AUGUST 2001 TROWBRIDGE AND ELGAR 2403
Turbulence Measurements in the Surf Zone*
JOHN TROWBRIDGE AND STEVE ELGAR Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
(Manuscript received 14 August 2000, in ®nal form 4 January 2001)
ABSTRACT Velocity measurements within1mofthebottom in approximately 4.5-m water depth on a sand beach provide estimates of turbulent Reynolds shear stress, using a dual-sensor technique that removes contamination by surface waves, and inertial-range estimates of dissipation. When combined with wave measurements along a cross-shore transect and nearby wind measurements, the dataset provides direct estimates of the terms in simpli®ed equations for alongshore momentum and turbulence energetics and permits examination of semiempirical relationships between bottom stress and near-bottom velocity. The records are dominated by three events when the measurement site was in the outer part of the surf zone. Near-bottom turbulent shear stress is well correlated with (squared correlation coef®cient r2 ϭ 0.63), but smaller than (regression coef®cient b ϭ 0.51 Ϯ 0.03 at 95% con®dence), wind stress minus cross-shore gradient of wave-induced radiation stress, indicating that estimates of one or more of these terms are inaccurate or that an additional effect was important in the alongshore momentum balance. Shear production of turbulent kinetic energy is well correlated (r2 ϭ 0.81) and consistent in magnitude (b ϭ 1.1 Ϯ 0.1) with dissipation, and both are two orders of magnitude smaller than the depth-averaged rate at which the shoaling wave ®eld lost energy to breaking, indicating that breaking-induced turbulence did not penetrate to the measurement depth. Log-pro®le estimates of stress are well correlated with (r2 ϭ 0.75), but larger than (b ϭ 2.3 Ϯ 0.1), covariance estimates of stress, indicating a departure from the Prandtl±von KaÂrmaÂn velocity pro®le. The bottom drag coef®cient was (1.9 Ϯ 0.2) ϫ 10Ϫ3 during unbroken waves and approximately half as large during breaking waves.
1. Introduction and mathematical models of surf zone processes, direct measurements of surf zone turbulence have been rare. Turbulence is believed to have a dominant effect on Separation of turbulence from waves is dif®cult because ¯ows and sediment transport in the surf zone. For ex- turbulent velocity ¯uctuations typically are two to three ample, often it is assumed that a balance between bottom orders of magnitude less energetic than wave motions. stress and cross-shore gradient of wave-induced radia- As a result, previous surf zone turbulence measurements tion stress controls the alongshore ¯ow driven by break- have focused on velocity ¯uctuations at frequencies far ing waves (e.g., Battjes 1988) and that the effect of higher than those of the waves, resulting in estimates bottom stress is transmitted through the water column of dissipation, but not stress (George et al. 1994), or by turbulence with scales much smaller than the water they have have been obtained in laboratory basins (e.g., depth (e.g., Svendsen and Putrevu 1994). Breaking-in- Ogston et al. 1995; Ting and Kirby 1996), where phase- duced turbulence dominates wave energy dissipation in averaging techniques appropriate for monochromatic the surf zone (Thornton and Guza 1986). Many models waves, which are not applicable in random ocean waves, of sediment transport are based on the assumptions that are used to separate waves from turbulence. Most es- near-bottom turbulent shear stress controls the entrain- timates of nearshore turbulence statistics have been ob- ment of sediment from the sea¯oor (e.g., Glenn and tained indirectly from measurements of waves and Grant 1987) and that turbulence maintains suspended winds. For example, depth-integrated dissipation has sediment in the water column (e.g., Fredsoe and Dei- been estimated as the residual in a wave energy balance gaard 1992; Nielsen 1992). (Thornton and Guza 1986; Kaihatu and Kirby 1995; Despite the importance of turbulence in conceptual Elgar et al. 1997; Herbers et al. 2000), and bottom stress has been estimated as the residual in an alongshore mo- * Woods Hole Oceanographic Institution Contribution Number mentum balance (Thornton and Guza 1986; Whitford 10275. and Thornton 1996; Feddersen et al. 1998; Lentz et al. 1999). Corresponding author address: Dr. John H. Trowbridge, WHOI, Here, surf zone measurements of near-bottom veloc- Woods Hole, MA 02543. ity designed to produce direct covariance estimates of E-mail: [email protected] turbulent Reynolds shear stress and inertial-range esti-
᭧ 2001 American Meteorological Society 2404 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31
FIG. 1. Array of near-bottom acoustic Doppler current meters deployed in about 4.5-m water depth approxi- mately 300 m from the shoreline. ADVF and ADVO are SonTek ®eld and ocean probes, respectively.
mates of turbulence dissipation are presented. The tur- alongshore momentum equation, in which wind stress bulence measurements were obtained within a cross- and cross-shore gradient of wave-induced radiation shelf transect of arrays of pressure sensors, which de- stress balance near-bottom turbulent Reynolds shear termine the wave ®eld, and near an anemometer and a stress (e.g., Mei 1983): pro®ling velocimeter, which determine wind forcing and dSxy vertical structure of currents. The analysis focuses on wy ϪϭϪ0 ЈwЈ. (1) dx estimation of terms in four equations that are commonly used to describe surf zone processes. The ®rst is an The second is a hybrid of a balance between dissipation
FIG. 2. Locations of instruments (labeled) and contours of water depth (in m below mean sea level every 0.5 m) for yearday 226. Each compact array (CA6 and CA7) contained six accurately (Ϯ0.5 m) surveyed bottom-mounted pressure sensors (small ®lled circles) and one current meter located about 0.5 m above the bottom near the center of the array (not shown) (Herbers et al. 2000, submitted to J. Phys. Oceanogr.). The shoreline location was centered approximately at cross-shore location ϭ 125 m, with Ӎ Ϯ10 m cross-shore ¯uctuations caused by the 1-m tidal range. The bathymetry seaward of about 3.5-m depth did not change signi®cantly during the experiment. AUGUST 2001 TROWBRIDGE AND ELGAR 2405
and shear production of turbulent kinetic energy, known gust (yearday 236) and 21 November (yearday 324) of to hold in the wall region of turbulent boundary layers 1997 on a sandy Atlantic beach near Duck, North Car- (e.g., Tennekes and Lumley 1972), and a simpli®ed olina, at the U.S. Army Corps of Engineers Field Re- model of surf zone energetics (Battjes 1975), in which search Facility. An array of ®ve upward-looking Sontek dissipation balances the depth-averaged rate at which acoustic Doppler velocimeters (ADVs) was mounted on breaking extracts energy from the shoaling wave ®eld: a low-pro®le frame (Fig. 1). The ADVs measure the three-dimensional velocity vector in a sample volume 1 dFxץ ⑀ ϭϪЈwЈϪ . (2) with a spatial scale of approximately 0.01 m (e.g., Voul- z (h ϩ ) dxץ 0 garis and Trowbridge 1998) and perform well in the surf The third is the Prandtl±von KaÂrmaÂn representation of zone (Elgar et al. 2001). The array included three of the the velocity pro®le in a turbulent boundary layer (e.g., ®eld version of Sontek's acoustic Doppler velocimeter Tennekes and Lumley 1972): (ADVF) and two of the more rugged ocean version 221/2 (ADVO), one of which was ®tted with pressure, tem- ץ ץ uץ 22 00(z ϩ h) ϩϭϪЈwЈ, (3) perature, pitch, and roll sensors, as well as compass. z The ADVOs shared a common logger that sampled theץ zץz ץ[] which has been used in the surf zone to estimate stress two sensors simultaneously at 7 Hz in one burst of 25 from measurements of horizontal velocity (Garcez-Faria minutes each hour. The ADVFs shared a separate com- et al. 1998). The fourth is a quadratic drag law: mon logger that sampled the three sensors simulta- c (u2ϩ 2) 1/2 ϭϪЈwЈ, (4) neously at 25 Hz in one burst of 10 minutes each hour. d 00The ADV frame was approximately 300 m from the commonly used to represent bottom stress in models of shoreline and 300 m north of the Field Research Facility nearshore processes (e.g., Bowen 1969; Longuet-Hig- (FRF: Fig. 2). The ADVs occupied an on±offshore line gins 1970). along the frame's northern edge, upstream of the frame In (1)±(4), x, y, and z are coordinates in a right-handed itself, relative to the predominantly southerly wind- and system with x positive onshore, y positive alongshore, wave-driven ¯ows. The bathymetry near the measure- and z positive upward, where z ϭϪh and z ϭ rep- ment site is approximately uniform in the alongshore resent the sea¯oor and water surface, respectively. The direction on scales of kilometers (Birkemeier et al. 1985; quantity 0 is a ®xed reference density, (u, ,w) is the Lentz et al. 1999), although there is a cross-shore chan- velocity vector, primes denote turbulent ¯uctuations, nel approximately 100 m wide and 1 m deep beneath Ϫ 0ЈwЈ is the alongshore component of the turbulent the pier (Fig. 2). Reynolds shear stress, ⑀ is the turbulence dissipation Other instrumentation included 12 Setra pressure sen- rate, Fx is the cross-shore component of the wave-in- sors making up ``compact arrays'' 6 and 7 (CA6 and duced energy ¯ux, Sxy is the off-diagonal component of CA7), centered approximately 40 m onshore and 80 m the wave-induced radiation stress tensor, wy is the offshore of the ADV frame, respectively (Herbers et al. alongshore component of the wind stress, is the em- 2000, submitted to J. Phys. Oceanogr.) two Marsh± pirical von KaÂrmaÂn constant (approximately equal to McBirney two-axis electromagnetic current meters, lo- 0.40), and c is an empirical drag coef®cient. An overbar d cated near the centers of CA6 and CA7; a Solent sonic represents a time average over a period long compared anemometer, located on a mast at the end of the FRF with the timescales of waves and turbulence, but short pier approximately 20 m above the water surface; and relative to the timescales of variability of the forcing. a Sontek 3-MHz acoustic Doppler pro®ler (ADP), lo- The wind stress, energy ¯ux, radiation stress, and dis- cated approximately 170 m north of the ADV frame sipation are averaged over the same time period. The statistical properties of the ¯ow are assumed to be sta- (Fig. 2). The pressure sensors and two-axis velocimeters tionary over the averaging period and independent of were sampled nearly continuously at 2 Hz, the sonic anemometer was sampled continuously at 21 Hz, and alongshore position. The quantity Ϫ 0ЈwЈ is evaluated at a height above bottom that is small compared with the ADP sampled continuously in 0.25-m range bins at the water depth, but large relative to the scale of the 25 Hz. bottom roughness elements, so that it represents the The ADV array experienced several problems. Esti- stress transmitted to the sea¯oor. The stress-transmitting mates of wave angles indicate that the head of the off- turbulence is assumed to be distinguishable from other shore ADVF began rotating intermittently immediately motions because of its small spatial scales, believed to after deployment. Approximately six days after deploy- be a small fraction of the water depth (e.g., Svendsen ment, the offshore ADVO began malfunctioning, and and Putrevu 1994). intermittently produced invalid data, characterized by intervals with velocities of precisely zero or poor cor- relation with velocities measured by the other ADVO. 2. Methods The ADVO measurements of acoustical backscatter in- a. Measurements tensity are clipped (i.e., do not exceed a ®xed maximum The turbulence measurements, described in detail by value) during strong ¯ows, which precludes use of this Fredericks et al. (2001), were acquired between 25 Au- measurement to infer sediment concentration. Approx- 2406 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31
FIG. 3. (a) Alongshore wind stress, (b) signi®cant wave height, (c) cross-shore gradient of wave- induced energy ¯ux, and (d) cross-shore gradient of wave-induced radiation stress vs time. imately ten days after deployment, the two onshore produced by the ADVOs during intervals when the off- ADVFs were destroyed by large waves and strong cur- shore ADVO produced valid data. To identify valid seg- rents. Divers reported that a scour hole with a depth of ments in the record produced by the offshore ADVO roughly 0.2 m formed beneath the ADV frame within [see Fredericks et al. (2001) for details], the ADVO a few days of deployment and that sand worms colo- records were divided into 100-sample blocks, and the nized the frame during the ®nal two weeks, terminating squared correlation coef®cient r 2 between cross-shore useful data on approximately 27 October (yearday 299). velocities at the two ADVOs was calculated for each block. Data from the intermittently functioning ADVO were assumed valid for blocks with r 2 Ͼ 0.90. Thresh- b. Analysis olds of r2 ϭ 0.90 and r 2 ϭ 0.80 produced nearly iden- The analysis focuses on data produced by the ADVOs tical results. and the two onshore ADVFs during the ®rst six days Separate estimates of the alongshore component of of the measurement period, when these four sensors turbulent Reynolds shear stress were obtained from the functioned well, and on the remainder of the records pair of ADVOs and from the onshore pair of ADVFs, AUGUST 2001 TROWBRIDGE AND ELGAR 2407
by means of the dual-sensor procedure described by ADVO, V, , and were set equal to the magnitude of Trowbridge (1998): the mean horizontal velocity, the standard deviation of the horizontal velocity, and the angle between the prin- ϪЈwЈϭϪ(1/2) cov(⌬, ⌬w), (5) 00 cipal axes of the horizontal velocity ¯uctuations and the where ⌬ and ⌬w are differences between alongshore direction of the mean velocity, respectively. Burst-av-
and vertical velocities measured at two sensors, and cov eraged velocity spectra Puu, etc., were obtained by com- is covariance. Equation (5) is based on the assumption bining spectra from Hanning windowed 100-sample that the stress-carrying turbulence has scales much segments. The noise level in (6) was estimated as the
smaller than scales of other motions, including in par- average of Puu ϩ P over cyclic frequencies greater ticular surface waves, and it requires that the sensor than 3 Hz. Two independent estimates of burst-averaged separation be such that the wave-induced velocities at dissipation were obtained by averaging estimates of 5/3 5/3 the two sensors are nearly identical, while the corre- [Puu() ϩ P() Ϫ noise level] and Pww() over sponding turbulent velocities are uncorrelated. In ad- cyclic frequencies between 1 and 2 Hz, and solving (6) dition to sampling variability, stress estimates based on and (7) for ⑀. The two estimates are well correlated (r 2 (5) have a wave bias, caused by imperfect cancellation ϭ 0.95) and approximately consistent in magnitude (re- of wave-induced velocities by the differencing opera- gression slope b ϭ 0.78 Ϯ 0.1 at 95% con®dence). The tion, and a turbulence bias, caused by nonzero corre- two estimates were then averaged to produce a single lation between turbulent velocity ¯uctuations at the two estimate of dissipation. sensors. Estimates based on the analysis of Trowbridge The estimates of dissipation are based on several con- (1998) indicate that the wave bias in the alongshore siderations. Measured spectra are consistent with the stress is an order of magnitude smaller than the along- model assumptions that P is noise-free and that P shore stress itself in the present case. In contrast, wave ww uu ϩ P has a constant noise level (section 3; see also biases in estimates of the cross-shore stress are similar Elgar et al. 2001). The model ®t is restricted to fre- in magnitude to the cross-shore stress so that the mea- quencies between 1 and 2 Hz because the measurements surements provide only order-of-magnitude estimates of indicate applicability of (6) and (7) in this range (section the cross-shore stress. The turbulence bias is discussed 3) and because estimates of the spatial scales of tur- in section 4. bulence corresponding to this range of frequencies, Estimates of dissipation were obtained from the on- combined with atmospheric velocity spectra (Kaimal et shore ADVO data by applying an inertial-range tur- al. 1968), indicate applicability of the inertial-range bulence model to velocity spectra. The model, which model. Equations (6) and (7) were not applied to ADVF describes frozen inertial-range turbulence advected past data because, in contrast to the ADVO data, 5/3 times a ®xed sensor by a steady horizontal current and random spectral density minus noise for ADVF measurements waves (Lumley and Terray 1983), was specialized (see was not constant at high frequency, instead exhibiting the appendix) for frequencies large compared with the one or more peaks suggesting input of energy by ¯ow dominant wave frequency, unidirectional waves, and disturbances produced by sensor supports and electron- conditions near the sea¯oor, where wave-induced ver- ics, which were not far below the ADVF sample vol- tical velocities have a negligible effect on advection of turbulence. The model representation of the sum of the umes (Fig. 1). High-frequency noise problems similarly spectra of the two horizontal components of velocity is precluded use of (6) and (7) with the offshore ADVO data. 21 Hour-averaged estimates of sea surface elevation var- P () ϩ P () ϭ ␣⑀2/3V 2/3 Ϫ5/3I , iance, energy ¯ux, and radiation stress in the wind wave uu 55 V frequency band (0.04 Ͻ f Ͻ 0.31 Hz) were obtained ϩ constant noise level, (6) from the arrays of bottom-mounted pressure sensors us- ing linear wave theory and a directional-moment-esti- and the model representation of the spectrum of vertical mation technique (Elgar et al. 1994; Herbers et al. 1995; velocity is Herbers et al. 1999). Estimates obtained from the pres- 12 sure arrays are well correlated and consistent in mag- 2/3 2/3 Ϫ5/3 Pww() ϭ ␣⑀ V I , , (7) nitude with independent estimates using linear theory 55 V from collocated measurements of pressure and velocity. ϩϱ where Puu, etc., are spectra, de®ned so that #Ϫϱ Puu() Records of Fx and Sxy at CA6 and CA7 were differ- d ϭ variance; is radian frequency, equal to 2f, enced to determine dFx/dx and dSxy/dx. Estimates of where f is cyclic frequency; ␣ is the empirical Kol- dSxy/dx are sensitive to errors in orientation, so a sim- mogorov constant, approximately equal to 1.5 (Grant et pli®ed procedure was used to estimate and correct for al. 1962); V is the magnitude of the current; 2 is the misalignment. The compact arrays CA6 and CA7 were variance of the wave-induced horizontal velocity; is assumed to be in coordinate systems rotated about the the angle between waves and current; and I(/V, )is z axis by small angles ␥ (6) and ␥ (7) with respect to the de®ned in the appendix. For each burst from the onshore isobath-aligned coordinate system. The radiation stress 2408 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31
Sxy at compact array n in isobath-aligned coordinates is currents (Fig. 4a), particularly during three events cen- then (e.g., Jeffreys 1974) tered on days 246, 263, and 292. During these events, -x had the same sign (Figs. 4b and 4c), sugץ/ ץ and S (n)(SÃÃÃn)(n)([S n)(S n)2] O( ), (8) xyϭ xyϩ ␥ xxϪ yy ϩ ␥ gesting that the ADV site was in the outer part of the à (n) whereSxx , etc., are components of the radiation stress surf zone, because the strength of the alongshore current tensor at array n in the slightly misaligned coordinate increased with increasing distance onshore. The fraction system. It follows from (8) that of valid data from the malfunctioning ADVO was high ÃÃà during the three strong ¯ow events (Fig. 4d). ⌬Sxyϭ⌬S xyϩ (S xxϪ S yy)avg⌬␥ Velocity spectra indicate energetic wave motions at ϩ⌬(SÃÃϪ S )␥ ϩ O(␥2), (9) cyclic frequencies f between approximately 0.05 and xx yy avg 0.5 Hz (Fig. 5a). At higher frequencies, spectra of hor- where ⌬ and subscript avg denote the difference and izontal and vertical velocity are consistent with isotropic the average, respectively, of the values at CA6 and CA7. inertial-range turbulence, as described by (6) and (7), With the assumption that the misalignments are of sim- 5/3 5/3 because f Pww and (12/21) f (Puu ϩ P Ϫ noise lev- ilar magnitude but unequal (i.e., ⌬␥ is order ␥avg), while el) are nearly independent of f and nearly equal to each differences between radiation stresses at CA6 and CA7 other (Fig. 5b). Cospectra of differences between along- are small compared with the radiation stresses them- shore and vertical velocities measured by the two AD- à à à à selves (i.e., ⌬[Sxx Ϫ Syy] K [Sxx Ϫ Syy]avg), (9) approx- VOs (Fig. 5c), which illustrate the frequency content of imates dual-sensor estimates of turbulent Reynolds shear stress ⌬S ϭ⌬SÃÃÃϩ (S Ϫ S ) ⌬␥. (10) based on (5), indicate signi®cant contributions to stress xy xy xx yy avg between approximately 0.01 and 1.0 Hz. The spectra To estimate ⌬␥, ⌬Sxy was assumed equal to zero when and cospectra in Fig. 5 are representative of the ®rst six the signi®cant wave height Hs (four times the standard days of the record, when four ADVs functioned well, deviation of sea surface displacement) at CA7 was less waves were unbroken, and ¯ows were weak. Spectral than 1 m, corresponding to nonbreaking waves. For shapes for periods with breaking waves are similar. Cos- à à these conditions, ⌬Sxy was regressed against Ϫ(Sxx Ϫ pectra of ⌬ and ⌬w during breaking waves were not à Syy)avg, yielding ⌬␥ ϭ 3.1 Ϯ 0.1 degrees at 95% con- computed because of short records caused by intermit- ®dence. Equation (10) with this value of ⌬␥ was then tent malfunctioning of the offshore ADVO (section 2). used to compute ⌬Sxy at all times, including those when Measurements from the beginning of the record, when Hs Ͼ 1 m at CA7. both ADVOs and the onshore pair of ADVFs functioned Hour-averaged estimates of wy were determined from well, permit a comparison of redundant stress estimates. covariances of horizontal and vertical velocities com- The four single-sensor estimates Ϫ 0ЈwЈ ϭϪ 0 cov(, puted from demeaned and detrended 10-min records w) from individual ADVs (Fig. 6a) are contaminated by from the sonic anemometer. These estimates are well waves, and thus are uncorrelated with each other and correlated (r 2 ϭ 0.85) and consistent in magnitude (b vary over a range much larger than the wind and wave ϭ 1.35 Ϯ 0.04) with independent estimates from a near- forcing during this period (Figs. 3a and 3d). In contrast, by mechanical anemometer (Fredericks et al. 2001). stress estimates from the pair of ADVFs and the pair ,(z at approximately the of ADVOs (Fig. 6b) are well correlated (r 2 ϭ 0.62ץ/ ץ z andץ/uץ Estimates of measurement height of the ADVOs were obtained by consistent in magnitude (b ϭ 1.0 Ϯ 0.2 at 95% con®- differencing hour-averaged velocities in ADP bins 1 and dence), and much smaller than the single-sensor esti- 3, 0.63 and 1.13 m above the sea¯oor, respectively. The mates. ADP data begin at yearday 262, so there are no estimates Time series of the terms in (1)±(4) are dominated by of velocity gradient for the ®rst 26 days of the ADV a few brief events when strong winds and waves forced record. energetic ¯ows. Estimates of the left and right sides of (1) (Figs. 7a and 7b) are well correlated (r 2 ϭ 0.63), 3. Results but the turbulent Reynolds stress is approximately half the sum of the wave and wind forcing (b ϭ 0.51 Ϯ During most of the measurement period, winds were 0.03). In (2), dissipation and shear production are well weak, waves were small, and the 4.5-m-depth ADV site correlated (r 2 ϭ 0.70) and consistent in magnitude (b was seaward of the surf zone. During a few events, ϭ 1.1 Ϯ 0.1), even though dissipation was large while however, the alongshore wind stress approached 0.5 Pa shear production was small during an event centered on (Fig. 3a), the signi®cant wave height reached or ex- day 270 (Figs. 8a and 8b). Dissipation and shear pro- ceeded 2 m (Fig. 3b), and shoreward-propagating waves duction are two orders of magnitude smaller than the lost energy ¯ux (Fig. 3c) and produced a cross-shore depth-averaged rate at which the shoaling waves lose gradient of wave-induced radiation stress (Fig. 3d), pre- energy to breaking (Fig. 8c). Estimates of Ϫ 0uЈwЈ ob- sumably because of breaking. The strong winds and tained from an expression analogous to (5) indicate that -z, which has been neץ/uЈwЈ uץbreaking waves forced strong alongshore currents (Fig. the production term Ϫ 4b), in some cases accompanied by strong cross-shore glected in (2), is an order of magnitude smaller than AUGUST 2001 TROWBRIDGE AND ELGAR 2409
FIG. 4. (a) Cross-shore velocity (negative is offshore-directed ¯ow), (b) alongshore velocity (positive is ¯ow toward the south), (c) cross-shore gradient of alongshore velocity, and (d) percent of valid data from the malfunctioning ADVO vs time. z. Estimates of the terms on the right and left the same isobath as the ADV frame (Hanes and Alymovץ/ ЈwЈ ץϪ sides of (3) (Figs. 9a and 9b) are well correlated (r2 ϭ 2000, submitted to J. Geophys. Res.) indicate short rip- 0.75), but log-pro®le estimates of alongshore stress are ples, with heights up to 0.02 m and lengths up to 0.25 approximately twice as large as estimates based on (5) m, and long ripples, with heights up to 0.06 m and (b ϭ 2.2 Ϯ 0.1), indicating a departure from the Prandtl± lengths up to 2 m. The heights and inverse wavenumbers von KaÂrmaÂn logarithmic velocity pro®le. A quadratic of these bedforms are smaller than the ADV measure- drag law with a best-®t drag coef®cient of cd ϭ (0.71 ment heights, indicating that local ¯ow distortions by Ϯ 0.03) ϫ 10Ϫ3 (Figs. 9a and 9c) is well correlated with bedforms had a negligible effect on the ADV measure- stress estimates based on (5) (r2 ϭ 0.80). Corresponding ments. estimates during periods with unbroken waves and weak The results in Figs. 7±9 did not depend sensitively ¯ows ( Ͻ 0.2 m sϪ1) are weakly correlated (r 2 ϭ 0.39) on the rate at which the malfunctioning ADVO produced and indicate a larger drag coef®cient of cd ϭ (1.9 Ϯ valid data (Fig. 4d), because the correlation coef®cients 0.2) ϫ 10Ϫ3. and regression slopes are not systematic functions of Acoustical measurements of bedform geometry on percent of valid data collected. 2410 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31
FIG. 5. Energy density spectra from (a) velocity time series and (b) modi®ed according to (6) and (7). Solid
curves are Puu ϩ P and dashed curves are Pww. (c) Integral of cospectrum of velocity differences vs frequency. Data are from 25-min-long ADVO records obtained at 12:00 (EST) on yearday 236. The burst-averaged cross- shore and alongshore currents were Ϫ0.02 and 0.10 m sϪ1, respectively, and the signi®cant wave height was 1.0 m.
4. Discussion with Fig. 3d) and because of the consistency of wind stress estimates from sonic and mechanical anemome- a. Momentum balance ters (section 2). Inaccurate estimates of dSxy/dx might The factor of 2 difference between the left and right result from use of linear wave theory to relate bottom sides of the momentum balance (1) (Fig. 7) might be pressure to Sxy or from the approximate procedure used caused by inaccurate estimates of dSxy/dx or Ϫ 0ЈЈ. to resolve uncertainties in sensor orientation (section 2).
Inaccurate estimation of wy is unlikely to be responsible Estimates of Ϫ 0ЈwЈ based on (5) are too small, pos- because wy was not a dominant term (compare Fig. 3a sibly explaining the observed differences between the AUGUST 2001 TROWBRIDGE AND ELGAR 2411
FIG. 6. Estimates of near-bottom stress using (a) single and (b) dual sensors vs time. (b) Estimates from the pair of ADVOs and ADVFs are shown as solid and dashed curves, respectively.
FIG. 7. (a) Sum of wind and wave forcing and (b) near-bottom turbulent Reynolds shear stress vs time. 2412 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31
FIG. 8. (a) Shear production of turbulent kinetic energy, (b) dissipation of turbulent kinetic energy, and (c) depth-averaged rate of energy extraction from the shoaling wave ®eld vs time. The vertical scale in (c) is different than in (a) and (b).
left and right sides of (1) if the distance between the events that dominate Fig. 7, because of advection of two ADVOs was smaller than the correlation scale of turbulence by large wave-induced velocities. the turbulence. To assess this effect, stress estimates The differences between the left and right sides of based on (5) were computed using all pairs from the (1) might also be caused by omitted terms in the along- four ADVs that functioned during the ®rst-six days of shore momentum balance, including divergence of a the record, providing spatial separations ranging from shoreward ¯ux of alongshore momentum by shear 0.6 to 2.3 m. No signi®cant increase in magnitude of waves (Oltman-Shay et al. 1989; Bowen and Holman stress estimates occurs with increasing sensor separation 1989), an alongshore pressure gradient, or advective during this period. However, turbulence correlation acceleration of the hour-averaged ¯ow. Velocity ¯uc- scales might have increased during the more energetic tuations at frequencies below the wind wave band con- AUGUST 2001 TROWBRIDGE AND ELGAR 2413
FIG. 9. (a) Covariance, (b) log pro®le, and (c) drag law estimates of turbulent Reynolds shear stress vs time. tributed negligibly to shoreward ¯uxes of alongshore curred at smaller scales, for example, because of along- momentum at the ADV frame, indicating that shear shore variability in wave statistics produced by along- waves had a small effect on the alongshore momentum shore-nonuniform bathymetry. A typical advective term
-y). To produce dynamiץ/ ץbalance. The pressure term neglected in (1) is neglected in (1) is O( 0h y), where g is gravitational acceleration. To cally signi®cant values of O(1 Pa) at h ϭ 4.5 m duringץ/ץO( 0gh y must be O(2ץ/ ץ ,produce dynamically signi®cant values of O(1 Pa) at h strong ¯ows with ϭ O(1) m sϪ1 (y must be ϫ 10Ϫ4)sϪ1, corresponding to differences ⌬ of O(0.04ץ/ץ ϭ 4.5 m, the alongshore surface slope O(2 ϫ 10Ϫ5). Measurements suf®cient to resolve slopes msϪ1 at an alongshore separation of 200 m. Velocity of this magnitude were not obtained. Previous pressure measurements from electromagnetic sensors 100 m measurements at the same site at alongshore separations north and south of the ADV frame indicate ⌬ exceed- of tens of kilometers (Lentz et al. 1999) indicate that ing this value, so that advective acceleration possibly peak alongshore surface slopes were an order of mag- is signi®cant. A quantitative assessment, dif®cult be- nitude smaller than 2 ϫ 10Ϫ5 (S. Lentz 2000, personal cause of measurement uncertainties, is beyond the scope communication), but larger gradients might have oc- of this study. 2414 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 b. Energetics, velocity pro®le, and drag laws events. Concentration measurements suf®cient to test this computation were not obtained at the ADV frame. Figure 8 indicates that a balance between shear pro- A model of the vertical distribution of suspended sed- duction and dissipation controlled the energetics of near- iment in turbulent shear ¯ows [Eq. (15) of Taylor and bottom turbulence at the ADV frame and that breaking- Dyer 1977] suggests that concentrations of this mag- induced dissipation associated with the shoreward decrease nitude are unlikely because rapid decay ofc with z of wave-induced energy ¯ux did not penetrate to the depth requires c ϭ O(1) (pure sediment and no water) at z Ӎ of the ADVs. These observations are similar to results in 0.2 m, which clearly is implausible, to produce c ϭ O(4 wall-bounded turbulent shear ¯ows in which shear pro- ϫ 10Ϫ5)atz ϩ h ϭ 0.85 m. However, this calculation duction balances dissipation near the wall (e.g., Tennekes is sensitive to the poorly constrained turbulent Prandtl and Lumley 1972). The observations are inconsistent with number for sediments, and the model of sediment dis- the simplest model of surf zone turbulence (Battjes 1975), tribution has not been tested in the surf zone. in which breaking-induced dissipation is assumed to be The drag coef®cient estimated from (4) for unbroken distributed uniformly over the water depth, and they also Ϫ3 waves [cd ϭ (1.9 Ϯ 0.2) ϫ 10 ] is consistent with a are inconsistent with empirical results for locally forced previous nearshore estimate seaward of the surf zone waves in deep water (e.g., Terray et al. 1996), which in- (Feddersen et al. 1998). However, the smaller drag co- dicate penetration of breaking-induced turbulence to ef®cient determined for breaking waves [cd ϭ (0.71 Ϯ depths of a few times the signi®cant wave height, which 0.03) ϫ 10Ϫ3] is smaller than other estimates obtained exceeds the water depth in the surf zone. However, the in the surf zone (Thornton and Guza 1986; Whitford observations are qualitatively consistent with Svendsen's and Thornton 1996) and contradicts studies that pro- (1987) analysis of laboratory measurements, which indi- duced larger estimates of drag coef®cients under break- cates that only a small fraction (2%±5%) of the dissipation ing waves than under nonbreaking waves (Feddersen et associated with breaking waves in the surf zone occurs al. 1998; Lentz et al. 1999). A reduced cd under breaking below trough level. waves is plausible because large wave-induced veloci- Possible explanations of the factor of 2 differences be- ties destroyed bedforms (Hanes and Alymov 2000, sub- tween the left and right sides of (3) (Fig. 9), indicating mitted to J. Geophys. Res.), and breaking-induced tur- departure from the Prandtl±von KaÂrmaÂn velocity pro®le, bulence did not penetrate to the measurement depth (Fig. include inaccurate estimates of Ϫ0ЈwЈ, discussed in sec- 8). In addition, stable strati®cation by suspended sedi- tion 4a, and stable strati®cation owing to temperature, sa- ments may have occurred near the sea¯oor, increasing z and thus for a ®xed bottom stress (e.g., Taylorץ/ץ linity, or suspended sediment (e.g., Monin and Yaglom 1971). When combined with ADP estimates of velocity and Dyer 1977). Alternatively, the small cd observed gradient, estimates of buoyancy frequency N obtained here under breaking waves could result from inaccurate from daily measurements of conductivity, temperature, and estimates of 0ЈwЈ (section 4a). If the true 0ЈwЈ under depth (CTD) at the end of the pier (Fig. 2) yield gradient breaking waves is twice as large as indicated by esti- z)2 less than roughlyץ/ץ)/Richardson numbers Ri ϭ N2 mates based on (5), cd would double and the momentum 0.01 during the strong events that dominate in Fig. 9, balance (1) would close. However, a doubling of indicating a negligible effect on the structure of the time- Ϫ 0ЈwЈ would cause a factor of 2 more shear produc- averaged velocity pro®le (e.g., Businger et al. 1971). A tion (Fig. 8a) than dissipation (Fig. 8b). commonly used model of strati®cation by suspended sed- iment (e.g., Glenn and Grant 1987) is 5. Summary and conclusions z ϩ hץ ((z ϩ h ϭ 1 ϩ  , (11) Measurements of turbulence, waves, currents, and u* zL -wind permit examination of 1) an approximate along ץΗΗ
1/2 shore momentum balance between wind stress, cross- where u* ϭ ||ЈwЈ is shear velocity,  is an empirical shore gradient of wave-induced radiation stress, and constant approximately equal to 5 (Hogstrom 1988), and near-bottom turbulent Reynolds stress; 2) an approxi- L is the Monin±Obukhov length, de®ned for strati®ca- mate turbulence energy balance, in which dissipation tion by sediments in steady state by balances the sum of shear production and the depth- u3 averaged rate at which breaking extracts energy from L ϭ * , (12) the shoaling wave ®eld; 3) the Prandtl±von KaÂrmaÂn rep- g(s Ϫ 1)wc s resentation of the velocity pro®le in a wall-bounded where s and ws are speci®c gravity and settling velocity shear ¯ow; and 4) a quadratic drag law. The results are of the sediments, respectively, andc is time-averaged dominated by three events when large winds and waves sediment concentration by volume. With s ϭ 2.65 and forced strong ¯ows, and the sensors were in the outer Ϫ1 ws ϭ 0.02 m s , typical of beach sands at Duck (Stauble part of the surf zone. Estimates of near-bottom turbulent 1992), (11) and (12) imply thatc must be O(4 ϫ 10Ϫ5) Reynolds shear stress are smaller than the sum of wind at the ADVO measurement height z ϩ h Ӎ 0.85 m to and wave forcing by a factor of approximately 2. In the balance the left and right sides of (3) during strong turbulent energy balance shear production balances dis- AUGUST 2001 TROWBRIDGE AND ELGAR 2415
sipation, and both are two orders of magnitude smaller Scripps Institution of Oceanography, deployed, main- than the depth-averaged rate of energy loss by the shoal- tained, and recovered the instruments in harsh surf zone ing wave ®eld, suggesting that breaking-induced tur- conditions. R. T. Guza, T. H. C. Herbers, B. Rauben- bulence did not extend to the measurement depth. Near- heimer, and W. C. O'Reilly helped design and manage bottom velocity gradients are larger by approximately the ®eld experiment and provided high-quality pro- 50% than indicated by the Prandtl±von KaÂrmaÂn loga- cessed measurements of wave statistics and currents. P. rithmic velocity law, given estimates of bottom stress. Howd planned, obtained, and processed the ADP mea- The bottom drag coef®cient for unbroken waves is sim- surements. J. Fredericks, G. Voulgaris, and A. Williams ilar to a previous nearshore estimate outside of the surf obtained the ADV measurements, and J. Fredericks pro- zone, but the drag coef®cient for breaking waves is cessed the ADV data. J. Edson obtained and processed smaller than values from previous results based on in- the wind measurements. S. Lentz and F. Feddersen pro- direct estimates of bottom stress. vided useful comments on the manuscript. Funding was The reasons for the imbalances in the momentum equation and the Prandtl±von KaÂrmaÂn velocity pro®le provided by the Of®ce of Naval Research, the National are not known. However, turbulent Reynolds shear Science Foundation, the Mellon Foundation, and the stress was central to these balances, and the technique Woods Hole Oceanographic Institution's Rinehart for estimation of stress is based on the explicit as- Coastal Research Center. sumption that the spatial scales of the stress-carrying near-bottom motions are smaller than the height above APPENDIX bottom. The present measurements are not suf®cient to test this assumption. Determining the spatial scales of Model of High-Frequency Turbulent Velocity the stress-carrying motions in wave-driven nearshore Spectra ¯ows is an important question for future research. Other unresolved questions include the adequacy of linear the- The purpose of the appendix is to derive the models ory to estimate the cross-shore gradient of wave-induced (6) and (7) for the high-frequency spectra of near-bottom radiation stress in breaking waves, the role of nonlinear velocities measured by a point sensor past which in- advective terms and pressure gradient in the alongshore ertial-range turbulence is advected by a mean current momentum balance, the effect of bedforms and sus- and wave-induced velocities. It is convenient to use in- pended sediments on nearshore ¯ows, and the vertical dicial notation, in which the coordinates are (x , x , x ), structure of dissipation in the surf zone. 1 2 3 where x3 is vertical, and the corresponding turbulent
Acknowledgments. Staff from the Field Research Fa- velocity vector is(u123Ј, uЈ, uЈ). The starting point is Eq. cility, Duck, NC, and the Center for Coastal Studies, (A11) of Lumley and Terray (1983):
11ϩϱ P () ϭ d exp(Ϫi) (k) exp(ik ´ U) exp Ϫ (k 22 ϩ k 22 ϩ k 22 ) 2d 3k. (A1) ij 2 ͵͵ij 2 11 22 33 Ϫϱ k []
2/3 Ϫ5/3 Here is radian frequency, Pij is the measured turbu- E(k) ϭ ␣⑀ k (A3) lence spectrum, de®ned so that #ϩϱ P () d ϭ uЈuЈ, Ϫϱ ij ij (e.g., Batchelor 1967). Here E(k) is the isotropic energy is a dummy variable of integration, ij is the spectrum tensor of the turbulence (e.g., Batchelor 1967), k is the spectrum, k is the magnitude of k, ␦ij is the Kronecker wavenumber vector, U is the mean velocity vector, and delta, ␣ is the empirical Kolmogorov constant, and ⑀ 22 2 is the dissipation rate. In addition, attention is restricted 12, , and 3 are the nonzero elements of the covari- ance tensor of the wave-induced velocity vector, which to cases with 3 ϭ 0 and U 3 ϭ 0, which is appropriate has been diagonalized. Equation (A1) is an asymptotic for near-bottom applications, and to cases with waves result for frequencies large in comparison with the dom- that are narrowbanded in direction so that 2 ϭ 0, where, without loss of generality, x is de®ned to be inant wave frequency. Of particular interest are P11() 1 the direction of wave propagation. Substitution of (A2) ϩ P 22() and P33(), which are invariant under coor- and (A3) into (A1), and use of standard results to eval- dinate rotations about the x3 axis. Several idealizations simplify the analysis. The tur- uate the integral (e.g., Gradshteyn and Ryzhik 1965), bulence is assumed to be homogeneous and isotropic: yields E(k) kk ϭ ␦ Ϫ ij, (A2) ij4kk22 ij ␣⑀2/3d 3k kk P ϭ ␦ Ϫ ij ij 3/2 11/3 2 2 1/2ij 2 2(2) ͵ k (k 11 ) k and to have a Kolmogorov spectrum: k 2416 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31
[kVcos() ϩ kVsin() Ϫ ]2 [kVcos() ϩ kVsin() Ϫ ]2 ϫ exp Ϫ 12 , (A4) ϫ exp Ϫ 12 . (A5) 22 22 Ά·2k 11 Ά·2k 11 where V cos() ϭ U1 and V sin() ϭ U 2, with V Ն 0. For the contraction Pii, the k3 integral similarly can be For the special case i ϭ j ϭ 3, the k3 integral in (A4) evaluated in terms of gamma functions, and it follows can be evaluated in terms of gamma functions (e.g., that Abramowitz and Stegun 1972), leading to 7 P () ϩ P () ϭ P () (A6) 11 224 33 2͙2 ⌫(1/3) 2/3 so that, once P33 has been determined, P11 ϩ P 22 is also P33 ϭ ␣⑀ 55 ⌫(5/6) known. Therefore attention is restricted to P33. Two sets of transformations of (A5) are necessary.
ϩϱ ϩϱ The ®rst is substitution of y ϭ k 2/k1, with special at- dk12 dk → ϫ tention to the behavior near k1 ϭ 0 and near k1 Ϯϱ, 224/3221/2 ͵͵(k 12ϩ k )(k 11 ) Ϫϱ Ϫϱ followed by substitution of x ϭ /(k1V), which leads to
2͙2 ⌫(1/3) ␣⑀2/3VV 5/3ϩϱ ϩϱ 2[cos() ϩ y sin() Ϫ x]( 2x 2) 1/3 dx dy P ϭ exp Ϫ . (A7) 33 5/3 2 2 4/3 55 ⌫(5/6) 11͵͵ 2 (1 ϩ y ) Ϫϱ Ϫϱ Ά·
The second is substitution in (A7) of yЈϭy Ϫ x/sin(), followed by substitution of xЈϭx/sin(), which yields
2͙2 ⌫(1/3) ␣⑀2/3V 5/3sin 5/3 () ϩϱ ϩϱ V 2[cos() ϩ y sin()] 2 (x 2) 1/3 dx dy P ϭ exp Ϫ , (A8) 33 5/3 2 2 4/3 55 ⌫(5/6) 11͵͵ 2 [1 ϩ (x ϩ y)] Ϫϱ Ϫϱ Ά·
where primes have been omitted. 4 ⌫(1/3) ␣⑀2/3V 2/3sin 2/3 () lim P ϭ 33 5/3 Equation (A8) can be reduced to a single integral by → 1 0 55͙ ⌫(5/6) considering two methods of obtaining the pure current limit, → 0. In this limit, P does not depend on , ϩϱ (x21/3) dx 1 33 ϫ . (A10) so can be set to zero in (A7). The y integral in (A7) ͵ {1 ϩ [x Ϫ cot()]24/3 } then can be evaluated in terms of gamma functions. In Ϫϱ the x integral, the exponential function behaves like a Equations (A9) and (A10) must be equal, so that
Dirac delta function centered at x ϭ 1 in the limit 1 → 0, so x can be set to unity except in the exponential, ϩϱ (x21/3) dx 3͙⌫(5/6) ϭ . (A11) which permits straightforward evaluation. The result of ͵ {1 ϩ [x Ϫ cot()]2 } 4/3⌫(1/3) sin 2/3 () these operations is Ϫϱ 12 Substitution of y ϭ cot() gives 2/3 2/3 Ϫ5/3 lim P33 ϭ ␣⑀ V , (A9) → 55 1 0 ϩϱ (x21/3) dx ⌫(5/6) ϭ 3͙ (1 ϩ y21/3), which is consistent with results for steady ¯ows (Batch- ͵ [1 ϩ (x ϩ y)]24/3 ⌫(1/3) → Ϫϱ elor 1967). Alternatively, as 1 0 in (A8), the ex- (A12) ponential acts like a delta function centered at y ϭ Ϫcot() so that y can be replaced by Ϫcot() in all terms which is an evaluation of the x integral in (A8). Sub- except the exponential. The y integral can then be eval- stitution of (A12) into (A8) and use of a change of uated, yielding variables leads to (7), where
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