Surface-Referenced Current Meter Measurements. Markku Juhani Santala

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1 Surface-Referenced Current Meter Measurements ~L 1-* J.s:2b by Markku Juhani Santala B.S., University f Massachusetts, Amherst (1985) S.M., Massachusetts Institute f Technlgy (1987) Submitted in partial fulfillment f the requirements fr the degree f Dctr f Philsphy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY and the WOODS HOLE OCEANOGRAPHIC INSTITUTION September 1991 Markku J. Santala, 1991 The authr hereby grants t MIT permissin t reprduce and t distribute cpies f this thesis dcument in whle r in part. ~ Signature f Authr..... Jint Prgram in Oceangraphic Engineering Massachusetts Institute f Technlgy Wds Hle Oceangraphic Institutin ~ September, 1991 ", Certified by,, ~.Q., ~ :.Alb~;~illia:~;""i1i Senir Scientil. Wds Hle Oceangraphic Institutin /LA:1IlU".'~~~ T~e.ffi...U~~.iS~~ 1i, VV. Kendall Melville njruttee fr Oceangraphic Engineering I

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3 Surface-Referenced Current Meter Measurenlents by Markku Juhani Santala Submitted t the Massachusetts Institute f Technlgy Wds Hle Oceangraphic Institutin Jint Prgram in Oceangraphic Engineering n September, 1991, in partial fulfillment f the requirements fr the degree f Dctr f Philsphy Abstract A general discussin f pssible techniques fr bservatin f near-surface currents indicates that the surface-fllwing frame f reference will prvide several advantages ver the Eulerian r Lagrangian frames. One prblem with surface-fllwing measurements is the biasing effects f the waves. A technique fr making unbiased measurements is develped. This technique requires that bth the sens~r velcity and the fluid velcity be measured. A sensr platfrm, the Surface Acustic Shear Sensr (SASS), which makes the required measurements is described. The prcessing scheme fr interpreting the measurements frm the SASS is described at length. The data that SASS has btained frm tw deplyments in the Shelf MIxed Layer Experiment (SMILE) is presented. This data shws clearly that the biasing effects f waves can nt, in general, be ignred. In the summary f the data we find suprisingly little shear in the dwmvind directin in the tp 4m f the waterclumn. In the crsswind directin bserved, bserved shear seems t be indicative f an acrss shelf pressure gradient and intense near-surface mixing. Thesis Supervisr: Albert J. Williams, III Title: Senir Scientist, Wds Hle Oceangraphic Institutin 3

4 Acknwledgments This thesis is dedicated t the memry f my father, Teuv Juhani Santala. Twenty-tw years after him, I t receive my dctrate frm M.LT. I wuld like t thank the members f my thesis cmmittee; Arthur B. Baggerer, Rbert C. Beardsley, Eugene A. Terray, Rbert A. Weller and Albert J. Williams, III; fr serving n my cmmittee. Particular thanks must g t Albert J. (Sandy) Williams, III and Eugene A. (Gene) Terray. Thrugh Sandy's guidance I received a master degree in 1985, and then was intrduced int experimental ceangraphy with the cnstructin and deplyment f the Surface Acustic Shear Sensr (SASS). Many f the ideas presented in this thesis came t fruitin after discussins with Gene; his willingness t freely exchange ideas has been a large factr in the grwth f my scientific capability. The wrk n the SASS buy was a multipersn effrt. Besides the verall directin f the prject by Sandy, a great deal f the reasn the SASS made it int the cean were the effrts f Chris Dunn and Ellyn Mntgmery. Steven Lentz and Carl Alessi were bth helpful in making the C3 mring data accessible and understandable t me. Mark Grsenbaugh has been bth a gd friend, cunselr and tennis partner. Andy Trivett and Judy White are tw peple wh, thrughut my years at Wds Hle, have helped t preserve my sanity thrugh their gracius friendship. I thank Deep-Sea Research fr granting permissin reprint and paraphase prtins f Santala and Terray (1991) (fund in Chapter 3). Financial supprt fr my wrk was frm NSF grant OCE

5 Cntents 1 INTRODUCTION 7 2 MEASUREMENT TYPES 2.1 Lagrangian Measurements 2.2 Eulerian Measurements 2.3 Surface Referenced. '. 3 WAVE BIAS 3.1 Measurements frm a mving sensr 3.2 Estimatin f the bias 3.3 Verificatin. 4 THE SURFACE ACOUSTIC SHEAR SENSOR (SASS) 4.1 Design f SASS. 4.2 Crdinate transfrmatins 4.3 Estimatin f directinal spectra 5 MAKING MEASUREMENTS FROM THE SASS 5.1 Deplyment histry. 5.2 Prcessing strategy. 5.3 Dynamic behavir f the SASS 5.4 Errrs. 6 RESULTS

6 6.1 The C3 mring 6.2 The data... 7 DISCUSSION 7.1 Summarizing the data. 7.2 Cnclusins A /L-FILTER INTEGRALS 239 B PARAMETERIZATION OF DRAG 242 C BASS FLOW DISTURBANCE 245 D WAVE DECAY 253 E COMPASS SENSITIVITY 262 F VERTICAL VELOCITIES 272 6

7 Chapter 1 INTRODUCTION In this thesis I will examine the nature f the mean shear current in the upper 2m f the cean. T a great extent the viewpint taken will be bservatinal. But, we will find that an apprpriate bservatin can nt be made withut a brad theretical understanding f the dynamics that gvern the near-surface envirnment. Chapter 2 will be entirely devted t a discussin n which frame f reference is the mst apprpriate (and practical) t make ur measurements. We will find that a surface-fllwing frame f reference prvides sme distinct advantages ver the thers prpsed. In Chapter 3 an extended study f the wave bias - ne f the utstanding prblems assciated with making upper-cean current measurements - is undertaken. It is the fact that the measurements t be made are near a mving bundary that requires that ur bservatin technique be discussed s carefully. The dynamic prblem that the current measurements will address directly is thaff the frm and size f the wind-induced shear current. The measurements will be time averaged ver perids n the rder f thirty minutes t btain mean current prfiles (the thirty minutes averaging was selected s as t be lng enugh t average ut the wave signals but shrt enugh s as t btain a statinary sample f the current). Wu(1969,1975,1983) has emphasized sme very practical cnsequences f the mean drift currents set up by the wind stress; the dispersin f man-made pllutants in the cean and the hydrdynamic lading f ffshre structures. Beynd these bvius direct implicatins, there exists a variety f diverse prblems which will benefit frm a mre accurate understanding f near-surface 7

8 currents. Fr example, the efficiency f the air-sea mmentum exchange, which is deduced frm the shear current prfile, is an imprtant input parameter fr the mdeling f large scale flws in bth the ceans and atmsphere. On a smaller scale, the mixing prcesses which gvern the mmentum transfer als transprt small marine rganisms and thus effect bilgical prductivity. Studies f the near-surface currents have been made many times in bth the labratry and the cean. With the SASS (Surface Acustic Shear Sensr), a newly develped instrument platfrm discussed in Chapter 4, we will be able t make unbiased measurements f the currents at distances as small as 1m away frm the cean's surface. The SASS essentially cnsists f a vertical array f acustic current meters and a mtin sensing package. In additin t making measurements f the current, we are able t estimate the directinal spectrum f the wavefield frm the sensrs munted n the SASS. In the final analysis, the wave measurements prvided invaluable infrmatin in the data interpretatin. Nt nly is this data used t estimate the wave bias, it is used t interpret the instrument perfrmance and t prvide cmplementary envirnmental data t the current recrds. The SASS was deplyed ff the cast f nrthern Califrnia as part f SMILE (Self MIxed Layer Experiment). The ther cmpnent f the SMILE prgram f interest here is a current meter mring which was named the C3 mring. This mring prvides windspeed and current meter data which will be essential in the final interpretatin f the SASS data. That we need s much infrmatin besides just the near-surface current recrds t understand the shear current in the upper cean is nt a surprise when we cnsider dynamics which give rise t the shear current. The driving frce behind the near-surface shear current is the wind. Mmentum is transferred frm the wind t the water either by pressure frces r thrugh viscus "skin-frictin" frces. The mmentum that is transferred thrugh pressure frces must undubtedly g int wave generatin (Stewart, 1961). Thrugh wave breaking (Melville and Rapp, 1985; Mitsuyasu, 1985), dissipatin f waves (Csanady, 1984) r sme ther mechanism, sme f the wave mmentum may be transferred t the shear current. Mmentum transferred 8

9 by skin-frictin may enter directly int the shear current r may enter the shear current indirectly via highly dissipative wavelets (Csanady, 1985; Okuda, 1982). Whatever the input path f mmentum frm the air int the current may be, it then diffuses dwnward int the the deeper cean. It is widely believed that the diffusin f mmentum dwn frm the surface is well described by the same dynamics which gvern the bundary layer flw relative t a flat plate. In this view, the shear stress in the fluid is assumed t be cnstant. Cupled with the assumptin that there exists an eddy viscsity which varies linearly frm the bundary, ne arrives at the the familiar "lg-law f the wall". Csanady (1984) succinctly states the assumptins under which this may be dne and supprts his cnjecture with a cmpilatin f field data frm varius investigatrs. Because the lg-layer has becme smething f a paradigm fr near-surface mmentum transfer, and Csanady's 1984 paper seems t well summarize why this is s, I will frequently reference this paper. Nt all measurements agree with the lg-layer mdel. Fr instance, Kitaigrdskii et al (1983) have fund dissipatin rates in the near-surface which are incnsistent with a cnstant stress (lg-law) mdel. Grdn (197) has argued that if Langmuir circulatin is present then the helical trajectries which define thi_s flw pattern will act as a very efficient mechanism by which mmentum can be transferred t the deeper flw. In his review f Langmuir circulatin, Pllard (1977) ntes that the theretical understanding f the mechanisms which give rise t Langmuir circulatin are still prly understd. He cncludes, nnetheless, that if Langmuir circulatin is present, it will ttally dminate the dwnward flux f hrizntal mmentum. The data btained frm the SASS and C3 buys is pltted ut file-by-file in Chapter 6. When the results are summarized in Chapter 7 we will find that at great depths (deeper than 1m) a regin reasnably well described as a cnstant stress layer may exist. It is cncluded that the frm f the shear current in the upper 1m must be a cnsequence f intense near-surface mixing. 9

10 Chapter 2 MEASUREMENT TYPES 2.1 Lagrangian Measurements The Lagrangian frame f reference is ne ~n which we study the trajectries f an individual fluid particle. In a sense, this is the natural frame f reference in which t study mass transprt. It is nt, perhaps, the best frame in which t understand the physics f air-sea mmentum exchange. Cnsider that we can rughly break the ttal mass transprt int three cntributins (nt necessarily independent cntributins!): (1) the wind-induced shear flw, (2) the wave-induced flw (Stkes' drift), and (3) the ther cntributins. Other cntributins being pressure gradients, the crilis frce, bttm frictin effects etc. Lagrangian techniques measure all three cntributins. Theretical prblems arise with the Lagrangian techniques when we seek t understand the physics gverning the flw. Because t d this it is necessary t understand what the relative cntributins frm each cmpnent are. This is ne reasn why Lagrangian techniques are well suited t the labratry. In the labratry the factrs giving rise t each parameter can, in thery, be independently varied ("ther cntributins" always being a prblem). In fact, in the labratry the wave-induced drift current can, in general, be ignred. This is because the waves generated in tanks are generally s small (due t fetch limitatins) that the wave cntributins are usually much smaller than the thse due t the wind. Fr instance, Wu(1975) fund that in his 22rn lng 1

11 wave tank that the Stkes' transprt at the surface was nly 1% f the ttal transprt (wind speeds varied frm 2m/s t 14m/s). In subsequent analysis he therefre ignres wave drift. Hwever, he des estimate that fr fetches n the ceanic scale that the wave-induced transprt will exceed the wind-induced transprt. In Figure 2-1 I have reprduced the drift cntributins fr a fully develped sea where the 1m windspeed, U lo, is 1m/s using Wu's expressins. It is imprtant t recgnize that the estimates d nt include nnlcally generated swell ~ 2.5 '-"... ~ ;;:l 1.5 ~~- ---_ L---'--.L...l-L..LU.!.l----J...-'--'~'-UL----' L.~'_' '_"' ' '_'_~ Z 1 3 fetch (km) Figure 2-1: Wind-induced (-) and wave-induced (- - -) cntributins t surface drift as a functin f fetch fr U lo = 1m/s. Calculated using equatins frm Wu(1975). Of curse, if it is truly the air-sea mmentum exchange that is f cncern t us then the magnitude f the current at the surface is f little relevance. It is the magnitude f the mean near-surface shear that indicates the flux f mmentum. In a deep water setting, it is usually nly the wind-induced and wave-induced cntributins that lead t depth dependent flws near the air-sea interface. Ideally, we wuld attempt t develp a plt similar t Figure 2-1 fr the magnitude f the shear instead f the magnitude f the current. Here 11

12 we'll settle fr a bit less. The mre mdest gal set here is t determine whether r nt it is pssible t ignre wave-induced drift in interpreting Lagrangian measurements. T examine the relative imprtance f the wave-induced drift we will again lk at cnditins expected with a windspeed f U ld = 1m/s. If atmspheric cnditins are neutral then a reasnable drag cefficient might be CD = 1.2 X 1-3 This implies that the air frictin velcity u*a is 34.6cm/s. If the stress in the fluid were equal t the stress in the air we wuld have that the water frictin velcity U*w wuld be equal t O.355u*a' But, because sme f the wind stress is supprted by wave drag, the shear in the fluid is less than that in the air and we have that: (2.1) Let's just stick with Wu's numbers and say U*w = O.241u*a =.83cm/s. If we assume that the lgarithmic prfile des indeed describe near-surface flw then we assume a prfile f the frm u(z) = ~ lg (=-) 8.5, u*w K, r and the slpe f the velcity defect is determined slely by the frictin velcity U*w' assume the Vn Karman's cnstant K, is cnstant and has a value f.4. The rughness scale r nly affects the prfile's ffset. We'll chse a value f r typical fr mdest windspeeds and let r = 1cm. The velcity defect fr the lg-prfile with these parameters is shwn as the slid line in Figure 2-2. (2.2) We On February 27, 1989, the SASS recrded data in cnditins f neutral stability and UlO = 1.2m/s (SASS file 1). The Stkes' drift fr that case was cmputed frm the measured directinal wave spectrum. The cmbined theretical wind-induced and measured wave-induced current prfile is pltted in Figure 2-2 as a dtted line. The Stkes' drift calculatin was cut-ff at wave frequencies f 1Hz, the highest frequency t which the SASS estimated the spectra. The message shuld be clear; if true Lagrangian drifters were released n February 27, then accrding t ur cmputatins, the greater prtin f the shear measured wuld be due t wave effects and nt wind effects. Fr ur purpses then, wave effects are nt negligible. In the past, sme investigatrs have chsen t ignre wave effects. It may be that in fetch-limited r yung wavefields, that mst 12

13 f the wind-stress is supprted by the shear current and that the wave-induced current is in fact negligible. Here we nly shw that this is: (a) nt always the case and (b) will prbably never be the case with the SMILE data, where large, nnlcally generated swell was always present l'-l '--" ;:l I ;:l * 14 fl d l: ;:l 12 Figure 2-2: Surface defect velcity using a theretical frm fr the wind-induced current (-). When the ttal drift current is predicted (...) the shear is apprximately twice as great. The ttal drift was cmputed frm actual SASS measurements f waves (file 1) when the windspeed was 1m/s. The sensitivity t high frequency waves was gauged by cutting ff the Stkes' drift cmputatin at O.5H z (- - -) and at.25hz (_._._). Scaling the distance using the mlecular viscsity II is the traditinal wall-layer apprach. The cut-ff frequency t which the Stkes' drift shuld be calculated is anything but a reslved issue. In Figure 2-2 the wave-induced drift was cmputed fr three different cut-ff frequencies; l.ohz, O.5Hz and O.25Hz (wavelengths lnger than 16cm, 62cm and 25cm, respectively). Nte, hw~ver, that because f the scaling f the prblem, the cmputatin cut ff at O.5Hz gives abut the same result as the cmputatin dne ut t 1.Hz. This is because the shallwest depth shwn n the plt crrespnds t 1cm. Since the wave spectrum falls ff at higher frequency and the shrter waves decay rapidly with depth, the 13

14 waves f frequency.5hz t 1.Hz make almst n cntributin t the wave drift at depths 1cm r mre. The waves f frequency O.25Hz t O.5Hz definitely make a cntributin. But, even cnsidering nly waves f frequency belw.25hz, the shear measured by a Lagrangian drifter wuld still be apprximately twice the wind-induced shear. Thugh there is sme dispute as t where t cut ff the Stkes' calculatin, mst investigatrs wuld prbably nt argue that there is a wave-induced cntributin frm waves up t frequency.5 Hz. What is in dubt is whether r nt the types f drifters rutinely deplyed wuld measure their cntributins. The near-surface envirnment is ften well mixed. With n density gradient, it becmes necessary t use drgue-type drifters. In this case there is a small surface flat t which a drgue is attached via a line. The drgue may take different frms but the area f the drgue is usually large cmpared t the surface flat. This gemetry is necessary t ensure that the drifter fllws the fluid at depth rather than the fluid at the surface. There are several cnsequences t using this type f drifter. First f all, if the drgue is set t a fixed depth, then the fllwer is nt Lagrangian in that fluid particles d nt remain at a cnstant depth beneath the air-sea interface. Csanady (1984) talks abut wind lading and the effects f wave breaking n flats with surface expressin. One thing that generally remains ignred, is that these types f flats have nn-negligible size. This is true particularly when ne cnsiders the length scale frm flat t drgue. \Ve usually assume that drifters measure bth the wind- and wave-induced drift. If the mtins f the drgue are nt crrelated t the higher frequency mtins f the fluid then it will nt measure the Stkes drift at thse frequencies. Therefre, even if waves f frequency.25hz t.5hz d cntribute t the Stkes' drift, if all the factrs leading t flat "slipage" cause the drifter mtin t be imperfectly crrelated with the fluid at higher frequencies, the drifter will nt measure the wave drift at thse frequencies. Suppse that a flat is cnstructed s that it fllws waves f frequency.25hz r lwer perfectly but abve this frequency, the mtins are essentially uncrrelated. Then, the measured drift wuld be given by the O.25Hz cut-ff curve f Figure 2-2 irrespective f hw much the higher frequency cmpnents cntribute. There are then tw distinct questins. First, "T what frequency d waves cntribute t 14

15 the Stkes' drift?" And secnd, "T what frequency des a drifter measure Stkes' drift?" The secnd questin implies that drifters need t be carefully calibrated. Calibratins f drifters fr near-surface flats are crude. Bth Niiler et al (1987) and Geyer(1989) have attempted t calibrate drugued drifters. Hwever, these drifters were typically much larger than thse we wuld find useful in the upper 5112 f the cean. In fact, Niiler's apprach was t try t mechanically decuple the drgue frm the wave mtin and hence t attempt t avid measuring the Stkes' drift at all. There is als the questin f what the drifter is fllwing. When releasing the drifters, if the flats diverge, then as the distance between flats becmes larger there arises the questin f if the results frm the tw drifters can be cmpared. Spatial cnvergence may be a prblem f equal difficulty. Drifters prbably seek regins f cnvergence (Richman et al (1987)). Cnvergence znes due t Langmuir cells can have wildly different flw characteristics than that f the surrunding fluid field. If flats are released and cnverge int a dwn-welling zne then the resulting drift measurements are prbably nt representative f the verall flw field characteristics. Deplyment f drifters can als be trublesme. Tracking f drifters in a labratry envirnment is simplified by the tw dimensinal nature f the flw and the ease with which subsurface drifters can be sptted in a glass-walled tank. Further, cnditins can always be repeated s that multiple runs can be averaged ver. These luxuries are nt available t the field experimenter. Tracking schemes can be smewhat elabrate (see Churchill and Csanady(1983)). Tracking is usually limited t a relatively small number f buys. S, while the drifter apprach remains ppular due t the relative ease with which an experiment can be cnducted, an accurate drifter study invlves much mre than thrwing sme flats ver the side f a bat. Perhaps mst difficult is interpretatin f drifter results. Churchill and Csanady(1983) measured, in lw wind-speed and high swell cases, anmalusly large shears. This they attributed t wave effects. The real questin we must nw keep in the back f ur minds is that, since mst field bservatins f lg-layer results 15

16 have been perfrmed with drifters, t what extent are f thse results measurements f the wind-induced shear current and t what extent are thse results measurements f the depth varying wave-induced drift. 2.2 Eulerian Measurements Eulerian measurements are bservatins f time series made at fixed pints in space. Therefre, sensr mtins are nt a cncern. There d arise, hwever, many ther measurement issues in this frame f reference - especially when making measurements in the upper cean. The mst imprtant issues are: deplyment prblems, flw disturbance, and advectin f the flw field. Clearly, fixing a prbe in a labratry tw tank is a trivial prblem. What is nt s clear is hw ne is t fix prbes in the deep cean. In intermediate depths, lake measurements have been made by Dnelan(1978) and thers in Lake Ontari at the CCIW twer in water f 12m depth. Cavaleri and Zecchett(1987) have made twer measurements in 16m f water; their twer was lcated in the Adriatic Sea. Shnting(1968,1967) made measurements in 2m f water in Buzzards Bay, Massachusetts. Other measurements have undubtedly been made in these relatively shallw, O(1m), depths. There is, hwever, a decided prblem in erecting a stable platfrm in deep water. The SMILE study was t be made in 9m f water and, hence, it was impracticable t build a twer specifically fr this applicatin. If chice f site is unimprtant t a study then fixed platfrms can be fund in waters much deeper than 9m; e.g., the Bullwinkle il prductin platfrm, wned by Shell Oil, is lcated in the Gulf f Mexic in 412m f water. Oil platfrms are nt, f curse, designed with a view twards making ceangraphic measurements. The flw disturbance in the vicinity f these platfrms makes their ptential use f limited value. The fiw disturbance prblem is the first cncern with twer based measurements. A twer fr ceangraphic measurements must be strng enugh t suffer the lading f the wave envirnment and at the same time must have as little structural elements as pssible 16

17 t minimize the flw disturbance. In nrmal situatins this cmprmise f design cannt be fully achieved. Usually, an investigatr making twer based measurements munts his instruments such that they are minimally disturbed by wake effects in prevailing cnditins. In sme cases, as cnditins change, instruments might be mved r steered s that they are nt dwnstream f any wake generating members. Even in such cases, there is cause fr cautin. The first cncern is that in a wave envirnment that the rbital velcities will, even when sensrs are placed dwnstream frm structural elements, pull the wake back acrss sensrs during the return flw. In a typical deplyment frm sme f the smaller twers, O(1m) depth, it wuld nt be unusual t experience 3cmjs rms velcities and wavefields with peak perids f 3 secnds. In such a case, the rms excursin f the wave velcity is abut 29cm. Clearly, there is n difficulty in using bms t munt instruments utside this advectin range even under mre harsh' cnditins. There will still be the wake f the instrument itself t cntend with. The instrument wake prblem is trublesme but is nt peculiar t the Eulerian measurement frame. In measurements f currents frm twers it is predminately the wake effects which are f cncern t us. In studying wave effects frm twers, the prblem f reflected waves shuld als be cnsidered. The ptential functin fr a wavefield, ll>, in the vicinity f an bstructin can be written as the sum f an incident ptential, ll>i, and a scattered ptential, ll>s, as (2.3) In radial crdinates, the ptential fr a mnchrmatic wave may be expressed as (2.4) In additin t satisfying the free surface kinematic and dynamic bundary cnditins and the bttm bundary cnditins, if there is a cylindrical bstructin in the fluid then, the ttal ptential, ll>, must satisfy 8<I> = O. 8r 17 (2.5)

18 n the cylinder surface T = b. Mrever, the scattered wave must satisfy the Smmerfield radiatin cnditin lim T2" 1 (OiI>S -- - zkii>s ) -; O. r-- r When we slve fr the scattered ptential, subject t the abve cnstraints we find a scattered ptential f the frm (2.6) where the H's are Hankel functins. The mst imprtant features t nte f the scattered ptential are the scaling f the ptential, (kb)2, and that the scattered waves prpagate in all directins (thugh there is an azimuthal dependence). Further nte that the scattered ptential falls ff n the length scale f the wave, (i. e. ii>s = ii>s (kr)). This is in cntrast t flw separatin effects, whse decay is usually scaled by the diameter f the flw bstructin (intensity f disturbance rv f). The cnsequence is that the scattering effect is felt even at relatively large distances frm the twer. It is true that the scattered ptential may be minimized by reducing the radius f the twer supprt b but cnsider that even thugh the scattered ptential is scaled by (bk)2, the waves will interact with the first rder incident waves. Calling the incident wave amplitude a, and the radian frequency a, the velcities f the incident wave scales as aa. The velcities f the radiated ptential scales as ab 2 ak 2 Fr 5cm amplitude waves with radian frequency a = 1rad/s scattering ff a cylinder f radius b = 5cm, the Reynlds stress T due t the resulting standing wave cmpnents is T = ~puw* rv pa 2 b 2 ak 2 = O(lN/m 2 ). These waves are much smaller than thse we lked at in the last sectin. Even s, fr the wind speed examined (1mls), we expected a wind-induced shear stress, T = pu;w' f nly.7n / m 2 This effect may therefre entirely mask the wind-induced shear stress when measurements are made via a velcity crrelatin. In higher windspeed cnditins (12m/s t 17m/s) than thse we have just discussed, Cavaleri and Zechett (1987) used a theretical mdel t predict a wind stress f apprximately O.3N1m 2 They measured shear stresses in the fluid, using velcity crrelatins, as large as 3N/m 2 The authrs cnsidered several surces f errr but did nt cnsider wave scattering frm the twer. Using a cmputer mdel, and inputting a spectrum f waves, I 18

19 fund that stresses as large as 3N1m 2 culd be explained slely in terms f waves scattered frm the single twer leg clsest t the sensrs. Obviusly, inclusin f the entire supprt structure and a mre accurate wave mdel wuld lead t different results. The pint is, whether r nt Cavaleri and Zechett have fund sme unknwn physical mechanism, as they seem t claim; it is dubtful that stresses f rder O.3N/ m 2 culd be measured using velcity crrelatins s near a wave scatterer withut carefully cnsidering these effects N~ 4 ~... ~ '-" 2 VJ VJ..., ~... VJ -2 VJ." " -4 l': >. ~ :: ~... kb=o kb=.5 kb= kr Figure 2-3: Reynlds stress due t a single mnchrmatic wave scattered frm a ~ircular cylinder. The incident wave amplitude is 1m, the radian frequency is 1radls, r ~s the upwave radial distance frm the scatterer and k is the wavenumber. The result is p1tted fr three different cylinder radii, b. Figure 2-3 shws the effect f wave scattering frm a single wave as a functin f radial distance frm a twer. Only the upwave directin (the directin sensrs wuld be steered t avid flw separatin effects) is shwn. Obviusly, the field experimenter must wrry abut a spectrum f incident waves. Fr a mre detailed discussin n the derivatin f scattered ptential, Mei(1989) prvides an excellent discussin. 19

20 The last issue is that f the apprpriateness f making an Eulerian measurement near the air-sea interface. The standard lg-law mdel fr wind-induced shear (Csanady, 1984) presumes that the free-surface is analgus t the wall in the classic wall-flw type experiment. In his paper, ne f the central questins Csanady tries t answer is that f what t call the wall. Thugh subsequent investigatrs have refined the thery, Van Drn's (1953) cnclusin that mst f the drag n the wind ver water is due t shrter waves is still generally held t be true. It is suppsed that the wind stress is supprted by the shrter waves and lnger waves grw thrugh nnlinear wave-wave interactins. The grwth f the shear current may be due directly t "skin-frictin" <)-nd als due t the vrticity f the smaller waves. The lnger waves, hwever, are nearly irrtatinal and, Csanady asserts, are dynamically unimprtant with respect t the shear current except in that the shear current is advected by the rbital mtins f the lng waves. The wind-induced shear current is, frm this pint-f-view, the steady cmpnent f the current relative t the surface that is defined by the lnger waves (Csanady argues that waves f wavelength 1m r greater culd be cnsidered "lng"). Hw des this affect near-surface Eulerian measurements? In small fetch situatins the effect f shear flw advectin is minimal because the energy in lnger waves is such that the advectin is small cmpared t the decay scale f the shear current. Fr the type f waves we encunter at the SMILE site the bundary layer advectin wuld be serius. T make an Eulerian measurement, ur shallwest sensrs shuld be lcated beneath the trughs f the waves t avid exiting the water. The significant waveheights H 1 by the SASS during its deplyment varied frm 1.6m t 3.m. 3 measured The waveheight spectra tended t be heavily weighted by swell. Let's examine the implicatins f wave advectin by cnsidering a wavefield where H 1 = 2m. Here, ur shallwest sensrs wuld have t be 3 lcated mre than a meter belw the mean free-surface t avid exiting the water. Suppse we put ur shallwest sensr 2m belw the mean free-surface. The bundary layer wuld be advected ±lm relative t this sensr. If the windspeed is 1mjs and we again assume that U*w =.83cmjs, then the gradient f the wind-induced shear current wuld be given 2

21 by au az 2.75cmJ8 z At a distance f 2m frm the bundary we expect t find a shear f abut (2.8) As the bundary layer advects up and dwn, we expect ur sensr t measure shears which vary frm t Hence, the variatin in shear is abut 3% greater than the mean shear we expect t measure. It seems lgical that t imprve ur reslutin, we need t make ur measurements frm sensrs which fllw the surface. Mrever, in a surfacefllwing mde, sensrs may be placed very clse t the bundary with little danger f exiting the water. This type f quasi-lagrangian surface-fllwing measurement, typically made frm a buy will be intrduced in the fllwing sectin. Befre mving n t discuss quasi-lagrangian measurements, hwever, we mentin a special case f what might be aptly termed.a quasi-eulerian measurement. In the case f the research vessel FLIP, the ntable spar buy, measurements may be made which are nearly Eulerian. Even with its enrmus size (draft::::: 91m) the FLIP des mve. But, with pitch and rll resnant frequencies f.21hz and a heave resnance f.37hz (Rudnick, 1967), we dn't expect the cupling between the wave and buy mtins t be strng. Thugh the FLIP des respnd t lw frequency mtins, it can be steered (see Weller, 1985) t keep current meters upstream f the hull-wake. Furthermre, the fact that FLIP drifts with the large scale flws wuld further reduce flw disturbance effects (ur earlier remarks abut scattered waves need be cnsidered; FLIP's hull diameter is 3.8m 'at the surface and gradually increases t 6.1m). A FLIP type platfrm wuld nt, in any case, have been apprpriate at the SMILE site, nly abut 5km frm shre, in a strng current envirnment and having a water depth f nly 9m. 2.3 Surface Referenced Usually, when slving a fluid mechanics prblem, we attempt t find ur slutin using either a Lagrangian r an Eulerian apprach. Each apprach has its advantages fr studying 21

22 near-surface dynamics. The Lagrangian measurement is a measure f ttal transprt and is a cnceptually easy experiment t design. Fr the Eulerian measurement wave effects are nt an issue and, in twer-based measurements, the experimenter usually has the equipment verhead (e.g., pwer supplies and data strage capabilities) necessary t make extensive measurements. But, as we discussed in the last tw sectins, each apprach has its prblems t. It IS difficult t make a truly Lagrangian measurment near the air-sea interface. Als, the tracking prblems assciated with Lagrangian measurements make cllectin f meaningful amunts f data difficult. The Eulerian apprach is ften nt pssible in deep water. And, even if a deep water twer can be fund r cnstructed, there remains the prblem f the bundary layer advecting past the sensr. The surface-fllwing sensr might be seen as a cmprmise between the Eulerian and Lagrangian appraches. Our archtypical surface-fllwing prbe is a current meter hung beneath a buy. Using this apprach is cnsistent with the idea that the near-surface bundary layer is advected with the lnger waves f the sea surface (Csanady, 1984). While making a bundary layer measurement it makes sense t use the bundary as a reference pint. Buy systems are relatively easy t deply and can be left unattended. The surface-referenced measurement is nt a panacea. One f the better knwn difficulties with this apprach is the fact that the measurements are biased in the dw-nwave directin. Pllard (1973) develped a thery t explain the "wave bias" in terms f linearized ptential thery. Here I'll repeat the simplest frm f the argument Pllard gave as an intrductin t the wave bias. Fr a mnchrmatic wave, the surface elevatin 'T7 and hrizntal velcity u may be written as 17 acs(kx-at). (2.9) aacs(kx - at)e kz. 22

23 If the buy heaves with the surface, but des nt mve hrizntally, then the mtin f a sensr munted a fixed distance Z beneath the waterline f the buy is given by [X, Z] = [x, -Z 1]]. (2.1) The hrizntal velcity measured by the sensr is (2.11) Expanding the expnential gives (2.12) and the resulting nnzer time average (2.13) is the "wave bias." The wave bias fr ur surface fllwing sensr is seen t have a frm similar t the theretical frm f the Stkes' drift. In the next sectin we'll shw that the derivatin f the wave bias and the Stkes' drift can be simply related. Here, I'd like t shw what type f effects this bias might have n the data. Figure 2-4 shws the expected shear current in a lom/s wind (the cnditins are assumed t be the same as thse fr Figure 2-2). Als shwn are the indicated shear fr a.perfect wave-fllwer and the indicated shear when the bias is that measured by the SASS. 23

24 ~\" O~ 1 5 ZU* lj Figure 2-4: Surface defect velcity (--) using a theretical frm fr the wind-induced current in a 1m/s wind. The shear indicated frm a sensr which perfectly fllws the v ;rtical excursins f the surface (...) includes the wind-induced and a wave-induced cntrib-utin. The SASS fllws bth vertical and hrizntal fluid displacements. If the shear were given by slid line we expect that SASS wuld verestimate (- - -) the shear if the wave-induced prtin f the measurement is nt cnsidered. 24

25 Chapter 3 WAVE BIAS 3.1 Measurements frm a mving sensr In the last chapter we fund that there were several advantages t making near-surface measurements in a surface-fllwing frame. We als admitted current meters which mve cherently with wave rbital velcities suffer frm a dwnwave bias. This has lng been recgnized and yet is a prblem which the ceangraphic cmmunity has still nt adequately addressed. We lked at the simplest frm f Pllard's (1973) analytical mdel t shw hw the bias arises. In his paper Pllard cncluded that: If the directinal spectrum is knwn, Kenyn 's(1969) technique can be used t calculate that part f 11 caused by vertical mtin. Hwever, even if the hrizntal mtin f the current meter were measured it wuld be difficult if nt impssible t make an acceptable estimate f the errr caused by hrizntal mtin. Nnetheless, Santala and Terray(1991) develped a thery wherein the bias due t bth vertical and hrizntal mtin culd be estimated with reasnable accuracy. Befre develping this thery in a frm sphisticated enugh t be applied t field data I'd like t lk again at the different measurement types, again using a single mnchrmatic wave, but nw using cmplex analysis. This will allw us t lk at the cnsequences f sensr-fluid crrelatins in a cmpact way. 25

26 Cnsider a tw-dimensinal mnchrmatic wave f amplitude a and frequency (7. ptential functin ip fr a deep water wave (k := (72/g) may be written as: The ip = _zafie'(kx-ctt)e kz. a The surface elevatin 7] and the hrizntal and vertical fluid velcities (u, w) are given by (3.1) T/(X, t) u(x, z, t) w(x, z, t) iPI -_ ae '(kx-ctt), g 8t z= _ 8iP = a(7e,(kx-ctt)ekz, 8x 8iP,(kX-CTt) kz 8z = -wae e. (3.2) The sensr trajectries may be described as being cmprised f a mean psitin (x, -z) and a time varying cmpnent (xe(t), ze(t)) as [X, ZJ = [x xe(t), -Z ze(t)j. (3.3) As the sensr traces ut its time histry it nt nly measures the time histry f the field velcity but als samples its spatial variability. Substituting the trajectries (3.3) int the expressins fr velcity (3.2) gives the expressin fr the measured velcity (U, W) as a functin f time (withut lss f generality X may be set t zer) U(t) Wet) aae-'ctte-kz[l _ 1.kx kz...j _ dxe e _ e dt ' _wae-lctle-kz[l _ zkx e kz e...j _ d~e. (3.4) In the abve expressin, the expnentials have been expanded int a pwer series with nly thse terms greater than O(a 3 ak 2 ) being explicitly shwn. If the respnse f the b~y is linear, then the time varying part f the sensr's trajectry may be written as -ICTt X e (t) = xme, (3.5) with X m and Zm being the cmplex cefficients describing the sensr mtin. The time average f the hrizntal and vertical velcity is then given by the real part f the fllwing expressins: (3.6) 26

27 where * dentes a cmplex cnjugate. The sensr mtin has been assumed t average t zer. The preceeding expressin, where we have yet t specify the frm f the sensr mtins, prvides us with a cmpact way t examine measurement principles in general. Fr instance, if the sensr is still, then the measurement is Eulerian and equatin 3.7 reduces t U Eulerian = a (Xm,Zm) = (,) =? vv Eulerian = a (3.7) If a sensr des nt drift, but t first rder fllws the particle trajectry at a certain depth, then we btain (t first rder) the Stkes' drift velcity (3.8) Fr the idealized case f a velcity sensr hf.nging frm a buy which is perfectly cupled t the surface, bth in hrizntal and vertical excursin, (X m, Zm) = (a,a) ==? UPllard = aa*ake- kz lvpllard = (3.9) In the case where the buy respnse is nt ideal but lags the wave excursins by a phase angle, 1./.) (r a time delay f tl ag = *) ( ) ( tv' tv' ) Xm,Zm = ae, ae =:> _(3.1) The abve case is mst interesting because it shws that there can als be a bias in measuring vertical velcities. What all the preceeding examples pint ut is that the final time-averaged utput f a mving current meter is nt nly dependent n the velcity field in which the current meter is immersed, but is als dependent n the mtin f the current meter itself. Furthermre, the size f this mtin dependent mean is rder O( a 2 ak) and the apparent shear is rder O((ak?a). These are the same rder f magnitude as the expected current and shear in 27

28 the wind-driven flw (Bye,1967; Wu,1975). The nly averaging prcedure which des nt measure the wave effects is the Eulerian. As was discussed in the last chapter, the Eulerian-type measurement is nearly impssible t realize and difficult t interpret in the near-surface envirnment in deep water. What we desire is t have an averaging prcedure that will measure the wind-induced shear current and the mean flw withut being biased by wave/mtin effects. While it is pssible t develp such a prcedure, the derivatin is nt entirely straightfrward. T pave the way fr the full derivatin, the cncept f the unbiased estimatr will be mtivated by the use f a simple mathematical cncept; that f a line integral in a ptential field. line integratin interpretatin Since the waves are irrtatinal and peridic (in space), the line integral f the wave velcity,. u(x), alng any trajectry, at any instant f time vanishes f dx \7if;(x) == O. (3.11) This integral can be regarded as an averaging prcedure that results in zer fr irrtatinal mtins. Frm a practical standpint, equatin 3.11 is f little use. It can be explited if we recgnize the time-space duality f water waves; i.e., the spatial variatin f a wave may be sampled by a fixed bserver wh samples a wave as it passes (in time). Cnceptually, this is mst easily visualized fr the case f a tw-dimensinal mnchrmatic wave, which is viewed in a frame freference mving at the wave celerity, c, s that the wave prfile is steady in time. Here, the trajectry f the sensr is described by the vectr (-ctx(t), Z Z(t)). Equatin 3.11 then becmes 1 it{ 1 1.} - U--UX--HlZ dt=o, T c C (3.12) where time derivatives have been indicated by dtting variables, T is the wave perid and, as abve, U(t) dentes the measured velcity. The case in which XCi) ~ c is depicted in 28

29 Figure 3-1(a), where the dashed line shws the integratin path. The lwer "return path" is taken t be a line at sme fixed depth belw the mean water level. The integral alng the return path is equivalent t an Eulerian measurement at that depth and, as such, will nt measure a wave cmpnent. The net cntributin ver that segment will be due t the advectin velcity, c, that is required when equatin 3.11 is examined in the steady frame and will precisely cancel the advectin cmpnent n the frward part f the path. In the case f infinite depth, the return path may be taken at z = -. a -_..- '" "' ~-..~'--..-~_..--l , b '----..~~_..-~-.--l , Figure 3-1: The line integral cncept. In a frame f reference which travels at the wavespeed, the fluid velcities are steady. The dtted lines shw indicate cnstant ptential surfaces. Paths are shwn which crrespnd t surface-fllwing measurements. A line integral (a) thrugh the wavefield will return a result f zer. In (b), the equatin f a nrmal time average f hrizntal velcity ~ f Udt has been transfrmed t this frame. Here, the differential distance vectr dx des nt lie alng the path f integratin and a zer wave cntributin is nt guaranteed. Equatin 3.12 defines a "trajectry average" f U, which will be dented as U (the ntatin is chsen in analgy t the cnventinal use f an verbar t indicate a simple 29

30 time r space average). The first term in equatin 3.12 reprduces the wave bias, while the remaining terms cnstitute "crrectins" that estimate the bias. Fr the tw-dimensinal case just shwn the cancellatin is exact. When the definitin f U is generalized in the next sectin t three-dimensinal, randm waves we will find this is n lnger true. With a spectrum f dispersive waves present, a steady reference frame des nt exist. Still, it will be shwn that while the cancellatin is n lnger exact, the bias remaining in U is reduced by a factr f wave slpe, ak, ver that in U. While the crrectins cmpensate fr the wave cntributins t the time average, they will nt unduly affect the estimate f the nnwave cntributins. When a rtatinal velcity (U', W') is als present, the crrectin in equatin 3.12 als includes terms such as U'X Ic and W'ZIc. Since X and Z are bth f rder f the wave rbital velcity u, the terms XIc and Zic are at mst O(ak), althugh in general they are expected t be much smaller since it is likely that bth X and H7 are prly crrelated with U'. S, u~ U' O(u(ka)2) O(kaU'), (3.13) where U, the net bserved velcity, is nw the sum f irrtatinal (wave) and rtatinal cmpnents. The line integral interpretatin can als be used t illustrate the surce f the wave bias. The time average f U(t) is prprtinal t the integral f the hrizntal velcity cmpnent u(x,z,t) evaluated alng the sensr trajectry. This situatin (again in the steady-frame f reference) is shwn in Figure 3-1(b). Since the differential distance elements d nt lie alng the path f integratin, we cannt guarantee that this integral vanishes (actually perfrming the integral will, f curse, yield the wave bias). 3

31 3.2 Estimatin f the bias general case When a spectrum f waves is present, it is nt pssible t find a reference frame in which the waves are steady and the develpment f the last sectin cannt be repeated exactly. Instead, it is assumed that equatin 3.12 may be generalized t (3.14) (repeated latin subscripts run frm 1 t 3 and are summed). The unknwn functins p(t) replace the factrs 1/c appearing in equatin Since bth the directinal spreading and phase speed f the waves are frequency dependent, the /1's are intrduced as filters hence the cnvlutin, dented by * in equatin The prblem is t find /1(a),s that reduce the bias in Va by an rder f magnitude ver that in 1/, Mathematically we express this as a requirement n the expected value f V as (3.15) On taking the expected value f equatin 3.14, \ve then require that the fllwing be satisfied t rder O(a 3 ak 2 ) : (3.16) Bth the left- and right-hand sides f equatin 3.16 can be evaluated using linear wave thery. The utput f a mving sensr is the relative velcity Vi - Xi, where Vi is related t the Eulerian velcity Vi by '~(t) = Vi(X(t), t). (3.17) In the fllwing it is assumed that the additive cntributin Xi f the sensr mtin t the relative velcity has been subtracted and V; is referred t as the bserved velcity. 31

32 Expanding equatin 3.17 t secnd rder gives ( ) ) () OVi(X(), t) Vi ( t ) = Vi ( X,t (Xj(t) - X.) J OX'... J (3.18) where X() dentes the average sensr psitin. The ptential functin f a directinally spread wavefield is cnveniently described using the Furier-Stieljes ntatin. (3.19) With this ntatin, the expressin fr the vectr velcity is (3.2) If we expand ut the velcities, as we did fr the tw-dimensinal case, and substitute in the sensr's trajectry cmpnents X, we find that the measured velcity V is v =1/, da(",oj"1::,: " )e,(kx(o'-u,)e- [1,k es BX 1,k sin X, kx, j (3.21 ) T first rder, sensr and fluid velcities are related thrugh the buy transfer functin Hij. The transfer functin belw relates the sensrs \'elcities t the fluid velcity at the surface..(3.22) Ifwe make the reasnable assumptin that mtin in a particular directin is predminately frced by fluid velcities in that directin (i.e. crss-cupled mtins between rthgnal directins are relatively weak), the transfer functin matrix is diagnal and the sensr trajectries and velcities are expressed as lh ll cs e IHn sin e H 33 (3.23) 32

33 Substituting in the sensr trajectries f equatin 3.23 int the expressin fr the measured velcity (equatin 3.21) and taking the expected value f the time average gives a general result fr the biased time average. H1, dada'h;,ak1 cs3 8 sin 2 8 cs 8-2 cs 2 8 ) e- b, H1, dada'h;,ak1 cs 2 8 sin 8. -2sin 2 8 ~II ros9 dada*h33 crk sin 8 ) e- kz sin 3 8 ) e- b, (3.24) -2 The right hand side f equatin 3.16 is ur bias estimatr and is rewritten in frequency space as (3.25) Using ur first rder expressins fr Xi and Ui this becmes {3.26) T find the I-l filters the /h cmpnent f equatin 3.25 is equated t equatin The bias arises frm the beating f like frequencies in sensr mtin and fluid velcity. Since the J.L'S are t mdel this prcess we will equate equatins 3.25 and 3.26 n a frequency bin-by-frequency bin basis. In ding this, let's rewrite the wave cefficients, da, in terms f a waveheight spectrum, STJTJ(cr), and a directinal spreading functin, f(j,8), (nrmalized s that I:1r f( cr, 8)d8 = 1) as 1 (dada*) ST)T)(J)f(cr,8) = 2 dad8 (3.27) 33

34 Then, fr a = 1 we will find the II'S frm the fllwing: 5'1).kr )crke- kz J{Hi1(cr )f(cr, ) COS 3 H22(a )f(a, ) cs 2 esin B H33(cr )f(a, ) cs O} do (3.28) =5'1)1]( cr )a 2 e- kz J{H;l (cr )f(cr, ) cs 2 H2Z(cr )f(cr, ) sin 2 H33(cr )f(cr, e) } de. The sensr transfer functin can be cmpletely remved frm the slutin by equating the abve n a term-by-term basis. Slving in this way als allws the i th crrectin term f equatin 3.14 t be interpreted as the bias riginating frm mtin in the i th directin. If this is dne, we find that the fl~l) filters are given by: (1) k f f(cr, e) cs 3 ede III ~ f f(cr, ) cs 2 OdO' (1) k f f( cr, ) cs 2 sin OdO liz cr f f( cr, ) sin 2 Ode (1) k f f(a, B) cs Ode fl3 cr f i(cr, O)de (3.29) Slving in the same way, the II~Z) filters are fund t be (2) ~f f(cr,e)sin 2 OcsOde fll cr f f( cr, 8) cs 2 ede (2) k f f(a, e) sin 3 ede flz p1 2 ) -;; f f(cr,e)sin 2 ede' k f f(cr, e) sin-odo a f f( a, O)de (3.3) Finally, the filters fr estimating the vertical bias fl~3) are fund t have the simple frm (3.31) If the spreading functin f( cr, e) is fund using a pitch and rll type technique, nly the first tw trignmetric mments f f( cr, ) are btained directly frm measurements (Lnguet Higgins et al, 1963). This means that the abve filters which require third mments f the spreading functin will be smewhat dependent n the extraplatin used t frm the final spreading functin estimate. Fr the SASS data, the wave spreading is estimated using the maximum entrpy methd (see sectin 4.3). 34