Spatial Scales of Stress-Carrying Nearshore Turbulence*

Transcription

1 1122 JOURNAL OF PHYSICAL OCEANOGRAPHY Spatial Sales of Stress-Carrying Nearshore Turbulene* JOHN TROWBRIDGE AND STEVE ELGAR Woods Hole Oeanographi Institution, Woods Hole, Massahusetts (Manusript reeived 5 November 2001, in final form 27 November 2002) ABSTRACT Measurements from a horizontal array of veloity sensors indiate that the alongshore sales of turbulene ontributing to the near-bottom Reynolds stress just seaward of the surf zone on an oean beah range from 0 to approximately 4 times the height of the measurements above the seafloor, with shorter sales during stable stratifiation than during neutral or unstable stratifiation. The dependene of alongshore turbulene sales on the stratifiation, Reynolds stress, and height above the bottom is onsistent with semiempirial results from the atmospheri surfae layer, implying similar dynamis of near-boundary turbulene in the atmosphere and the shallow oastal oean. 1. Introdution Spatial sales of turbulene in the atmospheri boundary layer have been haraterized for a range of onditions (Kaimal et al. 1972), but observations of turbulene sales in oeani boundary layers have been limited to unstratified tidal flows (Soulsby 1977; Gross and Nowell 1983). Spatial sales of weak turbulene in a strong mean flow an be estimated from time series measurements at a single point by means of the frozenturbulene hypothesis (Taylor 1938). In many oeani environments, surfae waves ompliate the interpretation of point measurements (Lumley and Terray 1983), and determination of turbulene sales requires a spatial array of sensors. In this study, observations from an array of veloity sensors are used to estimate sales of turbulene just seaward of the surf zone on an oean beah. The array was designed to estimate a spatial struture funtion D w (y), defined by D (y) [(x jy, t) (x, t)][w(x jy, t) w(x, t)]. (1) w Here, x and (x, y, z) denote the position vetor, with x positive onshore, y alongshore, and z positive upward and equal to zero at the bottom; y is the alongshore separation; j is a unit vetor in the y diretion; t is time; (U, V, W) is the veloity vetor; (u,,w) is the departure of the veloity vetor from its hour-averaged value; and an overbar denotes an hour average. The analysis is based on the assumption that veloity flutuations with timesales of less than 1 h are aused by superposed surfae waves and turbulene. At the sales resolved by the array, the operation of subtration in (1) largely removes effets of waves (Trowbridge * Woods Hole Oeanographi Institution Contribution Number Corresponding author address: John Trowbridge, Woods Hole Oeanographi Institution, Woods Hole, MA trowbridge@whoi.edu 1998). In priniple, D w (y) inreases from zero to 2 w, where primes denote turbulent flutuations, as y inreases from zero to values greater than the orrelation sale of the turbulene. Thus, D w (y) desribes the alongshore sales of the turbulene ontributing to the Reynolds shear stress 0 w, where is the density and the subsript 0 denotes a fixed referene value. This omponent of the Reynolds stress is a dominant term in the alongshore momentum balane in shallow water (Longuet-Higgins 1970; Feddersen et al. 1998; Lentz et al. 1999). The null hypothesis is that the dynamis of turbulene in the shallow oastal oean, at heights above bottom that are small relative to the water depth, but larger than the thikness of the osillatory boundary layer produed by surfae waves, are onsistent with loal appliation of Monin Obukhov saling (Monin and Yaglom 1971). In this saling, the statistial properties of the veloity and density depend only on z, 0, the vertial buoyany 2003 Amerian Meteorologial Soiety

2 MAY 2003 TROWBRIDGE AND ELGAR 1123 flux B (g/ 0 ) w, where g is gravity, and the vertial flux of streamwise momentum, here approximated by 0 w, beause hour-averaged urrents at the measurement position were predominantly shore parallel, so that U K V and uw K w. Dimensional onsisteny requires that properly saled statistis of the veloity and density fields be funtions of z/l, where L w 3/2 /(B) is the Monin Obukhov length, with the empirial von Kármán onstant Monin Obukhov saling implies that L zn, (2) z w where L is a length haraterizing the alongshore orrelation sale of the stress-arrying turbulene, is a funtion that an be derived from semiempirial results obtained in the atmospheri surfae layer, and N 2 (g/ 0 ) /z is the squared buoyany frequeny, whih an be positive or negative. Here, oeani measurements of D w and z are fit to a model to determine w and L, and these estimates are ombined with measurements of N 2 to test (2). 2. Methods a. Measurements Four aousti Doppler veloimeters (ADVs) were deployed in an alongshore array on a nearly planar sand beah, with a slope of approximately 0.03, about 60 m north of the pier at the Sripps Institution of Oeanography in La Jolla, California (Fig. 1). The ADVs were mounted on a low-profile, bottom-mounted frame, in an upward-looking orientation, with sample volumes about 0.75 m above the bottom. The ADV array permits estimates of D w at six nonzero equally spaed separations ranging from y 0.5 m to y 3.0 m. A pressure sensor was buried 0.3 m beneath the seafloor approximately 3 m from the enter of the ADV array. A pair of sonar altimeters and a pair of temperature sensors were deployed approximately 10 m from the enter of the ADV array (Fig. 1). The temperature sensors were 0.10 m above the ADV sample volumes. Ten-minute time-lapse video images obtained hourly (Holman et al. 1993) allowed the extent of the surf zone to be determined during daylight hours. Sequential 1-h time series sampled at 8 Hz were obtained from eah sensor between 19 Otober 2000 (yearday 295) and 7 January 2001 (yearday 369), shortly before the ADVs were buried by sand on 14 January Observations by divers indiate that the ADVs were oasionally fouled by kelp, and omparison of the hour-averaged temperature reords indiates that the temperature sensors oasionally drifted. Hour-long reords from a partiular ADV were identified as having been fouled by kelp if the mean or standard deviation of any veloity omponent from that sensor differed by more than 0.01 m s 1 from the orresponding array FIG. 1. Plan view of sensor array. The four ADVs (SonTek/YSI, In., OCEAN probes) were deployed in an alongshore array in approximately 4-m water depth, about 60 m north of the Sripps pier at La Jolla, CA. The aousti altimeters provide estimates of the seafloor elevation, and the pressure gauge (buried in the sand to redue flow noise) measures bottom pressure, whih an be onverted to sea surfae elevation using linear wave theory. The temperature measurements were used to estimate the squared buoyany frequeny, as desribed in the text.

3 1124 JOURNAL OF PHYSICAL OCEANOGRAPHY average. The results presented here are based on h reords for whih all four ADVs were above sand level, none of the ADVs was fouled by kelp, and the two hour-averaged temperature measurements were within 0.1C of eah other. Video images indiate that the array was seaward of the surf zone during these reords. b. Models of D w,l /z, and N 2 A representation of D w follows from a semiempirial expression for the ospetrum of streamwise and vertial veloities in the atmospheri surfae layer, given by kyco w(k y) 7 3 kl y sin. (3) 7/3 w k L y The oordinate system is defined so that the y axis is aligned with the mean veloity; Co w is the ospetrum of and w, defined so that # Co w (k y ) dk y w; k y is the y omponent of the radian wavenumber; and L is given by L 1.2 for z/l 0 (4) 3/4 z 1.2(1 7.9z/L) for 0 z/l. The Fourier transform of Co w (k y )ise w (y), defined to be the even part of (x, t)w(x jy, t) (Monin and Yaglom 1975). By definition, D w (y) 2[E w (0) E w (y)]. It follows that (3) is equivalent to 7 3 y z d 7/3 [ ] D w(y) 2w 1 sin os, (5) 6 7 z L 1 where is a dummy variable of integration. Equations (3) and (4) are similar to a model proposed by Kaimal et al. (1972) for neutral and stable stratifiation (B 0), and they are onsistent with theory and measurements at large k y for both B 0 and B 0 (Kaimal et al. 1972; Wyngaard and Cote 1972). The integral on the right side of (5) depends only on the ratio of y/z to L /z and an be evaluated numerially. Aording to (5), D w inreases monotonially with y and is within 10% of 2w for y 3L. Standard results for the atmospheri surfae layer lead to an expression for the squared buoyany frequeny (Hogstrom 1988): 2 Nz 0.95z/L(1 11.6z/L) for z/l 0 w z/l( z/L) for 0 z/l. (6) Equations (4) and (6) define in (2).. Analysis Struture funtions were omputed for eah reord at eah of the six nonzero spatial separations by removing the hour average of alongshore and vertial veloity, omputing produts aording to (1), and averaging over the 1-h reord. Estimates of z and the mean water depth h were obtained from the altimeter and pressure measurements. Estimates of wave statistis were obtained from the pressure and veloity measurements using linear wave theory. Estimates of T/z, where T is the temperature, were obtained by regressing array-averaged estimates of T/t, obtained from entered differenes of the temperature time series, against array-averaged measurements of w. This proedure is based on a linearized model of temperature flutuations given by T T w 0. (7) t z Both the variane of w and the ovariane of T/t and w were dominated by surfae wave flutuations at frequenies between approximately 0.05 and 0.5 Hz. The standard deviation of w was muh larger than the bottom slope times the standard deviation of u, whih suggests that estimates of T/z were not affeted signifiantly by terms negleted in (7), suh as ut/x, or by unertainties in the orientation of the ADVs. The squared buoyany frequeny was estimated from T/z by assuming a linear relationship between temperature and density, with a slope of 0.24 kg (m 3 C) 1, determined from previous temperature and salinity observations near the measurement site (Lerzak 2000). Estimates of w and L /z were obtained by fitting measurements of D w to (5), with the measured z/y, using iterative least squares regression. First, the measurements of D w were regressed against (5) to produe 854 preliminary estimates of w and a single estimate of L /z, representative of the entire measurement period. The measured z and N 2 and the preliminary estimates of w were then used to ompute 2 N 2 z 2 / w for eah 1-h reord. To exlude observations in whih w was too small to be estimated aurately, 54 1-h reords for whih 2 N 2 z 2 / w 2 were omitted from the analysis. The remaining 800 reords were sorted into four equally sized groups in order of inreasing 2 N 2 z 2 / w. The measurements of D w within eah group were then regressed against (5) to produe 200 final estimates of w and a single estimate of L /z for that

4 MAY 2003 TROWBRIDGE AND ELGAR 1125 FIG. 2. Time series of (a) signifiant wave height H s, (b) std dev u of the ross-shore veloity, and () alongshore veloity V. All statistis are hour averages. Only the reords with valid data, as defined in the text, are inluded. Signifiant wave height is defined to be 4 times the standard deviation of the surfae displaement and was estimated from pressure measurements by means of linear wave theory. Gaps in the time series are aused by fouling owing to kelp, a power failure, and drift in the temperature sensors. All reords used in the analysis have idential gaps. group. The preliminary and final estimates of w were not signifiantly different. The result of this proedure was four estimates of L /z, ordered by 2 N 2 z 2 / w, and 800 estimates of w. Error bounds for L / z were obtained by a standard proedure based on Gaussian statistis and linearization of the nonlinear regression about the best-fit values of the parameters (e.g., Draper and Smith 1966). In the alulation of error bounds, the six measurements of D w obtained during eah 1-h reord were assumed to have only 2 degrees of freedom, beause onurrent measurements of D w at different spatial separations were not statistially independent (e.g., Cressie 1993). 3. Results During the 70-day measurement period, the signifiant wave height ranged from approximately 0.5 to 1.5 m, the standard deviation of the ross-shore veloity ranged from approximately 0.2 to 0.5 m s 1, and the hour-averaged alongshore veloity usually was less than 0.1ms 1, with maximum values of about 0.2 m s 1 (Fig. 2). Hour-averaged ross-shore veloities were muh smaller than hour-averaged alongshore veloities. Wave inidene angles were within 10 of shore normal. The mean and standard deviation of T/z were 0.01 and 0.03C, respetively. Stable stratifiation usually ourred during southward alongshore flows ( V 0), and unstable stratifiation usually ourred during northward alongshore flows ( V 0). The hour-averaged water depth varied between approximately 3.0 and 5.5 m, primarily owing to tidal flutuations. The height of the veloity measurements above the bottom ranged between 0.75 and 0.93 m as the seafloor eroded and areted. The measurement height z was larger than the thikness of the osillatory boundary layer produed by surfae waves, estimated at a few entimeters (Grant and Madsen 1979) but was small in omparison with the hour-averaged water depth h (0.15 z/h 0.30). Approximately 60% of the variane of the alongshore urrent was at frequenies below 2 yles per day, approximately 20% was at semidiurnal frequenies, and the remaining 20% was at frequenies greater than semidiurnal. Visual observations obtained nearly daily by divers suggest that the seafloor was smooth, with no signs of wave-orbital or larger-sale ripples or bedforms. The seafloor elevation varied by entimeters on timesales of hours, but the differenes between the two altimeters indiate there were no bedforms with horizontal sales smaller than the altimeter spaing and vertial sales larger than the altimeter auray of about 0.01 m (Gallagher et al. 1998). The qualitative dependene of the measured struture funtions on spatial separation and stratifiation is onsistent with expetations (Fig. 3). The magnitude of D w

5 1126 JOURNAL OF PHYSICAL OCEANOGRAPHY FIG. 3. Struture funtion D w for different spatial separations vs 2 w. Units: 10 4 m 2 s 2. is smaller than 2w at small y, and it asymptotially approahes 2w as y inreases. The approah of D w to 2w with inreasing y is more rapid for positive w, orresponding to V 0 and stable stratifiation, than it is for negative w, orresponding to V 0 and unstable stratifiation. The alongshore turbulene sales suggested by Fig. 3 are roughly 1.0 m for stable stratifiation and are between about 1.5 and 2.5 m for unstable stratifiation. Estimates of w are approximately onsistent with a quadrati drag law, given by w d(u V ) V, (8) FIG. 4. Estimates of w vs (U V ) V. The estimate of the drag oeffiient, defined to be the slope of the best-fit straight line through the relationship between w and (U V ) V, is ( ) 10 3 at 95% onfidene. It is not known why the observed relationship between w and (U V ) V has a nonzero interept. with a drag oeffiient d ( ) 10 3 at 95% onfidene (Fig. 4), whih is similar to previous estimates of the drag oeffiient in nearshore flows under unbroken waves (Feddersen et al. 1998; Trowbridge and Elgar 2001). The dependene of L /z on 2 N 2 z 2 / w indiates suppression of turbulene sale by stable stratifiation and agreement with the atmospheri results in (4) and (6) (Fig. 5). Corresponding estimates of the drag o-

6 MAY 2003 TROWBRIDGE AND ELGAR 1127 FIG. 5. Dimensionless length-sale parameter L /z vs dimensionless stratifiation parameter 2 N 2 z 2 / w. The irles are estimated from the measurements, and the dashed urve is the atmospheri result given by (4) and (6). The horizontal error bars represent the range of stratifiation parameter that was used to obtain the averaged estimates of L /z. The vertial error bars represent approximate 95% onfidene limits, estimated as explained in the text. The measurements indiate a redution in length sale with inreasing stratifiation parameter, in agreement with the atmospheri results. The stratifiation parameter hanges sign beause N 2 an be positive or negative. effiient indiate no signifiant dependene on 2 N 2 z 2 / w. 4. Disussion The observations indiate that turbulene length sales in the shallow oastal oean, at heights above the bottom that are small in omparison with the water depth but are larger than the thikness of the wave boundary layer, are influened by stratifiation in a manner onsistent with loal appliation of Monin Obukhov saling (Fig. 5). This result implies that the dynamis of near-boundary turbulene in the shallow oastal oean and the atmosphere are similar, even though the environments differ. In the atmospheri surfae layer, both w and B are nearly independent of z. At water depths of a few meters seaward of the surf zone in the oean, saling and measurements (Feddersen et al. 1998; Lentz et al. 1999) indiate that Coriolis aeleration is negligible in the hour-averaged alongshore momentum balane and that the dominant terms in the depth-integrated balane are wind stress and bottom stress, implying that w is nearly independent of z, as in the atmospheri surfae layer. However, the seafloor usually is idealized as nononduting for both heat and salt, so that B 0 as z 0, and thus B must vary with z if B 0. The present results indiate that the dependene of B on z does not have a dominant effet on the loal dynamis of the turbulene. For a fixed turbulent Reynolds shear stress, inreasing stable stratifiation redues the eddy visosity and inreases the vertial derivative of the hour-averaged veloity. Measurements in the atmospheri surfae layer indiate that (Monin and Yaglom 1971) V 1 z 1/2 w sign(w) 1, (9) z z L where is an empirial onstant, approximately equal to 6.0 (Hogstrom 1988). For the observations used here, z/l was as large as 0.3, whih orresponds in (9) to a redution in eddy visosity and an inrease in V/z by a fator of 2.5. Measurements of V(z) often are fit to (9) to infer w. In the present ase, this proedure would result in overestimates of w by fators of roughly 6 during stable stratifiation. The measurements indiate a 50% redution in length sale during stable stratifiation (Fig. 5) but no signifiant dependene of the drag oeffiient on stratifiation (setion 3). This result is onsistent with loal appliation of Monin Obukhov saling. A simple model for near-bottom flow in the oean, where B 0asz 0, is w onstant and B zb/z, where B/z onstant. For this ase, integration of (9) with respet to z gives 1 z z 1/2 V w sign(w) log, (10) z 2 L [ ] 0 where z 0 is the bottom roughness length, whih inludes the effet of the bottom boundary layer produed by surfae waves (Grant and Madsen 1979). For the data disussed here, the seond term in square brakets in (10) is substantially smaller than the first, so that stratifiation has a small effet on the loal relationship between the hour-averaged veloity and the turbulent Reynolds shear stress, even though it has an important effet on the vertial gradient of the hour-averaged veloity in (9). Divergene from Monin Obukhov saling at small k y is observed in atmospheri measurements during unstable onditions (B 0) and is attributed to large-sale, onvetive irulations driven by the unstable density struture (Kaimal et al. 1972), whih presumably depend on sales in addition to B, w, and z. Here, the manifestation of similar proesses would be L /z larger than predited by (4) and (6) at large, negative 2 N 2 z 2 / w. This behavior is suggested by the measurements (Fig. 5), but the unertainty is suffiiently large that the observations are not neessarily inonsistent with (4) and (6). The alongshore turbulene sale during unstable stratifiation was the same order of magnitude as the water depth h (Figs. 3 and 5), but h is unlikely to have plaed an important onstraint on turbulene sale at the measurement height. The quantity z/h, a dimensionless measure of the importane of h, was small, and the measurements indiate no systemati dependene of L /z on z/h. This result is onsistent with laboratory measurements in open hannels indiating that the water depth

7 1128 JOURNAL OF PHYSICAL OCEANOGRAPHY affets the turbulene sales only for z/h 0.5 (Nezu and Nakagawa 1993). Also, muh of the energy appearing at long spatial sales in turbulene statistis referened to a single oordinate diretion, suh as D w (y) and Co w (k y ), is aliased from shorter sales, beause wavelengths of individual Fourier omponents appear longer if the wavenumber is not aligned with the oordinate diretion of interest. Thus, the atual sale of the stress-arrying turbulene was smaller than 3L, the value of y for whih D w is within 10% of 2 w. The wave-indued strain rate (of order k u, where k is the wavenumber of the surfae waves and u is the standard deviation of u), was larger than the strain rates of the hour-averaged veloity field and the stress-arrying turbulene (both of order w 1/2 /z), but straining by the wave-indued veloity field is unlikely to have had an important effet on the turbulene at the measurement position. The measurements indiate no systemati dependene of L /z on k u z/ w 1/2, the ratio of wave-indued to turbulent strain rates. The wave-indued strain rate is osillatory and therefore is unlikely to have had an important effet when averaged over many wave periods. Also, an order-of-magnitude argument adapted from Phillips (1961) indiates that wave-indued straining of turbulene had a negligible effet on the hour-averaged budget for the turbulent vortiity variane. Although it does not diretly address stress-arrying sales, the argument of Phillips (1961) suggests that wave-indued straining had a minor effet on the turbulene. 5. Conlusions Measurements from an alongshore array of near-bottom veloity sensors loated approximately 0.8 m above the seafloor in 4-m water depth on an oean beah were used to estimate the alongshore sales of the turbulent Reynolds stress just seaward of the surf zone. The analysis is based on a spatial struture funtion that rejets ontributions from waves. Estimates of Reynolds stress are onsistent with a quadrati drag law, with a drag oeffiient similar to previous observational estimates. Turbulent length sales were approximately 2 m during unstable stratifiation and were redued by a fator of roughly 2 during stable stratifiation, in a manner onsistent with atmospheri results based on Monin Obukhov saling. Effets of finite water depth and straining of turbulene by the wave-indued veloity field were small. The results imply that turbulene in the shallow oastal oean, just above the osillatory bottom boundary layer produed by surfae waves, is dynamially similar to turbulene in the atmospheri surfae layer. Aknowledgments. Support was provided by the Offie of Naval Researh (Marine Geology and Geophysis and Oean Modeling and Predition programs) and the National Siene Foundation. Britt Raubenheimer helped to design, deploy, and maintain the sensor arrays, ontributed signifiantly to the data analysis, and provided valuable omments. Staff from the Center for Coastal Studies at the Sripps Institution of Oeanography deployed the arrays and kept them operational despite nearly ontinuous attaks by kelp in old water with large waves. Janet Frederiks did the preliminary data proessing. Steve Lentz and two anonymous reviewers provided insightful omments on an early draft. REFERENCES Cressie, N. A. C., 1993: Statistis for Spatial Data. John Wiley and Sons, 900 pp. Draper, N. R., and H. Smith, 1966: Applied Regression Analysis. John Wiley and Sons, 407 pp. Feddersen, F., R. T. Guza, S. Elgar, and T. H. C. Herbers, 1998: Alongshore momentum balanes in the nearshore. J. Geophys. Res., 103, Gallagher, E., S. Elgar, and E. B. Thornton, 1998: Megaripple migration in a natural surfzone. Nature, 394, Grant, W. D., and O. W. Madsen, 1979: Combined wave and urrent interation with a rough bottom. J. Geophys. Res., 84, Gross, T. F., and A. R. M. Nowell, 1983: Mean flow and turbulene saling in a tidal boundary layer. Cont. 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