UBC Theses and Dissertations

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UBC Theses and Dissertations

Microscale pressure fluctuations measured within the lower atmospheric boundary layer Elliott, James Arthur 1970

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MICROSCALE PRESSURE FLUCTUATIONS MEASURED WITHIN THE LOWER ATMOSPHERIC BOUNDARY LAYER by JAMES ARTHUR ELLIOTT B.Sc, University of Saskatchewan, 1962 M.Sc., University of British Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics I n s t i t u t e of Oceanography We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 19 70 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e 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 s t u d y . 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g 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 a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Physics  Institute of Oceanography, The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada ABSTRACT i i An i n s t r u m e n t was d e v e l o p e d t o measure the s t a t i c p r e s s u r e f l u c t u a t i o n s w i t h i n t h e t u r b u l e n t f l o w o f t h e a t m o s p h e r i c b oundary l a y e r . T h i s i n s t r u m e n t was used t o measure some o f the p r o p e r t i e s o f p r e s s u r e f l u c t u a t i o n s o v e r a f l a t b o u ndary and o v e r w a t e r waves and has p r o v i d e d t h e f i r s t r e l i a b l e p r e s s u r e d a t a w i t h i n a t u r b u l e n t b oundary l a y e r . F o r a l l o b s e r v a t i o n s o v e r a f l a t b o u ndary t h e ro o t - m e a n - s q u a r e p r e s s u r e p r o d u c e d by the boundary l a y e r t u r b u l e n c e was about 2.6 t i m e s t h e mean s t r e s s . The s p e c t r a h a d a power law b e h a v i o u r w i t h a mean s l o p e o f -1.7 f o r s c a l e s above t h e peak o f t h e v e r t i c a l v e l o c i t y s p e c t r u m . P r e s s u r e f l u c t u a t i o n s were a p p r o x i m a t e l y s p h e r i c a l i n shape, and p r o p a g a t e d downstream a t a r a t e e q u a l t o the ' l o c a l ' mean w i n d . Above t h e b o u n d a r y , t h e l a r g e s c a l e p r e s s u r e f l u c t u a t i o n s were a p p r o x i m a t e l y i n phase w i t h the downstream v e l o c i t y f l u c t u a -t i o n s ; a t s m a l l s c a l e s t h e r e was a l a r g e phase d i f f e r e n c e (-135°). These phase d i f f e r e n c e s were i n t e r p r e t e d t o be t h e r e s u l t o f t h e l a r g e p r e s s u r e p r o d u c i n g s c a l e s i n t e r a c t i n g w i t h t h e e a r t h ' s s u r f a c e , w h i l e t h e s m a l l s c a l e s were ' f r e e ' o f t h e s u r f a c e . P r e s s u r e f o r c e s r e s u l t e d i n an e n e r g y f l u x o u t i o f t h e downstream v e l o c i t y f l u c t u a t i o n s o f about 0.45 o f t h e t o t a l e n e r g y s o u r c e f o r t h e t u r b u l e n c e w i t h i n t h e band o f 0.05 < k z < 20. The p r e s s u r e t e r m i n t h e n e t en e r g y b u d g e t was f o u n d t o be about 1/10 o f the en e r g y f e e d -i n g t e r m . P r e s s u r e measurements n e a r w i n d g e n e r a t e d waves showed a l a r g e s p e c t r a l hump a t t h e wave f r e q u e n c i e s . The a m p l i t u d e o f t h i s hump i n c r e a s e d , and i t s v e r t i c a l r a t e o f decay d e c r e a s e d , as t h e mean w i n d s p e e d i n c r e a s e d . The phase d i f f e r e n c e between p r e s s u r e and waves d u r i n g a c t i v e g e n e r a t i o n was f o u n d t o be about 135°, p r e s s u r e l a g g i n g waves. T h i s d i d n o t change v e r t i c a l l y . i i i TABLE OF CONTENTS page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS x i i INTRODUCTION 1 BACKGROUND 5 PRESSURE INSTRUMENT 12 P r o b e 12 T r a n s d u c e r et at 19 A m p l i t u d e and Phase Response 23 IN SITU CALIBRATION OF THE PRESSURE INSTRUMENT 24 S u r f a c e P r e s s u r e Measurement 24 Compar i s o n o f Measurements: S u r f a c e and A i r 26 MICROSCALE PRESSURE FLUCTUATIONS OVER A FLAT BOUNDARY 29 N o n d i m e n s i o n a l i z i n g o f P r e s s u r e S p e c t r a 30 Shape and I n t e n s i t y o f t h e S p e c t r u m 32 Some K i n e m a t i c s o f t h e P r e s s u r e F l u c t u a t i o n s 37 P r e s s u r e - V e l o c i t y R e l a t i o n s h i p 43 Energy T r a n s f e r by P r e s s u r e F o r c e s 48 MICROSCALE PRESSURE FLUCTUATIONS OVER WIND GENERATED WAVES 52 Example S p e c t r a 54 D a t a 56 A. Runs 60/4, 119/1, 119/2, 119/3 56 B. Runs 167/1/1, 167/1/2, 167/2, 167/3 57 i v page C. Runs 164/1, 164/2 59 D. Runs 80/3, 6 0 / 1 , 60/2 59 D i s c u s s i o n 60 SUMMARY OF RESULTS 72 BIBLIOGRAPHY 75 APPENDIX A: EXPERIMENTAL SITES, INSTRUMENTS AND TECHNIQUES 78 E x p e r i m e n t a l S i t e s 78 ( i ) S p a n i s h Banks S i t e 78 ( i i ) L a d n e r S i t e 79 ( i i i ) B oundary Bay S i t e 80 I n s t r u m e n t s and O b s e r v a t i o n a l T e c h n i q u e s 81 ( i ) A n a l o g D a t a R e c o r d i n g 81 ( i i ) S o n i c and U-wire 81 ( i i i ) Cup Anemometers 82 ( i v ) Wave P r o b e 83 (v) Water H e i g h t and C u r r e n t 84 ( v i ) A i r and Water Temperature 84 APPENDIX B: ANALYSIS OF DATA 86 APPENDIX C: DATA SUMMARY 9A SYMBOL TABLE . 192 V LIST OF TABLES Table page I Data f o r p l o t t i n g d i s k c r o s s - s e c t i o n s 16 I I Pressure propagation v e l o c i t y , U , as a f r a c t i o n of u | L 42 P p I I I V e r t i c a l pressure gradient at the s u r f a c e 47 IV The pw term i n the boundary l a y e r energy budget 49 V Mean data f o r Runs 96 v i L I ST OF FIGURES 1 P r e s s u r e i n s t r u m e n t used t o measure t h e s t a t i c p r e s s u r e f l u c t u a t i o n s w i t h i n t h e t u r b u l e n t f l o w (a) a s s e m b l e d 102 (b) w i t h c y l i n d e r removed 102 2 P r o b e d e v e l o p e d f o r m e a s u r i n g s t a t i c p r e s s u r e f l u c t u a t i o n s w i t h i n t h e t u r b u l e n t f l o w 103 3 C r o s s - s e c t i o n s o f t h e d i s k s o f probes E, F, G 104 4 Dynamic p r e s s u r e n o i s e t e s t f o r P r o b e E a t d i f f e r e n t w i n d speeds 105 5 Dynamic p r e s s u r e n o i s e t e s t f o r P r o b e F a t d i f f e r e n t w i n d speeds 106 6 S c h e m a t i c o f t h e B a r o c e l t r a n s d u c i n g s y s t e m 107 7 B a r o c e l p r e s s u r e t r a n s d u c e r and r e f e r e n c e volume i n t h e i r c o n t a i n e r 108 8 R e s u l t s o f w i n d t u n n e l t e s t f o r t h e dynamic p r e s s u r e d i s t r i b u t i o n i n f r o n t o f t h e t r a n s d u c e r case 109 9 Arrangement used f o r c a l i b r a t i n g t h e p r e s s u r e i n s t r u m e n t f o r a m p l i t u d e and phase r e s p o n s e 110 10 D e t a i l o f t h e drum used t o c r e a t e a s i n u s o i d a l l y v a r y i n g p r e s s u r e i l l 11 C i r c u i t d i a g r a m f o r power a m p l i f i e r used t o d r i v e t h e v i b r a t i o n g e n e r a t o r 112 12 Sample f r e q u e n c y c a l i b r a t i o n o f the p r e s s u r e i n s t r u m e n t ( p r o b e and t r a n s d u c e r ) 113 13 Arrangement used f o r c a l i b r a t i n g t h e p r e s s u r e i n s t r u m e n t in situ 114 14 Sample f r e q u e n c y c a l i b r a t i o n o f t h e s y s t e m used f o r t h e s u r f a c e p r e s s u r e measurement 115 15 S p e c t r a l c o m p a r i s o n of t h e s t a t i c p r e s s u r e measured i n t h e a i r and a t t h e s u r f a c e ; t h e s e p a r a t i o n was 40 cm v e r t i c a l l y . These measurements were t a k e n a t t h e L a d n e r s i t e 116 16 S p e c t r a l c o m p a r i s o n o f t h e s t a t i c p r e s s u r e measured i n t h e a i r and a t t h e s u r f a c e ; t h e s e p a r a t i o n was 32 cm v e r t i c a l l y . These measurements were t a k e n a t t h e L a d n e r s i t e 117 v i i F i g u r e 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 page Coherence and phase between t h e s t a t i c p r e s s u r e measured i n the a i r and a t t h e s u r f a c e . These a r e t h e L a d n e r Runs .... 118 S p e c t r a l c o m p a r i s o n of t h e s t a t i c p r e s s u r e measured i n t h e a i r and a t t h e s u r f a c e ; t h e s e p a r a t i o n v e r t i c a l l y , i n c e n t i -m e t e r s , i s g i v e n i n b r a c k e t s a f t e r t h e Run number. These measurements were t a k e n a t t h e Boundary Bay s i t e 119 Coherence and phase between t h e s t a t i c p r e s s u r e measured i n t h e a i r and a t t h e s u r f a c e . These a r e t h e Boundary Bay Runs 120 Comparison o f p r e s s u r e s p e c t r a measured s i m u l t a n e o u s l y a t two d i f f e r e n t h e i g h t s . Az i s the d i f f e r e n c e i n h e i g h t , g i v e n i n meters 121 N o n d i m e n s i o n a l i z e d p r e s s u r e s p e c t r a . O b s e r v a t i o n s t a k e n o v e r w a t e r 122 N o n d i m e n s i o n a l i z e d p r e s s u r e s p e c t r a . O b s e r v a t i o n s t a k e n o v e r (a) w a t e r 123 (b) l a n d 123 Summary o f t h e n o n d i m e n s i o n a l i z e d p r e s s u r e s p e c t r a . V a l u e s p l o t t e d a r e k l l ( k ) / ( p 2 ^ 4 ) a t a k o f 1 0 - 2 c m - 1 124 N o r m a l i z e d p r e s s u r e s p e c t r a n o r m a l i z e d by t h e i r v a r i a n c e .... 125 N o n d i m e n s i o n a l i z e d u and w s p e c t r a 126 N o n d i m e n s i o n a l i z e d v s p e c t r a 127 N o n d i m e n s i o n a l i z e d uw s p e c t r a 128 Comparison o f t h e s p e c t r a l s l o p e o f p r e s s u r e s p e c t r a 129 Comparison between the n o n d i m e n s i o n a l i z e d v a r i a n c e o f t h e p r e s s u r e and o f t h e v e l o c i t y components o f Run 120/1 f o r d i f f e r e n t f r e q u e n c y bands 130 N o n d i m e n s i o n a l i z e d p r e s s u r e s p e c t r a . The c u r v e i s t h e mean o f d a t a g i v e n i n F i g u r e 2 1 ; t h e dashed l i n e s a r e e x t r a p o l a t e d f r o m t h e s o l i d c u r v e 131 Coherence and phase between two p r e s s u r e measurements w i t h v a r i o u s v e r t i c a l s e p a r a t i o n s 132 Coherence and phase between two p r e s s u r e measurements w i t h v a r i o u s c r o s s s t r e a m s e p a r a t i o n s 133 v i i i F i g u r e page 33 Coherence and phase between two p r e s s u r e measurements w i t h a downwind s e p a r a t i o n (a) c o h e r e n c e 134 (b) phase 135 34 F i x e d c o h e r e n c e s between two p r e s s u r e s i g n a l s f o r v a r i o u s p r o b e s e p a r a t i o n s . The v a l u e s p l o t t e d a r e f o r a c o h e r e n c e o f 0.14 136 35 Coherence between two v e l o c i t y measurements w i t h d i f f e r e n t s e p a r a t i o n s • 137 36 F i x e d c o h e r e n c e s between two v e l o c i t y s i g n a l s f o r v a r i o u s s e n s o r s e p a r a t i o n s . The v a l u e s p l o t t e d a r e f o r a c o h e r e n c e o f 0.14 138 37 Coherence and phase between p and u, u measured w i t h a s o n i c . H e i g h t o f o b s e r v a t i o n s r a n g e d f r o m 1.5 t o 5.5 meters 139 38 Coherence and phase between p and w, w measured w i t h a s o n i c . H e i g h t o f o b s e r v a t i o n s r a n g e d f r o m 1.5 t o 5.5 meters 140 39 Coherence and phase between p and u, u measured w i t h a h o t - w i r e . H e i g h t o f o b s e r v a t i o n s r a n g e d f r o m 1.5 t o 3 meters 141 40 Coherence and phase between p and u , u measured w i t h a h o t - w i r e . H e i g h t o f o b s e r v a t i o n s was 2 meters 142 41 Coherence and phase between u and w, v e l o c i t y components measured w i t h a s o n i c . H e i g h t o f o b s e r v a t i o n s r a n g e d f r o m 1.5 t o 5.5 meters 143 42 W a v e l e n g t h o f the p r e s s u r e f l u c t u a t i o n s a s s o c i a t e d w i t h t h e p-u phase t r a n s i t i o n , as a f u n c t i o n o f o b s e r v a t i o n a l h e i g h t . The b r o k e n l i n e i s t h e measured s c a l e s i z e 144 43 Coherence between downstream v e l o c i t y , u, and two p r e s s u r e measurements. One p r e s s u r e s e n s o r was b e s i d e the u s e n s o r , one was a t t h e s u r f a c e , 30 cm b e l o w 145 44 S p e c t r a o f pw 146 45 R a t i o o f t h e pw and uwU terms o f t h e i n t e g r a t e d n e t e n e r g y b u d g e t e q u a t i o n 146 46 S p e c t r a l d i s t r i b u t i o n o f t h e energy f l u x , by p r e s s u r e f o r c e s , from t h e u v e l o c i t y component. The i n t e g r a l g i v e n i s f o r k z f r o m 0.05 t o 20 147 i x Figure page 47 S p e c t r a l d i s t r i b u t i o n of the energy f l u x , by pressure forces, from the u v e l o c i t y component. The i n t e g r a l given i s for kz from 0.05 to 20 148 48 Pressure, v e l o c i t y and wave spectra f o r Run 173/3 149 49 Wave spectra of Data Group A. The time of s t a r t and end of each Run i s given i n the brackets 150 50 Pressure, u v e l o c i t y and wave spectra f o r Run 60/4 151 51 Pressure, u v e l o c i t y and wave spectra f o r Run 119/1 152 52 Pressure, u v e l o c i t y and wave spectra f o r Run 119/2 153 53 Pressure, u v e l o c i t y and wave spectra f o r Run 119/3 154 54 Coherence and phase between the lower pressure sensor and the waves: Data Group A. P ^ - * ! phase p o s i t i v e means p^ leads n 155 55 Coherence and phase between the two pressure sensors: Data Group A. PL " " P U P N A S E p o s i t i v e means p^ leads p^ 156 56 Coherence and phase between the u v e l o c i t y and waves: Data Group A. u-n phase p o s i t i v e means u leads n 157 57 Wave spectra f o r Data Group B. The time of s t a r t and end of each Run i s given i n brackets 158 58 Pressure and wave spectra f o r Run 167/1/1 159 59 Pressure and wave spectra f o r Run 167/1/2 160 60 Pressure and wave spectra f o r Run 167/2 . 161 61 Pressure and wave spectra f o r Run 167/3 162 62 Amplitude of the Fourier c o e f f i c i e n t s f o r pressure and waves of Run 167/3 163 63 Coherence and phase between the pressure and waves: Data Group B. p-n phase p o s i t i v e means p leads r\ 164 64 Pressure, u v e l o c i t y and wave spectra f o r Run 164/1 165 65 Pressure, u v e l o c i t y and wave spectra f o r Run 164/2 166 66 Coherence and phase between the pressure and the waves: Data Group C. p-r| phase p o s i t i v e means p leads n 167 F i g u r e page 67 Coherence and phase between t h e u v e l o c i t y and t h e waves: D a t a Group C. u-n phase p o s i t i v e means u l e a d s n 168 68 P r e s s u r e , u v e l o c i t y and wave s p e c t r a f o r D a t a Group D 169 69 Coherence and phase between t h e p r e s s u r e and t h e waves: D a t a Group D. p-ri phase p o s i t i v e means p l e a d s n, 170 70 Coherence and phase between t h e u v e l o c i t y and t h e waves: D a t a Group D. u-r| phase p o s i t i v e means u l e a d s ri 171 71 R a t i o o f measured t o p r e d i c t e d p r e s s u r e a m p l i t u d e f o r p r o p a g a t i n g waves w i t h no w i n d 171 72 p (n) a t v a r i o u s c o n s t a n t f r e q u e n c i e s f o r d i f f e r e n t W v a l u e s o f U*| /C. The v a l u e s p l o t t e d a t U^/C = 0 a r e f o r t h e p o t e n t i a l f l o w s o l u t i o n 172 73 P /p f o r d i f f e r e n t u|r/C a t c o n s t a n t k z 173 w o '5 74 p /p f o r d i f f e r e n t k z a t c o n s t a n t u|c/C 174 w o '5 75 R a t i o o f t h e p measured a t two l e v e l s . The l i n e s drawn a r e t h e p r e d i c t e d * r a t i o f r o m e q u a t i o n 17 175 76 Comparison between e q u a t i o n 16 and Dobson (1969) 176 77 Phase s h i f t between p r e s s u r e and waves a t v a r i o u s v a l u e s o f u|,-/C. Phase p o s i t i v e means p r e s s u r e l e a d s waves 177 78 Wave a m p l i t u d e and c r i t i c a l h e i g h t f o r c o n s t a n t u|,. p l o t t e d f o r d i f f e r e n t wave f r e q u e n c i e s 178 79 S p e c t r a l d i s t r i b u t i o n o f t h e a p p r o x i m a t e energy f l u x t o t h e waves, c a l c u l a t e d u s i n g t h e p r e s s u r e measured above th e wave c r e s t s 179 80 Coherence and phase between p r e s s u r e and u v e l o c i t y measured n e a r waves. Phase p o s i t i v e means p r e s s u r e l e a d s v e l o c i t y . . 180 81 Coherence and phase between p r e s s u r e and u v e l o c i t y measured n e a r waves. Phase p o s i t i v e means p r e s s u r e l e a d s v e l o c i t y . . 181 82 W a v e l e n g t h a s s o c i a t e d w i t h the p-u phase t r a n s i t i o n 182 83 N o n d i m e n s i o n a l energy f l u x f rom t h e u v e l o c i t y component, measured n e a r waves 183 84 N o n d i m e n s i o n a l energy f l u x f rom t h e u v e l o c i t y component, measured n e a r waves 184 x i F i g u r e page 85 Map o f t h e S p a n i s h Banks s i t e 185 86 P l a t f o r m and i n s t r u m e n t masts a t t h e S p a n i s h Banks s i t e (a) P l a t f o r m and masts l o o k i n g E a s t 186 (b) I n s t r u m e n t e d mast 186 87 Map o f the L a d n e r s i t e 187 88 Box i n p o s i t i o n f o r s u r f a c e p r e s s u r e measurement ( L a d n e r s i t e ) 188 89 I n s t r u m e n t s s e t up a t t h e L a d n e r s i t e , l o o k i n g NNE 188 90 Map o f t h e Boundary Bay s i t e 189 91 T y p i c a l wave probe c a l i b r a t i o n s 190 92 Cj^. e v a l u a t e d f r o m t h e d i r e c t and $ ^ e s t i m a t e o f t h e s u r f a c e s t r e s s 191 ACKNOWLEDGMENTS This work was done as part of the Air-Sea I n t e r a c t i o n program at the 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 Columbia. The research has been supported p r i n c i p a l l y by the United States O f f i c e of Naval Research under Contract N00014-16-C-0047. Addtional support came from the National Research Council of Canada, Defence Research Board, and Department of Transport (Meteorological Branch). I have been personally supported with a.National Research Council Studentship and a MacMillan Family Fellowship, while on educational leave from the A t l a n t i c Oceanographic Laboratory, Bedford I n s t i t u t e , Department of Energy, Mines and Resources. I wish to thank Dr. R.W. Stewart, Dr. R.W. Bu r l i n g , and Dr. M. Miyake fo r t h e i r guidance during the course of t h i s work. I also wish to thank a l l others who have a s s i s t e d with t h i s p r o j e c t : the graduate students, i n p a r t i c u l a r G.E. E l l i o t t , F.W. Dobson, G.A. McBean and J.R. Wilson, and the technicians of this i n s t i t u t e . F i n a l l y I thank my wife G i l l i a n f o r her part i n the preparation of t h i s report. 1 INTRODUCTION This study was centered around the E u l e r i a n measurement of turbulent s t a t i c pressure f l u c t u a t i o n s within the atmospheric boundary la y e r . The ' s t a t i c ' pressure i s the normal s t r e s s associated with motions w i t h i n the f l u i d (Hinze, 1959, p.27). Increased knowledge of the ro l e of s t a t i c pressure f l u c t u a t i o n s w i t h i n turbulent f l u i d flow i s of great i n t e r e s t to many engineering and geophysical studies. There i s a lack of a v a i l a b l e information, not from a lack of e f f o r t but rather a lack of a b i l i t y to measure t h i s v a r i a b l e r e l i a b l y w i t h i n the body of the f l u i d . The importance of t h i s measurement i s evident from the following postulated properties of the s t a t i c pressure f l u c t u a t i o n s . Pressure f l u c t u a t i o n s are credi t e d with being the 'isotropy producing' force; that i s , they are expected to t r a n s f e r energy among v e l o c i t y components ( d i r e c t i o n s ) . Or, as has been stated by Batchelor (1960, p.88), "... the pressure i s non d i r e c t i o n a l and the probable consequence i s that i t b u i l d s up the weaker v e l o c i t y component at the expense of the stronger." There was l i t t l e known of the d e t a i l of t h i s process whereby a n i s o t r o p i c turbulence, the form i n i t i a l l y generated i n most turbulent flows, was transformed toward the much studied ' i s o t r o p i c turbulence' f u r t h e r down the energy cascade. The pressure-v e r t i c a l v e l o c i t y c o r r e l a t i o n , which enters as a f l u x divergence term i n the net energy budget of a boundary layer, had not been measured e i t h e r . This term was usually assumed to be small (Lumley and Panofsky, 1964, p.121); a recent numerical study, (Deardorff, 19 70) agreed with t h i s assumption. Experimental v e r i f i c a t i o n was required. The study of wave generation i s another example of an act i v e area of research i n which pressure f l u c t u a t i o n s play an important part i n energy t r a n s f e r . Knowledge of the s t a t i c pressure 2 d i s t r i b u t i o n over waves would further our understanding of the wave generation process. In a l l these examples, d i r e c t measurement of the s t a t i c pressure w i t h i n the flow was required. Accordingly, i n this present study, s t a t i c pressure measurements were made and the above aspects were investigated. Most of the present t h e o r e t i c a l knowledge on turbulent pressure f l u c t u a t -i o n s i s f o r i s o t r o p i c turbulence. Some of the important p r e d i c t i o n s are Batchelor's estimate of the i n t e n s i t y and Obukhov's dimensional argument for the s p e c t r a l slope (Batchelor, 1960). In contrast Kraichnan (1956) has shown t h e o r e t i c a l l y that f o r nonisotropic turbulent boundary l a y e r flow the primary c o n t r i b u t i o n to pressure f l u c t u a t i o n s near the surface r e s u l t s from i n t e r a c t i o n between the turbulence and the mean shear. He estimated, using experimental v e l o c i t y data, that the magnitude of the rms pressure f l u c t u a t i o n s was greater than, but of the order of, the w a l l shear s t r e s s . Previous experimental observations of boundary la y e r pressure f l u c t u a t i o n s have been mainly confined to the measurement of these f l u c t u a t i o n s at the surface, e i t h e r of the earth or of a wind tunnel. Two Russian authors, G o l i t s y n (1964) and Gorshkov (1967, 1968) have analysed atmospheric spectra and some pressure v e l o c i t y cross-spectra from such surface observations. Gossard (1960) showed atmospheric pressure spectra f o r a wide frequency range. He had a few examples of microscale spectra that were obtained from an 'instrument' located on a tower. For the microscale region, these studies 2 2 found a mean slope of about -2 f o r a p l o t of s p e c t r a l density ((dynes/cm ) /Hz) against frequency. The pressure spectra shown by Gossard d i d not e x h i b i t the mid-frequency minimum found i n v e l o c i t y spectra as reported by Van der Hoven (see Lumley and Panofsky, 1964, p.43) but generally decreased continuously i n i n t e n s i t y from the low frequency synoptic pressure f l u c t u a t i o n s to the higher 3 f r e q u e n c y p r e s s u r e f l u c t u a t i o n s a s s o c i a t e d w i t h the b o u n d a r y l a y e r t u r b u l e n c e . T h i s m i d - f r e q u e n c y ' f i l l i n g - i n ' i s t h o u g h t t o be due t o m e s o s c a l e phenomena w h i c h do n o t d i r e c t l y p r o d u c e v e l o c i t y f l u c t u a t i o n s a t t h e e a r t h ' s s u r f a c e ; f o r example, i n t e r n a l g r a v i t y waves a t h i g h e r e l e v a t i o n s ( H e r r o n et aZ,1969). T h i s s u g g e s t s t h a t m i c r o s c a l e p r e s s u r e o b s e r v a t i o n s n e a r t h e s u r f a c e may i n c l u d e some low f r e q u e n c y energy t h a t i s n o t a s s o c i a t e d w i t h t h e l o c a l t u r b u l e n t v e l o c i t i e s . There have been a t t e m p t s t o e v a l u a t e t h e p r o p e r t i e s o f t h e s t a t i c p r e s s u r e w i t h i n the f l u i d by r e l a t i n g i t t o the v e l o c i t y ( H i n z e , 1959, p . 2 3 9 ) . Because t h e s t a t i c p r e s s u r e f l u c t u a t i o n s i n a t u r b u l e n t f l o w a r e , g e n e r a l l y s p e a k i n g , t h e r e s u l t o f t h e a i r m o t i o n s i n t e r a c t i n g w i t h each o t h e r , t h e y a r e n o t an i n d e p e n d e n t q u a n t i t y b u t a r e d i r e c t l y r e l a t e d t o a f i e l d o f v e l o c i t y . I n v e s t i g a t i o n s o f t h i s r e l a t i o n s h i p have s o f a r been r e s t r i c t e d t o t h e c o n s i d e r a t i o n o f s i m p l e p r o p e r t i e s s u c h as t h e rms p r e s s u r e i n i s o t r o p i c t u r b u l e n c e , b e c a u s e a g e n e r a l c o n s i d e r a t i o n o f t h e r e l a t i o n s h i p r e q u i r e s c o m p l i c a t e d v e l o c i t y measurements. W i l l m a r t h and W o o l d r i d g e (1962) g i v e a t h o r o u g h summary o f , as w e l l as new, d a t a f r o m o b s e r v a t i o n s o b t a i n e d t o t h a t d a t e i n w i n d t u n n e l s t u d i e s o f s u r f a c e p r e s s u r e f l u c t u a t i o n s . C o n c l u s i o n s drawn f r o m t h e i r p a p e r a r e 1) t h a t the measured rms w a l l p r e s s u r e i s f a i r l y w e l l e s t a b l i s h e d a t a b o u t 2.5 t i m e s the w a l l s h e a r s t r e s s , 2) t h a t t h e a d v e c t i o n s p e e d o f t h e p r e s s u r e f l u c t u a t i o n s i s about 0.6 t o 0.85 t i m e s the s t r e a m s p e e d , 3) t h a t t h e p r e s s u r e - p r o d u c i n g e d d i e s o f w a v e l e n g t h X decay a f t e r t r a v e l l i n g a d i s t a n c e o f a few A, and 4) t h a t t h e t r a n s v e r s e s c a l e s and l o n g i t u d i n a l s c a l e s o f p r e s s u r e f l u c t u a t i o n s measured a t t h e w a l l a r e a p p r o x i m a t e l y t h e same s i z e . P r e v i o u s t o t h i s p r e s e n t s t u d y r e l i a b l e e x p e r i m e n t a l knowledge o f t u r b u l e n t s t a t i c p r e s s u r e f l u c t u a t i o n s was l i m i t e d t o s u c h o b s e r v a t i o n s made 4 at the surface. It was decided to concentrate on the measurement of pressure fluctuations with scales equal to the velocity scales that carry the turbulent momentum flux. It i s in this range that important energy transfers by pressure forces are expected (see Background). Instrumentation that could measure the st a t i c pressure fluctuations in the body of the f l u i d had to be developed and tested (see Pressure Instrument and In Situ Calibrations of the Pressure Instrument). This instrumentation was then used to obtain data related to the 'description' of the measured pressure fluctuations as well as to obtain estimates of some of the energy fluxes by the pressure forces (see Microscale Pressure Fluctuations over a Flat Boundary). Observations taken over wind generated waves are used to describe some of the properties of the pressure fluctuations associated with wave generation (see Microscale Pressure Fluctuations over Wind Generated Waves). The data analysed was collected at both land and over-water sites near the Institute of Oceanography, U.B.C. (I.O.U.B.C.). In making observations, other variables, such as, fluctuating wind and wave height, were obtained using instruments developed for atmospheric boundary layer research. In some cases these had been developed at I.0.U.B.C. A description of the sites and the equipment used, other than the pressure measuring instrument, is given in Appendix A. Since most of the data presented are put into nondimensional form the actual operating conditions (surface stress, mean wind) may not be given ex p l i c i t l y in the text but are included i n a table i n Appendix C. A l l the data are permanently labelled with a 'Run' number (e.g. 120/1). The data were analysed d i g i t a l l y ; details on the analysis methods are given i n Appendix B. 5 BACKGROUND It is the purpose of this section to present for later use some aspects of a turbulent boundary layer, especially s t a t i c pressure fluctuations, that can be predicted from physical arguments. Many of the predictions are based on the Navier-Stokes equation. In the usual manner (cf. Hinze, 1959), equations can be written to represent the momentum balance for the mean and for the fluctuating part of the flow. The equation for the fluctuating components in an incompressible, viscous, constant density f l u i d , written in Cartesian tensor notation is v2 3u. 3U. du 3u. du. 1 dp 3 u. — + u. — + U. — + u. — - u. — = - + V — (1) 3t 2 3x. 2 3x. 2 3 X j 2 3x. p 3x. 3x.3x. J 3 j 3 1 3 3 where U^ and u^ are the i * " * 1 components of the mean and fluctuating f l u i d velocity respectively, p is the fluctuating pressure, p i s the mean density, and v is the kinematic viscosity. The bar over a variable indicates an ensemble average; u^ and p have zero averages. When analysing observations, i t i s assumed that the data are measurements of a stationary, random process, and thus that the time averages used are ensemble averages (Batchelor, 1960, p.17). The right-handed Cartesian coordinate system to be used has x^ positive i n the direction of the mean motion in the boundary layer, and x^ vertica l l y upward. The notation x^, x^, ; u^, u^, u^; U^ is used interchangeably with x, y, z; u, v, w; U respectively. The close relationship between the pressure and velocity fluctuations can be seen by taking the divergence of equation (1). This gives 6 1 82_p_ ^ = — ~ ( u.u. - u.u. - U.u. - u.U. ) (2) p d x . d x . 9x. 8 x J i j x j x i x i K x x x j J J J J Thus t h e p r e s s u r e i s d e t e r m i n e d by t h e v e l o c i t y f i e l d and i s n o t an i n d e p e n d e n t v a r i a b l e . T a k i n g t h e i n t e g r a l o f (2) o v e r a l l s p a c e ( e . g . , Townsend, 1955, p.27) g i v e s the p r e s s u r e a t a p o i n t i n terms o f v e l o c i t y . P = " 4^ F r 2 ( u.'u.' - u.'u.' - U 'u.* - u.'U.') — d Y (3) dx.'dx.' 1 J 1 J 1 J 1 J Ixl _2 / • . . » . . I . . t TT » , , » , , t T T H where x = x ' - x; p i s measured a t x and t h e v e l o c i t i e s a t x'. T h i s shows t h a t the p r e s s u r e a t a p o i n t can be e x p r e s s e d i n terms o f t h e a p p r o p r i a t e l y w e i g h t e d v e l o c i t y g r a d i e n t p r o d u c t s from t h e s u r r o u n d i n g f l u i d and n o t j u s t a t the measurement p o i n t i t s e l f . T h i s i s sometimes c a l l e d t h e ' i n t e g r a l e f f e c t ' . The ' i s o t r o p y p r o d u c i n g ' c h a r a c t e r i s t i c o f the p r e s s u r e f o r c e s can be s e e n i n t h e e n e r g y b u d g e t o f t h e i n d i v i d u a l v e l o c i t y components. These t h e q u a t i o n s can be o b t a i n e d by m u l t i p l y i n g the i f o r m o f e q u a t i o n (1) by u^, and a d d i n g t o i t t h e j f o r m o f e q u a t i o n (1) m u l t i p l i e d by u^, and t h e n a v e r a g i n g . A s u i t a b l e a p p r o x i m a t i o n t o t h e s e e q u a t i o n s w h i c h w o u l d a p p l y t o an a t m o s p h e r i c b oundary l a y e r (Lumley and P a n o f s k y , 1964, p.71) i s o b t a i n e d by assuming the f l o w i s s t e a d y s t a t e and t w o - d i m e n s i o n a l , w i t h v a r i a t i o n o f mean q u a n t i t i e s i n t h e x ^ d i r e c t i o n o n l y . W i t h t h e s e a p p r o x i m a t i o n s the e n e r g y b u d g e t s f o r t h e i n d i v i d u a l v e l o c i t y components a r e where the f i r s t t erm on the r i g h t hand s i d e , w h i c h i s n o n - z e r o o n l y i n t h e ~uT/2 e q u a t i o n , i s t h e 'energy f e e d i n g ' t e r m r e p r e s e n t i n g e x t r a c t i o n o f e n e r g y from the mean f l o w and p u t t i n g i t i n t o the downwind component o f t h e t u r b u l e n c e ; t h e s e c o n d term r e p r e s e n t s t h e t r a n s f e r o f t u r b u l e n t e n e r g y by t h e t u r b u l e n t v e l o c i t i e s , t h e t h i r d t e r m r e p r e s e n t s the t r a n s f e r o f e n e r g y by t h e p r e s s u r e g r a d i e n t - v e l o c i t y c o r r e l a t i o n , and t h e f i n a l t e r m t h e t o t a l v i s c o u s e f f e c t on energy t r a n s f e r w h i c h i s assumed t o be e n t i r e l y d i s s i p a t i o n 'e'. The i n t e g r a l o f t h e sum o f t h e s e p r e s s u r e terms o v e r a l l s p a c e i s z e r o , t h u s t h e p r e s s u r e a c t s t o t r a n s f e r e nergy between components and i s n o t a n e t s o u r c e o r s i n k . S i n c e i t i s e x p e c t e d t h a t t h e n e t v i s c o u s e f f e c t i s t o d i s s i p a t e energy and s i n c e t u r b u l e n t a d v e c t i o n c a n n o t p r o d u c e a n e t t r a n s f e r o f energy between components (Lumley and P a n o f s k y , 1964, p.67) t h e o n l y terms r e m a i n i n g t o t r a n s f e r t h e e n e r g y b e i n g f e d i n t o t h e u-component ( i n e q u a t i o n 4a) i n t o t h e o t h e r two v e l o c i t y components a r e t h e p r e s s u r e t e r m s . Energy t r a n s f e r by t h e s e terms had n o t been measured p r e v i o u s t o t h i s s t u d y and thus i d e a s about t h e p r o p e r t i e s o f t h i s t r a n s f e r were o n l y s p e c u l a t i v e . Though the p r e s s u r e can t r a n s f e r e n e r g y between t h e d i f f e r e n t v e l o c i t y components, i t does n o t t r a n s f e r e nergy between d i f f e r e n t F o u r i e r components (wave numbers). T h i s can be s e e n i n t h e s p e c t r a l e n e r g y t r a n s f e r e q u a t i o n s ( B a t c h e l o r , 1960, p.87) where the p r e s s u r e t e r m drops o u t as a r e s u l t o f t h e 8 i n c o m p r e s s i b i l i t y condition. Thus any energy transferred from a v e l o c i t y component v i a the pressure term must appear i n another v e l o c i t y component at the same wave number. The t o t a l energy budget for the boundary la y e r can be represented by the sum of the i n d i v i d u a l equations (4); t h i s gives 9 U 1 1 3 , , 1 3 0 = - u..u - — ( u.u.u ) - — pu„ + V u.V u. (5) 1 3 „ 2 „ i x 3 p „ r 3 i x dx^ dx^ dx^ Three of these terms ( a l l except the pressure term) had previously been measured f o r the atmospheric boundary la y e r (Lumley and Panofsky, 1964, p.119 f f . ) . For most cases i t had been found that the energy feeding term was approximately balanced l o c a l l y by the viscous d i s s i p a t i o n and that the turbulent t r a n s f e r term was small. Because of the approximate balance between l o c a l production and d i s s i p a t i o n , speculation had been that the f l u x divergence by the pressure forces was also small. However the inaccuracy of such observations makes this method of approximating the f l u x divergence un s a t i s f a c t o r y and d i r e c t measurement i s d e s i r a b l e . The measurements required to evaluate the r e l a t i v e importance of the terms i n this equation can be s i m p l i f i e d by comparing the terms i n modified form. I f equation (5) i s integrated from to z, where z^ i s a lower l e v e l f i x e d near the ' t r a n s i t i o n region' (see Hinze, 1959, p.465) where the turbulence i s influenced by the viscous e f f e c t s of the surface, this gives u nu„ ( U - U ) - -T- ( u.u.u - u.u.uJ ) 1 3 'z 1 z^. 2 i x 3'z x x 3'z^ z r - p ( P u 3 l z - P ^ ' z ^ + 2 V u V u± dz = 0 (6) Z l 9 The notation ul means the wind at the l e v e l z. I t i s assumed that the 1 z ' t r a n s i t i o n region' i s thin, and there i s n e g l i g i b l e turbulent energy f l u x through i t , the stress being c a r r i e d by viscous forces. This assumption would only be applicable to observations over a s o l i d surface and not over water when waves are being generated. The approximations are that the terms evaluated at z^ are small compared with those evaluated at z. These assump-tions give - u.u.U.I ; - 4 u.u.u I ; - — p n l 1 3 1 1 z 2 I I 3'z p ^ 3 ' z as approximations for the f i r s t three terms which are terms i n the budget of turbulent k i n e t i c energy i n the space between the boundary and height z. The f i r s t term represents the net rate of working per unit area on the surface z by the Reynolds s t r e s s , - u^ u-j> the second the upwards f l u x of turbulent energy and the t h i r d the rate of working per unit area by the pressure force. A comparison of these three terms would i n d i c a t e the r e l a t i v e importance of each as a net energy source of turbulent k i n e t i c energy per unit area f o r the a i r below the l e v e l z. Dimensional arguments, i d e n t i c a l to those used to p r e d i c t the -5/3 region fo r the v e l o c i t y spectrum i n the i n e r t i a l subrange, have been used by Obukhov to p r e d i c t the shape of the pressure spectrum, II(n) (see Lumley and Panofsky, 1964, p.84). For this r e s t r i c t e d scale range with no production and no d i s s i p a t i o n , he obtained n(k) = K p p 2 e 4 / 3 k " 7 / 3 (7) where K i s some u n i v e r s a l constant. The predicted power law was f o r a 10 condition of ' l o c a l isotropy'. Batchelor evaluated the expected rms pressure for t h i s condition of isotropy. His c a l c u l a t i o n , which used measurements of the v e l o c i t y a u t o c o r r e l a t i o n (see Batchelor, 1960, p.182), gave p 2 = 0.58 p u 2 (8) Since i t i s not c e r t a i n that a condition of l o c a l isotropy occurs at any l e v e l i n the lower few meters of the atmospheric boundary layer (Stewart, 1969), i t i s doubtful whether these r e s u l t s can be compared with measured values, even though v e l o c i t y spectra have a -5/3 region. The mean wind p r o f i l e i s assumed, f o r a l l c a l c u l a t i o n s , to have the 'predicted' logarithmic form u Ul - — — In — (9) z K z o where u. = J - uw , K i s von Karman's constant (0.4) and z i s the v i r t u a l * o height where the mean v e l o c i t y goes to zero. The use of the logarithmic p r o f i l e assumes n e u t r a l s t a b i l i t y , a point to be discussed l a t e r . T y p i c a l winds at the experimental s i t e s were 3 to 10 m sec ^ . For comparison purposes, t h i s corresponds to a Reynolds number Re = ^ m ^ ^  5 of order 10^. V When nondimensionalizing the data presented below, two parameters are often — 2 used. One, the surface s t r e s s , x = -puw = pu A , i s evaluated i n one of three d i f -ferent ways: d i r e c t measurements of -uw, using the '^-^ method' or from the mean wind speed using a drag c o e f f i c i e n t . D e t a i l s on these methods are contained i n Appendix B. The method used f o r a p a r t i c u l a r Run i s given i n the Data Summary i n Appendix C. The other parameter i s the turbulent 'energy feeding' term 11 — 9U u * 3 - uw -r— . I t i s evaluated i n the form , where th i s l a s t step assumes dz Kz r a logarithmic p r o f i l e (equation 9) for U. I t was decided to take observations of pressure over the range of scales c o n t r i b u t i n g s i g n i f i c a n t l y to the t o t a l shear s t r e s s . For observations below nz U 5 meters, t h i s occurs f o r the range of nondimensional frequencies, f = -2 0 from approximately 10 to 10 (McBean, 1970). Thus the frequency range _3 required i n the observations i s from about 3 x 10 to 10 Hz. Another important frequency i s that at the peak of the w spectrum. When pl o t t e d i n i n t e g r a b l e logarithmic form, the peak occurs at about f = 4 x 10 ^ At frequencies above t h i s value, a l l spectra of the v e l o c i t y components have approximately equal i n t e n s i t y and s h o r t l y t h e r eafter a -5/3 slope. The observational technique used was to obtain observations at a f i x e d l e v e l above the surface. In the usual manner, the 'frozen f i e l d ' (Taylor's) hypothesis i s assumed. This gives the r e l a t i o n s h i p between frequency n and wave number k as k = 2 ; n d o where U i s the mean advection wind. The same assumption allows phase s h i f t corrections to be made and h o r i z o n t a l gradients to be ca l c u l a t e d from the r e l a t i o n i r - - F i r -An attempt was made to take observations only f o r a steady mean wind speed and d i r e c t i o n , n e u t r a l s t a b i l i t y and a homogeneous, f l a t t e r r a i n , since these are the assumptions used i n analysing the data. 12 PRESSURE INSTRUMENT Since no s u i t a b l e technique existed f or measuring the s t a t i c pressure fl u c t u a t i o n s i n a turbulent boundary layer, instrumentation was developed. The main d i f f i c u l t y associated with this measurement i s i n e l i m i n a t i n g the ef f e c t s of dynamic pressure; dynamic pressure i s the normal stress associated with d e f l e c t i n g flow around a s o l i d body, i n this case a sensor. As the a i r v e l o c i t y f l u c t u a t e s , so does the dynamic pressure. When measuring the s t a t i c pressure f l u c t u a t i o n s such dynamic pressure fluctuations are noise. A probe was s p e c i a l l y designed to reduce the dynamic pressure v a r i a t i o n to an acceptable l e v e l . Pressure f l u c t u a t i o n s sampled by the probe were converted i n t o an e l e c t r i c a l s i g n a l by a transducer. Through the use of pneumatic f i l t e r i n g , only the frequency range of i n t e r e s t was retained. Figure 1 shows the assembled instrument package. Probe Since i t i s not pos s i b l e to p r e d i c t the dynamic pressure d i s t r i b u t i o n over a streamlined body with s u f f i c i e n t accuracy, the shape of the probe was developed e m p i r i c a l l y . Testing was done i n a wind tunnel. I t was considered desirable to design the probe such that the s i g n a l to noise r a t i o was about 10:1. The an t i c i p a t e d f l u c t u a t i n g s i g n a l l e v e l was 2 -1 taken to be pu A ( l a t e r found to be a good guess). Thus f o r a 5 m sec wind -2 over water, the s i g n a l would be approximately 0.3 dynes cm and the desired -2 maximum noise l e v e l would be 0.03 dynes cm . This noise can be given i n 1 2 terms of a f r a c t i o n of the stagnation pressure pu where p i s the density 13 of a i r and U i s the mean wind speed. The desired noise l e v e l i s 0.001 of the of the stagnation pressure. Thus the task was to construct a pressure sampling probe which could operate at the various angles of incidence of flow that would be expected i n the atmospheric boundary layer, and have a maximum dynamic pressure v a r i a t i o n of only 0.001 of the stagnation pressure. A s u i t a b l e probe would be a shaped streamlined body, small with respect to the scales of i n t e r e s t , which has some point on i t s surface where there i s s u f f i c i e n t l y small v a r i a t i o n i n the dynamic pressure; the sampling port would be located at that point. The mean dynamic pressure at some p o s i t i o n away from the stagnation point i s minimized when the d i s t o r t i o n of the n a t u r a l flow as i t passes a probe i s minimized. W.W. Willmarth (personal communication to R.W. Stewart) suggested that a t h i n streamlined disk with a dipping i n the c e n t r a l region might be a s u i t a b l e method f o r f l a t t e n i n g the streamlines. A few hundred d i f f e r e n t shapes, v a r i a t i o n s on streamlined disks s i m i l a r to a planetary e l l i p s o i d , had to be tested before a promising one was found. The f i n a l probe was a th i n , c i r c u l a r , streamlined disk attached to a long, t h i n tubular stem. The disk was s l i g h t l y dipped i n the middle, with two sampling ports located one on each side at the center, Figure 2. This probe was designed to be used with the plane of the disk i n the h o r i z o n t a l . The thinness and c e n t r a l depression were designed such that the dynamic pressure at each of the two ports i s close to zero f o r the wind speeds of i n t e r e s t . This helped eliminate pressure v a r i a t i o n s from a r i s i n g from 'u 1, the downstream v e l o c i t y f l u c t u a t i o n s . C i r c u l a r symmetry was maintained i n order to eliminate 'v', the crossstream e f f e c t s . E l i m i n a t i o n of dynamic pressure changes due to 'w', the v e r t i c a l wind f l u c t u a t i o n s , was more d i f f i c u l t . For t h i s the shape of the cross-section of the disk was most important. The p r o f i l e was such that 'w' caused equal changes of opposite s i g n i n the dynamic 14 p r e s s u r e a t t h e two p o r t s . By c o n n e c t i n g the p o r t s , one on each s i d e o f the d i s k , t o a s i n g l e s m a l l n e a r l y r e c t a n g u l a r c h a n n e l c o i l e d i n t h e d i s k , c a n c e l l i n g was o b t a i n e d a t t h e m i d - p o i n t a l o n g the c h a n n e l . The p r e s s u r e s i g n a l was sampled from t h i s m i d - p o i n t . B ecause the r e q u i r e m e n t s f o r e l i m i n a t i n g t h e 'u' and 'w' e f f e c t s c o n f l i c t e d , a compromise was n e c e s s a r y i n o r d e r t o keep t h e s t a l l i n g a n g l e as l a r g e as p o s s i b l e . F o r t h e shapes d e v e l o p e d , t h e s t a l l i n g a n g l e was about 10°. T h i s a n g l e was c o n s i d e r e d l a r g e enough f o r t h e e x p e c t e d 'w'. I f t h e w i n d came from a d i r e c t i o n g r e a t e r t h a n the s t a l l a n g l e , t h e measured dynamic p r e s s u r e jumped by an o r d e r o f magnitude. The d i s k p a r t o f t h e p r o b e was c o n s t r u c t e d o f b r a s s . I n i t i a l l y i t 1 3 c o n s i s t e d o f two h a l v e s ( F i g u r e 2) , each 0.050 i n c h e s t h i c k and about 1 i n c h e s i n d i a m e t e r . I n each o f the h a l v e s a 0.031 i n c h w i d e , 0.020 i n c h deep s l o t was m i l l e d t o p r o v i d e t h e c h a n n e l i n g between t h e p o r t s and f o r t h e e x i t t u b i n g . I t was n e c e s s a r y t h a t t h e two ends o f t h e c h a n n e l n o t i n t e r s e c t , b u t be c l o s e t o t h e c e n t e r o f t h e d i s k f o r c o n n e c t i o n t o t h e p o r t s . To a c c o m p l i s h t h i s , t he ends o f t h e c h a n n e l were s t a g g e r e d t o l e a v e 0.0175 i n c h e s between th e w a l l o f t h e c h a n n e l and t h e c e n t e r . . A 0.020 i n c h p o r t was d r i l l e d a t an a n g l e from t h e c e n t e r o f t h e o u t e r s u r f a c e o f e a c h h a l f , t o i n t e r s e c t w i t h an end o f t h e c h a n n e l . When the two h a l v e s were p u t t o g e t h e r , t h i s formed a 1.50 i n c h l o n g c h a n n e l (0.031 i n c h e s w i d e , 0.040 i n c h e s deep) c o n n e c t i n g t h e two p o r t s . The two h a l v e s and a 3/4 i n c h (2 cm) l e n g t h o f s t a i n l e s s s t e e l s t e m (0.032 i n c h i n t e r n a l d i a m e t e r ) were g l u e d t o g e t h e r w i t h epoxy t o f o r m t h e i n i t i a l s t a g e o f t h e d i s k . T h i s d i s k u n i t was t h e n shaped on a l a t h e and t e s t e d f o r p e r f o r m a n c e i n a w i n d t u n n e l . F u r t h e r s h a p i n g and t e s t i n g e i t h e r p r o d u c e d a f a i l u r e o r a 1 s t a n d a r d e n g i n e e r i n g u n i t s 15 disk which had the desired q u a l i t i e s . The p r i n c i p a l reason f o r a f a i l u r e was i n a b i l i t y to match the two sides of the disk; this was necessary f o r e l i m i n a t -ing w e f f e c t s . The shapes (measured with a micrometer) of s u i t a b l e disks are shown i n Figure 3. Table I, pages 16 and 17, contains the values used to p l o t these curves. I t appears that i t i s necessary to keep the maximum and minimum thicknesses i n the cros s - s e c t i o n accurate to 0.001 inches, though the thi c k -ness between these points may vary by a few thousandths of an inch. This can be seen by comparing the disk shapes of probes E, F, and G, Figure 3. In reproducing these probes each h a l f may be shaped completely before assembly, now that the appropriate shape i s known. The good disks were s i l v e r soldered to s t a i n l e s s s t e e l tubing (Figure 1) of i n c r e a s i n g outer diameter ( f o r r i g i d i t y ) to make an o v e r - a l l stem length of approximately 22 inches (56 cm); the reason f o r t h i s length i s discussed i n the next subsection. The i n t e r n a l diameter of t h i s a d d i t i o n a l stem was 0.069 inches. The disk and stem together were the 'probe'. Dynamic noise t e s t i n g of the probes was done i n a low speed, low turbulence wind tunnel with a 90 cm by 70 cm t e s t s e c t i o n . The wind tunnel i s i n the Mechanical Engineering Department of U.B.C. I t i s a return type' tunnel i n which the a i r speed can be var i e d from 1 to 15 m sec \ The measured turbulence l e v e l i s approximately 0.1%. Even i n t h i s low turbulence wind tunnel, other background noise necessitated working on quie t , windless nights. The probe to be c a l i b r a t e d was mounted near the middle of the t e s t s e c t i o n ; a s p e c i a l clamp was constructed to hold the probe at any preset angle r e l a t i v e to the a i r flow. The dynamic noise l e v e l was measured by comparing the pressure observed by the probe to the pressure at the s t a t i c r i n g of the tunnel. The dynamic pressure noise l e v e l s f o r the two probes used f o r most of 16 TABLE I DATA FOR PLOTTING DISK CROSS-SECTIONS PROBE E PROBE F side 1 side 2 side 1 side 2 X X 0.000 0.069 0.143 0.225 0.295 0.364 0.449 0.519 0.599 0.667 0.722 0.766 0.789 0.855 0.911 0.973 1.029 1.089 1.149 1.202 1.242 1.307 1.374 1.432 1.489 1.528 1.578 0.0000 0.0285 0.0379 0.0421 0.0435 0.0436 0.0429 0.0410 0.0413 0.0403 0.0394 0.0387 0.0388 0.0387 0.0397 0.0402 0.0408 0.0418 0.0432 0.0435 0.0422 0.0424 0.0399 0.0345 0.0285 0.0235 0.0000 0.000 0.106 0.184 0.263 0.360 0.432 0.507 0.589 0.669 0.719 0.784 0.790 0.857 0.913 0.978 1.038 1.124 1.190 1.235 1.307 1.364 1.415 1.482 1.578 0.0000 0.0339 0.0381 0.0422 0.0437 0.0451 0.0422 0.0402 0.0389 0.0373 0.0366 0.0371 0.0368 0.0381 0.0385 0.0390 0.0421 0.0424 0.0422 0.0411 0.0387 0.0345 0.0277 0.0000 Diameter = 1.578 inches T o t a l Thickness = 0.087 inches 0.000 0.086 0.132 0.179 0.240 0.292 0.341 0.388 0.435 0.479 0.533 0.578 0.624 0.686 0.731 0.772 0.789 0.827 0.879 0.930 0.989 1.063 1.108 1.151 1.193 1.240 1.277 1.347 1.383 1.422 1.468 1.580 0.0000 0.0349 0.0404 0.0446 0.0458 0.0446 0.0416 0.0404 0.0395 0.0385 0.0373 0.0370 0.0365 0.0359 0.0358 0.0359 0.0358 0.0359 0.0359 0.0364 0.0372 0.0375 0.0389 0.0397 0.0405 0.0417 0.0435 0.0448 0.0439 0.0407 0.0361 0.0000 0.000 0.073 0.134 0.189 0.237 0.284 0.331 0.389 0.459 0.504 0.566 0.610 0.657 0.705 0.755 0.790 0.856 0.900 0.951 0.989 1.050 1.086 1.147 1.196 1.242 1.285 1.334 1.390 1.445 1.484 1.535 1.580 0.0000 0.0336 0.0414 0.0453 0.0458 0.0458 0.0458 0.0436 0.0413 0.0408 0.0392 0.0382 0.0371 0.0361 0.0362 0.0362 0.0362 6.0373 0.0380 0.0381 0.0398 0.0408 0.0421 0.0433 0.0455 0.0450 0.0450 0.0445 0.0404 0.0358 0.0247 0.0000 Diameter = 1.580 inches T o t a l Thickness = 0.091 inches TABLE I (continued) PROBE G side 1 side 2 X 0.000 0.115 0.225 0.335 0.415 0.515 0.625 0.715 0.790 0.900 1.015 1.100 1.195 1.295 1.390 1.485 1.581 Z 0.0000 0.0339 0.0426 0.0442 0.0459 0.0436 0.0411 0.0394 0.0379 0.0383 0.0399 0.0426 0.0434 0.0425 0.0374 0.0338 0.0000 X 0.000 0.110 0.200 0.325 0.435 0.525 0.615 0.705 0.790 0.900 0.995 1.095 1.185 1.290 1.375 1.470 1.581 Z 0.0000 0.0307 0.0394 0.0430 0.0435 0.0415 0.0399 0.0387 0.0367 0.0383 0.0393 0.0421 0.0434 0.0427 0.0391 0.0300 0.0000 Diameter = 1.581 inches T o t a l Thickness = 0.088 inches 18 the experimental work can be seen i n Figures 4 and 5. Each f i g u r e i s four graphs. Three of the graphs, l a b e l l e d as d i f f e r e n t constant wind speed, have axes of p i t c h and yaw which represent the alignment of the probe with respect to the mean wind. P i t c h i s r o t a t i o n about a h o r i z o n t a l axis perpendicular to the mean flow; t h i s simulated 'w'. Yaw i s r o t a t i o n about a v e r t i c a l a x i s ; t h i s simulated 'v'. Zero angles represent the i d e a l alignment of the probe with respect to the mean wind: the stem p a r a l l e l to the mean wind and the plane of the disk h o r i z o n t a l . The fourth graph shows the e f f e c t of a change i n mean wind speed when the probe had the i d e a l alignment; this simulates 'u'. The values p l o t t e d are the r a t i o s of the measured dynamic pressure to the c a l c u l a t e d stagnation pressure m u l t i p l i e d by - 1000. For example, i n Figure 4 the value p l o t t e d at 0° p i t c h and 0° yaw when U = 6.1 m sec ''"is 10. Thus the measured dynamic pressure f o r this alignment and wind speed was 10 P / -1000 = -0.01 P where P i s the stagnation pressure. An * p l o t t e d on s s s the graphs means that the probe s t a l l e d . I t had been decided above that an acceptable change i n th i s r a t i o between the measured dynamic pressure and the ca l c u l a t e d stagnation pressure f o r t y p i c a l v e l o c i t y f l u c t u a t i o n s to be expected i n an atmospheric boundary l a y e r i s about 0.001. This i s a change of 1 between the values p l o t t e d on the f i r s t three graphs or a change of 1 along the v e r t i c a l axis of the fourth graph. Superimposed on the dynamic noise p l o t s , by means of dashed curves, are the s t a t i s t i c a l l i m i t s f o r the v e l o c i t y f l u c t u a t i o n s expected to occur over a smooth t e r r a i n with small roughness elements, such as over water. The outer curves contain 95% of the expected f l u c t u a t i o n s i n the wind, the inner curves 68%. For these calcula-" t i o n s , the v e l o c i t y f l u c t u a t i o n s were assumed to have a Gaussian d i s t r i b u t i o n . From the graphs i t can be seen that f o r 95% of the expected angular or down-stream wind v a r i a t i o n s the dynamic pressure noise of a probe operating i n 19 a wind between 3.5 and 9 m sec i s about 0.001 to 0.002 of the stagnation pressure (a change of 1 to 2 on the graphs). Misalignment of the probe, mean wind s h i f t s and non-Gaussian, low frequency v e l o c i t y changes could cause the l i m i t curves shown to s h i f t to an area on the graphs where the dynamic pressure v a r i a t i o n i s l a r g e r than that shown. Thus a more r e a l i s t i c noise fi g u r e would be about 0.002 of the stagnation pressure giving a minimum s i g n a l to noise r a t i o of 5:1. These probes were considered s u i t a b l e f o r i n i t i a l studies i n a turbulent atmospheric boundary la y e r . Transducer et al The transducer and pneumatic f i l t e r i n g apparatus, which are connected to the end of the probe stem, are enclosed i n a streamlined container (Figure 1). Once the s i z e of t h i s container was known, the stem length of the probe was chosen (by c a l c u l a t i o n and experimentation) to keep the sampling ports beyond any dynamic pressure noise due to blockage. The e l e c t r i c a l output from the probe-transducer assembly (see Figure 1) was transmitted by cable to conditioning and recording equipment often up to 100 meters away. The transducer system used to convert the pressure f l u c t u a t i o n s i n t o e l e c t r i c a l s i g n a l s was the 'Barocel Modular Pressure Transducing System 1 (Datametrics Inc., Watertown, Mass., U.S.A.). I t consists of three u n i t s : a pressure sensor (Type 511), a s i g n a l conditioner (Type 1015), and a power supply (Type 700) (see Figure 6). The pressure sensor was a 0 to 10 mm Hg 4 -2 (0 to 1.3 x 10 dynes cm ) low i n t e r n a l volume model (0.1 cubic inch) with quick disconnect f i t t i n g s . A diaphragm i n the pressure sensor i s d e f l e c t e d by any pressure d i f f e r e n c e between two inputs. One input i s connected to the 20 p r o b e , t h e o t h e r t o a r e f e r e n c e volume w h i c h a c t s as a r e f e r e n c e p r e s s u r e . S i n c e the diaphragm i s the common p l a t e f o r two c a p a c i t o r s w h i c h a r e b o t h e x c i t e d a t 10 kHz, i t t a k e s on a v o l t a g e p r o p o r t i o n a l t o t h e p r e s s u r e d i f f e r e n c e . T h i s e l e c t r i c a l s i g n a l i s f e d t o t h e s i g n a l c o n d i t i o n e r by c a b l e . Long c a b l e s , up t o 150 m e t e r s , p e r m i t t e d remote o p e r a t i o n . The s i g n a l c o n d i t i o n e r a c c e p t s the a m p l i t u d e m o d u l a t e d 10 kHz s i g n a l and c o n v e r t s i t i n t o a 0 t o ±5 v o l t DC s i g n a l . D i f f e r e n t f u l l s c a l e s e n s i t i v i t i e s a r e o b t a i n e d by means o f a r e s i s t a n c e v o l t a g e d i v i d e r . A n u l l a d j u s t , f u l l s c a l e a d j u s t , q u a d r a t u r e a d j u s t , and s e n s i t i v i t y a d j u s t a l l o w t h e s y s t e m t o be c o r r e c t e d f o r o f f s e t s , l o a d i n g , g round l o o p s , and c a l i b r a t i o n r e s p e c t i v e l y w h i l e c o n n e c t e d i n t h e r e c o r d i n g mode. I n o r d e r t o make t h e s e a d j u s t m e n t s e a s i l y , a v a l v e between the two i n p u t s o f t h e p r e s s u r e s e n s o r can be opened. T h i s v a l v e was c l o s e d w h i l e t a k i n g measurements. The q u o t e d a c c u r a c y f o r the ' B a r o c e l ' s y s t e m i s about 0.5% o f the r e a d i n g ; n o i s e l e v e l i s a p p r o x i m a t e -l y 5 mv; and t h e t r a n s i e n t r e s p o n s e i s l e s s t h a n 2 m i l l i s e c o n d s . These s p e c i f i c a t i o n s a r e more t h a n adequate f o r t h e p r e s e n t r e q u i r e m e n t s . B e f o r e t h e p r e s s u r e s i g n a l i s c o n v e r t e d i n t o an e l e c t r i c a l s i g n a l , i t i s s u b j e c t e d t o p n e u m a t i c f i l t e r i n g . T h i s f i l t e r i n g g i v e s a band-pass c h a r a c t e r -i s t i c t o the e l e c t r i c a l o u t p u t . I n o r d e r t h a t most o f t h e R e y n o l d s s t r e s s r ange c o u l d be o b s e r v e d , t h e f r e q u e n c y range chosen f o r t h i s s t u d y i s about 0.003 t o 10 Hz. The h i g h f r e q u e n c i e s a r e damped by v i s c o s i t y i n t h e s m a l l p a s s a g e s o f t h e d i s k and p r o b e stem. Through e x p e r i m e n t a t i o n on the i n t e r n a l d i a m e t e r and t h e l e n g t h s o f t h e c o n n e c t i o n s t o the d i s k , i t was p o s s i b l e t o keep the f r e q u e n c y r e s p o n s e f l a t t o a p p r o x i m a t e l y 20 Hz. The low f r e q u e n c i e s a r e e l i m i n a t e d by a l l o w i n g the r e f e r e n c e volume t o f o l l o w t h e low f r e q u e n c y p r e s s u r e f l u c t u a t i o n s . T h i s f l o a t i n g i s a c c o m p l i s h e d by a s m a l l s l o w l e a k between the s i g n a l s i d e and t h e r e f e r e n c e s i d e o f the t r a n s d u c e r . 21 A 27 gauge, 1/2 inch hypodermic needle was found to be a s u i t a b l e leak. The pos i t i o n s of the high and low pass cutoffs are by design, not def a u l t , and are used to obtain optimum s i g n a l l e v e l s f o r the frequency range of i n t e r e s t . The frequency c a l i b r a t i o n i s described i n the next subsection. Since temperature or volume changes w i t h i n the probe and transducing system can also cause a pressure change (PV/T = constant), care i s necessary, e s p e c i a l l y on the reference s i d e , to keep these noise sources below the desired noise l e v e l . The l i m i t a t i o n s are a net temperature f l u c t u a t i o n of -4 -5 le s s than 10 C° and volume f l u c t u a t i o n s of less than 3 x 10 %. To meet these r e s t r i c t i o n s , the e n t i r e transducing system and probe package has to be r i g i d and adequately i n s u l a t e d . A h a l f l i t r e vacuum f l a s k (Figure 7) provides a pressure reference with volume and thermal s t a b i l i t y . This i s connected to one s i d e of the transducer by 6 mm diameter, i n s u l a t e d , copper tubing. A l l other interconnecting passages are formed by d r i l l i n g a s o l i d block of a c r y l i c p l a s t i c . The container f o r the Barocel sensor and reference volume i s a c y l i n d r i c a l aluminum pipe with a streamlined cone on the upwind, probe s i d e , and a f l a t p l a t e on the downwind side (Figure 1). Inside this pipe, and attached to the front cone i s a rack on which the transducer and reference volume are s o l i d l y mounted. When assembled, the back p l a t e b o l t s i n t o p o s i t i o n forming a r i g i d , watertight container. The e l e c t r i c a l connections are through the back of the case and watertight plugs are used so that the case can be disconnected from the long instrument cable. To make the attachment of the probe simple, a 'quick disconnect' f i t t i n g i s provided on the front cone. To help maintain thermal s t a b i l i t y , the outer surfaces of the case and probe are kept highly r e f l e c t i v e . These precautions are s u f f i c i e n t to reduce temperature and mechanical noise sources to the l e v e l required. Once the s i z e of the transducer case was known, the length of the stem 22 of the probe had to chosen j u s t s u f f i c i e n t l y long enough to keep the disk away from the dynamic pressure f i e l d produced by e i t h e r the case or the brackets used f o r holding the system. P o t e n t i a l flow theory was used to ca l c u l a t e the upwind pressure perturbation f o r a sphere and f o r an i n f i n i t e c y l i n d e r . I t was assumed that most of the objects producing blockage could be approximated by one of these. For a p o s i t i o n x, upwind of a sphere, the pressure, p, i s given by where a i s the radius of the sphere, U i s the mean a i r v e l o c i t y , and p i s the density of a i r . For the noise l e v e l required, the measurements would need to be about 8a away. Thus, assuming the case looked l i k e a sphere to the a i r f l o w , the length of the stem needed to be about 50 cm. The s i z e of the dynamic pressure perturbation f o r a model of the act u a l container was checked i n a wind tunnel. The res u l t s are i l l u s t r a t e d i n Figure 8 and show that the stem length of about 8a was adequate. A s i m i l a r c a l c u l a t i o n f o r an i n f i n i t e c y l i n d e r , of radius a, perpendicular to the mean flow gives P P = P x For the same noise l e v e l , the probe disk must be 20a to 25a ahead of a cy l i n d e r . These two values were used as c r i t e r i a i n evaluating blockage by the case and c y l i n d r i c a l supports during measurements. 23 Amplitude and Phase Response The pressure instrument (probe and transducer) was c a l i b r a t e d f o r amplitude and phase response. I f the response i s l i n e a r , as was assumed, then data can be corrected using the c a l i b r a t i o n s on the basis of the convolution theorem (e.g., Lee, 1960, p.28). The arrangement used for c a l i b r a t i n g the instrument i s shown i n Figure 9, A s i n u s o i d a l l y varying pressure was produced i n a closed 5 g a l l o n drum by o s c i l l a t i n g a latex rubber diaphragm stetched over one end. The drum i s shown i n more d e t a i l i n Figure 10. The diaphragm was o s c i l l a t e d by contact with a c i r c u l a r a c r y l i c p l a s t i c p l a t e that was attached to a Pye-Ling V45 v i b r a t i o n generator which was i n turn driven by an amplified voltage from a s i g n a l generator. Figure 11 i s the c i r c u i t diagram f o r the power a m p l i f i e r . A Barocel transducer (see Figure 9) referenced to the atmosphere was used to d i r e c t l y measure the magnitude and phase of the pressure i n the drum. Using a s t r i p - c h a r t recorder and an o s c i l l o s c o p e , t h i s measured drum-pressure was compared with the pressure recorded by the pressure instrument when the probe was sealed i n s i d e the drum. A sample c a l i b r a t i o n i s shown i n Figure 12; the measured dropoff i s about 6 db/octave at the high frequency end and 3 db/octave at the low frequency end. I t i s representative of a l l c a l i b r a t i o n s done f o r that probe and transducer and i s t y p i c a l of those obtained f o r the d i f f e r e n t probes. The accuracy of the c a l i b r a t i o n s i s approximately ±2% i n amplitude and ±3° i n phase. To ensure that these c a l i b r a t i o n s were maintained, the instruments were c a l i b r a t e d before and a f t e r each group of f i e l d observations. 24 IN SITU CALIBRATIONS OF THE PRESSURE INSTRUMENT Even though the probes had been tested i n a wind tunnel and found to meet the i n i t i a l requirements, much more confidence would be obtained i f they were c a l i b r a t e d i n t y p i c a l atmospheric turbulence. This was done by measuring the turbulent pressure f l u c t u a t i o n s i n the a i r , with the developed probe, and comparing the r e s u l t s with simultaneous measurements of the pressure f l u c t u a -tions at a surface port d i r e c t l y below. A surface pressure measurement does not have the problem of dynamic pressure contaminations provided the Reynolds Number of the port i s s u f f i c i e n t l y small (Shaw, 1960); a 0.025 inch diameter port was used to meet this requirement. I f the same s t a t i c pressure s i g n a l s are measured by both the surface and a i r systems then s u i t a b l e comparisons w i l l show any dynamic pressure noise associated with the probe. The usual distance between a i r and surface measurements was 30 to 40 cm. This height was necessary to ensure that a t y p i c a l mean wind and turbulence l e v e l were encountered. I t was also s u f f i c i e n t l y high to include any important v e r t i c a l v e l o c i t y e f f e c t s i n the frequency range to be analysed. Thus those pressure fl u c t u a t i o n s with a v e r t i c a l s c a le much greater than 30 to 40 cm could be r e l i a b l y compared. Surface Pressure Measurement To obtain a surface pressure measurement which does not include any unwanted dynamic pressure f l u c t u a t i o n s , the area surrounding the port should be smooth and l e v e l . There should be no small s c a l e blockage i n the v i c i n i t y of the port and any long, surface undulations should have scales large compar-ed to the separation of the a i r and surface instruments. The transducer and 25 reference volume used for the surface measurement were i d e n t i c a l with that used with the probe and were enclosed i n a wooden box that could be positioned below ground l e v e l , Figure 13. Part of the top of this box was a 20 cm by 15 cm f l a t aluminum p l a t e . By c a r e f u l l y p o s i t i o n i n g t h i s p l a t e and the r e s t of the box f l u s h with the l o c a l t e r r a i n , a l l dynamic e f f e c t s were eliminated over a surface port i n the aluminum p l a t e . Since i t was necessary to c a l i b r a t e the surface system, the 0.025 inch port was not d r i l l e d d i r e c t l y i n t o the aluminum, but instead i n t o a 1 inch (2.5 cm) diameter brass plug. This plug could be removed from i t s t i g h t l y f i t t i n g p o s i t i o n i n the aluminum p l a t e so that the port and the 4 inch (10 cm) long s t e e l tubing which l e d from port to transducer could be c a l i b r a t e d without modification. The method of c a l i b r a t i n g f o r amplitude and phase response was i d e n t i c a l to the one used f o r the a i r system. A t y p i c a l c a l i b r a t i o n s i s shown i n Figure 14. The connections between surface port, transducer and reference volume were s i m i l a r to those used f o r the a i r pressure measurement. On some occasions the transducing system normally used for the a i r measurement was used for the surface measurement and v i c e versa. The in situ f i e l d c a l i b r a t i o n s were done at two s i t e s : Ladner and Boundary Bay (see Appendix A). At the Ladner s i t e winds blew from the west perpendicular to a 40 meter wide asphalt runway. Upwind the t e r r a i n was mainly thick grass 5 to 10 cm high. The surface measurement was made 34 meters from the leading (windward) edge of the runway. Fine sand made a smooth t r a n s i t i o n between the box and the surrounding surface. At the Boundary Bay s i t e , the t e r r a i n was more inhomogeneous. At t h i s s i t e the box containing the transducer f o r the surface pressure measurement was conveniently placed f l u s h with the sand surface. However only an area w i t h i n 2 to 3 meters of the box could be considered f l a t and smooth. Patches of grass, w a t e r - f i l l e d potholes, 26 and t h e o c c a s i o n a l l o g were sometimes on the upwind s i d e . Some e f f o r t was made t o e n s u r e t h a t any t u r b u l e n c e g e n e r a t e d by t h i s inhomogeneous t e r r a i n was o f a s c a l e l a r g e compared t o t h e s e p a r a t i o n o f t h e s e n s o r s . C omparison o f Measurements: S u r f a c e and A i r The power s p e c t r a , c o h e r e n c e , and phase a r e u s e d t o compare t h e p r e s s u r e s i g n a l s measured by the probe i n t h e a i r and s u r f a c e p o r t . I f t h e two s i g n a l s a r e t h e same, w h i c h i s e x p e c t e d f o r l a r g e s c a l e s , t h e y have t h e same power s p e c t r u m and phases and have u n i t y c o h e r e n c e . A c o m p a r i s o n o f power s p e c t r a , II(n) , f o r f i v e o f t h e t e n Runs t a k e n a t t h e L a d n e r s i t e i s shown i n F i g u r e s 15 and 16. I n c l u d e d i n t h e s e c o m p a r i s o n s a r e t h r e e d i f f e r e n t p r o b e s and two d i f f e r e n t t r a n s d u c i n g s y s t e m s . A l l t h e d a t a were c o r r e c t e d f o r a m p l i t u d e and phase r e s p o n s e . The v e r t i c a l b a r s on t h e l o w e r c u r v e s denote t y p i c a l 95% c o n f i d e n c e l i m i t s ( c l . ) f o r e a c h e s t i m a t e p l o t t e d . The v e r t i c a l e x t e n t o f t h e dashed l i n e on t h e low f r e q u e n c y s i d e o f e a c h c u r v e i n d i c a t e s the amount by w h i c h each s p e c t r a l p a i r i n a g i v e n c o m p a r i -son were s h i f t e d v e r t i c a l l y r e l a t i v e t o t h e a c t u a l measured v a l u e s . T h i s s h i f t i n g was done t o accommodate more t h a n one p a i r o f s p e c t r a on t h e same v e r t i c a l s c a l e w i t h o u t h i n d e r i n g c o m p a r i s o n o f p a i r s o f c u r v e s . Thus t o r e t u r n a p a i r o f s p e c t r a t o t h e i r t r u e p o s i t i o n w i t h r e s p e c t t o t h e v e r t i c a l a x i s , t h e e n t i r e c u r v e must be s h i f t e d v e r t i c a l l y u n t i l t h e dashed l i n e becomes p a r a l l e l w i t h the a b s c i s s a o r f r e q u e n c y a x i s . The two c u r v e s i n F i g u r e 15 a r e f r o m measurements w i t h a v e r t i c a l s e p a r a t i o n o f 40 cm. The same p r o b e was used b u t t h e t r a n s d u c i n g systems were i n t e r c h a n g e d . F o r t h e t h r e e i n F i g u r e 16, t h e v e r t i c a l s e p a r a t i o n was 32 cm. The o n l y p a r t changed i n t h e s e t h r e e runs was the probe ( p r o b e s E, F, and G were u s e d ) . Each p a i r of s p e c t r a l estimates agree to ±20% (±10% i n amplitude) for frequencies from 0.010 to 1 Hz. Differences i n this frequency range appeared to be random from Run to Run. The values at the lowest frequency p l o t t e d i n the spectra often d i f f e r e d by more than 20%. This was considered to be an e f f e c t of the a n a l y s i s . The c o n s i s t e n t l y lower surface pressure i n the 1 to 10 Hz range was thought to be due to the i n t e r n a l boundary layer over the runway that r e s u l t e d from the large surface roughness (Z q) t r a n s i t i o n (grass, Z q - 2 cm; asphalt, Z Q - 0.1 cm). Using Panofsky and Townsend (1964), the predicted thickness of the i n t e r n a l boundary l a y e r at the point of measurement i s only 7 meters. When measurements were made with the pressure probe moved successively downwind from the surface port, the d i f f e r e n c e at high frequency became les s d e f i n i t e , as would be expected. For the frequency range analysed, t h i s set of observa-tions showed that the amplitude of the pressure f l u c t u a t i o n s w i t h i n the flow could be measured to approximately ±10%. The other requirement f o r the two s i g n a l s to be the same i s that the phase d i f f e r e n c e be zero for frequencies at which the coherence i s high; that i s , f o r v e r t i c a l scale lengths large with respect to the v e r t i c a l separation of the probes. The phase and coherence f o r the same 5 Ladner Runs are p l o t t e d i n Figure 17. The coherence i s approximately 0.95 f o r large s c a l e f l u c t u a t i o n s , f a l l i n g o f f at the higher frequencies. This f a l l o f f i s expected due to the v e r t i c a l separation. The coherence less than 1 i s an i n d i c a t i o n of the non-coherent noise between the two systems, i n c l u d i n g dynamic, thermal, volume f l u c t u a t i o n , and e l e c t r i c a l noise. The phase di f f e r e n c e for the scales with high coherence i s w i t h i n ±5°, which i s as close as can be expected. A s i m i l a r comparison was made f o r some of the Boundary Bay observations. In most cases the pressure measurements i n the a i r were 30 cm above the surface port; a few were 1 m. Four spectra from this s i t e are shown i n 28 Figure 18; the v e r t i c a l separation i s given i n brackets a f t e r the Run number. The curves were s h i f t e d v e r t i c a l l y with respect to the v e r t i c a l axis i n the same manner as done i n Figure 15 (described above). Again each p a i r of s p e c t r a l estimates are the same to ±20% i n power, i n c l u d i n g the 1 to 10 Hz range. No s p e c i f i c i n t e r n a l boundary layers were expected at t h i s s i t e , however, from some wind d i r e c t i o n s a high frequency d i f f e r e n c e s i m i l a r to that at Ladner was observed. The phase and coherence shown i n Figure 19 are also s i m i l a r to the Ladner r e s u l t s ; coherences are 0.8.to 0.9 and phases are the same to ±10°. I t i s f e l t that the instrument has been thoroughly tested i n a turbulent a i r flow t y p i c a l of atmospheric boundary layers with n e u t r a l s t a b i l i t y and has proven to be s u i t a b l e f o r making s t a t i c pressure measurements i n the a i r . 29 MICROSCALE PRESSURE FLUCTUATIONS OVER A FLAT BOUNDARY Some aspects of the k i n e t i c s and kinematics of pressure f l u c t u a t i o n s i n a turbulent boundary la y e r were measured using the developed pressure sensing probe. The observations were made at two s i t e s : Spanish Banks and Ladner. The s i t e s are described i n Appendix A. The Spanish Banks' s i t e s i s located on a t i d a l f l a t with a t i d a l range of about 4 meters. However most of the observa-tions used i n this section were taken when the water was e i t h e r absent or very shallow, leaving a surface of sand or water with small waves of wavelength less than h a l f a meter. Four data Runs (Appendix C) taken when the water was deeper and l a r g e r waves were present are also used i n t h i s s e c t i o n . These four Runs were taken at a height s u f f i c i e n t l y above the waves that the spectra did not show any influence from the waves s i m i l a r to that found i n observa-tions taken c l o s e r to the waves. The wave influence on spectra i s described i n the next s e c t i o n . A l l observations were taken below a height of 6 meters. For both s i t e s winds at the 5 meter l e v e l during observations were about 3 to 10 m sec \ The wind s t r e s s was assumed to be constant with height and f o r -2 the surface roughnesses encountered was generally between 0.1 and 1 dynes cm . The method used to evaluate the s t r e s s i s given i n Appendix B. The l o c a l s t a b i l i t y at the Spanish Banks s i t e was usually near n e u t r a l to s l i g h t l y un-s t a b l e during observations (see Appendix C), with a Gradient Richardson Number of 0 to -0.1. For t h i s range v e l o c i t y spectra show l i t t l e dependence on s t a b i l i t y (McBean, 1970). The s t a b i l i t y was not measured at the Ladner s i t e , however the observations were taken on cloudy (but dry) days when l o c a l buoy-ancy e f f e c t s would be at a minimum. Thus s t a b i l i t y was probably not important for most of these pressure observations, except f o r those s p e c t r a l estimates 30 dependent on any s t a b i l i t y at a higher e l e v a t i o n which, through the ' i n t e g r a l e f f e c t ' , could influence low frequency values. These s i t e s were considered s u i t a b l e f or making pressure measurements t y p i c a l of a turbulent atmospheric boundary l a y e r . Results obtained from observations at these s i t e s are given i n the following subsections. Nondimensionalization of Pressure Spectra Since the pressure i s c l o s e l y t i e d to the v e l o c i t y , as shown by equation 2, the same nondimensionalization parameters should be v a l i d . The importance of each of the p o s s i b l e parameters i s considered i n terms of the pressure spectra obtained i n t h i s study. The parameters used to nondimensionalize v e l o c i t y spectra are usually a 2 time scale of z/U and an i n t e n s i t y of u^ . One other parameter that might be used i s the s t a b i l i t y , which would be more important f o r the pressure due to the ' i n t e g r a l e f f e c t ' (equation 3). As has been previously mentioned a l l data were c o l l e c t e d under conditions estimated to be of nearly n e u t r a l s t a b i l i t y . Recent work (McBean, 1970) shows that the nondimensionalizing of v e l o c i t y i s r e l a t i v e l y i n s e n s i t i v e to s t a b i l i t y under these conditions. Since the s t a b i l i t y was nearly n e u t r a l during a l l Runs, i t i s assumed to be of secondary importance. The i n f l u e n c e of the ' i n t e g r a l e f f e c t ' on pressure spectra i s seen i n observations taken simultaneously at two l e v e l s . Figure 20 shows spectra, II, f o r 0, 1.8, 3.75, and 5.5 meter v e r t i c a l separations. As i n Figures 15 and 18 the dashed l i n e i n d i c a t e s the v e r t i c a l s h i f t of each curve (as discussed on page 26). Figure 20 shows that there i s no large systematic d i f f e r e n c e i n s p e c t r a l l e v e l s due to the v e r t i c a l separation of the measurements. In contrast, spectra of the v e l o c i t y components show a d i r e c t dependence on z 31 ( s e e McBean, 1970) and can be n o n d i m e n s i o n a l i z e d u s i n g the n o n d i m e n s i o n a l f r e q u e n c y f = . The ' i n t e g r a l e f f e c t ' appears t o have removed any s t r o n g z dependence from t h e p r e s s u r e s p e c t r a a t t h e s e l o w e r l e v e l s . O t h e r o b s e r v a -t i o n s w i t h v e r t i c a l s e p a r a t i o n s g i v e s i m i l a r r e s u l t s . I t i s d i f f i c u l t t o d e f i n e a U , o r p r o p a g a t i o n v e l o c i t y , f o r use i n t h e P n o n d i m e n s i o n a l i z i n g . F o r e x ample, an o b s e r v a t i o n made a t the b o t t o m o f t h e b o u n d a r y l a y e r , where the mean v e l o c i t y i s z e r o , has a p r e s s u r e power s p e c t r u m s i m i l a r t o , i f n o t i d e n t i c a l w i t h , t h a t a t , s a y , 1 m e t e r , where t h e mean v e l o c i t y i s n o n - z e r o . S i n c e t h e p r e s s u r e s p e c t r a w i l l have some d i r e c t dependence on t h e mean w i n d b u t have been shown above t o have l i t t l e on z, f r e q u e n c y i s n o t nondimen-s i o n a l i z e d i n t h e a n a l y s i s b u t i s changed i n t o a wave number d e f i n e d by k = 0 ) / U c , where U i s the mean w i n d a t 5 m e t e r s , p _> j The i n t e n s i t y o f t h e p r e s s u r e f l u c t u a t i o n s was n o n d i m e n s i o n a l i z e d by 2 4 p u A . S i n c e t h e same k i n d s o f d a t a were n o t a v a i l a b l e f o r each Run, t h r e e 2 d i f f e r e n t methods o f e v a l u a t i n g u A were u s e d : d i r e c t measurement u s i n g a s o n i c anemometer ( t h e most r e l i a b l e method) ; c a l c u l a t i o n u s i n g the $ ^ method; -3 and use o f a d r a g c o e f f i c i e n t o f = 1.2 x 10 . F u r t h e r d i s c u s s i o n on t h e methods i s c o n t a i n e d i n A p p e n d i x B. Examples o f p r e s s u r e s p e c t r a w h i c h a r e n o n d i m e n s i o n a l i z e d i n t h i s way a r e shown i n F i g u r e s 21 and 22. F o r v a l u e s o f wave number k g r e a t e r t h a n -3 -1 3 x 10 cm , the s p e c t r a have a p p r o x i m a t e l y t h e same s l o p e , -0.7. A t l o w e r wave numbers t h e s l o p e i s l e s s s t e e p and t h e r e i s more s c a t t e r , s i m i l a r t o t h a t 4 w h i c h o c c u r s i n low f r e q u e n c y v e l o c i t y s p e c t r a . The range o f u^ i n v o l v e d i s 1 x 10^ t o 50 x 10^. A f t e r n o n d i m e n s i o n a l i z i n g , extreme v a l u e s o f k l l ( k ) , a t a g i v e n k ^ now d i f f e r by a f a c t o r o f 2 whereas the d i m e n s i o n a l s p e c t r a d i f f e r e d by a f a c t o r o f about 50. The v a r i a n c e between s p e c t r a c o u l d n o t be i m p r o v e d 32 2 2 4 with the present data since that of is about ±20% and of p u^ about ±40%. The levels of the nondimensionalized pressure spectra scatter within 2 this value; therefore the scatter might be due entirely to the error in u^ rather than to dependence on some other variable. A l l spectra which were nondimensionalized i n this manner are summarized P kll(k) -2 -1 in Figure 23. The value of — 7 ; — r at k =10 cm is plotted as a function 2 4 p P u* of z. The mean is 3.5 ± 1. As noted earlier there is no noticable dependence on z. The data given by Gorshkov (1967) do not seem to agree with these present 4 results; they do not exhibit the U dependence shown here. As the wind speed increased the intensity of his pressure spectra did not necessarily increase. It i s not known however whether a l l of his measurements were taken with a nearly constant surface roughness. From the present study, i t appears that the pressure spectra below 5 meters in an atmospheric boundary layer of nearly neutral s t a b i l i t y are 2 adequately nondimensionalized in terms of the stress, pu^ t and a wave number k = to/ U L . p '5 Shape and Intensity of the Spectrum - _ As with any turbulent variable, the shape and intensity of the power spectrum for pressure fluctuations gives some clue of the role of pressure at different frequencies. 2 - l " v >-Normalized pressure spectra, II(n)/a (Hz ), are plotted i n Figure 24. 2 They are normalized by their integrals, which equal their variances a , -2 0 between 2.7 x 10 and 7.4 x 10 Hz, the frequency range plotted. These 33 s p e c t r a , p r e v i o u s l y shown i n n o n d i m e n s i o n a l f o r m i n F i g u r e 21, were chosen because a s s o c i a t e d v e l o c i t y s p e c t r a a r e a v a i l a b l e . They a r e c o n s i d e r e d t o be r e p r e s e n t a t i v e o f a l l t h o s e o b t a i n e d o v e r a f l a t b oundary. The o b s e r v a t i o n s were t a k e n a t h e i g h t s o f 3 t o 5 m e t e r s ; s u r f a c e w i n d s t r e s s was a p p r o x i m a t e l y 0.8 dynes cm . The s p e c t r a show a r e g u l a r power law b e h a v i o u r above 0.3 Hz, most o f t h e v a r i a t i o n s between d i f f e r e n t s p e c t r a o c c u r i n g a t t h e low f r e q u e n c y end. V e r t i c a l l i n e s i n d i c a t e t h e 95% c o n f i d e n c e l i m i t s ( c l . ) o f i n d i v i d u a l es t i m a t e s . The v e l o c i t y s p e c t r a a s s o c i a t e d w i t h the p r e s s u r e s p e c t r a i n F i g u r e 24 a r e shown i n F i g u r e s 25 and 26. ^ s t* i e f r e q u e n c y s p e c t r a l d e n s i t y f o r t h e i * " * 1 v e l o c i t y component. These v e l o c i t y s p e c t r a c o n f o r m w i t h t h e ' u n i v e r s a l c u r v e s ' o b t a i n e d by McBean ( 1 9 7 0 ) . The s l o p e s o f t h e v e l o c i t y and p r e s s u r e s p e c t r a i n i s o t r o p i c t u r b u l e n c e have been p r e d i c t e d f r o m d i m e n s i o n a l arguments ( s e e B a c k g r o u n d p.9) t o be -5/3 and -7/3 r e s p e c t i v e l y . The -5/3 f o r the v e l o c i t y s p e c t r a i s a r e a s o n a b l e f i t t o t h e d a t a f o r f r e q u e n c i e s above t h e peak o f t h e w s p e c t r u m , F i g u r e s 25 and 26; the s t r a i g h t l i n e has t h i s p r e d i c t e d s l o p e . F o r t h e same f r e q u e n c y r a n g e , t h e p r e s s u r e s p e c t r a , F i g u r e 24, has a mean s l o p e o f about -1.7, s i g n i f i c a n t l y d i f f e r e n t f r o m t h e -2.3 p r e d i c t e d . The a p p r o x i m a t e p o s i t i o n o f t h e peak o f t h e w s p e c t r u m i s i n d i c a t e d by t h e a r r o w . D e s p i t e t h e -5/3 s l o p e o f t h e v e l o c i t y s p e c t r a i n t h i s f r e q u e n c y r a n g e , t h e t u r b u l e n c e may n o t be c o m p l e t e l y i s o t r o p i c ( S t e w a r t , 1969). The s u g g e s t i o n i s t h a t t h e p r e s s u r e i s more s e n s i t i v e t h a n the v e l o c i t y t o any s u c h a n i s o t r o p y . T h r e e o t h e r i n d i c a t i o n s t h a t c o m p l e t e i s o t r o p y i s n o t p r e s e n t i n t h i s f r e q u e n c y range a r e a l s o f o u n d . F i g u r e 27 i s a p l o t o f the u and w c o s p e c t r u m , $ . T h i s has s i g n i f i c a n t v a l u e s n e a r f = 0.6; t h a t i s , a t t h e l o w e r 34 frequencies of the -5/3 slope region of the v e l o c i t y spectra. Thus this part of the flow cannot be described as i s o t r o p i c . Also (to be shown l a t e r i n this s e c t i o n , see Figure 46) s i g n i f i c a n t energy f l u x out of the u v e l o c i t y component by the pressure forces occurs through most of the frequency range 1 to 10 Hz. I t i s a l s o shown l a t e r that throughout the higher frequencies, the turbulence cannot be described as completely i s o t r o p i c since the pressure-v e l o c i t y coherences do not become i n s i g n i f i c a n t u n t i l near the highest frequencies p l o t t e d . Thus for these nondimensional frequencies studied (les s than 10) the data seem to i n d i c a t e a lack of complete isotropy. Because of the lack of isotropy these data are not a c r i t i c a l t e s t of the -7/3 behaviour f o r pressure spectra i n i s o t r o p i c turbulence. Even though the wave number range cu/u|^ > 10 has been used to obtain the Kolmogoroff constant for v e l o c i t y , i t i s questionable whether a rough estimate of the u n i v e r s a l constant, K , i n Obukhov's formulation (p.9) can P be made since the pressure spectra do not have a -7/3 slope. A f u r t h e r d i f f i c u l t y i n t r y i n g to evaluate with the present data i s the lack of any z dependence i n the pressure spectra, kll(k) (see Figure 21). I f the rate of d i s s i p a t i o n , £, i s given by u A / K Z (the rate of energy production), K - k ! ? ( k ) A ( Kkz ) 4 / 3 . p 2 4 P U * 4/3 Thus f o r the data obtained K would vary as z and not be a constant. P A d i f f e r e n t t h e o r e t i c a l argument p r e d i c t i n g the slope of the pressure spectrum has been given by H. Charnock (personal communication to R.W. Stewart). Based on s i m i l a r i t y asssumptions, he predicted a -1 slope. This i s also not 35 present i n the data for any s i g n i f i c a n t range of frequencies. The mean shape of the pressure spectra i n Figure 24 i s used for comparison with previously published pressure spectra, Figure 28. The l a t t e r were s h i f t e d v e r t i c a l l y to l i e near the mean curve from Figure 24. A l l of these spectra were obtained i n the atmospheric boundary layer at comparable wind speeds. The observations by the two Russian authors, G o l i t s y n (1964) and Gorshkov (1967), are of surface pressure measurements; those by Gossard (1960) were taken i n the a i r . The i n t e n s i t y of the Russian observations i s of the same order of magnitude as that obtained using the developed probe, while Gossard's r e s u l t s are an order of magnitude larger. Nevertheless the mean slopes of these curves are s i m i l a r to that obtained i n the present study. At low frequencies some of the differences may have been due to sources other than the boundary l a y e r turbulence; such as, i n t e r n a l gravity waves at higher elevations (Herron et a l , 1969). Variations from the w e l l defined slope shown i n Figure 24 were occasional-ly observed i n this study; they were s i m i l a r to those found by Gorshkov (1967). They could be a t t r i b u t e d to a lack of s t a t i o n a r i t y and non-uniformity of the t e r r a i n . A more d e t a i l e d comparison between the pressure and the v e l o c i t y i s shown i n Figure 29. P l o t t e d against frequency are two curves representing the variance w i t h i n narrow frequency bands of the nondimensional pressure and of the nondimensional sum of the three v e l o c i t y components from Run 120/1. The values p l o t t e d are / II(n) An /( pu^ 2) and ( § ^ + $ 2 2 + $ 3 3 ) An / u^ where An i s the bandwidth. The v e l o c i t y components and the s t r e s s were measured with a sonic anemometer at the same l e v e l as the pressure probe. The values have been p l o t t e d i n this form to show how the r e l a t i o n s h i p between the pressure variance and v e l o c i t y variance changes for d i f f e r e n t s c a l e ranges. 36 As can be s e e n b o t h c u r v e s e x h i b i t a s i m i l a r shape. A t h i g h f r e q u e n c i e s the p r e s s u r e c u r v e s i s r o u g h l y t w i c e the v e l o c i t y c u r v e ; a t low f r e q u e n c i e s , f < 1, the p r e s s u r e c u r v e i s more t h a n t w i c e t h e v e l o c i t y c u r v e . S i n c e the v e l o c i t y c u r v e i s a f u n c t i o n o f z and t h e p r e s s u r e c u r v e i s n o t (p.30) t h i s r e l a t i o n -s h i p has some z dependence w h i c h w o u l d be most n o t i c e a b l e a t t h e h i g h f r e q u e n -c i e s , n e a r and above the peak o f the w s p e c t r u m . At l o w e r f r e q u e n c i e s t h e maxima and minima w o u l d n o t have a z dependence s i n c e they a r i s e f r o m u and v f l u c t u a t i o n s o f s c a l e s much l a r g e r t h a n the o b s e r v a t i o n h e i g h t . The o t h e r Runs i n t h i s group (p.33) were a t s i m i l a r h e i g h t s and have t h e same g e n e r a l p r o p e r t i e s . B a t c h e l o r e v a l u a t e d , from t h e v e l o c i t y c o r r e l a t i o n , a v a l u e f o r the rms a m p l i t u d e o f t h e p r e s s u r e f l u c t u a t i o n s i n i s o t r o p i c t u r b u l e n c e . H i s r e s u l t was / p 2 = 0.58 p u 2 ( p . 1 0 ) . I t i s q u e s t i o n a b l e w h e t h e r t h i s r e s u l t can be compared w i t h t h e p r e s e n t measurements s i n c e , as shown above, i n t h e range a n a l y s e d the t u r b u l e n c e was n o t c o m p l e t e l y i s o t r o p i c . A t the h i g h e r n o n d i m e n s i o n a l f r e q u e n c i e s , f > 1, F i g u r e 29, where some s e m b l a n c e o f i s o t r o p y may have e x i s t e d , t h e root-mean-square p r e s s u r e i s about t e n t i m e s t h a t e x p e c t -ed from B a t c h e l o r 1 s r e l a t i o n s h i p . T h i s h i g h e r p r e s s u r e m i g h t be due t o i n t e r -a c t i o n between the t u r b u l e n c e and t h e mean s h e a r as s u g g e s t e d by K r a i c h n a n ( 1 9 5 6 ) . He p r e d i c t e d t h a t t h e m a j o r t e r m w o u l d be o f t h e f o r m p(3U/3z)(3w/3x). The s c a l e s o f t h e t u r b u l e n c e a t f r e q u e n c i e s n e a r f = 1 a r e o f t h e o r d e r o f 50 cm, s u f f i c i e n t l y l a r g e t h a t an i n t e r a c t i o n w i t h the mean s h e a r w o u l d be e x p e c t e d . The s p e c t r a p l o t t e d i n F i g u r e 21 show t h a t , a t l e a s t f o r t h i s group o f d a t a , t h e r e i s no s t r o n g i n f l u e n c e f r o m t h e low f r e q u e n c y m e s o s c a l e phenomena w h i c h were f o u n d i n p r e s s u r e s p e c t r a by G o s s a r d ( 1 9 6 0 ) . These s p e c t r a a p p e a r t o drop o f f a t t h e s e l o w e r f r e q u e n c i e s i n a manner s i m i l a r t o t h e v e l o c i t y s p e c t r a . These d a t a a r e used t o e v a l u a t e t h e r e l a t i o n s h i p b e t w e en t h e p r e s s u r e v a r i a n c e and the s u r f a c e s t r e s s assuming t h a t t h e p r e s s u r e f l u c t u a t i o n s 37 a r i s e e n t i r e l y from boundary layer turbulence. A l i n e a r p l o t showing the average of the nondimensionalized pressure of Figure 21 as a function of wave number i s given i n Figure 30. The curve i s considered to be representative of a l l the data c o l l e c t e d . The nondimension-a l i z e d pressure, kll(k) /(p 2u^ 1*), i n Figure 30 was integrated with respect to —5 -2 —1 In k from a wave number, k = a)/u|c, of 10 to 2 x 10 cm . The dashed p 'b l i n e s i n d i c a t e the curve used near the l i m i t s of the i n t e g r a t i o n . This i n t e g -r a t i o n gives the root-mean-square pressure i n terms of the surface s t r e s s , pu A : = 2.6 p u A 2 (12). This r e l a t i o n s h i p i s almost i d e n t i c a l to that obtained from surface measurements i n wind tunnels (Willmarth and Wooldridge, 1962). Since, as w i l l be shown l a t e r , the nondimensionalized curve integrated to get the root-mean-square pressure i s not a function of height, the r e l a t i o n s h i p would be expected to hold at the surface. The magnitude given by equation (12) i s i n the range predicted t h e o r e t i c a l l y by Kraichnan (1956). In summary, the pressure spectrum i s s i m i l a r i n shape to the v e l o c i t y spectrum and has a variance of about 6.5 p 2 ^ 1 * . Some Kinematics of the Pressure Fluctuations From the simultaneous measurement of s t a t i c pressure f l u c t u a t i o n s at two points i t i s p o s s i b l e to deduce some properties of the s t r u c t u r e of these f l u c t u a t i o n s . Observations were spaced i n each of the three p r i n c i p a l d i r e c t i o n s . Phase and coherence are used to evaluate the o r i e n t a t i o n , scale s i z e and propagation v e l o c i t y of the f l u c t u a t i o n s . Coherence and phase between d i f f e r e n t pairs of points with purely 38 v e r t i c a l (Az) and p u r e l y t r a n s v e r s e (Ay) s e p a r a t i o n s a r e shown i n F i g u r e s 31 and 32. The phases p l o t t e d a r e f o r f r e q u e n c i e s a t w h i c h t h e c o h e r e n c e was g r e a t e r t h a n 0.2. The s e p a r a t i o n s u s e d a r e s u i t a b l e t o o b s e r v e any phase s h i f t s t h a t may be p r e s e n t i n t h e p r e s s u r e f l u c t u a t i o n s i n t h e f r e q u e n c y range o f i n t e r e s t . F o r v e r t i c a l s e p a r a t i o n s o f up t o 5 m e t e r s , t h e a v e r a g e phase d i f f e r e n c e i s n e a r z e r o . Thus t h e r e i s no p r e f e r r e d v e r t i c a l o r i e n t a t i o n f o r a p r e s s u r e f l u c t u a t i o n i n t h i s r a n g e . As e x p e c t e d , the a v e r a g e r e l a t i v e phase f o r t r a n s v e r s e s e p a r a t i o n s i s n o t s i g n i f i c a n t l y n o n - z e r o . From a c o m p a r i s o n o f t h e c o h e r e n c e and phase f o r two s e n s o r s w i t h a down-s t r e a m s e p a r a t i o n i t i s p o s s i b l e t o e s t i m a t e t h e 'decay r a t e ' o f t h e p r e s s u r e f l u c t u a t i o n s . I d e a l l y , t o c o n f o r m w i t h t h e ' f r o z e n f i e l d h y p o t h e s i s ' , t h e c o h e r e n c e s h o u l d be 1 a t a l l f r e q u e n c i e s and a t a l l downstream s e p a r a t i o n s and a phase d i f f e r e n c e s h o u l d r e s u l t , t h e s i z e o f w h i c h depends on t h e downstream s e p a r a t i o n o f the p r e s s u r e s e n s o r s . A c o m p a r i s o n o f phase and c o h e r e n c e , F i g u r e 33, shows t h a t the s i g n a l s become e s s e n t i a l l y i n c o h e r e n t f o r phase s h i f t s o f a b out 360° o r a p p r o x i m a t e l y a f t e r one w a v e l e n g t h . W i l l m a r t h and W o o l d r i d g e (1962) f o u n d f r o m w i n d t u n n e l s t u d i e s u s i n g t i m e l a g c o v a r i a n c e s t h a t s i g n i f i c a n t e nergy h a d b e e n l o s t by a p r e s s u r e p a t t e r n t h a t h a d t r a v e l l e d a d i s t a n c e o f a b o u t two t o t h r e e w a v e l e n g t h s ( s e e t h e i r F i g u r e 1 0 ) . Time l a g c o v a r i a n c e s w o u l d be e x p e c t e d t o y i e l d a l a r g e r and c o r r e c t e s t i m a t e o f 'decay r a t e ' . From t h e c o h e r e n c e between two p r e s s u r e s i g n a l s a t a g i v e n s e p a r a t i o n , a s c a l e f o r t h e p r e s s u r e f l u c t u a t i o n s can be d e t e r m i n e d . When m e a s u r i n g t h e p r e s s u r e w i t h two p r e s s u r e s e n s o r s a t a f i x e d s e p a r a t i o n p e r p e n d i c u l a r t o the mean f l o w , p r e s s u r e f l u c t u a t i o n s w i t h a ' s c a l e ' s i z e l a r g e compared t o t h e s e p a r a t i o n o f t h e s e n s o r s w i l l o f t e n o c c u r s i m u l t a n e o u s l y i n t h e two s i g n a l s p r o d u c i n g a h i g h c o h e r e n c e , w h i l e f l u c t u a t i o n s w i t h a ' s c a l e ' s i z e s m a l l compared t o t h e s e p a r a t i o n cannot o c c u r s i m u l t a n e o u s l y . Those f l u c t u a t i o n s w i t h a ' s c a l e ' c o m p a r a b l e t o t h e p r o b e s e p a r a t i o n can o c c a s i o n a l l y p r o d u c e 39 some c o h e r e n t s i g n a l a t the two p r o b e s . Thus t h e p r e s s u r e s c a l e l e n g t h a t f r e q u e n c y n, L ^ ( n ) , i s d e f i n e d as t h a t s e p a r a t i o n a t w h i c h the c o h e r e n c e f a l l s t h r o u g h some low b u t m e a s u r a b l e v a l u e . The f r e q u e n c y chosen i s t h a t a t w h i c h where IT^^ r e p r e s e n t s the c o h e r e n t energy between two s i g n a l s o f e n e r g y II and I I 0 , and e = 2.72. I n o t h e r w o r d s , where the c o h e r e n c e = = 0.14 . ( F o r s e p a r a t i o n s p e r p e n d i c u l a r t o t h e mean f l o w , a s c a l e b a s e d on t h e c o h e r e n c e i s e q u i v a l e n t t o a s c a l e b a s e d on t h e c o r r e l a t i o n c o e f f i c i e n t ) . The c o h e r e n c e o f 0.14 r e p r e s e n t s a d e f i n i t e b u t low common e n e r g y between two p r e s s u r e s i g n a l s . R e p r e s e n t a t i v e c o h e r e n c e s f o r t h e t h r e e d i f f e r e n t s e p a r a t i o n d i r e c t i o n s a r e g i v e n i n F i g u r e s 3 1 , 32, and 33. S i n c e the f r e q u e n c y n ^ a t w h i c h t h e c o h e r e n c e f a l l s t o 0.14 depends on t h e p r o p a g a t i o n v e l o c i t y , i t i s c o n v e r t e d i n t o a w a v e l e n g t h , A , w h i c h i s i n d e p e n d e n t o f the mean w i n d u s i n g P ( 1 3 ) . I t w i l l be shown l a t e r t h a t u|^ i s a c l o s e a p p r o x i m a t i o n t o t h e a c t u a l p r o p a g a t i o n v e l o c i t y o f t h e p r e s s u r e f l u c t u a t i o n s f o r d a t a a n a l y s e d i n t h i s manner. A i s t o be compared t o L . I f c o h e r e n t noise''' i s p r e s e n t b e f o r e 1 T h i s can a r i s e f r o m wow and f l u t t e r i n t h e a n a l o g t a p e r e c o r d e r . 40 the c o h e r e n c e f a l l s t o 0.14 the c u r v e i s e x t r a p o l a t e d by a s t r a i g h t l i n e . P r o b e s e p a r a t i o n s ( e q u i v a l e n t t o L f o r n = n ) a r e p l o t t e d as p o i n t s a g a i n s t P L f o r c o h e r e n c e s o f 0.14 on the l o g - l o g p l o t i n F i g u r e 34. The s o l i d l i n e drawn among the p o i n t s has a s l o p e o f 1; t h a t i s , L v a r i e s d i r e c t l y as A . P P A l l d i r e c t i o n s y i e l d e d s i m i l a r f o r a g i v e n A^ ( i n c l u d i n g downstream s e p a r a t i o n , s i n c e the decay time i s about one c y c l e ) . Wind t u n n e l o b s e r v a t i o n s ( W i l l m a r t h and W o o l d r i d g e , 1962) a l s o f o u n d t h e two h o r i z o n t a l s c a l e s t o be c o m p a r a b l e i n s i z e . Thus, as a f i r s t a p p r o x i m a t i o n , the mean shape o f a p r e s s u r e f l u c t u a t i o n i s s p h e r i c a l . An L o f 100 cm c o r r e s p o n d s t o a A o f P P 210 cm. T h e r e f o r e the a c t u a l s i z e o f t h e p r e s s u r e f l u c t u a t i o n s ( L ^ ) i s a p p r o x i m a t e l y 1/2 o f a w a v e l e n g t h , A . The c h o i c e of d i f f e r e n t c o h e r e n c e s t o P d e t e r m i n e t h e s c a l e s i z e changes o n l y t h e p r o p o r t i o n a l i t y c o n s t a n t between L and A . The v a l u e o f 0.14 i s c l o s e t o t h e p r a c t i c a l l i m i t . P P F i g u r e 34 a l s o shows a f a m i l y o f c u r v e s o f c o n s t a n t c o h e r e n c e p l o t t e d f r o m t h e same Runs used i n e v a l u a t i n g L . These a r e used t o e v a l u a t e t h e P i n f l u e n c e o f p r o be s e p a r a t i o n on d a t a . As can be s e e n i n F i g u r e s 31, 32, and 33, a t a g i v e n s e p a r a t i o n , c o h e r e n c e f a l l s o f f r o u g h l y l i n e a r l y when p l o t t e d a g a i n s t I n n. T h i s f a l l o f f i s a p p r o x i m a t e l y t h e same i n a l l c a s e s and can be r e p r e s e n t e d by c o h e r e n c e ( l n n) . T h i s s l o p e was used t o p l o t l i n e s o f c o n s t a n t c o h e r e n c e i n F i g u r e 34. The c o h e r e n c e a t a g i v e n s e p a r a t i o n , F i g u r e s 31 and 32, a l s o has a p l a t e a u w h i c h d e c r e a s e s as the s e p a r a t i o n i n c r e a s e s , as i n d i c a t e d on F i g u r e 34. T h i s p l a t e a u i s n o t w e l l d e f i n e d and t h e c o r r e s p o n d i n g p r o b e s e p a r a t i o n v a l u e s shown, f o r w h i c h a g i v e n p l a t e a u o c c u r s , v a r y by about ±100%. The drop i n c o h e r e n c e a t t h e l o w e r f r e q u e n c i e s i s t h o u g h t t o be due t o n o i s e . These r e s u l t s o f s c a l e s i z e and'decay r a t e 1 can be u s e d t o e s t i m a t e the e x p e c t e d magnitude o f t h e p r e s s u r e f l u c t u a t i o n s . Assuming t h a t the p r e s s u r e 41 fl u c t u a t i o n s r e s u l t from the complete a c c e l e r a t i o n or deceleration of a v e l o c i t y f l u c t u a t i o n , the order of magnitude of — w i l l be of the order 3 6 p Az of Au./At, where A ind i c a t e s the rms f l u c t u a t i o n s from the mean. For l Run 120/1 (Figures 24 and 25) at 0.7 Hz (bandwidth = 0.21 Hz), w - u - v --1 -2 10 cm sec ; Ap - 1 dyne cm . The gradient of p w i l l act over a distance of about Lp/2. Thus for the example, L U U Az = — = — 1 ~ 250 cm. 2 4 n These values give At = p Au_^  Az/Ap - 3 sec. This i s comparable to the ca l c u l a t e d 'decay time' of about 4 seconds assuming the distance required f o r appreciable decay of a pressure f l u c t u a t i o n i s about 3 wavelengths (see p.38) The propagation v e l o c i t y i s evaluated from the phase difference between two simultaneous pressure measurements with a downwind separation. Figure 33(b) shows the phase difference, 0, f o r downwind separation (D) of about 0, 1, 2, and 4 meters. The'measurement upwind was of surface pressure and that downwind was i n the a i r (at 32 cm) . The propagation v e l o c i t y i s cal c u l a t e d from UP = ~ l F D n e (14)> where i s the frequency at which the phase diffe r e n c e i s G. The v e l o c i t y , U , i s then compared with the mean wind, u|L , at the l e v e l L appropriate P P P to the frequency UQ, since this height i s more representative of the mean wind at which the pressure f l u c t u a t i o n s of frequency TIQ o r i g i n a t e d . The mean wind at l e v e l L i s i n t e r p o l a t e d from observed cup p r o f i l e s . The height L i s P P c a l c u l a t e d by taking u|^/n'g = L ' as a f i r s t approximation and then l e t t i n g 42 L = U i/n„ . The s e c o n d s t e p adds l e s s than a 10% c o r r e c t i o n . T a b l e I I p L W P g i v e s t h e r e s u l t s o f the v e l o c i t y c o m p a r i s o n . F o u r d i f f e r e n t phase d i f f e r e n c e s were u s e d : 90°, 180°, 270° and 360°. S i n c e the s c a l e s i z e L i s d i f f e r e n t f o r P each phase d i f f e r e n c e , t h e U"| used f o r c o m p a r i s o n w i t h U- i s e v a l u a t e d f o r P P each phase s h i f t . The L a d n e r o b s e r v a t i o n s , c o n s i d e r e d t o be the most a c c u r a t e b e c a u s e o f low i n s t r u m e n t n o i s e and s t e a d y s t a t e mean f l o w , g i v e v a l u e s o f TABLE I I P r e s s u r e P r o p a g a t i o n V e l o c i t y , U , as a F r a c t i o n o f u| S i t e L L L L B.B. B.B. S.B. S.B. S.B. Run 320/2 425/1 425/2 426/1 137/2 142/1 196/1 196/2 196/3 m/sec 7.7 6.1 5.2 6.1 4.1 4.3 3.7 3.4 3.7 D (m) 0.96 2.0 3.1 4.1 0.56 0.56 4.3 2.7 1.5 U * t 1 p 0.90 0.91 0.96 1.00 0.97 0.93 0.91 0.86 1.49 1.17 1.04 0.98 1.08 1.04 0.99 0.93 0.96 0.99 1.10 1.30 0.95 0.99 1.05 1.20 1.20 1.30 1.10 1.05 1.15 1.15 1.14 1.10 1.10 1.00 1.05 Av. 0.94 0.92 1.17 1.01 1.07 1.05 1.16 1.14 1.05 1.0 ± 0.1 ( f r o m 4 Runs) f o r U / uL . S i m i l a r o b s e r v a t i o n s a t S p a n i s h Banks P L P g i v e 1.1 ±0.15 ( f r o m 3 Runs) and a t Boundary Bay, 1.05 ±0.1 ( f r o m 2 R u n s ) . Thus t h e p r e s s u r e f i e l d t r a v e l s a t about t h e l o c a l mean w i n d s p e e d , the h i g h e r f r e q u e n c i e s o r s m a l l e r s c a l e s t r a v e l l i n g s l o w e r t h a n t h e l o w e r f r e q u e n c i e s o r l a r g e r s c a l e s . C a l c u l a t i o n s by o t h e r i n v e s t i g a t o r s w o r k i n g i n w i n d t u n n e l s ( s e e W i l l m a r t h and W o o l d r i d g e , 1962) g i v e an a s y m p t o t i c v a l u e o f a b o u t 0.8 u| f o r the p r o p a g a t i o n v e l o c i t y . T h i s d i f f e r e n c e i s c o n s i d e r e d t o be due t o the c h o i c e o f t h e mean w i n d used f o r t h e c o m p a r i s o n . The v a l u e o f u| f o r t h e a t m o s p h e r i c b o u n d a r y l a y e r c o u l d e a s i l y be 15 t o 20% h i g h e r than t h e w i n d a t h e i g h t L , w h i c h w o u l d a c c o u n t f o r the d i f f e r e n c e . The d i f f e r e n c e i n 43 propagation velocities as a function of frequency has also been observed in wind tunnels. To compare the pressure fluctuations with the velocity fluctuations, similar analysis was done on some velocity data. The data comes partly from observations taken by others at the institute, using sonic and hot-wire anemometers, and partly from the hot-wire anemometer observations at Ladner. Figure 35 shows examples of coherences for two velocity sensors, and Figure 36 a composite of a l l such graphs. Though the data are more scattered i t is assumed that the velocity and pressure behave similarly and the best f i t line is drawn for separations proportional to A^. The scale L^ corresponding to X = U/n = 100 cm is about 30 cm, where L is defined as before from the v L v frequencies at which the coherence is 0.14. Therefore the size of the velocity fluctuations is approximately 1/3 of that given by X^ . The coherences —0 8 f a l l off approximately as (In n) * , about the same as for the pressure f i e l d . Velocities are also in phase at points separated verti c a l l y and across the stream, and the propagation velocity is about U, as expected. These results indicate that the pressure and velocity fluctuations are similar i n geometry and are advected at the same rate. Pressure-Velocity Relationships In this subsection a number of pressure measurements taken in conjunction with velocity measurements are used to establish some of the relationships bet-ween the two. The results apply to the pressure-velocity relationship for most of the scale range for which there is an active Reynolds stress. A priori, there is no specific relationship expected. If the pressure were passive, the phase 2 might be Bernoulli type (p varying as -u /2). If active, and responsible for 44 the t r a n s f e r of energy between components, such as i s required f o r a tendency to isotropy, a quadrature component i n the pu product would be required i n turbulent shear flow (see equations 4). I f pressure i s doing work across a plane, then the v e l o c i t y component normal to the plane and i n the d i r e c t i o n towards the f l u i d r e c e i v i n g energy would on the average be i n phase with the pressure. I f the energy were being extracted from a v e l o c i t y component, the v e l o c i t y would on the average be d i r e c t e d up the f l u c t u a t i n g pressure gradient, or decelerating; the sig n would be reversed f o r a gain of energy. T y p i c a l measured coherence and phase r e l a t i o n s h i p s between p and the v e l o c i t i e s u and w are p l o t t e d i n Figures 37, 38, 39 and 40. P o s i t i v e phases l a b e l l e d p-u, p-w, etc., means that p leads u, w, etc., r e s p e c t i v e l y . Figures 37 and 38 use data from the sonic, with the pressure probe placed about 25 cm upwind of the center of the sonic paths; t h i s p o s i t i o n i n g of the probe was checked i n a wind tunnel to ensure that no s i g n i f i c a n t dynamic pressure noise would be present. The pressure s i g n a l was corrected f o r i t s r e s u l t i n g phase lead using Taylor's hypothesis. For comparison, a p l o t of u-w phase and coherence i s shown i n Figure 41. Data obtained from the pressure probe and a hot-wire, 4 cm to the s i d e and 5 cm behind the probe, are p l o t t e d i n Figures 39 and 40. In a l l the figures the approximate peak of the w spectrum i s marked by an arrow f o r ease of comparison with standard v e l o c i t y spectra. These plo t s show that the pressure i s ' i n phase' with u at low frequencies (the opposite of B e r n o u l l i type) with coherences up to 0.8, while at high frequencies the phase di f f e r e n c e becomes about 135° ( i n d i c a t i n g that some energy t r a n s f e r was taking place) with coherences of about 0.1 to 0.2. The phase t r a n s i t i o n i s associated with a loss of coherence i n pu. The d i v i s i o n between the ' i n phase' and 'large phase d i f f e r e n c e ' i s at a frequency some-what higher than at the peak of the w spectrum. The coherence i n the pw 45 r e l a t i o n s h i p shows a gradual decrease from about 0.5 at low frequencies to near zero at high frequencies; the corresponding phase change i s gradual from about 180° to near 0°. Some of this loss of coherence i n the pw r e l a t i o n s h i p at the highest frequencies may be due to probe separation. As would be expected the measured pv coherence i s near zero f o r the same frequency range. The reason for the w e l l defined change i n the p-u phase can be seen from a p l o t , Figure 42, of wavelength, A , at this t r a n s i t i o n , as a function of z (height of observations above the surface). The wavelength i s c a l c u l a t e d using the f i r s t data point a f t e r the phase t r a n s i t i o n to near 135°. Also shown i s a dashed l i n e i n d i c a t i n g the s c a l e s i z e , , of the pressure fl u c t u a t i o n s f o r the t r a n s i t i o n frequency (L - -y A as evaluated i n a p z. p previous subsection, p.37). A l l points f a l l near or below the l i n e = z. That i s , a l l pressure f l u c t u a t i o n s with a scale s i z e l a r g e r than z have a d i f f e r e n t phase r e l a t i o n s h i p from those with scale s i z e less than z. This i s a t t r i b u t e d to the l a r g e r s c a l e f l u c t u a t i o n s ' f e e l i n g ' the bottom. A p o s i t i v e pressure at large scales i s associated with a negative (downward) w (see Figure 38). Since uw on the average i s negative (Figure 41), the p o s i t i v e pressure i s here associated with a p o s i t i v e u. Thus there i s a zero phase r e l a t i o n s h i p at large s c a l e s - between p and u. For turbulence scales smaller than the measurement height, a d i f f e r e n t r e l a t i o n s h i p r e s u l t s from t h e i r independence from the bottom. Even though the major source of the low frequency pressure f l u c t u a t i o n s i s a t t r i b u t e d to a i r motions i n t e r a c t i n g with the surface, not a l l w f l u c t u a -tions produce corresponding pressure f l u c t u a t i o n s . The coherence between the pressure and w i s lower than between pressure and u at these scales (see Figures 37 and 38). Since i t i s f e l t that this i s r e a l and not due to the instrumentation, the flow must contain s i g n i f i c a n t w f l u c t u a t i o n s which do not 46 produce corresponding u or p f l u c t u a t i o n s . As can be seen by comparing Figures 41 and 38, the u-w and p-w coherences are both about 0.5 at these sc a l e s . Further evidence that most of the pressure at low frequencies i s associated with deceleration of v e r t i c a l v e l o c i t i e s near the surface can be seen from the coherence between v e l o c i t y and two pressure measurements 0 near the surface, Figure 43. One pressure sensor was at the surface and a hot-wire anemometer and a pressure probe were together at 30 cm. The surface pressure s i g n a l has a higher coherence with downstream v e l o c i t y f l u c t u a t i o n s than does the pressure s i g n a l from the pressure measurement i n the a i r . The mean dif f e r e n c e i n coherence i s about 0.1 with the 30 cm separation. The two pressures are i n phase for frequencies less than 5 Hz, Figure 33(b). Since the two pressure s i g n a l s have the same s p e c t r a l l e v e l and 0° phase with respect to the v e l o c i t y , the d i f f e r e n c e i n coherence i s the r e s u l t of the in-phase pressure at the surface being on the average l a r g e r than the in-phase pressure at the l e v e l of the v e l o c i t y sensor. This means that the pressure gradient i s on the average d i r e c t e d upward. A c a l c u l a t i o n , Table I I I , page 47, shows this f o r Run 319/1. Thus at large scales the majority of the pressure f l u c -tuations were associated with the turbulence i n t e r a c t i n g with the surface. The coherences between u and p, at t h i s low l e v e l , Figure 43, are s i g n i f i c a n t l y smaller than those t y p i c a l l y measured at higher l e v e l s as shown i n Figures 37, 39, and 40. This i s thought to be an example of the ' i n t e g r a l e f f e c t ' (p.6) where the a i r motions co n t r i b u t i n g to uw may be decelerated by the boundary without the associated v e l o c i t i e s a r r i v i n g simultaneously at the boundary. Data c o l l e c t e d , using a hot-wire f o r measuring u and the surface pressure measuring technique to obtain p, were compared to the r e s u l t s of Gorshkov 47 TABLE I I I V e r t i c a l P r e s s u r e G r a d i e n t a t t h e S u r f a c e D a t a from Run 319/1 ( s e e F i g u r e 43) — 1 2 2 • u| = 10 m s e c u, - 0 . 4 m s e c f r e q u e n c y Hz B.W. Hz < "1/2 cm s e c Hz A P ( 2 > , -2 dynes cm Ap/Az dynes cm 0.026 3.1x10" 2 - 6 . 3 x 1 0 3 4 . 6 x l 0 2 2.4 - 7 . 7 x l O ~ 2 0.059 3 . 1 x l 0 ~ 2 - 4 . 2 x l 0 3 3 . 6 x l 0 2 2.1 - 6 . 8 x l 0 " 2 0.090 3 . I x l O - 2 - 2 . 2 x 1 0 3 3 . 0 x l 0 2 1.3 - 4 . 2 x l 0 " 2 0.12 3 . I x l O " 2 - 1 . 2 x l 0 3 2 . 4 x l 0 2 0.87 - 2 . 9 x l 0 ~ 2 0.17 6.1x10" 2 - 6 . 0 x l 0 2 1 . 8 x l 0 2 0.81 - 2 . 7 x l O " 2 0.23 6 . I x l O " 2 - 5 . 8 x l 0 2 1. 7 x 1 0 2 0.83 - 2 . 8 x l 0 " 2 0.41 1 . 2 x l O - 1 - 2 . 5 x l 0 2 l . l x l O 2 0.79 - 2 . 6 x l O ~ 2 0.54 1 . 5 x l O _ 1 - 2 . 0 x l 0 2 l . l x l O 2 0.73 - 2 . 4 x l 0 ~ 2 0.72 2 . 1 x l 0 - 1 - 1 . 5 x l 0 2 9 . 5 x 1 0 1 0.73 - 2 . 4 x l 0 ~ 2 0.97 2 . 7 x l O _ 1 - 9 . 5 x 1 0 1 7 . 2 x 1 0 1 0.62 - 2 . I x l O " 2 3 (1) T h i s v a l u e was o b t a i n e d by v e c t o r i a l l y s u b t r a c t i n g the c r o s s s p e c t r a l v a l u e s between the v e l o c i t y and t h e two p r e s s u r e s i g n a l s . The u n i t s < -2 -1 -1 a r e (dynes cm ) (cm s e c ) (Hz ) . -1/2 „ T 7 -1/2 (2) Ap = ( Apu $ B.W. ) i s a magnitude f o r t h a t p a r t o f t h e u p r e s s u r e d i f f e r e n c e (p . - p . ) w h i c h i s c o h e r e n t w i t h u and l i e s r Aground r a i r w i t h i n t h e a p p r o p r i a t e b a n d w i d t h . 48 (1968). When s i m i l a r spacings are used, the c o r r e l a t i o n c o e f f i c i e n t (defined as the cospectrum divided by the square root of the product of the two s p e c t r a l densities) i s s i m i l a r i n magnitude to the values he obtained, but of the opposite s i g n . The only suggestion that can be put forward f o r the diff e r e n c e i n the re s u l t s i s the d i f f e r e n t method used to obtain a surface pressure free of dynamic noise. Because the general features of h i s r e s u l t s can not be reproduced, they are not compared further with the present data. Energy Transfer by Pressure Forces Two aspects of energy t r a n s f e r by pressure forces are evaluated. The f i r s t i s the 3pw/3z term, representing the net e f f e c t of pressure forces i n the t o t a l turbulence energy budget of the boundary la y e r . The second i s the u3p/3x term by which pressure forces t r a n s f e r energy to or from the u v e l o c i t y component. For the evaluation of the r e l a t i v e s i z e of the 3pw/3z term , the equation f o r the boundary l a y e r energy budget was integrated (see Background, p.8), the term -uw 3U/3z becoming -uw u| z and - — pw becoming - — pw| z» These two terms represent the rate of working by Reynolds st r e s s (the energy feeding term) and the pressure forces on a column of height z and unit area. These terms are compared f o r f i v e Runs, each of approximately 1/2 hour duration. The spectra of p, u, w, v, uw, and pw are p l o t t e d i n Figures 28, 22, 22, 23, 24, and 44 re s p e c t i v e l y . The i n t e g r a l of pw i s negative; that i s , the tra n s f e r of energy i s downwards. The ca l c u l a t e d terms are summarized i n Table IV, page 49. The r a t i o 49 1 — - — pw R -- uw U is plotted in Figure 45 as a function of z. As can be seen, the ratio R is approximately equal to 0 ,1 for z between 2 and 6 meters. The Run at 1.5 meters has some wave generation present, therefore the assumption of no energy flux through the bottom boundary is not completely valid. TABLE IV The pw Term i n the Boundary Layer Energy Budget Run u| z -uw O /u^ -pw/p - — q w -uw U R = m sec m cm sec cm sec cm sec cm sec 110/1 7.1 5.5 698 1.32 6 . 3 x 1 0 4 - 6 . 5 x 1 0 4 5 0 . 5 x l 0 4 0.125 110/2 7.4 4.0 729 1.25 5 . 5 x l 0 4 - 5 . 5 x l 0 4 5 2 . 5 x l 0 4 0.105 120/1 6.5 3.4 580 1.40 3 . 7 x l 0 4 - 2 . 4 x l 0 4 3 9 . 5 x l 0 4 0.095 120/2 6.2 4 .8 443 1.47 2 . 7 x 1 0 4 - 3 . 9 x l 0 4 2 9 . 0 x l 0 4 0.094 121/1 6 .4 1.5 463 1.40 2 . 3 x 1 0 4 - 6 . 5 x l 0 4 2 7 . 0 x l 0 4 0.084 Another term in the boundary layer turbulent energy budget is - ^  — ^ — 2 dz 2 2 2 2 where q = u + v + w . When integrated between the surface and height z this is approximated by - ^  q^ w| , which (when positive) represents v e r t i c a l ^ z energy flux into the unit column by the turbulence. The ratio of this flux to - uwU i s about - 0 . 1 (roughly balancing the -pw term). The energy transferred by the - u 3 p / 3 x term in the energy budget of the 50 u v e l o c i t y component was calculated from the simultaneous measurements of u and p. A hot-wire positioned about 7 cm to the side and 4 cm to the back of the pressure probe was used to measure u. Necessary phase corrections were applied using Taylor's hypothesis. The energy f l u x was c a l c u l a t e d using the quadrature spectrum between u and p. This technique was checked by d i f f e r e n t i a t i n g the pressure term to obtain = - 7 ; 7 ^ - and c a l c u l a t i n g i t s 0 r dx U dt cospectrum with u. The two c a l u c l a t i o n agree wi t h i n ±10%. Figures 46 and 47 show the re s u l t s i n nondimensional form. As can be seen when integrated between values of kz from 0.05 to 20, the rate of energy loss (per u n i t 3 volume) from the u f l u c t u a t i o n s i s about 0.3 to 0.7 pu^ / ( K Z ) , where K i s von Karman's constant. As shown i n the Background s e c t i o n , i t would be expected that i f data were a v a i l a b l e f o r the e n t i r e range of turbulent s c a l e s , the i n t e g r a l would have a maximum value of about 0.67, provided that the turbulence eventually becomes i s o t r o p i c . Both w and v f l u c t u a t i o n s are possible sinks f o r t h i s energy, however f o r the scale range observed the w f l u c t u a t i o n s are expected to be the major sink, since the w gains energy i n t h i s s c ale range, and v has already acquired s i g n i f i c a n t energy at scales l a r g e r than those observed. Since the turbulence has not become f u l l y i s o t r o p i c w i t h i n the s c a l e range observed, fur t h e r energy f l u x i s expected at kz l a r g e r than that observed. For these reasons the i n t e g r a l (the energy loss from the u component) should at a maximum be less than 0.67. The r e l a t i v e l y large v a r i a t i o n i n the measure-ed i n t e g r a l i n c l u d i n g two l a r g e r than 0.67 could have resu l t e d from the lack of complete s t a t i o n a r i t y and the length of the runs being too short f o r s t a t i s -t i c a l r e l i a b i l i t y . The observed energy loss from the u v e l o c i t y component has 3 a mean value of about 0.45 pu A / K Z ; t his i s s u f f i c i e n t to account f o r that acquired by w f l u c t u a t i o n s i n t h i s same frequency range. The energy t r a n s f e r by -udp/9x can be discussed i n terms of two s c a l e 51 ranges. The f i r s t i n which energy i s l o s t i s at low values of kz where the w spectra f i r s t have appreciable energy. The r a t i o $ w/$ u i s shown i n Figures 46 and 47 for comparison. The second range which contains most of the energy loss measured occurs j u s t a f t e r the peak of the w spectrum; that i s , where the turbulence becomes free of the surface (p.45). The energy t r a n s f e r i n most cases drops o f f toward zero at the high frequencies (that i s , high k z ) . This i s considered to be r e a l rather than due to probe separation although values were not p l o t t e d beyond those shown since probe separation may be becoming important i n the c a l c u l a t i o n s by this point. This can be seen by evaluating the appropriate s c a l e s i z e f o r the f l u c t u a t i o n s (see Figure 34). From these energy f l u x measurements i t appears that, i n the t o t a l energy budget below 5 meters, under n e u t r a l conditions, the assumption of small c o n t r i b u t i o n by the pressure term i s reasonable. Not only i s the term small but i t i s also p a r t i a l l y bablanced by the turbulent f l u x term. However f o r the energy budget of the i n d i v i d u a l v e l o c i t y components near the frequency range where the turbulence i s carrying the s t r e s s , the pressure terms are very important as expected and as has been shown f o r the u component. 52 MICROSCALE PRESSURE FLUCTUATIONS OVER WIND GENERATED WATER WAVES When a w i n d b l o w s o v e r w a t e r , waves can be g e n e r a t e d t h r o u g h t h e a c t i o n o f t h e s u r f a c e p r e s s u r e f l u c t u a t i o n s . Measurements were t a k e n t o e v a l u a t e some p r o p e r t i e s o f t h e s t a t i c p r e s s u r e i n t h i s r o l e . The r e g i o n o f i n t e r e s t i s t h e a i r l a y e r c l o s e t o t h e w a t e r s u r f a c e where t h e t u r b u l e n t v e l o c i t y c o r r e l a t i o n uw, w h i c h n o r m a l l y c a r r i e s t he boundary l a y e r s t r e s s , t r a n s f e r s e n ergy t o a p r e s s u r e f i e l d w h i c h i s g e n e r a t i n g t h e waves. The e x a c t mechanism o f t h i s t r a n s f e r , t h e s t u d y o f wave g e n e r a t i o n i s an a r e a o f a c t i v e r e s e a r c h (Dobson, 1969; Manton, 1970). Recent measurements (Dobson, 1969) i n d i c a t e d t h a t a l a r g e f r a c t i o n o f t h e momentum t r a n s f e r r e d t o t h e w a t e r was v i a t h e waves. Dobson measured t h e t r a n s f e r d i r e c t l y f r o m p r e s s u r e measurements on the w a t e r s u r f a c e . I n c o n t r a s t , t h e p r e s e n t measurements were E u l e r i a n , t a k e n a s h o r t d i s t a n c e above t h e c r e s t o f t h e waves and a r e c o n c e r n e d w i t h t h e n a t u r e o f t h e a i r f l o w o v e r t h e waves. The d a t a were c o l l e c t e d a t the S p a n i s h Banks s i t e ( s e e A p p e n d i x A ) . Most o f t h e o b s e r v a t i o n s were t a k e n when t h e w i n d b l e w f r o m an e a s t e r l y d i r e c t i o n . The s t a t i s t i c s o f t h e wave f i e l d a t t h i s s i t e f o r w i n d s f r o m t h i s d i r e c t i o n have been t h o r o u g h l y s t u d i e d by G a r r e t t ( 1 9 7 0 ) . H i s measurements i n d i c a t e d t h a t t h e n o n - u n i f o r m f e t c h w h i c h p r o d u c e d an asymmetry t o t h e d i r e c t i o n a l s p e c t r a d i d n o t p r e v e n t t h e wave f r e q u e n c y s p e c t r u m f r o m a t t a i n i n g t h e ' e q u i l i b r i u m form' ( s e e P h i l l i p s , 1966, p . l 0 9 f f . ) . I n t h i s s t u d y t h e waves fr o m b o t h e a s t and w e s t were checked f o r the e x i s t e n c e o f an e q u i l i b r i u m r a n g e ; t h a t i s , where t h e h i g h f r e q u e n c y s p e c t r u m i s o f t h e fo r m n . The c o n d i t i o n s a t t h i s s i t e a r e c o n s i d e r e d s u i t a b l e f o r o b t a i n i n g measurements t y p i c a l o f t h e wave g e n e r a t i o n mechanism. D u r i n g most o f t h e o b s e r v a t i o n s t h e w a t e r d e p t h was 3 t o 3.5 meters and the t i d a l c u r r e n t s m a l l . The i n f l u e n c e of water depth and t i d a l c urrent on wave frequency and amplitude was considered; the c o r r e c t i o n was u s u a l l y l e s s than 10%. For example, waves of 3 second p e r i o d were t y p i c a l f o r the peak of the wave spectrum at t h i s s i t e . These waves have a wave number k = 2lT /X , -3 -1 -3 -1 of about 4.9 x 10 cm i n 3 meters of water as compared to 4.5 x 10 cm assuming i n f i n i t e depth. A wave probe, pressure probe and v e l o c i t y sensor were mounted on one of the instrument masts near the p l a t f o r m (see Appendix A f o r d e t a i l ) . The v e l o c i t y measurements r e f e r r e d to i n t h i s s e c t i o n were made w i t h a s i n g l e 'u-wire' probe which measured the downstream f l u c t u a t i o n s . These three sensors were pl a c e d as n e a r l y as p r a c t i c a l i n a l i n e p e r p e n d i c u l a r t o the mean wind d i r e c t i o n ; t h a t i s , along wave c r e s t s . Phase d i f f e r e n c e s due to non-alignment were c o r r e c t e d u s i n g the 'frozen f i e l d ' h y p o t hesis. The s p a c i n g crosswind between sensors was of the order of 5 to 10 cm w i t h the wave probe normally o p e r a t i n g at the mid-point. A minimum h e i g h t of the a i r sensors was t y p i c a l l y 30 to 50 cm above mean water l e v e l . The ' v i s u a l ' wave amplitude d u r i n g a c t i v e generation by east winds ( i n which the waves are f e t c h l i m i t e d ) was about 15 to 20 cm. On one occasion the sensors were w i t h i n 10 to 15 cm from the tops of the h i g h e r waves. In t h i s p o s i t i o n the case f o r the p r e s s u r e transducer was o c c a s i o n a l l y h i t by the waves. More than twenty Runs were made under a v a r i e t y of wind c o n d i t i o n s (see Data Summary, Appendix C). Most of the u s e f u l i n f o r m a t i o n comes from f o u r groups of Runs; the Runs w i t h i n each group were taken s e q u e n t i a l l y . These four groups l a b e l l e d A, B, C and D are used as r e p r e s e n t a t i v e of the r e s u l t s . Since a l l the data have some common p r o p e r t i e s , a s i n g l e Run i s used as an example f o r d e t a i l e d d e s c r i p t i o n . The s p e c t r a , coherences and phase r e l a t i o n -ships f o r the f o u r groups of data are then d e s c r i b e d , f o l l o w e d by a d i s c u s s i o n 54 which summarizes the information and presents conclusions. Example Spectra A plot i l l u s t r a t i n g the typical characteristics of spectra over waves, Figure 48, is for data obtained from two pressure sensors, a hot-wire measuring u and a wave probe. The two pressure probes were separated verti c a l l y . The lower pressure sensor and the hot-wire anemometer were at a height of 90 cm above mean water level; the upper pressure sensor was 50 cm higher. The wind was from the west at 3.6 m/sec, slightly slower than the phase speed, C, of the waves at the peak of the wave spectrum. The pressure spectra, Figure 48, always contain a 'hump' superimposed on a spectrum which i s similar to that found over a f l a t boundary. The extent in bandwidth of the pressure hump is denoted by a double arrow labelled 'H'. This hump corresponds closely with the wave spectrum, though not exactly proportionally. For example, the slope on the high frequency side of the hump is not always -4.5 to -5 as i t is for the waves. The humped portion of the spectrum for the higher level pressure sensor, p^, i s similar to that for the lower sensor, p , but is smaller i n amplitude, particularly at high L i frequencies, as would be expected. At higher frequencies than at the 'humped' portion, the two pressure spectra do not align, as they do at the frequencies below the hump, the lower level having a higher intensity. Observations at similar levels over land and at higher elevations over water do not show this high frequency difference, as may be seen from Figures 18 and 20. The spectral slopes at higher frequencies than at the hump are the same as those observed at similar heights over a f l a t boundary. To compare the wave induced pressure, p , observed i n different Runs and 55 a t d i f f e r e n t f r e q u e n c i e s and h e i g h t s , some measure o f t h i s p r e s s u r e i s r e q u i r e d . The q u e s t i o n a r i s e s as t o what f r a c t i o n o f t h e p r e s s u r e a m p l i t u d e i n t h e hump i s a s s o c i a t e d d i r e c t l y w i t h waves, and what i s a s s o c i a t e d w i t h random t u r b u l e n c e . S i n c e , as w i l l be shown l a t e r , t he c o h e r e n c e between t h e p r e s s u r e and t h e waves a t h i g h e r f r e q u e n c i e s t h a n a t t h e hump i s e s s e n t i a l l y z e r o , t h i s r e g i o n o f the s p e c t r u m can be s a i d t o r e s u l t f r o m random t u r b u l e n c e . I t i s assumed t h a t t h e s p e c t r a l s l o p e a s s o c i a t e d w i t h the random t u r b u l e n c e i n t h e f r e q u e n c y range o f t h e hump w i l l be s i m i l a r t o t h a t o b s e r v e d o u t s i d e o f i t . Thus as a f i r s t a p p r o x i m a t i o n t h e p r e s s u r e a m p l i t u d e , p , i n t h e w hump, a s s o c i a t e d d i r e c t l y w i t h t h e waves may be t a k e n t o be t h e d i f f e r e n c e b etween t h e i n t e n s i t y measured and t h e s t r a i g h t l i n e p r o j e c t i o n s f r o m t h e h i g h f r e q u e n c y t a i l ( s e e dashed l i n e s i n F i g u r e 4 8 ) . The s p e c t r u m f o r t h e u - v e l o c i t y measured n e a r t h e waves, F i g u r e 48, does n o t h ave as dominant a hump as t h a t o b s e r v e d i n t h e p r e s s u r e s p e c t r a . A t f r e q u e n c i e s on e i t h e r s i d e o f t h e range marked by 'H', t h e v e l o c i t y s p e c t r u m i s s i m i l a r t o t h a t o b s e r v e d i n t h e absence o f waves; f o r e x a m p l e , t h e -5/3 r e g i o n e x i s t s a t t h e h i g h e r f r e q u e n c i e s . The wave s p e c t r a f r o m o b s e r v a t i o n s a t t h e s i t e d i d n o t a l w a y s h a v e t h e e q u i l i b r i u m f o r m a t a l l f r e q u e n c i e s . G e n e r a l l y , t h e h i g h f r e q u e n c i e s e x h i b i t e d t h e -4.5 t o -5 s l o p e , however t h i s s l o p e o f t e n d i d n o t c o n t i n u e t o v e r y n e a r t o t h e peak o f t h e s p e c t r u m . The l o w e s t f r e q u e n c y t o w h i c h t h e e q u i l i b r i u m wave s p e c t r u m was c o n s i d e r e d t o e x i s t i s marked b y an arrow l a b e l l e d 'eq. p e a k 1 , see F i g u r e 48. A l l d a t a a s s o c i a t e d w i t h wave g e n e r a t i o n h ave been t r e a t e d i n t h e same way. As a check on t h e a n a l y s i s methods, t h e d a t a were o c c a s i o n a l l y r e a n a l y s e d t o i n c l u d e h a n n i n g . H a n n i n g does n o t a l t e r t he r e s u l t s s i g n i f i c a n t l y . 56 Data A. Runs 60/4, 119/1, 119/2, 119/3 This group of data consists of four sequential Runs recorded during an east wind. The instruments used were the wave probe, two pressure sensors spaced vertic a l l y 50 cm apart, and a hot-wire measuring downstream fluctua-tions. The lower pressure sensor and the velocity sensor were about 30 cm above mean water level. Figure 49 shows the wave spectra recorded during each of the Runs. The time of the Run is given i n brackets after the Run number. Visually the highest wave was about 10 cm i n amplitude; the root-mean-square amplitude is about 3 cm. The spectra have a secondary peak at a frequency near 0.25 Hz, thought to be due to waves generated elsewhere and propagating into the region (Garrett, 1970). The sequence 60/4 to 119/3 show that the wave f i e l d was growing i n time under almost steady but slowly increasing winds (see Table V, Appendix C). The mean wind, was about 4.5 m sec \ giving a value for (u|^ - C), where C is the phase speed of the waves, of about 2 m sec at the main peak ('eq. peak') of the wave spectrum. Figures 50 to 53 show spectra for the four Runs. A l l exhibited properties similar to those described for the example shown in Figure 48. The phase and coherence between the lower pressure (p T) and the wave amplitude (r) ), and L a between the two pressures (p and p ) are plotted i n Figures 54 and 55 i-i u respectively. A phase s h i f t i n the p-r) relationship occurred at the wave peak and continued out over the equilibrium wave spectrum. As w i l l be shown later, this s h i f t of about 30 to 50° from 180° is associated with the wave generation. The corresponding coherences are about 0.3 to 0.4. At lower frequencies than at the 'eq. peak' coherences are higher and phase differences are near 180°. This is associated with the residual longer waves. The phase information i n d i c a t e s that these low frequency waves were not being a c t i v e l y generated. Figure 55, a p l o t of the coherence and phase of the two pressure s i g n a l s shows that they are i n phase to ±5°.throughout the extent of the hump. Data not associated with waves at n < 0.1 are not p l o t t e d . Thus most of the phase s h i f t between p and T) occurs below the lower pressure sensor, with almost no s h i f t i n g occuring i n the next 50 cm v e r t i c a l l y . Figure 56 shows the observed u-r| coherence and phase. Coherences are low at a l l frequencies. The non-zero coherence at frequencies outside the range marked 'H' would be expected simply from random coherence because of the f i n i t e length of sample. The highest coherence, about 0.3, occurs f o r waves of frequencies lower than the frequency at the e q u i l i b r i u m peak. Within the frequency range where ac t i v e generation i s occuring, coherences are small, less than 0.1. The cross 2 c o r r e l a t i o n between u and r| gives r e s u l t s s i m i l a r to that shown f o r u and n. The phase shows no d e f i n i t e pattern, with grouping near both 140° and 40°. This group of Runs (A) represents the most d e t a i l e d observations taken. B. Runs 167/1/1, 167/1/2, 167/2, 167/3 This group of four data Runs was also recorded during east winds. The wave spectra f o r a l l four are shown i n Figure 57. As can be seen, the wave spectra are almost i d e n t i c a l , implying steady s t a t e , f e t c h - l i m i t e d conditions. The root-mean-square wave amplitude f o r these Runs i s about 6 cm. The 5 meter mean wind was 7 to 8 m sec ^ g i v i n g a value of (u|,_ - C) at the peak of the wave spectrum of about 4.5 m sec \ The only instrument used besides the wave probe was a s i n g l e pressure probe, positioned about 40 cm above mean water l e v e l . Figures 58 to 61 show the spectra f o r these Runs. They are s i m i l a r to the previous group of Runs (A), except that the peak of the pressure hump i s at a lower frequency than the peak of the wave spectrum. 58 This is thought to be due to waves travelling against the mean wind, which had been refracted to the site from the other side of Point Grey, see Figure 85. Such waves, travelling from the west during a SE wind have been measured (Garrett, 1970) by means of a directional wave array. In order to substantiate the above reason for this abnormal p-r] relationship, the amplitude of the Fourier coefficients for the pressure and wave signals of Run 167/3 was plotted, Figure 62. This shows that the energy in the pressure signal at frequencies lower than the peak of the wave spectrum is directly associated with wave energy. This is true for a l l the other Runs in this sequence. To show that these pressure magnitudes are in accord with this hypothesis the pressure amplitude was calculated assuming that the a i r responds i n a potential flow manner to the measured wave amplitude. For example, for Run 167/1/2, at 0.27 Hz, bandwidth An = 0.077 Hz, / 211 (n)An _2 = 5.3 dynes cm and / 2$^(n)An =1.3 cm. From a potential flow calculation using the mean wind at the pressure observation level / 211 (n)An = p (U + | c | ) k • 2* (n)An e " k z -2 = 4.9 dynes cm with the usual definitions (Lamb, 1932, section 231). The calculated and measured values agree to within the accuracy of the assumptions. Thus in this group of data the pressure signal at frequencies lower than the frequency at peak of the wave spectrum is not that normally encountered in wave generation. The phase and coherence between pressure and waves for these Runs are shown in Figure 63. As before, the phase sh i f t at the wave generation frequencies is about 30 to 50°, with coherences of 0.5 to 0.6. For the frequency range where the waves were propagating against the wind, the coherence is higher and phase differences are about 180°, as would be expected for a potential type of flow. 59 This group of Runs have the highest values of (U|^ - C) for the data obtained. C. Runs 164/1, 164/2 This group consists of two examples i n light WNW or NNW winds of 2.5 and 4 m sec \ The waves at the peak of the wave spectrum were travelling about 2 m sec ^ faster than the wind. They had been generated by strong winds in Howe Sound and had then fanned out from the mouth, part travelling into the English Bay region. The root-mean-square wave amplitude is about 9 cm. The waves travelled at a small angle to the wind, arriving from a direction of about NW. The instruments used were the wave probe, a single pressure sensor and a 'u-wire'. The latter two were located about 50 cm above the mean water level. The spectra for these Runs are shown in Figures 64 and 65. The p-n. phase and coherence are given i n Figure 66; coherences are high; phase d i f f e r -ences are near 180°. Figure 67 shows phase and coherence for u-n. At this low wind speed, the coherence is higher than for a l l other Runs and the phase differences are about 180°. D. Runs 80/3, 60/1, 60/2 This group of three Runs is typical of the data obtained when the value of (u|^ - C) at the peak of the wave spectrum is near zero. In the examples used here, the wind was from the west. The wave, pressure and u-velocity spectra are shown in Figure 68. The high frequency end of the wave spectrum, partially cut off in the graph, follows the -5 slope, and therefore can be considered as part of the equilibrium spectrum. The root-mean-square wave amplitude for these Runs i s about 6 cm. The mean wind was 4 to 4.5 m sec ^. Observations were taken at about 50 cm above mean water level. The phase and coherence for p-n and u-n, Figures 69 and 70 are similar to previous examples. 60 p-H phase shifts are near 180° when ( u | ^ - C) is small and negative, increas-ing as ( u | ^ - C) increases. For waves near the peak of the wave spectrum, where u | ^ / C < 1, the u-r| phase difference is near 180°. This f i n a l group are representative of the eight Runs for which ( u | ^ - C) is near zero at the.peak of the wave spectrum. Pis cussion Compared to the pressure spectrum over a f l a t boundary, that measured close to a wavy water surface is greatly modified. The most prominent change is the 'hump' in the pressure spectrum that is associated with the wave spectrum. The intensity i n this hump is up to 10 times larger than the spectral intensity expected for a f l a t boundary under similar wind conditions. Remnants of the hump are observed during normal wave conditions up to heights z between X /2 and X where X is the wavelength of the waves. This can be w w w estimated from the plots of the pressure spectra by using the highest frequency at which the hump is definable. For Run 173/3 in Figure 48, this occurs for p at a frequency n - 0.9 Hz. Waves of this frequency have a u wavelength X - 200 cm; the measurement height was 140 cm. For Run 167/1/1, w Figure 60, corresponding" figures are% ; ,= 1.5 Hz, X - 70 cm and z = 30 cm. w A hump, similar to that observed, would be expected even i f no mean wind or turbulence was present, and the wave f i e l d was simply propagating past the pressure sensor. This could be described by a potential flow solution (Lamb, 1932, section 231). The pressure (phase and magnitude (Pp)) predicted by the potential flow solution were checked in the f i e l d . Figure 71 shows the results obtained on s t i l l , windless days when a swell was propagating past the instrument mast. The pressure and wave height were 61 recorded on a strip-chart recorder instead of the usual method on magnetic tape. The values plotted each came from the average of about 100 estimates of individual amplitudes. The measured and predicted values agree within about 10%. Phase differences for p-n were 180°, within the accuracy of the method. Thus potential flow theory adequately predicts the pressure f i e l d (and hence supposedly the velocity field) for propagating waves in the absence of wind. When there was a wind, i t might have been expected that the pressure could have been approximated by using a velocity of (U|,- - C) i n the potential flow solution; that i s , PW * - p n a k ( u | 5 - c ) 2 e " k z , where n is wave amplitude. This behaviour was not found for the present data. For example, for the (u|^ - C) = 0 between 0.3 and 0.4 Hz i n Run 164/2, Figure 65, and near the peak of the wave spectrum i n Runs 80/3 and 60/2, Figure 68, there is no indication of a drop i n the measured p . w Two cases have already shown that the measured pressure was closely approximated by potential flow calculations; one for u|^/C = 0 and the other for u|^/C negative. However when u|^/C > 0 the data does not have such a simple interpretation. The non-dimensional variables which might cause variations in the pressure (p ) associated with the waves are: kz, representative of the fractional height of the observations in terms of wavelength; kn representative of the wave slope; and u|^/C, the ratio of the mean wind at 5 meters to the phase speed of the waves. r) a can be approximated by / 2<J>^ (n)An where $ n( n) I s the wave spectral density and An is the -^-octave bandwidth for a narrow band of 62 frequencies. When the wave spectrum has the equilibrium form, the product kn =; k/(0 (n)An) can be taken as a constant. For simplicity, pressure data a f) from one 'fixed height' above mean water level and only for the frequency range where the waves have the equilibrium form are i n i t i a l l y considered. In practice this 'fixed height' ranged from 30 to 50 cm. Thus, for this data, kz and kn are constant at any fixed frequency. In Figure 7 2 , P w(n) = (/ 2H(n)An - / 2IT o(n)An ) i s plotted against u| 5/C with frequency as a parameter. ^ Q(n) ^ s t n e background pressure spectrum ill u s t r a t e d by the dashed lines in Figure 4 8 . The points on the ordinate in Figure 72 are derived from the limiting cases of u|^/C = 0 which are assumed to be given at each frequency by the potential flow solution -kz p (n) = p / 2 $ (n)An k C e ( 1 5 ) w r) These calculations use the mean values from the measured wave spectra and the 'fixed height' z of 40 cm. The values of k and C at a given frequency were obtained from the solution for small amplitude gravitational waves: C 2 = ^  tanh kh and n = As can be seen, Figure 7 2 , the data tends to k 2TT ' 6 ' group along lines which could be approximately extrapolated to the potential flow calculations. As U|,./C increases, the pressure intensity at a l l frequencies also increases. There is no indication of a distinctly different behaviour near u | ^ = C. With this plot (Figure 72 ) as background, data from a l l heights were considered in terms of the nondimensional variables kz, ll|^/C, k/2$^(n)An , and p /p where p = pk/20 (n)An C . The product k/2$ (n)An was chosen *w *o *o K n . ^ n to be constant although i t is not exactly so for this data, varying by about 2 0 % . This is not important provided the role of kri a iri p w is the same as i n p . Two plo t s are used, Figures 73 and 74, i n which p /p i s shown as a o ° *w *o function of u|,_/C at constant kz and v i c e versa. The res u l t s i n Figure 73 are s i m i l a r to those already shown i n Figure 72, with l i t t l e change r e s u l t i n g from" the a d d i t i o n a l data a v a i l a b l e f o r heights other than 40 cm. Figure 74 shows a less d e f i n i t e dependence of nondimensional pressure on kz than on u|^/C as shown i n Figure 73. S t r a i g h t l i n e s representing constant values of kz have been drawn by hand among the data p l o t t e d i n Figure,73. I t i s f e l t that there i s i n s u f f i c i e n t data to warrant a c l o s e r f i t t i n g of 'curves' and there are no t h e o r e t i c a l p r e dictions to act as guidel i n e s . Acting as a f i r s t approximation, the data i n Figure 73, as represented by the l i n e s shown are summarized by the formula p (n) = p k • 2* (n)An C 2 exp(0.27 uL/C - k z ( l - 0.08 uL/C)) w n -> -> (16) The l i n e s shown i n Figure 74 representing constant values of u|,- /C are cal c u l a t e d from t h i s formula, and agree very c l o s e l y with those drawn independently on the basis of the data alone. The l i m i t i n g case of u|^/C = 0 i s the p o t e n t i a l flow s o l u t i o n as given by equation 15. There i s s u f f i c i e n t accuracy to show that'as (u|^ / C) increases, the slope of the l i n e s decreases. Thus as the wind increases, the pressure decay v e r t i c a l l y at a given wave number i s i n c r e a s i n g l y less than the exponential decay i n p o t e n t i a l flow. Equation 16, used to r e l a t e the observations to an em p i r i c a l formulation, becomes p h y s i c a l l y u n r e a l i s t i c i f extended to large values of u|^/C, beyond the data p l o t t e d . I f Tj|,./C = 12.5 i s sub s t i t u t e d i n t o equation 16 a l l z dependence disappears and at U|^/C > 12.5 equation 16 has the pressure i n c r e a s i n g v e r t i c a l l y ; the opposite of that which would be expected to occur. This 12.5 value, f o r u|c/C could represent a wavelength of 10 cm and a u| of 5 m sec . Therefore the formula given should not be extrapolated to regions outside that covered by the data, approximately 0 < u|5/C < 7 and 0.5 < kz < without caution, particularly at larger values of u|^/C. It is noticed that calculated pressures at frequencies where the equilibrium wave spectrum did not exist have magnitudes similar to those predicted by equation 16. It is possible to check the ve r t i c a l dependence shown in equation 16 using the simultaneous measurements at two levels described i n Data Group A (p.56). By taking the ratio of the pressure, p (n), at the two levels, a w Az dependence would be l e f t . In terms of equation 16 for z^ > z^. Measured values of this ratio and those calculated from equation 17 are plotted in Figure 75 against kAz. The f i t i s reasonable. In summary, the pressure hump has a magnitude which is similar to the potential flow solution i n very low winds; i t increases monotonically as U increases and decays vertically at a rate less than exponential, the higher the mean wind, the slower the decay. The surface pressure spectra obtained by Dobson (1969) have a similar type of pressure hump, but in general i t is not as well defined as those obtained in the present study (his low frequency intensities were in general an order of magnitude larger). There was some question as to how to compare Dobson's data with the present results since his instrument followed the water surface, and hence his measurements were not Eulerian. Nevertheless i t seems reasonable to compare Dobson's spectra with values predicted by U = exp( - kAz + 0.08 kAz ) (17) 65 equation 16, setting z = 0. Thus n(n) = p 2 k 2 C 4 * (n) exp(0.54 u|5/C) A comparison for three cases is shown in Figure 76 for which the values of $^(n) were taken from data i n Dobson's thesis. One of high wind speed runs (4b) and two low wind speed runs (2a and 2b) were chosen. Since no extrapolation of the pressure spectra, similar to that represented by the dashed lines in Figure 48 was obvious, no attempt was made to remove the energy associated with random turbulence. At frequencies near the peak of the wave spectrum, see Figure 76, this should not give any large error since the hump pressure is expected to be about an order of magnitude larger than the pressure associated with random turbulence. For each run three curves of pressure intensity are compared: Dobson's total spectral estimate, Dobson's spectral estimate with pgn removed, and the spectral estimate predicted by equation 16. As can be 3. seen the data collected by Dobson appear to agree reasonably well (within a factor of about 2) with the values predicted by the formula. Therefore equation 16 may predict the hump pressure, P w( n)» down to the wave surface. The pressure which generates the waves is the component of the pressure, P w(n), which is in quadrature with the wave. Even though the pressure fluctuations associated with the waves are up to one order larger than that expected from the random turbulence, the coherence with the waves is only about 0.4 to 0.6. This suggests that the process of wave generation is intermittent and hence that the phase difference between pressure and waves fluctuates significantly. It was shown in earlier diagrams that the average phase difference observed at about 50 cm above the waves during active wave generation is such 66 that pressure lags the waves about 140 to 120°. In the absence of active generation the phase is near 180°. Figure 77 is a composite of average p-n. phase differences plotted against u|^/C. This plot includes a l l of the data from Data Groups A to D (page 56 to 60) for frequencies within the extent of the equilibrium wave spectrum ($^(n) « n . The data are labelled with a different symbol to indicate the data group. Most of the large angles associated with wave generation occur for u|,-/C > 2 and do not show any further s h i f t from 180° for increasing u|^/C; i n fact, i f anything, the opposite appears to be true. For a l l other data either those outside the range of the equilibrium wave spectrum for which 0 < u|,-/C < 2 or those travelling against the wind for which u|^/C < 0 the phases were 180° ± 10° with no definite trend. The fact that the large p-n phase difference occurs at values of u|,./C greater than about 2 i s thought to be due to the relative height of the c r i t i c a l level and wave amplitude n for these particular groups of data. The 3. ' c r i t i c a l height', z^, is the height at which U = C (Phillips, 1966, p.91). For the data plotted in Figure 77 the c r i t i c a l height i s well above the waves for u|^/C near 1. Values of U"|^ /C - 2 are necessary before the c r i t i c a l height approaches a value equal to the wave amplitude. This can be seen from the use of Figure 78. In deriving the curves in this figure a logarithmic wind pr o f i l e and a roughness length of 0.01 cm"'" are assumed. The plot is to show which wave frequency has an amplitude equal to the c r i t i c a l height, given a fixed mean wind at 5 meters. Plotted are a family of curves representing the c r i t i c a l height for different 5 meter winds at various wave frequencies. The curve of r| = / 2$ (n)An against frequency is for the measured equilibrium 1 This value of roughness length is approximately equivalent to a drag coefficient C„ = 1.2 x 10 . wave spectrum described above in Data Groups A, B, C and D. Once a value of u|,. i s known, there w i l l be a different z^ for every wave frequency, but only one which has a z = X] . For example, in Data Group A, page 56, u| is about C 3. 3 - l i - l 5 m sec . In Figure 78, Z = n at 0.62 Hz when U _ = 5 m sec . This c a 5 frequency of 0.62 Hz is near the frequency at which the large p^ - n phase shifts occur as shown in Figure 54. For waves of this frequency u|^/C - 2. Therefore when replotting the entire set of phase differences from Figure 54 into Figure 77, the large phase s h i f t from 180° occurs at about u|,-/C = 2. This can also be shown for a l l the other data groups. Thus the large p-r\ phase s h i f t from 180° occurs only for those waves which have a z < n .. G cl Dobson's (1969) results, before pgr) was removed, gave a comparable SL phase distribution when plotted against u|^/C (M.J. Manton, personal communica-tion), although the scatter is larger than for the present data and the s h i f t from 180° appeared to increase continuously with increasing Uj/C. When pgn j a was removed from his signal, the phases between surface pressure and waves were larger by 20 to 50°. The energy flux to the waves by the action of surface pressure, noting that wl = -r- , can be represented by 'ri dt r J En = pw(n)| • Z — Tj = Z(to. Quad(pn)| n )An.| z = ^ (18) Values of En were approximated using the pressure measured above the waves. Since active wave generation was present for group A and group B (page 56) one example was taken from each. The results are shown in Figure 79. Using the pressure measurement above the waves shows maximum energy flux at the peak of the wave spectrum labelled 'peak' in the figure. The integrals shown on the graphs are approximately one f i f t h of the values obtained 68 • by Dobson (1969) at a comparable wind speed. Cases in which the waves were moving faster than the wind, or the waves were in the opposite direction to the wind; that i s , there was no z^; did not have the phase shifts necessary for such energy transfer, see Figures 54, 63, 66 and 69. It i s surprising that there is not an obvious 'hump' of energy i n the velocity spectra through the frequencies near the peak of the wave spectra i n view of the large increase in energy observed i n the pressure spectra. It is d i f f i c u l t to make an estimation of the expected amplitude of the velocity fluctuations associated with the waves or with the observed pressure since the relationships between them are not known. A rough estimation of the amplitude of the expected velocity fluctuations can be made using data from group A, page 56. The two verti c a l l y spaced pressure measurements can be used to evaluate a v e r t i c a l pressure gradient. Assuming that this gradient is accelerating or decelerating the a i r vertically, an approximate velocity may be inferred from ~ f^" ~ T T • For example, in Run 119/2, Figure 52, at n = 0.55 Hz (bandwidth = 0.15 Hz) the Ap vertically i s approximately -2 2.5 dynes cm over a distance of Az = 50 cm; this can be easily obtained from Figure 52 since at this frequency the coherence between the two pressure signals is 1 and the phase difference is 0° (see Figure 55). Since a typical Aw must be accelerated (or decelerated) during one quarter of a cycle, -1 -3 -3 At - 0.45 sec. This gives Aw - 18 cm sec , assuming p - 1.2 x 10 gm cm The corresponding observed value of / 2$u(n)An in this frequency band is -1 3 2 - 2 -1 17 cm sec ( $^(0.55 Hz) = 10 cm sec Hz ) which is comparable with that calculated for Aw. However this is only a rough agreement, since the relationship between u and w is not known near waves, and the u was measured next to the lower pressure sensor rather than at a position midway between the pressure sensors where the prediction would be most valid. Nevertheless the velocities observed may be sufficiently high to account for the observed 69 vert i c a l pressure gradients which are required to produce them. It is also possible that some of the observed pressure i n the frequency range referred to as the 'hump' may result from the 'integral effect' (p.6) with motions near and below the wave crests requiring the pressure observed at the higher level above the wave crests. The potential flow solution (p.62) predicts a phase difference between u and n of 180° for either no wind, or a mean wind constant with height and slower than the phase speed of the waves. This u-r| phase difference of 180° and the restriction on the wind speed are approximately those shown for the coherent part of the observed u-r) signals. The phases between velocity u and wave height n plotted in Figures 56, 67 and 70 for the Data Groups A, C and D respectively are generally near 180°. The few points near 40° in Figure 56 are for waves travelling from an unknown direction. When the wind speed is faster than the phase speed of the waves, the coherence is low, <0.1, and phases are random. The 180° phase difference occurs only for those frequencies when most or a l l of the a i r below 5 meters has a speed less than the wave speed, as required by the potential flow calculations. A comparison of Figures 55 and 56, 66 and 67, and 69 and 70 shows that the u-r) 180° phase difference occurs only when the p-T| phase differences are also near 180°. It is shown above (p.66) that this range of p-r) phases near 180° corresponds to a c r i t i c a l height z well above the wave amplitude (r) ); that i s , when the wind C 3. at the observation levels is less than, the phase speed of the waves (z_ » n). An example of the strong influence the wave f i e l d has on the turbulence in the a i r above is seen in the p-u cross correlation. Over a f l a t boundary i t was found that the p and u were in phase for those scales (L ) larger than the l ocal height of the observations, shifting to the neighbourhood of 135° at smaller scales. The coherence and phase of p and u over waves are shown in 70 Figures 80 and 81. The broad features appear to be the same as those obtained from observations over a f l a t boundary, Figures 39 and 40, except for the lower coherence through the region of the 'hump'. However the frequency of the phase 'transition' is entirely different, being much lower than over land. Figure 82 shows the wavelength A^ of the pressure at the phase transition compared to those obtained over a f l a t boundary. The scales, L = A /2, at P P the transition are several times larger than those expected over a f l a t boundary at a similar height. In every case i t was found that this phase transition over waves occurred at the frequency of the peak of the pressure hump when the pressure spectra were plotted as nll(n). In most cases this also corresponded to the peak of the n$^(n) wave spectrum. The behaviour of the p-u coherence and phase at frequencies above the phase transition i s similar to that found over a f l a t boundary, except the transition occurred at a lower frequency. The wavelength of the waves appears to introduce through the pressure f i e l d a new length scale with which the turbulence interacts. In p a r a l l e l with the observations over a f l a t boundary there was a large energy loss from the u-velocity component at frequencies above the phase transition. Using the same method as was used earlied, the nondimensional energy flux from the u-component, was calculated for the observations over waves. The results are plotted against kz i n Figures 83 and 84. The distribution of energy flux i s similar to that found over a f l a t boundary. Most of the flux occurred at scales immediately above the transition. For those values of kz associated with the peak of the wave spectrum, the sign of the flux i s often positive and the » magnitude is highly variable. The integrals under the curves are similar t those found for the data collected over a f l a t boundary being typically between 0.3 and 0.4. Thus the energy loss from the u-component to the othe velocity components is similar in magnitude to that found over the f l a t boundary layer but occurs at a nondimensional height kz which is lower by an order of magnitude than over land. 72 SUMMARY OF RESULTS An instrument was developed to measure the st a t i c pressure fluctuations within the turbulent flow of the atmospheric boundary. From an in s i t u calibration, which compared the pressure measured by the developed instrument with that measured by a reliable surface measuring technique, the accuracy of the new instrument was found to be about ±10% i n amplitude and ±5° in phase. This instrument was used to measure some of the properties of the st a t i c pressure fluctuations found over a f l a t boundary and over water waves. The data included the mean square pressure, spectral intensity and shape, coherence and phase between two pressure measurements separated in each of the three coordinate directions, pressure-velocity relationship for a l l three velocity components, horizontal pressure gradient-velocity relationships, and pressure, wave, downstream-velocity relationships. For a l l observations over a f l a t boundary the root-mean-square pressure resulting from the boundary layer turbulence i s about 2.6 times the mean stress; thus the pressure can be nondimensionalized by the stress. The spectral distribution was found to be only weakly dependent on height, in contrast to the velocity which is directly dependent. At frequencies above the peak i n the ver t i c a l velocity spectrum, the pressure spectra have a mean slope of about -1.7; the slope is less steep at lower frequencies. A scale was defined for the pressure fluctuations based on the coherence between two simultaneous pressure measurements. This scale was found to be the same in . a l l three coordinate directions; thus, to a f i r s t approximation, the individual pressure fluctuations are spherical. The phase measurements agree with this interpretation. From near-surface, simultaneous measurements with a down-stream separation, the propagation velocity of a pressure fluctuation was 73 estimated. When this rate was compared to the mean wind at a level corresponding to the size of the pressure fluctuations they were found to be about equal. " In this same measurement, consistent phase differences i n the cross-spectrum could not be traced further than about 360° before the signals became incoherent. The simultaneous measurement of pressure and velocity showed that the downstream velocity fluctuations are approximately in phase with pressure at low frequencies, while at higher frequencies there is large phase difference of about 135°. This phase difference is a function of the height of observa-tions, the in-phase portion occuring for pressure scales larger than the measurement height. At these larger scales the pressure appears to be inter-acting directly with the surface; smaller scales are effectively free of the surface. Measurements at the surface support this interpretation. Pressure measurements were used to calculate the energy flux by pressure forces in two cases. In the f i r s t , the energy flux out of the downstream velocity fluctuations was found to be about 0.35 of the net energy source to the downstream component. A possible sink for this energy is into the v e r t i c a l and crossstream velocity fluctuations; the v e r t i c a l velocity fluctuations are developing i n this same frequency range. In the second case, the pressure divergence term was found to be a small fraction, about 1/10, of the energy feeding term i n the net energy budget of a boundary layer. The terms were compared i n integrated form. Pressure measurements near wind generated water waves showed a large hump in the spectrum at the wave frequencies. The amplitude of this hump increased, and the rate of i t s decay vertically decreased, as the mean wind speed increased. The phase difference between pressure and waves during active wave generation i s about 135°, pressure lagging waves, and does not change vertically for measurements at heights greater than the wave crests. In non-generating conditions the phase difference is near 180°. The active generation occurs only when the c r i t i c a l height is low enough to be near crest heights or lower. Wave generation, inferred from these observations above the surface, occurs most actively at the peak of the wave spectrum. The pressure-downstream velocity relationship over waves is different from that found for similar observations over a f l a t boundary. Instead of the phase transition occuring at a frequency dependent on the size of the pressure producing scales which are directly proportional to the height, i t occurs at the frequency of the peak of the wave spectrum. Energy transfer out of the downstream velocity component measured near the waves is similar to that found for observations over a f l a t boundary, only i t is shifted to larger scales at a given height. The measurements suggest some strong inter-action between the normal boundary layer turbulence and the more organized flow over the waves. These measurements made with the developed instrument have provided the f i r s t reliable pressure data within a turbulent boundary layer. 75 BIBLIOGRAPHY Ampex Co. (1966) Instruction Manual, FR-1300 Recorder/Reproducer. Calif., U.S.A. Batchelor, G.K. (1960) The Theory of Homogeneous Turbulence. Cambridge University Press, 197pp. Blackman, R.B. and J.W. Tukey (1959) The Measurement of Power Spectra. Dover Publications, Inc., New York, 190pp. C.F. Casella and Co. Ltd., Instruction Leaflet - //3034/RA, Sensitive Anemometer. London, England. Deardorff, J.W. (19 70) A three-dimensional numerical investigation of the idealized planetary boundary layer. J. Fluid Mech. hX pt.2: 453-480. DISA Elecktronik A/S (1967) Instruction and Service Manual, Type 55D05 Battery Operated C.T.A. Herlev, Denmark. Dobson, F.W. (1969) Observations of normal pressure on wind-generated sea waves. Ph.D. dissertation, University of B.C., 240pp. Garrett, J.F. (1970) Field observations of frequency domain st a t i s t i c s and nonlinear effects in wind-generated ocean waves. Ph.D. dissertation, University of B.C., 176pp. Golitsyn, G.S. (1964) On the time spectrum of micropulsations in atmospheric pressure. Izvestiya, Geophysical Series, No.8:1253-1258. (Am. Geophys. Union translation:761-763) Gorshkov, N.F. (1967) Measurements of the spectrum of pressure micro-pulsations in the near-earth layer of the atmosphere. Izvestiya, Atmospheric and Oceanic Physics, 3^ (4):447-451. (Am. Geophys. Union translation:255-257) Gorshkov, N.F. (1968) On micropressure-fluctuations in the near-earth layer. Izvestiya, Atmospheric and Oceanic Physics, 4^  (4):460-462. (Am. Geophys. Union translation:259-261) Gossard, E.E. (1960) Spectra of atmospheric scalars. J. of Geophys. Res., 65(10) .-3339-3351. Herron, T.J., I. Tolstoy and D.W. Kraft (1969) Atmospheric pressure background fluctuations in the microscale range. J. of Geophys. Res., 74(6):1321-1329. Hinze, J.O. (1959) Turbulence, An Introduction to Its Mechanism and Theory. McGraw-Hill Book Co., Inc. Toronto. 586pp. 76 Hume, D. (1967) I n s t r u c t i o n Manual, Wind P r o f i l e System. I n s t i t u t e o f Oceanography, U n i v e r s i t y o f B.C., V a n c o u v e r . Hume, D. (1969) I n s t r u c t i o n Manual, C a p a c i t i v e Wave P r o b e . I n s t i t u t e o f Oceanography, U n i v e r s i t y o f B.C., Vancouver. I.E.E.E. T r a n s a c t i o n s on A u d i o and E l e c t r o a c o u s t i c s . S p e c i a l I s s u e on F a s t F o u r i e r T r a n s f o r m and i t s A p p l i c a t i o n t o D i g i t a l F i l t e r i n g and S p e c t r a l A n a l y s i s . June 1967, AU15(2). K a i j o D e n k i Co. L t d . (1967) I n s t r u c t i o n M a n u a l , Model PAT-311 U l t r a s o n i c Anemometer thermometer. Tokyo, Japan. K r a i c h n a n , R.H. (1956) P r e s s u r e f l u c t u a t i o n s i n t u r b u l e n t f l o w o v e r a f l a t p l a t e . J . o f t h e A c o u s t i c a l Soc. o f A m e r i c a , 2 8 ( 3 ) : 3 7 8 - 3 9 0 . Lamb, S i r H. (1932) Hydrodynamics ( 6 t h e d . ) . Cambridge U n i v e r s i t y P r e s s ( r e p r i n t e d by Dover P u b l i c a t i o n s , I n c . , New Y o r k , 1945), 738pp. Le e , Y.W. (1967) S t a t i s t i c a l T heory o f Communication. John W i l e y and Sons, I n c . , New Y o r k , 509pp. Lumley, J . L . and H.A. P a n o f s k y (1964) The S t r u c t u r e o f A t m o s p h e r i c T u r b u l e n c e . I n t e r s c i e n c e P u b l i s h e r s , New Y o r k , 239pp. M a k i n o Co., L t d . , I n s t r u c t i o n M a n ual, M a k i n o ' s P h o t o e l e c t r i c Anemometers. Tokyo, J a p a n . Manton, M.J. (19 70) T h e o r e t i c a l s t u d i e s o f t h e g e n e r a t i o n o f s u r f a c e waves and t h e p r o p a g a t i o n o f i n t e r n a l waves i n t h e s e a . Ph. D. d i s s e r t a t i o n , U n i v e r s i t y o f B.C., 175pp. McBean, G.A. (19 70) S i m i l a r i t y o f t u r b u l e n t t r a n s f e r s n e a r t h e s u r f a c e . Ph.D. d i s s e r t a t i o n , U n i v e r s i t y o f B.C., 150pp. P a n o f s k y , H.A. and A.A. Townsend (1964) Change o f t e r r a i n r oughness and t h e w i n d p r o f i l e . Q u a r t . J . Roy. M e t e o r o l . Soc. 90:147-155. P h i l l i p s , O.M. (1966) The Dynamics o f the Upper Ocean. Cambridge U n i v e r s i t y P r e s s , London, 261pp. Shaw, R. (1960) I n f l u e n c e o f h o l e d i m e n s i o n s on s t a t i c p r e s s u r e measurements. J . F l u i d Mech., _7 = 550-564. S t e w a r t , R.W. (1969) T u r b u l e n c e and waves i n a s t r a t i f i e d a tmosphere. R a d i o S c i e n c e , 4(12):1269-1278. Townsend, A.A. (1955) The S t r u c t u r e o f T u r b u l e n t S h e a r Flow. Cambridge U n i v e r s i t y P r e s s , London. 77 Weiler, H.S. and R.W. Burling (1967) Direct measurements of stress and spectra of turbulence in the boundary layer over the sea. J. Atmos. Sci., 24(6):653-664. Willmarth, W.W. and CE. Wooldridge (1962) Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer. J. Fluid Mech., 14:187-210. APPENDIX A EXPERIMENTAL SITES, INSTRUMENTS, AND TECHNIQUES The primary objective when collecting data i n the f i e l d was to obtain recorded data in raw form for later analysis. Data were collected at one of three different sites; over water or sand at the SPANISH BANKS SITE, over asphalt and cut grass at the LADNER SITE, or over sand at the BOUNDARY BAY SITE. Even though a typical length of recorded data, or a 'Run', was about 30 minutes each expedition providing several Runs lasted from one day to a few weeks, chiefly dependent on the weather. Analog signals from instruments responding to sta t i c pressure, wind velocity and wave height were recorded on magnetic tape. Auxiliary data were logged manually; typically these included the mean wind profile and direction at both land and sea sites, with wet and dry bulb temperatures, water temperature, currents and mean water level added at the Spanish Banks si t e . The method and instruments used to obtain these measurements, along with a description of the sites, i s given in the following paragraphs. Experimental Sites (i) Spanish Banks Site A l l over-water observations in this thesis are from the Spanish Banks sit e . It is located on a t i d a l f l a t on the south side of English Bay, Figure 85, 1/2 km from shore. The site has two common wind directions, easterly or westerly, both suitable for recording data. Wind speeds up to 10 m/sec are not uncommon, 79 5 m/sec is more typical. An easterly wind has an asymmetric fetch of about 7 km; this asymmetry was considered when choosing recording conditions. Winds from the west usually occurred when the wind was from the north-west in the Strait of Georgia. Thus even though the fetch to the west is about 50 km, the wave and wind f i e l d would not necessarily be uniform over this dis tance. A hut on pilings, called the platform, provided, liv i n g and working space, Figure 86 (a) and (b). It was accessible by walking at mean low tides, by boat at higher tides. The maximum t i d a l range i s approximately 4 m. Vertical aluminum masts located about 50 m to the seaward side of the platform were used as mounts for instrument sensors. These masts, about 7 m high and 15 cm in diameter, rested on the sand and were held r i g i d by a tripod bracing arrangement. A carriage on the masts could be raised or lowered hydraulically and turned e l e c t r i c a l l y by controls at the platform to accommodate changes i n tide and wind. Sensors mounted on brackets were placed such that there was negligible interference from the masts. Cables, weighted to the bottom, connected the sensors to the hut where signals were conditioned and then recorded. The AC e l e c t r i c a l power for operating the equipment was supplied by a 3 kw ONAN generator, fed through a S0RENS0N AC power regulator. The voltage and cycles of the AC power were monitored to ensure that they remained within instrument requirements. ( i i ) Ladner Site The Ladner s i t e is located on an asphalt runway at an abandoned airport, Figure 87, an area now part of the Canadian Forces Station, Ladner. F a c i l i t i e s at the site were arranged by a fellow graduate student, G.A. McBean. Grass growing between the runways was normally cut, providing a reasonably 80 uniform terrain for about 1 km i n a l l directions. Typical roughness elements were about 10 to 30 cm high in the grassed area and less than .5 cm on the runways. The station is surrounded by farmland and t i d a l f l a t s ; the nearest buildings or other large obstructions upwind (west) were always more than 2 km away. The instrument sensors were mounted at fixed levels on a 5 m mast which was aligned with the wind by rotating i t manually. Some surface pressure measurements were made at this site i n winds up to 10 m/sec. A hole was dug in the asphalt runway to contain the box used for the surface measurements, Figure 88. Cables led downwind to the signal conditioners and tape recorder that were i n the back of an I.0.U.B.C. truck, Figure 89. B.C. Hydro e l e c t r i c a l power, 115 VAC, was available from a nearby outlet. ( i i i ) Boundary Bay Site The Boundary Bay site i s located approximately 75 m from the high tide line on the mud flats of Boundary Bay, Figure 90. The flats to the south of the s i t e were free of obstruction except for the occassional log or patches of grass. Changes in surface elevation were about 10 cm; the potholes which occurred were f i l l e d with water. The area to the north is farmland. Along the high tide line is a 2 m high dike. Since this site was used for instrument calibration only, uniform terrain was not c r i t i c a l . The instrument box used for surface pressure measurements was placed flush with the surrounding surface in an area which was uniform and f l a t for a few meters. Mean wind profile measurements were made on a 4 m aluminum mast. Other instruments were attached to a 2 m aluminum stand. As at the Ladner s i t e , the electronics were kept in an I.0.U.B.C. truck and e l e c t r i c a l power was obtained from a nearby outlet. 81 Instruments and Observational Techniques (i) Analog Data Recording The analog data were recorded on 14-track 1" magnetic tapes using an AMPEX Model FR-1300 Recorder/Reproducer (Ampex Co., 1966) portable tape recorder. Signals were recorded FM using the IRIG scheme. An input level of ± 1 volt rms produces ± 40% deviation from the center frequency. Most data were recorded at 7 — ips. At this speed, the frequency response was f l a t 2 (within 1.0 db) from 0 to 2.5 kHz, and the rms signal to noise ratio was 44 db; adequate for the purpose. One feature of this tape recorder which is very useful is the two sets of tape heads; one set for recording and the other for simultaneous monitoring. A separate switching box permitted any two of the signals being recorded to be viewed on the dual beam oscilloscope. ( i i ) Sonic and U-Wire The turbulent velocity components were measured with a KAIJO-DENKI three dimensional ultrasonic anemometer-thermometer (Kaijo-Denki Co. Ltd., 1967) referred to as the 'sonic' and the downstream component using a DISA Type 55D05 battery operated constant temperature hot-wire anemometer (Disa Elecktronik A/S, 1967) referred to as a 'u-wire' or 'hot wire'. The sonic used has a probe with a path length of 20 cm. Thus for typical mean wind speeds, less than 10 m/sec, velocity fluctuations from DC to greater than 10 Hz could be measured before the effect of averaging over the 20 cm path became important. The junction box and probe were mounted at a well exposed position on the mast, usually more than 1 m from the main mast . A 100 m cable connected the junction box to the remaining electronics,,in the hut or 82 truck. The analog output of the sonic is ± 1 v peak to peak with offsets to adjust for the mean wind. Accuracies are about ± 3% of the f u l l scale ranges of 1 m/sec, 3 m/sec and 10 m/sec. The u-wires were mounted either separately from, or attached to, the pressure measuring instrument. Coaxial signal cable and compensating cable led to the Disa u-wire electronics which give an analog signal output. The signal had the DC level and gain adjusted before recording; this additional equipment was b u i l t by E. Jerome, a fellow graduate student. The accuracy of the instrument is limited by the calibration of the probes. Most probes used were calibrated in a wind tunnel i n i t i a l l y , then checked against cup anemometers or the sonic -in the f i e l d . The calibration was probably known to ± 15%. ( i i i ) Cup Anemometers The mean wind speed profile was measured with cup anemometers, normally positioned with a logarithmic spacing at levels between .5 and 5 m. The cups were mounted on arms up to 50 cm from the main mast. For most Runs more than one cup was i n working order. A different set was used at each of the sites: I.0.U.B.C. Wind Profile System at the Spanish Banks si t e (Hume, 1967), MAKINO system at the Ladner s i t e (Makino Co. Ltd.), and CASELLA system at the Boundary Bay site (CF. Casella and Co. Ltd.). A l l these systems are of the counter type, where either a photocell or a reed switch i s pulsed by the turning cups; the pulses are then modified to drive a counter which counted the total number of pulses. The I.0.U.B.C. and Makino systems include an rms meter for instantaneous visual monitoring. The total counts from each cup were obtained at timed intervals, usually spanning the period of analog data recording; the mean wind at fixed levels was read from the calibration curves. The calibration of the cups was checked in a wind tunnel. The mean wind for any other level was obtained graphically using a best line f i t to the cup readings plotted on a log-linear plot. For observations over water, the winds have been referenced to a coordinate system moving with the mean current. Results are accurate to .1 m/sec. Observations were analysed only when the mean wind speed and direction were later found to be reasonably steady; i f a 5 minute average wind speed changed by more than 20 to 30% from the previous 5 minute average, conditions were considered to be non-stationary and the Run was not used. (iv) Wave Probe Two different systems were used for wave height measurements, both employed a capacitive type of wave probe. The probe consisted of a brass rod 150 cm long and .635 cm (1/4 inch) in diameter covered with a sleeve of teflon .0254 cm (1/100 inch) thick which was sealed at one end and attached by a water tight connection to a coaxial cable at the other. The brass rod formed one plate of a capacitor, the teflon the di e l e c t r i c and sea water the other plate; as the water height changed so did the capacitance. This rod was held i n a bracket arranged so that i t could be mounted vertic a l l y on the masts. The two systems used different methods of measuring the capacitance. One system is the equipment as modified by a fellow graduate student (Dobson, 1969) i n which the wave probe capacity is part of the frequency controlling network of a blocking oscillator; the FM signal was returned to the platform, demodulated with a VETTER Model 3 FM Recording Adaptor and then recorded on magnetic tape. The other system (Hume, 1969) used a constant current source to charge the capacity of the probe and measured the time taken to charge to -4 a given voltage level; the ratio of this time to the cycling time of 10 sec 84 was converted to an analog signal. The resolution is 1/1000 of the range; a sensitivity adjustment allowed use of different wave height ranges. The system has very l i t t l e d r i f t . The probes were calibrated before and after each f i e l d trip by holding the probe at different depths in a tank of salty water. The probe was frequently wiped with an oily cloth to keep the wetting characteristics constant; this improved the reproducibility of the calibrations. Figure 91 shows typical calibrations for the two systems. Both systems should give wave height to ±10%. (v) Water Height and Current Mean water height was measured by reference to bands of reflecting (for working at night) tape on one of the masts. The tape interval was 1/2 meter; the mean could be estimated to ±10 cm. The surface current was obtained by measuring with a stop watch the time taken for a piece of tissue paper to travel 7.5 m between two of the support-ing members of the platform. Direction was by reference to topographic features. The t i d a l current normally flows in an east-west direction with speeds up to 30 cm/sec. The measuring technique gave results to ±1 cm/sec. Some current speeds at various depths were measured with a calibrated Savonius rotor current meter. The speed did not vary with depth within the top meter. (vi) Air and Water Temperature Wet and dry bulb temperatures were obtained to ±0.1 C° from mercury in glass thermometers. These thermometers were mounted i n a sun screen that could be moved to a well exposed location on the platform. Water temperature was obtained by immersing a similar thermometer in the upper few cm of water below the platform. A l l thermometers were calibrated against a laboratory standard. 86 APPENDIX B ANALYSIS OF DATA The data analysis can be considered in two parts; that concerned with the s t a t i s t i c a l description of variables such as turbulence or waves and that concerned with the general description of the flow by mean values such as stress or s t a b i l i t y . The s t a t i s t i c a l analysis was done d i g i t a l l y , using an IBM 7044 or 360 computor at the U.B.C. Computing Center. Since the data was i n i t i a l l y recorded analog, conversion to d i g i t a l form was necessary. This was done using an analog to d i g i t a l converter (10 binary bits) designed and b u i l t at I.0.U.B.C. Up to ten channels of information could be sampled sequentially i n a cross-channel sweep, with a delay of approximately 45 micro-seconds between channels. The time between cross--2 -4 channel sweeps could be varied from about 3 x 10 to 2 x 10 sec. Provided that data is band limited to frequencies less than n , and max the data is digitized at a minimum rate of 2n , Blackman and Tukey (1959, max J p.117) show that a l l such data can be represented in frequency space with no aliasing. To reduce aliasing the signals were f i l t e r e d with matched linear phase s h i f t f i l t e r s before digitizing. Since most of the data were recorded at tape speeds of 7— ips and reproduced for digit i z i n g at 60 ips (which 2 increased a l l frequencies by a factor of 8), the cutoffs of the f i l t e r s were designed accordingly. In real time the f i l t e r cutoff (3 db down) was at about 20 Hz and the folding frequency, n^, for most of the data was set at 31 Hz (i.e., data were digitized at 500 Hz). The output of the analog to d i g i t a l converter was written on an IBM compatible magnetic tape using a Control Data Corporation 8092 Teleprogrammer at the U.B.C. Computing Center. The method used to obtain s t a t i s t i c a l information from the data on the d i g i t a l tape requires two steps. The f i r s t step uses a main program, 'FTOR', which takes the time series d i g i t a l data and creates a second magnetic tape containing the corresponding complex Fourier coefficients of the data. This step makes use of the 'Fast Fourier Transform' algorithm, 'FFT' (I.E.E.E. Transactions, Special issue on Fast Fourier Transform, June 1967). The second step takes these Fourier coefficients and produces spectra, etc., from programs b u i l t around a main program called 'SCOR'. These basic programs were written by J.R. Wilson and J.F. Garrett of this institute (see J.F. Garrett, 1970). In 'FTOR', the maximum number of data points per block that can be handled is 10,240. The actual number of points used i n each block is the number of channels times the number, N, of data points from each channel, where the l a t t e r has to be a power of 2. The FFT technique then produces complex Fourier coefficients for each channel for harmonic frequencies from the fundamental frequency of 1 cycle per block to n^, the folding frequency. This is done sequentially for a l l blocks in the entire Run. For a half hour Run, with 4 to 6 channels of data, the total number of data blocks, M, was typically 30 to 50. When analysing a time series of length NAt, where At is the number of seconds between data points, the Fourier coefficients are calculated for 2frr specific angular frequencies, to^ = ^ ^ J T J where r is an integer. If the time function to be analysed contains a frequency, to^, not equal to an to^, this produces a set of Fourier coefficients that peak near the frequency tu^  and f a l l off asymptotically as l/(to - to ) (Blackman and Tukey, 1959, p. 33) or as 2 l/(u) - tu^ ) for the spectral estimates. This is essentially the spectral 88 window for the i n i t i a l part of the analysis. To improve the window, the Fourier coefficients could be 'hanned' using the formulation — A' + — A' k r - l 2 r and f r - l 1 B' 2 r tt r+1 „ Br+1 (19) where A^ and B^ are the cosine and sine coefficients, respectively. The window for the hanned coefficients gives a spectral f a l l o f f rate of l/(w - W Q ) * Hanning was used only in a few special cases. The program 'SCOR* uses the coefficients from 1FTOR' to obtain the desired s t a t i s t i c s (see J.F. Garrett, 1970). The complex Fourier coefficient, til ttl C , for the r harmonic i n the m block, can be written as rm C = A + i B , (20) rm rm rm where i = / -1,. The spectral energy density, S, is then obtained for the frequency, n^ = r/NAt, by multiplying by i t s complex conjugate and dividing by 2. The values of S are then grouped into bands. For this work, half octave bands are used for the frequency range where there are more than 5 spectral estimates (5 harmonics) per half octave. However, since the individual spectral estimates at the lower frequencies are more than half an octave apart, the f i r s t 9 bands are preset to a larger bandwidth so that at least one coefficient is included i n each band. When averaged over a l l blocks the spectral energy density for a band, S, , can be written as 89 " 2 M A 2 + B 2 C / N N At \ 1 \ rm rm W ' 1 + r 2 - r x 2 • ( 2 1 ) r=r^ m=l w h e r e -W °b N At i s the geometric mean of the end frequencies and the 1 + r 2 - r± bandwidth, B . W . = N~At " Typically, the number of bands in each _2 analysis i s 23, with a minimum frequency of 2.7 x 10 Hz and a maximum of 3.0 x 10 1 Hz. The cospectrum between, say, data channels 1 and 2 was estimated from r2 M A, A„ + B, B r n s \ N At \ 1 \ lrm 2rm lrm 2rm ,„„,. C ° 1 2 ( n b ) = 1 + r 2 - r M _____ 2 ( 2 2 ) 5 r=r^ m=l the quadrature spectrum from r2 M i A„ B, A, B, n f \ N At \ 1 \ 2rm lrm lrm 2rm / 0 0 , Q U12 (V = 1 + r 2 - r M _____ 2~ ( 2 3 ) 5r=r^ m=l the coherence from / C o 1 2 2 ( n b ) + Q u12 2 ( nb>' Coh (n ) = - — • (24); / S b l ( V Sb2<V 90 and the phase from ©-^(i^) = tan -1 12 ( nb ) (25). Confidence limits were calculated directly by assuming each block to be an independent sample of the data. If the difference between the spectral estimate for each block and the mean over a l l blocks is assumed to have a Gaussian distribution, then these differences can be used to find the 95% confidence limits. The assumption is reasonable only i f M, the number of data blocks, is quite large, which is true for most of the Runs. Information on the spectral shape at lower frequencies was obtained from an additional program (G.A. McBean, 1970) which uses the same techniques as already discussed but with the block averages as data points or by digitizing data at a lower frequency. The data had to be corrected for instrument and f i l t e r response. A program written by Dobson (1969) was used for phase correction; i t also appropriately adjusted the Co- and Quad-spectra. Amplitude corrections were done by hand. Since some wave data were collected when there was a significant mean water current, the frequency had to be corrected. In order to do this the influence of the water current on the wave frequency, (rad/sec), was removed by means of the equation n (26) w w w 91 where U i s the mean current and co is the measured wave w m In addition to the spectral description, the flow was described i n terms of mean conditions including the surface stress and s t a b i l i t y . The surface stress was estimated by three different methods: direct measurement, the ' m e t h o d ' , or using a drag coefficient. The direct measurement method involved estimating the stress from the variance of uw ( T = -puw ); the velocity components were measured with a sonic anemometer. Coordinate rotation was used to correct for non-alignment with the mean wind by appropriately adjusting the calibration coefficients to be used i n the SCOR program. This method for finding the stress i s considered the most accurate. The '®2_i m e th°d' estimates the stress from a knowledge of the magnitude of downstream velocity fluctuations i n the i n e r t i a l subrange (Weiler and Burling, 1967). For this range, the spectral density of the downstream velocity fluctuations ($,,) at wave number k is approximately where e i s the rate of energy dissipation and K' is the Kolmogoroff constant, taken to be 0.5. Assuming that the rate of production of turbulent energy i s equal to the rate of dissipation and that the wind profi l e is logarithmic, *„(k) = K' £ 2/3 k-5/3 (27) and using equation (27), the str e s s ( T = pu^ ) can be w r i t t e n : 2 = 3.86 p ( £ ) z .2/3 5/3 (28). The value of 1(n) at the frequency n was taken from a plot of log $ 11 92 against log n for which a best f i t -5/3 line was drawn. For observations over water a drag coefficient, , was occasionally used to relate the 5 meter mean wind, i n cm/sec, to the surface stress by the formula T = P u * 2 = CD5 P U l 5 2 ( 2 9 ) -3 where = 1.2 x 10 . A plot of the drag coefficient as a function of wind speed, Figure 92, uses values of stress estimated from direct measurement and the $ method. As can be seen, a l l the methods are compatible and each suitable for estimating a value for the surface stress. The s t a b i l i t y of the ai r over water was estimated in terms of the gradient Richardson Number, Ri,-,. It can be written as R i r = r f - ^ Z 3 | (30) for 9 = T (1 - 0.61 h ), where g is the gravitational acceleration, T^ is the ai r temperature, U is ai r velocity, z is height, and h is specific humidity. 6 is a vi r t u a l temperature, which includes the effect of both a i r temperature and humidity on the buoyancy. Ri was calculated i n a difference form: the difference between the value at some height z and at z = .01 cm. It is assumed that at this lower height ul rt1 = 0 and T = T (surface water • U I . U I W temperature). Then 980 ( 9 z " ^ O l * R i G = -f^- —z-. ~ - A O T z (ln z + 4.61) (31) mean ^ l z ^ where the 6's were calculated from 93 6 = T A (1+5.0 x 10 4 h e l ) z A re s a' z (32) and e = T (1 + 5.0 x 10 -2 .01 w The relative humidity, h , and the saturation vapour pressure, e , were X*6 S 3. obtained from handbook tables using the mean wet and dry bulb and surface water temperatures observed during a Run. It is intended that this method of estimating the s t a b i l i t y should give at least an order of magnitude value for comparison within this study and with other studies. 94 APPENDIX C DATA SUMMARY TABLE V (see below) l i s t s the 'mean' conditions under which each of the Runs mentioned i n t h i s report were taken. Some data from u n l i s t e d Runs were used i n the more general summary p l o t s . They were taken under conditions s i m i l a r to those shown. The table i s presented i n two sections: part A gives data for measurements associated with a f l a t boundary, and part B gives data for measurements associated with waves. The Runs are i n numerical order w i t h i n each s e c t i o n . The s i t e abbreviations used were: 'S.B.' f o r Spanish Banks, 'L' f o r Ladner, and 'B.B.' f o r Boundary Bay. Measurements at the Spanish Banks s i t e are used i n both parts since observations were taken both when the water was shallow or absent and when the water was deeper and a c t i v e wave generation was present. The 'duration' i s the t o t a l time, i n minutes, of d i g i t i z e d data. Most of the Runs were o r i g i n a l l y about h a l f an hour i n duration, however instrument s a t u r a t i o n or d r i f t often necessitated using shorter pieces of data f o r a n a l y s i s . The value f o r the s t r e s s , -3 2 T = 1.25 x 10 u * » i s followed by a number, 1, 2, or 3, i n brackets. This number i n d i c a t e s the method used to evaluate the s t r e s s : (1) f o r d i r e c t measurement, (2) f o r the $ method, and (3) for the drag c o e f f i c i e n t . I f two instruments at d i f f e r e n t l e v e l s were used, the heights above the surface (or mean water l e v e l ) a r e given i n succeeding l i n e s . The water current at the Spanish Banks s i t e generally flowed i n an east-west d i r e c t i o n . L i s t e d a f t e r the magnitude of the current i s the d i r e c t i o n s given as E for a current flowing from the east, and W for flow from the west. As mentioned i n Appendix A, for observations over water, the winds have been referenced to a coordinate system moving with the mean current. For Runs i n s e c t i o n B of the table, the 95 difference between the 5 meter wind and phase speed of the waves at the peak of the wave spectrum, ( u| - C ), is given; the frequency of the peak is given in brackets after this velocity difference. TABLE V MEAN DATA FOR RUNS A. FLAT BOUNDARY RUN DATE SITE t t (1) U l 5 (2) WIND DIR (3) T (4) z (5) U (6) h (7) U w (8) 72/1 Mar 21/69 S.B. 18 6.8 270° -0.01 0.650(2) 2.0 6.5 1.0 0 72/2 Mar 21/69 S.B. 19 6.2 270° -0.02 0.511(2) 2.0 6.0 1.0 0 73/1 Apr 3/69 S.B. 16 8.2 100° - 0.974(2) 2.0 7.8 1.0 0 73/2 Apr 6/69 S.B. 10 3.9 260° - 0.287(2) 3.5 3.9 0 -73/3 Apr 6/69 S.B. 11 5.5 260° - 0.221(2) 1.00 4.5 0 -110/1 Aug 8/68 S.B. 27 7.2 285° - 0.870(1) 5.5 7.1 0 -110/2 Aug 8/68 S.B. 26 7.2 285° - 0.910(1) 4.0 7.4 0 -120/1 Aug 8/68 S.B. 22 6.8 270° - 0.724(1) 3.4 6.5 0.75 0.40W 120/2 Aug 8/68 S.B. 30 6.5 280° - 0.553(1) 4.8 6.2 1.2 0.40W 121/1 Aug 8/68 S.B. 27 5.8 275° - 0.479(1) 1.5 5.4 2.2 0.63W 132/2/1 Sept 4/68 B.B. 9 3.7 280° - - 0.30 3.4 - -133/3/1 Sept 17/68 B.B. 26 6.1 270° - - 1.00 5.1 - -133/3/2 Sept 17/68 B.B.' 17 7.0 270° - - 1.00 5.9 - -137/1 Sept 17/68 B.B. 18 5.8 270° - - 0.30 5.0 - -A. (continued) RUN DATE SITE t t "Is. 137/2 Sept 27/68 B.B. 25 4.1 141/2/1 Apr 3/69 S.B. 18 7.8 141/2/2 Apr 3/69 S.B. 9 7.8 141/3 Apr 3/69 S.B. 19 7.6 142/1 Sept 27/68 B.B. 16 4.3 165/2 Mar 12/69 S.B. 13 4.4 172/1 Mar 20/69 S.B. 12 5.3 172/2 Mar 20/69 S.B. 20 5.9 173/1 Mar 20/69 S.B. 19 6.8 173/2 Mar 20/69 S.B. 22 7.4 186/3 Apr 7/69 S.B. 15 3.6 186/4 Apr 7/69 S.B. 9 3.7 186/5 Apr 7/69 S.B. 8 2.7 TABLE V (continued) WIND DIR ^G T z U h U w 295° - - 0.30 3.4 - -120° -0.01 0.900(2) 2.0 7.5 1.0 0 120° -0.01 0.900(2) 2.0 7.5 1.0 0 130° -0.01 0.900(2) 2.2 7.3 1.0 0 280° - - 0.30 3.5 - -280° -0.02 0.277(2) 2.5 4.3 2.0 0 270° -0.04 0.367(2) 1.5 5.0 1.5 0 270° -0.04 0.518(2) 3.0 5.7 1.0 0 280° -0.03 0.611(2) 2.5 4.6 6.5 6.8 0.7 0 280° -0.02 0.941(2) 2.0 2.5 7.0 7.1 1.0 0.2; 280° -0.02 0.156(2) 2.5 3.6 0.5 0 270° - 0.196(2) 3.0 5.1 3.6 3.7 0 -270° - 0.136(2) 1.25 6.75 2.6 2.7 0 -TABLE V (continued) A. (continued) RUN DATE SITE t t "Is WIND DIR T z U h U w 196/1 Jul 15/69 S.B. 13 3.7 270° -0.12 0.189(3) 0.46 1.05 -0.20 0 196/2 Jul 15/69 S.B. 20 3.4 270° -0.29 0.160(3) 0.46 1.05 -0.20 0 196/3 Jul 15/69 S.B. 8 3.7 270° -0.13 0.189(3) 0.20 0.76 - 0.30 0 200/2 Jul 17/69 S.B. 23 4.7 260° -0.01 0.302(3) 4.0 4.5 4.6 4.6 2.0 0.64W 205/1 Jul 19/69 S.B. 30 3.8 260° -0.04 0.197(3) 5.0 3.8 1.0 0.30W 205/2 Jul 19/69 S.B. 23 3.8 260° -0.03 0.197(3) 5.0 3.8 1.0 0.46W 318/1 Aug 23/69 L 30 4.8 270° - 1.23 (1) 0.40 3.5 - -318/2 Aug 23/69 L 31 4.8 270° - 1.17 (1) 0.40 3.3 - -319/1 Jan 27/70 ' L 24 9.9 270° - 3.84 (2) 0.32 6.7 - -319/2 Jan 27/70 L 22 9.8 270° - 3.84 (2) 0.32 6.7 - -320/1 Jan 27/70 L 32 8.8 270° - 1.84 (2) 0.32 6.0 - -320/2 Jan 27/70 L 29 7.7 270° - 1.16 (2) 0.32 5.3 - -425/1 Jan 27/70 L 24 6.1 270° — 0.862(2) 0.32 4.1 • — — A. (continued) RUN DATE SITE 425/2 Jan 27/70 L 426/1 Jan 27/70 L TABLE V (continued) tt UL WIND Ri T z U h U 5 DIR 23 5.2 260° - 0.756(2) 0.32 3.5 32 6.1 270° - 0.862(2) 0.32 4.1 vo B. NEAR WAVES RUN DATE SITE' tt -Is WIND DIR 60/1 Mar 27/69 S.B. 16 4.3 250° 60/2 Mar 27/69 S.B. 12 4.7 250° 60/4 Apr 2/69 S.B. 22 4.1 130° 80/3 Mar 27/69 S.B. 17 3.9 260° 119/1 Apr 2/69 S.B. 15 4.1 90° 119/2 Apr 2/69 S.B. 19 4.7 120° 119/3 Apr 2/69 S.B. 1 17 4.8 120° 164/1 Dec 18/68 S.B. 14 2.6 290° 164/2 Dec 18/68 S.B. 16 4.1 340° 167/1/1 Dec 14/68 S.B. 19 8.1 115° 167/1/2 Dec 14/68 S.B. 18 7.2 120° 167/2 Dec 14/68 S.B. 14 7.9 100° TABLE V (continue--0.01 0.274(3) 0.001 0.331(3) -0.05 0.252(3) -0.01 0.228(3) -0.05 0.252(3) -0.04 0.331(3) -0.04 0.344(3) -0.19 0.101(3) -0.09 0.252(3) 0.02 0.985(3) 0.02 0.776(3) 0.03 0.935(3) z U h 0.40 3.5 2.5 0.90 3.8 0.50 3.9 2.5 1.00 4.1 0.30 2.9 3.0 0.80 3.3 0.30 3.3 2.5 0.80 3.5 0.40 3.5 3.0 0.90 3.7 0.30 4.1 3.0 0.80 4.3 0.30 3.8 3.0 0.80 4.2 0.50 2.2 3.5 0.60 3.5 3.5 0.50 6.3 3.5 0.50 5.7 3.5 0.30 5.8 3.5 U U L - C w ' 5 (9) 0.22E 0.2(0.32) 0.22E 0.6(0.32) 0 1.7(0.65) 0.22E -0.2(0.32) 0 1.8(0.70) 0 2.3(0.65) 0 2.2(0.60) 0 -2.3(0.25) 0 -1.5(0.18) 0.24E 5.0(0.50) 0.24E 4.1(0.50) 0.24E 4.8(0.50) TABLE V (continued) B. (continued) RUN DATE SITE t t u L WIND Ri T z U h U u| - C 5 DIR G W 5 167/3 Dec 14/68 S.B. 6 6.4 125° 0.05 0.615(3) 0.30 4.8 3.5 0.24E 3.3(0.50) 173/3 Mar 20/69 S.B. 14 3.6 240° 0.02 0.194(3) 0.90 3.1 2.5 0.30W -0.5(0.30) 1.40 3.2 (1) Duration (minutes) (2) Mean wind at 5 meters (m sec ^) (3) Gradient Richardson Number -2 (4) Surface Stress (dynes cm ) (5) Instrument height (m) (6) Mean wind at the instrument height (m sec ^) (7) Water depth (th) (8) Current (m sec ^) (9) u| _ - C at the peak of the wave spectrum (m sec 102 Figure 1. P r e s s u r e Instrument used to measure the s t a t i c pressure fluctuations within the turbulent flow (a) assembled ( b ) ' • . ' i t i cylinder removed Figure 2. Probe developed for measuring stat i c pressure fluctuations within the turbulent flow o O J 104 PROBE 'E' .04 • x to a> .c o c .02 M / .2 * x x PORTS .6 .8 1.0 X (inches) 1.2 1.4 4~ 1.6 .04 JZ a c .021 PROBE 'F' / t PORTS •X \ i.6 .4 6 .8 1.0 X (i nches) 1.2 1.4 PROBE 'G' .04 OJ / .02 / rsi * i * PORTS .2 .6 .8 1.0 X ( i n c h e s ) 1.2 1.4 x SIDE 2 • SIOE I 1.6 Figure 3. Cross-sections of the disks of probes E, F, G IO°r u K o ° | CL -5<>\ - I 0 ° L IB 15 l<0 i£ to 16 23 U=3.6 m-sec"' 25" lie 68%^-\ '8 -95% •10° 18 _5f 0° YAW 28 /a it-16 21 10° 10° 5° u I—o° CL -5° - I O o L h7 * 5 U = 8 . 9 m - s e c -3 4 68%^ /- > -95% 4 5" 7 9 8 c 9 -* *-• - 1 0 ° 0 ° 1 0 ° YAW Figure 4. Dynamic pressure noise test for Probe E at different wind speeds IO#f U 19 - I O o L 15 14 IS U=3.7m-sec'' /4 14 68%-v ^  Jl^f95% • ~ is-16-ZD 23 -io« IS IS 19 2 8 i _ N \ <* ) y / o° YAW zo lS~ 12 17 I0« zo 10° 5° u h- 0° CL - 5 ° - I 0 ° L 3 U=8.9 m-sec'i 68% 8 « --10-I r 8 95% 0° YAW 8 to II 10s 10°-5° ' U -5« -10° IS U=6.2m-sec-' IS IS ll io ll 68%-v i i / 9 5 % / ^ V O N / ' • \ \ /5-18 19 I i /3 -10" 0 ° YAW IS~ 12 II 12 is-ll —i io-/2 20 Q O , o o P R O B E F H II I' lb>-95% ! ' l - ^ > 6 8 % I-I' It II II 4 6 8 VELOCITY (m-sec") io Figure 5. Dynamic pressure noise test for Probe F at different wind speeds o ON POWER SUPPLY (in recording hut) lOKhz Reference £ to probe ^ Ipressure | f" Inputs to reference Diaphragm PRESSURE SENSOR (mounted with probe) ± 5 V D C SIGNAL CONDITIONER (in recording hut ) 1—i Figure 6. Schematic of the Barocel transducing system o Figure 7. Barocel pressure transducer and reference volume in their container r-» o C O X I I I I I 1 1 0 20 40 60 Distance from Transducer Case (cm) Figure 6. Results of wind tunnel test for the dynamic pressure distribution in front of the transducer case g Reference B a r o c e l Drum w o > Probe V i brat ion Generator Transducer et a l -cables CD o B a r o c e l S y s t e m Strip-chart Recorder Osci l loscope o o o o Power Amplifier Signal Generator Figure 9. Arrangement used for calibrating the pressure instrument for amplitude and phase response Al p late- 5 g a l l o n d r u m rubber d i a p h r a g m r e f e r e n c e b a r o c e l ^ p r o b e •'r i v ibra t ion g e n e r a t o r t n J n n / n J ) ) / ) ) ) ) ) ) n I 0 cm Figure 10. Detail of the drum used to create a sinusoidally varying pressure 112 I2VDC i/p o f f on - I 2 V 0 C IW 2 5 & IOW 2 N 4 5 7 4 2NI53I I20& 2 W 5$, IW 2 N 3 0 5 4 A 6 C 2 o/p 5 & A6C2 2 N 3 6 I S I20& 2W Figure 11. Circuit diagram for power amplifier used to drive the vibration generator Figure 12. Sample frequency calibration of the pressure instrument (probe and transducer) Figure 13. Arrangement used for calibrating the pressure instrument in s i t u 1.0 .8 u CO o co .6 UJ or UJ Q L 4 _J • Q. A M P L I T U D E R E S P O N S E j — i i 1 1 1 1 - i — t — i < P H A S E R E S P O N S E i i i 1 1 1 J — i » 1 1 1 1 o / 20' 0° UJ CO o Q. CO UJ rr UJ co < 20° J o a> n (Hz ) Figure 14. Sample frequency calibration of the system used for the surface pressure measurement 116 ure 15. Spectral comparison of the st a t i c pressure measured in the a i r and at the surface; the separation was 40 cm vertically. These measurements were taken at the Ladner s i t e . IO3 1 0 2 N X 'e i o ' u CM* c >» " O 1 0 ° io-» 117 * AIR •SURFACE I 95 % c l . -I L . 0 1 \ * x •1 \ . \ A \ \ x" v I M I j—• • i • • i \. j — i — i — i i 11 iii n(Hz) io F i g u r e 16. S p e c t r a l c o m p a r i s o n o f t h e s t a t i c p r e s s u r e measured i n t h e a i r and a t t h e s u r f a c e ; t h e s e p a r a t i o n was 32 cm v e r t i c a l l y . These measurements were t a k e n a t the L a d n e r s i t e . 118 .8 LU O -Z LU or LU X 0.4 CJ + A o $ ** * + + + + O A "319/1 -319/2 --320 / I •318/1 + 318/2 , i i P-P o X o X o A X J — I I I I 11 n ( H z ) 1 i i i i i io •20°r LU I * ° ? . - * * • ; io °- n(Hz) 2 0 o l Figure 17. Coherence and phase between the s t a t i c pressure measured i n the a i r and at the surface. These are the Ladner Runs. .* 137/1 (30cm) 119 " A I R • S U R F A C E I 9 5 % c. I. n ( H z ) ure 18. Spectral comparison of the st a t i c pressure measured ln the a i r and at the surface; the separation vertically, in meters, is given in brackets after the Run number. These measurements were taken at the Boundary Bay s i t e . 120 I.Or .8 p-p e A K X O 6-4 OX * X 0 UJ o UJ or UJ X 4 i o 0 132/2/1 * 133/3/1 A 133/3/2 x o A o .01 - I — I — ' • 1 1 1 n(Hz) J_I_L X X I I I 10 - 2 0 ° UJ CO < 0 ° X a. 2 0 o L * ° a JA 10 n(Hz) Figure 19. Coherence and phase between the stati c pressure measured in the a i r and at the surface. These are the Boundary Bay Runs. / \ 165/2 ° U P P E R x L O W E R A Z = 0 A Z = l . 8 m 121 n ( H z ) Figure 20. Comparison of pressure spectra measured simultaneously at two different heights. Az is the difference in height, given in meters. IO 1 10° X O A + o X * 110/1 0 110/2 * 120/1 + 120/2 * 121/1 1 9 5 % c . l . 0A io- J I ' • ' M i l J L i i . ' • ' I 10" io- IO"3 IO" 2 kp = t u / U l 5 ( c m " 1 ) 10" Figure 21. Nondimensional pressure spectra. Observations taken over water. 123 10 10 IO"2 kp = cu/UI 3 (cm- ') ( a ) 10 -I 10' 10° + + * 319/1 * 319/2 * 320/1 * 3 2 0 / 2 * 425/1 * 4 2 5 / 2 I 9 5 % c . I . _i i i • • • ' I i i J L I I 10" IO' 3 IO' 2 k p = o j / U J 5 ( c m - 1 ) ( b ) 10° ure 22. Nondimensionalized pressure spectra. Observations taken over (a) water (b) land 5 -CM ^ 4 O II t = 2 <X X X • 9 • X 9 X x using Sonic s tress » using Cp or </>n method 01 i , i • • i i i 0 I 2 3 4 5 6 7 Z (m) Figure 23. Summary of the nondimensionalized pressure spectra. Values plotted are k I T ( k ) / ( p 2 u . 1 1 ) at a k of M IO" 2 cm"1. 125 ure 24. Normalized pressure spectra normalized by their variance. CM * •6F 1 0 ° 3'Z 2 z e s 4 4 A S 3 * 5 • » - ! 2 i* 3 ICH 4 4 3 * 5 12 ^ 3 4 X . 2i x +- o o o X ° o s l * 5 x 34 % + * § 4 * lo-+ X , . 4 u-w x HO/1 > - < » 1 1 0 / 2 a - - 1 2 0 / 1 4-* 1 2 0 / 2 * - + 1 2 1 / 1 io-1 0 -3 - l i_ • I I 1 0 ,-2 10 ,-l -I l_ f = n z / U • * • i io ( 1 1 1 0 ' F igu re 25 . Nond imens iona l i zed u and w s p e c t r a i - -ON 127 A 10° CM * AX ° X X ° ° + * X a 'OH to 10 -Il x + o " 110/1 « 1 1 0 / 2 • 1 2 0 / 1 A I 2 0 / 2 +121/1 o 10-21 • 1 • — • — i i i i I 1 1 1—i—i i i i I i i i i i i i i I i i _ i i I 0 " 3 IO" 2 10"' 10° f=nz /U Figure 27. Nondimensionalized uw spectra re 28. Comparison of the spectral slope of pressure spectra f = nz/U Figure 29. Comparison between the nondimensionalized variance of the pressure and of the velocity components for Run 120/1 for different frequency bands M U> O k = CL>/ UL (cm-1) ure 30. Nondimensionalized pressure spectra. The curve is the mean of data given in Figure 21; the dashed lines are extrapolated from the solid curve. 132 I.Or UJ o z UJ or ui o o * 173/2 * 2 0 0 / 2 + 173/1 * 186/4 <-186/5 A Z = . 5 m AZ=.56m k p = c u / U I 6 ( cm- 1 ) 2 0 ° 0 - 2 0 ° ui co < n o CL o 1 t M t + - A A. IO"3 t ' 0 • ' ' ° " _1_ I I I I 1 IO'1 kp = o j / U I 5 ( c m " ' } Figure 31. Coherence and phase between two pressure measurements with various v e r t i c a l separations Phase positive means p upper leads p lower. 20« Ixl to < 0°' X Q_ -20O L 10 -3 » x 10 -2 10 - i k =o>/UI5 (cm"') Figure 32. Coherence and phase between two pressure measurements with various crossstream separations 135 20°r 70 120 170' 140' -90« -40* LU co 10° < X °- 6 0 110' 160° 150° I00o1 -50° 0 ° r " 319/1 • 319/2 * 320/1 + 3 2 0 / 2 • 425/1 * 426/1 all have AZ=.3m D = 2m D=4m D=0m &! & *> t * A 4 4, [ $ .1 JL r] ( H z ) ( b ) io Figure 33(b). Phase between two pressure measurements with a dowEnwind separation I 0 4 r 136 IO3 -£ o c \ in " ? I 0 2 CL 10' D a t a p l o t t e d for C 0 H E R E N C E = .I4 Xp(x) Xp(y) Xp (Z) IQ° _L • • i • • • • 10' 10' 10* PROBE SEPARATION (cm) I 0 ; Figure 34. Fixed coherences between two pressure signals for various probe separations. The values plotted are for a coherence of 0.14. .8 o .6 A A A A A A u-u w-w * * 146/1 AZ=.9m ° • 139/1 AY = Im ° • 130/1 AY=2 .4m UJ rr UJ 8 .41 • a o • 0 1 — i — i i i i 11 .1 J 1 1 1 I I I I I o ° ° S o „ • • D * a i OB D e> ~ o • • • r I f n i l n(Hz) 10 Figure 35. Coherence between two velocity measurements with different separations I 0 4 r I O 3 E o 3 II > I 0 2 r -10' I I C O H E R E N C E . 8 / 138 D a t a p l o t t e d for C O H E R E N C E ^ . 1 4 U-U w-w Mx) O A My) X » ( z ) i o 0 ' -i i i i_ • • 1 1 1 1 1 I0 V io 1 i o 2 S E N S O R S E P A R A T I O N (cm) 10* Figure 36. Fixed coherences between two velocity signals for various sensor separations. The values plotted are for a coherence of 0.14. .9 .8 .7 LU .6 O LU .5 or LU X .4 o o . 3 .2 .1 0 •120° A X X t o p-u J I L + A xo W peak * 120/2 • 121/1 * 110/2 + 110/1 • 120/! © X 139 J l _ I I + • A X 0 AX Ay 1 * * 4 J I I I i l l 0 - g I io- 10 -2 k=u ) /U (cm-1) 10 •I •160° 160° I 2 0 e 8 0 e LU C/)40« < X CL 0° • 4 0 * 4- * + 4* 1 I I I _ l I I I I I I I k = w / U (cm-1) -2 10' Figure 37. Coherence and.phase between p and u, u measured with a sonic. Height of observations ranged from 1.5 to 5 . 5 meters. Phase p o s i t i v e means p leads u. .8 LU O .2 -p-w r * 120/2 o 121/I A * + 110/1 + * 110/2 120/1 •o A* o + A + + LU o . • OC X J ? A AX „ , X LU 4 - o S ' + * + A O A + • O + A + f * * AX X o e 0 < X . A A * « + X • v. ^ J T + x V + A 140 • A* W peak ' % I I + i * * # J — i i i i i I i i i i i i i i I i i i i * i i i i , IO"3 IO"2 10"' k=w/U (cm*1) 0° -40° LU £-80* X CL -120° -160° 160° X + A*. + A *X t ° O ° *0 i° A * Ajf. H»X 4* A. ^ * • + +X + A O J I I I L_l_ -I • • • ' J _ • • • I0" 3 I0" 2 k = a j / U (cm-1) 10" Figure 38. Coherence and phase between p and w, w measured with, a sonic. Height of observations ranged, from 1.5 to 5.5 meters. Phase positive means p leads w. . 8 -U J o .6 U J or ui X .4 O o 0 p-u o * o X c + e x 172/1 o 172/2 + 173/1 • 173/2 o o x T « © 0«X 141 O e l I I W peak + + I I X o» J • • i i.. I i i i i io"3 , / , . I 0 " 2 k = c u / U (cm-1) 10" -160° 160° 120° U J 8 0 ° to < X 4 0 ° Q. - 4 0 0 ° A + + + • o X c x f .x . f * © + + * ° + • ° o + „+ + •x O X J l _ • • • 1 * W X J I I ' ' ' • -I I I I I I I 10 10 k = a>/U (cm-1) -2 10 -i Figure 39. Coherence and phase between p and u, u measured with a hot-wire. Height of observations ranged from 1.5 to 3 meters. Phase positive means p leads u. .8 LU .6 o LU or LU X O .4 o Ho p-u 0 X +• 0 x x° • o • + X 4. 0 f X + • * 141/2/1 ° 141/2/2 • 1 4 1 / 3 + 73/1 * X o X 'W peak ' 4& J • • i • • > I I I I i . f -160° 160° 120° LU if) 80° < X CL 4 0 ° 0 ° - 4 0 o L 10" 10" 10' k=CL)/U ( c m " ' ) + • o +x + x + x *S > g +8 £ - * — « ? - » - r x T— •nr T — I I I I 10" 10 -2 10" k = C U / U ( cm- ' ) Figure 40. Coherence and phase between p and u, u measured with a hot-wire. Height of observations was 2 meters. Phase positive means p leads u. .8 u-w 143 UJ L * O rr .41 o o .2 * 1 2 0 / 2 A+ + o 121/1 * 110/2 + 110/1 4 _° • 120/1 X + >A • 4 • X o • + + I A « » * a o a x • * *» X A A X o A O * X 4- o W peak x x A • + ' i i i 1 1 1 1 1 i i i i i 11 . f 4 .. i IO" 3 I O - 2 10"' k=a>/U ( c m ' 1 ) 160* UJ < ! 8 0 e X Q. -I60°h o J I I, •A • a f ml A I I lAt o -I I I I I + • x A + X A £ X A * • • 1 A . ° O x * * • o 10"' °. k=«/U ( c m l • I 4 0 O L Figure 41. Coherence and phase between u and w, velocity components measured with a sonic. Height of observations ranged from 1.5 to 5.5 meters. Phase positive means u leads w. 144 Z (m) Figure 42. Wavelength of the pressure fluctuations associated with the p-u phase transition, as a function of observational height. The broken line is the measured scale size. 145 3 2 0 / 1 * 3 0 c m A S u r f a c e n(Hz) 319/1 o 3 0 c m • Surface n(Hz) Figure 43. Coherence between downstream velocity, u, and two pressure measurements. One pressure sensor was beside the u sensor, one was at the surface, 30 cm below. •10 £ o i E u to a> c >» T 3 c - 2 0 CL ok-,. - 3 0 • 4 0 L T — i — r .01 n(Hz) 146 I I I I A „-\ I \ x 110/1 ° 110/2 • 120/1 * 120/2 +121/1 Figure 44. Spectra of pw , 5 R = pw//)uwU .10 .05h Z ( m l Figure 45. Ratio of the pw and uwU terms of the integrated net energy budget equation. 1A7 10 * 51 - I 1 I T ~ l -r-l • .1 -i i i kz • • • • io I I I I T 1 1 I I I I I kz i — i — i — I I I I I io = --il c -.21 -.4 * 7 3 / 3 * 72/1 o 141/3 * 73/1 + 141/2/2 Figure 46. Spectral distribution of the energy flux, by pressure forces, from the u velocity component. The integral given is for kz from 0.05 to 20. 148 i.Or •0-V to • T - ' T I -i—i ' • • ' I kz - i 1 — i i ' i i i i 10 T 1 I I I I 1 1 — I I I I kz T 1 1 — I I I I 10 - .2 -.3 ^° x \ \ V - f t * 1 6 5 / 2 ° 172/2 4 172/1 + 173/2 • 141/2/1 - udp/dx/pul/KZ Figure 47. Spectral distribution of the energy flux, by pressure forces, from the u velocity component. The integral given is for kz from 0.05 to 20. Figure 48. Pressure, velocity, and wave spectra for Run 173/3. 10 x 10° CM £ 10 10 -2 10' 150 eq.peok J 6 0 / 4 (1650-1712) "119/1 ( 1 7 4 5 - 1 8 0 0 ) "119/2 ( 1 8 2 0 - 1 8 4 0 ) * 119/3 ( 1 8 5 0 - 1 9 0 8 ) .0) • • • • n ( H z ) io ure 49. Wave spectra of Data Group A. The time of start and end of each Run is given in brackets. Figure 50. Pressure, u velocity and wave spectra for Run 60/4. Figure 52. Pressure, u velocity and wave spectra for Run 119/2 Figure 53. Pressure and wave spectra for Run 119/3 1.0 .8 LU O LU or LU x .4 O o 0 01 1 6 0 V 180° - 1 6 0 ° LU co < - l 4 0 ° CL -120* - I 0 0 ° L * 6 0 / 4 * 119/1 o 119/2 * 119/3 o A o x A o o A 0 • e o o eq. peak A o J 1 1 L J 1—I n(Hz) J — I — t I I I I •x 1 a * 'xa * l l I I 1 • x w • © X 0 X 155 J ' ' ' - l — L n(Hz) a O m o re 54. Coherence and phase between the lower pressure sensor and the waves: Data Group A. P L - H phase positive means p L leads n. 156 IO, .8 LU LU ce LU X o ° .4 .2 O e x P-P o .1 X H J I L eq. peak i i i i f . o X o X o * 6 0 / 4 * 119/1 ° 119/2 * 119/3 o o o n ( H z ) 10 20° LU £ c X CL - 2 0 ° J I L X x X AO - I — * I I I I o • « » n ( H z ) 10 Figure 55. Coherence and phase between the two pressure sensors: Data Group A. PT-P.. phase positive means p T leads p . u L r u 157 .4 u-V LU O LU or LU X o o .2h _2_U o J l_ 0 o X J L x c J_L X o .1 n(Hz) io 20* X o • 6 0 / 4 x 119/1 ° 119/2 100* LU CO < X Q_ eq.peak H • 180* J i • • • • J 1 i i J 1 I I L 10 n(Hz) • i o o e L Figure 56. Coherence and phase between the u velocity and waves: Data Group A. u-f| phase positive means u leads r\. eq. peak 158 * 167/1/1 (1209-1228) • 167/1/2 (1228-1246) 167/2 167/3 (1252-1306) {1312 - 1318 ) X a o A X \ I I I I i l l ' I I I I I I I O \ I I I I 10 .01 n ( H z ) ure 57. Wave spectra for Data Group B. The time of start and end of each Run is given in the brackets. Figure 58. Pressure and wave spectra for Run 167/1/1 Figure 59. Pressure and wave spectra for Run 167/1/2 Figure 60. Pressure and wave spectra for Run 167/2 167/3 162 Figure 61. Pressure and wave spectra for Run 167/3 163 F o u r i e r C o e f f i c i e n t i i i i i i i J i _ J 0 0.2 0 . 4 0 . 6 0 . 8 n ( H z ) Figure 62. Amplitude of the Fourier coefficients for pressure and waves of Run 167/3 I.Or .8 A 167/l/l •167/1/2 ° 167/2 "167/3 o p-77 • LU O LU or LU 4 X o o o o A A o » • 0 A .2 « o A X • o A A 0 • x X A e • o - I 1—' • 1 • • eq.peak - H I o A S p o 1 I a i n ( H z ) 1 0 160° LU 1 8 0 CO < X - I 6 0 ° Q_ -140° - I 2 0 ° L -1 L x A — " A - 1— # l 1—' ' • » i A? o m X A X -1 1 ' • ' • n(Hz) 1 0 Figure 63. Coherence and phase between the pressure and p-H phase positive means p leads T). Data Group B. Figure 64. Pressure, u velocity and wave spectra for Run 164/1 Figure 65. Pressure, u velocity and wave spectra for Run 164/2 P-7? *I64/I (1408-1422) ° l 6 4 / 2 (1612-1625) 167 .8 x o 160* o Ul rr UJ x o o o x o x o H .2 o X eq. peak o o 0 ' -i-iJ 1 1 — I io n ( H z ) w 180° < CL - I 6 0 ° f -i—i—IIII -lS 'X ' n • • • • I - I 1 1 l l l i l , O X x x o o n ( H z ) io •I40° L Figure 66. Coherence and phase between the pressure and the waves: Data Group C. p-n phase positive means p leads n 1.0 .8 .6 UJ o UJ cc UJ X o o .4 .2 :oi * 164/1 o 1 6 4 / 2 U-77 o « H -I OJ 1 I * 1 1 1 eq. peal^ x 1 -I 1 1 I I L. n ( H z ) 168 —i 1 0 UJ co < X CL 160° 180° -160° - I 4 0 O L j — 1 1 • • -sr o 6 • ' i i i i o X o x n ( H z ) 1 0 Figure 67. Coherence and phase between the u velocity and the waves: Data Group C. u-n phase positive means u leads n. . 169 * 8 0 / 3 (MI2-II29) ° 60 /1 (1140-1156) + 6 0 / 2 (1214-1226) n ( H z ) Figure 68. Pressure, u velocity and wave spectra for Data Group D I.Or .8 LU o .6 LU CC LU X o O .4 .2 A 1 7 3 / 3 - p L * 8 0 / 3 o 60/1 + 6 0 / 2 X J L . X A O _I_L X + o p-7/ o A + O + X 0 'H eq.peok X o ° * o X 170 o X x 1 • • • .+ I I I I n ( H z ) 1 0 160' LU co 180° < X °- -160° - 1 4 0 ° - 1 2 0 ° -i 1 i _ o + -o-A + X x + o A O A+ JLAj_i£i_iaL o x O O + n ( H z ) 1 0 Figure 69. Coherence and phase between the pressure and the waves: Data Group D. p-r) phase positive means p leads n 171 LU O LU rr LU X o o .2 u-77 o X + A t A o A ' • • • • I X X o X A H * t t eq. peak X i I I I I + X * 60/1 * 6 0 / 2 * 8 0 / 3 ^ 173/3 * X x o .01 n ( h z ) 140° LU CO < 180° Q_ - I 4 0 ° L 1 L 1 A i • + 1 1 1 • 1 1 I 1 B 1 9 .1 + 1 i> ' ' ' 1 ' ' * & A 1 * O X n (hz) Figure 70. Coherence and phase between the u velocity and the waves: Data Group D, u-n phase positive means u leads n. 1.0 x x ex CL ,5 CL 2 3 Z (m) Figure 71. Ratio of measured to predicted pressure amplitude for propagating with no wind. : & 5 waves CM I e o t/J Q> C E o 6 II N Q? -A0.4IHZ 0.55 Hz 0.73 Hz 0.98 Hz 1.30 Hz X X J I I L • .41 Hz + . 55 Hz * .73Hz ° .98 Hz x 1.30 Hz 4 UL/C F i g u r e 72. p (n) a t v a r i o u s c o n s t a n t f r e q u e n c i e s f o r d i f f e r e n t v a l u e s o f u],-/C. The v a l u e s p l o t t e d a t w u|^/C = 0 a r e f o r t h e p o t e n t i a l f l o w s o l u t i o n . 177 I 2 0 e Data Group x A o B • C * D o x x A O O x X A X OO - 1 4 0 ° LU CO < X -160°} Q_ x o ox OO 180* A • A A 160' J i_ - J 1 1 L . I 4 0 Ul 5/C ~5~ Figure 77. Phase s h i f t between pressure and waves at various values of u| /C Phase p o s i t i v e means pressure leads waves. I I I ! M l 1 \ 1 1 | I \ \ 3 rrvsec"' \ .6 .8 1.0 Wave Frequency ( Hz) Wave amplitude and c r i t i c a l height for constant u| 5 plotted for different wave frequencies 179 Figure 79. Spectral distribution of the approximate energy flux to the waves, calculated using the pressure measured above the wave crests. .8 LU O LU CC .4 LU X o o 0 o X a A p-u •»-A J l_ J L O O X CJ X H o A X • I A * 119/1 o 119/2 • 6 0 / 4 * 164/1 • 164/2 o 173/3 o o X 0 A s o X + o X* 180 n(Hz) io - 1 6 0 ° 160' 120° 8 0 ° LU co 4 0 ° X Q_ 0 ° 40* - 8 0 o L A _I3_ LAxb A 1-2. x a o X X o o X o X o _l—1,1 I I I •f n(Hz) io ure 80, Coherence and phase between pressure and u v e l o c i t y measured near waves Phase p o s i t i v e means pressure leads v e l o c i t y . .8 LU O 2 LU . cc .4 LU X o o A + X A X A • A A o p-u H i i i o i A O X I I hi X t X 4 ? + o X * 8 0 / 3 * 6 0 / 1 * 6 0 / 2 * 8 0 / 2 + 6 0 / 3 + o X A ct X A 181 A X J 1 L X OA t a? n(Hz) 10 - 1 2 0 ° - 1 6 0 ° 1 6 0 ° 1 2 0 ° 8 0 4 0 ° 0 ° -40<>L LU CO < X 0- one* + x A X * A + X I*I 1 1 *l - e - A - i a . — A A o 4 X A X o A O X o A A + X + I I I J L , ' ' • • 10 n(Hz) F igu re 81 . Coherence and phase between p ressu re and u v e l o c i t y measured near waves. Phase p o s i t i v e means p ressu re leads v e l o c i t y . I4r 12 10 182 • Over w a v e s * Over a f l a t s u r f a c e E 8 / / / / x x / / X X X Z=L •J 1 1 1_ ' 0 4 z(m) — i 8 Figure 82. Wavelength associated with the p-u phase transition . Nondimensional energy flux from the u velocity component, measured near waves. 184 I N T E G R A L x 8 0 / 3 . 2 2 ° 60/1 . 3 4 * 6 0 / 2 . 3 7 -udp/dx/pu$/KZ Figure 84. Nondimensional energy flux from the u velocity component, measured near waves. Figure 85. Map of the Spanish Banks site POINT A T K I N S O N E N G L I S H BAY STANLEY^ PARK SPANISH BANKS SITE Y"low tide line -cl i ffs POINT G R E Y 2 K M oo Figure 8 6 . P l a t f o r m and instrument masts at the Spanish Banks s i t e (a) p l a t f o r m and masts l o o k i n g east (b) instrumented mast Figure 87. Map of the Ladner site r— co 188 F i g u r e 89. i n s t r u m e n t s s e t up at the Ladner s i t e , l o o k i n g NNE Figure 90. Map of the Boundary Bay site 190 120 e i o o l u UJ L H U M E m I S Y S T E M o rr o_ 8 0 | LU > < o 601 h-X <£ LU X 4 0 | 20 J 1 • L L . . . . , 0 2 4 V O L T S ( D C ) Figure 91. Typical wave probe calibrations 4 r X I 0 " 3 * S O N I C V ^ O U - W I R E J ^ " ^ 0 * * S O N I C - D i r e c t * X X O X OX o * x * ° x e> ° o X x x o x y ~ * " "  : ° • x x o ° x* x x 3 4 5 6 7 8 M E A N WIND ( m s e c " ' ) Figure 92. evaluated from the direct and $ ^ estimate of the surface stress. 192 SYMBOL TABLE A r e a l part of the complex Fourier c o e f f i c i e n t C rm rm a radius B imaginary part of the complex Fourier c o e f f i c i e n t C rm r rm B.W. bandwidth b as a subscript denotes a p a r t i c u l a r frequency band C phase speed of the waves C Q drag c o e f f i c i e n t drag c o e f f i c i e n t r e l a t e d to wind at 5 meters Co cospectrum C°12 cospectrum f o r data channels 1 and 2 C o h ^ coherence f o r data channels 1 and 2 tti tii C complex Fourier c o e f f i c i e n t f o r the r harmonic i n the m block rm D downwind separation of instruments E east En energy f l u x i n t o the waves e e = 2.72 e s a t u r a t i o n vapour pressure S SL f nondimensional frequency g g r a v i t a t i o n a l constant ' '" -H denotes the hump, associated with waves, i n the pressure spectrum h water depth h r e l a t i v e humidity h s p e c i f i c humidity K' Kolmogoroff constant K Obukhov's un i v e r s a l pressure constant P k wave number 193 kp a pressure wave number, k = oo/U|^ L scale size of pressure fluctuations P L scale size of velocity fluctuations v 1 M number of data blocks per Run m block number N number of data points per channel per block n frequency in cycles per second An bandwidth n^ geometric mean of the end frequencies in a bandwidth n^ folding frequency P total pressure P g stagnation pressure p fluctuating pressure P w fluctuating pressure associated with the waves p^ fluctuating pressure predicted by the potential flow solution Qu quadrature spectrum Qu^2 quadrature spectrum for data channels 1 and 2 2 2 , 2 , 2 q q = u + v + w R ratio of work done by the pressure force to work done by the Reynolds stress Re Reynolds number Ri gradient Richardson number Cr Rp ratio of measured dynamic pressure to calculated stagnation pressure r denotes the harmonic r^, the harmonics at the ends of a frequency band S spectral energy density T temperature T a i r temperature 194 T| a i r temperature at height z T water temperature w ^ t time U mean velocity in the downwind direction U"|z mean velocity in the downwind direction at height z U. mean velocity in the i ^ 1 direction U pressure propagation velocity U mean water velocity w J u velocity fluctuations in the downwind direction th u. velocity fluctuations in the i direction 1 J u* u * 2 =-™ V volume v crossstream velocity fluctuations in the horizontal W west w ver t i c a l velocity fluctuations x downstream coordinate • T H A- + x. l coordxnate x y crossstream coordinate z height above the surface Z £ c r i t i c a l height Z Q surface roughness z^ lower level where turbulent energy transfer goes to zero Y dy incremental volume E rate of viscous energy dissipation n fluctuating water elevation resulting from waves X] wave amplitude 3L 0 phase between two data channels v i r t u a l temperature vir t u a l temperature at height z von Karman's constant wavelength wavelength of pressure fluctuations wavelength of velocity fluctuations wavelength of the waves kinematic viscosity pressure spectrum cospectrum between two pressure signals 1 and 2 fr = 3.14.. . density of a i r variance of pressure signal surface stress cospectrum between i and j velocity components cospectrum between p and w u spectrum w spectrum wave spectrum X = x' - x frequency i n radians/sec measured wave frequency in radians/sec nonharmonic frequency in radians/sec harmonic frequency i n radians/sec wave frequency in radians/sec 

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