Using a Broadband ADCP in a Tidal Channel. Part II: Turbulence

Transcription

1 1568 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 Using a Broadband ADCP in a Tidal Channel. Part II: Turbulence YOUYU LU* AND ROLF G. LUECK School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada (Manuscript received 16 July 1997, in final form 1 August 1998) ABSTRACT A four-transducer, 600-kHz, broadband acoustic Dopple current profiler (ADCP) was rigidly mounted to the bottom of a fully turbulent tidal channel with peak flows of 1 m s 1. Rapid samples of velocity data are used to estimate various parameters of turbulence with the covariance technique. The questions of bias and error sources, statistical uncertainty, and spectra are addressed. Estimates of the Reynolds stress are biased by the misalignment of the instrument axis with respect to vertical. This bias can be eliminated by a fifth transducer directed along the instrument axis. The estimates of turbulent kinetic energy (TKE) density have a systematic bias of m s due to Doppler noise, and the relative statistical uncertainty of the 0-min averages is usually less than 0% 95% confidence. The bias in the Reynolds stress due to Doppler noise is less than m. The band of zero significance is never less than m s due to Doppler noise, and this band increases with increasing TKE density. Velocity fluctuations with periods longer than 0 min contribute little to either the stress or the TKE density. The rate of production of TKE density and the vertical eddy viscosity are derived and in agreement with expectations for a tidal channel. 1. Introduction Turbulence is ubiquitous in coastal waters and plays important roles in the transfer of momentum and the mixing of water properties. Turbulence parameters and their statistics vary both in space and time. Conventionally, turbulent velocity fluctuations in the ocean are measured by two types of instruments: shear probes (e.g., Osborn and Crawford 1980) and point current meters (e.g., Gross and Nowell 1983, 1985; McPhee 1994). Turbulent measurement with acoustic Doppler current profilers (ADCPs) is a relatively new technique and possesses the advantages of remote sensing. Several approaches have been developed to estimate turbulent quantities with an ADCP by taking advantage of the rapid sampling of this instrument. Gargett (1988, 1994) reported a large-eddy approach that provides estimates of the turbulent kinetic energy (TKE) dissipation rate by using a narrowband ADCP with a true vertical beam. Van Haren et al. (1994) applied a directcorrelation approach to estimate the velocity covariance (Reynolds stress) in the internal wave frequency band by using a moored narrowband ADCP with a standard beam configuration. The third method is the variance technique that derives Reynolds stress and TKE density from the autocovariances of along-beam velocities. This method was first explored by Lohrmann et al. (1990) with a pulse-to-pulse coherent sonar and was recently applied by Stacey (1996) with a broadband ADCP. The advantages of using a broadband ADCP are that the noise level in velocity profiling is lower than that of a narrowband unit and the profiling range is much larger than that of a pulse-to-pulse coherent sonar. In this paper, we address the issue of obtaining reliable estimates of turbulent quantities with the variance technique and, thereby, extend the work on firstorder moments presented in the companion paper (Lu and Lueck 1999, hereafter Part I). Section provides a brief introduction of the computation algorithm. In sections 3 5, we present analyses of bias and error sources, statistical uncertainty of the estimates and its causes, as well as spectra. Section 6 addresses the measurement results of Reynolds stresses, the TKE density, the TKE production rate, and viscosity coefficient. Section 7 offers a summary. * Current affiliation: Department of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada. Corresponding author address: Dr. Rolf G. Lueck, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 1700, 3800 Finnerty Road, Victoria, BC V8W Y, Canada. rlueck@unic.ca. Deriving turbulent products from variances of along-beam velocities The transducer geometry and beam orientation of the ADCP, manufactured by RD Instruments (RDI), is illustrated in Fig. 1 of Part I. Each of the four beams is inclined by 30 from the axis of the instrument, 1999 American Meteorological Society

2 NOVEMBER 1999 LU AND LUECK 1569 which is nominally vertical. The planes containing each of the two opposing beam pairs are orthogonal to each other. The heading angle of the instrument is 1, and the pitch and roll angles are and 3, respectively. As shown in (1) of Part I, the measured velocity along the ith (i 1,...,4)beam, b i, is the sum of the horizontal and vertical components, u i, i, w i, at the position of b i. Each of these velocities is decomposed into a mean part and a turbulent fluctuating part, that is, b i b i bi, u i u i ui, etc. To derive the mean velocity vector, we need to assume that the mean flow is statistically homogeneous in horizontal space over the distances separating the beams, that is, u 1 u, etc. (Part I). To derive the Reynolds stress and the TKE density, we must further assume that all the second-order moments of turbulent velocity fluctuations are horizontally homo- geneous, that is, u1 u, uw 1 1 uw, etc. There are at least two tests of the assumption of horizontal homogeneity of the mean flow (Part I), but we are unaware of tests for the second moments. Ninetyfive percent of the 0-min mean velocity estimates pass both tests when the horizontal averaging length (ensemble time interval multiplied by the mean speed) is 55 times longer than the beam separation. We anticipate that at least a similar amount of averaging is required for the second moments to be consistent with the assumption of horizontal homogeneity. From the variances of along-beam velocities, that is,, two components of the Reynolds stress can be deb i rived, b b1 uw 3(u w ) u (1) sin and b4 b3 w ( w ) 3u, () sin and a quantity S (b b b b ) 4 sin (3uw w) 1. (3) tan The quantity S is related to the TKE density q / (u w )/ by 1 S 1 q /, (4) 1 tan where w /(u ) is a measure of turbulence anisotropy. Apart from the neglected second- and higher-order terms in and 3, (1) and () also contain undetermined terms u w, w, and u. Neglecting these terms brings a bias to the estimates of stress, which is examined in section 3a. The value of the anisotropy FIG. 1. Examples of the estimated stress (heavy dashed lines) and the distribution (histogram) of the zero covariance obtained by computing the covariance [Eq. (9)] 1000 times for random lags larger than 30 s. The solid vertical lines denote the 95% significance levels for zero covariance. (a) and (b) The along- and cross-channel stress, respectively, for a 0-min interval during the strong ebb; (c) and (d) the same components for a 0-min interval during the weak flood. ratio ranges from zero for extremely anisotropic turbulence to 0.5 for isotropic turbulence. Hence S (1.7)q / (for 30). In this analysis we use 0. (hence, S 1.8q /), which is the value estimated by Stacey (1996) from measurements in an unstratified tidal channel. The anisotropy ratio cannot be determined from an ADCP with only four beams. The addition of a fifth beam directed along the axis of the instrument would provide a measure of w and its variance. The variance of each component of horizontal velocity can then be determined and, thus, also the TKE, q /, and two of the three bias terms in the estimate of stress. Calculating the Reynolds stress and the quantity S with (1) (3) requires that the tilt angles and 3,as well as the heading angle 1, are time invariant. In other words, the instrument needs to be rigidly mounted. For a nonrigidly mounted deployment, the decomposition of along-beam velocity b i is meaningless because its variations can be caused by changes in the three angles. To illustrate the problem with tilt eddy correlation in estimating turbulent products, we examine the case when only the roll angle 3 changes with time, the heading angle is constant, and the pitch angle 0. By assuming that u1 3 u 3 u 3, etc., (1) changes to b b1 uw (uu3 ww 3) sin (u 3 w 3), (5) where the terms with magnitudes of O( 3 ) have been dropped. The term that is most likely to introduce a

3 1570 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 large error is u u 3, that is, the product of eddy tilt correlation and the mean horizontal velocity. The product of a rms tilt-angle fluctuation and a 0.1 m s 1 rms turbulent velocity fluctuation equals m s 1 if the two fluctuations correlate perfectly. In a mean current of 1 m s 1, the term u u 3 will cause a bias of m s. Even if the tilt eddy correlation is only 0.1, the bias in the Reynolds stress is still m s and too large for most practical flows. Turbulence can be measured with a nonrigidly mounted unit when the dominant energy-containing eddies are much larger than the beam separation. In that case, the velocity field is horizontally homogeneous and a pingby-ping transformation gives the true velocity vector. The turbulent velocity fluctuations can be obtained by applying the Reynolds decomposition, and the products of turbulent fluctuations can then be calculated. This direct-correlation approach was adopted by van Haren et al. (1994) to measure the Reynolds stress in the internal wave band over a sloping bottom on the continental shelf. 3. Error analyses In this and the two subsequent sections, we present detailed analyses of 1-day-long, four-ping averaged data. A sample of this data is shown in Fig. 4 in Part I. a. Bias in turbulence estimates The measurement of beam velocity contains an uncertainty, referred to as Doppler noise. By denoting the measured value of along-beam velocity as b* i, its true signal as b i, and noise as N i, one has b* i b i N i. (6) For four-ping averaged ensembles and 1-m vertical cells, the noise standard deviation of our unit was ms 1 up to a profiling range of 40 m (Part I). We use this value in the following analyses, although the noise level may vary slightly with changes in flow and water properties (e.g., the size and abundance of sound scatters in the water). The uncertainty of the estimated noise standard deviation is approximately 15%, and no difference was detected among the four beams at this level. Doppler noise systematically biases the estimate of S to a higher level S*, that is, 4 1 S* S var(n i). (7) 4 sin i1 For 30 this bias amounts to four times the variance of N i and equals m s for four-ping averaged data. In the remainder of this paper, a bias of m s is removed from the estimates of S, and this value corresponds to the lower bound of our noise estimate. Doppler noise may bias the Reynolds stress estimates if var(n i ) is not identical for opposing beams; that is, FIG.. The variation of (a) along- and (b) cross-channel components of the local friction velocities u * s and u * n (solid lines with circles) and the 95% significance levels (shaded areas) at z 3.6 mab. * var(n ) var(n 1) uw uw, (8) sin where we have used (1) and ignored the tilt-angle bias. It is difficult to estimate the noise variance and our values are uncertain at a level of 35%. Using (8) a pessimistic estimate of the bias is 0.35var(N)/ m s. Another upper limit to the noise-induced bias is provided by the data taken during the weak flood (day ). Because the current, its shear, and the TKE were all small, one may surmise that the stress was also small. The estimated stress is small (Fig. ). The along-channel component is marginally significant and reaches m s, while the cross-channel component is insignificant and fluctuates around zero with an amplitude of m s. The cross-channel shear is completely negligible (Part I, Fig. 11) and, thus, there is no indication of a noise-induced bias in the cross-channel stress (if we assume that no shear means no stress). It is unlikely that the along-channel stress was less than zero. The along-channel shear and stress are both positive and the flux of momentum is directed downgradient. A negative stress would produce an upgradient flux. Thus, if the noise-induced bias is positive, then it was less than m s. A negative bias is not so easily constrained; however, a bias of m s would make the stress during the weak flood highly significant and render the stress during strong ebbs as only marginally significant. This too is unlikely. Finally, a noise-induced bias must be spectrally white, and this would produce a nonzero spectrum at very high and low wavenumbers where the real spectrum would tend to vanish. The signature of a noise-induced bias is not evident in the spectra (e.g., Fig. 11). Thus, the noiseinduced bias is less than (and possibly, much less than) m s.

4 NOVEMBER 1999 LU AND LUECK 1571 In section, we have noticed that neglecting the terms associated with tilt angles introduces another type of bias to the Reynolds stress estimates. From (1) and (), this bias is proportional to the magnitudes of the tilt angles and the undetermined terms u w, w, and u. If turbulence tends to be isotropic, u w and w are small. For anisotropic turbulence, 3 (u w ) O( 3 q /). The magnitude of q / is approximately five times larger than the Reynolds stress (e.g., Gross and Nowell 1983). Hence, at most, the bias due to 3 (u w ) amounts to 17% of the magnitude of uw for 3.0, as in this experiment. Similarly, if the magnitude of u is not significantly larger than that of uw or w, then neglecting u and 3 u will not cause a significant bias. However, for pitch and roll angles exceeding a few degrees and for strongly anisotropic turbulence, this will be the largest bias in the estimates of Reynolds stress. b. Statistical uncertainties In this analysis, we choose 0 min as the ensemble averaging time to calculate the turbulent quantities. The low- and high-frequency velocity components are separated by a low-pass fourth-order Butterworth filter at zero phase. Variances of beam velocity fluctuations are then calculated and smoothed with the same filter and averaged over 0-min intervals to give estimates of Reynolds stress and the quantity S. To test if a stress estimate is statistically different from zero, we apply the nonparametric approach outlined by Fleury and Lueck (1994) and Lueck and Wolk (1999). Differences of the beam velocity variances in (1) and () are rearranged into a covariance form, for example, b b1 ( b b1)( b b1 ). (9) Hence, the Reynolds stress uw is proportional to the covariance of the two time series ( b b1) and ( b b 1 ) at zero lag. The decorrelation timescales of the beam velocities are typically 15 s during strong flows and 6 s during weak flows (see Fig. 5 in Part I). If we shift one time series, say ( b b 1), by a lag (or lead) larger than 30 s, then its statistical nature is unchanged, but it will have no correlation with the other time series, ( b b1), on average. A histogram of this zero co- variance is obtained by randomly choosing the lag many times (e.g., 1000) and then computing the covariances. If the zero-lag covariance, that is, the stress is outside of the 95% confidence levels of the zero covariance, then this estimate is accepted as statistically different from zero. Figure 1 illustrates the examples of applying this method to two 0-min intervals of data, one during the strong ebb around day 3.91 and one during the weak flood around day For each interval we calculate the 95% significance levels (denoted as 95 ) for both the along- and cross-channel components of the stress separately. During the strong ebb, the along-chan- TABLE 1. Estimates of Reynolds stress and S, the 95% significance level ( 95 ) for stress, and two estimates of the 95% confidence intervals [( 95 ) 1 and ( 95 ) ] for both stress and S (all quantities are in units of 10 4 m s ). The estimates are for two 0-min intervals at z 3.6 mab. Strong ebb Weak flood (uw) s (vw) n S (uw) s (vw) n S Mean 95 ( 95 ) 1 ( 95 ) nel stress is significantly different from zero, and the cross-channel stress is marginally different from zero, at the 95% significance levels. During the weak flood, the along-channel stress is comparable to the size of the 95% significance levels (and marginally significant), whereas the cross-channel stress is not different from zero. We have also applied a bootstrap method (Efron and Tibshirani 1993, chapter 13) to determine the statistical uncertainties for both the stress and S estimates. For each 0-min interval, time sequences of the vectors ( b1, b, b3, b4) are randomly resampled at identical time indices to construct a new matrix of beam velocity fluctuations. In the first calculation, the new matrix has the same length as the original data, which is tantamount to assuming that all original samples are independent. In the second calculation, the length of the resampled matrix is reduced by a factor corresponding to the decorrelation time of the original data: 15 s for the interval during the strong ebb and 6 s for the interval during the weak flood. The stress and S for the resampled data are calculated according to Eqs. (1) (3). By repeating the above calculations 1000 times, distributions of the stress and S are constructed. The 95% confidence intervals for the stress and the quantity S, denoted as ( 95 ) 1 for the first calculation and ( 95 ) for the second calculation, are derived from the distributions. Table 1 compares the 95% zero-stress significance levels ( 95 ) against the 95% confidence intervals [( 95 ) 1, and ( 95 ) ] for the two 0-min intervals. The first confidence interval ( 95 ) 1 is smaller than ( 95 ) because of its inflated degrees of freedom and it provides a lower limit to the confidence interval. The second confidence interval ( 95 ) is based upon the appropriate degrees of freedom and usually equals the interval of zero significance, 95. The second confidence interval for S is approximately 0% of S. In the following, we take 95 as a measure of both statistical significance and uncertainty for the stress estimates. Time variations of the Reynolds stresses at z 3.6 mab, as well as the corresponding estimates of 95, are shown in Fig.. For clarity in linear coordinates, we plot the local friction velocities,

5 157 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 FIG. 3. Variations of stress magnitude (heavy solid lines), the size of its 95% significance intervals (dashed lines), and TKE density S (thinner solid lines) at (a) z 7.6, (b) z 15.6, and (c) z 3.6 mab. (uw) u s s * 1/ and (10) (uw) s (uw) u n n * 1/, (11) (uw) where (uw) s and (uw) n are the along- and crosschannel components of the Reynolds stress (the alongand cross-channel directions are parallel and normal to the depth-mean flow, respectively). At z 3.6 mab, estimates of the along-channel stress are generally significantly different from zero. The magnitudes of the cross-channel stress are marginally significant for half the time and not significant for the remainder. Figure 3 shows, at three heights (z 3.6, 15.6, and 7.6 mab), the magnitudes of the Reynolds stress, uw, the quantity S, and the 95% significance levels of the stress magnitude, denoted as 95. The three quantities vary with time in accordance with the changes in beam velocity variances and the magnitude and direction changes of the time-mean flow (Fig. 4, Part I). The correlation between the changes in the three quantities is good. At z 3.6 mab and z 7.6 mab, the magnitudes of the stress are generally larger than 95, but by less than a factor of 10. At middepth, where the stress is the weakest, 95 and uw are comparable. There are apparently two regimes in the 95 versus S diagram (Fig. 4). For S 10 4 m s, 95 increases with S according to n S 3/4 95, (1) 80 whereas for S 10 4 m s, it is constant, that is, m s. (13) FIG. 4. Scatter diagram (open circles) of 95 vss at three levels (z 3.6, 15.6, and 7.6 mab). The two solid lines represent (1) and (13). Figure 5 is the scatter diagram of uw versus S. Changes in uw are approximately proportional to S according to uw 10 1 S. (14) Combining (1) and (14) for S 10 4 m s gives 3/ ( uw ). (15) Equation (15) shows that the band of zero significance, FIG. 5. Scatter diagram of uw vss at three levels (z 3.6, 15.6, and 7.6 mab).

6 NOVEMBER 1999 LU AND LUECK 1573 FIG. 7. Four 0-min intervals of velocity data at z 3.6 mab from beam 1 (b1) and beam (b). The four intervals are centered on (a) day 3.91, (b) day 4.03, (c) day 4.15, and (d) day FIG. 6. Two components of Reynolds stress (a) (uw) s, (b) (uw) n, and (c) S (all in m s ) calculated from fluctuations with periods shorter than 0 min (solid lines) and from those with periods of 0 10 min (dashed lines) at z 3.6 mab. (d) The stick diagram of the 0-min mean flow at the same height. 95, increases (and that its relative size decreases) with increasing stress magnitude uw. For example, 95 is one, two, and three times smaller than uw when the stress magnitude is (0.4, 3.84, 19.0) 10 4 m s, respectively. Both Doppler noise and turbulent fluctuations contribute to the uncertainty in an estimated quantity. If the Doppler noise is steady, then the statistical uncertainty should also be steady in a weakly turbulent environment. This state appears to be reached for S 10 4 m s [Fig. 4; Eq. (13)]. Even if the Doppler (or process) noise is identical for opposing beams, the sample variance is not identical, and there will be a lower limit to the level of significance and confidence. For S 10 4 m s, the turbulence determines the level of significance of the estimated stress. c. Contributions from low-frequency fluctuations We have not fully justified the choice of 0 min as the cutoff period to separate turbulence (high frequencies) from low-frequency variations. We notice from Fig. 4 in Part I that filtering with a 0-min cutoff period still leaves fluctuations with scales that are longer than 0 min but shorter than the tidal period. We will examine if these low-frequency variations contribute to the estimates of Reynolds stress and kinetic energy, S. Figure 6 compares, at z 3.6 mab, the magnitudes of two components of the stress and S calculated from fluctuations with periods less than 0 min and their counterparts calculated from fluctuations with periods between 0 and 10 min. For the along-channel stress, the contribution from low frequencies is insignificant compared to that from high frequencies, except for intervals around the turning of the tide. During flow reversal, the velocity fluctuations are nonstationary and all estimates are meaningless. The same holds for the cross-channel stress with the addition that this stress is weak during the flood. During the flood, the two frequency s ranges make comparable contributions to the cross-channel stress, but the stress is insignificant. The low-frequency contribution to S is also small compared to that from high frequencies. During the fluctuations of current direction that occurred on the ebb of day 4.4 and 4.5, the contribution from the two ranges is equal. These two intervals of large low-frequency current fluctuation did not, however, provide a significant (low frequency) stress and may be considered as nonturbulent. Thus, 0 min is a reasonable boundary separating turbulence from low-frequency fluctuations. 4. The causes of statistical uncertainty The statistical uncertainty of an estimated quantity can be due to several reasons. First of all, to get a statistical estimate for a stochastic process requires the process to be stationary. Figure 4 in Part I shows that the intensity of the high-frequency fluctuations is not stationary throughout the tidal cycle. Our concern is on the stationarity of turbulence (and mean flow) over the time intervals we conduct statistical analysis. Figure 7 shows four 0-min intervals of velocity data along beams 1 and at the lowest cell (z 3.6 mab). The four intervals are chosen from different phases of the tide, with interval (a) centered at day 3.91 during the strong ebb, (b) at day 4.04 during the turning of the tide, (c) at day 4.16 during the strong flood, and (d)

7 1574 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 FIG. 8. Variations of b b 1 (thinner lines) and 10 times its cumulative mean (thicker lines) at z 3.6 mab. Each panel corresponds to one of the 0-min intervals shown in Fig. 7. FIG. 9. Variations of b1 b b3 b m s (thinner lines) and five times of its cumulative means (thicker lines) at z 3.6 mab. Each panel corresponds to one of the 0-min intervals shown in Fig. 7. at day 4.66 during the weak flood. By visual inspection, during intervals (a), (c), and (d), the low-frequency flow appeared to be nearly steady, and the intensity of highfrequency velocity fluctuations stationary. An extremely large velocity anomaly happened during interval (b) and lasted about min, and neither the flow nor the highfrequency fluctuations were stationary during the turning of the tide. The uncertainty in the estimate of the mean of a time series x i (i 1,...,N) can be examined by looking n at the cumulative mean of this series, defined as i1 x i / n (n 1,..., N). As pertinent to the estimates of Reynolds stress and S, we examine the cumulative means of b b1, b4 b3, and b1 b b3 b m s (where m s 4 is the bias in S due to Doppler noise). Note that b b 1 and b4 b3 are approximately proportional to the alongand cross-channel stress, respectively. Figures 8 and 9 show the time series of b b1 and b1 b b3 b m s 4, as well as their cumulative means, at z 3.6 mab, for the four 0-min intervals shown in Fig. 7. Each 0-min interval contains many events of different strength. During intervals of strong flow (Figs. 8 and 9, panels a and c), the cumulative means become nearly constant after 5 min, and then fluctuate with magnitudes smaller than 10% of their final values. During the weak flood (panel d), the cumulative mean of b1 b b3 b m s becomes constant after 1 min, whereas the cumulative mean of b b1 fluctuates with an amplitude equal to its final mean. This implies that the mean may be insignificant. For the interval during the turning of the tide (panel b), the cumulative means of both time series do not become independent of time and they are dominated by the single strong event. The turning of the tide is not a stationary interval, while the weak flood is stationary. To illustrate the variations in uncertainty with height above the seabed, we show the cumulative means of b b1, b4 b3, and b1 b b3 b m s at seven heights for the 0-min interval during the strong ebb (Fig. 10). Note that the cumulative means of b b1 are nearly constant after 10 min at all levels (Fig. 10a). The cumulative means of b 4 b 3 are also nearly constant except at z 11.6 mab, where it changes sign at 10 min (Fig. 10b). Note that z 11.6 mab is also the height at which the vertical profile of the cross-channel stress changes sign, and thus its true value may be zero. For b1 b b3 b m s, the cumulative means become constant after 5 min at all levels (Fig. 10c). Note that during this interval b1 b b3 b m s is nearly independent of height. 5. Turbulence spectra The spectra of Reynolds stress and S are obtained by subtracting and adding the spectra of beam velocities according to (1) (3). Spectra were computed for seven heights using 90 min of stationary data taken during the strong ebb on day 3.91 (Fig. 11). The horizontal axis in this figure is the wavenumber k calculated with the streamwise velocity at each height using Taylor s frozen turbulence hypothesis. Each spectrum, calculated with a fast Fourier transform (FFT) length of 0 min and 50% overlap, is averaged over 0 evenly distributed intervals in the log 10 k domain. A uniform level of m s 1, corresponding to a bias of m s in the estimates of S, is removed from the spectra of S.

8 NOVEMBER 1999 LU AND LUECK 1575 FIG. 10. The cumulative means of (a) b b1, (b) b4 b3, and (c) b1 b b3 b m s for a 0-min interval during the strong ebb. In each panel the lowest curve represents the height of 3.6 mab. Higher levels are labeled in (a) and the zero line is drawn for each level. FIG. 11. The spectra of the (a) along- and (b) cross-channel components of the stress and (c) S, for a 90-min interval during the strong ebb. The height of the measurements is marked in (a) and the zero line is drawn for each level. The shading represents the 95% confidence interval and is shown only for the lowest level.

9 1576 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 The signs of both components of the Reynolds stress spectra do not change over the resolved wavenumber range, except and only for the cross-channel stress at 15.6 mab where the spectrum is weak. The signs of the cross-channel stress spectra change from positive below middepth to negative above middepth, and this change corresponds to the change of sign of the transverse shear (Fig. 11, Part I). The wavenumber ranges contributing to the Reynolds stress spectra are fully resolved (Figs. 11a,b). The spectra of S are also resolved, except at the high-wavenumber end for heights less than 7.6 m. Below middepth, there is a noticeable shifting of the peak of the spectra toward smaller wavenumbers with increasing distance from the seabed. The analysis of another 90 min of stationary data, the strong flood starting at day 4.15, provides spectra of the along-channel stress and S that are similar to those shown in Fig. 11. However, the spectra of the cross-channel stress are weak and noisy due to the small cross-channel shear during the flood (Fig. 11, Part I). 6. Estimates of turbulent quantities In this section, we present the estimates of Reynolds stress and TKE density (obtained with the variance technique) and two more quantities, the TKE production rate P and the vertical viscosity coefficient A (derived from stress and the mean flow shear). Figure 1 shows the depth time sections of these quantities over 1-day period, covering two cycles of the dominant tidal constituent, M. Changes in the direction of the tidal flow correspond to sign changes of the along-channel stress (Fig. 1a) at levels close to the bottom, positive (uw) s during the flood and negative during the ebb. Estimates from the remaining.5 days of data collected during later days (with decreasing strength of semidiurnal tide and increasing strength of diurnal tide) show similar patterns of depth time variations as in Fig. 1. a. Reynolds stress The two upper panels in Fig. 1 show the local friction velocities u * s and u * n, which are related to the along- and cross-channel components of the Reynolds stress by (10) and (11). First of all, we note that the extremely large stresses obtained during the flow turning in direction, for example, the events around day 4.05, 4.35, 4.55, are unreliable because neither the flow nor turbulence is stationary (section 4). In fact these large stresses usually have opposite signs to the shear, as indicated by the negative production rate (blank areas in Fig. 1c). During the weak flood between day 4.6 and 4.8, the magnitudes of both u * s and u * n are small, less than 0.01 m s 1 in general. Excluding the extreme events during the tide turning in direction, the along-channel stress intensifies at levels close to the seabed, where the tidally varying signal is evident. At z 3.6 mab, u * s reaches 0.04 m s 1 at peak flow during the ebb, and 0.03 m s 1 at peak flow during the flood. Away from the bottom, the magnitudes of the stress decrease. Figure 1a also shows the height of log layer obtained by taking a least squares fitting to the streamwise velocity profiles (Lueck and Lu 1997). Note that the signs of u * s are uniform below the log-layer height but may change at the top of the log layer. The cross-channel stress is small during the flood, with the magnitude of u * n about 0.01 m s 1 in general throughout the water column. During the ebb, the magnitude of u * n reaches 0.03 m s 1 at strong flows. The cross-channel stress changes sign at middepth during the ebb, pointing to the headland in the upper layer and away from it in the lower layer. This flood ebb asymmetry in terms of cross-channel stress corresponds to the asymmetry we found for the transverse shear, which is related to variations in the strength of the transverse flow (Part I). b. TKE production rate The TKE production rate is calculated from the two components of the Reynolds stress (uw) s,(uw) n, and the corresponding mean flow shear u s /z, u n /z, according to us un P (uw) s (uw) n. (16) z z The depth and time variations of P are shown in Fig. 1c. Negative estimates of P are blank areas and occur mostly during flow turning (when the stress estimates are unreliable) and the weak flood. In general, the TKE production rate intensifies toward the seabed, bearing the character of wall-bounded turbulence. The major region of TKE production is within the log layer. However, during the ebb, there are events of large production rate that occur at heights above the log layer. These events correspond to the sign reversal of stress (and shear) above middepth. The magnitudes of P span about three decades, ranging from 10 4 m s 3 (W kg 1 ) near the bottom to 10 7 m s 3 (which may close to the noise level of P) at all levels during the weak flood. The variations with tidal flow are most evident in the lower quarter of the channel. c. TKE density The quantity S, which relates to the TKE density q / by (4), is illustrated in Fig. 1d. A uniform bias of m s (section 3a) has been removed from S calculated with (3). The blank area around day 4.75, during the weak flood, corresponds to negative S. The noise bias is a statistical quantity, and a few samples may actually be smaller than the bias in a weakly turbulent flow. Here S is smallest during the weak flood, about m s (close to the 95% confidence interval of S, see Table 1). The largest estimates of S are obtained at the beginning and end of the ebb, reach-

10 NOVEMBER 1999 LU AND LUECK 1577 FIG. 1. Depth time sections of the 0-min mean local friction velocities (a) u * s, (b) u * n (m s 1 ), (c) log 10 P (m s 3 ), (d) log 10 S (m s ), and (e) log 10 A (m s 1 ). The blank areas in (c), (d), and (e) represent negative values. ing m s. During the strong flood between day 4.1 and 4.3, and in the middle of the ebb around day 4.45, S is about m s. The variation of S with depth for a single profile does not exceed one decade. The magnitude of S increases slightly toward the seabed between day 4.1 and 4.3 during the strong flood. The magnitudes of S are proportional to those of the Reynolds stress by a factor of 10 [Eq. (14)]. From (4), we know that for a moderate level of isotropy 0.,

11 1578 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 S 1.8q /. If this is the case, then from (14) we have q / 5.6 uw, which is close to what Gross and Nowell (1983) got from measurements in the near-bottom region of tidal boundary layer. d. Turbulent viscosity In principle, an estimate of the vertical viscosity coefficient A can be readily obtained from one component of Reynolds stress and the corresponding shear, that is, or s u A (uw) s, (17) z n u A (uw) n. (18) z Alternatively, A can be calculated from the TKE production rate and the magnitude of shear, that is, [ ] un us A P. (19) z z By using (19), the sign of A is the same as P. In the optimal case the estimates from (17), (18), and (19) should be consistent, and this is found to hold statistically for the present dataset. The estimates of A are subject to large uncertainties, particularly when the magnitudes of the shear are small. Figure 1e indicates a variation of A with tidal flow, ranging from about 10 3 m s 1 during the weak flood to m s 1 during the ebb. During the strong flood between day 4.1 and 4.3, A is about 0.03 m s 1. The turbulent viscosity increases with increasing height in the lower half of the water column, and reaches a maximum near middepth. 7. Summary The variances of the along-beam velocities from a standard ADCP with four beams provide estimates of two components of the Reynolds stress and a quantity proportional to the TKE density. To apply the variance technique one needs to assume that the mean flow and the second-order moments of turbulent velocity fluctuations are statistically homogeneous in horizontal space over the distances separating the beams. The ADCP needs to be rigidly mounted for turbulence measurements to avoid contamination due to correlation of instrument motions and turbulent velocity fluctuations. Doppler noise brings a bias to the estimate of TKE density, but not to stress estimates if the Doppler noise in opposing beams is identical. For our unit, this bias is less than m s. For a four-beam unit, the bias due to pitch and roll angles cannot be removed and may be significant for tilt angles larger than a few degrees in anisotropic turbulence. The uncertainty levels of the stress estimates are mainly due to turbulent fluctuations and increase with increasing turbulent intensity. The influence of Doppler noise emerges when turbulent intensities are low. The highest signal-to-noise ratio is obtained at levels close to the seabed, where the stress magnitudes are large. Contributions from velocity fluctuations at periods longer than 0 min add little to the estimates of Reynolds stress and TKE density. The spectral ranges of the Reynolds stress and TKE density were usually resolved with our sampling interval of 3 s. The high wavenumber end of the spectrum of TKE density was not resolved near the seabed. The combination of the Reynolds stress and mean shear provides estimates of the rate of production of TKE density and the vertical eddy viscosity. Excluding intervals during the turning of the tide, the measurement captures the depth and time variations of turbulence in the channel. The Reynolds stress shows a clear tidal signal in the near-bottom layer for the along-channel component and presents a clear ebb flood asymmetry in the cross-channel component. The TKE production rate is bottom enhanced, ranging from 10 4 m s 3 (W kg 1 ) during strong flows to 10 7 m s 3 during weak flows. The TKE density is approximately proportional to the stress magnitude. The eddy viscosity increases with increasing flow speed, and also with increasing height in the lower half of the water column. Acknowledgments. We would like to thank D. Newman and J. Box for their technical support to the field program, and A. 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