO oY Table F .1 - The s p e c t r a l r o u t i n e used to e s t ima te the v e r t i c a l shear spectrum and sample o u t p u t . 121 FREO<O-I./DT/N 122 IN1-IN-1 123 DO 2T 127*1.INI 124 FRE0(2-I27)-0F*FL0AT(I2T) 129 2T FRE0(2-I27*1)«FRE0<2M27) 128 FREQ(BI3)-DF«2S6. 127 C 158 DO 26 128-1,312 129 AVP(I28)-0. 130 26 CONTINUE -131 C 132 DO «B 188-1,IFLA1 133 NUP-NL0.15 134 USUB-O. 135 00 124 H24-NL0.NUP 13S 124 USUB«USUB+US(I124)«100. 137 USUB-USUB/16. 138 NLO-NLO+16 139 C 140 C USUB - VELOCITY OF SUBMARINE (M/SEC) 141 IFfI77.E0.1.0R.I77.E0.2)FACT-(B.»PI«A/<2O4B.»S0RT(2.)-R0 142 »»S»USUB««2«0F))««2/.a390 143 IF(I77.EQ.3)FACT-4.3E-6/USUB»«3 144 IFU77.EQ.4.0R. I77.E0.B)FACT-< 1 .7/FL0AT(3048)>«-2 145 IF(I77.EQ.6)FACT-(I./FLOAT(3048))««3 146 CALL PAWEL2(C) 147 C 148 00 1 H-I.N 149 1 0ATA{I1)-C(I77.I1*1)-C(IT7,I1) 160 CALL AVER(DATA.N) 161 00 13 113-1,N 152 13 0ATA(I13)-SCB(I13)«REAL(0ATA(113)) 153 C 154 CALL F0UR2I0ATA.N,1.1,1) 155 C 156 00 2 I2-2.NDIM 157 A1-REAL(0ATA(I2>) 158 A2-AIMAG(DATA(12) ) 159 P(12)-FACT»(A1*A1*A2«A2)«DT/RN/(4.-SIN(PI»(I2-1)/4086.)««2) 160 2 CONTINUE 161 C 162 H-0. 163 00 3 13-2.8 164 H-H»P(I3) 169 3 CONTINUE 166 AVP(3)-»VP(2)*H/7./FL0AT(IFLA1) 167 ' AVP(1)-AVP(2) 168 00 4 14-2,IN 169 H-0. 170 00 6 IB-1.8 171 H-H*P(16*(I4-1)»B> 172 9 CONTINUE 173 AVP(2 'I4)-AVP(2M4)*H/8./FL0AT(IFlA1) 174 AVP(2-I4-1)-AVP(2M4) 175 4 CONTINUE 176 C 177 88 CONTINUE 178 C 179 00 84 194-1,N8 180 84 AINT(I94)-0. Table F.I 181 0 0 85 193-2.IN 182 AINT(2*I9S)-AtNT(2*!93-2)+0F*AVP(2*I9S) 183 93 AINr(2-I95-1)-.S»(AINT(3*I85)-AINT(2<I8B-3)) 184 00 86 I86-1.N3 tas 96 WRITE(6.G010)(K,rRE0(2*K),AVP(2*K).AINT(2«K).K-I86.IN,N3) 186 6010 FORMAT(IX, 3(0PI3, 3X, F6.2, JX,1PE9.3,2X,E8.3,6X)) 187 WRITE(6.6123)USUB 188 6123 FORMAT{// 5IHAVG VALUE OF FALL RATE USED TO CALCULATE SHEAR 189 4.F10.4.9H CM/SEC) 190 C 191 CALL PLOUREO.AVP.S^BO.WT) 182 c 193 RETURN 194 c 195 END 196 FUNCTION SCB(K) 197 DATA PI/3.1415927/ 198 1F(K.OT.200)GO TO 1 199 SCB-.S«(1.-COS(P1«K/2CO.)) 200 RETURN 201 1 IF(K.GT.3B96)G0 TO 2 202 SCB-1. 203 RETURN ' 204 2 5CB-.5-(l.-C0S(PI«(4O96-K)/2OO.)) 205 RETURN 206 END 207 C 208 SUBROUTINE AVER(A.N) 209 COMPLEX'S A(ai82) 210 SUM-O. 211 00 1 I-1.N 212 1 SUM-SUM*REAL(A(I)) 213 SUM-SUMZFLOAT(N) 214 DO 2 K-1.N 219 2 A(K)-REAL(A(K))-SUH 216 RETURN 217 END 21B c 2 19 SUBROUTINE PAWEL2CC) 220 INTEGER'S C(10.4097),A(3B60).LEN 221 INTEGER LMJM 222 c 233 00 1 11-1. 17 224 CALL READ(A,LEN.0.LNUM,B,630) 229 DO 3 13-1.256 226 K-2S6-(I1-1)+12 227 IFfK.QT.4087)00 TO 30 2 28 DO 7 17-1. 10 229 C(I7.K)-A(IO-(I2-l)*I7) 230 7 CONTINUE 231 2 CONTINUE 232 1 CONTINUE 233 2 0 RETURN 234 c 335 30 STOP 30 336 END 237 SUBROUTINE AXL0G(XO.YO.IFLAG.NMIN.DN. I) 338 DIMENSION TNUM(IS) 239 DATA TNUM/2H-1.2H-2.3H-3.2H-4,2H-5.2H-6.2H-7.2H-8.2H-8, 24Q » 3H-10.3H-11,3H-12,3H-13,3H-14.3H-18/ cont'd 301 CALL AXCTRLC'YORI'.Y0RI+YS12E) 303 CALL AXPL0T( ' ; ' .0..XSIZE.0..XSlZE ,DX) 303 CALL AXCTRLC'SIDE',-1) 304 CALL AXCTRLC *X0RI',X0RI*XSIZE) 305 CALL PLOTCXORI»XSIZE,YORI,3) 306 IF(USUB.GE.1.)G0 TO 3 307 AL0"AL0G1O(1./USUB«»2) 308 MM'NMIN-I-INT(ALO) 309 Y0RIR*Y0RI*(1.-AM0D(AL0.1.))/DY 310 GO TO 4 311 3 AL-AL0G10(USUB"2) 312 MM'NNIN-MNT(AL) 313 YORIR*YORI»AMO0(AL,1.)/0Y 314 4 CALL PL0T(X0RI»XSIZE,Y0RIR,2) 315 CALL AXCTRL('YORI'.YORIR) 316 CALL AXPLOTC'i'.8O..YSIZE-1./0Y.O..CYSIZE-1./DY)*0Y) 317 CALL PL0TCX0RI*XSIZE.Y0R1R»YSIZE-1./DY,3) 318 CALL PL0T(X0RI*XSIZE,YSIZE*Y0RI,2) 319 KR*(YSIZE-(Y0RIR-Y0RI)>«DY*1 330 CALL AXLOGtXORH-XSlZE,YORIR.-I,MH,DY.KR) 33 1 CALL PSYM(XDRI*XSIZE*2.,Y0RI»2./3.*YSlZE.HTX.UNITS(2>.-80.,4, 333 CALL PSYM(-0.,-0.,HTX.UNITS!3).-DO..4,0) 333 CALL P5YMC-0.,-0.,HTX.UNITS(4),-00.,4,0) 334 C DRAW INQ NUMBERS 1,10,100,1000 UNOER OX(LOO) AX 333 2 YX*Y0RI-3.«HTX 326 CALL SYMB0L(X0Rl-4./7.*HTX.YX,HTX,1H1.0.,1) 337 CALL SYMB0L(X0RI*1./DX-6./7.«HTX,YX,HTX.3H10.0.,3) 32B CALL SYMB0L(X0RI*2./0X-10./7.»HTX.YX.HTX.3H100.0..3) 329 CALL SYMB0L(XORI*3./OX-12./7.»HTX,YX,HTX,4H1OO0.0..4) 330 CALL P5YM(XORI+XSIZE°2./3.,-HTX,HTX,UNITS(1) ,0 . .4,0) 331 KL-(YSIZE-(YORIL-YORI))»OY*l 333 C 333 CALL AXL0G(XORI.YORIL.1,NMIN.DY,KL) 334 C 335 DO 1 I1-N1.N3 336 X(I1*1-N1)«AL0010(X(I1))/0X»XORl 337 I Y(I1*1-N1)-(ALOG10(Y(I1))«FLOAT(NMIN))/DY+YORI 338 C 339 , CALL LINE(X.Y.K.1) 340 CALL PLOT( 13. ,0 . .-3) 341 C 342 RETURN 343 END 344 SUBROUTINE AXLOG(XO.YO,IFLAG.NMIN.DN.I) 345 DIMENSION TNUMC15) 346 DATA TNUM/2H-1. 2H-2.2H-3. 2H-4. 2H-5.3H-6. 3H-T. 2H-8.2H-8, 347 ft 3H-10,3H-11.3H-I2.3H-I3,3H-14,3H-15/ 348 DATA HTYL/.S/ 349 HTYS*.4»HTYL 330 C IFLAO- I COUNTERCLOCK SIDE 331 C IFLAG'-1 CLOCK SIDE 352 IF(I FLAG.EQ.DXI"XO"10./7.*HTYL-18./7.'HTYS-.2 353 IF(IFLAG.EO.-1)X1*XO*.2 354 C 355 00 1 11*1,1 356 Y 1*Y0-.4*HTYL*1./ON*FLOAT(11-1) 357 CALL SYMB0L(X1,Y1,HTYL,2H10..0.2) 356 N-NMIN»1-I1 339 J-3 360 . IF(N.GT.9)J-3 Table F.1 -341 OATA HTYL/.S/ 342 HTYS*.4*HTYL 343 C IFLAG* 1 COUNTERCLOCK SIOE 344 c IFLAG--I CLOCK SIOE 245 IF(I FLAG.EQ.1)X1*X0-10./7.*HTYL-18./7.•HTVS-.2 246 IF(IFLAG.E0.-1)X1*XO*.2 247 c 248 DO 1 11*1.1 249 • Y1 *Y0- . 4*HTYL+'l ./DN*FL0AT( 11-1) 250 CALL SYM80L(X1,VI.HTYL.2H10,.0.2) 231 N*NM1N»1-I1 . 232 d-2 253 IFCN.GT.8)0*3 234 IF(N.LT.1) RETURN 255 CALL SVMS0L(X1*1O,/7.•MTYL*2./7.*HTYS.V1+HTYL-MTYS.HTYS. 236 ft TNUM(N),0.,0) 267 1 CONTINUE 238 RETURN 259 END 260 SUBROUTINE PLO(X.Y,N1,N2,J) 261 REAL*4 X(312),Y(312) 262 OATA XDRI,YORI,VORIL,YORIR,XSIZt.VS1Z6,OX,DY 363 264 INTEOER C(6)/ZE6O87D4O,ZtBO87040.Z7BE3404O.ZC138A84O.ZC138A7.4O t,ZC13BA940/ 363 366 INTEGER UNITS(9)/ZAD88A9BO.ZA0A28583.ZO86OF238.Z6188A88b, 367 •ZAD409461,ZA2BS8309,ZO9F23B50.Z08F23861.Z88A9B040/ 268 INTEGER UNIT3(41/ZA04DC3A1.Z61848008,Z09F23888,ZASBD4040/ 369 INTEGER LEN(6)/4,4,3,4,4.4/ 270 COMMON/SUB/USUB 371 c 272 HTX*.33 373 IFCU.LE.3)NMIN*8 374 If(J.CE.3)NMIN*10 275 K*N3-N1*1 376 CALL AXCTRLC'SYMISIZE'..137) 277 CALL AXCTRL('SIDE'.0) 278 CALL AXCTRLC YORI', YORI) 379 CALL AXCTRL('XORI'.XORI) 280 CALL AXCTRLC'LABELS',0) 381 CALL AXCTRLC'LOGS', 1) 383 CALL AXPLOTI'; ' .0..XSIZE.O..XSIZE'OX) 383 CALL AXPLOT(' ; ' . 90. . VSIZE ,0 . . VSI ZE*OY ) 384 CALL PSYM(HTX.V0RI«YSlZE/3. .HTX.C(O) .80. .LEN( J ) .O) 383 IF(J-3)10.11,12 286 10 CALL PSYMC-O.,-0.,HTX,UNITSC3).90..4.0) 287 CALL PSYM(-0..-0..HTX,UNITS(3),90..4.0) 288 CALL PSYM(-0. ,-0. ,HTX,UNITS(4 ),90. .4.0) 388 GO TO 2 290 11 CALL PSYM(-0..-O..HTX,UNIT3<1),80..4.0) 291 CALL PSYM(-0.,-0.,HTX,UNIT3(2),90.,4,0) f 393 CALL PSYM(-0.,-0..HTX.UNIT3C3).90.,4.0) 293 CALL PSYM(-0.,-0..HTX.UN1T3C4).80..2.0) 294 GO TO 2 393 12 CALL PSYM(-0.,-0.,HTX:UNITS(5),80..4.0) 296 CALL PSYM(-0.,-0..HTX,UNITS(6),90..4.0) 397 CALL PSYM(-0..-0.,HTX,UN1TS(7).B0..4,0) 398 CALL PSVM(-0.,-0.,HTX.UNITS(8),80.,4,0) 299 CALL PSYM(-0.,-0..HTX,UNITS(8).80.,3,0) 300 CALL AXCTRLC SIDE ' . D c o n t ' d 361 363 363 364 365 366 367 368 368 370 371 373 373 374 375 376 f n 0 o f P M * IF(N.LT.1) RETURN CALL SVMaOL(X1»10./T.-HTVL*J./7.-MTYS.V1*HTVL-HTyS,HTy5. 8 TNUM(N),0.,U) 1 CONTINUE RETURN END BLOCK DATA COMMON/NUM/TNUM COMMON/TAPE/MOUN COMMON/FILE/POSN COMMON/RECORO/FSR LOGICAL* 1 POSN(8)/'P'.'0'.'S'.'N', '.'•'/ LOGICAL*1 TNUM(10)/'1'.'a'.'3'.'4'.'B'.*«'.'7'.'8'.'9','0'/ INTEGER FSR(S)/'FSR '.* '/ INTEGER'S MOUN(30)/30*' '/ END CO Table F . 1 - c o n t , d 1,317, I.I.I..023,1.028..4330000E-04.17.,512, 7 B a 10 11 12 13 14 13 IC 17 10 13 30 21 -23 23 24 25 28 27 38 30 30 31 33 33 34 35 30 37 38 30 40 41 42 43 44 45 48 47 48 49 50 31 83 33 54 55 56 57 58 59 60 CO) 4096 DATA POINTS .0030 TINE INTERVAL (SEC) AVERRAOED BV 1 POSN. FREOU. VAR. 1 O.08 6.00IE-04 3 1 .83 4.333E-03 3 3.93 1.004E-04 4 3.91 4.834E-09 8 4.88 3.37GE-05 6 3.00 3.327E-03 7 6.84 4.654E-05 8 7.01 9.77IE-OS 8 8.79 9.780E-03 10 9.77 5.705E-O3 11 10.74 3.281E-09 13 11.72 3.S04E-09 13 12.70 I.644E-03 14 13.07 3.3SOE-03 15 14.65 7. 03(31-oa (6 15.63 1.373E-03 17 16.60 2.200E-0S 18 17.58 1.070E-03 19 10.53 7. 130E-0G 30 10.53 1.I03E-03 31 30.91 1.279E-00 33 31.40 S.3D0E-07 33 22.40 7.039E-07 34 33.44 1.05 IE-OG 35 24.41 1.80 IE-06 20 25.39 2.34GE-0fi 27 26.37 I.2D0E-0G 2A 27.34 3. 103E-00 30 30.33 1.743E-00 30 30.30 1.53GE-00 31 30.37 2.393E-06 32 31.33 4.1S6E-06 33 32.33 0.547E-07 34 33.30 3.3O1E-O0 33 34. 18 2.5I8E-06 36 39. 10 4.333E-0G 37 36. 13 1.515E-06 38 37. 1 1 5.087E-0G 39 30.09 5.f)62E-06 40 39.06 4.421E-OG 41 40.04 2. I32E-OG 43 41.03 1.fil9E-0G 43 41.09 I.G7SE-0S 44 42.87 3.850E-08 CUMULATIVE POSN. FREOU 0.0 07 84.86 4.334E-0S 80 03.94 1.403E-04 89 80.91 1 .075E-O4 90 87 .83 2.22SE-04 9 1 BB.87 3.550E-04 82 on.84 3.004E-04 93 90.82 3.5GOE-04 94 81.00 4.324E-04 95 92.77 3.08 IE-04 00 93.78 5.40IE-04 87 94.73 3.743E-04 98 95.70 5.904E-04 99 98 .GO 8.I35E-04 100 07.66 6.2I3E-04 101 08.63 6.347E-04 102 00.61 6.SG3E-04 103 100.59 6.745E-04 104 101.GG G.ni5E-01 103 102.54 G.033E-OI 106 103.53 0.033E-04 107 104.49 6.04 IE-04 100 103.47 6.047E-O4 109 106.45 6.0G0E-O4 110 107.43 6.003E-O4 1 1 1 100.40 7.00GE-04 112 103. 38 7.0IDE-04 113 110.33 7.040E-04 1 14 111.33 7.0B7E-04 118 1 12.30 7.072E-04 1 10 113.28 7.000E-04 117 114.2G 7.I3GE-04 118 115.33 7.I4GE-04 119 1 1G.2 1 7.1G8E-04 130 117. 19 7.192E-04 131 118.16 7.234E-04 122 119.14 7.249E-04 123 120.13 7.299E-04 124 131.00 7.357E-04 12S 132.07 7.401E-04 138 123.03 7.421E-04 127 124.02 7.437E-04 138 123.00 7.G0IE-04 12S 128.90 7.G29E-0-I 130 136.95 VAR. 6.414E-06 3.730E-00 2.130E-06 3.SOOE-06 1.3G5E-OG 2.477E-OG 3.263E-06 6.879E-06 1.804E-05 3.084E-03 7.IS3E-0S 2.068E-04 2.040E-04 S.04SE-09 6.724E-0S 0.3GOE-OS 2.704E-05 2.BGOF-05 3.7741:-05 2.334C-03 3.63IE-03 1.053C-OS 4.741E-0G 7.17DE-0G 1.047E-05 8.222E-OG 0.I5SE-0G 3.780E-O0 2.374E-08 3.SAVE-06 3.37IE-O0 3.837E-06 3.3I7E-06 1.20-IE-OG 1 .03IE-0G 3.073E-06 9.233E-06 3.3G4E-0G 7.87SE-07 3.501E-06 3.540E-06 I.636E-0G I.7I3E-OG 1.37 IE-06 CUMULATIVE POSN. FREOU. VAR. CUMULATIVE t. I09E-03 I.107E-03 1 IOE-03 1I3E-03 1ISE-03 II7E-03 I20E-03 137E-03 I45E-03 183E-03 1.353E-03 1.454E-03 I.634E-03 1.713E-03 1.7686-03 1.849E-03 1.87GE-03 I.90IE-03 1.9JDE-03 I.030E-03 1.07GE-03 I.093C-03 1.998E-03 3.005E-03 2.015E-03 2.023E-03 3.033E-03 2.O3GE-03 3.03nE-03 3.047E-03 2.044E-03 3.048E-03 2.05IE-03 2.0S3E-03 2.053E-03 3.05GE-03 3.0G5E-03 3.0G8E-03 3.060E-03 2.O73E-03 2.07SE-03 2.077E-03 3.O70E-O3 2.080E-03 173 •174 175 176 177 170 179 180 181 183 183 184 183 180 187 ISO 188 100 191 193 103 104 103 106 107 100 100 200 201 203 203 204 205 20G 207 200 209 210 211 212 213 214 215 218 168.05 109.92 170.80 171.88 172.85 173.83 174.80 175.78 176.76 177.73 178.7 I 178.69 180.66 1(11.64 103.62 183.50 184.37 185.55 106.52 107.GO 188.48 1HD.45 100.43 19 1.41 192.38 103.36 104.34 193.31 100.29 197.37 198.24 199.22 300.30 301. 17 202.15 203.13 204. 10 205.00 206.03 207.03 208.01 308.98 203.96 210.94 4.7846-07 2.426E-07 3. BODE-07 7.042E-07 1.618E-07 5.237E-07 8.5S8E-07 8.976E-07 B.751E-07 4. BI7E-07 1.0B7E-06 3.B67E-07 5.2S0E-07 3.347E-07 3.GG0E-O7 2.571E-07 2.777E-07 6.373E-07 5.711E-07 3.733E-07 4.01GE-O7 5.447E-07 6.5G2E-07 0. 122E-07 4 .313E-07 I.072E-O7 3.0AGC-0? S.00GE-07 7.707E-O7 4.230E-07 8.070E-07 I.981E-07 4. I01E-07 3.304E-O7 2.921E-07 4.2I6E-07 5.137E-07 3.571E-07 7.088E-07 3.0I1E-O7 5.962E-07 B.579E-07 7.-674E-07 3.O0OE-O7 3. I38E-03 3.130E-03 3.130E-03 2. I39E-03 I39E-03 HOE-03 14 1E-03 I42E-03 I43E-03 143E-03 144E-03 144E-03 145E-03 2. MSE-03 2.I45E-03 2.143E-03 3.I46E-03 3.14GE-03 3. 147E-03 2. I4 7E-03 3. 148E-03 3. 148E-03 2.149E-03 3. 149E-03 2.IS0E-03 2. 150E-O3 2. ISOE-03 2.IS1E-03 3.IS 11-03 2.1321-03 3.1521-03 3.IS3E-03 . 2.I53E-03 3.153E-03 154E-03 154E-03 I54E-03 155E-03 I55E-03 15GE-03 2. 1S6E-03 3. 1S7E-03 3.150E-O3 2. I58E-03 O Tab le F.1 - cont'd 191 S33-SS33SSS83?S3?S8SS83 l i i i l i i i iUHH ????S?SSSS8SfSSS8SS333S3SS8S88^ S S 5 S 3 S r : £ S ? S S 3 3 g S 5 S 5 5 ? 5 S S S S 8 S S 5 » 2 2 ^ = 8 S 8 3 8 S S 3 8 3 3 3 S S 3 8 8 3 3 3 3 8 8 S 3 8 3 3 3 8 3 3 3 8 8 3 S 3 3 8 3 3 5 3 S 8 S 8 8 8 8 8 8 8 8 S 8 8 8 8 8 8 S 8 S S S S S S ^ s 2SB2S8R8SS;5555H|58sSSS^ I ???????????????????????????????????????? I .88888888888838888888888888888888^ -S 3 § S S S g S S S ^ S S S S S K S S g 5 ! S ? ^ S 8 S o S J ; 5 S S c s = 23S35S 8 SSS33SS8S2^SS2SSt:?SS = 3SSSS5SSgSSSSS8£SS8oS35So 192 As a means of check ing the c a l c u l a t i o n per formed by t h i s r o u t i n e , a t ime s e r i e s was genera ted and input to the r o u t i n e . 4 The form of t h i s t ime s e r i e s was g ( t ) = Z a sinw t , where a i=1 i i i r ep resen t the ampl i tudes of the s p e c i f i c s i n u s o i d , co = 2itn /NAt i i i s the f r equency , N i s the t o t a l number of data p o i n t s , and At i s the i nve r se of the sampl ing r a t e . For the t e s t f u n c t i o n , a sampl ing ra te of 500 Hz was used , so tha t At = .002 seconds . The parameters used were a , = 25 b i t s n, = 100 a 2 = 5 0 n 2 = 150 a 3 = 75 •— n 3 = 200 a 4 = 100 n„ = 300 These va lues of n g i ve f = n /NAt va lues of i i i f , = 12.2 Hz f 2 = 18.3 Hz f 3 = 24.4 Hz f , = 36.6 Hz T - The v a r i a n c e i s a2 = _]_/ g 2 ( t ) d t . Us ing i n t e g e r v a l ues fo r T 0 the n reduces the form to a2 = Ea 2 / 2 , which i s = 9375 b i t s 2 i . i i f o r the parameters g iven above. 1 93 f (Hz) numerical cumulative f(Hz) a n a l y t i c a l cumulative variance, o' variance, o' 11 .72 12.70 13.67 ' 3.4 312.0 314.8 12.2 312.5 17.58 18.55 19.53 322.9 1547. 1 563. 18.3 1562.5 23.44 24.41 25.39 26.37 1575. 1619. 4367. 4377. 24.4 4375.0 36. 13 37. 1 1 38.09 39.06 4424. 9310. 9358. 9362. 36.6 9375.0 f (Hz) 12.2 18.3 24.4 36.6 ( ( a 2 - o J ) / < r z ) x l 0 0 N A A  0.7% 0.03% 0.05% 0.14% Tab l e F.2 - Compar ison of c u m u l a t i v e v a r i a n c e s c a l c u l a t e d u s i n g the s p e c t r a l r o u t i n e wi th the expected v a l u e s ( u n i t s a re b i t s 2 ) . 194 The d i r e c t l y c a l c u l a t e d output i s l i s t e d in Tab le F.2 a long w i th the v a r i a n c e c a l c u l a t e d n u m e r i c a l l y by the s p e c t r a l r o u t i n e , which i s 9363 b i t s 2 . The d i s c r e p a n c y i s due to the f a c t tha t the generated t e s t f u n c t i o n i s not random but i s equa l to ze ro at the beg inn ing of the b l o c k , and on l y a sma l l number of c y c l e s occur in the tapered r e g i o n . In t h i s r e spec t the t e s t s i g n a l i s not r e p r e s e n t a t i v e of the a c t u a l shear s i g n a l to which the r o u t i n e i s a p p l i e d . The r e l i a b i l i t y of the program in c o n v e r t i n g the data in b i t s to a p p r o p r i a t e shear u n i t s was con f i rmed by a p p l y i n g the a p p r o p r i a t e c a l i b r a t i o n f a c t o r s to 9u/9z = (2.5/4)(5/2048)X/(2/2KkSpW 2 ) where K, k, S, p and W have been d e f i n e d in Appendix E . X i s the magnitude in b i t s of the s p e c t r a l v a l u e , (5/2048) conve r t s from b i t s to v o l t s and (2.5/4) i s the FM g a i n . For the f o l l o w i n g va lues of the c a l i b r a t i o n c o n s t a n t s : k = 2.7 seconds S = 4 x 1 0 ' 5 v o l t s / ( d yne/cm 2 ) K = 1 p = 1 .028 gm/cm3 W = 48 cm/sec i t f o l l o w s tha t 9u/9z = 2. 11x10" 3 X The a n a l y t i c a l and numer i ca l c a l c u l a t i o n s are shown in Tab le 1 9 5 f ( H Z ) n u m e r i c a l c u m u l a t i v e f ( H z ) a n a l y t i c a l c u m u l a t i v e v a r i a n c e , o' v a r i a n c e , o' 11.72 12.70 13.67 17.58 18.55 19.53 23.44 24.41 25.39 26.37 36. 13 37. 1 l 38.09 39.06 1 .493x10" 1.382x10" 1.394x10" 1.430x10" 6.853x10" 6.923x10" 6.976x10" 7.173x10" 1.935x10" 1.939x10' 1.959x10" 4.125x10" 4.146x10" 4.147x10" 12.2 18.3 24.4 36.6 1.379x10" 1 6.897x10" 1 1.931x10" 4.138x10" 2 f ( H z ) 12.2 18.3 24.4 36.6 ((o2-o2)/o2)xl00 N A A  0.80% 0.38% 0.40% 0.20% T a b l e F . 3 - Compar ison of cumu la t i v e v a r i a n c e s c a l c u l a t e d u s i n g the s p e c t r a l r o u t i n e w i th the expected v a l u e s ( u n i t s are s e c - 2 ) . 196 APPENDIX G - RESOLUTION OF THE DISSIPATION MEASUREMENT The r e s o l u t i o n of a measur ing techn ique i s determined by the c o n t r i b u t i o n to the no i se of the v a r i o u s components of the sys tem. The fundamental measurement made in t h i s case i s of sma l l s c a l e shear u s i ng the a i r f o i l p robe . The probe i s mounted on an inst rument hous ing and the s i g n a l i s p rocessed e l e c t r o n i c a l l y . Sources of no i s e a s s o c i a t e d w i th the measurement a r e : 1) i nheren t no i s e in the a i r f o i l probe i t s e l f 2) e l e c t r o n i c no i se 3) ins t rument v i b r a t i o n p i c k e d up by the p robe . A number of t e s t s were made to determine both the predominant source of no i se to the measurement and the a c t u a l no i s e l e v e l or r e s o l u t i o n of the d i s s i p a t i o n c a l c u l a t i o n . The v a r i o u s no i se sources w i l l be d i s c u s s e d in t u r n . I n i t i a l t e s t s made to determine the no i se l e v e l of the probe i t s e l f were conducted in the anecho ic chamber of the Mechan i ca l E n g i n e e r i n g Department at the U n i v e r s i t y of B r i t i s h Co lumb ia . The measurement was a f f e c t e d , however, by an un fo r tuna te m i s l a y i n g of an e l e c t r i c a l condu i t which se rved to coup le the anecho i c chamber to the b u i l d i n g , and i t appeared tha t the s i g n a l p i c k e d up by the probe was c h i e f l y b u i l d i n g n o i s e . An a l t e r n a t e t e s t was conducted in the Oceanography h u t s . A probe was p o t t e d up in epoxy r e s i n to r e s t r a i n i t s motion and suspended by a rubber band (which performed the r o l e of a r a the r i 97 c rude v i b r a t i o n i s o l a t i o n system) in a meta l p a i l s h i e l d e d from 60 Hz us i ng aluminum f o i l . The s i g n a l was p r e a m p l i f i i e d and d i f f e r e n t i a t e d and the output spectrum (measured wi th the HP3582A spectrum ana l y se r ) i s shown in F i gu re G . 1 . The da ta and the c a l c u l a t i o n made to determine the no i se are shown in Tab le G . 1 . and w i l l be c o n s i d e r e d to be r e p r e s e n t a t i v e of the o ther c a l c u l a t i o n s in t h i s s e c t i o n . The no i se s p e c t r a l d e n s i t y i n t e g r a t e d to 40 Hz i s 62 M v o l t s . Conver ted to u n i t s of d i s s i p a t i o n us i ng the nominal c i r c u i t v a l u e s , the probe s e n s i t i v i t y of 3X10~ 5 vo l t s / (dyne/cm 2 ) and a f a l l r a te of 75 cm/sec , t h i s i s 2 x 1 0 " 1 0 W/m 3. The r e l a t i v e c o n t r i b u t i o n to the no i s e due to FM t r a n s m i s s i o n of the s i g n a l and the tape r e c o r d i n g were measured in the l a b . To measure the FM c o n t r i b u t i o n , the a i r f o i l probe connec t i ons were s h o r t e d , and the VCO outputs fed d i r e c t l y i n t o the Sonex FM demodulat ion sys tem. The outputs of the d i s c r i m i n a t o r s used fo r the shear s i g n a l s (3 .9 and 5.4 kHz) as w e l l as f o r the a m p l i f i e d shear s i g n a l s (1.3 and 1.7 kHz) were input to the HP3582A and the no i s e s p e c t r a l d e n s i t y f u n c t i o n measured and i n t e g r a t e d to 40 Hz . To determine the e f f e c t of the tape r e c o r d i n g system the mixed FM s i g n a l was r eco rded on one channe l of the i n s t r u m e n t ' s Ph ideck r e co rde r wh i l e a tape speed r e f e r ence s i g n a l was r eco rded on a second channe l to p rov ide a means of a c coun t i ng f o r wow and f l u t t e r of the tape d r i v e . T h i s tape was p l a yed back , the s i g n a l demodulated and the no i s e s p e c t r a l d e n s i t y c a l c u l a t e d . 861 199 date March 26, 198l time 2000 PST bandwidth(BW) = 0.6 Hz f(Hz) V ( n v o l t s ) V/VBW 0-5 3. 4. 5-10 7. 9. 10-15 6. 8. 15-20 8. 10. 20-25 7. 9. 25-30 9. 12. 30-35 8. 10. 35-40 11. 14. V = l /U(V/,/BW) IAf) = 62<iV NOISE p r e a m p l i f i e r g ain = .85 seconds S = 3x10" 5 v o l t s / ( d y n e / c m J ) W = 75 cm/sec Ou/3z) = V /(W2SW*/v) = 1 . 6 x l 0 " * s e c ' 1 NOISE NOISE t = 7 . 5 K U U / 3 Z ) j = 2 x l O " 1 0 W/mJ NOISE NOISE I n t e g r a t i o n of the no i s e s p e c t r a l d e n s i t y f u n c t i o n and c a l c u l a t i o n of e q u i v a l e n t d i s s i p a t i o n due to the i nhe ren t n o i s e of the shear p robe . 200 The r e s u l t s of these t e s t s are summarized in Tab l e G .2 , in terms of e q u i v a l e n t d i s s i p a t i o n u n i t s . Use of the a m p l i f i e d shear s i g n a l s reduces the no i se l e v e l by a f a c t o r of 10 and the a d d i t i o n of the tape r e c o r d i n g system s u b s t a n t i a l l y i n c r e a s e s the n o i s e . The very best we can do , us ing the tape r e c o r d i n g system and both a m p l i f i e d shears i s 2 . 2 x 1 0 " 1 0 W/m 3, which r i v a l s the inhe ren t no i s e l e v e l of the p robe . S ince the s m a l l e s t measurements made have been no sma l l e r than 5X1 0 " 8 W/m3 and g e n e r a l l y a re more l i k e 1-3X10~ 7 W/m3 in q u i e t s t r e t c h e s of the data r e c o r d , the probe n o i s e and e l e c t r o n i c no i s e are not l i m i t i n g f a c t o r s . Arguments have been made fo r the e x i s t e n c e of a background tu rbu l ence l e v e l r e q u i r e d to d i s s i p a t e the t i d a l energy (Lambeck(1977)) and to ma in ta in the energy ba lance in the i n t e r n a l wave f i e l d (O lbe r s ( 1983 ) ) . The e s t ima tes g i ven by Lambeck and O l b e r s are i n the range of the s m a l l e s t d i s s i p a t i o n measurements made to d a t e . T h i s prompted an i n v e s t i g a t i o n of h yd rodynamica l l y induced no i se sources which i s r e p o r t e d in Mourn and L u e c k ( l 9 8 4 ) . From a s e r i e s of p r o f i l e s made in a l o c a l B r i t i s h Columbia i n l e t noted fo r i t s t u r b u l e n t qu i escence and f o r which a c ce l e rome te r measurements were compared to shear probe measurements, i t was determined tha t v e h i c l e a c c e l e r a t i o n s which must be hyd rodynamica l l y induced by f low over the ins t rument body r e s u l t in an upper l i m i t to the no i se of 3X10~ 7 W/m 3. I t i s s i g n i f i c a n t tha t t h i s no i se l e v e l can be reduced f u r t h e r by at l e a s t a f a c t o r of 4 by removal of the rear r ecove ry r i n g , due to the replacement of broadband v i b r a t i o n a l c h a t t e r (1-10HZ ) which 201 contaminates the d e s i r e d s i g n a l by low f requency t i l t i n g of the inst rument which can be f i l t e r e d o u t . T h i s i m p l i e s that any background tu rbu l ence l e v e l s must be l e s s than the no i se l e v e l of the Camel . It a l s o p r o v i d e s g u i d e l i n e s f o r the ongoing r edes ign of Camels . 202 date August 10, 1 9 8 1 n o i s e s p e c t r a l d e n s i t y f u n c t i o n i n t e g r a t e d to 40 Hz S = 3x10" 5 v o l t s / ( d y n e / c m J ) W = 75 cm/sec e * e e -S± _S_2_ SA 1 SA2 FM 1.3x10"' 1.2x10-' 1.1x10"" 1.4x10"" FM/TR 2.3x10"' 2.7x10'' 2 . 2 x l O " 1 0 7.4x10"" Comparison of n o i s e l e v e l s due to FM and tape r e c o r d i n g systems on a l l of the shear c h a n n e l s . 203 APPENDIX H ~ ERRORS IN THE DISSIPATION CALCULATION A d i s c u s s i o n of the e r r o r s in the c a l c u l a t i o n of e i s necessa ry so tha t we may be w e l l aware of i n d i v i d u a l c o n t r i b u t i o n s and may, at some fu tu r e t ime , be ab l e to c o r r e c t the f a u l t s tha t e x i s t in the system and improve the accu racy of our e s t i m a t e s . In no p a r t i c u l a r o rder of importance the r e c o g n i z e d e r r o r s are due t o : 1) the i naccu racy in the measurement of the ga in of the e l e c t r o n i c s used to p roces s the s i g n a l 2) the es t imate of the f a l l r a te 3) the es t imate of the shear probe s e n s i t i v i t y 4) the l a ck of c o r r e c t i o n fo r the f requency response of the e l e c t r o n i c s 5) the assumption of i s o t r o p i c t u rbu l ence 6) the l i m i t e d s p a t i a l response of the probe 7) the s p e c t r a l v a r i a n c e missed s i n c e the i n t e g r a t i o n scheme does not t r a ck the v i s c o u s c u t o f f wavenumber e x a c t l y . The f i r s t four sources of e r r o r on the l i s t are the most obv ious and are more e a s i l y d i s c u s s e d . Measurement of e l e c t r o n i c s ga in us ing the HP3582A spectrum ana l y se r i s no l i k e l y worse than 1%. As a squared term in the es t imate f o r e (Equat ion 2.6) t h i s r e s u l t s in a 2% e r r o r i n e. From Appendix C, the f a l l r a t e i s known w i t h i n ±1.5 cm/sec. S ince the ins t rument s lows c o n s i d e r a b l y as the d e n s i t y i n c r e a s e s , the s m a l l e s t f a l l r a t e s of about 50 cm/sec occur at dep th , g i v i n g a 204 3% e r r o r in the f a l l r a t e , and .hence a 12% e r r o r in e, s i n ce e a W * . Lueck , Crawford and Osborn ( l983) es t imate tha t n e g l e c t i n g the f requency response of the e l e c t r o n i c s causes a d i r e c t 5% e r r o r , or 10% in e. The c a l i b r a t i o n of the shear probes i s a c cu ra t e to 10% ( R . N i n n i s , pe r sona l communica t ion ) . T h i s r e s u l t s in an e r r o r of 20% in e. The problem of how f a r to i n t e g r a t e the spectrum to recover the a p p r o p r i a t e v a r i ance f o r computing e i s more s u b t l e and hence warrants more a t t e n t i o n . A c l o s e l y r e l a t e d problem i s the cho i c e of h i g h or low ga in shear channe l used in the computa t ion . These d e c i s i o n s were made a u t o m a t i c a l l y by my r o u t i n e 'DISS IPATION' . The c r i t e r i a upon which the d e c i s i o n s were based are d i s c u s s e d h e r e . F i g u r e H.1 shows four shear s p e c t r a which range over approx imate l y four decades in e and are from a p r e l i m i n a r y t e s t drop made in Qua ts ino Sound on Vancouver I s l and in A p r i l , 1981. These have been computed us ing the r o u t i n e d i s c u s s e d in Appendix F and are l a b e l l e d in the f i g u r e in descend ing magni tude . A l s o computed i s the Kolmogorof f m i c r o s c a l e , rj = (v3/e)^, which i s conve r t ed to a c y c l i c f requency us i ng the f a l l r a te of 50 cm/sec, f = W/2TTT>. The expected downward and l e f t w a r d s h i f t in s f requency of the peak of the shear s p e c t r a as the energy d i s s i p a t i o n dec reases i s seen . The cons tancy ( in d i m e n s i o n l e s s wavenumber k/k , where k = 1/TJ) of the peak in the d i s s i p a t i o n s s spectrum has been g e n e r a l l y accepted due to i n v e s t i g a t i o n s which have found t h a t : i ) k/k = 0.1 in a t i d a l channel (Grant et s 205 1 10 100 1000 L O G F (Hz) Four shear s p e c t r a from Qua t s ino Sound on Vancouver I s l a n d taken i n A p r i l , 1981. The d i s s i p a t i o n s c a l c u l a t e d from each spectrum a r e : 1 )6 .2x10-* ; 2 ) 1 . 0 x 1 0 - ' ; 3 ) 1 . 3x 1 0'" 5 ; and 4 ) 1 . 3 X 1 0 ' 6 W/m 3. The f a l l r a te was 48 cm/sec. The arrows r e f e r to k/k = 0.15 fo r each spec t rum. s 206 a l . ( l 9 6 2 ) ) and i i ) k/k » 0.15 in a g r i d produced t u rbu l ence s (Stewart and Townsend(1951)) . For compar i son , an arrow i s drawn over the spectrum at k/k =0.15 and the agreement w i th the peak s i s q u i t e good - excep t i ng spectrum 4, which i s q u i t e l i k e l y ove res t ima ted us ing the low ga in shear which was used fo r t h i s c a l c u l a t i o n , thereby s h i f t i n g the es t imated peak to the r i g h t . Spectrum 4 in F i gu re H.1 i l l u s t r a t e s the n e c e s s i t y of h igh ga in a m p l i f i c a t i o n of the shear s i g n a l p r i o r to r e c o r d i n g . The peaks at 6 and 12 Hz a lone of spectrum 4 account f o r 30 percent of the v a r i a n c e i n t e g r a t e d to 20 Hz and are d i r e c t l y a t t r i b u t a b l e to unwanted s i g n a l added by the tape r e c o r d i n g system in the form of incomple te compensat ion fo r tape speed wow and f l u t t e r by the FM d i s c r i m i n a t o r system used . The e f f e c t of incomplete tape speed compensat ion i s e a s i l y demonstra ted by examining the spectrum of the output of a tape r e co rded s i g n a l w i th grounded i n p u t . The e f f e c t i v e n e s s of the h i g h ga in channe l in overcoming t h i s problem w i l l be d i s c u s s e d f u r t h e r . The ambient h igh f requency no i se i s cons tan t among the four s p e c t r a in F i g u r e H.1 and becomes of i n c r e a s i n g importance f o r sma l l e r d i s s i p a t i o n s . Spectrum 1 may be i n t e g r a t e d to 60 Hz (120 cyc/m) be fo re no i se i s encoun te red . However, i n t e g r a t i o n of spectrum 3 pas t 15 Hz l e ads to the i n c l u s i o n of no i s e in the e s t i m a t e , and spectrum 4 must be r e p l a c e d by the h i g h ga in shear s i n ce i t i s dominated by n o i s e . The s p e c t r a l i n t e g r a t i o n c u t o f f f r e q u e n c i e s used are shown in F i g u r e H .2 , p l o t t e d w i th the i n t e g r a t e d u n i v e r s a l d i s s i p a t i o n 207 v < o L U C J CC CC Q ' o C O L U C h i o o •t _±_ _2_ e ( W / m 3 ) • I O ' 2 4> 1 0 " 3 X I O ' 4 + 1 0 - 5 A 1 0 - 6 © I O " 7 • X 0 . 0 0 . 4 K / K S 0 . 8 F i g u r e H.2 - Percentage of the v a r i a n c e r e s o l v e d by the shear probe vs n o r m a l i z e d wave number. The cu rve i s the i n t e g r a t e d u n i v e r s a l d i s s i p a t i o n wavenumber spec t rum. Symbols at the very bottom r ep resen t the probe c u t o f f (70 cyc/m) wh i le s m a l l e r symbols above r e p r e s e n t i n t e g r a t i o n c u t o f f s . 208 c u r v e . The o r i g i n a l energy spectrum i s from Nasmyth(1970), conve r t ed to the t r ans ve r s e spectrum and d i s s i p a t i o n spectrum by Oakey(1982) and i n t e g r a t e d to show the cumula t i ve v a r i a n c e by Lueck , Crawford and Osborn (1983) . It shows the v a r i a n c e r e s o l v e d by i n t e g r a t i n g to d imens i on l e s s wavenumber k/k . The s upper l i m i t to the i n t e g r a t i o n must account f o r the s p a t i a l r o l l o f f of the p robe , e s t ima ted to be approx imate l y 70 cyc/m by N i n n i s ( 1 9 8 4 ) . The e q u i v a l e n t d imens i on l e s s wavenumber k/k fo r s 70 cyc/m i s shown ad jacent to the k/k a x i s fo r v a l ues of e s denoted by the symbols in the key and a k inemat i c v i s c o s i t y , v = .01 c m 2 / s e c . At q u i t e h igh d i s s i p a t i o n r a t e s of 1 0 " 2 W/m 3, b a r e l y 50% of the v a r i ance i s r e s o l v e d by i n t e g r a t i n g to the probe c u t o f f , wh i le at 1 0 " 3 W/m 3, 80% i s r e s o l v e d . S ince no va lues were measured e i t h e r from PEQUOD or WESPAC which were g r e a t e r than 1 0 " 2 W/m 3, and on l y a dozen (of more than ten thousand) were g rea te r than 1 0 " 3 W/m3 (and the l a r g e s t of these was 3 X 1 0 - 3 ) which were i n t e g r a t e d to 56 Hz (70 cyc/m at 80 cm/sec ) , the s p a t i a l r e s o l u t i o n problem was not c o n s i d e r e d to be a s e r i o u s one . Below 1 0 " 3 W/m 3, i n t e g r a t i o n to the probe c u t o f f l eads to the i n c l u s i o n of n o i s e in the v a r i a n c e computa t ion , and hence the i n t e g r a t i o n i s t e rm ina ted at 30 Hz (38 cyc/m at 80 cm/sec ) . Below about 5 x 1 0 " 5 W/m 3, however the probe c u t o f f wavenumber i s g r ea t e r than the v i s c o u s c u t o f f wavenumber, and does not a f f e c t the measurement. For va lues of e < lO " 5 W/m 3 , the i n t e g r a t i o n was t e rm ina ted at 15 Hz (19 cyc/m at 80 cm/sec ) . The o ther cho i c e to be made, bes i des the f requency at which 209 to t e rmina te i n t e g r a t i o n of the spectrum i s the use of low or h igh ga in s h e a r . F i g u r e s H.3 and H.4 show a con t inuous t ime s e r i e s w i th r e s p e c t i v e s p e c t r a from four ad jacent data b l o c k s computed from a drop made in Monterey Bay in November, 1981. Shown are the two low ga in time s e r i e s w i th one of the h i g h ga in shear t ime s e r i e s and a l l of the s p e c t r a from the four s i g n a l s fo r each s e c t i o n (numbered 57-60) . The d i s s i p a t i o n computed from each low or h igh ga in p a i r i s shown in H .3 . The t ime s e r i e s have not been c a l i b r a t e d and are on ly i n c l u d e d f o r q u a l i t a t i v e compar i son . Noise in the low ga in shear channe l i s a t t r i b u t e d to incomplete tape speed compensat ion of the s i g n a l r e co rded on the i n t e r n a l c a s s e t t e r eco rde r of Camel I I I . At d i s s i p a t i o n s l e s s than about 1 0 " 5 W/m3 (as computed from the h igh ga in shear channe l ) t h i s no i se dominates the s i g n a l and the d i s s i p a t i o n es t ima ted from the low ga in shear channe ls i s c o n s i d e r a b l y h ighe r (a f a c t o r of 20 at 4 x 1 0 " 7 W/m 3). High ga in shear s p e c t r a i n d i c a t e the r e d u c t i o n of the h i g h f requency no i s e by about 2 decades in shear u n i t s . Spec t r a fo r b lock 57, where the computed d i s s i p a t i o n i s 1 .5x10" 5 W/m3 agree much b e t t e r due to the h ighe r ( t u r b u l e n t v e l o c i t y s i g n a l ) / ( t a p e r e co rde r no i s e ) l e v e l . On the o ther hand, b lock 60 shows the predominance of the no i se above the tu rbu l ence s p e c t r a in the low ga in c h a n n e l s , wh i le the h i g h ga in channe ls e x h i b i t a t t enua ted no i s e peaks . These s p e c t r a a l s o i n d i c a t e the agreement between the two o r t h o g o n a l l y mounted shear p r o b e s . At h ighe r d i s s i p a t i o n s , not shown h e r e , the a m p l i f i e d s i g n a l i s c l i p p e d by the VCO to which 210 S2 S A i 57 E-2-2x10-5W/m3 E A «. .5* l ( r 5 58 E = l-O*10~5 EA-2.5xK)H 59 E-87K10-6 E A«96xl0- 7 60 E»82*Kr 6 E A-4.1xi(r 7 F i g u r e H.3 - Example of shear probe outputs from drop made i n Monterey Bay i n November, 1981. The s i g n a l on the r i g h t i s the h i g h ga in a m p l i f i e d s i g n a l of tha t on the l e f t wh i l e the midd le s i g n a l i s the h i g h ga in a m p l i f i e d shear s i g n a l from a probe which i s mounted p e r p e n d i c u l a r to the f i r s t . These were d i v i d e d i n t o b locks of 1024 p o i n t s f o r which the d i s s i p a t i o n was c a l c u l a t e d from both r e g u l a r and h i g h ga in s i g n a l s ( s u b s c r i p t A ) . s i s2 sA1 s A 2 I I I ! 1 100 F i g u r e H.4 - Spec t ra from the four ad jacen t b l o c k s (57-60) of the t ime s e r i e s shown in F i g u r e H .3 . The h o r i z o n t a l a x i s i s l o g f requency (Hz) and the v e r t i c a l a x i s i s shear s p e c t r a l d e n s i t y ( s e c " 2 / H z ) . These are s c a l e d in lower l e f t hand c o r n e r . The r e f e r ence dot on each spectrum i s . l o c a t e d at (1Hz, 1 0 " 5 s e c ~ 2 / H z ) . 212 i t i s input and hence underes t imates the d i s s i p a t i o n and the low ga in channe l must be used . A p l o t of the r a t i o of the d i s s i p a t i o n computed by the low and h igh ga in shears p l o t t e d a g a i n s t that computed by the low ga in shear i n d i c a t e s that a t h r e s h o l d l e v e l of 1 0 " 9 W/m3 shou ld ~be""used when the a m p l i f i e r ga in i s 10 (as fo r PEQUOD), above which the low ga in shear i s u sed . When the ga in i s 30 (WESPAC), a t h r e s h o l d l e v e l of 1 0 " 5 W/m3 was used . I t i s very d i f f i c u l t to es t imate the e r r o r due to a poor c h o i c e of h igh or low ga in shea r . O b v i o u s l y , i t i s c r i t i c a l away from the t h r e s h o l d l e v e l but s i n ce the range over which the t h r e s h o l d l e v e l can be chosen appears to be r e l a t i v e l y b road , we can be reasonab ly we l l a ssu red tha t the best c h o i c e has been made and accept the computed es t imate from the chosen s i g n a l . The remain ing c o n t r i b u t i o n to the e r r o r i s the i n i t i a l assumption of i s o t r o p i c t u rbu l ence which i s made to enable the use of Equa t ion (2.6) fo r computing e. We have no i n f o r m a t i o n on the v e r t i c a l shear component, 9w/9z, and can on l y compare the two h o r i z o n t a l components, 9u/9z and 9v/9z. The va lue of the two h o r i z o n t a l components can vary by up to a f a c t o r of 2, a l t hough not f r e q u e n t l y , and a v a r i a t i o n of about 40% i s more u s u a l . As d i s c u s s e d in Chapter 2, however, a recent paper by Garge t t et a l . ( l 9 8 4 ) i n d i c a t e s that the assumpt ion of i s o t r o p y at d i s s i p a t i o n s c a l e s may be a good assumpt ion . E r r o r s 1)~4) c o n t r i b u t e 44% when the a b s o l u t e v a lues a re summed, r e p r e s e n t i n g the worst c a s e . D i s s i p a t i o n s l e s s than 1 0 " 9 W/m3 are r e s o l v e d to 80% or b e t t e r , a l though l a r g e r but 213 i n f r e q u e n t v a lues are more p o o r l y r e s o l v e d due to s p a t i a l r o l l o f f of the p robe . 10."* W/m3 r ep r e sen t s the worst i n t e g r a t i o n e r r o r s i n ce the spectrum becomes l i m i t e d by no i se at h i gh f requency and i n t e g r a t i o n cannot be con t i nued to the probe c u t o f f wavenumber. The h i g h ga in shear improves the s i g n a l to n o i s e r a t i o here but o c c a s i o n a l l a r g e t u r b u l e n t b u r s t s may be e l e c t r o n i c a l l y c l i p p e d , r e s u l t i n g in p o s s i b l y worse e r r o r s . These e r r o r s r e p r e s e n t , t hen , a worst case of 64%. C o n s i d e r i n g these and the assumpt ion of i s o t r o p y made, I b e l i e v e tha t the es t imate of e i s good to w i th i n a f a c t o r of 2. 214 APPENDIX I - UNITS OF e There are a number of commonly quoted u n i t s used fo r e. The u n i t s used in t h i s t h e s i s are W/m 3, which r ep resen t r a te of change of energy per un i t volume, and the assumpt ion of cons tan t d e n s i t y of seawater has been made (1028 kg/m 3 ) . T h i s assumption w i l l cause a worst case e r r o r of 4% in the e s t ima te fo r e, which i s sma l l compared to the f a c t o r of 2 which I b e l i e v e i s the a c c u r a c y . T h i s u n i t i s u s e f u l in comparing energy t r a n s f e r r a t e s , which are commonly in u n i t s of J / m 3 . U n i t s of d i s s i p a t i o n r a te per u n i t mass a r e , however, more fundamenta l l y c o r r e c t i f the c o r r e c t d e n s i t y i s not i n c l u d e d in the c a l c u l a t i o n . These are c m 2 / s e c 3 and m 2 / s e c 3 . I f one chooses to put the d e n s i t y i n t o the c a l c u l a t i o n i t i s e a s i l y shown that m 2 / s e c 3 i s approx imate l y equa l to W/kg. S ince a l l of these u n i t s have been used , and are c o n t i n u i n g to be used by v a r i o u s r e s e a r c h e r s , the f o l l o w i n g t a b l e i s p resen ted f o r the convenience of the r e a d e r . 1 m 2 / s e c 3 = IO" 4 c m 2 / s e c 3 1 W/m3 10" 1 c m 2 / s e c 3 1 W/kg « 1 m 2 / s e c 3 1 e r g . s e c " 1 /cm 3 l c m 2 / s e c 3 1 e r g . s e c " 1 /gm 1 c m 2 / s e c 3 . 215 APPENDIX J - TREATMENT OF WHITE HORSE VELOCITY AND CTD DATA White Horse v e l o c i t y data were r e c e i v e d from J . L u y t e n and G .Neede l l of WHOI. A magnetic tape was p repared of the upper 1000 m of each White Horse p r o f i l e which cor responded to a Camel III p r o f i l e . The v e l o c i t i e s were c a l c u l a t e d at 25 m i n t e r v a l s . White Horse p r o f i l e s shown in t h i s t h e s i s are of these d a t a . In order to c a l c u l a t e s h e a r s , however, the p r o f i l e s were smoothed us i ng a 3 p o i n t runn ing mean and the shear c a l c u l a t e d from f i r s t d i f f e r e n c e s of the smoothed d a t a , S = A U / A z ,where U 2 = ( u 2 + v 2 ) , and u, v are the E-W and and N-S v e l o c i t y components. Some of the p l o t s in t h i s t h e s i s r e q u i r e a d i s c u s s i o n of e r r o r s in the es t imate of the parameters in order to unders tand the s i g n i f i c a n c e of t r e n d s . A d i s c u s s i o n of the l i m i t a t i o n s of the White Horse measurements i s in Luy t en , N e e d e l l and Thompson(1982). They i n f e r tha t the v e l o c i t y i s r e s o l v e d to 4 cm/sec us ing a 25 dbar s p a c i n g . The amount of smooth ing, however, u s i ng the 3 p o i n t runn ing mean i s s u b s t a n t i a l and , f o r the purpose of c a l c u l a t i n g d i f f e r e n c e s in v e l o c i t y (as opposed to the abso lu t e va lues of the v e l o c i t y ) , I have used an e r r o r in the es t imate of the v e l o c i t y of 5u = 1 cm/sec. The e r r o r in the magnitude of the v e l o c i t y U , then i s S U = |9U/9u|6u + |9U/9v|5v S U = |u/U|8u + |v/U|5v. But 6u = 5v so 8U = ( Iu|+|v | )5u which i s l a r g e s t f o r the U sma l l e s t v a l ues of u, v . S ince the l i m i t i n g va lues are 1cm/sec, f o r which U = y/2 cm/sec, I w i l l take a worst case e s t i m a t e . o f 5U 216 = /28u = \J2 cm/sec. The accu racy of the p r e s su re measurement i s quoted as 0.1 percent of f u l l s c a l e (0.1/100x6500 dbar = 6.5 d b a r ) . However, the r e s o l u t i o n i s the c r i t i c a l parameter in de te rm in ing d i f f e r e n c e s and must be c o n s i d e r a b l y s m a l l e r . The 12-bit system of the White Horse r e s u l t s in an e r r o r of 6z = 6 5 0 0 / 2 1 2 1.6 mete rs . The worst case e r r o r in d i f f e r e n c i n g two ad jacent p o i n t s i s twice the e r r o r of each i n d i v i d u a l v a l u e , 6(AU) = 25U, 6(Az) = 28z, where 8 r e f e r s to the e r r o r and A to the d i f f e r e n c e in ad jacen t v a l u e s . The e r r o r in the shear e s t i m a t e , S = AU/Az, i s 8S = 9S 6(AU) + as 6(Az) 3(AU) 3(Az) g i v i n g 5S/S = 2|6U/AU| + 2 | 6 z / A z | . Az i s f i x e d at 25 dbar so the e r r o r in the shear es t imate depends c r i t i c a l l y on the v e l o c i t y d i f f e r e n c e between the two ad jacen t p o i n t s . In the e q u a t o r i a l s u r f a c e c u r r e n t -undercur ren t i n t e r f a c e r eg ion where v e l o c i t y d i f f e r e n c e s over 25 dbar may reach 50 cm/sec, the r e s u l t i n g e r r o r i s 5S/S = 2/2/50 + 2 (1 .6 )/25 = 0 . 2 . However, where v e l o c i t y d i f f e r e n c e s are s m a l l e r , say 1 cm/sec, the s i g n a l to no i s e r a t i o i s c o n s i d e r a b l y worse, 5S/S = 3. White Horse CTD data from the PEQUOD t r i p was a l s o p r o v i d e d by J . L u y t e n and G .Neede l l a long wi th the v e l o c i t y d a t a . P . N i i l e r of S c r i p p s I n s t i t u t e of Oceanography p rov ided the CTD data from the WESPAC t r i p . In both cases the data was f u l l y c a l i b r a t e d and g i ven at 2 meter i n t e r v a l s . 217 The in s i t u d e n s i t y i s c a l c u l a t e d us i ng the 1980 UNESCO i n t e r n a t i o n a l equa t ion of s t a t e (as g i ven by Pond and P i c k a r d ( 1 9 8 3 ) ) . The speed of sound used to c a l c u l a t e the Brunt-V a i s a l a f r equency , N, i s c a l c u l a t e d from the form g i ven by De l G rosso (1974 ) , and N i s c a l c u l a t e d from N 2 = -gAp/pAz - g 2 / c 2 , where g i s the a c c e l e r a t i o n due to g r a v i t y , c i s the speed of sound in seawater and p = p ( S ,T ,p ) i s the in s i t u d e n s i t y . S ince e s t i m a t i n g the e r r o r in N c a l c u l a t e d from CTD data i s a l o n g , t ed i ous p r o c e s s , s i n ce I do not have a l l of the s p e c i f i c s of the i n s t rumen ta l behav iour and s i n ce i t has a l r e ady been done in a q u i t e gene ra l manner by G r e g g ( l 9 7 9 ) , I r e l y on h i s i n t e r p r e t a t i o n of the e r r o r in N f o r t h i s work. The CTD mounted on the White Horse i s a 12-bi t system and a l though i t i s not at a l l c e r t a i n tha t the l i m i t a t i o n i s q u a n t i z a t i o n n o i s e , I s h a l l use F i g u r e 11 of Gregg ( l979 ) as a lower bound on the no i se in the c a l c u l a t i o n of N from PEQUOD and as an upper bound to the e r r o r in N from the 16-bi t system used fo r WESPAC. N ranges from 1 0 " 3 rad/sec to 1 0 " 2 rad/sec in most of the ocean w i th s p i k e s , f o r example, i n the e q u a t o r i a l undercur ren t co re at g r ea t e r than 3 x 1 0 " 2 rad/sec o c c u r r i n g on ly r a r e l y . For Az = 20m which r e s u l t s in a s l i g h t l y g r ea t e r e r r o r than f o r the Az = 25m used in t h i s s t udy , these v a l ues of N are a s s o c i a t e d w i th the f o l l o w i n g e r r o r s . N ( rad/sec ) 8N/N 1 0 " 3 1.0 1 0 " 2 .03 218 3x10- 2 .01 A bulk R i cha rdson number i s c a l c u l a t e d from these parameters as Ri = N 2 / S 2 , and the e r r o r i n the es t imate of R i can be es t imated from 6S and 6N, 8Ri = |9R i / a S |6S + |3Ri/aN|6N, o r , 5Ri/Ri = 2|6S/S| + 2|5N/N| . S ince Ri depends on independent v a l ues of N and S, I have e s t ima ted the ranges of each f o r a number of va lues of Ri to g i ve a f e e l f o r the e r r o r in R i . Ri N( rad/s ) S ( s- 1 ) 8N/N 8S/S 5Ri/Ri . 1 5 ( .002-.02) ( .005-.03) .05 .15 .40 .50 ( .0005-.01) ( .0005-.01) .25 .30 1 . 1 2.0 ( .0004-.01) ( .0002-.005) .30 1 .0 2.6 10. ( .0003-.003) ( .0001-.001) .50 2.0 5.0 It i s f o r t u n a t e tha t sma l l e r and more i n t e r e s t i n g va lues of Ri have a b e t t e r s i g n a l to no i se r a t i o . 219 APPENDIX K - PEQUOD DROPS T h i s appendix c o n t a i n s i n d i v i d u a l p l o t s of the l oga r i t hm to base ten of t u r b u l e n t k i n e t i c energy d i s s i p a t i o n f o r each of the drops of the PEQUOD c r u i s e . These are p l o t t e d as h i s tograms w i th the r i g h t end of each bar r e p r e s e n t i n g the d i s s i p a t i o n in u n i t s of W/m3 c a l c u l a t e d over approx imate l y 2 meter i n t e r v a l s . The l og e a x i s spans 5 decades from 1 0 " 7 - 1 0 " 2 W/m 3. The v e r t i c a l a x i s has a t i c every 200 dba r . P r eced ing the p l o t s i s a drop l og which l i s t s r e l e v a n t data p e r t a i n i n g to each d rop . The White Horse net code i s l i s t e d under WH NET. For those drops which were accompanied by White Horse d r o p s , v e l o c i t y , t empera tu re , s a l i n i t y and B r u n t - V a i s a l a f requency p r o f i l e s are p l o t t e d . Temperature i s a s o l i d l i n e wh i le s a l i n i t y i s dashed. The ranges on the h o r i z o n t a l s c a l e s are 0 - 3 0 ° C , 34.5 - 35.7 p p t , and 0. - 0.024 r a d / s e c . The v e l o c i t y p r o f i l e s show zona l v e l o c i t i e s as s o l i d l i n e s (wi th eastward f low > 0) and m e r i d i o n a l v e l o c i t i e s dashed (northward f low >0). F u l l s ca l e on the p l o t s i s ± 100 cm/sec. Drops made p r i o r to drop 13 were s t a r t e d approx imate l y four hours a f t e r the White Horse had been d ropped . T h i s procedure gave the crew the o p p o r t u n i t y to recover the ins t rument be fo re l aunch ing the Camel . However, drops 13 and f o l l o w i n g were made n e a r l y s imu l t aneous l y w i th White Horse d r o p s , as the crew ga ined c o n f i d e n c e i n our o p e r a t i o n a l p r o c e d u r e s . Due to the much s h o r t e r drop time of the Camel (the White Horse goes a l l the way to bo t tom) , we were ab le to launch and recover the Camel w i th i n one White Horse c y c l e . 220 DROP DATE TIME POSITION GOOD DATA(m) WH NET 2 02/07/82 1900 2N,138W 20-1540 B 3 02/08/82 1502 .50N,138W 20-1245 D 4 02/09/82 1210 ON,138W 20-900 E 5 02/09/82 2014 .50S,138W 20-365 F 6 02/10/82 1 247 1 .25S,138W 20-920 G 7 02/13/82 1236 ON,144.7W 20-835 none 8 02/13/82 1449 ON,144.7W 20-130,270-820 none 10 02/15/82 0835 ON,145W 200-565,590-810 K 12 02/17/82 1 108 .25N,144.7W 20-900 none 1 3 02/20/82 0417 . 50S,145W 20-920 L 14 02/20/82 1054 0N,145W 20-920 K 15 02/20/82 1947 . 50N,145W 20-900 none 16 02/21/82 0604 1.25N,145W 20-930 0 1 7 02/21/82 1618 0N,145W 20-930 K 18 02/22/82 1 355 ON,148W 200-280,390-815 none 19 02/24/82 0845 ON,153W 20-220,550-925 Q TOTAL DATA 12335m 20-300m 3780 300-I000m 8015 >!000m 540 Tab l e K.I - PEQUOD drop l o g . 221 222 TEMPERATURE (^C) BUOYANCY FREQUENCY (r»d/sec) 10 20 30 0 0.00S 0.016 0.024 -1 • 1 1 ^ 3 4 . 5 3 4 . 9 3 5 . 3 3 5 . 7 SALINITY ( p p t ) 223 224 TEMPERATURE (*C) BUOYANCY FREQUENCY (r»cV»«c) 0 10 20 SO B F0 0.006 O.OU 0.02« _ l 1 , , f. 3 4 . 5 34.9 35.3 35.7 SALINITY ( p p t ) 225 226 227 LOG € (W/m') HORIZONTAL VELOCITY (cm/Bee) PEQUOD NET F 09/02/82 228 TEMPERATURE ( * C ) BUOYANCY FREQUENCY ( r a d / t e c ) 0 10 20 30 0 0 . 0 0 8 0 . 0 1 6 0 . 0 2 4 34.5 34.9 35.3 35.7 SALINITY (p p t ) 229 LOG t (W/mM HORIZONTAL VELOCITY (cm/sec) i o - ' io-* io-» -too 0 100 230 TEMPERATURE C O BUOYANCY FREQUENCY ( r « « / » t c ) 0 10 20 30 0 0.00B 0.016 0.024 - r - 1 1 1 1-34.5 34.9 35.3 35.7 SALINITY (ppt) 231 LOG e (W/m») PEQUOD DROP 7 I 11 M i l l — i 11 m i l — i 111 m l — i i m m — i i m i l l 232 233 234 235 LOG t (w/m») PEQUOD DROP 12 n — i 11nm i i Mini—i i n u n— i i Mini 2 3 6 237 024 1 , , 1 34.5 34.9 35.3 35.7 SALINITY ( p p t ) 238 LOG t (W/m») HORIZONTAL VELOCITY (cm/sec> i i i u m — i i m i n — I 1 1m i l — i I m i n — I 111HIT/ "1 I T' 239 240 LOG t ( W / m » ) PEQUOD DROP 15 i 11 nm—i i nun—i i mm—i mini—i i nmr 241 LOG * (W/m») HORIZONTAL VELOCITY (cm/sec) i O " ' I O " ' 10"» - 1 0 0 0 100 242 TEMPERATURE C O BUOYANCY FREQUENCY (r«<S/*ec) -i • —i 1 r 3 4 . 5 3 4 . 9 3 5 . 3 3 5 . 7 SALINITY ( p p t ) 243 LOG t (W/m') HORIZONTAL VELOCITY (cm/sec) 1 0 " ' IO"' 10-* - ' '00 0 100 244 245 246 LOG * (W/m') HORIZONTAL VELOCITY (cm/sec) 1 0 - ' 10"» IO'' - 1 0 0 0 100 i i ium—i 11 nm—i 11nm i 11 inn—i 11iiiif I r 247 TEMPERATURE (*C) BUOYANCY FREQUENCY ( r * d / » e c ) 10 20 30 0 O.OOB 0 . 0 1 6 0 . 0 2 4 SALINITY (p p t ) 248 APPENDIX L - WESPAC DROPS V e r t i c a l p r o f i l e s of e and a s s o c i a t e d tempera ture , s a l i n i and B r u n t - V a i s a l a f requency p r o f i l e s from the WESPAC c r u i s e a long w i th a drop l og are i n c l u d e d h e r e . The s c a l e s are the same as f o r Appendix K, except f o r the s a l i n i t y which ranges from 33.8 - 35.0 p p t . Note tha t the deepest drops 11-13 have been p h o t o g r a p h i c a l l y reduced in order to accomodate them on s tandard page s i z e . Drops were t imed so tha t the Camel broke s u r f a c e s h o r t l y a f t e r the CTD had been brought back on deck . 249 DROP DATE TIME POSITION GOOD DATA(m) CTD 1 05/24/82 1010 22.7N.149E 20-840 none 2 05/26/82 01 54 27.7N,152E 20-1470 3 4 05/27/82 0833 28.5N,152E 20-1400 5 5 05/28/82 0847 30N,152E 20-670 7 6 05/28/82 1830 30.7N,152E 20-975 8 8 05/30/82 1447 32.5N,152E 20-135,195-1400 11 9 05/31/82 1240 34N,152E 20-1655 1 3,1 4 10 06/01/82 1239 35N,152E 420-1510 16 1 1 06/03/82 0121 37.5S,152E 930-2240 none 12 06/03/82 0837 38.25N,152E 1380-2270 21 13 06/05/82 0213 41N,152E 20-340,990-2240 24 TOTAL DATA 13070m 20-300m 2180 300-l000m 5085 >l000m 5805 T a b l e L.1 - WESPAC drop l o g . 250 LOG * (W/V) WESPAC DROP 1 i i m m — i n i n n — i t nun—i 11nm—i i IIIIII 251 LOG t (W/m») ' 0 - ' 10"» 1 0 " ' WESPAC DROP 2 i m m — i i i i n n — i m i n i i m n n — i m m « 252 253 LOG f (VJ/oiM WESPAC DROP 4 1600 11mil—i i m i n — i i i u m — i i m m — i i nmr 254 TEMPERATURE (*C) BUOYANCY FREQUENCY (r«cV«ec) , , _^  , r 33.8 3«.2 34.6 35.0 SALINITY (ppt) 255 LOG t ( W / m » ) WESPAC DROP 5 800 TTTinn—i 111fm—i' i n m * — r r r r n n— i i in i ir 256 TEMPERATURE (*C) BUOYANCY FREQUENCY (r«d/«ec) -i 1 1 1-3 3 . 8 34 .2 3 4 . 6 3 5 . 0 SALINITY (ppt) 257 LOG * (w/m») 258 TEMPERATURE (»C) BUOYANCY FREQUENCY ( r a d / s e c ) 33.8 34.2 34.6 35.0 SALINITY (ppt) 259 LOG e (W/»») 1600 WESPAC DROP 8 l M i l l i<—l I l l l l l l — I 111 nil—i 11 mil—l I 11 III 260 TEMPERATURE CO BUOYANCY FREQUENCY (r«a/«ec) _ L _ J , J , [. R 33.8 34.2 34.6 35.0 SALINITY (ppt) 261 LOG c ( W / m * ) WESPAC DROP 9 i 11 mn i i MIIII i i n i i i i i 1 1 m n — i i I 262 TEMPERATURE BUOYANCY FREQUENCY ( r » d / » e c ) 0 10 20 30 0 0 . 0 0 8 0 . 0 1 6 0 . 0 2 4 i — 1 1 ' 1 1 33.8 34.2 34.6 35.0 SALINITY (ppt) 263 TEMPERATURE ( * C ) BUOYANCY FREQUENCY ( m d / s e c ) H 1 1 : 1 r H : 1 r 33.8 34.2 34.6 35.0 SALINITY (ppt) 264 265 TEMPERATURE <*C) 10 20 BUOYANCY FREQUENCY (rad / sec ) 30 0 0 . 0 0 8 0 . 0 1 6 0 . 0 2 4 J L 34.2 34.6 SALINITY (ppt ) 35.0 266 267 LOG t (W/»») 10 04 -» 10" J  i o - » LUU1 I I I I III* 200 J 400 A ~ 600 m •o W B S 1/1 I A U « a 800-^ 1000 H 1200H WESPAC DROP 12 2400 — I I l i l i " — ' 11HUB—i M M I H — i i Hum i i iimf 268 TEMPERATURE (°C) BUOYANCY FREQUENCY ( r a d / s e c ) 0 10 20 30 0 0.008 0.016 0.024 > 0 33.B 34.2 34.6 35.0 SALINITY (ppt) 269 2400 WESPAC DROP 13 I I nun—i 111inn—i i Minn—i 1 1 nun—i m m 270 

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