• Garrett, C. J. R., and R. H. Loucks, 1976: Up-welling along the Yarmouth shore of Nova Scotia. J. Fish. Res. Board Can.,33, 116–117.

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  • Geyer, W. R., 1993: Three-dimensional tidal flow around headlands. J. Geophys. Res.,98, 955–966.

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  • Kalkwijk, J. P. T., and R. Booij, 1986: Adaption of secondary flow in near-horizontal flow. J. Hydraul. Res.,24, 19–37.

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  • Lhermitte, R., and U. Lemmin, 1990: Probing water turbulence by high frequency Doppler sonar. Geophys. Res. Lett.,17, 1549–1552.

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  • Lohrmann, A., B. Hackett, and L. P. Roed, 1990: High resolution measurements of turbulence, velocity and stress using a pulse-to-pulse coherent sonar. J. Atmos. Oceanic Technol.,7, 19–37.

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  • Lu, Y., and R. G. Lueck, 1999: Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Oceanic Technol.,16, 1568–1579.

  • Lueck, R. G., and Y. Lu, 1997: The logarithmic layer in a tidal channel. Contin. Shelf Res.,17, 1785–1801.

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  • Wolanski, E., J. Imberger, and M. L. Heron, 1984: Island wakes in shallow coastal waters. J. Geophys. Res.,89, 10 553–10 569.

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    Transducer geometry and beam orientation of the ADCP. The beams are inclined by θ = 30° from the axis of the instrument, which is nominally vertical. Here, ϕ1, ϕ2, and ϕ3 are heading, pitch, and roll angles.

  • View in gallery

    Area chart showing the bathymetry (depth in meters) of Cordova Channel, the location of the ADCP, and the positions of the moored microstructure instrument during its two deployments (TAMI1 and TAMI2). A current meter (CMI), transmitter (T), and receiver (R) of an acoustic scintillation system were also deployed during the experiment.

  • View in gallery

    One-day-long vertical velocity at middepth (15.6 mab), calculated for roll angle ϕ3 = −2° (thick lines) and ϕ3 = −3° (thin lines). The two upper curves are calculated from 20-min smoothed beam velocities and the lower curves are 3-h smoothed. The lower panel is a stick diagram of the horizontal current at middepth with a north flow directed straight up.

  • View in gallery

    A sample of one day of data collected at middepth (15.6 mab) from a single beam that was oriented in the downstream direction during ebb tide. The upper curve is the raw data of four-ping averages collected every 3.05 s. The middle curve is the same data with 20-min smoothing and offset by −0.5 m s−1. The lower curve is the beam velocity fluctuation formed by taking the difference of the upper two curves and offset by −1.5 m s−1. The stick diagram in the lower panel is the 20-min mean horizontal velocity at middepth derived from the velocity along all four beams.

  • View in gallery

    The lagged autocorrelation of the high-passed velocity from a single beam that was directed along the flow. (a) and (c) The correlation at 15.6 mab, (b) and (d) the correlation at 3.6 mab. (a) and (b) Conditions during strong ebbing starting at day 23.91; (c) and (d) give the correlation during the weak flood starting at day 24.6. The horizontal dashed lines represent the confidence bounds for large lags. The thick curves are the correlations for all lags up to 20 min (labels on lower axis); thin curves are the correlations for lags shorter than 45 s (labels on upper axis).

  • View in gallery

    (upper panel) Twenty-minute mean vertical (solid lines) and error velocities (dashed lines) at the heights of 15.6, 3.6 (offset by −0.05 m s−1), and 27.6 mab (offset by 0.05 m s−1). The sign of the error velocity is made identical to the sign of the vertical velocity to facilitate their comparison. The shading spans ±1 standard deviation of the vertical and error velocities (3 × 10−3 m s−1). (lower panel) The ratio of horizontal averaging length to beam separation, M = /L, at 3.6, (thin, upper curve), 15.6 (thick, middle curve), and 27.6 mab (dashed, lower curve).

  • View in gallery

    (a) A scatterplot of the magnitude of the ratio of error velocity to its standard deviation vs the nondimensional averaging length M. Points that fall below the ratio of 1 (horizontal line) pass the first test of the assumption of horizontal homogeneity of the mean. (b) Upper panel: The percentage of samples with velocity ratios less than 1 as a function of M. The 95%–100% range is shaded. Lower panel: histogram of M.

  • View in gallery

    (a). A scatterplot of the magnitude of the ratio of error to along-channel velocity vs the nondimensional averaging length M. Points that fall below the ratio of 0.01 (horizontal line) pass the second test of the assumption of horizontal homogeneity of the mean. (b) Upper panel: The percentage of samples with velocity ratios less than 0.01 as a function of M. The 95%–100% range is shaded. Lower panel: histogram of M.

  • View in gallery

    Polar diagram of the 20-min depth mean flow for all 4.5 days of data with respect to true north (0°).

  • View in gallery

    (upper panel) Time series of the direction of the 20-min mean velocity at 3.6 mab (solid line) and 27.6 mab (dashed line) and the shear at 3.6 mab (circles). (lower panels) Typical profiles of current direction during (a) flood and (b) ebb and of the shear direction during the (c) flood and (d) ebb.

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    Cordova Channel depth–time sections of 20-min mean (a) streamwise, (b) transverse, and (c) vertical velocities (all in m s−1); (d) streamwise and (e) transverse shear (both in s−1). The solid line marks the log-layer height. For the streamwise flow, red and blue shades are floods and ebbs, respectively. Values smaller than one standard deviation are coded in shades of gray.

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Using a Broadband ADCP in a Tidal Channel. Part I: Mean Flow and Shear

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  • 1 School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
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Abstract

This paper discusses the principles of measuring the mean velocity and its vertical shear in a turbulent flow using an acoustic Doppler current profiler (ADCP), and presents an analysis of data gathered in a tidal channel. The assumption of horizontal homogeneity of the first moments is fundamental to the derivation of the mean velocity vector because the velocity is never homogeneous over the span of the beams in a turbulent flow. Two tests of this assumption are developed—a comparison of the mean error velocity against its standard deviation and against the mean speed. The fraction of the samples that pass these tests increases with increasing spatial averaging and exceeds 95% for distances longer than 55 beam separations. The statistical uncertainty of the velocity and shear vector, averaged over 10 min and longer, stems from turbulent fluctuations rather than Doppler noise. Estimation of the vertical velocity requires a correction for the bias in the measured tilt.

The mean velocity and shear estimates from this natural tidal channel show more complex depth–time variations than found in idealized one-dimensional channel flow, which seldom occurs in nature. The ADCP measurements reveal the secondary circulation, bursts of up- and downwelling, shear reversals, and transverse velocity shear.

* 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 Rd., Victoria, BC V8W 2Y2, Canada.

Email: rlueck@uvic.ca

Abstract

This paper discusses the principles of measuring the mean velocity and its vertical shear in a turbulent flow using an acoustic Doppler current profiler (ADCP), and presents an analysis of data gathered in a tidal channel. The assumption of horizontal homogeneity of the first moments is fundamental to the derivation of the mean velocity vector because the velocity is never homogeneous over the span of the beams in a turbulent flow. Two tests of this assumption are developed—a comparison of the mean error velocity against its standard deviation and against the mean speed. The fraction of the samples that pass these tests increases with increasing spatial averaging and exceeds 95% for distances longer than 55 beam separations. The statistical uncertainty of the velocity and shear vector, averaged over 10 min and longer, stems from turbulent fluctuations rather than Doppler noise. Estimation of the vertical velocity requires a correction for the bias in the measured tilt.

The mean velocity and shear estimates from this natural tidal channel show more complex depth–time variations than found in idealized one-dimensional channel flow, which seldom occurs in nature. The ADCP measurements reveal the secondary circulation, bursts of up- and downwelling, shear reversals, and transverse velocity shear.

* 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 Rd., Victoria, BC V8W 2Y2, Canada.

Email: rlueck@uvic.ca

1. Introduction

In this paper we report on using an acoustic Doppler current profiler (ADCP) in a fully turbulent tidal channel to estimate mean velocity and shear profiles and examine the bias and statistical uncertainty of these estimates. A companion contribution (Lu and Lueck 1999, hereafter Part II) deals with the estimation of turbulence parameters (Reynolds stress, turbulent kinetic energy, etc.). The ADCP is increasingly applied to oceanic measurements because its spatial resolution and profiling range are usually adequate to measure the flow and shear throughout a large portion of the water column in coastal seas. The ability of this instrument to sample data rapidly lends it to estimating turbulence quantities.

Accurate estimates of mean velocity and shear are essential for a study of the flow structure; however, there is a significant difference between the measurement principle of an ADCP and that of a current meter. A current meter measures the “instantaneous” velocity vector at its position. For an ADCP, the directly measured velocities are the radial speed of the flow along its inclined acoustic beams, and the “true” velocity vector is derived from these along-beam velocities. The derivation assumes that the flow is homogeneous in the horizontal plane over the distances separating the beams (e.g., Lohrmann et al. 1990). At best, this assumption holds only statistically in a turbulent environment. A test of statistical homogeneity is required to justify its assumption and to determine the bias of the estimated mean velocity vector. Flows that have a Reynolds stress are anisotropic, and the variances of the velocity along the beams are different and cannot be used to test statistical homogeneity.

In section 2 we examine the algorithm used to derive the velocity vector, the assumption of statistical homogeneity in the horizontal plane, and the effect of platform motion. In section 3 we use data collected in a tidal channel to estimate the statistical uncertainty of the velocity vector and the contribution by tilt-angle bias, and we develop two tests of the assumption of horizontal homogeneity. In section 4 we examine some of the complicated flow patterns revealed by the ADCP including burst of vertical flow, secondary circulation, and the tidal asymmetry of the mean flow and its shear. The results are summarized in section 5.

2. Deriving the velocity vector

a. Calculation algorithm

We assume that the direction of all four beams deviates by the angle θ = 30° from a single axis that forms the centerline of the instrument. We also assume that the beams are orthogonal when viewed from along the centerline. Rotation around the centerline, ϕ1, defines heading (Fig. 1). The tilt with respect to gravity of the planes containing opposing beam pairs defines the pitch and roll (ϕ2, ϕ3). We denote the velocity along the ith (i = 1, . . . , 4) beam by bi, and the horizontal and vertical components at the position of bi by ui, υi, wi. For small pitch and roll angles, correct to the first order in ϕ2 and ϕ3, the relationship between bi and ui, υi, wi is (Lohrmann et al. 1990)
i1520-0426-16-11-1556-e1
where a rotation of the heading angle ϕ1 should be added to (1) when ui and υi are defined as the eastward and northward components, respectively.
The along-beam velocities bi are directly measured by the ADCP. When the ADCP operates in the “earth coordinates” mode, the beam velocities are transformed to the “velocity vector” and the “error” velocity by
i1520-0426-16-11-1556-e2
and an additional compass rotation. The next section shows that u, υ̂, ŵ cannot be regarded as a true velocity vector for a single ping.

b. The assumption of homogeneity

From (1) and (2), the relationship between û, υ̂, ŵ, ê and ui, υi, wi, correct to the first order in ϕ2 and ϕ3, can be derived to read
i1520-0426-16-11-1556-e3
For û, υ̂, ŵ to form a true velocity vector, each velocity component at the different beams must be identical; that is, the velocity field must be homogeneous in the horizontal plane over the distances separating the beams. In a turbulent flow, this requirement is not satisfied due to the existence of eddies with scales comparable to and smaller than the beam separation. We may assume, however, that the statistical properties of the flow are horizontally homogeneous. In terms of the first-order moments, we assume that the three-dimensional time-mean flow (u, υ, w) is horizontally homogeneous over the spatial domain of the beams and that the fluctuations average to zero. Then from (3), we have
uûυυ̂wŵê
The expectation of the error velocity ê is zero under the assumption of statistical homogeneity, and we use this property (in section 3d) to form a test of the assumption.

The horizontal inhomogeneity of the flow over the distance separating the beam pairs, L, is caused by turbulent eddies with scales comparable to and smaller than L. As a result, the amount of averaging required to give a satisfactory estimate of the mean flow must span many such eddies, and this is realized with an averaging time τL/U, where U is the mean speed. The relevant parameter is M = /L, the ratio of horizontal averaging scale to beam separation.

If the assumption of homogeneity is validated by some test, then the statistical uncertainty of the mean flow estimates can be determined from knowledge of the Doppler noise and the turbulent fluctuations. If such a test is impossible, then a fundamental assumption of ADCP measurements remains questionable. Averaging over longer periods generally reduces the statistical uncertainty of the mean flow estimates. However, for tidal flows, a long averaging time may conflict with the requirement to resolve the phase of tidal variations. The quest for an ideal averaging time is linked to the existence of a “spectral gap” that separates turbulence from the tidal variations (this issue is addressed in Part II).

c. Rigid versus nonrigid deployments

An ADCP can be mounted either rigidly or nonrigidly. For a rigidly mounted ADCP, the tilt angles ϕi (i = 1, 2, 3) are constant, and the mean velocity components are the linear combinations of the averaged beam velocities:
i1520-0426-16-11-1556-e5
A nonrigidly mounted instrument experiences changes of orientation (reflected by the variations in tilt angles) and translation. Changes in beam orientation lead to changes of velocity along each beam even in a steady flow field, hence averaging a beam velocity time series is meaningless. Instead of using (5), the transformation from beam velocity to “velocity vector” must be done ping by ping using (2) and then taking the average of û, υ̂, ŵ to derive the mean velocity components.

Would the motions of a nonrigidly mounted instrument influence the mean velocity estimates? For shipboard measurements, the mean translational motions (equal to ship speed) should be monitored (e.g., by bottom tracking) and must be subtracted from the mean flow estimates. For a moored nonrigid deployment, the translational motions are seldom monitored, but they usually average to zero. Variations in tilt angles, however, may correlate with either the environmental velocity or the velocity of the ADCP and bias the velocity estimates [e.g., terms such as ϕ3(u2u1) in (3)]. There is no velocity–tilt correlation for eddies with spatial scales that are large compared to the beam separation because the velocity fluctuation difference is zero. For eddies with a very small scale compared to the beam separation, the tilt–velocity correlation at points 1 and 2 will be small. Intermediate-scale eddies, those with scales comparable to the beam separation, may have fluctuations of velocity difference (u2u1) that correlate with tilting motions. A 2° rms tilt-angle fluctuation and a 0.1 m s−1 rms velocity fluctuation difference would produce a bias of 4 × 10−3 m s−1 if the two fluctuations are perfectly correlated. Hence, this effect is not important for estimates of horizontal current but may be significant for estimates of the mean error and vertical velocity.

Bias in the tilt sensors introduces another difficulty for a nonrigidly mounted ADCP. Inaccuracy in the measurement of ϕ2 or ϕ3, as can be seen from (2), can contaminate the transformation from beam velocity to velocity vector. Again, the bias is of little practical consequence for the horizontal velocity estimates but will affect the estimated vertical velocity. To eliminate this contamination, one needs to determine the bias in tilt angles and recalculate the beam-to-earth coordinate transformation. The reprocessing of data can be conducted, using (2), if data are recorded ping by ping. If only the ensemble average of a large number of pings are recorded (which is usually the case), then the correction for tilt-angle bias using (5) is possible only for a rigidly mounted instrument. The horizontal direction of the biased tilt can rotate during the averaging interval for a nonrigidly mounted instrument.

Thus, for a nonrigidly mounted instrument, the variances of the measured vertical and error velocities will be larger than for a rigidly mounted ADCP, and testing the assumption of homogeneity may be more difficult. More significantly, a rigidly mounted ADCP permits the estimation of turbulent quantities with the “variance method,” whereas this is impossible with a nonrigidly mounted instrument.

3. Analysis of data from a rigidly mounted ADCP

a. Experiment setup

A broadband and 600-kHz ADCP, manufactured by RD Instruments (RDI), was rigidly mounted to the bottom of Cordova Channel during a multiinvestigator experiment conducted from 19–30 September 1994. The channel is a side passage among a series of waterways that link Juan de Fuca Strait to the Strait of Georgia (located between Vancouver Island and the mainland of North America). The ADCP was mounted on a quadripod near the center of the channel where the depth was about 30 m (Fig. 2). A cable connected the ADCP to a power supply and computer in a shore station, and the data were directly transferred to the computer via this cable. About 4.5 days of data were collected, of which 3.8 days of data were recorded at rapid sampling rates. The profiling range was broken into uniform segments (depth cells) with 1-m vertical spacing. The centers of useful cells ranged from 3.6 to 27.6 mab (meters above bottom). The ADCP pinged at the fastest rate possible, approximately 0.75 s per ping (1.3 Hz). The velocity was recorded in beam coordinates, and the number of pings averaged into ensembles by the ADCP ranged from 2 to 4. The ADCP was operated in mode 4, which is the default mode for firmware versions 3 and later. This mode produces the lowest noise of all modes available at the time of the measurements, provided that certain conditions are satisfied regarding range and signal strength. Mode 4 uses a single set of pulses with a long lag to determine the phase and, hence, the Doppler shift of the echo. The horizontal ambiguity velocity is 1.6 m s−1 and is not an issue here because it exceeds the maximum current in Cordova Channel. This mode is particular to the ADCP manufactured by RDI and may not be available on units produced by others.

b. Correction for tilt-angle bias

The measured values of ϕ2 and ϕ3 are 0.00° and −3.00°, respectively, throughout the experiment. However, the vertical velocity calculated with these tilt angles has a tidally varying signal (Fig. 3, thin lines) that is far greater than the vertical velocity attributable to the tides. The estimated vertical velocity is proportional to the along-channel velocity and must be due to a bias in the tilt measurement. By choosing combinations of different values of ϕ2 and ϕ3, we found that the vertical velocity at tidal frequency is minimized by changing only the roll angle from −3.0° to −2.0° (thick lines, Fig. 3). Hence the values of ϕ1, ϕ2, ϕ3 used in the transformation from beam to earth coordinates are 150.3°, 0.0°, −2.0°, respectively.

c. Measurement uncertainties

Figure 4 shows a sample of a 1-day-long, four-ping averaged beam velocity at middepth from beam 1, which is projected into the downstream direction during ebb tide. (The flow is southward during the ebb and northward during the flood.) The low- and high-frequency variations are separated with a zero-phase, low-pass and fourth-order Butterworth filter with a cutoff period of 20 min. The measured velocity contains uncertainties due to both Doppler noise and turbulence. The noise of this particular instrument was determined from measurements in Saanich Inlet, which is known for its extremely slow tidal current and low turbulence. The standard deviation of the beam velocity noise, for four-ping averaged ensembles with 1-m bin size, is 0.013 m s−1 and close to the specification of the manufacturer. The contribution to Doppler noise due to finite beamwidth is not included in this estimate because there was very little relative motion between the water and the ADCP. The contribution due to finite beamwidth (1°) is proportional to speed and equals 0.006 m s−1 in a current of 1 m s−1 (Lhermitte and Lemmin 1990). Hence, the total noise standard deviation is 0.014 m s−1, and it is significant only during the weak flood between days 24.6 and 24.8. During the other segments of this 1-day-long data, the high-frequency beam velocity fluctuations are dominated by turbulence.

Velocity variations due to Doppler noise are uncorrelated; therefore, by averaging N ensembles the noise standard deviation is reduced by a factor of N1/2. For 20-min averages, the uncertainty in beam velocity due to Doppler noise is 7 × 10−4 m s−1. Samples of the turbulent fluctuations, however, are not fully decorrelated. To determine how much the statistical uncertainty is reduced by time averaging, we need to know the decorrelation timescale of the turbulent fluctuations. The autocorrelation functions (ACFs) of beam velocity fluctuations for two 20-min intervals—one during strong ebb and one during weak flood—are shown in Fig. 5. During the strong ebb, both at middepth and near bottom (Figs. 5a,b), the beam fluctuations decorrelate slowly. By comparison, the velocity fluctuations during the weak flood decorrelate quickly, like white noise (Figs. 5c,d), possibly because the Doppler noise is comparable to the turbulent fluctuations during this interval.

We define the decorrelation timescale as the lag at which the ACF is reduced to the upper bound of its uncertainty level at very large lags (0.1). From an examination of all beams and various phases of the tide, we find that 15 s is the typical decorrelation time during strong flow. Thus, a 20-min average has 80 degrees of freedom and the standard deviation of the estimated mean along-beam velocity is typically 5.5 × 10−3 m s−1 during strong flow. During the weak flood between days 24.6 and 24.8, the standard deviation of the 20-min mean is 1.7 × 10−3 m s−1 and still larger than the Doppler noise.

By including terms representing the uncertainty in the mean beam velocity in (5), we deduce that the standard deviations of the horizontal and vertical velocity components are 2 times larger and 3 times smaller, respectively, than that of the beam velocity. Thus, the uncertainty due to turbulence of the 20-min mean velocities is typically about 8 × 10−3 m s−1 for u and υ and 3 × 10−3 m s−1 for w and ê during strong flow, and the level of uncertainty is three times smaller during the weak flood between days 24.6 and 24.8.

For the above estimates of the uncertainty, we assume that the turbulent fluctuations are stationary during the averaging time of 20 min. However, the ACF can fluctuate significantly at long lags (Fig. 5), and this indicates the presence of eddies with large time and long length scales. The time series of along-beam velocity (Figs. 4 and 7b of Part II) show that distinct events occurred during the turning of the tide and that they lasted for up to 4 min. Smoothing the along-beam velocity with a 20-min cutoff period keeps not only the tidal signal but some eddy fluctuations with periods longer than 20 min as well (Fig. 4). A more detailed analysis presented in Part II shows that the turbulent processes are stationary over 20-min intervals and that eddies with timescales longer than 20 min contribute negligibly to the turbulence kinetic energy, except during the turning of the tide. Thus, averaging the data over 20 min removes virtually all turbulent fluctuations, and the above estimates of statistical uncertainty are typical.

d. Tests of homogeneity

Averaging the data for 20 min reduces the standard deviation of the estimates of mean current to low levels. However, low statistical uncertainty is not the only requirement of fidelity. A test must be devised for the assumption of horizontal homogeneity of the first moments. We will examine two such tests. In the first test, we compare the mean error velocity against its standard deviation. The estimate of the standard deviation is based upon the standard deviation of the along-beam velocity and is derived using (5). The estimate is not based upon the sample standard deviation of the error velocity. Under the assumption of horizontal homogeneity of the first moments, the expectation of the error velocity is zero and its 20-min mean should be smaller than its standard deviation. The standard deviation of both the vertical and the error velocities is typically 3 × 10−3 m s−1.

In Fig. 6, we plot the time variations of the 20-min mean error velocity ê at three heights above the bottom. The corresponding nondimensional averaging distance M at the three heights is plotted in the lower panel of Fig. 6. The magnitude of ê increases, while M decreases, with height above the bottom. At the lowest level, ê is generally smaller than its standard deviation, while the magnitude of M is typically 300–400 (except near the turning of the tide, which is marked by the dips in the lower panel of Fig. 6). At middepth, M is typically 40–70 and some values of ê exceed its standard deviation, particularly near the turning of the tide. At the uppermost level, M is typically 20–40 and the error velocity is frequently comparable to and larger than its standard deviation.

Can M be used to predict the likelihood that the estimates of the mean velocity are consistent with the assumption of horizontal homogeneity for the first moments? A scatterplot of the ratio of mean error velocity to its standard deviation, |ê|/(3 × 10−3 m s−1), against the nondimensional averaging length M indicates that most ratios are less than 1 (Fig. 7a). The fraction of the samples with ratios less than 1 increases with increasing M and exceeds 95% for nondimensional averaging lengths greater than 50 (Fig. 7b, upper panel). Thus, 95% of the samples of the error velocity are indistinguishable from the expectation of zero, when M > 50, and they are consistent with the assumption of horizontal homogeneity of the mean. They pass the first test.

A second test compares the magnitude of the error velocity against the horizontal speed. The speed generally does not have a zero mean. The standard deviation of the 20-min mean horizontal velocity, due to turbulence and Doppler noise, is typically 8 × 10−3 m s−1 or about 1% of the mean speed. The error velocity can be considered negligible if it is less than the standard deviation of the speed because it implies an inhomogeneity that is smaller than the statistical uncertainty of the estimated speed. A scatterplot of |ê/U| against M shows that most of the samples have a ratio less than 0.01 and nearly all ratios larger than 0.01 come from M < 50 (Fig. 8a). The fraction of samples that have ratios smaller than 0.01 and, thus, pass the second test, increases rapidly with M, and exceeds 95% for M > 55 (Fig. 8b).

e. Vertical velocity

The vertical velocity averaged over a flood or ebb is extremely small (less than 3 × 10−4 m s−1). Are the 20-min averages significant? We accept all estimates as significant if their magnitude exceeds both the standard deviation and the magnitude of the error velocity. The vertical velocity is usually smaller than its standard deviation and, thus, it is insignificant (Fig. 6). The vertical velocity exceeds its standard deviation in bursts and all such events are significant at 3.5 mab. At middepth (15.6 mab) most bursts are significant, while at the highest level (27.6 mab) most bursts have magnitudes comparable to the error velocity and are insignificant. Large error velocities occur around the turning of the tide, and, for those times, most estimates of vertical velocity are insignificant or dubious on the grounds of horizontal inhomogeneity.

4. Mean flow and shear estimates in Cordova Channel

a. Results

Over a span of 8 days we collected 4.5 days of velocity data and by averaging these data over intervals of 20 min, we obtained a total of 306 mean velocity profiles, each consisting of 25 depth cells. Shear was estimated by taking the analytic derivative of a fifth-order polynomial fit to the velocity profiles. A harmonic fit, composed of the four major tidal constituents (M2, S2, K1, and O1) and the zero-frequency residue, explains more than 91% of the variances of the velocity time series at each cell. Thus, the flow in Cordova Channel during this experiment was mainly tidal.

The depth-averaged velocities (for each 20-min average) are asymmetric with respect to the flood and ebb (Fig. 9). The ebb is stronger than the flood and their directions are not exactly opposite (they differed by 170°). The current direction difference between bottom and top layers is typically about 15°, both for flood and ebb. The veering with height during the flood (Fig. 10a) is in the same direction as for a bottom Ekman layer, but the backing during the ebb (Fig. 10b) is in the opposite bearing. The vertical rotation of the shear (Figs. 10c,d) is about 90°, much larger than it is for the flow. At 3.6 mab, the directions of the shear and the flow are quite close during the flood, whereas the shear is about 30° counterclockwise from the flow during the ebb (top panel, Fig. 10). The large transverse shear in the lower layer during the ebb provides a transverse stress directed offshore from Cordova Spit (Fig. 2).

We define the streamwise s direction to be aligned with the depth-mean flow, and make the streamwise velocity us positive during the flood and negative during the ebb. The transverse n direction is normal to s and points away from Cordova Spit during the flood. Figure 11 shows the depth–time sections of 1-day-long streamwise (us) and transverse (un) flow, their shears (∂us/∂z, ∂un/∂z), and vertical velocity (w). The uncertainty owing to turbulence in the mean shear estimate is 8 × 10−3 s−1.

The streamwise flow (Fig. 11a) shows a strong tidal signal, with the flow in the lower layer diminished by bottom drag. The nontidal variations can be strong enough to distort the tidal variations. For example, the strong ebb between days 24.3 and 24.6 has two peaks with weaker flow in between. The slight slanting of the isotachs indicates that the upper layer leads the lower layer in the decelerating stage, whereas it lags during the accelerating phase. The maximum speeds of streamwise flow are frequently found at heights below the uppermost bin.

The transverse flow is strongest in the near-surface and near-bottom layers, and weakest at middepth (Fig. 11b), with magnitudes reaching O(0.15 m s−1) at the lowest cell during the ebb, and about O(0.05 m s−1) in both the upper and the lower layers during the flood. The transverse flow is usually directed toward Cordova Spit in the lower layer, and in the opposite direction in the upper layer, with exceptions around the reversal of the tidal and the weak flood between days 24.6 and 24.8. The transverse velocity profiles usually vary linearly with height up to middepth but are more complex in the upper half of the water column. During the ebb near day 23.9 and between days 24.85 and 25, the transverse flow has a three-layer structure.

Both the streamwise and the transverse shear are bottom-enhanced (Figs. 11d,e). The streamwise shear reaches a magnitude of O(0.05 s−1) in the lower layer and contains a strong tidal signal. During the ebb, the transverse shear reaches about half the magnitude of the streamwise shear near the bottom, but it is small during the flood. The streamwise shear frequently reverses sign near middepth because the maximum velocity is frequently below the surface. Reversals of the shear profile are also found in the transverse direction during the ebb when the transverse flow has three layers.

The mean vertical flow (Fig. 11c) is generally less than 0.02 m s−1 in magnitude, but several significant up- and downwelling events can be identified with some occurring during flow turning and some during strong currents. These intensified vertical flow intervals generally occur over the entire profiling range, and the maximum velocity is at middepth, reaching O(0.05 m s−1) at times. These bursts are significant, at least up to middepth (section 3e). Note that an O(0.05 m s−1) vertical flow lasting 20 min is enough to bring the bottom water to the surface and vice versa, invoking a significant vertical exchange of water.

The stratification was not measured at the site of the ADCP. CTD profiles were taken at the south end of Cordova Channel and indicate that the lower half of the channel is well mixed when currents exceed 0.3 m s−1. The upper half of the channel is frequently stratified. The vertical gradient of salinity and temperature was measured every 5 min at middepth using the tethered microstructure profiler TAMI (Fig. 2), which was approximately 90 and 150 m north of the ADCP. At middepth, the Richardson number was less than ¼ for 50% of the time, assuming that one can combine the 20-min means from the two instruments. Thus, the Richardson number must have been mostly less than ¼ in the lower half of the channel because the shear increased and the stratification decreased toward the bottom.

b. Discussions

The measurements reveal a complicated picture of tidal flow in this natural channel. The occurrence of strong transverse flow can be attributed to the curvature of the channel, especially the presence of Cordova Spit at the western side boundary (Fig. 2). The forms of transverse velocity profiles are derived theoretically by Kalkwijk and Booij (1986) as
i1520-0426-16-11-1556-e6
where unc denotes the contribution from Coriolis forcing and unb from the effect of curvature (bending), f is the Coriolis parameter, h is water depth, R is the local radius of streamline curvature, us is the depth-mean flow of us and α = C1/2D/κ (CD is the bottom drag coefficient referred to the depth-mean flow and κ = 0.4 is von Kármán’s constant). The two profile functions, fc and fb, are nearly linear in z except very close to bottom. In deriving (6) and (7), Kalkwijk and Booij assumed that a log-layer exists in the streamwise direction and extends to the surface. A log-layer extending to large heights above the seabed has not been convincingly observed in previous work. The streamwise velocity in Cordova Channel, however, has been accurately fitted to the logarithmic form to heights of over 20 mab during peak flows (Lueck and Lu 1997), and this analysis gives CD = 4 × 10−3 with respect to the depth–mean flow. Hence, α ≈ 0.15. The functions fc and fb are both O(0.5) in the near-surface and bottom layers. Using typical values of f, h, and us in Cordova Channel, (6) predicts a transverse flow of only unc ≈ 0.01 m s−1. Thus, the Coriolis force makes only a small contribution to the transverse flow, and the predicted direction is opposite to the observed direction during the ebb. To reproduce the correct magnitude of un, which is approximately 0.05 and 0.15 m s−1 during the flood and ebb, respectively, one requires a radius of curvature of 4 and 1.5 km. Both radii are reasonable considering the topography of the channel near the ADCP. Cordova Spit and Saanichton Bay provide an expansion/contraction that is asymmetric with respect to ebb and flood (Fig. 2). During the ebb, the water must turn sharply around Cordova Spit while passing the ADCP. During the flood, the flow curvature is weaker because the channel is fairly straight south of the ADCP, and the flow can separate from the western shore just north of the ADCP.

The shear reversals may be due to the inertia of the oscillatory flow. However, the greater abundance of shear reversals during the ebb, as compared to the flood, suggests that these reversals are produced by the entrainment of shallow water from Saanichton Bay. This entrainment would slow down the water above middepth in the mainstream.

The occurrence of intense up- and downwelling events lasting longer than 20 min is surprising and difficult to explain without some measure of the horizontal distribution of the flow. Flow separation and the production of eddies by a headland are possible (Geyer 1993) and were visually observed in Cordova Channel (D’Asaro 1995, personal communication). Vertical flow can be induced by transverse flow and flow separation via the mechanism proposed by Garrett and Loucks (1976) and Wolanski et al. (1984).

5. Summary

This paper discusses the measurement principles of an ADCP, and the data processing required to apply this instrument to velocity profiling in a turbulent environment. The ADCP was rigidly mounted to the bottom of Cordova Channel where the flow is mainly tidal with peak currents of 1 m s−1 and rms turbulent fluctuations of O(0.05) m s−1.

An ADCP is not equivalent to a chain of point current meters in the sense that only the time-mean velocity vectors, rather than the “instantaneous” ones, can be obtained with an ADCP. The velocity is never horizontally homogeneous over the span of the beams in a turbulent flow. The assumption of horizontal homogeneity of the first moments is fundamental to the derivation of the mean velocity vector. Two tests of this assumption are available. The first compares the mean error velocity against its standard deviation and the second compares the mean error velocity against the mean speed. More than 95% of the samples pass both of these tests when the averaging length exceeds 50 beam separations.

Turbulence was separated from the “mean” using 20-min averages. The statistical uncertainty of the velocity estimates stems predominantly from turbulence rather than Doppler noise and is typically about 8 × 10−3 m s−1 for the horizontal components and 3 × 10−3 m s−1 for w and ê during strong flow and less for weaker flows. Because the uncertainty depends on the characteristics of the turbulence rather than the instrument, its evaluation requires rapid sampling (and recording) to resolve the velocity fluctuations. That is, simply recording the ensemble means is not sufficient to assess statistical uncertainty.

Bias in the measurement of tilt angles (1° in our case) can easily contaminate the weak vertical flow signals. The effect of this bias can be removed for rigidly mounted deployments by reprocessing the data with the correct tilt angles, but this is not possible for a nonrigidly mounted instrument that records only the mean velocity vector. The bias due to a possible correlation of tilt and horizontal velocity is insignificant for estimates of the horizontal flow but may be important for the vertical velocity.

The depth–time variations of flow and shear observed in this natural tidal channel are complicated. Both the magnitude and the direction of the ebb and flood are asymmetric. Strong transverse flow is observed and is attributed to flow curvature produced by Cordova Spit on the western side of the channel. The streamwise shear is bottom enhanced and strong transverse shear is observed only during the ebb. The streamwise shear frequently reverses sign at middepth during the ebb, and this is attributed to the entrainment of water from the shallow bay to the north of the experiment site into the mainstream of the channel. Up- and downwelling occurs in bursts that are significant, last for up to 20 min, and reach speeds of 0.05 m s−1 at middepth.

Acknowledgments

We would like to thank D. Newman and J. Box for their technical support to the field program, and A. Adrian for the deployment and recovery of the ADCP. We thank the reviewers from their many thoughtful and helpful comments. This work was supported by the U.S. Office of Naval Research under Grant N00014-93-1-0362.

REFERENCES

  • Garrett, C. J. R., and R. H. Loucks, 1976: Up-welling along the Yarmouth shore of Nova Scotia. J. Fish. Res. Board Can.,33, 116–117.

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  • Geyer, W. R., 1993: Three-dimensional tidal flow around headlands. J. Geophys. Res.,98, 955–966.

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  • Kalkwijk, J. P. T., and R. Booij, 1986: Adaption of secondary flow in near-horizontal flow. J. Hydraul. Res.,24, 19–37.

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  • Lhermitte, R., and U. Lemmin, 1990: Probing water turbulence by high frequency Doppler sonar. Geophys. Res. Lett.,17, 1549–1552.

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  • Lohrmann, A., B. Hackett, and L. P. Roed, 1990: High resolution measurements of turbulence, velocity and stress using a pulse-to-pulse coherent sonar. J. Atmos. Oceanic Technol.,7, 19–37.

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  • Lu, Y., and R. G. Lueck, 1999: Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Oceanic Technol.,16, 1568–1579.

  • Lueck, R. G., and Y. Lu, 1997: The logarithmic layer in a tidal channel. Contin. Shelf Res.,17, 1785–1801.

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  • Wolanski, E., J. Imberger, and M. L. Heron, 1984: Island wakes in shallow coastal waters. J. Geophys. Res.,89, 10 553–10 569.

Fig. 1.
Fig. 1.

Transducer geometry and beam orientation of the ADCP. The beams are inclined by θ = 30° from the axis of the instrument, which is nominally vertical. Here, ϕ1, ϕ2, and ϕ3 are heading, pitch, and roll angles.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 2.
Fig. 2.

Area chart showing the bathymetry (depth in meters) of Cordova Channel, the location of the ADCP, and the positions of the moored microstructure instrument during its two deployments (TAMI1 and TAMI2). A current meter (CMI), transmitter (T), and receiver (R) of an acoustic scintillation system were also deployed during the experiment.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 3.
Fig. 3.

One-day-long vertical velocity at middepth (15.6 mab), calculated for roll angle ϕ3 = −2° (thick lines) and ϕ3 = −3° (thin lines). The two upper curves are calculated from 20-min smoothed beam velocities and the lower curves are 3-h smoothed. The lower panel is a stick diagram of the horizontal current at middepth with a north flow directed straight up.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 4.
Fig. 4.

A sample of one day of data collected at middepth (15.6 mab) from a single beam that was oriented in the downstream direction during ebb tide. The upper curve is the raw data of four-ping averages collected every 3.05 s. The middle curve is the same data with 20-min smoothing and offset by −0.5 m s−1. The lower curve is the beam velocity fluctuation formed by taking the difference of the upper two curves and offset by −1.5 m s−1. The stick diagram in the lower panel is the 20-min mean horizontal velocity at middepth derived from the velocity along all four beams.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 5.
Fig. 5.

The lagged autocorrelation of the high-passed velocity from a single beam that was directed along the flow. (a) and (c) The correlation at 15.6 mab, (b) and (d) the correlation at 3.6 mab. (a) and (b) Conditions during strong ebbing starting at day 23.91; (c) and (d) give the correlation during the weak flood starting at day 24.6. The horizontal dashed lines represent the confidence bounds for large lags. The thick curves are the correlations for all lags up to 20 min (labels on lower axis); thin curves are the correlations for lags shorter than 45 s (labels on upper axis).

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 6.
Fig. 6.

(upper panel) Twenty-minute mean vertical (solid lines) and error velocities (dashed lines) at the heights of 15.6, 3.6 (offset by −0.05 m s−1), and 27.6 mab (offset by 0.05 m s−1). The sign of the error velocity is made identical to the sign of the vertical velocity to facilitate their comparison. The shading spans ±1 standard deviation of the vertical and error velocities (3 × 10−3 m s−1). (lower panel) The ratio of horizontal averaging length to beam separation, M = /L, at 3.6, (thin, upper curve), 15.6 (thick, middle curve), and 27.6 mab (dashed, lower curve).

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 7.
Fig. 7.

(a) A scatterplot of the magnitude of the ratio of error velocity to its standard deviation vs the nondimensional averaging length M. Points that fall below the ratio of 1 (horizontal line) pass the first test of the assumption of horizontal homogeneity of the mean. (b) Upper panel: The percentage of samples with velocity ratios less than 1 as a function of M. The 95%–100% range is shaded. Lower panel: histogram of M.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 8.
Fig. 8.

(a). A scatterplot of the magnitude of the ratio of error to along-channel velocity vs the nondimensional averaging length M. Points that fall below the ratio of 0.01 (horizontal line) pass the second test of the assumption of horizontal homogeneity of the mean. (b) Upper panel: The percentage of samples with velocity ratios less than 0.01 as a function of M. The 95%–100% range is shaded. Lower panel: histogram of M.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 9.
Fig. 9.

Polar diagram of the 20-min depth mean flow for all 4.5 days of data with respect to true north (0°).

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

Fig. 10.
Fig. 10.

(upper panel) Time series of the direction of the 20-min mean velocity at 3.6 mab (solid line) and 27.6 mab (dashed line) and the shear at 3.6 mab (circles). (lower panels) Typical profiles of current direction during (a) flood and (b) ebb and of the shear direction during the (c) flood and (d) ebb.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

 Fig. 11.
Fig. 11.

Cordova Channel depth–time sections of 20-min mean (a) streamwise, (b) transverse, and (c) vertical velocities (all in m s−1); (d) streamwise and (e) transverse shear (both in s−1). The solid line marks the log-layer height. For the streamwise flow, red and blue shades are floods and ebbs, respectively. Values smaller than one standard deviation are coded in shades of gray.

Citation: Journal of Atmospheric and Oceanic Technology 16, 11; 10.1175/1520-0426(1999)016<1556:UABAIA>2.0.CO;2

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