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The turbulent fluxes of momentum and sensible heat over the open sea during moderate to strong winds Large, William George 1979

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THE TURBULENT FLUXES OF MOMENTUM AND SENSIBLE HEAT OVER THE OPEN SEA DURING MODERATE TO STRONG WINDS. by WILLIAM GEORGE LARGE B. A. Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS INSTITUTE OF OCEANOGRAPHY We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMEIA August, 1979 © William George Large, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the 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 ag ree 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 ag ree 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 . I t 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 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 . i on Department o f nrrawnCTAPHv 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 2075 Wesbrook Place Vancouver, Canada V6T 1WS Date NOVEMBER 1. 1Q7Q i i ABSTRACT Two systems f o r remote measurements of the a i r - s e a f l u x e s of momentum, s e n s i b l e heat and moisture during moderate to st r o n g winds are d e s c r i b e d . One employs the d i s s i p a t i o n method and the other the Reynolds f l u x or eddy c o r r e l a t i o n method. A modified G i l l p r o p e l l e r - v a n e anemometer i s the v e l o c i t y sensor and a method of r e s o l v i n g the v e r t i c a l v e l o c i t y component, t h a t accounts f o r the p r o p e l l e r ' s non-cosine behavior and avoids i t s n o n - l i n e a r o p e r a t i n g r e g i o n , i s d e r i v e d . The dynamic responses of the sensors are found from measurements i n the a c t u a l t u r b u l e n t c o n d i t i o n s of the f l u x measurements. The r e s u l t s of an experiment on the Bedford tower, a s t a b l e p l a t f o r m moored i n 59m of water 10 km o f f s h o r e , are presented. S p e c t r a , c o s p e c t r a , t u r b u l e n c e s t a t i s t i c s and t r a n s f e r c o e f f i c i e n t s are c a l c u l a t e d from the Reynolds f l u x v e l o c i t y and temperature data and found to be comparable t o p r e v i o u s l y r e p o r t e d v a l u e s . Simultaneous d i s s i p a t i o n and Reynolds f l u x e s t i m a t e s of both the momentum and s e n s i b l e heat f l u x e s i n up to 20 m/s winds are shown to be i n e x c e l l e n t agreement. Also presented are the r e s u l t s of a second experiment where the systems were deployed on the weathership CCGS Quadra. A comparison of s h i p and tower drag c o e f f i c i e n t s from the d i s s i p a t i o n system, demonstrates that the Bedford tower i s e s s e n t i a l l y an open ocean s i t e . The n e u t r a l drag c o e f f i c i e n t , CDN, i s found, on average, t o be ne a r l y constant a t 1.14x10 - 3 f o r winds between 4 and 10 m/s and to i n c r e a s e almost l i n e a r l y t o about 2.18x10 - 3 a t 26 m/s. No v a r i a t i o n with e i t h e r f e t c h i i i (greater than 10 km) or s t a b i l i t y i s observed. Dissipation estimates of the sensible heat flux from a wide range of conditions are presented. The neutral transfer c o e f f i c i e n t , CTN, i s found, on average, to vary from about 0.69x10 - 3 i n stable s t r a t i f i c a t i o n to 1.08x10-3 i n the unstable case. An increase i n CTN with increasing wind speed i s suggested by only some of the data. Time series of the fluxes are used to investigate additional sources of variation i n the transfer c o e f f i c i e n t s . Their s t a t i s t i c a l v a r i a b i l i t y about a running mean i s seen to be about 10%. Evidence i s presented that indicates that persistent departures from average values are related to sea surface conditions. CDN i s observed to be s i g n i f i c a n t l y smaller, on average, during r i s i n g winds than during f a l l i n g winds or after a change i n wind d i r e c t i o n . iv TABLE OF CONTENTS Page ABSTRACT . .. i i TABLE OF CONTENTS » i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENT - x i CHAPTER 1 INTRODUCTION . r r . P . . 1 CHAPTER 2 EXISTING THEORY AND EXPERIMENTAL RESULTS 2.1 Air-Sea Interaction 4 2.2 Monin-Obukhov S i m i l a r i t y Theory 9 2.3 Bulk Aerodynamic Parameterizations ............. 13 2.4 The Reynolds Flux Method 19 2.5 The Dissipation Method 21 2.6 Estimating The S t a b i l i t y Parameter Z/L 31 CHAPTER 3 THE INSTRUMENTATION AND EXPERIMENTAL PROGRAM 3.1 Introduction 37 3.2 The Sensors „ 38 3.3 The Reynolds Flux System 46 3.4 The Dissipation System 52 3.5 Sensor Response i . . . . . . . . . 56 3.6 The Experimental Program 63 V CHAPTER 4 REYNOLDS FLUX MEASUREMENTS FROM THE BEDFORD STABLE TOWER 4.1 I n t r o d u c t i o n 69 4.2 The S t a b i l i t y Parameter Z/L 70 4.3 Turbulence S p e c t r a And Cospectra 72 4.4 Turbulence S t a t i s t i c s 87 4.5 The Fluxes Of Momentum And S e n s i b l e Heat ....... 92 CHAPTER 5 INTERCOMPARISON OF THE REYNOLDS FLUX AND DISSIPATION METHODS 5.1 I n t r o d u c t i o n 101 5.2 The Momentum Flu x 103 5.3 The S e n s i b l e Heat Flux 117 CHAPTER 6 DISSIPATION MEASUREMENTS FROM THE BEDFORD STABLE TOWER AND CCGS QUADRA 6.1 I n t r o d u c t i o n 122 6.2 Bulk Aerodynamic P a r a m e t e r i z a t i o n Of The Momentum Flux 126 6.3 Bulk Aerodynamic P a r a m e t e r i z a t i o n Of The S e n s i b l e Heat Flux 150 CHAPTER 7 SUMMARY AND CONCLUSIONS 166 REFERENCES . ... 172 APPENDIX THE INTERCOMPARISON RESULTS 177 v i LIST OF TABLES TABLE I Summary of possible errors i n the velo c i t y measurement and t h e i r e f f e c t s on the Reynolds f l u x method 43 TABLE II Turbulent ve l o c i t y s t a t i s t i c s from the 196 Reynolds flux runs 88 TABLE I I I Standard deviations of the turbulence s t a t i s t i c s about th e i r s t a b i l i t y band means plotted in figure 18. ................. 89 TABLE IV Comparison of the d i f f e r e n t methods of integrating the u,w cospectrum. ............. 93 TABLE V Comparison of the d i f f e r e n t methods of integrating the w,t cospectrum. ............. 95 TABLE VI Ratio of d i s s i p a t i o n to Reynolds flux estimates of the momentum flux band averaged over 2 m/s wind speed i n t e r v a l s . ... 113 TABLE VII Ratio of dissipation to Reynolds flux estimates of the momentum flux band averaged over s t a b i l i t y ranges. ............. 114 TABLE VIII Variation of the neutral drag c o e f f i c i e n t with fetch for winds between 4 and 10 m/s 128 TABLE IX The s t a b i l i t y dependency of the drag c o e f f i c i e n t s from the 4 < U10 < 10 m/s data of table VIII 130 TABLE X Mean 103CDN of wind speed bands i n di f f e r e n t wind conditions. 140 TABLE XI Mean 103CDN of wind speed bands for di f f e r e n t months. 142 TABLE XII Parameterization of the sensible heat flux by band averaging 0 AT over ranges of <wt> . 158 TABLE XIII Averaged neutral Stanton number as a function of wind speed i n stable s t r a t i f i c a t i o n only. 161 TABLE XIV Averaged neutral Stanton number as a function of wind speed i n unstable s t r a t i f i c a t i o n only. . 162 v i i LIST OF FIGURES FIGURE 1: S t a b i l i t y adjustments of the four d i s s i p a t i o n methods of es t i m a t i n g the momentum f l u x . 28 FIGURE 2: S t a b i l i t y adjustments of the two d i s s i p a t i o n methods of esti m a t i n g the s e n s i b l e heat f l u x . . 30 FIGURE 3 A: The G i l l twin propeller-vane anemometer. B: The HAT sensor housing. C: D e f i n i t i o n of angles used i n r e s o l v i n g the v e l o c i t y components and c a l c u l a t i n g the t i l t angle 6 40 FIGURE 4: S i g n a l processing i n the Reynolds f l u x system, showing the sampling scheme and a l l parameters considered i n the a n a l y s i s . > .. 47 FIGURE 5: S i g n a l processing i n the d i s s i p a t i o n system. 53 FIGURE 6: Determination of the distance constants from the 0.8 : 1.6 Hz band-pass f i l t e r r a t i o s of A: the G i l l - u h o r i z o n t a l p r o p e l l e r , B: the G i l l - w t i l t e d p r o p e l l e r , <=<= 59.5°. 58 FIGURE 7: Determination of the microbead the r m i s t o r response from the 0.8 : 1.6 Hz band-pass f i l t e r r a t i o s . 60 FIGURE 8: Ratios of <uw> c a l c u l a t e d from d i f f e r e n t p a i r s of band-pass f i l t e r s as a f u n c t i o n of wind speed 62 FIGURE 9 A: The Bedford tower s i t e near H a l i f a x Nova S c o t i a . B: The instrumentation on the Bedford tower. ... 64 FIGURE 10 A: Ocean Weather S t a t i o n "PAPA", 50°N, 145°W, and the route of the weatherships. B: The instrumentation on the foremast of CCGS Quadra 67 V l l l FIGURE 11 Comparison of the most complete e x p r e s s i o n f o r the s t a b i l i t y parameter Z/L(u*,<wt>) t o : A: an approximate e x p r e s s i o n Z / L ( U * , A T ) . B: the bulk e s t i m a t e Z / L (AT) . • 71 FIGURE 12 Normalized s p e c t r a o f the downstream v e l o c i t y component from averages of f ^ u ( n ) / u * 2 from: A: 108 unstable runs B: 88 s t a b l e runs 75 FIGURE 13: The h o r i z o n t a l v e l o c i t y spectrum, f ( f ) / averaged over 4 runs from CCGS Quadra i n 22 m/s winds. 78 FIGURE 14 Normalized v e r t i c a l v e l o c i t y s p e c t r a from averages of f "rw (n) / u * 2 over: A: 108 u n s t a b l e runs B: 88 s t a b l e runs 80 FIGURE 15: The normalized temperature spectrum from . averages of f *rt(n)/(o*t) 2 over a l l 60 temperature runs 82 FIGURE 16: Normalized u n s t a b l e (A) and s t a b l e (B) u,w c o s p e c t r a from averages o f f ^uw(n)/u* 2. I n t e g r a t i o n under the s o l i d c u r v es g i v e s E f o r the 13 method. ..... 84 FIGURE 17: Normalized w,t c o s p e c t r a from averages of f ^wt(n)/<wt> i n (A) u n s t a b l e and (B) s t a b l e c o n d i t i o n s . I n t e g r a t i o n under the s o l i d curves gi v e s E f o r the 13 method 86 FIGURE 18: Non-dimensional t u r b u l e n c e s t a t i s t i c s as f u n c t i o n s o f s t a b i l i t y . See t a b l e I I I f o r standard d e v i a t i o n s about p l o t t e d means. 90 FIGURE 19: The n e u t r a l drag c o e f f i c i e n t vs wind speed f o r the 196 Reynolds f l u x momentum runs 97 FIGURE 20: <wt> vs. U10(TSFC-T10) f o r the 52 temperature runs with |AT| >0.5°C . 99 i x FIGURE 21: Comparison of u* i n m/s from the n e u t r a l d i s s i p a t i o n method and the Reynolds f l u x method f o r a l l 192 simultaneous Bedford tower runs .. 104 FIGURE 22 I n v e s t i g a t i o n of t h e "near n e u t r a l " momentum f l u x regime f o r ; A: 61 runs with 0.0 <Z/L< 0.5 B: 70 runs with -0. 1 <Z/L< 0.0 . 106 FIGURE 23 Comparison of 88 s t a b l e Reynolds f l u x runs t o A: d i s s i p a t i o n method 2 B: d i s s i p a t i o n method 3. 108 FIGURE 24 Comparison of 104 unstable Reynolds f l u x runs to A: d i s s i p a t i o n method 2 B: d i s s i p a t i o n method 4. ....................... 110 FIGURE 2 5: Intercomparison of u* from the "best" d i s s i p a t i o n method (2) and the "best" Reynolds f l u x method (13) f o r a l l 192 simultaneous runs 112 FIGURE 26: Comparison of u* from the d i s s i p a t i o n system and from the BIO eddy c o r r e l a t i o n systems f o r 20 runs on the Bedford tower. ... 116 FIGURE 27 Comparison o f the 60 simultaneous s e n s i b l e heat f l u x c a l c u l a t i o n s i n °Cm/s. A: <wt>DISS1 vs. <wt>FLUX B: <wt>DISS2 vs. <wt>FLUX. 118 FIGURE 28 The n e u t r a l drag c o e f f i c i e n t as a f u n c t i o n of wind speed from A: 1086 h o u r l y averages from the Bedford tower B: 505 h o u r l y averages from the CCGS Quadra. ... 125 FIGURE 29: Comparison of s h i p (pluses) and tower ( t r i a n g l e s ) n e u t r a l drag c o e f f i c i e n t s . ...... 127 FIGURE 30: The n e u t r a l drag c o e f f i c i e n t as a f u n c t i o n of wind speed. L i n e s show a Charnock r e p r e s e n t a t i o n with <=<•=• 0.0 144, K= 0.41 (dashed) and eguations 6.1 ( s o l i d ) . ...... 132 X FIGURE 31: Time s e r i e s of the momentum f l u x from run D33C. Time i s from 4:40 GMT December 7, 1976. . 138 FIGURE 32: Time s e r i e s of the momentum f l u x from run D29D. Time i s from 17:00 GMT September 26, 1976. 145 FIGURE 33: Momentum f l u x time s e r i e s from run D31F. Time i s from 4:00 GMT October 20, 1976. 147 FIGURE 34: Momentum f l u x time s e r i e s from run D38B. Time i s from 16:00 GMT March 12, 1977 148 FIGURE 35: P a r a m e t e r i z a t i o n of the s e n s i b l e heat f l u x (°Cm/s) i n unstable s t r a t i f i c a t i o n . .... 151 FIGURE 36: Time s e r i e s of the s e n s i b l e heat f l u x during run D33C ( f i g u r e 31) . Time i s from 4:40 GMT December 7, 1976. 153 FIGURE 37: P a r a m e t e r i z a t i o n of the s e n s i b l e heat f l u x i n °Cm/s f o r s t a b l e s t r a t i f i c a t i o n 156 FIGURE 38: The n e u t r a l Stanton number as a f u n c t i o n of s t a b i l i t y ... 159 FIGURE 39: S e n s i b l e heat f l u x time s e r i e s from run D38B ( f i g u r e 34). Time i s from 16:00 GMT March 12, 1977 164 x i ACKNOWLEDGEMENT I should l i k e to express my thanks t o the many persons who have c o n t r i b u t e d t o the success of t h i s work. The p r o j e c t was i n i t i a t e d by Dr. S. Pond, who, as research su p e r v i s o r , c o n t i n u a l l y provided h i s e f f o r t s , guidance and enthusiasm. A great deal i s owed t o E. Meyer {Meyer Systems I n c . ) , who designed the e l e c t r o n i c s and made them work. B. Walker developed much of the mini-computer software. I applaud the e f f o r t s of D. E n g l i s h , P. Merchant, H. Heckl and others of the IOUBC s t a f f i n b u i l d i n g and deploying the instruments. Throughout the Bedford tower experiment Dr. S. D. Smith and h i s a i r - s e a i n t e r a c t i o n group and others at the Bedford I n s t i t u t e provided both l o g i s t i c and moral support. E. Anderson and D. Hendsbee o f t e n s e r v i c e d the instrumentation i n my absence. The co-operation of Captain Dykes, the o f f i c e r s and the crew of CCGS Quadra was e s s e n t i a l t o the weathership operations. Some s e r v i c i n g was p o s s i b l e at PAPA, thanks t o a i d from Dr. M. Miyake and h i s s t a f f (IOS P a t r i c i a Bay) and from T. Neuhaus (Seakem Oceanography) . The p r o j e c t was generously supported by the United States O f f i c e of Naval Research (Contracts N 00014-66-C-0047 and N 0C014-76-C-0446 under P r o j e c t 083-207) and by the National Research C o u n c i l of Canada (Grant A8301). I pe r s o n a l l y received post graduate s c h o l a r s h i p s from NRC and a f e l l o w s h i p from UBC. 1 CHAPTER 1 INTRODUCTION This t h e s i s describes an experimental program designed to measure the t u r b u l e n t exchanges between the open ocean and the atmosphere i n moderate t o strong (5-50 m/s) winds. Measurements of the most important exchanges have been the subject of recent reviews, i n which they are parameterized by non-dimensional t r a n s f e r c o e f f i c i e n t s . Recent determinations of the drag c o e f f i c i e n t , which i s used to express the momentum f l u x i n terms of the sguare of the mean wind speed, have been reviewed by G a r r a t t , 1977. The se n s i b l e heat and moisture f l u x e s are parameterized by Friehe and Schmitt, 1976, i n terms of the surface - a i r temperature and humidity d i f f e r e n c e s , r e s p e c t i v e l y and the mean wind speed. There are s e v e r a l o b s tacles t h a t make open ocean measurements i n high winds d i f f i c u l t and that have, t h e r e f o r e , r e s t r i c t e d the ma j o r i t y to low winds and to near or onshore platforms. The most common methods of o b t a i n i n g the f l u x e s , the Reynolds f l u x (or eddy c o r r e l a t i o n ) and p r o f i l e , work best on s t a b l e platforms with minimal flow d i s t o r t i o n , but these c o n d i t i o n s are not easy t o s a t i s f y during storms a t sea. Adverse c o n d i t i o n s accompanying high winds, cause towers to c o l l a p s e and many sensors e i t h e r t o f a i l completely or t o lose t h e i r c a l i b r a t i o n . The a i r - s e a energy exchanges are i n v o l v e d i n a number of important processes, i n c l u d i n g the l a r g e s c a l e c i r c u l a t i o n s of the ocean and atmosphere and, at smaller s c a l e s , thermocline development and wave generation. By extending the e x i s t i n g data set to the open ocean and t o higher wind speeds, t h i s 2 experimental program should be r e l e v a n t to the study of these processes. Modelling and p r e d i c t i n g l a r g e s c a l e f e a t u r e s r e q u i r e the f l u x e s , which are too d i f f i c u l t and c o s t l y t o measure d i r e c t l y on t h i s s c a l e , to be c a l c u l a t e d from e a s i l y measured q u a n t i t i e s through parameterizations based on r e l a t i v e l y few d i r e c t observations. For t h i s purpose, extension of the measured t r a n s f e r c o e f f i c i e n t s t o 20 m/s ought to s u f f i c e , because higher winds r a r e l y c o n t r i b u t e very much to the f l u x e s averaged over a month or more ( F i s s e l et a l . , 1977). A f u r t h e r extension t o about 25 m/s should c l e a r l y r e v e a l any wind speed dependencies of the c o e f f i c i e n t s . At present there i s an opinion t h a t , i n view of the s c a t t e r , a constant drag c o e f f i c i e n t up to about a 14 m/s wind speed i s appropriate (Stewart, 1974), while Smith and Banke, 1975, and others f i n d a s i g n i f i c a n t increase with wind speed. The average s t r e s s computed from e i t h e r type of drag c o e f f i c i e n t f o r m u l a t i o n should be nearly the same, because the trend i s , at most, s m a l l , but the c u r l of the wind s t r e s s could be a f f e c t e d to a greater degree. The l a r g e amount of s c a t t e r , t y p i c a l of turbulence measurements, suggests, t h a t i n order t o a r r i v e at a r e p r e s e n t a t i v e p i c t u r e of the open sea, a great deal of data from a l l p o s s i b l e c o n d i t i o n s are r e q u i r e d . With a l a r g e data set i t would be p o s s i b l e to examine the e f f e c t s of s t a b i l i t y , the wave f i e l d and other sources of v a r i a b i l i t y i n the t r a n s f e r c o e f f i c i e n t s apart from the r e a l s t a t i s t i c a l s c a t t e r and systematic instrumentation e r r o r s . 3 Continuous records over a long period of time should include a variety of l o c a l and short-lived phenomena, such as f r o n t a l passages, for which a large scale parameterization may not be applicable. In some cases i t may be possible to find appropriate transfer c o e f f i c i e n t s and in others d i r e c t stress estimates may be the simpler approach. The c a p a b i l i t y to operate i n winds above 40 m/s should allow the en t i r e time h i s t o r i e s (winds, temperatures and fluxes) of most storms to be followed. Such time series should be useful f o r the investigation of many small scale processes. A modified G i l l propeller-vane anemometer proved to be a very suitable v e l o c i t y sensor for this study. The momentum flux and drag c o e f f i c i e n t were successfully measured i n 26 m/s winds. Temperature and humidity sensors were housed i n protective enclosures. The microbead thermistors were often broken by spray and contaminated by s a l t , however they did survive some high winds and heat transfer c o e f f i c i e n t s corresponding to large fluxes were obtained. No useful humidity data were ever recorded, because of the f a i l u r e of several types of sensors. For open sea work a ship i s the most convenient platform and i t s motion and flow d i s t o r t i o n can be tolerated by the dissipation method of measuring fluxes (Pond and Large, 1978) . This method was f i r s t tested on a stable offshore platform where i t s r e s u l t s compared favourably to the Reynolds flux method. It was then employed on a ship, allowing open sea data to be c o l l e c t e d . In order to gather as much data as possible, the instrumentation was designed to record continuously for a month or more, while operating remotely. 4 CHAPTER 2 MISTING THEORY AND EXPERIMENTAL RESULTS 2.1 Air-Sea Interaction Exchanges between the atmosphere and ocean are most e a s i l y measured i n the atmospheric surface layer where the transfer processes are dominated by turbulence. Viscous and d i f f u s i v e molecular transfers are n e g l i g i b l e i n t h i s layer, which begins a few centimeters above the surface and extends up to a l e v e l where the earth's rotation and the geostrophic pressure gradient become important. Detailed treatments of the turbulent flow i n the layer may be found in Lumley and Panofsky, 1964, Monin and Yaglom, 1965 and 1967, and Kraus, 1972. . This chapter and Busch, 1977, are s p e c i f i c a l l y concerned with the theory related to turbulent flux measurements and t h e i r i n t e r p r e t a t i o n . Here the sensible heat flux and the momentum flux are treated e x p l i c i t l y and the theory i s also extended to any passive atmospheric scalar guantity, R. Following Reynolds' convention, the turbulent properties are partitioned into a mean ( < > denotes a time average) and a fluctuation (lower case symbols). The components of the instantaneous wind vector, V = Oi+Vj+Wk, where i , j , and k are unit vectors of an x-y-z coordinate system, become U=<U>+u, V=<V>+v and W=<W>+w. The usual orientation of the axes puts k v e r t i c a l l y up and i along the mean horizontal wind vector such that the mean cross-stream and v e r t i c a l components, <V> and <W>, are both zero (Burling and Stewart, 1967). Si m i l a r l y any scalar f i e l d R becomes <R>+r (and the a i r temperature T=<T>+t and the a i r pressure P=<P>+p). By d e f i n i t i o n <u>, <v>, <w>, <t> and <r> are a l l zero. 5 I t i s the f l u c t u a t i n g v e r t i c a l v e l o c i t y which bodily t r a n s p o r t s f l u i d p r o p e r t i e s up and down, g i v i n g r i s e t o the Reynolds f l u x e s defined by: Momentum f l u x TT _ _^ <uw> Sensible heat f l u x Hs = ^  Cp <wt> (2. 1) any s c a l a r f l u x Hr = <wr> , where ^ i s the mean a i r density and Cp i s the s p e c i f i c heat at constant pressure. Since, <W>=<w>=0, the f l u c t u a t i n g q u a n t i t i e s i n 2.1 can be replaced by t h e i r instantaneous values U, T and R. In t h i s coordinate system <vw> should tend to zero with a long enough averaging p e r i o d , so ~~C represents the t o t a l momentum f l u x . Since i t gives r i s e to a f o r c e i n the d i r e c t i o n of the mean wind on a u n i t area of under l y i n g s u r f a c e , 77 i s a l s o r e f e r r e d t o as the Reynolds s t r e s s . Hs i s a tur b u l e n t heat t r a n s f e r ( p o s i t i v e up). The moisture f l u x , a l s o an important a i r - s e a exchange, i s expressed by simply s u b s t i t u t i n g absolute humidity f o r R. S i m i l a r l y gas f l u x e s such as carbon d i o x i d e may a l s o be considered. Often the terms momentum f l u x and s e n s i b l e heat f l u x r e f e r to the kinematic f l u x e s <uw> and <wt>. The Reynolds f l u x e s a r i s e i n the equation of motion and i n s c a l a r conservation equations where t h e i r surface values become important boundary c o n d i t i o n s f o r both the atmosphere and ocean. In the equation f o r <D> i n the boundary l a y e r , the C o r i o l i s f orce due to the cross-stream component, V, i s , on average, zero. The equations f o r the mean flow and mean temperature i n the surface l a y e r , assuming h o r i z o n t a l l y homogenous turbulence 6 (Busch, pp 74 and 75) , but r e t a i n i n g terms with the l a r g e s t mean h o r i z o n t a l g r a d i e n t s , are: P _<_> = ^jF - _<_> 3t 3z <3x (2.2) d<T> + <H> __<T> + _d (Hs_+_ET) = 0 , 3t dx dz ~£ Cp where RT i s the v e r t i c a l heat t r a n s f e r due t o r a d i a t i o n . Hith n e g l i g i b l e h o r i z o n t a l pressure, h o r i z o n t a l temperature and v e r t i c a l r a d i a t i v e f l u x gradients and a steady mean s t a t e , the tur b u l e n t f l u x e s are constant throughout the l a y e r . I t i s then p o s s i b l e t o measure the surface f l u x e s above wave i n f l u e n c e s at a convenient height, Z. However, away from the s u r f a c e , r o t a t i o n and the l a r g e s c a l e h o r i z o n t a l gradients e v e n t u a l l y become i n f l u e n t i a l . In steady flow, the measured s t r e s s , T"(Z) , i s l e s s than the surface s t r e s s , ~Co. I f the d i f f e r e n c e at a height, he, i s 10% (Lumley and Panofsky, 1964, a r b i t r a r i l y use 20%), then the flow below can be regarded as being a "constant f l u x " or "constant s t r e s s " l a y e r . E f f e c t i v e l y , he i s taken to be the upper l i m i t of the atmospheric surface l a y e r . In m i d - l a t i t u d e s winds above the surface l a y e r are governed by the geostrophic balance, f Ug = 1/£ d<P>/ dn , where f i s the C o r i o l i s parameter (about 1x10-* s _ 1 ) , Ug i s the geostrophic wind and n i s a h o r i z o n t a l coordinate perpendicular to the Ug d i r e c t i o n . Observations have shown Ug t o be about 7 1.3 <U> and about 16° to the r i g h t of the i d i r e c t i o n (Deacon, 1973). S u b s t i t u t i n g , d<P>/ dx = s i n 16° d<P>/ 3 n , i n t o the f i r s t of 2.2 y i e l d s . cku> S i 3 7 7 + 1.3 f <D> s i n 16° 5t ^ 5z In steady flow a 10% r e d u c t i o n i s found when 0.1 = To - T(Z) = 1.3 f <0> s i n 16° Z , To "£b/^ so he = _ 0 i j <uw> = 2790 seconds <uw>. (2.3) 1.3 f <U> s i n 16° <D> Measurements over the sea at 10m have almost always shown <uw>/<U> t o average more than 10— 3 <U> (drag c o e f f i c i e n t > 1x10 - 3, G a r r a t t , 1977), s e t t i n g 14m as a lower l i m i t of he, when <U>=5 m/s. In unsteady flow, 3<0>/ dt can e a s i l y be the same order o f magnitude as (1.3 f <U> s i n 16°) = 0.13 <0> /hour. On the r i s i n g wind the a c c e l e r a t i o n i s down the pressure g r a d i e n t and he r i s e s , because a s m a l l e r s t r e s s g r a d i e n t i s s u f f i c i e n t to balance 2.2. On the f a l l i n g wind the l o s s of f l u x with height i s enhanced by the d e c e l e r a t i o n and i n t h i s s i t u t a i o n he may be c o n s i d e r a b l y lower. These arguments are p o s s i b l y good to a f a c t o r of 2 d e s p i t e the n e g l e c t of h o r i z o n t a l a d v e c t i o n terms, r e l a t i v e t o 3<P>/dx. They serve t o p o i n t out t h a t , with the same s u r f a c e s t r e s s * the measured 10m s t r e s s may be g r e a t e r on the r i s i n g wind than during f a l l i n g winds, when i t c o u l d be l e s s than the s u r f a c e s t r e s s by 5 t o 10%. 8 The height to which the s e n s i b l e heat f l u x remains w i t h i n 10% of i t s surface value i s not obvious and i t may sometimes be below usual measurement heights. Since c-<T>/dx i s not simply r e l a t e d to the l a r g e s c a l e pressure gradient, i t i s not p o s s i b l e to s c a l e the v e r t i c a l divergence of Hs to a C o r i o l i s term as was done f o r the Reynolds s t r e s s . Neglecting r a d i a t i o n , a 10% change i n Hs i s found a t a height he given by he = 0.1 <wt> ( f i T / f i t ) - * , where 6T i s the change i n temperature due t o the v e r t i c a l heat f l u x divergence, during a time i n t e r v a l fit. Measurements over the sea show t h a t <wt> i s of order 10~ 3 <U> A T (Friehe and Schmitt, 1976), where AT i s the temperature d i f f e r e n c e between the sea surface and atmosphere. To keep he above 10m, with <0> AT as low as 10 °Cm/s requires 6T/ fit < 0.36 °C/hour, which may not n e c e s s a r i l y be s a t i s f i e d . When a,<T>/o,t over the sea i s l a r g e , presumably h o r i z o n t a l advection i s the major c o n t r i b u t o r with 6T/ fit h o p e f u l l y remaining s m a l l . When the temperature i s steady <D> 3<T>/3x must balance dHs/c-z i n 2. 2, g i v i n g he = 10-* AT ( 3<T>/dx )-» and implying that he i s only above 10m when the h o r i z o n t a l 9 temperature gradient i s rather small, l e s s than 0.01 °C/km for a AT of 1°C. A further complication arises during low winds when infrared absorption by water vapour may produce a radiative flux divergence that tends to cause Hs to increase with height (Busch, 1977). This effect enhances the positive flux gradient when _.T i s negative, but reduces the loss of sensible heat flux with height when AT i s positive. I t w i l l be assumed that Hs anywhere i n the "constant stress" layer over a temperate sea i s equivalent to the surface f l u x to within the accuracy of the measurement. Hopefully, t h i s assumption i s true on average, but v e r i f i c a t i o n would require a di r e c t measurement of Hs at two l e v e l s . 2.2 Monin-Obukhov S i m i l a r i t y Theory The understanding of the turbulent atmospheric surface layer i s largely due to Monin-Obukhov s i m i l a r i t y theory (Monin and Yaglom, p 425 f f ) . The theory assumes that turbulent c h a r a c t e r i s t i c s depend only on a few physical parameters, which f a c i l i t a t e s the application of dimensional analysis. Above the dir e c t influence of the bottom boundary the important parameters are the height (the only s p a t i a l variable l e f t i n the assumed horizo n t a l l y homogenous turbulence), the a i r density, the turbulent transports and the s t a b i l i t y of the a i r column* The supposed height independence of the fluxes naturally leads to the following scales which incorporate the transports through the layer and the density: 10 f r i c t i o n v e l o c i t y u* = (Ib/^ >) */2 = |-<uw>|*/2 temperature scale t * = -<wt>/ /<u* (2.4) scalar scale r * = -<wr>/ Ku* , where von Karman's constant, K, i s included to simplify l a t e r equations, but other scales T* = -<wt> / u* and E* = -<wr> / u* are sometimes used. An appropriate s t a b i l i t y parameter, Z/L, i s obtained from the r a t i o of the convective or buoyant turbulent kinetic energy, B, produced i n a non-neutral a i r column, to the purely mechanical production i n the equivalent neutral case, Po. It w i l l be seen i n section 2.5, that B = g <w Tv> / To and Po = u*V KZ , (2.5) where Tv, the v i r t u a l temperature i n degrees Kelvin, accounts for the fluctuating temperature and moisture contributions to the fluctuating density, g i s g r a v i t a t i o n a l acceleration and To i s the l o c a l average v i r t u a l temperature. The important scale i s the Monin-Obukhov length, L, whose magnitude gives the height at which Po = |B|, thus -Z = B_ and L = -u* 3 _To_ . L Po K g <w Tv> (2.6) In neutral s t r a t i f i c a t i o n <w Tv> and Z/L go to zero while L 11 approaches i n f i n i t y and the sign i s chosen to make L and Z/L positive in stable conditions. Dimensional analysis predicts that a l l turbulent functions non-dimensionalized by Z, u*, r* and L should be functions of the only possible dimensionless group, Z/L. Note that each additional scalar adds both a scale and a dimension to the problem. Ose of spectra introduces a frequency, f, making a further dimensionless group, fZ/u*, possible, but because of the d i f f i c u l t y i n obtaining u* the dimensionless natural frequency n = fZ/<U> i s usually substituted. Normalized spectra and cospectra can be regarded as non-dimensional turbulent functions and therefore should be functions of both n and Z/L, whereas t h e i r i n t e g r a l s such as (cru) 2 = <u2> and ( e f t ) 2 = <t 2> should depend only on Z/L. With the important parameters common to a l l surface layer flows, the structure of the turbulence, according to t h i s theory, must always be "s i m i l a r " , with any dependencies cn Z/L and n being universally applicable. An important consequence of s i m i l a r i t y theory i s the logarithmic p r o f i l e of the mean wind and mean scalars. Dimensional considerations lead d i r e c t l y to the forms: XZ_ _<0> = 0m (Z/L) u* dz ( 2 * 7 ) and Z_ d<R> = 9r(Z/L) , r* l>z where von Karman's constant sets ^m=1 at neutral s t a b i l i t y and has already been included in the d e f i n i t i o n of r* so that ^r(0 ) = 1 . Measured values of K vary between about 0.35 and 0.42 12 (Busch, 1977), so at l e a s t a 5% e r r o r i n i t s customary value of 0.40 must be allowed. In a review of f l u x - p r o f i l e r e l a t i o n s h i p s . Dyer, 1974, suggests that the best forms of the u n i v e r s a l f u n c t i o n s are 0 < Z/L < 0.2: -1.0 < Z/L < 0 : where the unstable case i s provided by Dyer and Hi c k s , Z, UZ and EZ, are found by i n r*m = r>r = 1 + 5 Z/L fm = (1 - 16 Z/L)-i/* r*r = (1 - 16 Z/L) - i / 2 , the Businger-Dyer r e p r e s e n t a t i o n 1970. The mean values at a height egrating 2.7, (Paulson, 1970), OZ = ( U * / K) • [ln(Z/Zo) - Ym(Z/L) ] EZ = BSFC + r * [ln(Z/Zor) - Y r (Z/L) ], (2.8) where Viz/L) = J^\^ ' * ( £ ) 3 / £ ^ s t a b l e : "Ym(Z/L) = Yf (Z/L) = -5 Z/L unstable: Ym (Z/L) = 2 l n [ ( 1 + X ) / 2 ] + ln[ ( 1 + X 2 ) /2 ] - 2 t a n - i X + 7T/2 Yr(Z/L) = 2 l n [ ( 1 + X 2 ) / 2 ] with X = ( 1 - 1 6 Z/L) i / * . At n e u t r a l s t a b i l i t y the i n t e g r a l vanishes, l e a v i n g V ( 0 ) = o -The constants of i n t e g r a t i o n Zo and Zor, assumed to be much 13 smaller than Z, are the roughness lengths, which f u l l y describe the surface as "seen" by the turbulence, but they are not simply related to sea surface parameters such as wave height and temperature. They need to be included i n the dimensional analysis only near the surface where they set the magnitude, but not the structure, of the turbulence throughout the layer.. 2.3 Bulk Aerodynamic Parameterizations The fluxes are parameterized i n terms of the mean wind, the sea surface-air temperature difference and surface-air mean scalar difference, AR, with the bulk aerodynamic formulae: <uw> = CD <D>2 <wt> = CT <0> AT (2.9) <wr> = CE <D> AR, with -AT=TSCF-TZ and AR=RSFC-RZ, where TSFC and RSFC are the mean surface temperature and scalar values, respectively. The non-dimensional transfer c o e f f i c i e n t s CD and CT are the drag c o e f f i c i e n t and Stanton number while the corresponding c o e f f i c i e n t of moisture transport, CE, i s the Dalton number. Their dependence on s t a b i l i t y , roughness and height i s evident from 2.8, but Zo over the sea has a complicated functional form (Burling and Stewart, 1967). However, they can also be determined experimentally from 2.9 using measured fluxes and bulk quantities and such calculated c o e f f i c i e n t s provide a Z~/£ = u* 2 Hs/^Cp = - K u* t* = Hr = - K u* r * = 14 convenient means of comparing f l u x measurements. To eliminate the variation with height they are commonly evaluated at 10m as: C10 = -<uw> / (U10) 2 CT10 = <wt> / [D10 (TSFC-T10) ] (2.10) CR10 = <wr> / [D10 (RSFC-R10) ] . . Equation 2.8 shows the wind speed, temperature and scalar means at 10m (U10, T10 and R10) to be: D10 = DZ - (u*/ X) [In (Z/10m) -"Vm(Z/L) + Yin (10m/L) ] T10 = TZ - t* [ln(Z/10m) - VI (Z/L) + "Vi (1 Om/L) ] (2.11) R10 = RZ - r * [ln(Z/10m) - Yt (Z/L) + "Vr (10m/L) ] . For comparative purposes i t i s convenient to eliminate the s t a b i l i t y dependence by evaluating the roughness lengths from 2.8 and using them to f i n d the c o e f f i c i e n t s i n the equivalent neutral case at 10m from: CDN = K z / [ln(10m/Zo) ] 2 CTN = X 2 / [ In (10m/Zot) • ln(10m/Zo)] (2.12) CRN = X 2 / [ln(10m/Zor) • ln(10m/Zo)] . The neutral c o e f f i c i e n t s should be constants over homogeneous te r r a i n where the roughness lengths can be regarded as constant. This prediction has been v e r i f i e d over land where Zo, for example, i s exclusively determined by topography and vegetation. It i s not unreasonable to expect t h i s concept to 15 work even b e t t e r over the sea where there i s only one type of surface. However, d i r e c t measurements* reviewed by G a r r a t t (1977) are very s c a t t e r e d and i n d i c a t e that the n e u t r a l drag c o e f f i c i e n t v a r i e s with wind speed and i s much smaller than that found over land. This r e s u l t i m p l i e s t h a t there are a d d i t i o n a l important parameters determining the roughness of the sea surface. An obvious d i f f e r e n c e between a land and sea boundary i s surface g r a v i t y waves, so i t seems appropriate to add the a c c e l e r a t i o n due to g r a v i t y , g, t o the problem. This leads to the Charnock (1955) dimensionless group f o r flow near the waves, Zo g / u*2 = <=<- (2.13) where =0.0144 i s suggested by G a r r a t t (1977). Stewart, 1974, notes that f o r winds below 10 m/s the Charnock rep r e s e n t a t i o n with °<- constant p r e d i c t s a more r a p i d i n c r e a s e i n CDN with wind speed than i s i n d i c a t e d by the r e s u l t s of Brocks and Kriiggermeyer (1970) and t h i s feature i s a l s o present i n G a r r a t t ' s review. I t appears, t h e r e f o r e , that more surfa c e parameters may be important t o t h i s aspect of tur b u l e n t flow. Stewart discusses the p o s s i b l e r o l e s of surface t e n s i o n ( c a p i l l a r y waves), the length and phase speed of the longest e x c i t e d waves, the wave slope and the t o t a l wind generation f o r c e which i s p r o p o r t i o n a l t o (wind speed - wave speed). K i t a i g o r o d s k i i and Z a s l a v s k i i , 1974, consider the phase speed of the dominant wave and a purely viscous momentum f l u x . B u r l i n g and Stewart, 1967, examine the i m p l i c a t i o n s of dependency on various moments of the wave spectrum. With the roughness 16 lengths possibly depending on many parameters i t appears that a rather detailed knowledge of the sea surface would be required before the turbulent fluxes over the sea could be found from Zo and Zor. The parameterizations can also be regarded as empirical formulae. An experimental formulation of the neutral c o e f f i c i e n t s at 10m, 2.12, for example* would allow the fluxes to be estimated from mean or bulk quantities, UZ, TZ and TSFC (AT = TSFC-TZ), with Z/L, i f known, providing a s t a b i l i t y correction. The momentum f l u x can be found from 2.9 by finding the drag c o e f f i c i e n t at the measurement height, Z, the wind speed, OZ, and the s t a b i l i t y , 2/L, from CDN. Elimination of Zo from 2.12 and 2.8 and substitution into 2-9 leaves, CD = CDN {1+ CDN1/2 K - 1 (In (Z/10m)-Ym (Z/L) ) } - 2, (2.14) where ln(Z/10m) and Yin (Z/L) describe the variation of the drag c o e f f i c i e n t with height and s t a b i l i t y , respectively. If CDN i s given as a function of the 10m wind* then DZ must f i r s t be shifted to 10m before a drag c o e f f i c i e n t can be determined. Substituting u*/UZ = CD*/2 = C10V2 010/DZ, i n t o 2.11 and solving for UZ/010 leaves, DZ/010 = 1+ C10V2 K~ l [ln(Z/10m)-Vin(Z/L)+Yln(10m/L) ]. The term i n square brackets, K(Z,Z/L), i s usually dominated by 17 ln(Z/10m), which, with C10 1/ 2/ = 0.1, makes 010 about 10% l a r g e r or s m a l l e r than UZ f o r a Z of 3.7m and 27m, r e s p e c t i v e l y . Throughout t h i s height range n e g l e c t of the s t a b i l i t y p o r t i o n of K(Z,Z/L) i n t r o d u c e s an e r r o r i n 010 of l e s s than 2% f o r -1.0 <Z/L< 0.08, however at Z=27m i t decreases 010 by about an a d d i t i o n a l 6% with Z/L=0.2 and 3% at Z/L=0.1, so i t i s not ign o r e d . C10 i s e q u i v a l e n t t o CD a t Z=10m, so 2.14 g i v e s C10V 2 = CDN*/2 {1 - K~l CDNi/2 Ym (10m/L) }~i . Since Ym(10m/L) ranges from about 1 a t Z/L= -1 to about -1 at Z/L=0.2, t a k i n g the term i n c u r l y brackets t o be 1.0 i n t r o d u c e s an e r r o r of only 1% i n 010, which i s w i t h i n usual measurement e r r o r , when i t i s c a l c u l a t e d from OZ, CDN and Z/L using 010 = UZ [1 + CDN*/2 K-1 K{Z,Z/L)]~i. (2.15) Should CDN i t s e l f depend on 010, 2.15 may have to be s o l v e d with an i t e r a t i v e technique. I f the s t a b i l i t y i s i n the range -1.0 <Z/L< 0.2, but i s unknown and assumed t o be n e u t r a l , e r r o r s a r i s e from an i n a c c u r a t e 010, i n f i n d i n g CDN from 010 and through 2.14, because CDN i s not s h i f t e d t o t h e proper s t a b i l i t y . The t o t a l e r r o r i n 010's found from 2.15 s h o u l d be l e s s than 10%. Smith and Banke, 1975, r e p o r t a drag c o e f f i c i e n t equal to 0.00061 + 0.000075 U10, so a 10% e r r o r i n 010 at 20 m/s, reduces to 7% i n CDN and the momentum f l u x , which i s l e s s than the e r r o r a s s o c i a t e d with d i r e c t measurements. With Y i n (Z/L) ranging from 18 1.0 to -1.0, t a k i n g i t to be zero could l e a d to a 20% e r r o r i n the momentum f l u x . Only i n the range -0.25 <2/L< 0.1 does the t o t a l e r r o r i n assuming n e u t r a l s t a b i l i t y remain l e s s than 10%. I f CD and CDN are a v a i l a b l e , the analogous procedure can be followed to f i n d the s e n s i b l e heat f l u x from TZ, TSFC, and 0Z, using 2.9. The Stanton number, CT, i s expressed as a f u n c t i o n of CTN, CDN, CD, and Z/L, by e l i m i n a t i n g Zo and Zot from 2.12 and 2.8, then s u b s t i t u t i n g i n t o 2.9: CT = CTN (CD/CDN) */2 . [ 1 + CTN K- 1 CDN-*/2 ( l n (Z/10m)-"Vl (Z/L) ) ] (2.16) The major e r r o r comes from the un c e r t a i n t y i n CTN. The e r r o r s i n CDN and CD due to U10 cancel i n (CD/CDN) and a 10% e r r o r i n CDN and CTN introduces a 2% e r r o r i n the denominator. Since Yt(Z/L) i s n o t very d i f f e r e n t from Yin (Z/L) , assuming n e u t r a l s t a b i l i t y causes about the same e r r o r i n CT as i n CD. However, t h i s assumption i s never needed, because a means of est i m a t i n g Z/L from OZ and AT, which are re q u i r e d i n 2.9, i s developed i n s e c t i o n 2.6. The un c e r t a i n t y of t h i s estimate ( s e c t i o n 4.2) should r e s u l t i n about a 5% er r o r i n CT and CD. The s e n s i b l e heat f l u x i s sometimes parameterized as <wt> = a 010 (TSFC-T10) + b , where a and b are experimentally determined f u n c t i o n s of s t a b i l i t y and wind speed. The a i r temperature at 10m, T10, can be found as f o l l o w s ; from 2.11 19 TZ - T10 = t * [ In (Z/10m) - (Z/L) +>t(10m/L)] , where the term i n square brackets, t o be denoted as Kt(Z,Z/L), behaves as K(Z rZ/L) and the s t a b i l i t y c o n t r i b u t i o n can again be ignored i n near n e u t r a l c o n d i t i o n s . S u b s t i t u t i n g f o r t * using 2.9 gives T10 = TZ • (CT/CDi/2) Kr1 (TSFC-TZ) K t ( Z f Z / L ) . (2.17) 2.4 The Reynolds Flux Method The Reynolds f l u x or eddy c o r r e l a t i o n method i s the most d i r e c t measurement of the f l u x e s and has been employed over the sea by Pond et a l . , 1971, H i c k s , 1972, Smith and Banke, 1975, and others. I t i n v o l v e s i n t e g r a t i n g the cospectra of w and e i t h e r u or r to obta i n the covariances and hence f l u x e s . The s p e c t r a l forms of the covariances are: <uw> = JVuw(f) df <wt> = J ^ w t f f ) df (2.18) <wr> = _/>vr(f) df. In p r a c t i c e the cospectra are determined by d i g i t a l f a s t F o u r i e r transform techniques, which give d i s c r e t e values of f ( f ) at i n t e r v a l s of f such t h a t f(f) Af gives the covariance i n a band centered at f of width Af. The highest frequency computed, the Nyquist frequency, fny, i s set by the d i g i t i z a t i o n period At: 20 (fny=1/2_t) . Contributions from higher frequencies can be allowed to a l i a s back below fny so that the e f f e c t i v e upper l i m i t of the integration i s increased ( to about 2 fny i n the system described i n section 3.3). The contributions to the covariances from natural frequencies, n=fZ/<0>, greater than 1 are only a few per cent of the t o t a l . With 20 m/s winds, measurements at 10 m w i l l therefore include most of the high frequency contributions i f At i s no longer than Z/<0>, about 0.5 seconds. The lowest calculated frequency, f 1 , i s the reciprocal of the duration of the measurement and ^(f1) includes the covariance down to f1 /2. Some cospectra are s t i l l non-zero at n=0.001, so i n order to be able to measure fluxes i n 5 m/s winds at 10 m the samples must be taken for about Z/(0.002<0>) = 1000 seconds or about 15 minutes. However ^(f1) i s not a s t a t i s t i c a l l y well determined quantity and i n practice at least 3 sequential determinations need to be averaqed, requiring the duration of a flux run to be at least 45 minutes. Unfortunately, measurements cannot be extended to more than about 1 hour because s t a t i o n a r i t y begins to be l o s t as a new flow situation develops. The low frequency contributions to the fluxes are often not well established by the Reynolds flux method. I t i s also clear that another disadvantage to this method i s the large amount of data required. For example, a sinqle <uw> estimate from a 45 minute run with _t=0.5 seconds requires about f i v e thousand d i g i t i z a t i o n s of each variable. Of course the spectra of the measured quantities may also be found from the same data and are often a great advantage in checking sensor performance. The s e n s i t i v i t y of the Reynolds flux method 21 to instrument orientation i s a great handicap on ships and buoys, whose motion and mean t i l t e f f e c t the measured covariances. A one degree error i n the mean t i l t cf an anemometer may induce errors i n <uw> i n the order of 10% (Pond, 1968). It i s possible to measure the instantaneous platform motion and to correct the v e l o c i t i e s point by point (Mitsuta and F u j i t a n i , 1974), but t h i s greatly increases the recording requirements and i s not a very p r a c t i c a l means of obtaining large amounts of open sea flux measurements. The Reynolds flux method i s not very applicable to remote open sea operation, but i t has become the standard to which other methods are compared either d i r e c t l y or through the calculated transfer c o e f f i c i e n t s . 2.5 The Dissipation Method Following Deacon's, 1959, suggestion the di s s i p a t i o n method has been employed in open sea conditions by Pond et a l . , 1971, Wucknitz, 1976, Denman and Miyake, 1973, and others. I t i s a very a t t r a c t i v e method because i t does not involve an e x p l i c i t measurement of the v e r t i c a l velocity* allowing moving platforms to be used and reducing measurement errors. In addition, flow di s t o r t i o n s that would hinder covariance measurements, can be tolerated. Instead, the major sources of error a r i s e i n the uncertainty of various constants and in the necessary assumptions. In the c i t e d studies i t s results have been 22 compared to d i r e c t Reynolds f l u x measurements up to moderate winds, but further comparison at high wind speeds i s s t i l l necessary. In the case of the momentum flux the method stems from a consideration of the balance of turbulent k i n e t i c energy per unit mass, e = (u 2 + v 2 + w 2)/2, i n horizontally homogeneous flow (Busch, 1977), cke> = u* 2 d<0> + g <w_Tv> - £ - 3 [<we> + 1, <wp>] dt dz To 5z £ P + B - € - D , (2. 19) where Tv i s the v i r t u a l temperature in degrees Kelvin and To i t s l o c a l average and p i s the flu c t u a t i n g pressure. Turbulent fluctuations are c h i e f l y produced by mechanical int e r a c t i o n s of the Reynolds stress with the mean flow represented by the f i r s t term, p=u*2 3<U>/dZ and lost at small scales to molecular d i s s i p a t i o n , £. From Lumley and Panofsky, 1964 p95, B i s recognizable as the lo s s or gain due to buoyancy referred to i n section 2.2. D i s the sum of two v e r t i c a l divergences: the f i r s t , of the turbulent k i n e t i c energy flux <we> and the second, of the work done per unit area by the flu c t u a t i n g pressure, <wp>/g. These are referred to as the turbulent and pressure transports of k i n e t i c energy, respectively. The complete divergence term has been investigated by McBean and E l l i o t , 1975. In t h i s work <we> and <wp> were measured over land at one height f o r a range of Z/L values. A f i t between -0.31 <Z/L< 0.12 gave 23 <wp> / u*3) = 2.3 Z/L - 0.20 . Their <we> r e s u l t s were p l o t t e d with those of G a r r a t t , 1972, and Banke and Smith, 1973. In view of the s c a t t e r i t i s not unreasonable t o assume a r e l a t i o n <we> / u* 3 = -2.3 Z/L + constant. Extensive measurements over land of the tu r b u l e n t t r a n s p o r t term have been made by Wyngaard and Cote", 1971, over a wider range of s t a b i l i t i e s . In unstable c o n d i t i o n s t h e i r r e s u l t s can be expressed as <we> / u* 3 = -2.5 Z/L + constant. The combined experimental evidence i n the range -1 <Z/L< 0.1 suggests t h a t t o a very good approximation <we> + <wp>/^ = a constant D i f f e r e n t i a t i o n by z i m p l i e s that on average, the k i n e t i c energy gained through pressure t r a n s p o r t n e a r l y balances that l o s t by turbu l e n t t r a n s p o r t , that i s , D=0- Wyngaard and Cote a l s o conclude that the e f f e c t s of h o r i z o n t a l inhomogeneities and n o n - s t a t i o n a r i t y are n e g l i g i b l e by more than two orders of magnitude, that i s , d<e>/dt = 0. 24 Combining 2.6 and the p r o f i l e equations, 2.7, with the remaining terms i n 2.19 r e s u l t s i n the simple set of equations: P = (u* 3 / KZ) 0m (Z/L) = Po fm(Z/L) B = - ( U * 3 / jcz) Z/L = -Po Z/L 6 = Po [0m (Z/L) - Z/L ] where Po, introduced i n s e c t i o n 2.2, i s the mechanical production i n the equ i v a l e n t n e u t r a l case. Thus u* can be simply expressed as a f u n c t i o n of € and Z/L: u* 3 = < Z € / [0m (Z/L) - Z/L] . (2.20) In a s i m i l a r f a s h i o n the problem of e v a l u a t i n g r * and the s c a l a r f l u x e s can be s i m p l i f i e d t o f i n d i n g u* and the d i s s i p a t i o n r a t e of s c a l a r f l u c t u a t i o n s , Nr. The analogue of 2.19 i s the simpler s c a l a r variance budget, (Busch, 1977), 1 _<r 2> = -<wr> d<R> - Nr - J. d<wr2> . 2 dt 6z 2 bz (2.21) The study of flyngaard and Cote (1971) i n v e s t i g a t e s t h i s equation thoroughly f o r temperature. The v e r t i c a l divergence term turns out t o be an order of magnitude smaller than the production term and the time r a t e of change and inhomogeneities are again n e g l i g i b l e . S u b s t i t u t i n g f o r d<E>/3z from 2.7 gives the st r a i g h t f o r w a r d r e l a t i o n s h i p r*z = Nr Z / [K u* fr(Z/L) ] . 25 (2.22) Direct measurements of 6 and Nr are d i f f i c u l t because they involve centimeter scales (frequencies well beyond 100 Hz). However, they can be inferred from the spectra of the scalars, ^r ( f ) , and downstream v e l o c i t y , ^ u ( f ) , at frequencies, f, i n the -5/3 region where the Kolmogoroff hypothesis predicts ^U(f) = K' 62/3 (27T/<0>)-2/3 f-5/3 ^r(f ) = Br' Nr € " V 3 (27r/<0>) - 2 / 3 f-sfc , (2.23) where Taylor's hypothesis i s used to replace the downstream radian wavenumber with (2TC f/<0>) . The form of these equations i s also based on dimensional analysis so the 1-dimensional Kolmogoroff constants K' and Br* may be functions of s t a b i l i t y , but they are not, as yet, well enough established f o r any dependency to be observable. Reasonable values are K' = 0.55 and Br' = 0.80 for both temperature and moisture (Paquin and Pond, 1971 and Busch, 1977), with a possible 10% error- In terms of the natural frequency, n=fZ/<D>, the -5/3 region has been found to be well developed by n=1 so that d i s s i p a t i o n estimates may be obtained from r e l a t i v e l y low frequencies (about 2 Hz, for 20 m/s winds at 10m heiqht). Several methods of calcula t i n g the momentum f l u x from measurements of 6 and Z/L are feasibl e . The simplest, method 1, used by Denman and Miyake, 1973, i s to employ the neutral form of 2.20 <UW>DISSI = ( K z e ) z/3 26 (2. 24) This equation i s al s o v a l i d i n non-neutral c o n d i t i o n s providing there i s an o v e r a l l balance between the v e r t i c a l divergences, the buoyant production and the s t a b i l i t y m o d i f i c a t i o n of the mechanical production. The experimental evidence over land suggests that the complete form of 2.20 should be t e s t e d over the sea. Method 2, t h e r e f o r e , assumes only t h a t the v e r t i c a l divergences balance and uses <uw>DISS2 = ( K Z € ) 2/3 • ( l _ (Z/L) - Z/L)-2/3 (2.25) This i s the method used by Khalsa and Businger, 1978, and by Wucknitz, 1976, but the l a t t e r uses a Eichardson number formulation and the former assume a balance, f o l l o w i n g Wyngaard and Cote, of buoyancy and t u r b u l e n t t r a n s p o r t with a pressure t r a n s p o r t of order -Z/L t o a r r i v e at 2.25. I t has ofte n been assumed that l o c a l production, P, balances d i s s i p a t i o n (Smith and Banke, 1975), i m p l y i n g an o v e r a l l balance between the two divergences and buoyancy. This i s the assumption of method 3, which s t a t e s <uw>DISS3 = {K Z 6)2/3 • [^m(Z/L) ]-2/3 . (2.26) Pond et a l . , 1971, found that i n unstable c o n d i t i o n s the momentum f l u x from the d i s s i p a t i o n and Eeynolds f l u x methods were i n the best agreement i f they assumed that the re d u c t i o n i n mechanical production due t o s t a b i l i t y m o d i f i c a t i o n of the 27 p r o f i l e was compensated by the net gain i n turbulent energy from v e r t i c a l divergences, that i s , 6 = Po + B. This fourth method i s expressed by <uw>DISS4 = ( n 6 ) ^  • (1 - Z/L)-2/3. (2.27) The general expression of the four d i s s i p a t i o n methods i s <uw>DISS = ( K Z 6 ) 2/ 3 • FX Method 1 F1 = 1 Method 2 F2 = [ ^ m(Z/L) - Z/L ]~2/3 (2.28) Method 3 F3 = [ ^m (Z/L) l " 2 / 3 Method 4 F4 = £1 - Z/L]- 2/ 3. The functions, FX, are plotted i n figure 1 over the range of s t a b i l i t i e s l i k e l y to be encountered over the sea at mid-latitudes. It i s apparent that a reasonable measure of s t a b i l i t y i s important to a l l but method 1. any one of the methods may be v a l i d over an i n d i v i d u a l run, but over any s t a b i l i t y range there should be one di s s i p a t i o n method that i s the most appropriate, on average. Equations 2.23 and 2.28 indicate that <uw> from a l l the dissi p a t i o n methods i s proportional to [ K Z] 2/ 3 K'-i ( f S i(f) <0>~2/3). (2.29) Even with no error i n the measurement of ^u (f) and <D>, the uncertainties i n K, 59S, in the measurement heiqht Z, say 0.5m i n 28 29 10m and in K', 10%, could combine to produce a 15% error i n <uw>. These errors and assumption errors are l i k e l y to be somewhat systematic, but fortunately they are not the same in the Reynolds flux method. Intercomparisons with Reynolds flux measurements are therefore es s e n t i a l , i n order to est a b l i s h the "best" d i s s i p a t i o n method and to ensure that there are no major systematic errors. There are fewer ways of "juggling" the terms of 2.21 to arr i v e at Nt, and only two methods of ca l c u l a t i n g the sensible heat flux are p r a c t i c a l . At neutral s t a b i l i t y and when the v e r t i c a l divergence term i s balanced by s t a b i l i t y modification of the temperature p r o f i l e , method 1, gives <wt>DISS1 = [ K u* Nt Z ]i/2 . (2.30) Again the experimental evidence over land suggests using 2.22 from which method 2 assumes <wt>DISS2 = [ K u* Nt Z]i/ 2 • [ ft (Z/L) ]~ 1IZ. (2.31) The general form of ca l c u l a t i n g the sensible heat f l u x i s simply <wt>DISS = [ K u* Nt Z]*/2 • FX Method 1 F1 = 1.0 (2.32) Method 2 F2 = [ ^ t (Z/L) ]-i/ 2 . F1 and F2 are shown i n figure 2. The methods d i f f e r considerably even near neutral s t a b i l i t y , therefore, the "best" FIGURE 2 S t a b i l i t y adjustments of the two d i s s i p a t i o n methods of estimating the sensible heat f l u x . 31 method should be easy to est a b l i s h , depending on the accuracy of the s t a b i l i t y measurements, since method 2 i s sensi t i v e to errors i n Z/L. Equations 2.32, 2.23 and 2.20 indicate that <wt> from both dis s i p a t i o n methods i s proportional to Again uncertainty i n K, Z and the Kolmogoroff constants together could produce a 15% error i n <wt>DISS. However, i t should be possible to substantially reduce systematic errors through Reynolds f l u x intercomparisons. 2.6 Estimating The S t a b i l i t y Parameter Z/L In section 2.2, the s t a b i l i t y of the a i r column i s characterized by a s t a b i l i t y parameter, Z/L, which plays a fundamental role i n the theory and measurement of turbulence. Unfortunately, i t i s d i f f i c u l t to obtain with 2.6, so three means of estimating i t from incomplete data w i l l now be investigated. The complete expression from 2.6 i s , ( K Z)2fr _ [ 0 t 0U]l/2 <U>"2/3 . [ Bt» K ' ]7/2 (2.33) Z = - KJL g L u* 3 < w_T v> . To Tv and To, the instantaneous and l o c a l average v i r t u a l temperatures, are defined as the temperatures reguired to give 32 dry a i r the same d e n s i t y as a c t u a l moist a i r at the same pressure.. Lumley and Panofsky, 1964, show t h a t ; Tv = T (1 + 0.61 mj , where Tv and the a i r temperature, T, are i n degrees K e l v i n and m i s s p e c i f i c humidity. Lumley and Panofsky (p 96) a l s o approximate the v i r t u a l temperature f l u x by <w Tv> = <wt> + 0.61 TZ <wm> , where TZ i s the mean a i r temperature a t the height Z. The convers i o n t o a b s o l u t e humidity, Q, i n g/m3, i s accomplished with (Phelps, 1971), Q = 1298 (273°K / To) m so Tv = T [1 + To Q 1.72 x 10-*] . Over temperate seas a t Z more than about 10m there i s not a l a r g e temperature g r a d i e n t and the v i r t u a l and a i r temperatures d i f f e r by l e s s than 2%, making To = <Tv> = TZ + T Z 2 QZ 1.72 x 10~6 Z = - K Z g <wt> • [1+ To 2 1.72x10 - 6 <w_> ] L u * 3 To <wt> (2.34) reasonable approximations. 33 Over temperate seas the moisture content of the atmosphere and the atmospheric pressure, a f f e c t the a i r de n s i t y by about 1% and 5% r e s p e c t i v e l y . For dynamic f l u x c a l c u l a t i o n s , i t i s enough t o only i n c l u d e the atmospheric pressure, PA i n JcPa, by c a l c u l a t i n g the density from Very often the absolute humidity and moisture f l u x , <wQ>, are unknown and Z/L must be approximated from measurements of u*, <wt> and the mean a i r and surface temperatures. The r a t i o of s e n s i b l e heat f l u x t o l a t e n t heat f l u x , the Bowen r a t i o , G, can be used to give i n g/m3/°C, where the pressure and moisture e f f e c t s on the density have been neglected. Phelps and Pond, 1971, report a value of 0.24 f o r G from t h e i r San Diego r e s u l t s . S u b s t i t u t i n g i n t o 2.34 gives £ = 1.29 (273/TZ) (PA/101) . (2.35) <w£> = p_Cp_ = 0.534 (273/ TZ) <wt> L G G Z = - K_Z g L u* 3 <wt> • [ 1 + 0.001 To], To which shows that the moisture f l u x may c o n t r i b u t e about on e - t h i r d as much to the s t a b i l i t y as does the s e n s i b l e heat f l u x . 34 An i n s i t u Bowen r a t i o can be estimated from the bulk aerodynamic parameterizations, s e c t i o n 2.3, v i z : G = £_Cj} <wt> = p_Cp_ <0>_CT AT . L <wQ> L <0> CE A Q The Stanton number, CT, and Dalton number, CE, have sometimes been found to be n e a r l y equivalent (Pond et a l . , 1971), but Francey and G a r r a t t , 1978, f i n d CT t o be 30% lower than CE. Because the humidity i s not the major c o n t r i b u t o r to Z/L and because of the large e r r o r i n AQ, CE and CT w i l l be assumed equal. The s a t u r a t i o n humidity as a f u n c t i o n of temperature, i s given by Hertzman et a l . , 1974, as QSAT (T) = C1 exp(C2/ T) with C1 = 6.4038 x 10» and C2 = -5107.4. This expression i s a f i t t o a t a b l e of s a t u r a t i o n h u m i d i t i e s over pure water at various temperatures so t h a t the surface humidity over s a l t water, QSFC, i s 0.98 QSAT(TSFC). In order t o reduce systematic e r r o r s , a r e l a t i v e humidity of 75% w i l l be assumed, which i s i n the middle of the humidity range expected over temperate seas. Since humidity c o n t r i b u t e s only about 30% to Z/L, t h i s assumption should introduce a random e r r o r i n Z/L of about 20% at worst and u s u a l l y of l e s s than 10%. The s e a - a i r humidity d i f f e r e n c e , AQ, and hence a Bowen r a t i o , can now be found from TSFC and TZ using G (AT) = (p Cp/L) AT [0.98 QSAT(TSFC) -0.75 Q SAT (T Z) ]~i . 35 S u b s t i t u t i n g i n t o 2.34 estimates Z/L from u*, <wt>, TZ and TSFC: Z(u*,<wt>) = - X_Z g <wt> • [1 + To 2.5x10-* 1 L u * 3 To G (AT) (2.36) In cases where <wt> i s al s o unknown, bulk parameterization replaces i t with CT OZ AT. The review by Friehe and Schmitt, 1978, i n d i c a t e s that CT i s about 1x10~ 3 i n unstable s t r a t i f i c a t i o n and about 0.86x10 - 3 i n s t a b l e . Therefore, a simpler estimate of Z/L i s given by Z/L (u*,AT) = - KJL g CT OZ__T [1 + To 2.5x10-* 1. u* 3 To G (AT) (2.37) Following Deardorff, 1968, the bulk formula, 2.9, replaces u* 3 with CD3/2 OZ 3 so tha t a s t a b i l i t y parameter Z/L (AT) can be determined soley from the bulk parameters, OZ, TZ, and TSFC. A po r t i o n of the Z/L expression i s i d e n t i f i e d as a bulk Eichardson number E i (AT) = -g _Z AT (1 + To 2_5x_10-*], OZ 2 To G (AT) so Z/L (AT) = K CT E i (AT) . CD CD*/2" A reasonable average CD i s 1.25 x 10~ 3 ( G a r r a t t , 1977), so Z/L (AT) = 11 E i (AT) (CT/ CD) (2.38) with CT/CD = 0.70 f o r AT < 0 = 0 . 8 0 AT > 0 36 i s a very p r a c t i c a l estimate of the s t a b i l i t y parameter. This d i f f e r s from Deardorff's f i n a l form of Z/L = 12 E i (AT) f o r unstable conditions, as i t r e f l e c t s more recent determinations of the bulk c o e f f i c i e n t s . However, i t ought to be compared to the more exact expression, 2.36, whenever <wt> and u* are both available* 37 CHAPTER 3 THE INSTRUMENTATION AND EXPERIMENTAL PROGRAM 3.1 I n t r o d u c t i o n In order to c o l l e c t the des i r e d amount of high wind speed data a Reynolds f l u x system and a d i s s i p a t i o n system have been designed f o r unattended operation. A d d i t i o n a l aspects of both systems i n c l u d i n g e r r o r a n a l y s i s , design c r i t e r i a , and sensor response requirements are given i n Pond and Large, 1978, and the d e t a i l e d a n a l y s i s of the v e l o c i t y measurement i s al s o i n Pond et a l . , 1979. The e s s e n t i a l c o n s i d e r a t i o n s are low power consumption and a l a r g e recording c a p a b i l i t y to keep the s e r v i c i n g period long and sensors able t o f u n c t i o n i n the hoped f o r 30-40 m/s winds and accompanying spray. When operating i n a h o s t i l e environment f o r long periods of time, sensors and e l e c t r o n i c s are l i k e l y to f a i l p e r i o d i c a l l y , so whenever p o s s i b l e the important measurements are e i t h e r d u p l i c a t e d or t h e i r sensors are c a l i b r a t e d i n s i t u . The r e s u l t s from two f i e l d operations are t o be presented i n t h i s study. The f i r s t was conducted on the Bedford tower near H a l i f a x Nova S c o t i a , which provided a s t a b l e enough platform to allow meaningful Reynolds f l u x measurements to be used to " c a l i b r a t e " the d i s s i p a t i o n system. Intercomparisons are a l s o p o s s i b l e with the a i r - s e a i n t e r a c t i o n system from the Bedford I n s t i t u t e of Oceanography, BIO, which was a l s o i n s t a l l e d on the tower. In a p r e l i m i n a r y experiment on Sable I s l a n d a l l systems were found t o be compatible when operating on the same platform. The r e s u l t s of that intercomparison and of a previous 38 BIO experiment on the i s l a n d are reported i n Smith e t a l . , 1976, and Smith and Banke, 1975, r e s p e c t i v e l y . The d i s s i p a t i o n measurements were extended to higher wind speeds and more open sea co n d i t i o n s i n a second experiment conducted from the CCGS Quadra during i t s p a t r o l s at ocean weather s t a t i o n "PAPA". 3.2 The Sensors The v e l o c i t y measurements are based on the G i l l p ropeller-vane anemometer (R.M. Young Co.), whose p r o p e l l e r s are c a r e f u l l y constructed h e l i c o i d s that t u r n a precise number of r e v o l u t i o n s f o r each meter of passing a i r (Baynton, 1970). This number was checked i n a wind tunnel and found to be w i t h i n 2% of the f a c t o r y c a l i b r a t i o n . This accuracy i s maintained from p r o p e l l e r to p r o p e l l e r and i s not a f f e c t e d by considerable a b l a t i o n of the l e a d i n g edge. A problem does a r i s e when the a x i a l wind f a l l s below about 1 m/s, because the i n e r t i a and f r i c t i o n begin to produce a non-linear output. Another problem i s that when the wind vector makes an angle -9- (angle of attack) greater than about 20°, to the p r o p e l l e r a x i s the apparent a x i a l v e l o c i t y component i s l e s s than the expected cos(^) times the magnitude of the wind, by a f a c t o r /3[$r) - For angles of attack between 35 and 75 degrees, t h i s non-cosine behavior i s approximated by -1. 1 03-0.27 -0-, f o r -9- i n radians (Pond and Large, 1978). Although these problems pose no s e r i o u s d i f f i c u l t i e s i n determining the h o r i z o n t a l v e l o c i t y components, they complicate the measurement of the v e r t i c a l v e l o c i t y . W i s derived from a propeller, Gill-w, whose axis i s t i l t e d at an angle, °<=60°, to the axis of a standard, G i l l - u , propeller, which i s generally t i l t e d at a small angle 8 from the horizontal (figure 3) . At an average wind speed greater than about 4 m/s the a x i a l component of the t i l t e d propeller always contains enough of the horizontal wind to avoid i t s non-linear regime. The propeller axes and instantaneous wind vector are kept i n e s s e n t i a l l y the same plane by the vane and i n t h i s way the geometry of figure 3B i s always maintained and corrections for the non-cosine behavior are possible. The twin propeller-vane anemometer of figure 3 i s described f u l l y i n the references cited, so only an outline of how the v e l o c i t y components are resolved follows. The defined angles of figure 3C and the following notation conform to these previous publications. The Gill-w (taking the ncn-cosine behavior at •d=°< into i t s calibration) and G i l l - U signals from the Reynolds flux system supply the v e l o c i t i e s : G i l l - u = V1 = Q cos 8 + w sin 8 and Gill-w = V2 = £Q cosH+fi)+w s in (*<+ S) }•£ 1-0..27 (6- tan- 1 (w/Q) ) ] with -e 8 - tan- 1 (w/Q) , (3.1) where Q and w are the horizontal and v e r t i c a l v e locity components, respectively. Because V2 contains a considerable contribution from the horizontal wind, the low frequency variations in the G i l l - u and Gill-w signals track one another very c l o s e l y , providing a check that everything i s working 4 0 FTGORE 3 A: The G i l l twin propeller-vane anemometer. B: The HAT sensor housing. C: D e f i n i t i o n of angles used i n r e s o l v i n g the v e l o c i t y components and c a l c u l a t i n g the t i l t angle o. properly. The t i l t angle 8 needs to be evaluated before Q can be removed from the Gill-w signal and w calculated. Since the average v e r t i c a l v e l o c i t y must eventually go to zero, a good estimate of 8 i s that angle of rotation needed to make the calculated <W> exactly 0, where < > denotes the averaging period. An average 8 i s measured over at least 15 minutes to give good averages of the r a t i o Y = <V1>/<V2> from which i t i s derived. Assuming tan-* (w/Q) = w/Q and <w>=0, <V1> = <Q> cos 8 and <V2> = <Q> {cos(<x+S) [1-0.27 5/^(^0 ] +<w2> <Q>-2 s i n « + S ) 0.27/ /${<<) } are correct to second order. The term i n <w2> <Q>-2 i s only about 0.2% of the previous term, leaving Y= {1-0.27 8/ /3{c<)} [cos<=< - s i n K tan 8 ] , which, assuming 8 - tan 8, gives a quadratic i n 8. It i s the negative square root of the quadratic formula that i s needed, so • 8 i s estimated from 8 = b - { b 2 + /0.27) (Y-coso<) /sin°< } b = 0.5 [cot ^ +/^(^0/0.27]. (3.2) This expression shows that o f f s e t errors, which enter Y, and errors i n °< and the/3 r e l a t i o n produce apparent t i l t s that can seriously affect Reynolds stress measurements. Now some 42 straight forward algebra y i e l d s : V 2 - Y V 1 = B w + (A/V1) w 2 A = £0.27/^(00 ] sin«+6) cos 6 (3.3) B = cos(«x+5) 0.27/^x) + {1-0.27 £//£(*)} si n <x/cos S, where Q has been replaced by V1/cos & and a term i n <w2>/<Q>2 and terms of order (w/Q) 3 have been neglected. On a fixed platform the instantaneous t i l t a c t u a l l y depends on the wind dir e c t i o n , but the use of the "average" 6 introduces very l i t t l e error (Pond and Large, 1978). The quadratic i n w i s solved using the positive square root of the quadratic formula, to give an estimate of w for each pair of recorded Gill-w and G i l l - u values i n the averaging period. The instantaneous horizontal v e l o c i t y components are then found from: Q = V1/cos £ - w tan 8 0 = Q cos (an) (3.4) V = v = Q sin (an) , where the instantaneous wind d i r e c t i o n , AN, eguals <AN>+an and the mean d i r e c t i o n , <AN>, i s chosen such that the average cross-stream velocity <V> = <Q sin(an)> =0. The possible errors associated with resolving 0,v#w and Q in t h i s manner are summarized i n table I (reproduced from Pond and Large, 1978) * together with t h e i r effects cn the calculated momentum flu x and drag c o e f f i c i e n t . Some errors should tend to cancel, so hopefully there i s no more than a ±10% error i n the average CD. 43 r — - — -j SOUECE T" |<uw>| i - -| CD r - — i | COMMENTS | i i I °< e r r o r o f ±1° j ±5% I ±5% 1 1 | The °< e r r o r i s | | b e l i e v e d t o be | | w i t h i n ±.5° w i t h | |2-3% e f f e c t s | I Offset at 5m/s | e r r o r s a t 10m/s ±2% ± U | ±3% | ±2% |Offset e r r o r s p a r t l y ! | c a n c e l r a t h e r t h a n | |add g i v i n g a b o u t | |1/2 t h e e f f e c t s | I±2% i n c a l i b r a t i o n ! ±4% I ±1% r e l a t i o n i ±3% I ±3% | S f l u c t u a t i o n s i i N e g l i g i b l e I N e g l i g i b l e |Non -cos ine | 8=0 I r e s p o n s e o f |£=±10° I V I p r o p e l l e r | 1 1 1 1 i • ±1% ±10% I ±1% I ±10% I For | S\ < 2-3° e r r o r | | i s s i m i l a r t o 6=0. | |For |6 | =5° i t i s | | l i k e l y w i t h i n ±5%. | I I I I I I TABLE I Summary of p o s s i b l e e r r o r s i n the v e l o c i t y measurement and t h e i r e f f e c t s on the Eeynolds f l u x method. Only the d i s s i p a t i o n method i s p r a c t i c a b l e on a moving platform, because the 8 derived from Y may be very d i f f e r e n t from the instantaneous t i l t needed to f i n d w. 0 i s a l s o not c a l c u l a t e d from t h i s type of data because i t i s not a simple matter t o separate the wind e f f e c t from the platform motion i n the vane's s i g n a l . The G i l l - u s i g n a l from the d i s s i p a t i o n system does give an average V1/cos £ (from which a ship's v e l o c i t y i s removed v e c t o r a l l y ) and approximate average values of Q (and 0, since <Q>=1.005 <0>) . In t h i s method the v e l o c i t y measurement i s not the major source of e r r o r ( s e c t i o n 2.5). A 44 + 2% c a l i b r a t i o n e r r o r only leads t o 2.7% and -1.3% e r r o r s i n |<uw>| and CD r e s p e c t i v e l y and a t 5m/s they are a f f e c t e d by the o f f s e t e r r o r by only 0.3% and 1.3%. However, only frequencies above those contaminated by the plat f o r m motion may be u t i l i z e d . The G i l l - w s i g n a l f u n c t i o n s as a check that the G i l l - u p r o p e l l e r and e l e c t r o n i c s are working properly and, i f necessary, as an inp u t t o the d i s s i p a t i o n method. The enclosure, HAT, of f i g u r e 3, serves as a r a d i a t i o n s h i e l d and o f f e r s p r o t e c t i o n against r a i n and spray f o r the temperature and humidity sensors that are mounted i n i t . Glass coated microbead t h e r m i s t o r s ( V i c t o r y Engineering Corp.) measure both the mean and f l u c t u a t i n g a i r temperature while g l a s s rod th e r m i s t o r s potted i n epoxy measure both the sea temperature and a mean a i r temperature.. A l l these transducers form part of s i m i l a r bridge c i r c u i t s whose n o n - l i n e a r i t y balances that of the th e r m i s t o r s , making the output of an o p e r a t i o n a l a m p l i f i e r detector l i n e a r over about a 25°C range. A l l probes were i n i t i a l l y c a l i b r a t e d i n a water bath against a standard mercury thermometer but l a t e r the rod th e r m i s t o r provides an i n s i t u c a l i b r a t i o n check of the microbead. The two temperature measurements should not d i f f e r by more than 0.1°C, when both the rod and microbead are working properly. Although no l a t e n t heat f l u x data are as yet a v a i l a b l e , a b r i e f d e s c r i p t i o n of the attempted humidity measurements f o l l o w s . On Sable I s l a n d the humidity f l u c t u a t i o n s were taken w i t h an aluminium-oxide sensor (Panametrics Corp.) and a Brady array (Thunder S c i e n t i f i c Corp.), but these f a i l e d because the 45 sensors d e t e r i o r a t e d i n the s a l t a i r environment. This pccurred l e s s r a p i d l y i n the case of the Brady so an attempt was made on the Bedford tower t o provide an i n s i t u c a l i b r a t i o n by r e p l a c i n g the aluminium-oxide probe with a second Brady covered with a 60 micron s t a i n l e s s s t e e l s i n t e r e d f i l t e r . The c a l i b r a t i o n d r i f t was much reduced, but s t i l l s e r i o u s , making both Brady arrays u n s u i t a b l e f o r long unattended operation. For up to a week or two the d r i f t of the f i l t e r e d Brady was not too bad, but i t s c a l i b r a t i o n was complicated by what may have been h y s t e r e s i s e f f e c t s and temperature s e n s i t i v i t y . In a d d i t i o n , the response of an open Brady i s marginal at best (Smith et a l . , 1976). I t t h e r e f o r e seemed best t o abandon the Brady array and t o regard a l l the data from i t as u n r e l i a b l e . For the ship operations where power reguirements are not r e s t r i c t i v e a Lyman-alpha humidiometer (Electromagnetic Research Corp.) has been employed to give the f l u c t u a t i n g humidity, while a Cambridge Systems (Model 2000) dewpoint system provided the i n s i t u c a l i b r a t i o n and average . The l a t t e r worked pr o p e r l y f o r over a month before needing s e r v i c i n g and i s promising. I t a l s o gives an a s p i r a t e d mean a i r temperature f o r checking the microbead. The Lyman-alpha, however, reguired constant a t t e n t i o n as i t s windows q u i c k l y became so d i r t y that i t s s i g n a l went o f f s c a l e before p r o v i d i n g any u s e f u l data. 46 3.3 The Reynolds Flux System This system i n c l u d e s the sensors, an e l e c t r o n i c s package and 6 d i g i t a l c a s s e t t e tape recorders t o sample, d i g i t i z e , and record the data needed to determine the t u r b u l e n t f l u x e s by the Reynolds f l u x method, s e c t i o n 2.4- I t processes each of i t s 6 channels i n the same manner as shown i n f i g u r e 4, which i l l u s t r a t e s the data flow and parameters considered i n converting stored data back to the o r i g i n a l p h y s i c a l q u a n t i t i e s sensed by the transducers (V1,V2,AN,R, where R i s any s c a l a r ) . Although the data processing i s , f o r convenience, shown f o r the spec t r a , i t i s a c t u a l l y performed on the Fo u r i e r c o e f f i c i e n t s from which the s p e c t r a and cospectra are derived. A Reynolds f l u x record c o n s i s t s of NG s e q u e n t i a l groups, each formed by sampling f i r s t the prewhitener, Vp, at 3Hz, NF times, then the low-pass f i l t e r , VL, at 1/SSP, NS times. The slow sampling period, SSP, i s made as long as p o s s i b l e to conserve power and tape. The high frequency variance and covariance l o s t because the low-pass f i l t e r prevents f u l l a l i a s i n g i s recovered by the f a s t subsamples. The 3 Hz r a t e i s f i x e d because i t i s as f a s t as 6 channels can be recorded by a p a i r of p a r a l l e l r e c o r d e r s , each r e c e i v i n g 3 channels. NF,NS,NG and SSP are programmed by means of thumbwheel switches and together they determine the record l e n g t h , the subsampling scheme and the p o r t i o n s of frequency space covered by each sampling r a t e . An a d d i t i o n a l switch s e t s the time i n t e r v a l between the s t a r t of rec o r d s . In order to prevent the c a s s e t t e s f i l l i n g with low wind speed data, a wind speed l i m i t can be set by another switch so that a scheduled record i s not taken i f the average wind speed i n the 47 | ENVIRONMENT | I D v w R fuvwr (f) | V1 V2 AN R r*2 (f) Transducer and Preamplifier | Gain=C Offset=B | Transfer Function Hs (f) | i Vs= B + C (V1,V2,AN,R) 4»s (f) =C2 <f>2 (f) | Hs (f) | 2 | Prewhitener I | dc gain=0 Offset=Bp | | Transfer Function Hp(f) | i , J VP <rp(f)= <rs(f) !Hp(f) | 2 i • 1 | Low-Pass F i l t e r I I dc gain=1 Offset=BL | | Transfer Function HL(f) | i VL=Vs+BL *L(f) = ^s (f) IHL(f) | 2 3 Hz Sample And Hold Multiplexer A To D Converter 12-bit Words 1/SSP NF Samples / Group NS Samples / Group | A pair of d i g i t a l cassette j | tapes store NG groups per | | fl u x record. j FIGURE 4: Signal processing i n the Reynolds flux system, showing the sampling scheme and a l l parameters considered in the analysis. 48 previous s i x minutes i s l e s s than the s e t l i m i t . The system continues t o c o l l e c t data u n t i l the three p a i r s of c a s s e t t e s are f u l l . The low frequency u,v,w and r F o u r i e r c o e f f i c i e n t s and spectra ^uvwr ( f ) , are a v a i l a b l e from the slow samples. The low-pass f i l t e r s have a 1 second time constant and t r a n s f e r f u n c t i o n , HL (f) (3db down at fL= 0.16 Hz). SSP has always been set to 3 seconds, g i v i n g a Nyguist frequency f s n , (1/6 Hz), almost equal t o f L , so the f i l t e r l o s s and a l i a s i n g should nearly cancel one another. The sensor t r a n s f e r f u n c t i o n , H s ( f ) , i s ignored because i t i s t y p i c a l l y 3db down at more than 5 times f s n . A f a s t Fourier transform of the slow samples produces s p e c t r a , ^ L ( f ) , which must be n e a r l y equivalent t o C 2 r*2(f) f o r f< f s n , where the g a i n , C, i n c l u d e s the dimensional conversion from p h y s i c a l u n i t s to voltage. Because of the n o n - l i n e a r i t y i n the w c a l c u l a t i o n , the v e l o c i t y data must f i r s t be converted to V1, V2 and AN = C-i (VL-BL-B) and then t o U, v and w with equations 3.3 and 3.4, before transforminq and o b t a i n i n g the ^uvw(f) spectra. NS has always been set to 256, so one group of slows l a s t s 12.8 minutes and i t s F o u r i e r c o f f i c i e n t s occur at frequencies from 0.0013 Hz to fns and c o n t a i n the variance and covariance from 0.00065 Hz to 0.167 Hz. Finding the high frequency c o f f i c i e n t s from the f a s t samples i s more complicated because the prewhitener. Hp (f) , and sensor responses are e x p l i c i t l y i n v o l v e d . The prewhitener c i r c u i t s behave as time d i f f e r e n t i a t o r s at low frequencies to increase the s i g n a l l e v e l s and e l i m i n a t e s p e c t r a l d i s t o r t i o n 49 from non-sampled fr e q u e n c i e s , then r o l l o f f as R-C low-pass f i l t e r s a t the higher frequencies. By design t h e i r response i s maximum and nearly f l a t at the f a s t sampling Nyquist frequency, 1.5 Hz, so that a l i a s i n g occurs without much l o s s of c o n t r i b u t i o n s below a few Hz. The F o u r i e r c o e f f i c i e n t s of the f a s t samples (corresponding to f*p (f)) are e a s i l y converted to 7*2 (f) = ^p(f) [ C | H s ( f ) | |Hp(f)| ]-2 . With the v e l o c i t y channels, the 02(f) are inverse F o u r i e r transformed and a dc l e v e l i s added to make the l a s t f a s t sample of V1, V2 and AN equal t o the f i r s t slow sample of each group. 0, v, and w are then c a l c u l a t e d using the mean coordinates axes as determined by the slow samples, and i t i s the Fo u r i e r c o e f f i c i e n t s of these time s e r i e s that are used to produce the desi r e d high frequency 0uvw(f) spectra and cospectra. Without the n o n - l i n e a r w c a l c u l a t i o n only one transform need be performed and t h i s p o s s i b i l i t y i s discussed i n the c i t e d references. NF has always been set to 128 making the lowest f a s t frequency 0.0234 Hz, so a l l the variance and covariance of the lower 6 and part of the seventh frequency bands are already contained i n the slow samples. Flux estimates and s t a t i s t i c a l q u a n t i t i e s are c a l c u l a t e d from runs of NG RP s e q u e n t i a l groups of a record. The t o t a l run time i s used as the averaging period i n the 8 c a l c u l a t i o n and coordinate determination. The group s p e c t r a and cospectra are averaged together over the run. An attempt has been made to use the overlapping frequency bands to adjust the high frequency 50 portion, however, t h i s has not been successful because the lower 6 estimates from the f a s t samples are not s t a t i s t i c a l l y very certain and often r a d i c a l l y different from one another and from the other high frequency estimates. The 128 low frequency values span 0.00065 <f< 0.167 Hz, but the eighth high freguency band does not begin u n t i l f= 0.176 Hz so an intermediate value at f= 0.1715 Hz (bandwidth 0.009 Hz) i s formed by i n t e r p o l a t i n g between the seventh and eighth values. The resulting 186 point spectra and cospectra are integrated i n three d i f f e r e n t ways. The f i r s t , the 11 method, i s to simply multiply each value by i t s bandwidth and sum, that i s , to integrate from f= 0.00065 Hz. There are two disadvantages with t h i s method: f i r s t , the lowest natural frequency, n= fZ/<0>, included in the integration decreases with <0> and t h i s could give r i s e to apparent wind speed dependencies; second, the lowest frequencies are not as s t a t i s t i c a l l y c ertain as one would l i k e and f a i l u r e to converge over a flux run could create a great deal of v a r i a b i l i t y i n observations. A second integration method, 12, a l l e v i a t e s the f i r s t problem by including the variance and covariance cf the i n d i v i d u a l group means about the o v e r a l l mean of a run. This e f f e c t i v e l y adds a further NGBP samples at a 13.5 minute period, which contain the contributions from 0.00062/NGEP to 0.00062 Hz. However, the s t a t i s t i c a l uncertainty i n t h i s additional contribution i s very high, enhancing the second 11 problem. Note that a small spectral gap from 0.00062 to 0.00065 Hz i s present in the 12 integration, because the slow sampling i s suspended during the f a s t sampling. The 13 method always begins 51 i t s integration at the same natural frequency, n= 0.004, where the spectral values are reasonably well established. The in t e g r a l i s then multiplied by a constant factor E to compensate for the excluded low freguency contributions. The value of E for each quantity i s found from the normalized spectrum or cospectrum, NT* (n) , which i s determined i n section 4.3 from averaqes over a l l available runs^ The non-dimensional (n), plotted against log{n) for convenience, should display a universal form, depending only on Z/L. In section 4.3, E i s calculated from the normalized wt and uw cospectra and i s found to be a function of s t a b i l i t y . Although the implied low frequency contribution may not be exact, the results should give representative averages and be subject to minimal v a r i a b i l i t y . The three methods are expressed by: f~ CO 11 = / ^(f) df J.00065 (~. 00062 1 2 = 1 1 + ^ (f) df (3.5) J .00062/NGEP J r- OO r*(f) df f=.004<D>/Z N^(n) d(logn) / j N^(n) d(logn). 0 70.004 The 13 method i s the most a t t r a c t i v e and i s others in section 4.5 to determine i f applicable. compared with the i t i s universally 52 3.4 The Dissipation System This system employs the same sensors and preamplifiers, but has i t s own electronics package and two d i g i t a l cassette recorders. It provides the estimates of 6 and Nr f o r the flux calculations (section 2.5) and the mean sea and a i r temperatures, humidity and wind for parameterization (section 2.3) and s t a b i l i t y c a l c u l a t i o n s (section 2.6). The architecture of the data flow i s shown i n figure 5. Between recording at i n t e r v a l s of Dt, each band-pass f i l t e r output i s sampled at 20 Hz, d i g i t i z e d , squared and summed NI times. The sums are stored i n i n t e r n a l memory units u n t i l they are written onto a cassette, at which time a l l the low-pass f i l t e r channels are also sampled and recorded. Switches proqram NI to be as large as possible while s t i l l allowing the summation to be completed before the st a r t of a tape write. Dt i s usually set to 4 or 5 minutes, making t h i s recording system very economical of tape. The low-pass f i l t e r s are single pole E-C c i r c u i t s with a 25 second time constant, followed by a unity gain operational amplifier, hence they are well suited f o r providing the mean velo c i t y and scalar values over the averaging period used for the £ ca l c u l a t i o n and vel o c i t y computations of section 3. 1. These averaged re s u l t s may be combined to give the means over longer time i n t e r v a l s i f desired. The band-pass f i l t e r s consist of double pole high and low pass stages centered nominally at fc= 0.4, 0.8 and 1.6 Hz. The prewhitener i s a d i f f e r e n t i a t o r at low frequencies that r o l l s off as a single pole R-C f i l t e r above 10-15 Hz. The two f i l t e r s 53 | ENVIRONMENT | |,. D v w R T*uvwr (f) | V1 V2 AN R •2 (f) I Transducer and P r e a m p l i f i e r | I dc gain=C Offset=B | | T r a n s f e r F u n c t i o n Hs (f) I 1 . 1 Vs = B + C (V1,V2,AN,R) I Prewhitener I | dc gain=0 Offset=Bd H*-| T r a n s f e r Function Hd(f) | ^s(f)=C2 <^2(f) |Hs(f) | 2 | Low-Pass F i l t e r - M dc gain=1 Offset=EL | T r a n s f e r F u n c t i o n HL (f) Vd = ^ s ( f ) |Hd (f) | 2 VL= Vs+BL | .4Hz | I Band-Pass | I F i l t e r H4(f) 1 I I . 8Hz Band-Pass | F i l t e r H8 (f) | J I 1.6Hz | | Band-Pass | | F i l t e r H16 (f) | 20Hz ± 1/Dt ± M u l t i p l e x e r A To D Converter I | Square, sum and s t o r e the | J Band-Pass channels | T o t a l Power P D i g i t a l c a s s e t t e , r e c o rds the | sum of NI squares and the | Low-Pass samples every Dt |~ seconds. | I FIGURE 5 : S i g n a l p r o c e s s i n g i n the d i s s i p a t i o n system. 54 together form band-pass f i l t e r s represented by combined transfer functions He (f) with center frequencies at about 0.55, 1 .05 and 2.1 Hz. Since the input spectrum f a l l s o f f rapidly ( f - 5 / 3 ) the frequencies of greatest power output are near the f c values. The stored data words divided by NI give the average power, <P>, passing through each band-pass f i l t e r . A -5/3 spectrum across He i s assumed i n order to get a discrete spectral value at each f c for a l l selected channels from <P> = RHO <r (fc) where RHO = C 2 j ( f / f c) -s/3 | H s ( f ) | 2 | He (f) 1 2 df with the i n t e g r a l taken over the range of frequencies passed by the f i l t e r s . Because RHO contains Hs (f) , a function of 0 (section 3.5), the i n t e g r a l must be evaluated at each wind speed for each band-pass f i l t e r channel. Equation 2.23 relates 0(fc) to the d i s s i p a t i o n of scalar f l u c t u a t i o n s , Nr, the molecular d i s s i p a t i o n , €, and the one-dimensional Kolmogoroff constants, Br' and K*. In the case of scalars 0 r ( f ) = 02(f) and Nr = <P> € _ _ (2TT/<0>) 2/ 3 f c S / 3 (3.6) RHO Br' i s available from each scalar band-pass f i l t e r once 6 has been determined. Calculating €. from the velocity signals i s more complicated because V2 always contains some w and a non-zero t i l t introduces some w into V1 as well. Some error i s thus 5 5 introduced i n t o EHOr because at the frequencies u t i l i z e d only 0u(f), and not 0w (f) , i s expected t o be p r o p o r t i o n a l to f ~ 5 / 3 and hence n e i t h e r V1 nor V2 should have e x a c t l y a -5/3 spectrum. Since the s p e c t r a l values of the h o r i z o n t a l v e l o c i t y component are n e a r l y equal t o those of the downstream component, 0u(f), (Pond and Large, 1978), 3.1 qives 02(f) = A 0u(f) S (8,f) f o r the V1 s i g n a l A = c o s 2 8 S(6,f) = [1+tan 2 S ^w(f)/^u(f) +2tan8 0uw (f) /0u(f) ]. S (8,f) should r e a l l y be placed i n s i d e the i n t e g r a l of the RHO expression and i n t e g r a t e d over the band-passed frequencies, but because 0w(f)/0u(f) and 0uw(f)/0u(f) can only be approximated, they are taken t o be constants over each f i l t e r and placed outside the i n t e g r a l to give an approximate S'(8). In the i n e r t i a l subrange 0w(f)/0u(f) = 4/3, but t h i s behavior i s not observed near the f c ' s and ins t e a d i t i s taken to be 0.81, 1.11 and 1.29 at f c = 0.4, 0.8 and 1.6 Hz r e s p e c t i v e l y . These are simply the averages of 14 so n i c anemometer observations from Sable I s l a n d which a l s o show 0uw(fc)/^u(fc) t o average -0.11, -0.16 and -0.15. S» (8) i s only a small c o r r e c t i o n (about 1.03 at 8=-5° and 0.98 at 8= + 5°) , but not accounting f o r i t introduces a systematic e r r o r , which on a lean i n g tower could turn out to be a f u n c t i o n of wind d i r e c t i o n and hence f e t c h . Therefore, S'(8) i s a p p l i e d to reduce the e r r o r and make i t more random. The c o r r e c t i o n i s l a r g e r and the e r r o r more s e r i o u s i n 56 the V2 case where 3-1 gives A = [ cos ^ ( ^ / ^ ( X ) ]2 S(6,f) ={1+tan2 K+£) 0w(f)/^u(f)+2tan (<<+£) u^w (f)/^u (f) } An approximate S'(6) can again be used, but -O' = £+ °< must be assumed since the data to c a l c u l a t e the instantaneous angle of attack at each d i g i t i z a t i o n i s not a v a i l a b l e . Therefore, whenever p o s s i b l e i t i s f a r more d e s i r a b l e to c a l c u l a t e the d i s s i p a t i o n from the G i l l - u rather than the G i l l - w data. With these approximations and r e s e r v a t i o n s each v e l o c i t y band pass f i l t e r g i v e s : 3.5 Sensor Response When c a l c u l a t i n g € and Nr, sensor response c o r r e c t i o n s are e s s e n t i a l and they are of some importance t o the Reynolds f l u x measurements. F o r t u n a t e l y , the d i s s i p a t i o n system provides a means of e s t a b l i s h i n g Hs (f) under the a c t u a l t u r b u l e n t c o n d i t i o n s encountered. Assuming the sensors behave as an R-C f i l t e r (3db down at fo : Hs(f)= (1 + j f / f o ) - * , j=/ rT ) , e i t h e r Nr or 6 values from any two band-pass f i l t e r s can be made equal by an appropriate choice of f o . Of course random departures of the input spectrum from i t s average -5/3 slope create s c a t t e r i n these f o ' s , but on average nearly the same response i s i n d i c a t e d € z / 3 = ( 2 7 T / < 0 » 2 / 3 <P> fcf/l A K • RHO S • (B) (3.7) 57 by a l l three p a i r s of f i l t e r s (the 0.4 and 0.8 Hz, the 0.4 and 1.6 Hz and the 0.8 and 1.6 Hz). The success of t h i s technique e s t a b l i s h e s t h a t the assumption of an E-C response i s appropriate and that there i s a c o n s i s t e n t average -5/3 region throughout the range of frequencies passed by the three f i l t e r combinations. The response of mechanical sensors, such as the G i l l p r o p e l l e r s , i s c h a r a c t e r i z e d by a distance constant, D = the wind passage r e q u i r e d f o r a 63% recovery from a step change i n v e l o c i t y . The E-C f i l t e r analogue gives 2ix fo=<U>/D. I t i s more important t o e s t a b l i s h fo during low winds where the response i s poor. In f i g u r e 6, the hourly averaged wind speed i s p l o t t e d against the 2 7Tfo required to make the G i l l - u c a l c u l a t i o n s of € from the 0.8 and 1.6 Hz f i l t e r s equal. Now i f D were t r u e l y a constant t h i s p l o t would be a s t r a i g h t l i n e through the o r i g i n of slope D. The data imply that below 12 m/s, D i s about 0.65m, a s i g n i f i c a n t l y b e t t e r response than the quoted value of 0.8m. At higher speeds the response appears to improve even more than expected (D decreases) and t h i s i s incorporated by using the s o l i d l i n e of f i g u r e 6A, D = 0.56m 0/(0-1.Om/s). The l i n e i s a f i t , f o r 0> 4m/s, to comparisons of a l l three band-pass f i l t e r outputs over the e n t i r e time that t h i s p a r t i c u l a r anemometer-propeller (19cm, 2-bladed) combination was i n use. The response changes only s l i g h t l y with a d i f f e r e n t anemometer, but depends s t r o n g l y on the type and weight of the p r o p e l l e r . Heavier p r o p e l l e r s of s i m i l a r c o n s t r u c t i o n , used on Sable I s l a n d , give D = 1.0m 0/(0-0.7m/s) and 19cm 4-bladed ones show D = 0.79m 0/(0-1.8m/s). 58 o CM CM 1 1 1 1 [ fi: G I L L - U C D i — i — I — I — I — I — I — I — I — I — I — I I I T 0. 5. 10. 15. 20 . 25 . 30. 35 1 1 — I — I — r B: GILL-W 1 — I — r +++ + — i — i — i — r o. 5 . 10 ~ i — i — i — i — r 15. 20 . 25 2TTf B (HZ) 30. 35 FIG DEE 6: Deteraination of the distance constants from the 0.8 t 1.6 Hz band-pass f i l t e r r a t i o s of A: the G i l l - u horizontal propeller, B: the Gill-w t i l t e d propeller, °< = 59.5°. 59 The data of f i g u r e s 6A and 6B come from the same time periods, but e v i d e n t l y more a i r must pass i n the h o r i z o n t a l (longer distance constant, DT) before the t i l t e d p r o p e l l e r responds. This may r e f l e c t the non-cosine behavior, because assuming DT=D//5?(°<) gives the s o l i d l i n e of f i g u r e 6B and DT=1.22 D for°<=60o, which i s an acceptable f i t f o r 0<12 m/s. The technique f a i l s at higher speeds because the w spectrum which forms a l a r g e p a r t of the input s i g n a l , V2, i s no longer -5/3 throughout the G i l l - w 0.8 Hz band-pass f i l t e r . This means that u s e f u l G i l l - w data are l e s s p l e n t i f u l as w e l l as more uncer t a i n than that of the G i l l - u , so i t seems pre f e r a b l e t o take the distance constant of the t i l t e d p r o p e l l e r as D//3{<K). Such an approach t u r n s out to l i e between Hicks (1972 B) and G i l l (1975), whose r e s u l t s f o r a 60° angle of attack give DT=1.41 D and 1.15 D, r e s p e c t i v e l y . A s i m i l a r procedure was used t o e s t a b l i s h the i n s i t u response of each microbead t h e r m i s t o r mounted i n the HAT. Reg r e t t a b l y , i t had t o be based on very l i t t l e data because each bead worked f o r only a r e l a t i v e l y short time and a l o t of recorded data has been r e j e c t e d because of c a l i b r a t i o n problems and suspected s a l t contamination of the microbead. Most of the r e l i a b l e temperature data from the Bedford tower i s inc l u d e d i n f i g u r e 7, i n which the hourly averaged wind speed i s p l o t t e d against the sensor response r e g u i r e d t o make Nt c a l c u l a t e d from the 0.8 and 1.6 Hz temperature band-pass f i l t e r s , equal. C l e a r l y the response improves with wind speed. The s o l i d l i n e , of slope 0.90m, i s an acceptable f i t to a l l the data, implying that the microbead response, i n winds up to at l e a s t 20 m/s, can F I G U R E 7: Determination of the microbead thermistor response from the 0.8 : 1.6 Hz band-pass f i l t e r r a t i o s . The s o l i d l i n e has slope 0.90m. 61 a l s o be described as an E-C f i l t e r with a distance constant, DB. Miyake et a l . (1970 B) quote a 26 Hz response f o r s i m i l a r microbeads at an a i r c r a f t speed of 70 m/s (distance constant about 0.4m). The lower wind speeds may qive the poorer response i n d i c a t e d by f i q u r e 7. It i s suspected, however, t h a t the response i s l i m i t e d by the HAT i t s e l f and the success of a distance constant d e s c r i p t i o n i s a consequence of the amount of a i r needed to f l u s h the enclosure. Accordingly* the data from i n d i v i d u a l microbeads agree with a DB of 0.90m t o w i t h i n ±5%. I f the response i s being t r e a t e d p r o p e r l y , the Eeynolds s t r e s s derived from each of the three band-pass f i l t e r s should, on average, be equal.. In f i g u r e 8 the r a t i o s of €zl3 (from 3.7) and e q u i v a l e n t l y (from equation 2.28) <uw> are p l o t t e d as a fu n c t i o n of wind speed f o r the same data as used i n f i g u r e 6. Ev i d e n t l y the chosen response i s reasonable. Very few i n d i v i d u a l r a t i o s d i f f e r from 1.0 by more than 10%. At the lower winds the averages of a l l t h r e e r a t i o s are n e a r l y 1. 0 and i n the case of the 0.8/1.6 r a t i o , t h i s i s true up to 20 m/s. Between 12 and 20 m/s, the 0.4 Hz band-pass f i l t e r appears to have about 5% smaller s t r e s s values ( l e s s output power) than expected. This may r e f l e c t a d e v i a t i o n from a purely E-C response, but more l i k e l y i t i s evidence of the lower p o r t i o n of the f i l t e r l y i n g , on average, below the f - 5 / 3 frequencies during these higher winds. At 16 m/s the distance constant formulation gives D= 0.6m, but i f i t were taken t o be a constant 0.65m the observed 0.4/0.8 r a t i o would be even l e s s , because the response c o r r e c t i o n at 0.8 Hz increases f a s t e r than at 0.4 Hz as the response i s made poorer. The c o r r e c t i o n i s h i g h l y non-linear FIGURE 8: Ratios of <uw> calculated from di f f e r e n t pairs of band-pass f i l t e r s as a function of wind speed. The s o l i d l i n e of f i g u r e 6fl gives the sensor response* 63 with wind speed and very d i f f e r e n t from f i l t e r to f i l t e r . Because the r a t i o s of f i g u r e 8 vary so l i t t l e , on average, with mean wind speed, i t i s u n l i k e l y that e r r o r s i n the sensor response w i l l produce any major spurious wind speed dependencies. A s i m i l a r check on the microbead response i s not f e a s i b l e because both the 0.4 and 0.8 Hz band-pass f i l t e r s are often not e n t i r e l y i n the -5/3 region of the temperature spectrum. The i n e v i t a b l e presence of contaminated data adds a f u r t h e r c o m p l i c a t i o n . 3.6 The Experimental Program The Bedford tower experiment l a s t e d from September 1976 to A p r i l 1977. A d e s c r i p t i o n of the tower and the r e s u l t s from the BIO system can be found i n Smith, 1979. The tower was a f l o a t i n g spar buoy moored i n 59m of water, which makes the s i t e e s s e n t i a l l y a deep water wave regime (Smith, 1979). The l o c a t i o n and a photogragh of the tower are shown i n f i g u r e 9. The s h o r t e s t f e t c h to the tower s i t e i s from the west and i s about 10 km, while open f e t c h c o n d i t i o n s extend over a 170° range. The e l e c t r o n i c s packages can be seen on the main deck, about 3m above the sea. The G i l l and HAT are at the very top alongside the BIO t h r u s t and aerovane anemometers and microbead t h e r m i s t o r . The t i d e t a b l e s f o r H a l i f a x harbour are used to f i n d the phase and amplitude (assumed equal t o h a l f the t i d a l range) at the beginning of each run, from which the measurement height Z i s c a l c u l a t e d assuming a purely M2 t i d e , which i s the FIGURE 9 A: The Bedford tower s i t e near H a l i f a x Nova S c o t i a . B: The i n s t r u m e n t a t i o n on the Bedford tower. 65 predominant t i d e i n t h i s area. The sea temperature sensor was t i e d to the tower about 10m below mean sea l e v e l . Surface meteorological observations, i n c l u d i n g the atmospheric pressure, PA, f o r a i r density c a l c u l a t i o n s , were r o u t i n e l y recorded at the Shearwater "A" land s t a t i o n of the Atmospheric Environment S e r v i c e , l o c a t e d about 15 km north of the tower s i t e ( f i g u r e 9). While on the tower, the Reynolds f l u x system recorded the three anemometer s i g n a l s plus a microbead t h e r m i s t o r , an open Brady humidiometer and a f i l t e r e d Brady from the HAT. The switch s e t t i n g s of s e c t i o n 3.3 were employed so that f l u x runs of e i t h e r 3 or 4 groups corresponding t o 40.5 or 54 minutes r e s p e c t i v e l y , can be processed. From September 15 a new record began every hour, but on the 23rd the i n t e r v a l was incre a s e d to 3 hours and the lower wind speed l i m i t was set to 10 m/s. On October 4 the l i m i t was changed to 8 m/s, then increased to 12m/s on October 6 and put back t o 10m/s on February 15 where i t remained u n t i l the end of the experiment. Meanwhile the d i s s i p a t i o n system ( f i g u r e 5) sampled and recorded the low-pass data from the same s i x s i g n a l s p l u s a rod thermistor from the HAT and the sea temperature. The G i l l - u , G i l l - w , microbead and open Brady s i g n a l s were s e l e c t e d t o be band-pass f i l t e r e d . The number of summations, NI, was set to 4500 t o comply with a 4 minute Dt, which enabled the p a i r of c a s s e t t e s t o l a s t f o r 52.5 days. 66 The weather ship experiment i n c l u d e d the four p a t r o l s of CCGS Quadra between J u l y 1977 and A p r i l 1978. During the t h i r d and f o u r t h p a t r o l s a s i n g l e p r o p e l l e r G i l l anemometer was used with the G i l l - u s i g n a l a l s o being processed as the missing G i l l - w s i g n a l . The t y p i c a l mode of operation of the s h i p when at "PAPA" ( f i g u r e 10) was to d r i f t with the wind then r e t u r n to s t a t i o n by steaming i n t o the wind at l e s s than 4 knots. During the l a t t e r operation the great bulk of good data were c o l l e c t e d , but some u s e f u l data were a l s o c o l l e c t e d as the ship steamed at 7 to 12 knots while en route t o "PAPA" ( f i g u r e 10). The l o c a t i o n of the sensors on the sh i p ' s foremast i s shown i n the photograph of f i g u r e 10. Cables were run to the e l e c t r o n i c s packages two decks below the base of the mast. With winds coming over the bow the measured t i l t angles are t y p i c a l l y only about +7°, i n d i c a t i n g t h a t the ship's d i s t o r t i o n of the mean flow i s not enough to upset the d i s s i p a t i o n method. However, winds more than 30° to starboard or 60° to port were found to have g r e a t l y perturbed high frequencies, which proved to be very unfortunate because the ship d r i f t e d f o r many hours i n such winds. In a d d i t i o n the foremast l o c a t i o n r e c e i v e d a great deal of spray so that the microbeads broke and the lyman-alpha windows became d i r t y soon a f t e r the f i r s t encounter with heavy seas. As a consequence very l i t t l e temperature data and no humidity data are a v a i l a b l e from the s h i p . The spray a l s o caused a great deal of p i t t i n g i n the l e a d i n g edges of the p r o p e l l e r s , which they can f o r t u n a t e l y t o l e r a t e . In a subsequent experiment (JASIN 1978), the same sensors were mounted forward of the bow of the FS Meteor so t h a t the wind FIGUEE 10 A: Ocean Weather Station "PAPA", 50°N, 145°w, and the route of the weatherships. B: The instrumentation on the foremast of CCGS Quadra. 68 c a r r i e d the spray away and a great deal of both temperature and humidity data were c o l l e c t e d . While on CCGS QUADRA, the d i s s i p a t i o n system low-pass f i l t e r e d and recorded the s i g n a l s from the f o l l o w i n g sensors: a rod and two microbead th e r m i s t o r s from the HAT, the G i l l anemometer, a Lyman-alpha humidiometer and the dewpoint system. The two G i l l v e l o c i t y , the two microbead and the Lyman-alpha s i g n a l s were a l l band-pass f i l t e r e d * . Dt was set a t 5 minutes, a l l o w i n g NI to be 5800 and up to 56 days of data t o be st o r e d on the two c a s s e t t e s . The Reynolds f l u x system was i n c l u d e d to provide v e l o c i t y and s c a l a r s p e c t r a . I t was set up as on the tower, but the Brady arrays were replaced by the dewpoint and Lyman-alpha s i g n a l s . The wind speed l i m i t was always set to at l e a s t 8 m/s so t h a t f l u x records were taken throughout most of a p a t r o l . With these switch s e t t i n g s i t was p o s s i b l e to t u r n the systems on p r i o r t o s a i l i n g and to r e t r i e v e the data cas s e t t e s upon return seven weeks l a t e r . A sea surface "bucket" temperature and the atmospheric pressure are a v a i l a b l e from the ship's three hourly meteorological observations. 69 CHAPTER 4 REYNOLDS FLUX MEASUREMENTS FROM THE BEDFORD STABLE TOWER 4.1 Introduction The Reynolds f l u x data set from the Bedford tower consists of 196 momentum flux runs with winds up to 20 m/s and the majority of s t a b i l i t i e s i n the range -0.4 <Z/L< 0.1. In most cases the fetch i s unlimited, but winds from a l l directions, except those which put the sensors i n the wake of the BIO thrust (from the east), are allowed, so some fetches are as short as 10 km. The runs are r e s t r i c t e d to the 5 m/s or greater winds necessary to keep the t i l t e d propeller in i t s l i n e a r operating range, which also ensures that the measurement height, Z = 13 m, i s i n the "constant f l u x " layer. I t i s very g r a t i f y i n g to find that i n each case where simultaneous d i s s i p a t i o n data e x i s t s , 192 runs, the band-pass data confirm the existence of a -5/3 region i n the vel o c i t y spectra. Other runs are rejected because the Gill-w and G i l l - u signals do not track each other properly. A l l 196 runs have been considered for sensible heat flux c a l c u l a t i o n s , because even 5 m/s winds are s u f f i c i e n t to flush the HAT. However, during many of these runs the temperature data are not available, because either the microbead was broken or very cold a i r drove i t s sig n a l off scale. A few runs are also rejected because the mean a i r temperatures from the rod and microbead thermistors do not agree to within ±0.1°C. In addition many more runs have not been processed due to what i s believed to be the s e n s i t i v i t y of a s a l t contaminated microbead 70 to humidity f l u c t u a t i o n s as reported by Schmitt e t a l . , 1978. This behavior i s recognized by r e l a t i v e l y l i t t l e variance i n the temperature spectrum below n=0.01 and i n many of these cases, but not a l l , the d i s s i p a t i o n data r e v e a l that the temperature spectrum i s not f a l l i n g as steeply as -5/3. Only 60 of the 196 runs have been found to s a t i s f y c r i t e r i a f o r temperature f l u x c a l c u l a t i o n s . A l l the Reynolds f l u x r e s u l t s are tabul a t e d i n the Appendix and are r e f e r r e d t o as runs T1 t o T196. 4.2 The S t a b i l i t y Parameter Z/L A s t a b i l i t y parameter Z / L ( A T ) i s c a l c u l a t e d from equation 2.38 f o r each of the 196 runs. The surface temperature, TSFC, i s approximated by the d i s s i p a t i o n system's sea temperature probe, TSEA, which i s assumed t o have f a l l e n l i n e a r l y over a 16 hour period t h a t was not recorded (runs T90-T93). The mean a i r temperature, TZ, at the measurement height, Z, u s u a l l y comes from the rod thermistor of the d i s s i p a t i o n system, but over the unrecorded gap and during periods when the rod's s i g n a l e i t h e r has e r r a t i c behavior (runs T47-T55, T74-T85 and T138-T149) or i s o f f s c a l e , the f l u x system's recordings of the microbead are used. Unfortunately, during runs T102-T110, T117-T122 and T131-T133 n e i t h e r temperature sensor was o p e r a t i o n a l and i t i s necessary t o use the meteorological observations from Shearwater. In f i g u r e 11B, Z / L ( A T ) i s p l o t t e d against the more 71 8"0 t 7 * 0 0 * 0 (IV ) 1/Z t 7 ' 0 -FIGURE 11 K 0 ( j v • Xfl) O'O 1/Z Comparison of the most complete expression f o r the s t a b i l i t y parameter Z/L (u* r <wt>) t o : A: an approximate expression Z/L(u* #AT). B: the bulk estimate Z/L (AT) . 72 exact Z/L(u*,<wt>) (equation 2.36) f o r the 60 temperature runs. The two c a l c u l a t i o n s tend t o agree on averaqe and seldom d i f f e r by more than ±0.05, but o c c a s i o n a l l y the d i f f e r e n c e i s s u b s t a n t i a l (more than 0.2). Since u* i s c a l c u l a t e d f o r a l l the runs i t can be used, i n s t e a d of <0>, to c a l c u l a t e a s t a b i l i t y parameter Z/L(u*,AT) from equation 2.37. However t h i s estimate does not agree as w e l l with Z/L(u*,<wt>) as shown i n f i g u r e 11A, where systematic departures from a 1:1 r e l a t i o n s h i p are evident. Since <wt> i s only a v a i l a b l e from the temperature runs i t often must be approximated by CT <D> AT, but e v i d e n t l y the associated e r r o r i s p a r t i a l l y compensated by the e r r o r i n r e p l a c i n g u* 2 with CD <0>2. The bulk estimate Z/L ( AT) i s to be used e x c l u s i v e l y , because i t i s the best estimate of s t a b i l i t y that i s always a v a i l a b l e , even though i t may not be very accurate for an i n d i v i d u a l run. 4.3 Turbulence Spectra And Cospectra The spectra of the flu c t u a t i n g v e l o c i t y components and fluctuating temperature, f*(f), provide a means of evaluating the performance of the Reynolds flux system (section 3.3). To find a value of E f o r each quantity integrated by the 13 method, eguation 3.5, normalized spectra and cospectra, (n), are established by averaging over a l l available runs. A l l spectral values and the i r corresponding natural frequencies, n=fZ/<0> (186 per run), are f i r s t calculated* The f(n) are then 73 m u l t i p l i e d by f t o produce a v a r i a n c e p r e s e r v i n g p l o t a g a i n s t l o g ( n ) . Next t h e f 0(n) a r e n o n - d i m e n s i o n a l i z e d by d i v i d i n g by: u * 2 i n t h e case of v e l o c i t y s p e c t r a and c o s p e c t r a , (at) 2 f o r temperature s p e c t r a and <wt> =-Xu*t* f o r w,t c o s p e c t r a . F i n a l l y , t he d i s c r e t e n o r m a l i z e d s p e c t r a l v a l u e s from M runs i n a p a r t i c u l a r s t a b i l i t y range are band averaged over n, such t h a t Alog (n) r e m a i n s c o n s t a n t , g i v i n g a mean, t a k e n t o be N 0 ( n ) , and a s t a n d a r d d e v i a t i o n , cr, f o r each band. A p l o t of NT1 (n) vs. l o g (n) s h o u l d , a c c o r d i n g t o s i m i l a r i t y t h e o r y , d i s p l a y a u n i v e r s a l form, depending on s t a b i l i t y . The n o r m a l i z i n g f a c t o r s u*2=-<uw>, <tt>=(at) 2 and <wt> a r e found f o r each run by i n t e g r a t i n g 0uw, 0 t and 0 w t from n=0.004 and a r e t h e r e f o r e as much as 10% s m a l l , b u t the e r r o r i s independent of wind speed. There i s a g r e a t d e a l of s c a t t e r i n t h e 0 ( f ) ' s , not o n l y from run t o r u n , but between nearby f r e q u e n c i e s o f t h e same r u n , due to the i n h e r e n t v a r i a b i l i t y o f the F o u r i e r c o e f f i c i e n t s . The l a t t e r e f f e c t i s reduced by a v e r a g i n g over t h e NG groups of a f l u x r u n . However, i t i s not reduced f u r t h e r , by a v e r a g i n g over F o u r i e r bands, because t h i s g r e a t l y i n c r e a s e s t h e band-width, which s h o u l d be kept as narrow as p o s s i b l e i n o r d e r t o keep N0 (n) r e p r e s e n t a t i v e of i t s n a t u r a l f r e q u e n c y , e s p e c i a l l y a t low f r e q u e n c i e s where t h e r e a r e o n l y a few p o i n t s i n each band. As a consequence, or i s e x t r e m e l y l a r g e and not i n d i c a t i v e o f t h e v a r i a b i l i t y o f t h e mean, N0 ( n ) . S i n c e the M r u n s a re independent, crm-oy/F i s taken as an e s t i m a t i o n o f t h e s t a n d a r d d e v i a t i o n o f t h e mean. T h i s e s t i m a t i o n assumes G a u s s i a n s t a t i s t i c s , which s h o u l d be approached w i t h a l a r g e number o f r u n s , and t h a t t h e mean and cr are good measures c f the 74 population s t a t i s t i c s , which requires that NP, the number of points i n a band, be much larger than M. This l a t t e r assumption i s not s t r i c t l y s a t i s f i e d as NP approaches M, but because each run continues to contribute at l e a s t one point, the s t a t i s t i c s should remain nearly Gaussian with o"m=o/'vrM a reasonable approximation. S i m i l a r l y , when NP becomes less than M, each point comes from a d i f f e r e n t run and oti=o/v/NP~ i s assumed. In the following normalized plots N^(n) from each band i s plotted in the middle of the log(n) band and shown by a diamond with v e r t i c a l bars extending up and down 1 o"m. In the logarithmic plots the means (sguares) are plotted and s o l i d l i n e s of -2 / 3 slope, indicating f (f) proportional to f - 5 / 3 , are drawn. The runs are sometimes s p l i t into stable and unstable groups which are averaged and plotted separately. There i s no attempt to average over smaller s t a b i l i t y ranges because the large majority of runs span only a narrow range and because of the large uncertainty i n Z/L (AT) . Velocity Spectra The normalized spectra of the downstream velocity component, N^u (n), are shown in figure 12 and there i s a marked dependence on s t a b i l i t y . The peak of the spectrum of the stable runs* figure 12B, occurs at a natural frequency more than a decade higher than that of the unstable runs (figure 12A), whose spectrum i n turn has a greater proportion cf i t s energy at lower frequencies. The spectral points below n=10~3 come from the highest wind speed runs and t h e i r average over the natural 75 O ' O » O T i-OI Q'Z FIGDRE 12 A: Normalized spectra of the downstream velocity component from averages of f 0u(n)/u*2 (pluses from slow samples only). Solid l i n e has slope -2/3. Ve r t i c a l l i n e s are ± 1 o*m. 108 unstable runs, B: 88 stable runs. 76 freguency band, log(n)= -3.21 to -3.00 i s about 0.75 i n both the s t a b l e and unstable case. I t does appear, t h e r e f o r e , that a s p e c t r a l gap i s emerging between the f l u c t u a t i n g motion and the mean flow. Some platform motion i s expected, but i t i s not evident i n the s p e c t r a , perhaps because i t i s obscured by averaging over l a r g e freguency bands. The l o g - l o g p l o t s of f i g u r e 12 d i s p l a y a s i m i l a r shape and s t a b i l i t y dependence as do the over land spectra of McBean, 1971. In c o n t r a s t , the measurements over land of Kaimal et a l . , 1972, suggest sharper peaks. C o n t r i b u t i o n s t o the bands between n=0.1 and n=0.4 come from both the slow and the f a s t samples depending on wind speed and the small pluses on f i g u r e 12A represent the band averages using only the slow samples. These are p l o t t e d at the average value of log (n) and not at the band center as are the o v e r a l l averages. In t h i s range (0.1 <n< 0.4) the f a s t sampled spectrum does, on average, match the more s t a t i s t i c a l l y c e r t a i n s p e c t r a l values found from the slow samples. There i s no evidence of a l i a s i n g i n the slow samples c o r r o b o r a t i n g the argument t h a t i t i s compensated by the 1 second time constant low-pass f i l t e r . S i m i l a r matching i s not done i n f i g u r e 12B because the points f a l l near the peak of the spectrum. I t happens that a l l three G i l l - u band-pass f i l t e r s may be u t i l i z e d under a l l c o n d i t i o n s encountered at the tower and at "PAPA". The l o g a r i t h m i c p l o t s of f i g u r e 12 show that both the s t a b l e and unstable spec t r a begin t o d i s p l a y a -5/3 region at about a n a t u r a l freguency n=0.2. Above n=1.0, the Nyguist 77 frequency when <0>=18 m/s, the N0u (n) f a l l above the -2/3 l i n e s ( f a l l l e s s r a p i d l y than -5/3), r e f l e c t i n g the expected s p e c t r a l d i s t o r t i o n due to a l i a s i n g . Figure 8 suggests t h a t , at Z=13m, the output of the 0.4 Hz band-pass f i l t e r c ontains freguencies below the -5/3 range with 12 m/s and higher winds, which sets the low frequency c u t - o f f of Hc(f) at about (0.2 12m/s/ 13m) = 0.2 Hz. However, f i q u r e 8 a l s o i n d i c a t e s that the s t r e s s c a l c u l a t e d from the 0.4 Hz f i l t e r i s only about 5% lower, on average, than that c a l c u l a t e d from the other f i l t e r s , f o r winds up to 20 m/s. Therefore, u t i l i z i n g a l l three f i l t e r s to f i n d the s t r e s s should introduce a systematic e r r o r l e s s than 2% and i n r e t u r n improve the s t a t i s t i c a l c e r t a i n t y of the estimate. The e f f e c t i v e low frequency c u t - o f f of the 0.4 Hz f i l t e r and prewhitener combination i s n <0>/Z = (0.2 20m/s/ 13m) =0.3 Hz, the 3db down frequency of H c ( f ) . The tower winds are always l e s s than 20 m/s and at "PAPA", the wind plus sh i p speed i s always below (0.3Hz 22m/ 0.2) = 33 m/s, t h e r e f o r e , the d i s s i p a t i o n c a l c u l a t i o n s always use the data from a l l three band-pass f i l t e r s . Of course, f i l t e r s cannot be used i n the d i s s i p a t i o n method i f they pass frequencies contaminated by the ship's motion. The h o r i z o n t a l v e l o c i t y spectrum, f 0 Q ( f ) , of f i g u r e 13, i s an average of four Reynolds f l u x records taken during very rough seas and s l i g h t l y unstable c o n d i t i o n s at "PAPA". The s p e c t r a l values are averaged over bands of l o g ( f ) i n the manner described f o r the normalized s p e c t r a . The number of runs, M, i s 4, so the v e r t i c a l bars extend ±1 ffm - o/2, where <r i s the standard d e v i a t i o n about the mean f f*Q(f) of a log (f) band. The 22 m/s 78 p r o o O i O - 6 -r r*si 7 t o T o 2'T 8 ' 0 O'O FIGURE 13 The horizontal v e l o c i t y spectrum, f 0Q (f) i n Hz and (m/s) 2, averaged over 4 runs from CCGS Quadra i n 22 m/s winds. V e r t i c a l bars extend ±1 estimation of the standard deviation of the mean. 79 winds make n and f very nearly equivalent and the spectrum i s , a c c o r d i n g l y , nearly the same as f i g u r e 12A except between f=0.1 to 0.3 Hz where the motion of CCGS Quadra i s conspicuous. T h a n k f u l l y , even under such extreme c o n d i t i o n s , very l i t t l e ship motion should be passed by the 0.4 Hz band-pass f i l t e r prewhitener combination. Nonetheless, i t s output while cn a ship i s always checked. Figure 14 shows the normalized v e r t i c a l v e l o c i t y spectrum, N0w(n). The peaks, near n equals 1, are barely reached with the 3Hz f a s t sampling r a t e , but there appears to be a s h i f t t o a higher frequency i n the s t a b l e case, which i s i n accord with the over land studies p r e v i o u s l y c i t e d . The s p e c t r a l shapes near the peaks must be d i s t o r t e d by the a l i a s i n g of a r e l a t i v e l y l a r g e amount of high frequency v a r i a n c e , some of which i s i n e v i t a b l y l o s t because there i s only p a r t i a l c o r r e c t i o n f o r sensor response. Again there i s no evidence of platform motion. At low frequencies am i s very s m a l l and the low s p e c t r a l l e v e l s and the observed s t a b i l i t y dependence have a l s o been found i n other s t u d i e s , i n d i c a t i n q that the h o r i z o n t a l v e l o c i t y i s being properly removed from the t i l t e d p r o p e l l e r s i g n a l . In s e c t i o n 3.5 the G i l l - w response i n v e s t i g a t i o n suggested t h a t the -5/3 range of 0w(f) begins above n= (0.8Hz 13m/12m/s) -0.9, and t h i s i s c o n s i s t e n t with the l o g a r i t h m i c p l o t s of f i g u r e 14. Therefore, some r e s t r i c t i o n s have to be imposed when the G i l l - w band-pass f i l t e r outputs are used i n the d i s s i p a t i o n method. On the tower, f o r example, the 0.4, 0.8 and 1.6 Hz outputs should d e f i n i t e l y not be used when the wind speed exeeds (0.3 Hz 13m/ 1.0) = 4 m/s, 8 m/s and 16 m/s, r e s p e c t i v e l y . 80 w o • CO H ZD CC LE • DD • CE CO 2: ZD CO O .—1 =-a O — .—1 -— T ^ =-— -0 1—1 i-OT j-01 8 ' 0 wy> I M FIGURE 14 Normalized v e r t i c a l v e l o c i t y spectra from averages of f f w (n)/u* 2 over: A: 108 unstable runs B: 88 stable runs. V e r t i c a l l i n e s are ±1 crm. 81 The Temperature Spectrum The normalized spectrum of the temperature f l u c t u a t i o n s , N 0 t (n), from a l l 60 temperature runs, i s shown i n f i g u r e 15. I n d i v i d u a l p l o t s f o r averages over the 27 unstable and 33 s t a b l e runs are not presented because they are not very d i f f e r e n t and, with so few runs, not very s t a t i s t i c a l l y c e r t a i n . However, the majority of runs have |Z/L| < 0.1, so f i g u r e 15 may be re p r e s e n t a t i v e of "near n e u t r a l " s t r a t i f i c a t i o n . Accordingly, the s p e c t r a l shape i s very much l i k e McBean's (1971) near n e u t r a l case, however the g e n e r a l i z e d temperature spectrum of Kaimal et a l . , 1972, i n d i c a t e s a sharper peak at a higher freguency. An average over the log(n)= -3.25 t o -3.0 frequency band gives a mean of 0.15, which, despite a l a r g e standard d e v i a t i o n * supports the e x i s t e n c e of a s p e c t r a l gap that seemingly would emerge i f more high wind speed runs were a v a i l a b l e . The two pluses p l o t t e d near n=0.2 and n=0.3 are the averages of 1524 and 402 p o i n t s , r e s p e c t i v e l y , from the slow samples only and t h e i r average f i t s the high frequency p o r t i o n of the temperature spectrum q u i t e w e l l . The r i s e of the higher frequency plus may be a consequence of combining the s t a b l e and unstable runs, or due to a l i t t l e a l i a s i n g , but when the microbead i s suspected of responding to humidity f l u c t u a t i o n s the peak of the spectrum i s found above n=0.1 and the i n c l u s i o n of only a small amount of such data could l i k e l y be the source of t h i s f e a t u r e . 82 o I o 6 0 R U N S • • • • E n I 111 n i | — r r r l O " 2 I O " 1 FXZ/<U> FIGOEE 15 The normalized temperature spectrum from averages of f ^ t ( n ) / ( o t ) 2 over a l l 60 temperature runs (pluses from slow samples only) . S o l i d l i n e has slope -2/3 and v e r t i c a l l i n e s extend ± 1 oTn. 83 The l o g a r i t h m i c p l o t does not e x h i b i t any evidence cf a -5/3 r e g i o n , which the San Diego r e s u l t s of Phelps and Pond, 1971, show to begin at about n=0.6. Here the spectrum i s d i s t o r t e d by a l i a s i n g before i t can develop, however one of the c r i t e r i a f o r s e l e c t i n g the 60 temperature runs i s th a t the r a t i o s of the band pass f i l t e r s r e v e a l the existence of a -5/3 region above n=0.6. The 3db down frequencies of the combined prewhitener and 0.4, 0.8 and 1.6 Hz temperature band-pass f i l t e r s are about 0.3, 0.55 and 1.0 Hz r e s p e c t i v e l y . The i m p l i c a t i o n i s that the use of the temperature data i n the d i s s i p a t i o n method ought to be r e s t r i c t e d i n the f o l l o w i n g manner: the 0*4 Hz t o winds l e s s than (Z 0.3Hz/0.6) = 6 m/s on the tower and 10 m/s on the ship, the 0.8 Hz to speeds l e s s than (Z 0.55Hz/0.6) = 12 m/s on the tower and 20 m/s on the Quadra and the 1.6 Hz t o <0> below (Z 1.0Hz /0.6) = 20 m/s on the tower and 37 m/s on the ship. U n l u c k i l y , at the higher wind speeds only data from the 1.6 Hz f i l t e r are u s e f u l , so there i s no check on the -5/3 r e g i o n . The uxw Cospectrum The normalized u,w cospectrum, N 0 u w ( n ) , from the s t a b l e runs i s s i g n i f i c a n t l y d i f f e r e n t from that found i n the unstable case, as shown by f i g u r e 16. The over land r e s u l t s of both McBean and Miyake (1972) and Kaimal et a l . (1972) are i n e x c e l l e n t agreement with these over sea spectra. When c a l c u l a t i n g E f o r the 13 method of i n t e g r a t i n g u,w and w,t cospectra (page 51), the N0(n) are averaged over bands of Alog(n) = 0.25. For c l a r i t y , the bands i n f i g u r e s 16 and 17 are 84 CO O CXI o 5 ° fl: 108 UNSTABLE RUNS 10 — i i i T i i i i | — i r -4 1 0 - a Tvm\—i i 111 n i | — io-2 io-» -FXZ/O 111 n i | — i r 111 in 1 0 ° 10 CQ O O O O B : 88 STABLE RUNS i i 1 1 1 i i t y — 1 1 1 1 1 1 1 — i i i i i n i | — i i 1111111—i r 10"4 10-3 lO" 2 10"1 1 0 ° -FXZ/<U> TTTTT 10 FIGURE 16 Normalized unstable (A) and stable (B) u,w cospectra ±1 Ola, from averages of f fuw(n) /u* 2. Integration under the s o l i d curves gives E f o r the 13 method. 85 0.40, but the s o l i d curves t r a c e out the approximate areas i n t e g r a t e d . From i t s peak at a n a t u r a l freguency n=0.03 the unstable cospectrum, f i g u r e 16A, f a l l s s t eeply t o lower frequencies leaving about 1% of the t o t a l covariance below n=10- 3. The more gentle f a l l t o higher freguencies i s usual, but here, as i n f i g u r e s 12 and 14, a l i a s i n g d i s t o r t s the shape above n=1. The highest frequency point i s p l o t t e d at n=4, the band center, but the average log(n) of t h i s band i s at n=3. 2, which i s near the high frequency c u t - o f f of the i n t e g r a t i o n over the Alog(n) = 0.25 bands. The t o t a l area under the curve was found to be 1.06 times the area from n=0.004. The s t a b l e cospectrum, f i g u r e 16B, shows more covariance at higher feguencies with a peak at about n=0.2- There i s no covariance, on average, below n=0.002, but i n d i v i d u a l runs of t e n d i s p l a y s i g n i f i c a n t amounts, both p o s i t i v e and negative. The t o t a l area under the s o l i d curve i s only about 1.005 times the area from n=.004. The r a t i o E, required f o r the 13 method of i n t e g r a t i n g the cospectrum, turns out to be a f u n c t i o n cf s t a b i l i t y and Eu=1-06 and Es=1.005 are r e a l i s t i c values f o r unstable and s t a b l e cases, r e s p e c t i v e l y . The w^t Cospectrum The normalized w Ft cospectrum, N^wt (n), behaves i n a s i m i l a r f a s hion as N^uw and a l s o agrees with the over land measurements. The unstable cospectrum, f i g u r e 17A, d i s p l a y s a broader peak at a s l i g h t l y lower frequency than does the s t a b l e , f i g u r e 17B. Again there i s a greater proportion of the covariance at the lower frequencies i n the unstable case, where o o FIGURE 17 10" TTTTT Normalized w,t 10"a 1 0 " fXZ/<U> cospectra ±1 T T TTTTTTj 1 0 ° rrm 10 Cm, from averages of f ^wt(n)/<wt> in (A) unstable and (B) stable conditions. Integration under- the s o l i d curves gives E for the 13 method. 87 the t o t a l area under the s o l i d curve i s about 1.10 times the amount from n=0. 004.. In the stable case t h i s r a t i o i s only 1.04. The suggestion i s that the 13 method of integrating w,t cospectra ought to use Eu=1. 10 (unstable) and Es=1.04 (stable). With only 33 stable and 27 unstable temperature runs these cospectra and integrations have more uncertainty, as r e f l e c t e d by the larger estimates of the standard deviation of the mean, than found for N0uw. Humidity s e n s i t i v i t y of the temperature sensor i s d i f f i c u l t to diagnose from the cospectrum, because i t results in f fwt(n)/<wt> being very s i m i l a r to the average stable case, figure 17B. 4.4 Turbulence S t a t i s t i c s The spectra and cospectra from each run are integrated from f=0.00065 Hz (the 11 method), to give estimates of the s t a t i s t i c a l quantities oil, oV, oV, at and the u,w c o r r e l a t i o n c o e f f i c i e n t r(uw) = <uw>/(cru oV) . The mean, standard deviation and wind speed dependence of some of these normalized quantities are presented i n table I I , however there may be some s t a b i l i t y dependence i n these r e s u l t s . I t i s evident from fiqure 12 that t h i s method of integrating 0u (f) i s l i k e l y to underestimate oil by more than 13% at <0>=6 m/s. This e f f e c t decreases with increasing wind speed and accounts for at le a s t h a l f of the observed increase i n au/<0> with <0>. A si m i l a r behavior i s expected for OV and 0"t, which also have s i g n i f i c a n t low frequency contributions. The mean and scatter of ov/<0> and 88 Means of 196 runs Standard d e v i a t i o n Linear r e g r e s s i o n i 1 Results cf Smith & Banke 33 runs OTi/<U> 092 018 .061 + .0026<O> 094 ± .014 0"V/<U> .080 .031 .036 + .0037<U> 084 ± .022 Cw/< U> .042 .006 027 + .0013<U> .048 ± .005 ffW/U* 1. 24 10 1.18+ .005 <0> 1. 47 ± .11 -r(uw) 31 ,06 34 ± .07 TABLE I I Turbulent v e l o c i t y s t a t i s t i c s from the 196 Reynolds f l u x runs. The means ± 1 standard d e v i a t i o n from Smith and Banke, 1975, are shown f o r comparison. 0"u/<U> are, t h e r e f o r e , q u i t e reasonable. The values cf o"w/<U> and ov/u* are rather l e s s than those of Smith and Banke, 1975, perhaps because the a l i a s e d frequencies are not completely co r r e c t e d f o r sensor response. Figure 16 shows that the 11 method should never underestimate CTuw by more than 5% and that the l o s s of high frequency covariance should be l e s s than h a l f the amount of variance l o s t by tfw. The combined e r r o r i n otiw may, t h e r e f o r e , be about h a l f the sum of the oru and o*w e r r o r s , which i s c o n s i s t e n t with the c o r r e l a t i o n c o e f f i c i e n t r(uw) being s i m i l a r to p r e v i o u s l y reported values. Figure 18, shows the means of o*u/u*, o"v/u* and ot/T* (?*=Kt* i s used to conform with McBean, 1971) band averaged over ranges of Z/L. Only 39 of the temperature runs are used i n the averaging, because many runs are near n e u t r a l where lar g e ot/T* 89 S t a b i l i t y range mid Z/L ±0.01 Number of runs Standard deviations about the plotted means o"u/u* tfv/u* tfw/u* -0.05 13 0.30 0.79 0.12 -0.03 17 0.25 0.41 0.09 -0.01 22 0.43 0.51 0.06 0.01 25 0.58 0.69 0.10 0.03 28 0.34 0.26 0.08 0.05 16 0.47 0.45 0.09 TABLE III Standard deviations of the turbulence s t a t i s t i c s about the s t a b i l i t y band means plotted i n figure 18. values occur as a r e s u l t of the "noise" i n ot discussed by McBean. With so few temperature runs available ort/T* at Z/L=0.077 i s plotted even though only 6 runs f a l l i n the band. Otherwise, the s t a b i l i t y ranges are selected so that at le a s t 10 runs f a l l into each band. For c l a r i t y , some standard deviations from the band averaging at small vaues of |Z/L|, are presented i n table I I I . T y p i c a l l y , the standard deviations about the mean cTt/T* values in figure 18 are about 0.5. The magnitude of the standard deviations i n table III are comparable to the scatter in McBean's r e s u l t s . o*w/u* exhibits the smallest s t a b i l i t y dependence and the least scatter, perhaps because low freguency contributions to Cw are minimal. Almost a l l the averages plotted in figure 18 f a l l within the scatter cf McBean's (197 1) 90 -0.3 FIGURE 18 Non-dimensional turbulence s t a t i s t i c s band averaged over s t a b i l i t y . See table III f o r standard deviations about plotted means. 91 p l o t s and the observed d i f f e r e n c e s are not unexpected. His variances are computed as i n t e g r a l s from n=0.01 to 10, g i v i n g even s m a l l e r CTu's, but b e t t e r ow's than the 11 method. Accordingly, h i s o"u/u* values tend t o be lower and h i s 0*w/u*'s higher than the corresponding means i n f i g u r e 18. However, the wind speed ranges of the two s t u d i e s are cons i d e r a b l y d i f f e r e n t (<0> <8 m/s f o r a l l of McBean1 s runs), so any wind speed dependencies, such as shown i n t a b l e I I , complicate the comparison. In a d d i t i o n * both u* and T* are c a l c u l a t e d from i n t e g r a l s over d i f f e r e n t frequencies. Despite these problems, oV/u* and cft/T* i n f i g u r e 18 are ge n e r a l l y i n e x c e l l e n t agreement with McBean. Measurements of the s e n s i b l e heat and moisture f l u x e s were used by McBean to f i n d Z/L. The o v e r a l l s t a b i l i t y dependence of a l l the s t a t i s t i c a l q u a n t i t i e s i s very s i m i l a r i n both s t u d i e s , which lends credence to the b e l i e f that Z/L(AT) i s , on average, a good estimate of the s t a b i l i t y parameter. There do not appear to be any unexpected d i s c r e p a n c i e s between the s t a t i s t i c a l q u a n t i t i e s and previous r e s u l t s . The l o s s of low frequency covariance can h o p e f u l l y be avoided with the 12 or 13 methods of i n t e q r a t i n g cospectra. The o"w values are probably too s m a l l , but because t h i s i s due to frequencies above 1.5 Hz there should be no se r i o u s consequences f e l t by <uw> and <wt>. In conc l u s i o n the sensors and Reynolds f l u x system seem to be performing as expected and guite capable of prov i d i n g r e l i a b l e estimates of the momentum and s e n s i b l e heat f l u x e s . 92 4-5 The Fluxes Of Momentum And Sensible Heat Three methods of i n t e g r a t i n g cospectra over the uncertain low frequencies were discussed i n s e c t i o n 3.3 and formulated i n equations 3.5. 11 i n c l u d e s a l l frequencies from the slow samples, 12 adds c o n t r i b u t i o n s from the lower frequencies represented by the group means and 13 i n t e g r a t e s from n=0.004 to which a s t a b i l i t y dependent f a c t o r , E, i s a p p l i e d to account f o r the low frequencies not i n t e g r a t e d . The r a t i o s of <uw> from the d i f f e r e n t methods are averaged over 2 m/s wind speed bands and tabu l a t e d i n t a b l e IV. The 13:11 r a t i o i s expected t o converge s y s t e m a t i c a l l y to 1.0 from greater values as the wind speed i n c r e a s e s . For the unstable runs t h i s should occur at about 13 m/s when the i n t e g r a l from f=0.00065 Hz begins t o cover a l l the n a t u r a l frequency range of the normalized cospectrum ( f i g u r e 16A). With Eu=1.06 t h i s does occur, but above 14 m/s the r a t i o i s again greater than 1 p o s s i b l y because these runs are nearer to n e u t r a l c o n d i t i o n s than the average and re q u i r e a lower Eu. However* the o v e r a l l average of 1.00 suggests that Eu should be increased to r e f l e c t t h a t the 11 method sometimes does miss some of the covariance. The uncertainty i n Z/L makes s t a b i l i t y adjustments to Eu i m p r a c t i c a l , but these would a f f e c t the f l u x by much l e s s than the measurement e r r o r . Keeping Eu=1.06 seems reasonable and the o v e r a l l 12:13 r a t i o of 0.99 i n d i c a t e s that using t h i s f a c t o r i n c l u d e s the average c o n t r i b u t i o n to the covariance from the group means, which i n f a c t u s u a l l y reduces the downward momentum f l u x as shown by 11:12 >1.0. 93 Wind speed ±1 m/s 108 unstable runs Eu=1.06 Points 11:12 12:13 13:11 88 stable runs Es=1.005 Points 11:12 12:13 13:11 13 1. 10 0. 97 1.02 1.02 0.92 1.18 17 1. 03 1. 00 0.98 15 1.01 0.79 1.04 10 18 1. 00 1.01 0.99 21 1.05 0.95 1.02 12 21 0. 98 1.06 0.99 21 1.03 0.96 1.02 14 12 1.04 0. 94 1.03 11 1.02 0.99 1.00 16 14 1. 03 0. 96 1.03 1.01 1.00 0.98 18 11 1. 03 0. 95 1.02 1.01 1.04 0.96 o v e r a l l average ±1o-1. 03 . 15 0.99 . 14 1 .00 .07 1.03 .22 0.94 .31 1.03 . 14 -J . U TABLE IV Comparison of the d i f f e r e n t methods of integrating the u,w cospectrum. At 0=11 m/s the 11 method begins to integrate over the entire stable normalized u,w cospectrum (figure 16B). When averaged over a l l runs above 11 m/s the 13:11 r a t i o i s 1.00 and the 12:13 r a t i o 0.98, showing Es= 1.005 to be appropriate. At lower speeds the 13 method does not include low freguency contributions which reduce the downward f l u x , making 13:11 >1 and 12:13 <1. These positive contributions to -<uw> were balanced in the normalized cospectrum by negative contributions (downward flux) from the higher wind speed runs. One pa r t i c u l a r 94 run i n the 6 m/s band gives 13:11 = 2.13 and there are not enough other runs to balance i t o f f . Excluding t h i s run from the band gives 13:11 <1. I t i s l i k e l y then that the o v e r a l l 13:11 would be reduced i f more runs were a v a i l a b l e f o r averaging. One s t a b l e run i n the 8 m/s band has such a l a r g e p o s i t i v e c o n t r i b u t i o n from the group means that the t o t a l i n t e g r a l , <uw>, becomes p o s i t i v e with 12:13= -1.56. Without t h i s one run both the 12:13 band average and o v e r a l l average increase to 0.96, so that most of the covariance from the group means i s , on average, included i n the 13 method. The decrease of 13:11 with wind speed i n the s t a b l e case, suggests t h a t i f more runs and a b e t t e r measure of Z/L were a v a i l a b l e Es should be made a f u n c t i o n of s t a b l i l i t y . The d i f f e r e n t methods of i n t e g r a t i n g the w,t cospectrum are compared i n t a b l e V. The strange r e s u l t s i n the 12 m/s, unstable band are caused by run T111 whose group mean c o n t r i b u t i o n , -0.018 °Cm/s, i s l a r g e r i n magnitude than and of opposite s i g n to the 11 i n t e g r a l , 0.015 °Cm/s. I f t h i s one run i s excluded, the band average and o v e r a l l average 11:12 become a more reasonable 1.06 and 0.94, r e s p e c t i v e l y , with the corresponding 12:13 r a t i o s going t o 0.99 and 1.11.. For the low speed unstable runs, 11:12 l e s s than one, suggests t h a t there i s a s i g n i f i c a n t c o n t r i b u t i o n t o the w,t covariance a t frequencies below f=0.00065 as do the r e s u l t s of McBean and Miyake, 1972. Without high wind speed runs i t i s not p o s s i b l e t o extend the normalized cospectrum ( f i g u r e 17A) below n=0.001, where i t i s s t i l l f a i r l y l a r g e . The cospectrum was made t o drop o f f t o 0 at n=0.0002 so that Eu=1.10, which i s a compromise. The McBean and Wind speed ±1m/s 27 unstable runs Eu=1. 10 Points 1 1 : 1 2 T 1 2 : 1 3 1 3 : 1 1 33 stable runs Es=1.04 95 — i Points 1 1 : 1 2 1 2 : 1 3 1 3 : 1 1 .79 1.36 1.01 1.08 1.04 .92 f-11 96 1 .06 1.05 .52 6. 28 1. 17 10 99 1.00 1 .01 1.19 0.95 1.03 12 -. 14 80 1 .06 1.06 0.92 1.03 14 1.03 0.91 1.06 16 96 1.07 C.99 18 73 52 1.0 1 ++ Overall average ±\<r .68 1.37 1.06 .39 1.04 . 12 99 47 1.22 1.89 1.03 .09 TABLE V Comparison of the d i f f e r e n t methods of integrating the w,t cospectrum. Miyake r e s u l t s indicate a more rapid drop, but the o v e r a l l 1 2 : 1 3 r a t i o of-1.06 hints that Eu should be larger. The wind speeds during unstable conditions were always too low for the 1 1 method to integrate over the entire normalized cospectrum making 1 3 : 1 1 >1 at a l l speeds and the overal l average 1.04. 96 At a l l wind speeds the 11 method includes the entire stable cospectrum, figure 17B, and accordingly the 13:11 r a t i o shows no systematic trend with wind speed and i t s v a r i a b i l i t y attests to the existence of random low frequency contributions to the sensible heat f l u x which are smoothed by the 13 method. It i s suspected that more runs at the lower speeds would give an average r a t i o of 1.0, the same as above 15 m/s. Again there are some very anomalous runs with large group mean contributions as shown by the 12:13 band averages of 0.52 and 6.28. Without such runs the 13 method again accounts for most of the average group mean contribution. In view of the uncertainty i n table V and in figure 17B, caused by the lack of runs, a value of Es=1.04 i s acceptable. Tables IV and V exhibit evidence of large random contributions to the fluxes from the uncertain low frequencies causing a great deal of scatter i n the fluxes calculated from the 11 and 12 methods. To avoid the r e s u l t i n g scatter, the 13 method w i l l be adopted as the means of integrating the cospectra of a l l the runs. The choices for 0uw of Eu=1.06 and Es=1.005 and for 0wt of Eu=1.10 and Es=1.04 have some uncertainty, but they are reasonable compromises and do not appear to cause any s i g n i f i c a n t systematic errors or apparent trends. The i n t e g r a l s w i l l be denoted by <uw>FL0X and <wt>FL0X and the derived parameters by u*FL0X, t*FL0X and so on. 97 i 1 1 r Z NOG OOOI o FIGURE 19 The neutral drag c o e f f i c i e n t vs wind speed from the 196 Reynolds flux momentum runs. Triangles are the stable runs and pluses the unstable. 98 The Reynolds f l u x r e s u l t s are presented i n f i g u r e s 19 and 20 and compared with the d i s s i p a t i o n method i n Chapter 5. A more complete view of the behavior of the f l u x e s i s o f f e r e d i n Chapter 6 by the more extensive d i s s i p a t i o n data s e t . For the present the measured wind, OZ, temperatures, TZ and TSFC, <uw>FLUX, <wt>FL0X and Z/L(AT) have been put i n t o eguations 2.4 and 2.8 to give the roughness lengths, Zo and Zot, and i n t o 2.11 and 2.10 t o give a wind speed, 010 an a i r temperature, T10, and a drag c o e f f i c i e n t , C10, at 10 meters. A n e u t r a l drag c o e f f i c i e n t , CDN, i s derived from Zo with equation 2-12. The p l o t of CDN vs. 010, f i g u r e 19, look s i d e n t i c a l to the BIO tower r e s u l t s i n Smith, 1979, from which a r e g r e s s i o n of 120 near n e u t r a l C10 values on 010, gives 0.44 + 0.063 U10 = 103C10 as compared to 0.46 + 0.069 010 = 103CDN from f i g u r e 19. Since there i s nearly an equal p a r t i t i o n between s t a b l e ( t r i a n g l e s ) and unstable (crosses) runs i n f i g u r e 19, a re g r e s s i o n of C10 on 010, 0.43 + .069 010 = 103C10, i s not very d i f f e r e n t . The higher c o e f f i c i e n t s at the higher wind speeds are commonly observed, but o v e r a l l these values are d i s t i n c t l y smaller than those at s i m i l a r wind speeds i n G a r r a t t ' s , 1977, review. In f i g u r e 19 the s t a b l e ( t r i a n g l e s ) and unstable (pluses) data do not separate i n t o d i s t i n g u i s h i n g p a t t e r n s . Throughout the small s t a b i l i t y range found over the sea, average s t a b i l i t y e f f e c t s appear t o be sma l l . In f i g u r e 20 <wt>FL0X i s p l o t t e d against 010(TSEA-T10) f o r the 52 temperature runs with |AT| > 0-5°C, so that a l i n e from any point t o the o r i g i n has a slope equal t o CT10, equation 2.10. The s o l i d l i n e represents the parameterization of Friehe 99 1 X X 1 1 <x x<$£ • x X X X 1 1 1 1 o CD O CvJ CD ZD a CD i o o ( S / W G o ) < 1 M > FIGURE 20 <wt> vs. U10 (TSFC-T10) i n ° C and temperature runs with | A T | ranges: A , z/L >0.05; X, -0.1 -0.2 <Z/L< -0.1; <J>, Z/L <-0.2. Friehe and Schmitt (1976). m/s for the 52 >0.5°C. S t a b i l i t y <Z/L< 0.05; + , Solid l i n e i s from 1 0 0 ana Schmitt, 1976, which f i t s quite well for -10 <U10 AT< 25°Cm/s. The data do not support an increase i n CT10 above this range, which Friehe and Schmitt suggest on the basis of Smith and Banke's, 1975, measurements on the beach at Sable Island. The BIO tower re s u l t s (Smith, 1979) span | <0> AT| < 150°Cm/s and indicate s l i g h t l y higher CT10 values than do either Friehe and Schmitt or figure 20, in both the unstable and stable cases. Smith (1979) finds 103<wt>= 3.2+ 1.10 010 AT, for AT>0 and -0.1 + 0.83 010 AT, for AT <0, from regressions of <wt> on 010 AT. A l l three studies show the stable c o e f f i c i e n t to be smaller than CT10 i n unstable s t r a t i f i c a t i o n . There i s not a great deal of scatter in figure 20, except from two runs (plotted as sguares), during which very warm a i r moved over a cold sea. Such conditions greatly influence the average sensible heat flux and are discussed i n section 6.3, when the corresponding d i s s i p a t i o n data i s presented. Near neutral runs (|Z/L| <0.05), plotted as crosses, are much the same as the more stable (triangles) , but the more unstable (pluses, Z/L 0.05 to 0.25 and diamonds, 0.25 <Z/L) seem to give smaller CT10s, however there are far too few data to be conclusive as there are wind speed, fetch and other e f f e c t s to consider. 101 CHAPTER 5 INTERCOMPARISON OF THE REYNOLDS FLOX AND DISSIPATION METHODS 5-1 Introduction The Bedford tower experiment provides an excellent opportunity to establish, by comparison with both the Reynolds flux and BIO r e s u l t s , a d i s s i p a t i o n method that i s v a l i d over a wide range of open sea conditions. The d i s s i p a t i o n system recorded data during 192 Reynolds flux momentum runs and a l l 60 temperature runs and the r e s u l t s are tabulated with those of th e i r corresponding f l u x runs i n the Appendix. The s t a b i l i t y , Z/L (AT), and 6 give a u*DISS = (<uw>DISS) 1/2 from each of the four methods of manipulating the turbulent k i n e t i c energy eguation and ve l o c i t y p r o f i l e , eguations 2.28. Si m i l a r l y <wt>DISS i s obtained both by including and excluding the s t a b i l i t y e f f e c t on the temperature p r o f i l e , equations 2.32. The "best" momentum and sensible heat flux d i s s i p a t i o n methods are to be determined by comparison with u*FL0X and <wt>FL0X from the direct eddy correlation measurements. However, the diss i p a t i o n and Reynolds flux systems are not e n t i r e l y independent, because they share the same sensors. In order to complete the intercomparison of the methods, u*DISS i s also checked with eddy cor r e l a t i o n measurements cf u* from the BIO system. 102 It i s es s e n t i a l to the intercomparison that the runs be as nearly simultaneous as possible. The d i s s i p a t i o n runs are chosen to begin between 0 and 4 minutes before the s t a r t of the f l u x run and in most cases the s t a r t s are within 2 minutes. They l a s t for 56 minutes i f the f l u x run consists of four, 13.5 minute, consecutive groups and for 44 minutes in the few cases that only three groups comprise the run. These times become the averaging periods, < >. The d i s s i p a t i o n , €, i s taken as the average of the three i n d i v i d u a l values obtained by substituting the average power, <P>, from each G i l l - u band-pass f i l t e r , into eguation 3.7. The separate values never d i f f e r from th e i r average by more than 15% and a difference of more than 10% i s found i n only 10 runs. With both (fu(fc) fc 5/ 3) and u* 2 proportional to €2/3 (eguation 2.23), t h e i r average values d i f f e r from those derived from a single band-pass f i l t e r by less than 10% and usually by less than 7%. The deviation i n (fu(fc) fc 5/ 3) i s probably due to the spectrum's fluctuations about a -5/3 line and the average € should be a good measure of the molecular d i s s i p a t i o n . Values of the d i s s i p a t i o n of temperature fluctuations are calculated from eguation 3.6 using only f i l t e r s that l i e e n t i r e l y i n the -5/3 region of f t ( f ) . Thus, Nt i s often an average of only 1 or 2 separate estimates and therefore may not be as r e l i a b l e as 6. In order for the calculated (ft(fc) fc 5/ 3) values to agree with i n d i v i d u a l band-pass f i l t e r s to within 10%, the average, Nt, must not d i f f e r from each i n d i v i d u a l estimate by more than 10% (eguation 2.23). Equation 2.33 shows that 10% deviations i n both f t (f) and fu(f) w i l l produce a 10% deviation i n the calculated <wt>. 103 5.2 The Momentum Flu x At n e u t r a l s t a b i l i t y a l l four momentum f l u x methods are equivalent and i n non-neutral c o n d i t i o n s methods 2 r 3 and 4 simply adjust the n e u t r a l approximation, method 1, by a f u n c t i o n of Z/L, f i q u r e 1. u*DISS1, equation 2.24, i s the simplest c a l c u l a t i o n and i t i s p l o t t e d against u*FLUX i n f i g u r e 21. There i s g e n e r a l l y good agreement between the two c a l c u l a t i o n s from the 192 simultaneous runs, f o r which Z/L i s u s u a l l y between -0.45 and 0.20. A 20% d e v i a t i o n i n the u* estim a t i o n s from a 1:1 r e l a t i o n s h i p i s i n d i c a t e d by the dashed l i n e s , which s a t i s f y the equation |x-y| / [ (x + y) /2] = 0.2 , (5. 1) with x=u*FLUX and y=u*DISS. In view of the e r r o r s i n both methods, d e v i a t i o n s of t h i s magnitude are expected. P o i n t s at the higher u* values r a r e l y f a l l o utside the dashed l i n e s . I t appears, t h e r e f o r e , t h a t the n e u t r a l d i s s i p a t i o n method provides a very good estimate of momentum f l u x e s greater than about u* 2 = 0.16 (m/s) 2, which occur at wind speeds above about 11 m/s. The smaller f l u x e s span a greater s t a b i l i t y range and the c l u s t e r of points l y i n g above the upper dashed l i n e , with u*<0.4 m/s, come from the most s t a b l e runs. Apparently, the assumptions of d i s s i p a t i o n method 1, cause a systematic e r r o r i n s t a b l e c o n d i t i o n s , which tends to make u*DISS1 s i g n i f i c a n t l y greater than u*FL0X. 104 FIGURE 21: Comparison of u* i n m/s from the neutral dissipation method and the Reynolds f l u x method for a l l 192 simultaneous Bedford tower runs. Dashed lin e s indicate a 20% deviation fiom the s o l i d 1:1 l i n e . 105 Method 1 should be v a l i d over a range of "near neutral" s t a b i l i t i e s where the buoyancy and two v e r t i c a l divergence terms of the turbulent k i n e t i c energy equation and the effects of Z / L on the v e l o c i t y p r o f i l e are either small or tend to cancel one another. The extent of such a reqime i s investigated by grouping the simultaneous runs according to Z / L { A T ) and seeing over what range u*DISS1 and u*FL0X agree on average. On the stable side i t i s observed to extend to Z/L=0.05 as shown by figure 22A. The s o l i d t r i a n g l e s , representing the runs with 0.04 <Z/L< 0.05, are s t i l l showing reasonable agreement. For Z/L>0.10 every run gives u*DISS1 greater, often by more than 20%, than u*FL0X and a s t a b i l i t y correction that reduces u * D I S S i s necessary. On the unstable side, figure 22B shows that down to Z / L =-0.10 use of the neutral eguation i s acceptable. The agreement i s evident i n the runs with -0.1 < Z / L < -0.05 (solid t r i a n g l e s ) . Runs in the -0.3 <Z/L< -0.1 range are not included because U * D I S S 1 tends to be smaller than u*FL0X by an average of about 10%. However, i n more unstable conditions the two calculations again tend to agree. A combination of figures 22A and 22B shows that i n a "near neutral" regime, -0.1 < Z/L< 0.05, u*DISS1 gives a very good estimate of the momentum flux without an e x p l i c i t knowledge of the s t a b i l i t y . Extension of the unstable l i m i t down to at l e a s t Z/L=-0.4, introduces very l i t t l e e r ror. This i s a useful r e s u l t , because open ocean conditions are often within t h i s range and because an accurate Z / L i s not always available. I t i s only i n rather stable, Z/L>0.0 5, and perhaps i n very unstable s t r a t i f i c a t i o n , that the neutral approximations cause appreciable errors. 106 8 ' 0 9'0 W O 2"0 0*0 I 9 S I Q m CO 8 ' 0 9"0 t7* 0 2*0 O'O issia m Investigation of the "near neutral" momentum flux (u* i n m/s) regime for; 61 stable runs (solid t r i a n g l e s 0.04 <Z/L< 0.05). 70 unstable runs (solid t r i a n g l e s -0. KZ/K-0.05) . 107 F o r t u n a t e l y , estimates of Z/L(AT) are a v a i l a b l e f o r a l l the simultaneous runs. Under s t a b l e c o n d i t i o n s the buoyancy term of the t u r b u l e n t k i n e t i c energy eguation ( 2 . 1 9 ) i s a s i n k , so there i s a c t u a l l y more production than i s l o s t through d i s s i p a t i o n alone and the l a r g e r production gives l a r g e r u* estimates. Another e f f e c t of s t a b l e s t r a t i f i c a t i o n i s to reduce the amount of f l u x a s s ociated with a given v e l o c i t y p r o f i l e , so the u* estimates become smal l e r . Method 4 (eguation 2.27) i n c o r p o r a t e s the buoyancy term and method 3 ( 2 . 2 6 ) the e f f e c t on the p r o f i l e , while method 2 ( 2 . 2 5 ) does both, with the p r o f i l e e f f e c t dominating. Thus, i n s t a b l e c o n d i t i o n s , u*DISS4> U*DISS1> u*DISS2> u * D I S S 3 , as i s c l e a r l y shown by f i g u r e 1. I t i s observed that i n the s t a b l e runs u*DISS1 tends to be greater than u*FLUX, so only methods 3 and 2 can provide the proper adjustment. The u* estimates from both these methods are p l o t t e d against u*FLUX f o r a l l 88 s t a b l e runs i n f i g u r e 2 3 . Both d i s s i p a t i o n methods appear to give equally good agreement, e s t a b l i s h i n g that the l o c a l production nearly eguals d i s s i p a t i o n , on average, with both the divergence and buoyancy terms of equation 2.19 being s m a l l . However, method 2 i s p r e f e r r e d because i t does account f o r buoyancy, i t s adjustment i s smaller and i t s assumption, that the two divergence terms balance, i s supported by McBean and E l l i o t ' s , 1 9 7 5 , observations. Comparison of 88 stable Reynolds f l u x runs to diss i p a t i o n method 2 (u* i n m/s) diss i p a t i o n method 3 (u* i n m/s). 109 During unstable conditions the roles of buoyancy and the s t a b i l i t y modified p r o f i l e , on the di s s i p a t i o n estimates are reversed, with u * D I S S 3 > u * D I S S 1 > u * D I S S 4 . Figure 1 shows that i n method 2 the eff e c t on the p r o f i l e s t i l l dominates down to Z / L = - 0 . 1 , when the buoyancy f i n a l l y becomes an important source of turbulent k i n e t i c energy. At about Z / L = - 0 . 4 , u * D I S S 2 becomes less than U * D I S S 1 . Throughout the unstable range of the runs, u * D I S S 2 d i f f e r s from U * D I S S 1 by les s than 1 0 % and g u a l i t a t i v e l y i t i s the only method that provides the appropriate s t a b i l i t y refinements. Method 3 c l e a r l y gives far too large a u* i n the more unstable cases. The u* estimates from methods 4 and 2 are plotted against u * F L 0 X for the 1 0 4 simultaneous unstable runs i n figure 2 4 . The agreement of method 2 i s excellent and the scatter i s very evenly distributed about the 1:1 l i n e . This resu l t indicates that the divergence terms also tend to cancel i n unstable conditions as suggested by McBean and E l l i o t ' s findings. Because u * D I S S 1 also gives a reasonable estimate throughout - 0 . 4 <Z/1< 0 . 0 5 , the e f f e c t of s t a b i l i t y on the p r o f i l e must nearly balance the buoyant production. Pond et a l . , 1 9 7 1 , calculated u* up to 0 . 3 m/s using method 4, which i s seen to be acceptable, but not applicable to higher u*values. FIGOEE 24 Comparison of 104 unstable Reynolds flux runs to A: di s s i p a t i o n method 2 (u* i n m/s) B: di s s i p a t i o n method 4 (u* i n m/s). 111 Whenever Z/L i s available i t i s evident that, over the s t a b i l i t y range of the intercomparison, method 2 gives the "best" estimates of u*, henceforth u*DISS and <uw>DISS w i l l be calculated using t h i s method. Figure 25 i s a plot of u*DISS vs u*FLDX from the 192 simultaneous runs and a comparison with figur e 21 i l l u s t r a t e s the ov e r a l l e f f e c t of the s t a b i l i t y correction.. In the region 0.15 <u*FL0X< Q.Hm/s, the reduction of u*DISS from the most stable runs greatly improves the agreement, but u*DISS s t i l l tends to be greater than u*FLUX. As a consequence, the average of regressions of u*DISS and u*FL0X, u*DISS = 0.96 u*FL0X + 0.025 m/s, has a positive o f f s e t and the ov e r a l l average U * D I S S:u*FLUX r a t i o , 1.03 (standard deviation 0.10), i s greater than 1.00. The two techniques d i f f e r by at most 28% and usually by less than 20% i n u*, which i s about the amount of scatter expected. A 20% error i n <uw>DISS and a non-compensating 20% error i n <uw>FL0X give a 20% deviation in u* estimates. It i s doubtful that the u* agreement would be so good i f a major systematic error had arisen from the propeller response correction because i t i s of fundamental importance to u * D I S S , but only secondary to u*FL0X. Conversely, the non-cosine behavior of the G i l l propellers i s a potential source of substantial error i n Reynolds flux measurements* but not to the dis s i p a t i o n estimates, thus the good agreement indicates that i t too i s being treated properly. S i m i l a r l y , i t appears as i f the low freguency covariances are being handled s a t i s f a c t o r i l y . 112 FIGURE 25: Intercomparison of u* i n m/s from the "best" dis s i p a t i o n (2) and the "best" Reynolds f l u x (13) method for a l l 192 simultaneous runs. 113 <uw>DISS <uw>FLUX 1 T Standard deviation Wind speed range (m/s) 6 - 8 8 - 1 0 Number of points 27 25 Mean 1- 10 1.02 0. 14 0. 21 Minimum Maximum 0.82 0. 68 1 .42 1 .49 10 - 12 54 1. 14 0. 23 12 - 14 32 1.05 0. 16 14 - 16 18 0.97 0. 13 16 - 18 17 1.00 0.09 18 - 20 1.01 0. 14 0. 68 0.79 0.76 0.79 0.83 1 .58 1 .34 1 .23 1.10 1 .36 Overall 4 - 2 0 182 1.05 0.17 0. 68 X L 1 .48 TABLE VI Eatio of diss i p a t i o n to Reynolds flux estimates of the momentum flux band averaged over 2 m/s wind speed i n t e r v a l s . The momentum flu x and drag c o e f f i c i e n t are proportional to <uw>, so trends i n the <uw>DISS:<uw>FL0X r a t i o are investigated by averaging the r a t i o over wind speed and s t a b i l i t y bands in tables VI and VII, respectively. In order that the band means are not unduly weighted by i n d i v i d u a l runs with atypical r a t i o s , only those runs whose r a t i o s f a l l within ±2 standard deviations about a complete band average are used to calculate the means, standard deviations and ranges i n tables VI and VII. In table VI, the ov e r a l l average of 1.05 and standard deviation, tr, of 0.17, are guite acceptable. A ar of 17% i s comparable to the 114 < U W > D I S S < U W > F L U X i r Standard deviation .j S t a b i l i t y range Z/L -.45 -.30 -.30 -.15 Number of points 15 Mean 1.08 0.97 0.23 0.14 Minimum Maximum 0.77 1. 48 0.72 1-31 -.15 74 1.04 0. 17 0.68 1.36 0 0.05 59 1.06 0. 17 0.76 1. 40 C.C5 0.10 18 1.16 0.17 0.92 1.53 I- 0.10 0.20 1.34 0.33 0.87 1.65 JL L. TABLE VII Ratio of dis s i p a t i o n to Reynolds flux estimates of the momentum flux band averaged over s t a b i l i t y ranges. expected error in each method and the mean i s less than o/3 greater than the desired o v e r a l l average of 1.00, despite the large r a t i o s that occur during lower winds when s t a b i l i t y has i t s greatest range. Otherwise there i s no systematic trend with wind speed, although the range of the r a t i o seems to decrease with increasing speed. At the higher wind speeds where s t a b i l i t y should always be near neutral the band means are about 1.00. Therefore, i t seems l i k e l y that any wind speed dependency of'the drag c o e f f i c i e n t observed i n d i s s i p a t i o n r e s u l t s would also be found i n corresponding eddy c o r r e l a t i o n measurements. Table VII shows that, as expected, the stable runs produce the largest average r a t i o s . Figure 1 shows that switching to di s s i p a t i o n method 3 would not greatly a l t e r t h i s r e s u l t (4% i n 1 15 the 0.10 <Z/L< 0.20 band). I t i s possible that the higher r a t i o s are caused by underestimating <uw>FL0X with the 13 method. As Z/L increases, the proportion of covariance at high frequencies and, hence, the amount l o s t through incomplete sensor response corrections also increases, but the e f f e c t should not exceed 5%. I t has been shown i n section 4.5, that, on average, Es=1.005 treats the low frequency covariance adequately, but the four runs that give such a large average to the most stable band of table VII occurred nearly seguentially (runs T174, T175, T176 and T178), so the band does not neccessarily r e f l e c t average conditions. Since there are no fluctuating temperature data from these runs, Z/L (AT) could be underestimating Z/L and causing overestimates of u*DISS. In order to complete the intercomparison of methods i t i s important to show that the observed agreement of figure 25 does not depend on using the same sensors. Dr. S.D. Smith of the Bedford I n s t i t u t e of Oceanography has kindly allowed some of his eddy co r r e l a t i o n measurements of u*, u*BI0, from the tower (Smith, 1979) to be compared to simultaneous di s s i p a t i o n measurements. The runs overlap as do the Reynolds flux runs and are t y p i c a l l y about 44 minutes in duration. The BIO mark 6.4 thrust anemometer operated from October 7 to December 8, 1976. In figure 26, u*DISS i s plotted against u*BI0 for a l l 20 simultaneous runs from t h i s period. On average, the agreement i s excellent and the scatter i s no larger than expected considering the addition of c a l i b r a t i o n errors and the separation of the anemometers by about 2m. A direct comparison of u*FLUX and u*BIO values i s not possible, because simultaneous 116 FIGUBE 26: Comparison of u* i n m/s from the d i s s i p a t i o n system and from the BIO eddy correlation system from 20 runs on the Bedford tower. 117 measurements are r a r e , however the d i s s i p a t i o n intercomparison gives a favourable i n d i r e c t one. These conclusions ought to be g u a l i f i e d by noting t h a t there are disagreements with the mean winds from l a t e r BIO data which are s t i l l to be r e s o l v e d . The u* comparison of these data shows u*DISS t o be greater than u*BIO (often by more than 20%) i n 28 of 30 runs. 5.3 The Sensible Heat Flux The s e n s i b l e heat f l u x i s estimated using the two d i s s i p a t i o n methods f o r the 60 simultaneous temperature runs and these are compared to <wt>FLUX i n f i g u r e 27. The dashed l i n e s s a t i s f y 5.1 with x=<wt>FL0X and y=<wt>DISS, i n d i c a t i n g a 20% de v i a t i o n of the a c t u a l f l u x e s from the s o l i d 1:1 l i n e . <wt>DISS, the n e u t r a l approximation, i s c l e a r l y , on average, more negative than <wt>FL0X ( f i g u r e 27A). . Method 2 a p p l i e s a rath e r l a r g e c o r r e c t i o n f o r the i n f l u e n c e of s t a b i l i t y on the temperature p r o f i l e , but f i g u r e 2 shows the adjustment to be i n the proper sense i n both s t a b l e and unstable c o n d i t i o n s . A regression of <wt>DISS2 against <wt>FL0X ( f i g u r e 27B) gives <wt>DISS2 = 1.04 <wt>FLUX, with a c o r r e l a t i o n c o e f f i c i e n t of 0.99. The agreement i s remarkably good c o n s i d e r i n g the e r r o r s i n both methods and the s e n s i t i v i t y of <wt>DISS2 to the r a t h e r u n c e r t a i n Z/L(AT). These r e s u l t s suggest t h a t the t u r b u l e n t t r a n s p o r t term i n the 1 1 8 frO'O O'O t 7 Q ' o -< l r t > i 1 1 1 r t ? 0 ' 0 O'O t 7 0 ' 0 -TSSIQ < l t t > FIGITBE 27 Comparison of the 60 simultaneous sensible heat flux calculations i n °Cm/s. A: <wt>DISS1 vs. <wt>FLUX B: <wt>DISS2 vs. <wt>FLUX. 119 temperature variance budget i s i t s e l f small and t h a t i t cannot compensate f o r changes i n the l o c a l production of temperature variance due to s t a b i l i t y . In the range -0.03 <Z/L< 0.05 the c o r r e c t i o n to <wt>DISSl i s l e s s than 10% and, i f necessary, method 1 could be used to estimate the s e n s i b l e heat f l u x without an an e x p l i c i t Z/L. However, method 2 gives the t e t t e r estimate and w i l l always be used t o c a l c u l a t e <wt>DISS. The two points i n the fo u r t h guadrants of f i g u r e s 27A and 27B i l l u s t r a t e an inherent d i f f i c u l t y i n f i n d i n g small f l u x e s with the d i s s i p a t i o n method. <wt>DISS i s c a l c u l a t e d frcm eguation 2.32 and the sig n of the sguare root i s chosen so as to forc e <wt> to have the same s i g n as AT = TSFC-TZ. However, i t i s commonly observed i n eddy c o r r e l a t i o n measurements t h a t <wt> i s between 0 and +0.0 5 °Cm/s when .AT i s s l i g h t l y negative. There have been very few measurements of l a r g e s e n s i b l e heat f l u x e s over the sea, but the e x c e l l e n t agreement i n f i g u r e 27B, when the magnitude of the f l u x i s l a r g e , s t r o n g l y suggests t h a t t h i s measurement may be done using the d i s s i p a t i o n method. The low and high freguency temperature f l u c t u a t i o n s a f f e c t the s e n s i b l e heat f l u x c a l c u l a t i o n s i n the same manner as the v e l o c i t y v a r i a t i o n s a f f e c t the momentum f l u x . S i m i l a r arguments regarding non-cosine behavior, sensor response and low freguency covariance lead to the conclusion that the temperature elements of both the Reynolds f l u x and d i s s i p a t i o n systems must be performing properly i n order to achieve the observed agreement. With no larg e systematic e r r o r s i n evidence, <wt>DISS and <wt>FLUX should both be g i v i n g r e p r e s e n t a t i v e estimates of the s e n s i b l e heat f l u x . 120 As previously noted many temperature runs have not been processed because of suspected s a l t contamination of the microbead. In a few of these cases the temperature band-pass f i l t e r s unexpectedly display a -5/3 drop to within 10%.. The band-pass f i l t e r check i s , therefore, a necessary, but i n s u f f i c i e n t test for this behavior. This i s unfortunate because i t r e s t r i c t s d i s s i p a t i o n temperature measurements to periods when the temperature spectrum can be obtained from the Reynolds flux system. A possible means of overcoming t h i s problem would be to average the output of another band-pass f i l t e r centered at about n=0.02 for comparison with the outputs of the f i l t e r s i n the -5/3 range. I t i s also i n t e r e s t i n g to note that <wt>DISS and <wt>FLUX are i n good agreement i n some unstable contaminated runs, al b e i t both seem to be somewhat high. It i s expected that the largest d i s s i p a t i o n errors are in the constants and the assumptions. The fact that there are no major persistent errors, suggests that the combined errors in the constants i s not very large. This i s supported by eguations 2.29 and 2.33 which show <wt>DISS to be proportional to ( KZ ) 1/3 (Bt ' ) - V 2 u*DISS . If , for example, the Kolmogoroff constant, Bt', was greatly in error, u*DISS would be expected to be more accurate than <wt>DISS and hence, in better agreement with Reynolds flux c a l c u l a t i o n s . The r e s u l t s , figures 25 and 27B, do not support t h i s . S i m i l a r l y , K and Z appear to be reasonable. Of course 121 there could be major o f f s e t t i n g errors, or the constants may be highly variable, but i t seems more l i k e l y that i t i s the assumptions which are the most uncertain aspect of the d i s s i p a t i o n method. It i s also probable that much of the scatter i n the u* and <wt> intercomparisons originates with the Reynolds f l u x values. On average, the Reynolds flux and d i s s i p a t i o n methods give nearly the same momentum and sensible heat fluxes. In addition, both the d i s s i p a t i o n and Reynolds flux systems appear to be functioning as expected. With the possible exception of u* from the most stable cases (Z/L> .05), the estimates of <uw> and <wt>, by both techniques, should be r e l i a b l e and free of major systematic errors. I t i s , therefore, possible to have confidence i n the dissipation system when i t i s operating on a ship where the Reynolds flux method i s not practicable. 122 CHAPTER 6 DISSIPATION MEASUREMENTS FROM THE BEDFORD STABLE TOWER AND CCGS QUADRA 6.1 I n t r o d u c t i o n The d i s s i p a t i o n system has provided a great deal more data from the Bedford tower than the Reynolds f l u x system, because i t continuously records and has proved t o be more r e l i a b l e . In a d d i t i o n , a considerable amount of d i s s i p a t i o n data has been c o l l e c t e d at "PAPA". I n t o t a l 1086 hours of momentum f l u x measurements from the tower and 505 hours from the weathership are found to s a t i s f y a v a r i e t y of c r i t e r i a f o r data r e l i a b i l i t y . Only 237 hours of tower data and 23 hours from CCGS Quadra are found to be s u i t a b l e f o r s e n s i b l e heat f l u x c a l c u l a t i o n s . There are s e v e r a l long periods of continuous recording which make i t po s s i b l e t o i n v e s t i g a t e the time h i s t o r i e s of the f l u x e s , winds and temperatures. Only data from the h o r i z o n t a l * G i l l - u , p r o p e l l e r are used to f i n d the d i s s i p a t i o n , €, and hence momentum f l u x . Because of the greater u n c e r t a i n t y , i t i s not worthwhile i n c l u d i n g the moderate wind speed measurements from the t i l t e d , G i l l - w , p r o p e l l e r that are a v a i l a b l e from the few periods when the h o r i z o n t a l p r o p e l l e r f a i l e d . The tower a n a l y s i s i s l i m i t e d to wind speeds greater than 4.0 m/s. At lower winds the G i l l - u p r o p e l l e r i s s t i l l i n i t s l i n e a r regime, but the sensor response c o r r e c t i o n s become very large and i t i s not impossible f o r the thickness of the "constant f l u x " l a y e r to f a l l below the anemometer height. On CCGS Quadra the measurement height i s 123 nearly twice as high, so the l i m i t i s set at 8.0 m/s. A further r e s t r i c t i o n requires the s t a b i l i t y to be i n the range -0.6 <Z/L< 0.15. At Z/L=0.15 dissipation momentum flux calculations require a correction for s t a b i l i t y of about 25% (figure 1). This adjustment i s supported mainly by the Reynolds flux intercomparison, where Z/L i s greater than 0.15 for only three runs, and also by McBean and E l l i o t t ' s , 1975, measurements, that include only three stable runs with the highest at Z/L=0.12. It i s , therefore, dangerous to extrapolate these findings beyond Z/L=0.15, where the correction becomes even larger. On the unstable side there are 21 runs i n the McBean and E l l i o t study and 104 runs in the section 5.2 intercomparison, with the most unstable at Z/L=-0.3 and -0.45, respectively. At Z/L=-0.6 the s t a b i l i t y adjustment i s s t i l l less than 10% so i t should be a l l right to extend d i s s i p a t i o n method 2 at least t h i s far. This c r i t e r i o n e f f e c t i v e l y sets the outer l i m i t s of sensible heat fl u x c a l c u l a t i o n s , because they require € from G i l l - u . The s t a b i l i t y corrections to <wt>DISS are 1.8 at Z/L=-0.6 and 0.75 at Z/L=0.15 (figure 2) and although these are substantial, they are strongly supported by the work of Wyngaard and Cote, 1971, which consists of runs from Z/L=-1.1 to Z/L=0.4, and the intercomparison of section 5.3. Temperature data are also rejected i f there i s any evidence of s a l t contamination of the microbead or i f the rod and bead thermistors do not agree, on average, to within ±0.1°C. 124 In processing a d i s s i p a t i o n momentum [temperature] run from e i t h e r the ship or tower, 20 minute averages of a l l G i l l - u [temperature microbead] band-pass f i l t e r s i n the -5/3 range of r"u(f) [ f t ( f ) ] are used to f i n d independent values of the d i s s i p a t i o n of t u r b u l e n t k i n e t i c energy [temperature f l u c t u a t i o n s ] from equation 3.7 [ 3 . 6 ] . These i n d i v i d u a l values are averaged together t o give the € [ N t ] necessary f o r the c a l c u l a t i o n of the momentum f l u x [ s e n s i b l e heat f l u x ] from eguation 2.28 [2.32]. However, i f any separate band-pass value d i f f e r s from 6 [ N t ] by more than 15% [10%] the run i s r e j e c t e d . This c r i t e r i o n ensures t h a t the s p e c t r a l values at the frequencies used f a l l w i t h i n 10% of the average -5/3 slope of the input spectrum and t h a t f l u x e s c a l c u l a t e d from i n d i v i d u a l f i l t e r s are w i t h i n 10% of the value found from the average. The 20 minute averages could be used t o produce time s e r i e s of the f l u x e s and r e l a t e d parameters, but i n p r a c t i c e hourly averages are employed because i t i s f e l t the longer averaging time increa s e s the r e l i a b i l i t y of the d i s s i p a t i o n method. In order to obtain the hourly averages, three s e g u e n t i a l 20 minute f l u x estimates are averaged and then used t o c a l c u l a t e the r e l a t e d parameters. Hereafter, these hourly averages from the d i s s i p a t i o n system w i l l not be s p e c i f i c a l l y denoted, but r e f e r r e d to simply as <uw>, <wt>, u* t * , CD, CT and so on. For dynamic f l u x c a l c u l a t i o n s the a i r de n s i t y £ i s found from equation 2.35, using the atmospheric pressure from the meteorological observations from Shearwater and CCGS Quadra. 125 C M CD L J CD CD' CD B: CCGS QUADRA + + + + ~i i i i r 10 14 18 U10 (M/S) CD 2 22 26 10 14 18 U10 (M/S) FIGOEE 28 The n e u t r a l drag c o e f f i c i e n t as a f u n c t i o n of wind speed from: A: 1086 hourly averages from the Bedford tower. B: 50 5 hourly averages from the CCGS Quadra. 126 6.2 Bulk Aerodynamic Parameterization Of The Momentum Flux Hourly averages of u* and UZ and Z/L are used to c a l c u l a t e U10 from eguation 2.11 and the n e u t r a l drag c o e f f i c i e n t , CDN, from 2.8 and 2.12. These are p l o t t e d separately f o r the tower and ship data i n f i g u r e 28. There are very few extreme values and most points are obscured by the concent r a t i o n about a mean CDN value. Between U10 =4 t o 11 m/s the s c a t t e r i s uniform with only a few values of 103CDN outside the range 0.65 to 1.5. At higher wind speeds a tendency for higher values develops and there appears to be l e s s s c a t t e r . Above U10=20 m/s there are few p o i n t s , so these data may not f u l l y r e f l e c t average c o n d i t i o n s . F o r t u n a t e l y , i n the region 8 <010< 18 m/s, there i s a very good overlap of tower and ship r e s u l t s which allows the two s i t u a t i o n s to be compared. In f i g u r e 29, CDN i s averaged over 2 m/s i n t e r v a l s of U10, provided there are a t l e a s t 10 runs i n a band, and the means pl o t t e d with v e r t i c a l bars extending up and down 1 standard d e v i a t i o n . For c l a r i t y , tower r e s u l t s ( t r i a n g l e s ) are j u s t to the l e f t of band center and the ship p o i n t s (pluses) to the r i g h t . Over the range 8.5 <010< 20.5 m/s, the band averaged CDN's from one platform are seen t o d i f f e r by l e s s than one standard d e v i a t i o n (but u s u a l l y l e s s than 1/2 o") from the other platform's mean. No systematic e r r o r s , such as i n the choice of Z, appear to have been introduced i n moving the d i s s i p a t i o n system from the tower t o CCGS Quadra. The drag c o e f f i c i e n t agreement a l s o e s t a b l i s h e s t h a t the tower s i t e i s , i n f a c t , r e p r e s e n t a t i v e of open sea c o n d i t i o n s , at l e a s t up to 20 m/s 127 FIGURE 29 Comparison of the ship (pluses) and tower (triangles) neutral drag c o e f f i c i e n t s . V e r t i c a l bars extend ± 1 o" about plotted band averages. 128 winds. This allows the tower and s h i p r e s u l t s t o be combined i n t o a s i n g l e data set of nearly 1600 hours from about 16 months of operation between September 1976 and A p r i l 1978. Fetch E f f e c t s The tower data of f i g u r e 28A i n c l u d e some runs from l i m i t e d f etches, whose i n c l u s i o n i n the o v e r a l l data set must be j u s t i f i e d . Up to about 10 m/s, CDN from the tower does not vary appreciably with wind speed, therefore data i n t h i s range are used to i n v e s t i g a t e the i n f l u e n c e of f e t c h on the drag c o e f f i c i e n t . The r e s u l t s are shown i n t a b l e V I I I , where the n e u t r a l drag c o e f f i c i e n t , CDN, from measurements about 13m above 1 T | Fetch | 1 (km) 1 • T Number | of | hours | — - — — T -Mean 103CDN | ±1 standard | d e v i a t i o n | • 103CDN | Min | 103CDN Max x — 1 | Azimuthall | range | I (° True) | | 1 0 - 2 0 | 200 | 1.14 ±.18 J 0.75 | 2.03 | 253 - 393| I I | 20 - 100 | I j 54 | 1.10 ±.22 I 0.73 | 1.87 | 246 - 253| | 33 - 65 | | 100 - 200 | 85 | 1.13 ±.24 | 0.64 | 1.76 | 235 - 246| | Unlimited | | tower | i i 263 | 1.13 ±.22 | 0.62 | 1.75 | 67 - 235| i i i i | Unlimited | I tower+shipl 291 | 1-14 ±-21 | 0.62 | 1.75 1 1 | Tower | | a l l f e t c h e s | • « 590 | i 1. 13 ±. 21 j • 0.62 | • 2.03 I j | 0 - 360| • • TABLE V I I I V a r i a t i o n f e t c h f o r of the n e u t r a l drag winds between 4 and 10 c o e f f i c i e n t with m/s. 129 the sea, does not e x h i b i t any dependency on fetches greater than 10 km and, i n view of the o v e r a l l s c a t t e r , the means are very c o n s i s t e n t . This conclusion depends on using CDN, since the mean C10 from the 10-20 km f e t c h range i s 17% l a r g e r than from the u n l i m i t e d f e t c h runs, presumably because the offshore winds tended t o be more unstable. The i n d i c a t i o n i s , that f o r winds below 10 m/s, f e t c h e f f e c t s are not important i f the aspect r a t i o , Z/fetch, i s at l e a s t as s m a l l as 10~ 3. There i s a l s o an i n d i r e c t i m p l i c a t i o n that the surface roughness, Zo, does not depend s t r o n g l y on surface wave parameters that are not f u l l y developed by 10 km during 10 m/s winds. The data compiled by Wiegel (1964, p 216), f o r f e t c h l i m i t e d waves, show the s i g n i f i c a n t wave height and the phase speed from a 1000 km f e t c h (fetch g/<D>2-105) to be, r e s p e c t i v e l y , 10 and 5 times greater than expected with a 10 km f e t c h - The n e c e s s i t y of e x t r a c t i n g the wind speed dependency from CDN complicates extending the i n v e s t i g a t i o n above 10m/s, but as the aspect r a t i o stays the same, there should not be a sudden f e t c h dependency. Tower data from a l l fetches are, t h e r e f o r e , grouped together i n a l l subseguent analyses. The bet t e r s t a t i s t i c s r e s u l t i n g from the a d d i t i o n a l data should more than compensate f o r any small f e t c h e f f e c t s . 130 S t a b i l i t y E f f e c t s Eguation 2.12 de f i n e s CDN as a f u n c t i o n only of Zo and 2.8 im p l i e s C10CDN f o r Z/L<0, C10<CDN f o r Z/L>0 and C10=CDN at n e u t r a l s t a b i l i t y . Table IX i n d i c a t e s that CDN and hence Zo are independent of Z/L i n the unstable case. At Z/L<-0.3, C10 values are c l e a r l y higher than the average, but re d u c t i o n to n e u t r a l s t a b i l i t y gives no systematic trend t o the mean unstable CDN's, which are reasonably c o n s i s t e n t i n view of the s c a t t e r . Extraneous e f f e c t s seem t o i n f l u e n c e the s t a b l e runs to a much greater degree. When reduced t o n e u t r a l , the average s t a b l e CDN * 1000 T r C10 * 1000 n Z/L (AT) range Number of hours Mean ±1o~ Max Min Mean ±1cr Max Min -.60 -.45 25 1.06 ±.12 0. 80 1. 31 + T 1.22 ±.14 0.91 1.52 - .45 - .30 35 1.12 ±.19 0.77 1.73 1.26 ±.22 0.86 1.98 -.30 -.15 77 0.98 ±.16 0. 67 1. 38 1.07 ±.18 0.70 1.54 -. 15 0 166 1.09 ±.20 0.68 1.74 1.12 ±.21 0.68 1.83 0 +.05 104 1.19 ±.18 0.62 1.85 + r 1.16 ±.18 0.60 1.81 +.05 +.10 122 1.21 ±.19 0.64 1.79 1.13 ±.17 0.60 1.64 +.10 +.15 61 1.23 ±.28 0.75 2.03 1.11 ±.24 0.68 1.80 - i J TABLE IX The s t a b i l i t y dependency of the drag c o e f f i c i e n t s from the 4 < 010 < 10 m/s data of t a b l e V I I I . 131 CDN's are l a r g e r than the o v e r a l l mean and they seem to increase with s t a b i l i t y . However, the l a r g e drag c o e f f i c i e n t s r e s p o n s i b l e f o r t h i s feature are found to be a s s o c i a t e d with r a p i d changes i n wind d i r e c t i o n f o l l o w i n g a storm. Such a sequence of events may p o s s i b l y i n f l u e n c e parameters important to Zo and merely c o i n c i d e with s t a b l e s t r a t i f i c a t i o n . A time s e r i e s showing the development of these high, low wind speed, s t a b l e drag c o e f f i c i e n t s i s presented l a t e r ( f i g u r e 31). The data suggest that the s t a b i l i t y dependence of the drag c o e f f i c i e n t i s w e l l described by s i m i l a r i t y theory with Zo independent of Z/L and t h a t sea surface c o n d i t i o n s are the source of the trend i n CDN with Z/L found i n t a b l e IX. Wind Speed Dependency In f i g u r e 30 the 1591 n e u t r a l c o e f f i c i e n t s from a l l the data are band averaged over 2 m/s i n t e r v a l s and p l o t t e d at the 010 mean ( v e r t i c a l bars show + 1 ff). The mean C10 values are s i m i l a r , because of the almost equal numbers of s t a b l e and unstable runs. A smooth curve (concave up) with CDN i n c r e a s i n g with 010 could be made to f i t these p o i n t s . However, below about 10m/s, the s l i g h t increase of the average CDN's should be adequately described by a constant. The higher wind speed po i n t s show a more r a p i d r i s e of CDN with U10, t h a t i s approximately l i n e a r from 10 t o 26 m/s. In order to guantify t h i s behavior, a l l 618 hourly CDN values with U10<10 m/s are averaged and a l l 973 with U10>10 m/s are regressed against 010 ( c o r r e l a t i o n c o e f f i c i e n t 0.74). A l l the band averages are very w e l l described by the r e s u l t i n g s o l i d l i n e of f i g u r e 30: 132 1 1 1 1 \ £ c! T 0 NOG OQOI FIGURE 30 The neutral drag c o e f f i c i e n t averaged over wind speed bands. V e r t i c a l bars indicate ± 1 or and the number of points i n band is shown below each average. Lines show the Charnock representation with °<= 0.0144, K= 0.41 (dashed) and eguations 6.1 (solid) . 133 I O 3 CDN 103 C10 = 1.14 4 < U10 < 10 m/s (6. 1) 103 CDN 0.49 + .065 U10 10 < 010 < 27 m/s 103 C10 0.46 + .068 010 where the C10 r e s u l t s are included f o r comparison. The exact numerical values of 6.1 depend s l i g h t l y on the somewhat a r b i t r a r y choice of a lower wind speed l i m i t f o r the r e g r e s s i o n . Conveniently* with the choice of 10 m/s, the l i n e segments happen t o match. The behavior of CDN described by 6.1 i s d i s t i n c t l y d i f f e r e n t from the dashed curve of f i g u r e 30, which i s the Charnock representation with <=< = 0.0144, X= 0.4 1 (or <=<=0.016, K= 0.4), as suggested by G a r r a t t (1977). Eguations 6.1 f i t Smith's (1979) eddy c o r r e l a t i o n data from the Bedford tower s i t e extremely w e l l . For winds below 10 m/s, h i s average 103CDN i s 1.11 from 14 runs (1976-1978) and about 1.24 from 11 near n e u t r a l runs (1968-1969). A r e g r e s s i o n of a l l 120 runs (1976-1978) between 6 and 22 m/s y i e l d s 0.44 + 0.063 010 = 10 3C10. However, the Sable I s l a n d r e s u l t s of Smith and Eanke (1975) are s i g n i f i c a n t l y higher, suggesting that the Sable I s l a n d s i t e with i t s surf zone and perhaps other shallow water l o c a t i o n s : Lough Neagh at 8 to 15 meters (Sheppard et a l . , 1972), Lake Flevo at 4 meters (Wieringa, 1974), the 10m Caspian Sea l o c a t i o n of K i t a i g o r o d s k i et a l . (1973) and the Spanish Banks s i t e of Miyake et a l . (1970 A) and Weiler and B u r l i n g (1967), may not be r e p r e s e n t a t i v e of the open ocean s i t u a t i o n . Unfortunately, almost a l l the measurements above 14 m/s (and many at lower speeds), that were a v a i l a b l e t c G a r r a t t came from 134 these shallow water l o c a t i o n s . A number of open sea measurements c o n t r i b u t e d to h i s 10 t o 11 m/s band average, which very nearly l i e s on 6.1. I t also seems l i k e l y t h a t excluding the r e s u l t s of Sheppard et a l . would give band averages i n the 3 to 7 m/s range t h a t would be adequately described by 6. 1. In the range 7 to 10 m/s, the band averages of G a r r a t t g e n e r a l l y run about 20% ( l e s s than 1 ar) higher than eguations 6.1. Open ocean measurements used by G a r r a t t (1977) are i n be t t e r agreement. From the Argus I s l a n d tower, a s i t e s i m i l a r t o the Bedford tower, the o v e r a l l average of 69, near n e u t r a l , Reynolds f l u x CD's at 7.5m i s 1.24x10~ 3 f o r wind speeds from 4 to 10 m/s (De Leonibus, 1971).. The corresponding value of C10 i s about 1.17x10~ 3. Brocks and Krugermeyer (1970) present C10's from 152, near n e u t r a l , 15 minute p r o f i l e s over the B a l t i c and North Seas and the E g u a t o r i a l A t l a n t i c measurements (787, near n e u t r a l , 10 minute p r o f i l e s ) of Hoeber (1969). Eguations 6.1 are w i t h i n the s c a t t e r , but ge n e r a l l y lower by 10 t o 15%. The o v e r a l l average 10 3C10 from Hoeber (5 t o 11 m/s) i s 1.23±0.25 and h i s band averages agree very w e l l with those of f i g u r e 30, except between 5 and 6 m/s where an average of only 11 po i n t s i s 1.48. With 2/3 of h i s p r o f i l e s u n s table, CDN may average a few percent l e s s . Averaging the B a l t i c and North Sea data (4 to 12 m/s) gives 10 3C10 = 1.30±0.18, but there i s a s l i g h t trend with wind speed. The Bass S t r a i t s i t e of Hicks (1972 A) ought t o be re p r e s e n t a t i v e of the open sea, however h i s wind speeds are reduced by u* to account f o r surface d r i f t . I f the eddy c o r r e l a t i o n drag c o e f f i c i e n t s are reduced by about 7%, they then conform to the CDN's of f i g u r e 28 and the 30 values between 3 135 and 8 m/s then average 1.13x10 - 3, however 6 runs from 8 to 10 m/s average about 1.4x10~ 3. Garratt a l s o uses the 18 eddy c o r r e l a t i o n runs (3 to 11 m/s) from Hasse (197 0), quoting an average of 10 3C10 = 1.21 ±20%. The two data sets used by Gar r a t t t h a t remain to be discussed are open sea st u d i e s from B/V FLIP, however, they are not independent. Because of flow d i s t o r t i o n due t o FLIP, Paulson et a l . (197 2) use an e m p i r i c a l c o r r e c t i o n f a c t o r chosen so that t h e i r average <uw> (from p r o f i l e s ) and the average of corresponding eddy c o r r e l a t i o n measurements from Pond S i * (1971) , are equal. The l a t t e r data (20 runs) give an average CD at about 8 meters of 1.52x10~ 3 i n winds between 4 and 8 m/s and s l i g h t l y unstable c o n d i t i o n s . The eguivalent average CDN i s expected to be about 1.3x10 - 3 with a 20% v a r i a b i l i t y . A c cordingly, Paulson e t a l . f i n d 103CDN = 1.32 f o r 19 runs. I t appears as i f the major source of the disagreement between the data of f i g u r e 28 and the data used by Gar r a t t (1977) i s that the l a t t e r i n c l u d e s measurements from onshore and shallow water ( l e s s than 15 meters) s i t e s . Eguations 6.1 are c o n s i s t e n t with most of the deep water (more than 50 meters) drag c o e f f i c i e n t s to w i t h i n p o s s i b l e measurement e r r o r . The disc r e p a n c i e s suggest a s l i g h t l y higher drag c o e f f i c i e n t at wind speeds below 10 m/s. In c o n t r a s t , recent measurements (10 hours) i n Bass S t r a i t (Antonia et a l . , 1978) i n winds between 5 and 10 m/s and -0.1 <Z/L< 0 give an average 103CD at 5 meters of 1.05 and 1.25 (about 0.9 and 1.1 at 10 meters) from the Beynolds f l u x and d i s s i p a t i o n methods, r e s p e c t i v e l y . Since a constant 136 CDN i s a good f i t t o most i n d i v i d u a l data s e t s , i t i s l i J c e l y to remain a good d e s c r i p t i o n of the o v e r a l l open ocean s i t u a t i o n below about 10 m/s. The "best" constant t o use i s debatable, but i t should be i n the range 1.1x10~ 3 t o 1.3x10 - 3. The average 10 3C10's (neutral or near neutral) from a l l the discussed open ocean; eddy c o r r e l a t i o n , d i s s i p a t i o n and p r o f i l e measurements i n winds l e s s than 10 t o 12 m/s are about; 1.17 (178 r u n s ) , 1.14 (626 hours) and 1.24 (958 p r o f i l e s , 188 hours), r e s p e c t i v e l y . The o v e r a l l average i s 1.20, but only 1.17 i f the r e s u l t s from Hoeber and from Brocks and Kruggermeyer are weighted by 1/6 and 1/4, r e s p e c t i v e l y , t o make them comparable to hourly values. At the higher wind speeds, the vast majority of the open sea drag c o e f f i c i e n t s come from Smith (1979) and f i g u r e 28, which are both w e l l described by 6.1. I t i s an i n t e r e s t i n g f e a t u r e of 6.1, that an e x t r a p o l a t i o n t o 50 m/s f i t s the hurricane and wind flume data compiled by Garratt (1977) as w e l l as a c o n t i n u a t i o n of the Charnock l i n e of f i g u r e 30. The V a r i a b i l i t y Of CDN And Zo I t may be p o s s i b l e to describe turbulence and the drag c o e f f i c i e n t s with a v a r i a b l e Zo, although t h i s i s expected t o be d i f f i c u l t and there are no measurements of surface c o n d i t i o n s a v a i l a b l e . The wind speed dependence of CDN above 10m/s shows tha t a constant Zo i s not a p p l i c a b l e . A Charnock r e p r e s e n t a t i o n withX=0.0123 (K=0.40) , passes w i t h i n +1 standard d e v i a t i o n of a l l the band means of f i g u r e 30, but i t s predicted low wind speed behavior i s not observed. A Charnock l i n e with <=< =0.008 f i t s the region from 010 = 11 t o 17 m/s f a i r l y w e l l , but then 137 f a l l s o f f much too q u i c k l y to account f o r the values up to 26 m/s. The CDN's t o 50 m/s i n Garratt (1977), r e q u i r e °< to be about 0.016. CDN i s very n e a r l y a constant, 0.00114, f o r 4 <O10< 10 m/s, as i s the average of C10. This behavior may be a consequence of h i g h l y v a r i a b l e sea surface c o n d i t i o n s tending to make Zo increase with decreasing wind speed and bala n c i n g , on average, the tendency of lower u* values t o make i t smaller. Above 10 m/s, the important parameters seem t o change ch a r a c t e r , so that Zo v a r i e s with wind speed, producing the observed near l i n e a r r i s e i n CDN with 010. I t may be p o s s i b l e t o describe Zo by a Charnock formul a t i o n with °< an experimentally determined f u n c t i o n of various surface parameters. To explore the i n f l u e n c e of the sea surface on the turbulence above i t , i t i s us e f u l to examine the time h i s t o r i e s of the winds and f l u x e s . An example, with hourly averages, i s run D33C, f i g u r e 31. The drag c o e f f i c i e n t i s seen to vary q u i t e smoothly with time, such that the random v a r i a b i l i t y about a running average over a few hours i s only about 10%. The increase with wind speed i s evident from hour 6 t o 21. In a d d i t i o n , CDN i s seen to behave very d i f f e r e n t l y on the r i s i n g wind than i t does when the wind speed i s dropping. The f a l l i n g wind i s a l s o accompanied by an almost complete r e v e r s a l i n d i r e c t i o n and i t i s t h i s s i t u a t i o n that gives r i s e to the high drag c o e f f i c i e n t s at low wind speeds. The p i c t u r e of lower than average drag c o e f f i c i e n t s on the r i s i n g wind and higher ones on the f a l l i n g wind i s a l s o supported by runs D29D and D31F, f i g u r e s 32 and 33. Denman and Miyake (1973) observed that drag c o e f f i c i e n t s tended to i n c r e a s e on the l e a d i n g s i d e of a storm. 138 —~t=>' P CM T—r 1—1—r i — r ++++++ ++ +++ + + o Q -I C J + + + + + + + + + + + ~ i — r i — 1 — r i — r 1—r 1 X 1 ° . _ | ++ + +++ ++++++ ++ + + + ++ +++++ 1 — r 1—r T—r m o ' J V a. r +s ++++++ 1—1—1—r i — 1 — 1 — 1 — 1 — r UJ +++ • ++" 0 FIGURE 31 -1—1—1—1—1—1—r 12 IB 24 30 TIME (HOURS) 36 1 — r 42 48 Time series of the momentum flux from run D33C, sta r t i n g at 4:40 GMT December 7, 1976. Momentum fluxes are from the dis s i p a t i o n (pluses), Reynolds flux (squares) and bulk ( s o l i d line) methods. 139 then either remain constant or decrease s l i g h t l y . This i s also a very apt description of the time series i n figures 31, 3 2 and 33. The scaling arguments of section 2.1 predict that the difference between the 10m and surface momentum flux should be greater (giving lower c o e f f i c i e n t s ) , on the f a l l i n g wind. Apparently the increase i n Zo in t h i s s i t u a t i o n i s more than enough to overcome t h i s loss. It i s possible to speculate that Zo depends on surface parameters that, as the wind speed f a l l s , remain near th e i r " o ld" values produced by the previously higher winds. If °< were dependent on the t o t a l wave generating force and hence 010-Cw (section 2.3), where Cw i s the component of the wave speed i n the i d i r e c t i o n , i t would be very sensitive to sudden changes i n wind d i r e c t i o n , because these would tend to make Cw smaller and perhaps even negative. I t appears l i k e l y that some of the observed scatter i n drag c o e f f i c i e n t s i s due to the influence of the past history of the wind as "remembered" by the sea surface. With access to wave spectra, Denman and Miyake (1973) concluded that the drag c o e f f i c i e n t was dependent on the nature of the wave f i e l d to the order of 20%. In order to examine their influence on the average CDN, tower data from various wind conditions are presented i n table X. Unfortunately, above 15 m/s there are too few data for meaningful averages. Winds of "constant speed" and "constant d i r e c t i o n " are allowed to have 4 m/s and 30 degree ranges respectively. Data used in the "after d i r e c t i o n changes" category are from the six hours following a change i n dir e c t i o n of at least a 60 degrees i n less than two hours. Peak winds are included in the r i s i n g winds. The suggestion i s , that either a 140 r • WIND SPEED BANDS (M/S) a • | Wind | | conditions j | (number of hours) | 1 Eegression | and | correlation | 4.5-7| 7-9 I 9-11 J 11-15| J A l l tower data | I (1086) J - 79 +.043 010| .59 | 1.11 | (176) | 1.13 (264) | 1.17 | I (259) | 1.33 (247) | Constant speed | J and direction | I (253) | .71 + .049 010| .60 | 1.03 | (31) I 1. 13 (68) I 1-14 I I (79) | 1.31 (59) • I Eising winds | I (167) J . 64 + .054 010| .70 | 1.01 | (22) | 1.11 (33) I 1-14 I I (31) I 1.25 (49) I • | Eising wind | | steady direction! 1 (81) | . 60 +.055 010| .67 | 1.08 | (3) | 1.11 (15) I 1-10 | I (18) | 1.24 (29) i I F a l l i n g wind | I (111) I . 94 +.036 010| .51 | 1.40 | (3) I 1.25 (12) | 1.28 | I (24) | 1.38 (49) | F a l l i n g wind | | steady d i r e c t i o n l I (80) | 1.05+.025 010| .36 | 1.31 | (2) I 1.28 (8) I 1-29 | I (18) | 1.34 (29) i I After d i r e c t i o n | | changes | I (60) | L _ . 1 • 1.18 | (22) | . J , 1.34 (12) • i 1.60 (16) i i TABLE X Mean 103CDN of wind speed bands i n d i f f e r e n t wind conditions. Bracketted values are the number of hours contained in an average or regression of CDN against 010. dire c t i o n change or a f a l l i n g wind tends to produce higher than average drag c o e f f i c i e n t s . Table X also indicates that i f observations were available only during r i s i n g winds with constant d i r e c t i o n , as may be the case i n some data sets, the average CDN value for 7 <010< 11 m/s, would be about 4% lower than given by 6 .1 . The observations of CDN during r i s i n g winds 141 show a systematic r i s e with wind speed and a r e g r e s s i o n of CDN against 010 gives 10* CDN= 0.64+ 0.054 010 with a c o r r e l a t i o n c o e f f i c i e n t of 0.7 0. Thus, i f only these data were a v a i l a b l e one could e a s i l y conclude t h a t the average drag c o e f f i c i e n t increases l i n e a r l y with wind speed even below 10 m/s. Such a trend may a l s o emerge even i f data with constant speed and d i r e c t i o n are in c l u d e d . S i m i l a r e f f e c t s are expected at the higher speeds too, so i n order to a r r i v e a t a r e a l i s t i c mean drag c o e f f i c i e n t f o r a l l wind speeds i t i s imperative that measurements be made during a l l l i k e l y wind c o n d i t i o n s . I t i s , t h e r e f o r e , p o s s i b l e that drag c o e f f i c i e n t formulations may vary with l o c a t i o n and time of year, with the break from a constant t o a l i n e a r l y i n c r e a s i n g CDN being q u i t e v a r i a b l e . The tower data i s al s o sorted i n t o monthly groupings i n t a b l e X I . There i s no trend i n the September data and the average CDN's are lower than those of the other months. There i s a reasonably s i g n i f i c a n t trend i n the October data throughout the e n t i r e wind speed range. The high CDN's at low winds are evident i n the December r e s u l t s , while March seems to r e f l e c t average c o n d i t i o n s , t a b l e X. The r e s u l t s of experiments conducted at d i f f e r e n t times of the year may, t h e r e f o r e , be considerably d i f f e r e n t , p o s s i b l y because wind c o n d i t i o n s vary throughout the year. I t would be d e s i r a b l e to i n c l u d e data from every month i n any drag c o e f f i c i e n t f o r m u l a t i o n . Onfortunately, no measurements from May through August are a v a i l a b l e from e i t h e r the tower or the weathership, but the high wind speed formulation should not be a f f e c t e d as these are not the months with the gre a t e s t winds. 142 i 1 | WIND SPEED BANDS (M/S) | r Wind conditions (number of hours) r - i -| Regression | I and | | cor r e l a t i o n | 5-7 | 7-9 | 9-11 | 11-15| I September (241) 11.05+.001 | .008 010| 1.07 | (94) | 1.04 | (79) | 1.05 | (47) | 1.06 | (21) I l October (131) |.83 +.047 | .65 D10| 1.11 | (28) J 1.16 | (25) | 1.30 | (28) | 1.45 | (34) | i December (251) |.78 +.044 | . 62 O10| I 1.22 | (20) | 1. 14 | ("2) | 1. 14 | (53) | 1.33 | (73) | l March (271) |.77 +.044 | .65 O10| 1.12 | (24) | 1. 18 | (49) | 1. 17 | (96) | 1.31 | (70) | i « 1 ,, J • i • TABLE XI Mean 103CDN of wind speed bands f o r d i f f e r e n t months. Bracketted values are the number of hours contained i n an average or regression of CDN against 010. The Empirical Drag C o e f f i c i e n t The investigation of the tower data has shown that neither s t a b i l i t y nor fetch (greater than 10 km) have a s i g n i f i c a n t e f f e c t on the average neutral drag c o e f f i c i e n t . I t has been demonstrated that much of the v a r i a b i l i t y i n CDN measurements, not only between i n d i v i d u a l data points, but between enti r e data sets, may be due to the influence of the surface wave f i e l d . However, i t appears as i f the measurement of the important surface parameters could possibly be more complicated than direct momentum flux measurements. An important parameter that can be p r a c t i c a l l y incorporated i n t o a neutral drag c o e f f i c i e n t formulation i s the wind speed. 143 The e m p i r i c a l CDN formulation of 6.1 gives bulk estimates of the momentum f l u x from the wind speed, OZ and s t a b i l i t y , Z/L(AT), f o l l o w i n g the procedure o u t l i n e d i n s e c t i o n 2.3. An i t e r a t i v e technigue i s used to so l v e eguation 2.15 f o r 010. For the f i r s t pass 010«= OZ/ [1+ 0.1 K(Z,Z/L) ] i s assumed (CDN*/2/ K -0.1), g i v i n g a CDN' from 6.1. On subseguent i t e r a t i o n s (never more than 3) 010' i s found from 2.15 using the CDN' cf the previous p a s s . u n t i l successive 010* values change by l e s s than 1%. CDN' from the f i n a l pass i s s u b s t i t u t e d i n t o 2.14 to give a CD, which g i v e s the momentum f l u x through 2.9. Bulk estimates are to be compared t o d i r e c t estimates i n l a t e r time s e r i e s t h a t i n c l u d e data with -1.0 <Z/L< 0.3. The s t a b i l i t y need not be included i n the 010 c a l c u l a t i o n , because the maximum s t a b i l i t y of the ship data i s only 0.08 and on the tower, where s t a b i l i t i e s are higher, the re d u c t i o n i s only from 13 to 10 meters, t h e r e f o r e , ( s e c t i o n 2.3) the worst e r r o r i s l e s s than 2% i n 010 and about 1% i n CD and <uw>. The e r r o r i n Z/L (AT) (section 4.1) introduces an error i n CD of about 5%.. A 2% e r r o r i n 0Z becomes 4% i n <uw>. The la r g e standard d e v i a t i o n s i n f i g u r e 30 and the range of previous measurements, suggest that any CDN for m u l a t i o n has at l e a s t a 10% u n c e r t a i n t y . The combined e r r o r i n bulk estimates from equation 2.9 are, th e r e f o r e , greater than 15% as are the e r r o r s i n d i r e c t estimates. Although i n d i v i d u a l bulk estimates may d i f f e r from d i r e c t d i s s i p a t i o n or Reynolds f l u x measurements by 50% or more, there should be reasonable agreement on average, a l b e i t the necessary averaging period may vary with time and place. 144 Comparison Of Methods In a l l the time s e r i e s presented, f i g u r e s 31 to 34, the momentum f l u x or Reynolds s t r e s s , ( i n N/m2 or pa s c a l s . Pa) i s c a l c u l a t e d from the d i s s i p a t i o n method ( p l u s e s ) , from the Reynolds f l u x method (squares) and from the bulk formula of 6.1 ( s o l i d l i n e ) . Run D29D, f i g u r e 32, i s a t y p i c a l September s i t u a t i o n . During the unstable c o n d i t i o n s of the r i s i n g wind (to hour 22) the bulk formula gives a t o t a l momentum inpu t of 2.23 N-hr/m2, which i s 34% higher than found from the d i s s i p a t i o n c a l c u l a t i o n s . Upon reaching the peak winds the two methods are i n much better agreement and the d e v i a t i o n over the whole run i s reduced to 18%. A f r o n t i s passing the tower i n D33C ( f i g u r e 31) and a s i m i l a r s i t u a t i o n occurs, but with the r i s i n g and f a l l i n g wind e f f e c t s b a l a n c i n g , the o v e r a l l d e v i a t i o n i s only 1%. In D29D the f a l l i n g wind and changing d i r e c t i o n e f f e c t s are not great enough t o make the average s t r e s s from the two methods equal. I t would be expected that these i n f l u e n c e s are s t r o n q l y f e l t a f t e r the passage of low pressure systems, because as centers pass the winds q u i c k l y f a l l , then regain v e l o c i t y as the d i r e c t i o n changes a f t e r which they drop o f f slowly. A low pressure center i s passing to the west of the tower, from south t o north, i n D31F, f i g u r e 33. Unfortunately the e a r l i e r winds from the east were obstructed by other instrumentation on the tower. S t a b i l i t y i s always very nearly n e u t r a l or s l i g h t l y s t a b l e and should not have been a f a c t o r i n causing the l a r g e observed s t r e s s e s that give a t o t a l momentum input over the period shown of 6.5 N-hr/m2, which i s 12% higher than given by the bulk c a l c u l a t i o n s . A run from March 1977, 145 Q • C J <=>r-1 1 i—i—r i—i—i—r i — r co^-ZD" +++++++++++ +++ +++ i—i—r i—i—i—r i — i — i — i — i — i — i — i — r i — i i — i — r CO LU 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 12 18 24 30 36 42 48 TIME (HOURS) FIGURE 32 Time series of the momentum flux from run D29D. Time i s from 17:00 GMT September 26, 1976, and symbols are the same as i n figure 31. 146 D38B, f i g u r e 34 shows t h a t a l l three methods can be i n e x c e l l e n t agreement over a long period of time provided the wind i s reasonably steady. Over t h i s period of n e a r l y two days, the cumulative momentum input from the d i s s i p a t i o n c a l c u l a t i o n s i s l e s s than 3% higher than t h a t found from the bulk formula. In general the Reynolds f l u x method v e r i f i e s the d i s s i p a t i o n c a l c u l a t i o n s , but an exception i s found at hour 28 of run D29D, f i g u r e 32. The f l u x run, T61, i s an average of only 3 groups (40 minutes) s t a r t i n g 15 minutes a f t e r the beginning of the hour 28 d i s s i p a t i o n average. In t h i s 15 minute i n t e r v a l there was a sudden 4 m/s increase i n wind speed, which the hourly averaging smooths. The simultaneous d i s s i p a t i o n run does agree with T61 (Appendix), i l l u s t r a t i n g that i d e n t i c a l time i n t e r v a l s are r e g u i r e d f o r intercomparisons. The f l u x run, T112, at hour 14 of run D33C ( f i g u r e 31) i s a l s o found to be i n much b e t t e r agreement with i t s simultaneous d i s s i p a t i o n run than with the hourly average. Run D33C d i s c l o s e s a sampling problem associated with the tower operation of the Reynolds f l u x system. Most of the low wind speed runs were c o l l e c t e d i n September when the drag c o e f f i c i e n t s tend to be low. L a t e r , when the more v a r i a b l e winds seem to produce some very high c o e f f i c i e n t s at wind speeds below 10 m/s, the Reynolds f l u x system was prevented from recording by the wind speed l i m i t s e t t i n g . Thus the Reynolds f l u x data set ( f i g u r e 19) tends t o be biased toward small CD's, r e l a t i v e t o 6.1, at the low wind speeds and to e x h i b i t a trend with wind speed over the whole range of measurements. Above the maximum wind speed l i m i t s e t t i n g (12 m/s) 6.1 f i t s the data of f i g u r e 19 very w e l l . 147 ol _4 CM i — r i — i o r\i" T 1 1 1 1 1 1 1 1 ++ + + + +++ + +++++ ++++ i — r i—i—i—r~ ++ i — r + + +++ + + + + + i — r rn .o" T—r i — r 1 1 1 i — r i — r co - i ++-HH-+++-r+++++-H-~1 1 1 42 48 U J i — i — i — i — i — i — i — i — i — r 6 12 18 24 30 TIME (HOURS) 36 FIGURE 33 Momentum flux time series f r c n run D31F. Starting time i s 4:00 GMT October 20, 1976. The s o l i d l i n e represents the bulk estimates of T. 148 O -++ +++ ++++ +++++++++++ ++.-*+++++ + + + + + + + + + + ++++ i — i — i — i — r i—r i—i—r lo!£H ZD" + ++++ + + + + + + + + + ++ ++++ T— i — r *•++++++++,H i — i o " J y a. i +++++++4.-,.+++++. +++++++++++++++++ i — i — i — r i — i — r CO H + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + FIGURE 34 - | 1 1 1 1 1 1 1 1 1 1 1 1 1 ) 6 12 18 24 30 ?6 42 48 TIME (HOURS) Momentum flux time series from D38Bi Time i s from 16:00 GMT March 12, 1977 and symbols are the same as i n figure 31. 149 It appears that the bulk aerodynamic method gives good estimates of the t o t a l momentum input or average stress over periods of a few days or more. The bulk estimate should improve i f measured or approximated s t a b i l i t y conditions are included. I t has been shown that i n steady winds the hourly bulk estimates are good measures of the stress, but with more variable winds they may consistently d i f f e r , over periods up to a day, from diss i p a t i o n estimates. With comparable error in both methods, i t i s not obvious which i s the more accurate. I t i s conceivable that the varying winds are producing systematic errors i n one or more of the dissipation method's assumptions and causing the discrepancies. However, i f t h i s occurred i t i s l i k e l y that the o v e r a l l scatter of figure 28 would be greater than the scatter i n drag c o e f f i c i e n t s calculated from other methods, which i s not observed. It i s f e l t that, since at least some of the scatter in measured drag c o e f f i c i e n t s at the same wind speed i s r e a l , d i s s i p a t i o n measurements, although subject to random errors, ought to follow changes i n the r e a l stress more closely than bulk estimates. A bulk formula based on long term averages may not be s t r i c t l y applicable to short term phenomena, such as wave development during a r i s i n g wind or the deepening of an i n l e t ' s upper layer following a switch from down to strong up i n l e t winds. In such cases, i t may be possible to devise an appropriate formulation cabable of providing good hourly stress values with minimal measurement error. A possible means of incorporating variable wind effects would be to allow the drag c o e f f i c i e n t to depend on the past history of the wind. 1 5 0 6.3 Bulk Aerodynamic Parameterization Of The Sensible Heat Flux The sensible heat f l u x i s parameterized i n terms cf a surface - a i r temperature difference. The mean a i r temperature at 10m, T10, i s obtained from the average temperature at the measurement height, TZ, through eguation 2.11. At the Bedford tower the surface temperature, TSFC, i s approximated by a sea temperature, TSEA, measured about 10m below the mean sea l e v e l , where the heating and cooling of the surface by radiation and heat exchanges with the atmosphere may not be followed exactly. On CCGS Quadra a surface "bucket" temperature was recorded as part of the 3-hourly meteorological observations. These are interpolated to give hourly TSEA values, which may be a problem, because the ship was steaming through large sea surface temperature gradients during the one run that useful temperature measurements were recorded. At any time then, TSEA may d i f f e r s ubstantially from a representative surface temperature. As a precaution that the Stanton number, CT (eguation 2.9), i s not too adversely affected, data used i n the parameterization are re s t r i c t e d to conditions with |TSEA-TZ| > 1.0°C, even though the flux measurements may s t i l l be r e l i a b l e . Figure 35 i s a plot of <wt> against 010 .AT (AT=TSEA-T 10) , for a l l 129 hours of unstable temperature data, and a regression l i n e (the average of <wt> against 010 AT and 010 AT against <wt>). The 23 hours of data from the weathership (plotted as pluses) show considerable scatter, but no systematic departure from the tower re s u l t s ( t r i a n g l e s ) . The regression i s 151 FIGURE 35 Parameterization of the sensible heat f l u x in °Cm/s i n unstable s t r a t i f i c a t i o n . Triangles represent tower data and pluses are from CCGS Quadra,. 152 <wt> = 0.00100 U10 AT + 0.0029 °Cm/s 103 CT10 = 1.00 + 2-9(°Cm/s) / (U10 A T ) , (6.2) with a c o r r e l a t i o n c o e f f i c i e n t of 0.86. This i s almost i d e n t i c a l t o the formula given by Friehe and Schmitt* 1976, f o r 0< 010 A T <25 °Cm/s only. R e s t r i c t i n g the parameterization to data with |<wt>| >0.004°Cm/s, 010 >4.0 m/s and | A T | >1°C e l i m i n a t e s low heat f l u x and small U10 A T s i t u a t i o n s , but a p o s i t i v e heat f l u x i s s t i l l p r e d i c t e d at 010 A T = 0, although the value of 0.002 °Cm/s given by Friehe and Schmitt, who used mostly small heat f l u x data, i s probably more r e a l i s t i c . The a d d i t i o n a l data at higher s e n s i b l e heat f l u x e s i n d i c a t e that a s i n g l e parameterization i s acceptable from 010 A T =0 to p o s s i b l y more than 100 °Cm/s. Again t h i s r e s u l t i s i n accord with the BIO tower data of Smith, 1979, but the average c o e f f i c i e n t i s much lower than found by Smith and Banke (1975) from a l i m i t e d number of Sable I s l a n d measurements. The s e n s i b l e heat f l u x time s e r i e s from run D33C, f i g u r e 36, shows that seemingly very small Stanton numbers a r i s e during a r a p i d i n crease i n a i r temperature, j u s t a f t e r A T f i r s t goes negative. F o r t u n a t e l y , Reynolds f l u x measurements (squares) are a l s o a v a i l a b l e over t h i s period and t h e i r qood agreement with the d i s s i p a t i o n method v e r i f i e s the treatment of equation 2.21. Temperature spectra from the Reynolds f l u x recordinqs do not d i s p l a y any evidence of s a l t contamination, so i t i s f e l t that the <wt> values from t h i s period are reasonably accurate. However, i t i s p o s s i b l e that they are s i g n i f i c a n t l y l e s s than t h e i r surface values. The l o s s of s e n s i b l e heat f l u x with 153 o ++ + + + + + + + + + i — i — i — i — i — i — i — i — i — i — i — i — r 0 24 +++++++++ + o in in. i k>n4 +++++ + +" 1 — — i — i — i 1 r 0 24 T — i — i — i — i — r + + + + + + + ^ r + " + 0 24 . ++++++ + + + + + ++ ++++ + CO "V. " ~ i — i — i — i — i — i — i — i — i — i — i — i — r 6 12 18 24 30 36 42 TIME (HOURS) FIGURE 3 6 Time series of the sensible heat flux from run D33C (figure 31). Time i s from 4:40 GMT December 7, 1976. Hs i s from the dis s i p a t i o n (pluses), Reynolds flux (squares) and bulk ( s o l i d line) methods;. 154 height was discussed i n s e c t i o n 2.1 and i f the 0.5°C/hour heating were due s o l e l y t o the v e r t i c a l f l u x divergence, then a 40% l o s s i n f l u x at 13m would be expected. Q u a l i t a t i v e l y the low CTN values observed can be explained by s p e c u l a t i n g that <wt> was measured above the "constant s e n s i b l e heat f l u x " l a y e r , s i n c e i f t h i s were the case, the percentage of the l o s s should decrease as the magnitude of the surface f l u x increases and no l o s s should be evident a f t e r the peak temperature. However, there i s probably a great de a l of heating due t o h o r i z o n t a l advection, which would mean l e s s l o s s of f l u x with height. There are other s i t u a t i o n s of r i s i n g temperature which do not seem t o be a f f e c t e d to such a great degree so p o s s i b l y the measured <wt>'s do not d i f f e r from the surface f l u x by as much as the low CTN's would suggest. Another p o s s i b i l i t y i s that TSEA a t a depth of about 10m i s not t r a c k i n g the surface temperature. I n the 12 hours p r i o r to run D33C there was very l i t t l e wind and the a i r temperature was below -15°C, g i v i n g a huge AT of about 20°C. I t appears as i f the surface temperature, TSFC, may have become much c o l d e r than TSEA during t h i s time, because as the a i r warmed and the wind increased, presumably mixing the unstable water column, TSEA a c t u a l l y decreased from 3.9 t o 3.5°C. I t i s not impossible that the surface heating produced by the l a t e r high a i r temperatures was prevented from mixing down to the sea temperature probe by the then s t a b l e upper water column* In t h i s s i t u a t i o n TSEA could become l e s s than TSFC by enough t o produce the observed behavior of D33C and to upset the parameterization of the s e n s i b l e heat f l u x . There was only one other s i m i l a r s i t u a t i o n at the tower 1 5 5 and even lower Stanton numbers were calculated, but these data were rejected on the basis of abnormal temperature spectra. Despite the preceding arguments, lower than average CT values may occur in the circumstances described, but they are not l i k e l y to occur over the open sea, so t h i s period of D33C i s not included in the parameterization and analysis of the stable sensible heat flux. Figure 37 i s a plot of -<wt> against -010 (TSEA-T10) from 131 hours of stable s t r a t i f i c a t i o n with the suspect D33C r e s u l t s plotted as crosses. The regressions of -<wt> against -010 AT and -010 A T against -<wt> for the 123 tr i a n g l e s have a correlation c o e f f i c i e n t of 0.93 and when averaged y i e l d <wt> = 0.00075 010 A T + 0.0020 °Cm/s 103 CT10 = 0.75 + 2.0(°Cm/s) / (010 A T ) , (6.3) which i s also plotted on figure 37. Again a positive heat flux of about 0.002 °Cm/s at 0 AT=0 i s indicated even though there i s no data with |0 A T | < 5°Cm/s. Inclusion of the suspect D33C data lessens the slope to about 0.00065.. Friehe and Schmitt only considered data with 0 A T > -15°Cm/s and t h e i r suggested formula d i f f e r s from 6.3 by only 10% at 0 A T = -15°Cm/s. There are only a few data points beyond 0 A T = -35°Cm/s that contribute to 6.3, which, therefore, cannot be expected to be representative of more negative 0 A T ' s . The BIO tower re s u l t s (Smith, 1979) contain more, large negative heat flux runs and suggest a larger CT10 of about 0.00083. Although the BIO regression d i f f e r s from 6.3 by about 13% at U A T = -100 °Cm/s, the data points overlap 156 FIGURE 37 Parameterization of the sensible heat flux (°Cm/s) in stable s t r a t i f i c a t i o n . Crosses are from D33C. S o l i d l i n e i s a regression of the t r i a n g l e s only* 157 very w e l l . A s i n g l e parameterization of the s e n s i b l e heat f l u x i n s t a b l e s t r a t i f i c a t i o n should adequately describe a l l the measurements, but i t appears t o d i f f e r from the unstable case. I t i s f e l t t h a t the u n c e r t a i n t y i n U10 Al i s perhaps qreater than i n <wt>, so band averaqinq i s c a r r i e d out over bands of <wt> and the r e s u l t s are ta b u l a t e d i n t a b l e X I I . The band averages are reasonably w e l l described by 6.2 and 6.3 over the e n t i r e range of measurement, thus l i n e a r f i t s to the data of f i g u r e s 35 and 37 are appropriate. The r a t i o of the averages gives a CT10 f o r each band and the unstable values are c l e a r l y higher than the s t a b l e ones. The i n d i v i d u a l points are f a r too few and sca t t e r e d f o r any trends to be seen. For example, the 103CT10 value of 1.21 i n the 0.05 t o 0.07 °Cm/s band, becomes 1.04 i f U10A.T i s increased by only 1/2 a standard d e v i a t i o n . S t a b i l i t y E f f e c t s The s t a b i l i t y dependence of the n e u t r a l Stanton number i s examined i n f i g u r e 38, where CTN, eguation 2.12, i s p l o t t e d against Z/L. The s o l i d l i n e s represent the o v e r a l l averages: 123 s t a b l e 10 3 CTN = 0.69 129 unstable 10 3 CTN = 1.08 , (6.4) about which the standard d e v i a t i o n s are 0.16 and 0.36, r e s p e c t i v e l y . There i s not a great deal of data on which to base d e f i n i t i v e c o n c l u s i o n s , but i t appears as i f CTN and Zot are reasonably independent of Z/L i n s t a b l e and unstable 158 r Range of <wt> °Cm/s i Average <wt> ±1 standard d e v i a t i o n T T~ I Average 010AT | I ±1 standard | | d e v i a t i o n | - -•r-103 CT10| r a t i o of| averagesi i Number | of | runs | 0.C9, 0.11 1 .0,98 +.007 | 84.9 ±30 | 1. 15 | 4 I 0.07, 0.09 l .080 ±.007 | 79.0 ±16 | 1.01 | 7 1 0.05, 0.07 1 .058 ±.005 | 47.9 ±16 | 1.21 | 15 | 1 J | 1 C.03, 0.05 1 .038 ±.007 | 35.5 ±13 | 1.07 | 38 | 0.01, 0.03 i .023 ±.005 | 22.0 ±8.2 | 1.05 | 58 | i 0.006, 0.01 l .007 ±.0C1 I 9.86 ±1.7 | 0.71 | 7 | l -.01, -.004 j .008 ±.002 | 13.7 ±3.6 | 0.59 I 43 | -.02, -.01 I .014 ±.003 | 22.4 ±8.4 | 0.62 | 55 | -.03, -.02 i .025 ±.003 | 34.0 ±6.0 | 0.74 | 19 | -.05, -.03 i .035 | 50.3 | 0.70 | 3 I -.07, -.05 l .057 I 71.6 | 0.80 | 3 I L JL- .J _ J. i • TABIE XI I Parameterization of the s e n s i b l e heat f l u x by band averaging 0 AT over ranges of <wt>. s t r a t i f i c a t i o n separately. However, the d i s c o n t i n u i t y i n the mean CTN at Z/L=0, i m p l i e s a dramatic change i n Zot. This feature could be inco r p o r a t e d i n t o the theory by r e l a t i n g Zot to a parameter that changes c h a r a c t e r a b r u p t l y at n e u t r a l s t a b i l i t y , but t h a t otherwise has very l i t t l e s t a b i l i t y dependence as i s i n d i c a t e d by the r e l a t i v e constancy of the FIGURE 38 The n e u t r a l Stanton number as a f u n c t i o n of s t a b i l i t y . S o l i d l i n e s represent the mean i n both s t a b l e and unstable s t r a t i f i c a t i o n . 160 average CTN away from Z/L=0. There i s a great deal of s c a t t e r i n f i g u r e 38 and i t should be r e i t e r a t e d t h a t microbead contamination by s a l t p a r t i c l e s could only be checked at the few, sometimes i n f r e g u e n t , times f o r which spectra were a v a i l a b l e from the Reynolds f l u x system. I t i s therefore p o s s i b l e t h a t some contaminated data has escaped n o t i c e and i s re s p o n s i b l e f o r some of the high CTN values at Z/L<0. S i m i l a r l y , because temperature and humidity are op p o s i t e l y c o r r e l a t e d i n s t a b l e s t r a t i f i c a t i o n , contaminated data may be re s p o n s i b l e f o r some low CTN's at Z/I >0. The high CT10 values f o l l o w i n g hour 40 of run D38B (figure 39) are guite l i k e l y a r e s u l t of bead contamination, because a Reynolds f l u x run at about hour 44 was r e j e c t e d on the b a s i s of a lack of low freguency variance i n the temperature spectrum and the run at hour 41 was a b o r d e r l i n e case. Win d Speed E f f e c t s Equation 2.12 r e l a t e s CTN t o Zo, i n d i c a t i n g t h a t i t should increase with wind speed above 10 m/s, i f Zot does not counteract the increase i n Zo. The s i t u a t i o n may be complicated by the observed dependency of Zot on Z/L, which c o r r e l a t e s with wind speed. The e f f e c t of wind speed on CTN i s i n v e s t i g a t e d f o r the s t a b l e and unstable case i n t a b l e s X I I I and XIV r e s p e c t i v e l y . In the s t a b l e case there i s no i n d i c a t i o n of an incr e a s e i n CTN with wind speed. In f a c t above 10 m/s where CDN begins to i n c r e a s e , there i s a h i n t , a l b e i t not s i g n i f i c a n t , t h a t CTN decreases. In t a b l e XIV the means are s c a t t e r e d , but 161 T 1 1 Minimum I Maximum Wind speed range (m/s) +-Number of runs Mean 103CTN ±1 standard d e v i a t i o n 6 to 8 19 0.71 ± .18 0. 43 1. 12 8 t o 10 39 C.73 + .17 0.47 1.61 10 to 12 25 0.65 ± .14 0.36 0.98 12 to 14 26 0.69 ± .13 0. 44 0.88 14 to 18.2 14 0.62 ± .20 0. 29 0. 88 TABLE X I I I Averaged n e u t r a l Stanton number as a f u n c t i o n of wind speed i n s t a b l e s t r a t i f i c a t i o n only. again there seems to be no obvious wind speed dependency. However, nearly a l l of the data i n t a b l e XIV above 18 m/s come from CCGS Quadra, which d i d not r e a l l y y i e l d enough data at the lower wind speeds to f u l l y intercompare the ship and tower r e s u l t s . There i s a suggestion i n t a b l e XIV that the unstable CTN increases from U10 = 10 to 18 m/s. Larger CTN values are reported over shallow water by Francey and G a r r a t t , 1978, who f i n d an increase with wind speed given by 103CTN = 0.083 010 + 0.48. The contention that CTN i n c r e a s e s with 010 i s apparently supported by run D33C, f i g u r e 36, where the increase i n 0Z from 10 to 13 m/s (hours 7 to 10) i s accompanied by a sharp r i s e i n CTN from about 1.0 to 1.4x10~ 3. The r i s e i n CTN with winds above 10 m/s appears to be g r e a t l y perturbed by the t r a n s i t i o n to s t a b l e s t r a t i f i c a t i o n and although the f o l l o w i n g s t a b l e data are suspect, the r i s e seems t o subsequently continue. No 162 1 - — " I Wind speed I range (m/s) - - T— Number | of runs | Mean 103CTN +1 standard d e v i a t i o n , „ T _ — _ I Minimum " i — - — | Maximum 1 5. 5 to 8 I 32 | 1.13 ± -47 | 0.56 | 2.28 1 8 to 10 ! 29 | 0.94 ± .28 | 0.54 | 1 .47 1 10 to 12 i 25 | 1.06 ± .34 | 0.64 | 2.21 I 12 to 14 i 11 | 1.13 ± .24 | 0.71 | 1 .38 i I 14 to 18 —! i 13 | 1.31 ± .24 | 0.43 | 2.01 . . . ^ • I 18 to 22 — i — 13 | 1.14 ± .28 | 0.78 I 1 .76 i • | 22 to 26 1 0.90 ± .06 | 0.85 | 0.99 i i , i • • • • TABLE XIV Averaged n e u t r a l Stanton number as a f u n c t i o n of wind speed i n unstable s t r a t i f i c a t i o n only. d e f i n i t e conclusion i s p o s s i b l e because of the l a c k of data and the strong i n f l u e n c e of s t a b i l i t y . For example, i f the suspect D33C data were i n c l u d e d i n the 14 to 18.2 m/s range of t a b l e X I I I , the band average would decrease g i v i n g a smaller CTN at the highest speed, even though these a d d i t i o n a l CTN's increase with wind speed. These r e s u l t s do not, t h e r e f o r e , r u l e out the p o s s i b i l i t y t hat CTN f o l l o w s the wind speed dependency of Zo. However, because t h i s trend i s not supported by the Quadra r e s u l t s , because of the l i m i t e d amount of data, some of which i s p o s s i b l y contaminated (run D38B), and because the s t a b i l i t y e f f e c t s are not f u l l y understood, the n e u t r a l Stanton number i s not 163 formulated as a f u n c t i o n of wind speed. I t i s an i n t e r e s t i n g observation t h a t , on average, CTN f o r Z/KO, 0. 00 108, i s very ne a r l y egual t o CDN f o r 4.0< 010 <10 m/s, 0.00114, implying Zo= Zot- 0.0058 cm. I t i s p o s s i b l e t h a t Zot remains constant above 10 m/s with CTN i n c r e a s i n g due t o Zo. I f t h i s i s t r u e , CTN could be obtained from an experimentally determined Zot and a Zo given from a CDN f o r m u l a t i o n , using equation 2.12. Method Comparison The bulk estimates of the s e n s i b l e heat f l u x shown by the s o l i d l i n e s of f i g u r e s 36 and 39, are c a l c u l a t e d from the s t a b i l i t y dependent CTN given by 6.4. As o u t l i n e d i n s e c t i o n 2.3, the n e u t r a l Stanton number i s converted to a CT at the measurement height, wind speed and s t a b i l i t y . The e r r o r i n bulk estimates found from <wt> = CT OZ AT, eguation 2.9, i s about 2% from 0Z, perhaps 10% from CTN and only 5% from Z/I, because the bulk s t a b i l i t y estimate i s always a v a i l a b l e . I n a d d i t i o n considerable e r r o r i s introduced through AT, which may sometimes be i n e r r o r by 0.5°C. Even with a l a r g e AT of 5°C, the e r r o r i s 10%, making the bulk s e n s i b l e heat f l u x c a l c u l a t i o n s more uncertain than those of the momentum f l u x . With a l a r g e p o s s i b l e e r r o r and problems with the data, i t i s d i f f i c u l t to compare the bulk and d i s s i p a t i o n estimates i n the time s e r i e s . The Reynolds f l u x c a l c u l a t i o n s (sguares) lend c r e d i b i l i t y to the d i s s i p a t i o n estimates ( p l u s e s ) , because of the g e n e r a l l y e x c e l l e n t agreement between the two methods. Figure 39 shows the s e n s i b l e heat f l u x time s e r i e s from run D38B and ( l i k e the momentum f l u x of f i g u r e 34) the b u l k , eddy c o r r e l a t i o n and the 164 2Z . - M l . i f i ± E*~ I C O o 1 2 . i — r i — r 24 CM o O A + ++++ +++ + + + ++ + + + + + i — i — i — r 24 CO 2ZoJ ZD + + + + ~ i — i — i — i — i — i — i — i — i — i — r 12 18 24 30 36 42 TIME (HOURS) FIGOEE 39 Sensi b l e heat f l u x time s e r i e s from run D38B (f i g u r e 34). Time i s from 16:00 GMT March 12, 1977 and symbols are the same as i n f i g u r e 36. 1 6 5 d i s s i p a t i o n estimates of the s e n s i b l e heat f l u x continue to agree f o r a considerable length of time. On the whole the s e n s i b l e heat f l u x parameterization i s compatible with the r e s u l t s of F r i e h e and Schmitt, 1 9 7 6 , and Smith, 1 9 7 9 , so a combination of a l l a v a i l a b l e data should give a CTN formulation capable of providing reasonable averages (over a few days or more) of s e n s i b l e heat f l u x values, depending on the accuracy of AT. However, as with the momentum f l u x , there may be short l i v e d s i t u a t i o n s , as p o s s i b l y seen i n D 3 3 C , where a param e t e r i z a t i o n , v a l i d for long term averages, does not s t r i c t l y apply. I t i s a l s o p o s s i b l e t h a t erronous A T values are causing the d i f f e r e n c e between the bulk and d i s s i p a t i o n c a l c u l a t i o n s between hours 2 and 1 0 of D 3 3 C , f i g u r e 3 6 and, as has been discussed, between hours 12 and 2 4 . 166 CHAPTER 7 SUMMARY AND CONCLUSIONS The experimental program described i n t h i s t h e s i s s u c c e s s f u l l y measured the momentum and s e n s i b l e heat f l u x e s over the sea at winds between 4 and 26 m/s. As hoped, a great many hours of momentum f l u x data were s u i t a b l e f o r a n a l y s i s , but much l e s s s e n s i b l e heat f l u x and no moisture f l u x data were found to be r e l i a b l e . Success depended c h i e f l y on the performance of the sensors and on the establishment of the d i s s i p a t i o n method as a v i a b l e means of measuring the f l u x e s of momentum and s e n s i b l e heat. The v e l o c i t y sensor worked very w e l l , t u t there were some problems with the temperature measurements and no humidity sensor was found to be s u i t a b l e f o r remote operation i n a s a l t - a i r environment. The G i l l twin propeller-vane anemometer operated f o r periods of more than a month i n adverse c o n d i t i o n s without s e r v i c i n g . I t provided the f l u c t u a t i n g h o r i z o n t a l and v e r t i c a l v e l o c i t i e s to the Reynolds f l u x system and responded to the lower frequencies of the downstream v e l o c i t y spectrum's -5/3 region s u f f i c i e n t l y f o r the molecular d i s s i p a t i o n to be i n f e r r e d , although the p r o p e l l e r responses f i r s t had to be determined. The distance constant was found to depend on the type and weight of p r o p e l l e r , the wind speed and the angle of at t a c k . The humidity s e n s i t i v i t y of s a l t contaminated microbeads was recognized by the l a c k of low freguency variance i n the temperature spectrum, but only sometimes were other c h a r a c t e r i s t i c s , such as the absence of a -5/3 r e g i o n , observed. 167 This behavior remains a major problem with the remote operation of t h i s type of sensor. The response of the microbeads seemed to be l i m i t e d by i t s p r o t e c t i v e enclosure and was described by a distance constant of about 0.90m, which i s adequate f o r both the Reynolds f l u x and d i s s i p a t i o n methods. Reynolds f l u x measurements from the Bedford tower were shown to be r e a l i s t i c by comparisons with s p e c t r a , cospectra and turbulence s t a t i s t i c s from previous s t u d i e s . In a d d i t i o n , the drag c o e f f i c i e n t s and Stanton numbers were g e n e r a l l y comparable to the r e s u l t s of Smith, 1979, however there i s probably a bias to small drag c o e f f i c i e n t s at low wind speeds as a r e s u l t of the Reynolds f l u x sampling. U n i v e r s a l shapes f o r the v e l o c i t y spectra and cospectra i n both the s t a b l e and unstable cases, were found from averages over the 196 momentum f l u x runs. With only 60 temperature runs a v a i l a b l e , there was considerable u n c e r t a i n t y i n the normalized temperature spectrum and w,t cospectrum. The i n t e g r a t i o n of a l l cospectra began at n=0.004, then the unstable and s t a b l e 0uw (f) and the unstable and st a b l e •wt (f) i n t e g r a l s were m u l t i p l i e d by 1.06, 1.005, 1.10 and 1.04, r e s p e c t i v e l y , i n order to account f o r the lower freguencies. This method was found t o preserve covariance, on average, and to reduce the s c a t t e r i n eddy c o r r e l a t i o n measurements caused by the u n c e r t a i n low frequency c o n t r i b u t i o n s to the f l u x e s . 1 6 8 R e l i a b l e Reynolds f l u x estimates were needed f o r comparison with simultaneous d i s s i p a t i o n c a l c u l a t i o n s and the d i s s i p a t i o n method was shown to give e s s e n t i a l l y the same r e s u l t s * on average. The agreement between the two methods was found to be best when the magnitude of the f l u x e s was l a r g e . In a l l but the most s t a b l e s t r a t i f i c a t i o n ( Z / L > 0 . 0 5 ) , U * D I S S 1 , the n e u t r a l d i s s i p a t i o n method, which does not re q u i r e an e x p l i c i t s t a b i l i t y parameter, was found to be i n q u i t e good agreement (to w i t h i n about 2055) with eddy c o r r e l a t i o n values of u*. The agreement between the two techniques improved, p a r t i c u l a r l y the s e n s i b l e heat f l u x c a l c u l a t i o n s , when the s t a b i l i t y m o d i f i c a t i o n of the l o g a r i t h m i c p r o f i l e s and the buoyant production were incorporated i n t o the d i s s i p a t i o n method ( u * D I S S 2 and <wt>DISS2). These c o r r e c t i o n s i n v o l v e d the s t a b i l i t y parameter, f o r which a bulk estimate Z/L(AT) was shown to be a reasonable approximation, on average. A l i n e a r r e g r e s s i o n gave u*DISS = 0 . 9 6 u * F L 0 X + 0 . 0 2 5 m/s , where the p o s i t i v e o f f s e t r e s u l t s from a tendency f o r <uw>DISS to be greater than <uw>FL0X by more than 3 0 % , i n the most s t a b l e runs ( Z / L > 0 . 1 0 ) . In near n e u t r a l c o n d i t i o n s (-0.45 <Z/L< 0.05 ) at a l l wind speeds the momentum f l u x c a l c u l a t i o n s agreed to w i t h i n 4%, on average. The agreement between the s e n s i b l e heat f l u x c a l c u l a t i o n s was very good, with a regression g i v i n g <wt>DISS = 1.04 <wt>FL0X. 169 The Bedford tower experiment e s t a b l i s h e d t h a t r e l i a b l e d i s s i p a t i o n estimates of both the momentum and s e n s i b l e heat f l u x e s and the bulk estimates of Z/L could be obtained from the CCGS Quadra. A favourable comparison of ship and tower d i s s i p a t i o n drag c o e f f i c i e n t s showed the Bedford tower to be e s s e n t i a l l y an open ocean s i t e , which allowed the combined 1591 hours of momentum f l u x and 260 hours of s e n s i b l e heat f l u x measurements t o be considered as a s i n g l e open ocean data s e t . The d i s s i p a t i o n data showed the n e u t r a l drag c o e f f i c i e n t to depend on wind speed, as approximated by 103CDN = 1 . 1 4 4 < 010 < 10 m/s 103CDN = 0.49 + 0.065 010 10 < 010 < 26 m/s. Below 10 m/s the v a r i a b i l i t y of CDN with wind speed, f e t c h , and s t a b i l i t y was minimal, < 5% on average. Time s e r i e s of c a l c u l a t e d CDN's displayed about a 10% random f l u c t u a t i o n . Bulk estimates of the momentum f l u x were obtained from the above formulat i o n by: f i r s t , s h i f t i n g the measured wind speed to 10m with an i t e r a t i v e technique i n v o l v i n g CDN; second, c a l c u l a t i n g CDN; t h i r d , s h i f t i n g CDN t o CD (the drag c o e f f i c i e n t at the measurement height, Z, wind speed, OZ, and s t a b i l i t y Z/L(AT) ); f o u r t h , a p p l y i n g the bulk aerodynamic formula -<uw> = CD 0Z 2. I t was shown that the neglect of Z/L would produce a minor e r r o r i n the c a l c u l a t e d 010 and CDN, but that the CD found i n step 3 could be a f f e c t e d by as much as 20%. However t h i s e r r o r was reduced to 5% by using the bulk s t a b i l i t y estimate. E r r o r s i n the CDN fo r m u l a t i o n , OZ and Z/L could add up to 15% i n the bulk 170 estimates, which were seen to give a good measure of the momentum f l u x averaged over a few days or more and a good hourly average when the wind was steady. Over periods of up t o a day, the bulk and d i s s i p a t i o n c a l c u l a t i o n s were seen to c o n s i s t e n t l y d i f f e r by as much as 30%. These d i s c r e p a n c i e s were found t o be associated with varying winds, w i t h the d i s s i p a t i o n estimate being smaller on the r i s i n g wind and l a r g e r on the f a l l i n g wind or a f t e r a change i n wind d i r e c t i o n . The c o n c l u s i o n drawn was t h a t the surface roughness and hence drag c o e f f i c i e n t depend on surface parameters which are a product of both past and present winds. A constant n e u t r a l 10m drag c o e f f i c i e n t was found to be an adequate d e s c r i p t i o n of almost a l l recent measurements over deep water, throughout the wind speed range, 4 t o 10 m/s. The value of the constant v a r i e d from about 1.1x10~ 3 to 1.3x10~ 3. This behavior has been found by the eddy c o r r e l a t i o n , d i s s i p a t i o n and p r o f i l e methods, but measurements from onshore or shallow water s i t e s often d i s p l a y e d a d i s t i n c t l y d i f f e r e n t wind speed dependency. A reasonable compromise over the open ocean would be t o use 103CDN = 1.2 i n the bulk aerodynamic method f o r winds up to 11 m/s. At 010 = 11 m/s, the high wind speed re g r e s s i o n of t h i s study, 103CDN = 0.49 + 0.065 010 = 1.2 . Since t h i s r egression f i t s the only other l a r g e set of open sea high wind speed data (Smith, 1979), i t should give s a t i s f a c t o r y CDN's at the higher wind speeds (to at l e a s t 26 m/s). 171 I t was necessary t o parameterize the s e n s i b l e heat f l u x d i f f e r e n t l y i n s t a b l e and unstable s t r a t i f i c a t i o n . The majority of the data were i n the ranges -40 <U10 AT< 60°Cm/s, -0.3 <Z/L< 0.1 and 6 <D10< 18 m/s, f o r which e i t h e r 103<wt> = 1.00 D10 A T + 2.9 °Cm/s 103<wt> = 0.75 010 A T + 2.0 °Cm/s sta b l e unstable or 103CTN = 1.08 103CTN = 0.68 , unstable s t a b l e described the r e s u l t s . More measurements of CTN were needed i n order t o confirm or deny a wind speed dependency, which was suggested by some of the data. Bulk estimates of the s e n s i b l e heat f l u x were obtained with <wt> = CT OZ AT, where CT was found from Z/L, CDN, CD and the s t a b i l i t y dependent CTN. 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J . : 511pp. Wucknitz, J . , 1976: Determination of turbulent f l u x e s of momentum and s e n s i b l e heat from f l u c t u a t i o n measurements and the s t r u c t u r e of the wind f i e l d over waves above the t r o p i c a l a t l a n t i c during ATEX. Meteor B, No- J J , : 25-50. Wyngaard, J . C. and 0. R. Cote, 1971: The budgets of tu r b u l e n t k i n e t i c energy and temperature variance i n the atmospheric surface l a y e r . J . Atrnos^ S c i . , 28: 190-201. 177 APPENDIX . THE INTERCOMPARISON RESULTS | 1 1 1 — 1 1 1 + RUN TIME DATE UZ (M/S) TZ TSFC Z/L u* (m/s) 10<tit> GMT 1976 °True °C • °Cm/s FLUX DISS FLUX DISS 1 •• -1 • 1 i —r —i i r r ... . T + T1 17:00 17/ 9 6. 4 232 16.8 17. •5 -. 12 0. 197 0. 193 T2 18: 00 17/ 9 6. 4 240 16.6 17. 5 -.13 0. 173 0. 171 0. 038 0. 043 T3 19:00 17/ 9 6. 1 243 16.6 17. 5 -. 16 0. 158 0. 166 0. 042 0. 036 T4 20: 00 17/ 9 6. 3 236 16.3 17. 5 -.17 0. 197 0. 167 0. 051 0. 031 T5 21:00 17/ 9 6. 3 235 16.3 17. 5 -.18 0. 195 0. 177 0. 042 0. 030 T6 3:00 18/ 9 6. 8 238 17.1 17. 5 -.09 0. 176 0. 224 T7 4:00 18/ 9 7. 7 232 17. 2 17. 5 -.05 0. 229 0. 244 T8 5: 00 18/ 9 7. 3 243 16. 9 17. 5 -.08 0.225 0. 240 T9 0 :00 19/ 9 5,. 8 233 17. 1 16. 7 0.00 0.185 0. 215 T 10 1: 00 19/ 9 6. 1 233 17.2 16. 7 0.01 0. 179 0. 211 T1 1 2:00 19/ 9 6. 2 228 17. 1 16. 7 -.01 0. 192 0. 207 T12 3: CO 19/ 9 6. 8 236 17.0 16. 7 -.01 0. 187 0. 217 T13 14:00 19/ 9 6. 8 167 16.7 16. 3 -.01 0.227 0. 250 T14 18: 00 19/ 9 6. 0 199 17.4 16. 0 0.09 0. 164 0. 170 • 0 22 — . 034 T15 19:00 19/ 9 6. 7 214 17.3 16. 0 0.07 0. 190 0. 192 • 022 • 028 T16 20: 00 19/ 9 8. 4 209 16.9 16. 0 0.02 0. 207 0. 235 - . 003 — • 017 T17 21:00 19/ 9 9. 5 213 16.9 16. 1 0.02 0.296 0. 291 "™ a 011 015 T18 22:00 19/ 9 10. 4 212 17.0 16. 2 0.01 0.349 0. 326 """* • 014 — . 019 T19 23:00 19/ 9 9. 0 217 16.9 16. 6 -.01 0. 324 0. 286 0. 010 030 T20 0: 00 20/ 9 8. 5 226 16.8 16. 8 -.03 0.287 0. 270 0. 021 "~ • 043 T21 1:00 20/ 9 9. 2 230 16.6 16. 8 -.04 0.326 0. 285 0. 058 0. 062 T22 2: 00 20/ 9 8. 6 234 16.3 16. 9 -.06 0. 266 0. 272 0. 063 0. 081 T23 3:00 20/ 9 7. 3 231 16. 2 16. 8 -.09 0. 242 0. 2 34 0. 056 0. 075 T24 4: CO 20/ 9 6. 8 236 16.2 16. 8 -.10 0. 220 0. 229 0. 040 0. 063 T25 5:00 20/ 9 6. 6 238 16. 2 16. 8 -. 10 0. 200 0. 247 0. 024 0.060 T26 6: 00 20/ 9 7. 0 233 16.2 16. 7 -.08 0.220 0. 238 0. 032 0. 058 T27 7:00 20/ 9 7. 4 238 15.7 16. 4 -.08 0.240 0. 229 0. 054 0. 058 T28 8: 00 20/ 9 7. 6 250 15.3 16. 3 -.11 0.221 0. 228 0. 048 0. 058 T2 9 9:00 20/ 9 7. 7 244 14. 8 16. 3 -.14 0. 202 0. 213 0. 054 0. 057 T30 14: 00 20/ 9 7. 3 220 16.7 16. 3 0.00 0.219 0. 204 T31 15:00 20/ 9 6. 9 220 16.8 16. 4 -.00 0. 187 0. 191 T32 16: 00 20/ 9 8. 0 221 16.3 16. 4 -.03 0.244 0. 239 T33 17:00 20/ 9 7. 5 220 16. 8 16. 6 -.02 0.240 0. 235 T34 18: 00 20/ 9 5. 9 220 17.3 16. 7 0.01 0. 168 0. 180 T35 5:00 21/ 9 7. 2 201 17. 1 16. 8 -.01 0. 246 0. 267 T36 6: 00 21/ 9 6. 8 204 17.3 16. 9 -.00 0.222 0. 242 T37 7:00 21/ 9 7. 5 198 17. 4 16. 9 0.01 0.242 0. 264 T38 8: 00 21/ 9 8. 7 197 17.5 17. 0 0.00 0.267 0. 290 T39 9:00 21/ 9 8. 3 193 17.6 17. 1 0.00 0. 258 0. 270 T40 10: 00 21/ 9 8. 2 190 17.9 17. 2 0.01 0. 290 0. 267 T41 11:00 21/ 9 8. 4 182 18.0 17. 2 0.02 0.259 0. 273 T42 12: 00 21/ 9 9. 1 178 18.3 17. 2 0.04 0.238 0. 253 T43 13:00 21/ 9 8. 6 182 18. 4 17. 2 0.04 0. 263 0. 277 Tuzj 14: 00 21/ 9 8. 6 180 18.4 17. 3 0.03 0. 278 0. 285 T45 15 :00 21/ 9 9. 0 183 18. 6 17. 4 0.04 0.295 0. 302 T46 16: 00 21/ 9 9. 5 183 19.0 17. 4 0.05 0.302 0. 317 T47 19:02 23/ 9 9. 9 231 16. 4 16. 9 -.04 0.362 0. 298 T48 20: 00 23/ 9 10. 1 230 16.6 16. 9 -.03 0.335 0. 318 T4 9 L R 23:00 23/ 9 11. 7 224 16. 1 17. 0 -.04 0.359 0. 359 i 178 I 1 1 1 1 T T 1 : EUN TIME DATE UZ °TEUE TZ TSFC Z/L U* (m/s) 10<Wt> GMT 1976 FLUX DISS FLUX DISS I 1 1 1 1 1 1 1 T50 2:00 24/ 9 11.9 228 15.5 17.0 -.06 0.396 0.367 T51 5:00 24/ 9 10. 2 229 15.6 16. 5 -.06 0.356 0.326 T52 11:00 24/ 9 9.6 215 16.1 16.3 -.03 0.321 0.298 T53 13:16 24/ 9 9.8 211 16.4 16.4 -.02 0.340 0.302 T54 14:32 24/ 9 10.3 208 16.7 16.6 -.01 0.303 0.311 T55 20:12 24/ 9 7.2 210 16.2 15.9 -.01 0.231 0.247 T56 11:00 27/ 9 11.4 162 14.7 16.0 -*06 0.394 0.349 T57 12:11 27/ 9 12.1 162 15.0 15.9 -.04 0.458 0.365 T58 13:38 27/ 9 13.5 166 15.8 15.8 -.01 0.442 0.438 T59 15:42 27/ 9 11. 9 167 16.8 15. 2 0.03 0.448 0.368 T60 16: 36 27/ 9 11. 1 194 16.6 14.5 0.05 0.338 0.360 -. 146 -. 160 T61 20: 14 27/ 9 15. 2 210 16.7 14.2 0.04 0.585 0.529 -.302 -.257 T62 23:00 27/ 9 10^7 228 15.8 14.1 0.04 0.331 0.352 -.121 -.144 T63 11:00 28/ 9 9.8 11 10.9 14.7 -.19 0.300 0.308 0.236 0.250 T64 12:48 28/ 9 8.5 10 10.4 15.1 -.29 0.253 0.264 0.233 0.260 T65 23:00 28/ 9 12.0 311 10.8 14.0 -. 12 0.379 0.413 0.284 0.352 T66 5:00 29/ 9 8.9 333 8.5 14.7 -.35 0.339 0.298 0.388 0.413 T67 17:00 29/ 9 10.0 230 12.4 16.0 -.16 0.341 0.321 0.324 0.294 T68 18:06 29/ 9 11.0 232 12.8 15.5 -.11 0.370 0.352 T69 19:48 29/ 9 10.2 226 13.3 14.3 -.06 0.356 0.321 T7C 23:00 29/ 9 11.8 222 13.9 13.7 -.01 0.392 0.410 T71 2:00 30/ 9 13.6 236 14. 1 13.2 0.01 0.420 0.437 T72 5:00 30/ 9 10.1 243 13.4 12.8 0.01 0.352 0.329 T73 11 :30 30/ 9 8. 3 239 12. 4 13. 1 -.07 0.270 0.256 0.059 0.075 T74 2:23 10/10 13.9 194 18.8 16.2 0.04 0.523 0.496 -.276 -.246 T75 5:23 10/10 17.1 191 19.4 16.3 0.04 0.638 0.670 -.368 -.374 T76 8:23 10/10 12.7 224 16.5 16.2 -.00 0.476 0.470 -.055 -.064 T77 11:23 10/10 15.8 223 17.1 16.2 0.01 0.660 0.580 -.043 -.067 T78 12:53 10/10 17.5 232 16.6 16.1 0.00 0.714 0.682 T79 14:01 10/10 17.8 234 15. 6 16.0 -.01 0.762 0.676 T80 17:23 10/10 19.2 238 13.8 15.1 -.02 0.786 0.755 T81 20:23 10/10 16.6 254 12.2 14.2 -.04 0.641 0.624 T82 22:23 10/10 14.0 268 11.5 13.6 -.05 0.523 0.504 T83 23:23 10/10 13.6 271 11.1 13.3 -.06 0.442 0.495 T84 2:23 11/10 12.3 265 10.1 12.4 -.07 0.396 0.417 T85 5:23 11/10 10.5 269 9.2 12.5 -.14 0.401 0.373 T86 16:01 14/10 7.7 177 13.3 7.7 0.30 0.173 0.203 -.027 -.052 T87 17:17 14/10 10. 2 174 13.6 8. 3 0.18 0.318 0.297 -.112 -.105 T88 18: 35 14/10 11.4 179 13.2 1.0-2 0.08 0.355 0.376 -.162 -. 158 T89 20: 17 14/10 6. 3 242 12. 1 10.7 0.09 0.224 0.237 T90 8:23 15/10 16.1 259 7.6 10.5 -.05 0.625 T91 11:23 15/10 15.7 263 7.9 10.1 -.04 0.692 T92 12:29 15/10 14.4 263 7.7 9.7 -.05 0.515 T93 14:23 15/10 14.6 259 8.2 9.3 -.03 0.559 T94 17:23 15/10 13.7 259 10.5 9.0 0.02 0.546 0.491 T95 19:00 15/10 13.2 260 11.4 8.8 0.04 0.535 0.496 -.186 -.187 T96 2:10 27/11 12.4 199 7.9 5.7 0.04 0.496 0.441 -.181 -.168 T97 2:10 28/11 11.3 215 8.1 5.4 0.07 0.377 0.416 -.173 -.229 T98 0:00 3/12 11.7 123 5.4 4.6 0.02 0.393 0.411 -.052 -.059 179 \ 1 1 1 1 1 1 1 + RON TIME DATE OZ °TROE TZ TSFC Z/L u* (m/s) 10<wt> GMT 1976 FLOX DISS FLOX DISS 1 T 1 1 ' I ' I I 1 + T99 6 :00 3/1 2 12i 4 216 6^9 4. 6 0.05 0. 424 0.447 180 — .185 T100 9: 00 3/12 17.9 250 3.1 4. 6 -.02 0.826 0.752 T101 12:00 3/12 18.6 264 -2. 5 4. 6 -.09 0^733 0.737 T102 15:00 3/12 18.5 274 -9.7 4. 6 -.19 0.746 0.724 T103 18:00 3/12 16. 9 278 -11. 4. 6 -.26 0. 672 0.690 T104 2-1:00 3/12 15. 0 270 -12. 4. 5 -.33 0. 552 0.588 T105 0 :00 4/12 16. 8 267 -11. 4. 5 -.25 0. 735 0.690 T106 3:00 4/12 19.3 269 -9.6 4. 5 -.17 0. 817 0.806 T107 6:00 4/12 15. 9 266 -10. 4. 4 -.27 0. 633 0.599 T108 9: 00 4/12 16. 0 261 -8.9 4. 3 -.22 0.551 0.559 T109 12:00 4/12 14. 4 264 -7.8 4. 3 -.26 0.500 0.495 T 110 15:00 4/12 10.7 261 -5.8 4. 2 -.34 0. 352 0.414 T11 1 15:00 7/12 11. 7 169 2.5 3. 5 -.04 0.418 0.380 0. 179 0 .200 T112 18: 00 7/12 14. 6 139 4.1 3. 5 0. 01 0.478 0.519 T113 21:00 7/12 17. 1 143 5. 8 3. 6 0.03 0.660 0.651 • 131 -.125 T11 4 0: 00 8/12 18.6 153 7.6 3. 5 0. 04 0.746 0.736 385 — .352 T115 3:00 8/12 18. 9 159 7.8 3. 6 0.04 0. 728 0.729 • 435 - .516 T116 6: 00 8/12 13.3 170 7.9 3. 6 0.08 0.518 0.506 372 -.452 T117 18:00 9/12 14. 6 300 -11. 3. 6 -.31 0.601 0.598 T118 21: 00 9/12 18.3 302 -14. 3. 5 -.27 0.640 0.739 T119 0:00 10/12 17. 1 302 -17. 3. 4 -.33 0. 675 0.695 T120 3: 00 10/12 15.6 292 -17. 3. 4 -.38 0.656 0.607 T121 6:00 10/12 16. 9 289 -15. 3. 3 -.31 0. 487 0.592 T122 9: 00 10/12 13.5 278 -14. 3. 3 -.45 0.463 0.485 T123 2:00 11/12 10. 5 211 5.2 3. 3 0.06 0.362 0.3 48 • 069 -.089 T124 8: 00 11/12 12. 4 220 5.9 3. 4 0.06 0.483 0.462 • 159 — . 152 T125 11:00 11/12 15. 3 230 6.3 3. 5 0. 04 0.661 0.578 • 228 — .176 T126 14:00 11/12 16. 0 231 6.5 3. 5 0.04 0.606 0.613 251 — . 186 T127 17:00 11/12 17. 2 291 1.8 3. 5 -.03 0.645 0.673 T128 20: 00 11/12 16. 0 290 -1.0 3. 5 -. 08 0.565 0.591 T129 23 :00 11/12 11. 9 308 -2. 6 3. 5 -.21 0.378 0.433 T130 14:00 12/12 12.6 201 1.6 3. 4 -.06 0.446 0.4 05 0. 265 0 . 296 T131 5:00 14/12 14. 2 317 -14. 3. 2 -.35 0.531 0.541 T132 8:00 14/12 14.6 312 -16. 3. 2 -.43 0.534 0.543 T133 23 :00 14/12 13. 2 218 -7.0 3. 0 -.27 0.493 0.456 T134 2: 00 15/12 18.2 220 2.2 3. 0 -.01 0.769 0.735 T135 5:00 15/12 17. 5 219 3.9 2. 9 0.01 0. 693 0.681 •> 023 -.047 T136 8: 00 15/12 15. 9 213 -4.8 2.. 9 0.03 0.663 0.602 o 132 — .082 T137 11 :00 15/12 1977 15.6 231 5. 4 2. 9 0.04 0.613 0.592 T138 0:48 1 3 / 3 10. 6 233 1.9 0. 0 0.06 0. 221 0.264 097 — .146 T139 9: 48 17/ 3 10. 4 236 1.8 0. 2 0.05 0.363 0.355 m 142 — . 104 T140 12:48 17/3 11. 2 240 1.6 0. 3 0.03 0.395 0.387 m 071 - .097 T141 15: 48 17/3 11.0 234 1.5 0. 2 0.04 0.356 0.359 m 079 — .061 T142 18:48 17/3 12. 6 259 1.0 0. 3 0.01 0.456 0.456 T143 21: 48 17/3 12.8 270 -0.3 0. 3 -.02 0.440 0.464 T144 0:48 18/3 12. 6 278 -1.3 0. 2 -.04 0.442 0.457 T145 3: 48 18/3 10.5 280 -1.9 0. 2 -.09 0.360 0.387 0. 214 0 .253 T146 6:48 18/3 11. 3 271 -2. 4 0. 2 -.09 0.389 0.371 0. 2 68 0 .237 T147 9: 48 18/ 3 11. 7 270 -2.2 0. 2 -.07 0.380 0.410 0. 227 0 .279 I T 1 1 1 1 1 1 + 180 1 T T T 1 1 1 1 + RUN TIME DATE UZ °TRUE TZ TSFC Z/L u* (m/s) 10<wt> GMT 1977 FLOX DISS FLUX DISS 1 T 1 1 1 1 1 ; 1 h T148 12:48 18/ 3 11. 5 282 -1. 7 0. 2 06 0. 405 0. 421 T149 15: 48 18/ 3 12.7 254 0. 4 0. 3 • 00 0. 398 0. 444 T150 6:48 19/ 3 17. 3 61 -1. 0 0. 3 02 0. 694 0. 685 T151 9: 48 19/ 3 12.8 16 -1. 7 0. 2 • 05 0. 458 0. 488 T152 12:48 19/ 3 16. 4 347 -2. 8 0. 2 05 0. 646 0. 645 T153 15: 48 19/ 3 17.6 325 -2. 9 0. 3 05 0. 685 0. 708 T154 18:48 19/ 3 14. 1 323 -2. 8 0. 3 • 08 0. 469 0. 530 T155 0: 48 20/ 3 10.7 296 -1. 9 0. 2 08 0. 347 0. 405 T156 21:48 24/ 3 12.3 358 -0. 7 0. 1 03 0. 369 0. 425 T157 0: 48 25/ 3 12.2 356 -1. 1 -0. 1 • 03 0. 359 0. 415 T158 3 :48 25/ 3 11. 3 357 -0. 9 -0. 2 03 0. 321 0. 369 T159 9: 48 25/ 3 10.7 353 -0. 5 -0. 2 • 02 0. 321 o. 360 T160 12:48 25/ 3 10. 3 3 0. 3 -0. 2 0. 01 0. 290 0. 329 T161 15: 48 25/ 3 10.6 359 1. 5 -0. 2 0. 05 0. 310 0. 342 T162 18:48 25/ 3 10. 5 358 2. 2 0. 0 0. 07 0. 276 0. 341 T163 21: 48 25/ 3 10.4 356 0. 9 0. 0 0. 03 0. 312 0.368 T164 3:48 26/ 3 10. 7 353 0. 1 0. 0 00 0. 303 0. 352 T165 6: 48 26/ 3 10. 5 350 0. 1 0. 0 0. 00 0. 259 0. 333 T166 9:48 26/ 3 10. 3 353 0. 5 0. 0 0. 01 0. 267 0. 333 T167 12: 48 26/ 3 11.4 352 1. 3 0. 0 0. 04 0. 328 0. 369 T168 15:48 26/ 3 10.6 351 0. 9 0. 0 0. 02 0. 323 0. 365 T169 18: 48 26/ 3 12. 2 17 1. 4 0. 0 0. 03 0. 388 0. 424 T170 21:48 26/ 3 12. 3 14 1. 0 0. 0 0. 02 0. 395 0. 421 T171 3: 48 27/ 3 10.8 10 1. 4 0. 0 0. 04 0. 341 0. 371 T172 12:48 27/ 3 11. 3 4 1. 6 0. 0 0.04 0. 315 0.370 T173 15: 48 27/ 3 10. 4 17 2. 4 0. 0 0. 07 0. 313 0. 350 T174 18:48 27/ 3 10. 9 3 3. 5 0. 0 0. 10 0. 266 0. 335 T175 21: 48 27/ 3 8. 0 1 2. 7 0. 0 0. 13 0. 235 0. 287 T176 21: 48 29/ 3 8.6 172 2. 2 0. 0 0. 10 0. 232 0. 298 T177 18: 48 31/ 3 11. 5 168 3. 0 0. 0 0. 08 0. 367 0. 413 T178 21:48 31/ 3 7.8 204 3. 1 0. 0 0. 16 0. 200 0. 242 T17S 12: 48 3/ 4 12. 6 147 0. 6 0. 0 0. 01 0. 451 0. 428 T180 15:48 3/ 4 14. 2 125 1. 2 0. 0 0.02 0. 562 0. 531 T181 18: 48 3/ 4 11. 4 170 3. 1 0. 0 0. 09 0. 407 0. 445 T182 3:48 4/ 4 19. 1 288 1. 8 0. 0 0. 02 0. 626 0. 762 T183 6: 48 V 4 18. 0 300 -0. 3 0. 0 • 01 0. 684 0. 723 T184 9: 48 V 4 16. 2 313 -1. 4 0. 0 03 0. 610 0. 628 T185 12: 48 4/ 4 15. 2 311 -0. 7 0. 0 -. 01 0. 511 0. 566 T186 15:48 4 13. 5 315 0. 4 0. 0 0. 01 0. 462 0. 519 T187 18: 48 4/ 4 10.8 321 1. 7 0. 0 0. 05 0. 311 0. 388 T188 21:48 V . 4 10. 9 311 1. 8 0. 0 o: 05 0. 316 0. 374 T189 3: 48 6/ 4 15. 1 189 6. 0 0. 0 0. 09 0. 566 0. 573 T190 6:48 6/ 4 13. 0 209 4. 7 0. 0 0. 11 0. 482 0. 471 T191 9: 48 6/ 4 10.7 223 1. 3 0. 0 0. 04 0. 346 0. 399 T192 12:48 6/ 4 12. 4 224 1. 3 0. 0 0. 03 0. 466 0. 460 T193 15: 48 6/ 4 12. 4 220 1. 2 0. 0 0.02 0. 460 0. 448 T194 18:48 6/ 4 Mi. 5 223 1. 9 0,. 0 0. 03 0. 5.45 0. 540 T19 5 21: 48 6/ 4 13.2 223 1. 5 0. 0 0. 03 0. 461 0. 494 T196 0:48 7/ 4 11. 6 217 0. 9 0. 0 0. 02 0. 406 0. 427 y—i 1 T 1 1 T ; — r - [ 

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