Variability of turbulent quantities in the tidal bottom boundary layer: Case study in the eastern English Channel

Continental Shelf Research 58 (2013) 21–31

Contents lists available at SciVerse ScienceDirect

Continental Shelf Research journal homepage: www.elsevier.com/locate/csr

Research papers

Variability of turbulent quantities in the tidal bottom boundary layer: Case study in the eastern English Channel Konstantin Korotenko a,b,n, Alexei Sentchev a, Francois G. Schmitt c, Nicolas Jouanneau a a

Laboratoire d'Océanologie et de Géosciences, UMR8187, Université du Littoral Côte d'Opale, Wimereux, France P.P Shirshov Institute of Oceanology, RAS, Moscow, Russia c Laboratoire d'Océanologie et de Géosciences, UMR8187, CNRS, Wimereux, France b

art ic l e i nf o

a b s t r a c t

Article history: Received 30 July 2012 Received in revised form 11 December 2012 Accepted 5 March 2013 Available online 16 March 2013

Tidal current structure and turbulent quantities within a tidal bottom boundary layer (BBL) have been examined using an upward-looking acoustic Doppler current profiler (ADCP). The instrument was deployed on the seafloor, off the north-eastern French coast in the eastern English Channel over 12 tidal cycles and covered the period of the transition from mean spring to neap tide. Forcing regimes varied from calm to moderate storm conditions during the deployment. For the study of turbulent quantities in the BBL, we have chosen a calm period, when an effect of surface waves on the velocity structure was negligible. Stresses were found to vary regularly with the predominantly semidiurnal tidal flow, with the along-shore stress being generally greater during the flood flow (∼3.0 Pa) than during the ebb flow (∼−1.5 Pa). The turbulent kinetic energy (TKE) production rate, P, TKE density, Q, and its dissipation rate, ε, followed a nearly regular cycle with close to quarter-diurnal period. Near the seabed, peak values of P, Q and ε were found to be 0.5 W m−3, 0.5 m2 s−2 and, 0.04 W m−3, respectively, during the flood while, during the ebb, these quantities reached lesser values: 0.1 W m−3, 0.1 m2 s−2 and 0.03 W m−3, respectively. Near the bottom, eddy viscosity, Az, peak ranged from about 0.1 m2 s−1 during the flood to 0.03 m2 s−1 during the ebb flow. Away from the bottom, Az increased to reach a maximum near the middepth. Time–depth variation of the P/ε ratio indicated that the turbulence in the BBL, most of the time, was at a non-equilibrium state (P/ε≠1). The largest deviation from the equilibrium occurred during the flood, when P/ε exceeded about one decade near the bottom. During the ebb, P/ε was close to the equilibrium state, slightly decreasing with height above the bottom. Results are found to be in a good agreement with those of the other researches working on direct measurements of turbulence in tidal flows. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Reynolds stress Bottom boundary layer Tidal current Variance method ADCP English Channel

1. Introduction A turbulent boundary layer is formed above the seabed by bottom friction. Within the boundary layer the flow is attenuated, while the shear and frictional force are enhanced, and the turbulent kinetic energy production and its dissipation are intensified. Mechanisms controlling the transport, erosion and deposition of fine sediments, and mechanisms controlling mixing in the bottom boundary layer (BBL) are directly affected by highly variable hydrodynamic conditions and turbulence properties near the seabed. Understanding these mechanisms in the tidal boundary layer is one of the key goals of coastal physical oceanography, since turbulent processes are crucial in controlling flow dynamics

n Corresponding author at: P.P Shirshov Institute of Oceanology, RAS, Moscow, Russia. Tel.: þ 7 499 129277; fax: þ 7 499 1245983. E-mail addresses: [email protected], [email protected] (K. Korotenko).

0278-4343/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.csr.2013.03.001

and the vertical exchange of momentum and scalars throughout the water column. Another aspect, where understanding of turbulence from direct measurements is important, is testing and verification of coastal ocean circulation models. Until the end of the 1980s, it was only possible to test the performance of such models against mean flow parameters (Davis and Flather, 1987). In these conditions, differences in the velocity profiles were effective in discriminating between the numerical schemes. However, in many cases, particularly in tidally dominated flows, the velocity profile is very sensitive to changes in the eddy viscosity profile and only direct measurements of the Reynolds stresses can help to validate model parameterization of vertical mixing due to turbulence (Simpson et al., 1996). The development of free-fall profilers for the measurement of finescale velocity shear has facilitated the estimation of the dissipation rate of turbulent kinetic energy (Dewey et al., 1987; Monin and Ozmidov, 1985; Gibson et al., 1993, Caldwell and Moum, 1995; Simpson et al., 1996, Paka et al., 1999). Recent

22

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

advances in technology, marked by greatly improved Acoustic Doppler Current Profilers (ADCP), became invaluable tools in studies of turbulence in vertically sheared flows. The popularity of ADCPs has considerably increased since Lohrmann et al. (1990) introduced a method, known as the Variance method (VM). In this method, Reynolds stress profiles are estimated from along-beam velocity measurements, using the difference between the velocity variances along opposing beams. Stacey et al. (1999), Lu and Lueck (1999a, b), and Williams and Simpson (2004) extended this work to analyses of the confidence of Reynolds stress estimations. Since that, the use of ADCP for direct measurements of turbulent quantities has had stunning success in coastal oceanography while the use of free-fall profilers near the bottom is still associated with certain difficulties and even instrument destruction and loss (Korotenko and Nabatov, 1987). The VM has been successfully used in a large number of studies of energetic tidal systems (Lu and Lueck, 1999a,b; Stacey et al., 1999; Seim, 2002; Rippeth et al., 2002, 2003; Fugate and Chant, 2005; Souza and Howarth, 2005; Nidzieko et al., 2006; Peters and Johns, 2006; Korotenko and Sentchev, 2011, Korotenko et al., 2012). It should be noted, however, that in the presence of energetic surface gravity waves the prediction of turbulent quantities with VM presents certain difficulties. The problem is that wind-induced waves can produce velocity variances of an order of magnitude larger than those associated with turbulence, and they often dominate the measured covariance between horizontal and vertical velocities. Since surface waves often occupy the same frequency range as marine turbulence, it is difficult to separate the latter from waveinduced velocity fluctuations using simple filtration. Therefore, over the past decade, a number of methods for reducing wave impact on turbulence measurements were developed (Shaw and Trowbridge, 2001; Trowbridge and Elgar, 2003; Whipple et al., 2006; Feddersen and Williams, 2007; Schmitt et al., 2009, Huang et al., 2010; Kirincich et al., 2010). Since each of these methods has certain disadvantages (Rosman. et al., 2008), the most appropriate way for studying turbulence in the BBL is to use ADCP either during nonwavy conditions or when the effect of surface waves is negligible. Given a long duration (one week) of measurements and a variety of meteorological conditions observed during the experiment in the EEC, we adopted the following approach to our study. Firstly, we perform the analysis of velocity variations and external forcing in order to choose a suitable period for estimating turbulent quantities, i.e. a period when the effect of surface waves on bottom turbulence was insignificant. Secondly, for the period chosen, we examine depth–time series of turbulent quantities in the tidal BBL and compare them with those obtained by the other researchers. This paper is organized as follows: In Section 2, we describe the region of interest, experimental settings, measurements and forcing data. In Section 3, we analyze characteristics of mean current for different intervals of the deployment period and we present velocity spectra for a nonwavy period chosen for subsequent analysis. We also test the Reynolds stresses with the Variance Fit (VF) method to show the negligibility of their contamination by waves in BBL for the period chosen. In Section 4, we present the estimation of turbulent quantities in the BBL: the Reynolds stresses, the TKE density, the turbulent viscosity and its production and dissipation rates. Discussion of these results and the comparison of those obtained by other researches are given as well. Section 5 summarizes the results. Appendix A provides a brief description of the methods used for estimating the turbulent quantities.

2. Observations and external forcing The study area, instrument orientation and deployment, and forcing have been already described in Korotenko et al. (2012).

Here, we remind some important information and add some clarification needed for further analysis of the results.

2.1. Study area and ADCP deployment Velocity measurements were performed in the eastern English Channel (EEC), approximately 6 km offshore, northwest of the port of Boulogne-sur-Mer (BLM), France (Fig. 1). The area of interest is characterized by a tidal range up to 7 m and current velocity amplitude close to 1.5 m s−1 at spring tide, and about 0.7 m s−1 during neap tide. Tidal currents have a predominantly semidiurnal period with a pronounced fortnightly modulation due to interference of the major semi-diurnal constituents (M2, S2, N2). A significant asymmetry of the sea surface elevation curve in the study area revealed the contribution of higher order non-linear harmonics (M4, MS4), which also generated larger velocity during the flood flow. A 1.2-MHz upward-looking four-beam broadband RDI ADCP was deployed on the bottom (18 m mean water depth) for one-week

Fig. 1. The eastern English Channel (upper panel) and location of the ADCP deployment site (black circle) in the Strait of Dover (lower panel). The gray circle denotes the location of the Boulogne-sur-Mer (BLM) lighthouse and tidal gauge. The bottom topography is also shown. The x–y plane and direction of ebb and flood flows are shown in the upper panel. Tidal ellipse at nearest point to ADCP deployment is shown in the inset.

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

period, from June 9 to 16, 2009, covering tide evolution from spring to neap. The instrument was operated in fast pinging mode 12, providing one instantaneous velocity profile per second. Each velocity record was an average of ten short pulse measurements over a second interval. Velocities were recorded in beam coordinates with 0.5 m vertical resolution (bin size), starting from 1.5 m above the bottom (the lowest bin). The orientation of the ADCP horizontal axes (heading) was chosen with respect to shoreline (stretched meridionally) and dominant current direction, so that the opposing beams 3 and 4, lying in the y–z plane (deflection from northward direction was 51), allowed us to estimate the along-shore component of current velocity and Reynolds stress. Beams 1 and 2, lying in the x–z plane, allowed estimating the cross-shore component of these quantities. Note that, at the site chosen, the axis of dominant tidal current is rotated at less than 151 in respect to shoreline direction, as shown in Fig. 1 (upper panel). In the area of interest, tidal ellipses are greatly elongated (the aspect ratio is about 1:6) and oriented along the coast. In Fig. 1 (upper panel), we inserted a tidal ellipse inferred from radar measurements at a nearest to the ADCP deployment point (Sentchev and Korotenko, 2004). 2.2. Forcing: wind and wave data Hydrodynamic conditions in the Eastern English Channel (EEC) can vary from a relatively simple ebb-and-flood tidal system to a very complex one in which tide, wind stress, freshwater influx, and wind waves have significant forcing effects on the system (Brylinski et al., 1996; Sentchev and Korotenko, 2004, 2005; Sentchev and Yaremchuk, 2007; Korotenko and Sentchev, 2004, 2008, 2011; Korotenko et al., 2012). In subsequent analyses, we complemented velocity measurements by wind and wave data to choose a proper period, when the influence of waves on current structure would be negligible. Wind speed and direction were recorded at the BLM lighthouse (Fig. 1). Significant wave height, HS, was extracted from ADCP measurements. Fig. 2 shows the wind record and time variation of HS in response to wind forcing. Two different wind regimes could be identified during the observation period. Winds blowing from southern and southwestern sectors, with moderate to strong speed (up to 10 m/s), were dominant during the measurements. Weaker winds (≤6 m/s) from northern and northwestern sectors represented the second characteristic regime of the regional atmospheric circulation. Rapid changes in wind direction back and forth, occurring on a time scale of the order of a day, were a noticeable feature of the local wind variability. Calm and stormy weather conditions followed each other during the experiment. As seen, two calm periods were characterized by weak winds that were from northern or southern sectors and low waves, the significant wave height of which, HS, was less than 0.4 m. In between, the storm (began at 2200 h GMT on June 10) lasted slightly more than one day, and peaked at the

Fig. 2. Time series of averaged wind speed (gray line) and significant wave height, HS, observed by ADCP (black line). Two distinct storm events and calm periods, identified based on the wind speed and wave height records, are annotated above the panel.

23

end of June 11, when HS reached 1.4 m. For the early hours of June 12, both wind speed and significant wave height dropped abruptly to 2 m s−1 and 0.3 m, respectively. More details on the forcing conditions are given in Korotenko et al. (2012). 2.3. Stratification The observational period was characterized by a homogeneous distribution of temperature and salinity throughout the water column. Although, during calm periods, a weak diurnal thermal stratification appeared in the upper 2 m-layer.

3. Data analysis For the whole period of ADCP measurements, detailed analyses of velocity and features of tidal current in the EEC were carried out by Korotenko et al. (2012). Here, we summarize some important results for tidal velocity structure and spectra that allowed choosing the right period for studying turbulent quantities in the BBL, when a wave impact on turbulent quantity estimates is negligible. 3.1. Time–depth variability of mean current and velocity shear Fig. 3 shows velocity variation during multiple tidal cycles in the Strait of Dover and indicates large velocity (41 m s−1) and significant asymmetry of the sea surface height and current. Stronger currents occurred on flood, weaker currents on ebb, and high values of current acceleration are typically observed during rising tide (Fig. 3a, b). The period of rising tide lasted less than that of falling tide. Because of the strong asymmetry of the SSH curve, the period of falling tide exceeds that of rising tide by approximately 2 h. Moreover, the tidal current lags the sea level by approximately 2.5 h. Therefore, the surface current reversal (S–N component) occurred 2.5 h or 3 h before the arrival of the high water or low water, respectively, in BLM. A counter-clockwise veering of current vector with depth could be also recognized in the point of the ADCP location. Ebb and flood periods could be easily defined using zero velocity contours in the field of the crossshore velocity component shown in Fig. 3a. Fig. 3 shows that, during the experiment, the mean current velocity reached maximal magnitudes at the surface while the highest values of velocity shear were found near the seabed. The along- and cross-shore components of velocities and their shears are tidally forced and exhibit semidiurnal variability. The alongshore velocity exceeded 1.1 m s−1 on the flood flow and −0.7 m s−1 on the ebb flow, while the across-shore velocity component was much weaker and its magnitude did not exceed 0.4 m s−1. Note that storm periods could be distinctly identified due to ‘knotted’ lines of zero-mean shear of both components of current velocity (Fig. 3c, d). During both storms, velocity shears significantly decreased near the sea surface, and gradually began to restore when winds ceased. This can be more clearly seen in Fig. 3e showing the sum of the velocity components shear squared: "

2 # ∂u 2 ∂v S2uv ¼ þ ð1Þ ∂z ∂z for three-day period following the first storm. During this period, the zero magnitude of S2uv appeared at the sea surface (the same for the second storm event), while during the calm period following the storm, minimum magnitudes of S2uv were found only along the vertical lines specifying time and position of the water slack, where current velocities and shears were weak. Hereafter, we conventionally refer to the water slack as a moment when the flow velocity passes through a minimum while the current vector draws an ellipse (Fig. 1). Asymmetrical tide, according to our

24

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

Fig. 3. Time–depth variability of (a) cross-shore u(m s−1) and (b) along-shore v(m s−1) mean current velocity components, their shears (c) log½du=dz10 (s−1) and (d) log½dv=dz10 (s−1), and (e) shear velocity squared logS2uv 10(s−2). The sea level is marked by bold solid line over the mean velocity components. Zero-mean velocity and shear components are marked by solid lines. The abbreviation ‘mab’ denotes meters above the bottom. The storm and calm periods correspond to those indicated in Fig. 2 (after Korotenko et al., 2012).

previous study (Sentchev and Korotenko, 2005; Korotenko and Sentchev, 2008) and observations (Prandle et al., 1993), creates a residual current, which is fairly strong (∼0.25 m s−1) and represents a northward jet-like flow directed along the French coast.

3.2. Velocity spectra For the analysis of velocity spectra, we used power spectral density estimate by a complex Fast Fourier Transform (FFT) with a Nyquist frequency of 0.5 cycles per second (cps). The essential requirement for applying a classical FFT method is the continuity of the data recorded. Therefore, the time series of the current velocity components were inspected for gaps in the initial 1 cps sampled data. Their analysis revealed that, in the layer ≤12 m above the bottom (mab), the data had no gap longer than 6 s. All gaps discovered, shorter than 6 s, were linearly interpolated. The data in the upper layer (412 mab) were excluded from subsequent spectral analysis. Fig. 4 shows the power spectral density of cross-shore, u, and along-shore, v, components, of the velocity vector at 7 mab, recorded during the first calm period. Spectra of both components have a maximum at the semidiurnal frequency 0.08 cycles per hour (cph), while the diurnal frequency, which peaked at 0.04 cph is not clearly pronounced because of a short FFT-length. Major tide constituents are indicated on the spectra in Fig. 4. Weak power peaks at 250–400 cph are associated with contributions of swells with the period of 10 s and wind-induced waves with the period of 5 s. At frequencies higher than 400 cph,

Fig. 4. Power spectral density for cross-shore, Euu (dotted) and along-shore Evv (solid) components of the horizontal velocity at 7 mab computed for the first calm period. Solid black lines represent the Kolmogorov −5/3 slope expected for an inertial subrange.

the spectra indicate a well-pronounced inertial subrange with the spectral slope of −5/3. Fig. 4 reveals that influence of wind-induced waves and swells is relatively small at 7 mab. To show their possible impact, we used

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

25

The cross- and along-shore components of Reynolds stress show regular variations with tidal flow (Fig. 6a and b). During the flood, both stresses are positive (warm shading) and their

magnitude generally decreases away from the bottom. During the ebb, both stress components are negative (cool shading) and their magnitude also decreases with the increasing height above the bottom. It is clearly seen during low and high water, when shading changes vertically from blue to light blue and from dark red to yellow, respectively. During current reversal, estimates of the Reynolds stresses are unreliable and often grew away from the seabed. The vertical structure and profiles of the Reynolds stress have been discussed in detail by Korotenko et al. (2012). Fig. 6a and b also reveals a pronounced asymmetry of stress magnitudes between the ebb and flood. The stress cycle is seen to be highly regular and dominated by the along-shore component, τy, which, at 2 mab, exceeds 3 Pa and −1.5 Pa on flood and ebb flows, respectively. The cross-shore stress, τx, at the same depth rarely exceeds 1.5 and −0.5 Pa on flood and ebb flows, respectively. Note that large stresses estimated during the flow reversal are often unreliable. In the upper panel of Fig. 6, flow reversal events are marked by black dots on the SSH line. The projection lines drawn down from the dots allow tracing turbulent quantity estimates corresponding to these events. As seen, large unreliable stresses obtained during the tide flow reversal mostly occurred at the beginning of flood periods, although, the along-shore stress indicates large magnitudes also at the beginning of ebb. Further, to elucidate the behavior of the Reynolds stresses near the seabed, we created a scatter plot of the along-shore stress versus velocity squared (Fig. 7). The plot indicates that the stress near bottom and the mean tidal currents are highly correlated and exhibit quadratic drag law behavior. The drag coefficient, C D ,

Fig. 5. Time series of Reynolds stresses calculated using the VF method (solid line) and uncorrected (dashed line) Reynolds stress time series at 7 mab (after Korotenko et al., 2012).

Fig. 7. Scatter plot of the along-shore velocity square at 2.5 mab versus near-bed Reynolds stress averaged within the layer 1.5–4.0 mab. Both quantities were 10 min averaged. Linear fit (dashed lines) for the flood flow yields a drag coefficient of 0.0025 and for the ebb flow yields a drag coefficient of 0.0012.

the VF method suggested by Whipple et al. (2006). This method allowed us to assess the contribution of wave-induced contamination of the Reynolds stresses and, then, perform wave-bias correction of them (Korotenko et al., 2012). Fig. 5 shows time series of de-meaned values of the along-shore Reynolds stress corrected with the VF method and that uncorrected (Eq. (A1)) at 7 mab for the first calm period. As seen, the values of the corrected and uncorrected Reynolds stresses are mostly tracking each other indicating that wave-induced contamination, for the period chosen, is very small. Note that for further study we have chosen the first calm period despite the second one being longer. A comparative analysis of the Reynolds stress with the VF method for the second calm period revealed a greater contamination of the Reynolds stresses by waves than the period chosen.

4. Results and discussion Fig. 6 shows the depth–time variation of the 10-min averaged estimates of turbulent quantities computed by applying the variance method (Appendix A). 4.1. Reynolds stresses

Fig. 6. Time–depth variations of the Reynolds stresses (a) τx (Pa) and (b) τy (Pa), (c) the angle ϕ (degree) between mean current and stress vectors, and (d) turbulent kinetic energy TKE (m2 s−2). The sea surface level (SSH) with black dots indicating turning of tidal current are shown at upper panel. Zero-stress components (panels a, b) are marked by solid black lines.

26

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

strongly depends on the phase of the tide. Its estimates based on the 10 min averaged velocities, for the first calm period, varied systematically between 0.0012 on ebb and 0.0025 on flood. Difference of the drag coefficient between flood and ebb is not often observed in natural tidal flow and its explanation is of great interest. In a tidal inlet, Seim (2002) has found similar dependence of the drag coefficient on phase of tide. C D was found to vary from 0.0013 on flood to 0.0021 on ebb that is very close to that we obtained. In Seim (2002), the discrepancy of C D between flood and ebb was explained by bottom dense water inflowing in the inlet only during the flood phase. The bottom stratification created by dense water presumably suppressed overturning as the tidal current changes direction and limits overturning scales throughout the flood. Since, in our case, there was no near-bottom stratification, we can assume that the dependence of the drag coefficient on the tidal flow direction was associated with the bottom topography, which, according the quadratic drag law, was likely rougher to the immediate south of the ADCP than it was to the north. However, to confirm this, we need additional data on the roughness of the bottom. In addition to the abovementioned study by Seim (2002), we should mention works by Howarth and Souza (2005) and Willes et al. (2006), in which dependence on the phase of tide was also traced, although rationale of it was not discussed. 4.2. Reynolds stress vector rotation In study of shear-induced turbulence in tidal flow, differences between direction of the mean tidal current and the Reynolds stress provides important information for understanding the internal structure of turbulent flow and its evolution. In our work, we study variations of the angle (ϕ) between mean current and stress vectors. Fig. 6c illustrates the time–depth distribution of ϕ throughout the bottom layer studied. Here, the cold shading denotes that the stress lags the mean current while the warm shading denotes that the stress advances the mean current. Note that during a tidal cycle, as it follows from Fig. 3, the vector of the mean current rotates counter-clockwise with the tidal current ellipse. The structure of the angle distribution indicates its slight variation with depth. During strong flood and ebb flows, the values of the angle ϕ are shaded mostly with dark blue color throughout the water column indicating that the stress acted at angles close to those of the mean flow. A deviation of the vectors from the exact alignment results in turning the shading from dark blue to dark red, as seen at different depths during flood. In contrast to periods of strong flood and ebb, in periods close flow reversal, one can see a considerable misalignment of vectors. It is remarkable that, in periods of the transition from ebb to flood, the mean current vector advances that of the stress (dark blue shading sharply turns to light blue) while in periods of the transition from flood to ebb, on the contrary, the stress advances the mean current (red shading gradually turns to light blue). 4.3. TKE density Fig. 6d shows the estimated quantity Q, which is related to TKE density, q2/2 by Eq. (A4). The magnitudes of Q spans about three decades, ranging from 0.05 m2 s−2 to 10−4 m2 s−2, the latter being close to the Doppler noise level of Q according to tests of Lu and Lueck (1999b). Similarly to the Reynolds stress, the magnitude of Q increases toward the bottom; however, its increase occurs more slightly than that for the stress. The increase is clearly seen during the strong flood while, on the ebb, such a tendency is undistinguishable in the bottom layer studied. Nevertheless, temporal variations of Q at fixed depth indicated that, at 2 mab, maxima of TKE density were found to be about 0.025 and 0.02 m−2 s−2 on

flood and ebb, respectively, while, at 5 mab, they were about 0.02 and 0.016 m−2 s−2, respectively. Detailed analysis showed that depth variations of Q along a single profile do not exceed one decade. Lu and Lueck (1999b) obtained similar estimates of Q and showed that the magnitudes of Q are proportional to those of the Reynolds stress by a factor of 10, i.e., Q ¼ 3:1j−v′w′j, which was close to the parameterization by Gross and Nowell (1983) inferred from their measurements in the near-bottom region of tidal boundary layer. Lu et al. (2000) noticed an interesting character of Q variations during the tide cycle. They found that the largest estimates of Q have occurred at the beginning and end of the ebb (in their case ebb was stronger than flood) and the smallest ones are obtained during weak flows. 4.4. TKE production rate As shown in Fig. 8a, the TKE production rate, P, was estimated from the product of the Reynolds stress and the velocity shear according to Eq. (A2). It indicates the amount of energy that is transferred from the mean flow to turbulent kinetic energy. For nonwavy conditions, in a tidal flow, P intensifies toward the seabed, bearing the character of wall-bounded turbulence. As with the stress, there is a clear asymmetry in evolution of P between flood and ebb. Near seabed, peak value of P, on strong ebb, is typically an order of magnitude less than that observed at strong flood. So that, at 2 mab, the magnitude of P spans about four decades, ranging from 0.2–0.5 W m−3 on flood to ∼10−5 W m−3 at all levels during weak flows. The latter is close to noise level of P estimated by Lu and Lueck (1999b). Negative estimates of P (blank areas in Fig. 8a) appear either due to round-off (and are usually small), or they are caused by unreliable stress estimates obtained during the turning of the tide when the stress and correspondent shear have opposite signs, as illustrated in Fig. 9. Similar variations and values of TKE production rate were obtained by Rippeth et al. (2002) for a tidal channel. For spring tide, they found that the magnitude of P spanned about five decades, ranging from about 1 W m−3 near the bottom to ∼10−5 W m−3 during weak flows while, for neap tide, it spanned four decades with a maximum about 0.5 W m−3 near the bottom. 4.5. TKE dissipation rate The depth–time evolution of TKE dissipation rate, ε, derived from the structure function method (Appendix A), reveals many common features with P (Fig. 8b). There is a quarter diurnal variation in both P and ε, which is clearly related to the phase of the tidal current. As P, ε decreases with height above the seabed with a reduction of about an order of magnitude at 7 mab. At 2 mab, estimates of dissipation rates were found to be about 0.04 and 0.03 W m−3 on flood and ebb, respectively, while, at 5 mab, they were about 0.015 and 0.01 W m−3. Our estimates of TKE dissipation rate derived from the structure function method and their variations are very close to those obtained by Willes et al. (2006) using the same method as well as close to direct measurements of dissipation rate with a free-fall profiler conducted by Simpson et al. (1996). 4.6. Turbulent viscosity As presented in Fig. 8c, the eddy viscosity coefficient Az, was calculated by dividing P by the squared shear (Appendix A). Eq. (A1) indicates that the sign of Az is the same as P. Fig. 8c illustrates pronounced depth–time variations of Az with tidal flow intensity. Generally, the eddy viscosity increases with increasing height above the bottom in the lower half of the water column,

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

27

Fig. 8. Time–depth variations of (a) log10 P (W m−3), obtained from the variance method, (b) log10 ε (W m−3) obtained from the structure function method, (c) turbulent viscosity Az (m2 s−1) and (d) log10 (P/ε). The sea surface height (SSH) and moment of current reversal (black dots) are shown at upper panel. Zero-values of log10 (P/ε) are marked by solid black contour lines. The blank areas in (a), (c), and (d) represent negative values.

Fig. 9. Time series of the along-shore components of Reynolds stress, τy (solid line), shear, Sv (dotted line) and TKE production rate, P (dashed line) at 1.5 mab. Gray shading indicates zones of negative P.

and reaches a maximum near the mid-depth (Korotenko et al., 2012). Within the layer we inspected, peak values of Az, at 5 mab, ranged from about 0.017 m2 s−1 during ebb flow to 0.1 m2 s−1 during flood flow. At 2 mab, these values ranged from 0.015 m2 s−1 to 0.08 m2 s−1. The estimates of Az are subject to large uncertainties, particularly when the magnitudes of the stress shear are small. Near the slack water, Az reached its resolved minimum value of 10−3 m2 s−1 that is rather large for calm water. The true minimum value of Az, however, remains unknown since estimates of turbulent viscosity, following Eq. (A3), directly depend on the choice of noise level for P. Negative values of Az, in Fig. 8c, associated with those of P were blanked. The magnitudes and range of variation of Az, obtained in our study, were found to be similar to estimates reported by Lu et al. (2000) where, for a tidal channel, the values of Az varied from 0.3 during strong flows to about 10−3 m2 s−1 during weak flows, and reached a maximum at mid-depth. 4.7. P/ε ratio The ratio of the turbulent kinetic energy production rate, P, to its dissipation, ε, indicates the state of turbulence and its variability throughout the water column. We estimated this ratio by calculating directly P/ε for each level and time interval. Note that P/ε ¼1 means that turbulence is in the equilibrium state, i.e., the rate of TKE production is equal to its dissipation. For clarity, the equilibrium value (log 10(P/ε) ¼0) is marked in Fig. 8d by the black contour separating ratios P/ε 41 (warm shading) from P/ε o 1 (cold shading). Fig. 8d shows that the ratio P/ε varies approximately between 0.1 and 10 during a tidal cycle.

Fig. 8d reveals an important property of turbulence, i.e., over the tidal cycle it significantly departs from the equilibrium state. So that, during the flood, TKE production rate is considerably higher than that of dissipation and, despite the fact that P/ε has a tendency to decrease away from the bottom, the ratio remains larger than the equilibrium value in the BBL. During ebb, the P/ε ratio is slightly higher than unity near the bottom. It gradually decreases with increasing distance from the bottom, approaching the equilibrium value P/ε ¼ 1 at some depth above the seabed (Fig. 8d). To examine the variability of P/ε ratio with a tidal cycle in detail, in Fig. 10, we present the time series of P/ε at 1.5, 4 and 6.5 mab. The most remarkable feature is an asymmetric variability of P/ε. The ratio increases slowly at the beginning of the flood, falls sharply at the end, and varies irregularly during the flood. In contrast, during the ebb, P/ε indicates more regular (nearly symmetric) variations. Fig. 10 also indicates a general tendency of P/ε to decrease with the height above the bottom for both flood and ebb periods. During flood, the P/ε ratio is higher than the equilibrium value at all the three depth levels shown. At 1.5 mab, the TKE production rate is larger than the dissipation rate by a factor of 12. Such a difference gradually decreases with the distance above bottom. During ebb, the behavior of P/ε is completely different. Globally, the ratio is close to equilibrium value at higher levels ( 44 mab) and exceeds 2 in the near-bottom layer (1.5 mab). Negative estimates of P/ε, in Fig. 10, associated with negative P are blanked. A comparison of our estimates of the P/ε ratio with those obtained by other researchers indicates that our estimates for flood phases significantly exceed those reported, for example, in Rippeth et al. (2003), where peaks of the P/ε ratio ranged from 1.17

28

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

Fig. 10. Time series of P/ε ratio at 1.5 (solid line), 4.0 (dashed line) and 6.5 (dotted line) mab. SSH is shown by solid line with its scale given in mab. The level of equilibrium turbulence is shown by the line P/ε ¼1. Blanking indicates zones of negative estimates of P/ε ratio.

Fig. 11. Mixing length lm (diamonds) and lz (solid line). Blanking indicates zones of negative estimates of lm.

to 3.3 for ebb and flood, respectively. Using the structure function method, Willes et al. (2006) reported on estimates of the P/ε ratio close to ours. According to Fig. 2 from Willes et al. (2006), on flood and ebb, TKE production rate exceeded dissipation rate by about a factor of 10.

ADCP) most of the time was between 0.3 and 1 m and rarely exceeded 1 m.

5. Summary 4.8. The mixing length As we showed above and in Korotenko et al. (2012), in the nearbottom layer, the turbulence structure bears the character of a wall-bounded turbulence and the mean velocity profiles are fitted accurately to a log-layer. For additional tests whether the observations fit to the mixing length theory, we derive the mixing length from our ADCP measurements. According to the Prandtl theory, in unstratified wall-bounded turbulent flows, the mixing length is proportional to the distance from the wall. Since in shallow waters the growth of eddies is constrained by the presence of both the seabed and surface, the mixing is determined as z-dependent function (e.g., Simpson et al. (1996)), namely  z 1=2 lz ¼ κz 1− , h

ð2Þ

where κ ¼0.4 is von Karman’s constant, h is the total mean water depth and z is the height above the seabed. To compare lz with the mixing length derived from the measurements, we use the following expression (Lu et al., 2000): !1=2 P lm ¼ , ð3Þ ρS3uv Fig. 11 shows the time variations of the mixing length lm at 1.5 mab (the lowest bin of the ADCP) vs lz, which is equal to 0.8612 m at 1.5 mab. As seen, for the first calm period, most of the time the values of lm are close to lz at 1.5 mab, although, slightly exceeding it on strong flood. Over strong ebb lm is slightly smaller than lz at 1.5 mab. During periods of slack water, lm sharply decreases approaching to 0.1 m (negative values were blanked). Note that similar temporal variations and close values of the mixing length were obtained by Lu et al. (2000). They have shown that the mixing length at 3.6 mab (their lowest bin of the

A bottom-mounted 1.2-MHz upward-looking four-beam broadband RDI ADCP “Workhorse” moored at the seabed measured current velocity in the Dover Strait over twelve tidal cycles. The flow in the strait is tidally dominated with along-shore velocity up to 1.1 m s−1 at the site of interest. The variances of the along-beam velocities recorded by the ADCP provided estimates of turbulent quantities. However to apply the variance technique, we (1) assumed that the mean flow and the second-order moments of turbulent velocity fluctuations were statistically homogeneous in horizontal space over the distances separating the beams, and (2) for the period of study chosen, the contamination of turbulent velocity fluctuations by surface wave, according to the tests performed with the VF method, was negligible within the bottom layer of 1.5–7 mab. The combination of the variance-inferred Reynolds stress and mean shear provided estimates of TKE production rate, the vertical eddy viscosity and the mixing length. Excluding intervals of the tidal current reversal, the measurements captured the depth–time variations of turbulence in the strait. The Reynolds stress in the near-bottom layer clearly demonstrated a strong semidiurnal variability for both along- and across-shore components and also revealed an ebb–flood asymmetry in their time variation. Both the Reynolds stress and TKE production rate were bottom enhanced, reaching peak magnitudes 3.0 Pa and 0.5 W m−3, respectively, during the flood, and −1.5 Pa and 0.1 W m−3, respectively, during the ebb. The TKE density was approximately proportional to the stress magnitude, ranging from 0.5 m2 s−2 during strong flows to 10–4 m2 s−2 during weak flows. The eddy viscosity increased with increasing flow speed, and also with increasing height in the lower half of the water column, ranging from 0.1 m2 s−1 during strong flows to 10−3 m2 s−1 during weak flows. The TKE dissipation rate also increased with increasing flow speed. Near the bottom, dissipation rate reached ∼0.04 W m−3 at peak flow during the flood, and ∼0.03 W m−3 at peak flow during

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

29

the ebb. Away from the bottom, the magnitude of dissipation rate decreased. Note that, for the estimation of P and ε, we used different methods. Their testing showed a good agreement between the patterns of time–depth variations of P and ε. Both parameters closely tracked each other and followed a semidiurnal cycle. Using estimates of P and ε, we scrutinized the state of the turbulence in tidal current. Time–depth variation of the P/ε ratio indicated that the turbulence in the near-bottom layer most of the time was at a non-equilibrium state (P/ε≠1). Large deviation from the equilibrium state occurred on strong flood, when P/ε exceeded one decade near the bottom. Away from the bottom, P/ε decreased but still remained 4 1. During the ebb, P/ε was close to the equilibrium state and slightly changed with height above the bottom. Estimating the drag coefficient, we discovered its strong dependence on the phase of tide. We have no explanation for this phenomenon yet and can only assume that such a dependence, in the absence of near-bottom stratification, might be associated with dissimilarity of dynamic conditions (e.g. roughness length, sand waves, etc.) for ebb and flood flows at the bottom. Theoretical analyses (e.g., Belcher et al. (1993)) have shown that the turbulent boundary layer over small-scale topographic features can be significantly distorted from the classical boundary layer over smooth walls. Therefore, if the horizontal inhomogeneity caused by bedforms is different for different phases of tide one should expect a discrepancy between turbulent characteristics of BBL obtained for ebb and flood including the C D . However, to verify this, we need more data on the bottom topography and its possible variations in the region of interest. It could be a topic for future research. The mixing length derived from our measurements showed a good agreement with mixing length theory slightly varying around the theoretically estimated value. In conclusion it should be noted that the present study is a continuation of our work on study of turbulence in a shallow water zone of the Dover Strait, off the Opal coast of France (Korotenko et al., 2012).

water density. The overbar denotes a time-averaged velocity at chosen interval (10 min). Note that to derive the mean velocity vector, we assume that the mean flow was statistically homogeneous in the horizontal space over distances separating the beams, that is, u 1 ¼ u 2 . To derive the Reynolds stress, it has to be assumed that all the second-order moments of turbulent velocity ′2 fluctuations were horizontally homogeneous, that is, u′2 1 ¼ u1 , u′1 w′1 ¼ u′2 w′2 , etc. In Eq. (A1), we omitted the terms describing noise errors due to pitch and roll of an ADCP. As was shown by Lu and Lueck (1999b), and Peters and Johns (2006), the contribution of such terms could be neglected even for relatively significant roll and pitch angles in the absence of surface gravity waves.

Acknowledgments

Appendix A. Variance method

Results from Eqs. (A1)–(A3) were sensitive to the averaging time interval chosen in the Reynolds decomposition. As was mentioned above, we used an averaging interval of 10 min, a choice justified by the examination of Reynolds stress spectra by Lu and Lueck (1999b) who revealed that comparatively low frequencies could also contribute to the stress. Technically, the high- and low-frequency velocity components were separated by the fourth-order Butterworth filter at zero phase. Variances of beam velocity fluctuations were then calculated and smoothed with the same filter and averaged over 10 min intervals to give estimates of the Reynolds stress.

(a) Reynolds stresses

The estimation of TKE density

This study was supported by a CNRS-RAS Partnership Program, Grant no. 21247. K. Korotenko acknowledges also a financial support by “Region Nord–Pas-de Calais”, France for an invited research grant. Two anonymous reviewers provided insightful comments, and we would like to thank them for their criticisms and suggestions on this manuscript improvement.

For the upward looking ADCP in a Janus configuration (Lu and Lueck, 1999a), a relationship between the velocity along the four beams, V i (positive toward the instrument) to those in Cartesian coordinates u, v and w allows inferring components of the Reynolds stress: ′2 τx =ρ ¼ −u′w′ ¼ ðV ′2 2 −V 1 Þ=2 sin 2θ,

′2 τy =ρ ¼ −v′w′ ¼ ðV ′2 4 −V 3 Þ=2sin 2θ,

ðA1Þ Here i¼ 1–4 represents the ADCP beam number, u′, v′ and w′ are the turbulent fluctuation components of velocity obtained after the decomposition of the raw velocity ðu, v, wÞ into a mean velocity ðu, v, wÞ and a turbulent part ðu′, v′, w′Þ, θ is the half angle between opposing beams (201 for the ADCP we used), and ρ is the

The estimation of TKE production rate The rate at which energy was transferred from the mean flow to the turbulent kinetic energy through the interaction of the turbulence with the shear was estimated from the scalar product (between matrices, often called double dot product) of the Reynolds stress and the mean velocity shear:   ∂u ∂v P ¼ −ρ u′w′ þ v′w′ ðA2Þ ∂z ∂z where both the stress and velocity shear were estimated from the ADCP data. Because of the alignment of the ADCP to the tidal flow, being itself globally oriented in S–N direction, we would have expected the main contribution to the rate of production to come from the second term on the right-hand side of Eq. (A2). The estimation of the vertical viscosity coefficient Az was calculated by dividing the TKE production rate, P, by the sum of mean velocity shear squared components. This yielded (Lu and Lueck, 1999b) "

 #−1 1 ∂u 2 ∂v 2 Az ¼ P þ ðA3Þ ρ ∂z ∂z

According to Lohrmann et al. (1990), the quantity Q obtained ′2 ′2 ′2 as the sum of the turbulent component, Q ¼ ðV ′2 1 þ V 2 þV 3 þ V 4 Þ= 2

4sin θ, is related to the TKE density q2 =2 ¼ ðu′1 þu′2 þ w′Þ=2 by Q ¼ γq2 =2, where the factor

ðA4Þ γ ¼ ð1 þ 2α tan−2 θÞ=ð1 þαÞ is determined by

′2 anisotropy, and α ¼ w′2 =ðu′2 1 þ u2 Þ. The value of γ ranges from 1 to 2.7, corresponding to α¼0 (extremely anisotropic turbulence) and to α ¼0.5 (isotropic turbulence). Similarly to Lu and Lueck (1999b), our analysis was based on γ ¼ 1:8, hence α¼0.2, which is the value estimated by Stacey et al. (1999) from measurements in an unstratified tidal channel.

30

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

(b) Structure function method for estimation of TKE dissipation rate In homogeneous turbulent flows with spatial and temporal constant background currents, the velocity difference between two points separated by a distance r is mainly determined by turbulent eddies of a spatial scale near to r. The mean square of the velocity fluctuation difference, defined as radial velocity structure function D(z, r) at depth z (Kolmogorov, 1941, Monin, Yaglom, 1965) is Dðz,rÞ ¼ ðu′ðzÞ−u′ðz þ rÞÞ2 ,

ðA5Þ

where u′ is the velocity component along the line connecting both measuring points. An assumption of proportionality of the velocity difference to an associated velocity scale u′s (Willes et al., 2006) gives Dðz,rÞ∝u′2 s :

ðA6Þ

In case of isotropic turbulence, the relationship between the characteristic length scale, the velocity scale of turbulent eddies in the inertial subrange and the TKE dissipation rate, ε, follows from the Taylor theory (Taylor, 1937): ε∝u′3 s =r

ðA7Þ

Eqs. (A5)–(A7) give a relationship between the structure function D(z, r) and the TKE dissipation rate ε (Monin and Yaglom, 1965): Dðz,rÞ ¼ Cε2=3 r 2=3 :

ðA8Þ

For isotropic turbulence, C is assumed to be a constant and, as was found empirically by Saddoughi and Veeravalli (1994), varies between 2.0 and 2.2. The variation depends on the ratio r/η, where η is the Kolmogorov microscale. In our work, we used C ¼ 2.1, which is a good approximation in the observed range of dissipation rate. As can be followed from the structure of Eq. (A6) the sets of D(z, r) were fitted to an equation of the form: Dðz,rÞ ¼ Ar 2=3 þp

ðA9Þ

where A ¼ Cε . The dissipation ε could be derived from the slope A, estimated with the use of a least-square algorithm. The offset parameter p may be interpreted as noise, but it is not the inherent Doppler noise as in the case of measurements with an ADCP. Instead p contains that part of velocity variance, which is not proportional to r2/3. The equations above are valid for isotropic turbulence with a spatially constant background flow. According to Eq. (A5), a constant shear in the background current contributed to the structure function, although it was not related to velocity fluctuations in the inertial subrange. Thus, technically the calculation of the structure function, D(z, r) was performed for the fluctuating part of the flow. The raw beam velocities were temporally averaged over the averaging interval that was used for the structure function calculation. 2=3

References Belcher, S.E.J., Newley, T.M., Hunt, J.C.R., 1993. The drag on an undulating surface induced by the flow of a turbulent boundary layer. Journal of Fluid Mechanics 249, 557–596. Brylinski, J.M., Brunet, C., Bentley, D., Thoumelin, G., Hilde, D., 1996. Hydrography and phytoplankton biomass in the eastern English Channel in spring 1992. Estuarine, Coastal and Shelf Science 43, 507–519. Caldwell, D.R., Moum, J.N., 1995. Turbulence and mixing in the ocean. Reviews of Geophysics 33, 1385–1394. Davis, A., Flather, R.A., 1987. Computing extreme meteorologically induced currents, with application to the north-west European Shelf. Continental Shelf Research 7, 643–683. Dewey, R.K., Crawford, W.R., Garrett, A.E., Oakey, N.S., 1987. A microstructure instrument for profiling oceanic turbulence in coastal bottom layers. Journal of Atmospheric and Oceanic Technology 4, 288–297.

Feddersen, F., Williams III, A.J., 2007. Direct estimation of the Reynolds stress vertical structure in the nearshore. Journal of Atmospheric and Oceanic Technology 24, 102–116. Fugate, D.C., Chant, R.J., 2005. Near-bottom shear stresses in a small, highly stratified estuary. Journal of Geophysical Research 110, C03022. Gibson, C.H., Nabatov, V.N., Ozmidov, R.V., 1993. Measurements of turbulence and fossil turbulence near Ampere seamount. Dynamics of Atmospheres and Oceans 19, 175–204. Gross, T.F., Nowell, A.R.M., 1983. Mean flow and turbulence scaling in a tidal boundary layer. Continental Shelf Research 2, 109–126. Howarth, M.J., Souza, A.J., 2005. Reynolds stress observations in continental shelf seas. Deep-Sea Research II 52, 1075–1086. Huang, Y.X., Schmitt, F.G., Lu, Z.M., Fougairolles, P., Gagne, Y., Liu, Y.L., 2010. Second order structure functions in fully developed turbulence. Physical Review E 82, 026319. Kirincich, A.R., Lentz, S.J., Gerbi, G.P., 2010. Calculating Reynolds stresses from ADCP measurements in the presence of surface gravity waves using the cospectra-fit method. Journal of Atmospheric and Oceanic Technology 27, 889–907. Kolmogorov, A.N., 1941. Dissipation of energy in locally isotropic turbulence. Proceedings of the USSR Academy of Sciences 32, 16–18. Korotenko, K.A., Nabatov, V.N., 1987. To processes of turbulent mixing along the section of NordCap—Bear Island. In: Ozmidov, R.V. (Ed.), Structure of Hydrophysical Fields in Norwegian and Barents Seas. IO USSR Academy of Sciences. Мoscow, pp. 52–58, In Russian. Korotenko, K.A., Sentchev, A.V., 2004. On the formation of anomalies in the ichthyoplankton concentration field along the French coast in the eastern English Channel. Oceanology 44, 644–653. Korotenko, K.A., Sentchev, A.V., 2008. Effects of the particle migration on the features of their transport by tidal currents in the region of freshwater influence. Oceanology 48, 672–684. Korotenko, K.A., Sentchev, A.V., 2011. Study turbulence in shallow tidal coastal zone. Oceanology 51, 1–14. Korotenko, K., Sentchev, A., Schmitt, F., 2012. Effect of variable winds on current structure and Reynolds stresses in a tidal flow: analysis of experimental data in the eastern English Channel. Ocean Science 8, 1025–1040, http://dx.doi.org/ 10.5194/os-8-1025-2012. Lohrmann, A., Hackett, B., Roed, L.P., 1990. High resolution measurements of turbulence, velocity, and stress using a pulse-to-pulse coherent sonar. Journal of Atmospheric and Oceanic Technology 7, 19–37. Lu, Y., Lueck, R.G., 1999a. Using a broadband ADCP in a tidal channel. Part I: Mean flow and shear. Journal of Atmospheric and Oceanic Technology 16, 1556–1567. Lu, Y., Lueck, R.G., 1999b. Using a broadband ADCP in a tidal channel. Part II: Mean flow and shear. Journal of Atmospheric and Oceanic Technology 16, 1568–1579. Lu, Y., Lueck, R.G., Huang, D., 2000. Turbulence characteristics in a tidal channel. Journal of Geophysical Research 30, 855–867. Monin, A.S., Yaglom, A.M., 1965. Statistical Hydromechanics: Part I. Nauka, Moscow. Monin, A.S., Ozmidov, R.V., 1985. Turbulence in the Ocean. In: D. Reidel, (Ed.), Dordrecht. Nidzieko, N.J., Fong, D.A., Hench, J.L., 2006. Comparison of Reynolds stress estimates derived from standard and fastping ADCPs. Journal of Atmospheric and Oceanic Technology 23, 854–861. Paka, V.T., Nabatov, V.N., Lozovatsky, I.D., Dillon, T.M., 1999. Oceanic microstructure measurements by BAKLAN and GRIF. Journal of Atmospheric and Oceanic Technology 16, 1519–1532. Peters, H., Johns, W.E., 2006. Bottom layer turbulence in the Red Sea outflow plume. Journal of Physical Oceanography 36, 1763–1785. Prandle, D., Losch, S.G., Player, R., 1993. Tidal flow through the Straits of Dover. Journal of Physical Oceanography 23 (1), 23–37. Rippeth, T.P., Williams, E., Simpson, J.H., 2002. Reynolds stress and turbulent energy production in a tidal channel. Journal of Physical Oceanography 32, 1242–1251. Rippeth, T.P., Simpson, J.H., Williams, E., Inall, M.E., 2003. Measurement of the rates of production and dissipation of turbulent kinetic energy in an energetic tidal flow: Red Wharf Bay revisited. Journal of Physical Oceanography 33, 1889–1901. Rosman., J.H., Hench, J.L., Koseff, J.R., Monismith, S.G., 2008. Extracting Reynolds stresses from acoustic Doppler current profiler measurements in wave dominated environments. Journal of Atmospheric and Oceanic Technology 25, 286–306. Saddoughi, S.G., Veeravalli, S.V., 1994. Local isotropy in turbulent boundary layers at high Reynolds number. Journal of Fluid Mechanics 268, 333–372. Schmitt, F.G., Huang, Y., Lu, Z., Liu, Y., Fernandez, F., 2009. Analysis of velocity fluctuations and their intermittency properties in the surf zone using empirical mode decomposition. Journal of Marine Systems 77, 473–481. Seim, H., 2002. Reynolds stress measurements in a short tidal inlet. In: Proceedings of the 2nd Meeting on the Physical Oceanography of Sea Straits, Villefranche, 15–19 April, pp. 199–202. Sentchev, A, Korotenko, K.A., 2004. Stratification and tidal current effects on larval transport in the eastern English Channel: observations and 3D modelling. Environmental Fluid Mechanics 4, 305–331. Sentchev, A., Korotenko, K.A., 2005. Dispersion processes and transport patterns in the ROFI of eastern English Channel derived from a particle tracking method. Continental Shelf Research 25, 2293–2308. Sentchev, A., Yaremchuk, M., 2007. VHF radar observations of surface currents off the northern Opal coast in the eastern English Channel. Continental Shelf Research 27, 2449–2464.

K. Korotenko et al. / Continental Shelf Research 58 (2013) 21–31

Simpson, J.H., Crawford, W.R., Rippeth, T.P., Campbell, A.R., Cheok, J.V.S., 1996. The vertical structure of turbulent dissipation in shelf seas. Journal of Physical Oceanography 26, 1579–1590. Shaw, W.J., Trowbridge, J.H., 2001. The measurement of near-bottom turbulent fluxes in the presence of energetic wave motions. Journal of Atmospheric and Oceanic Technology 18, 1540–1557. Souza, A.J., Howarth, M.J., 2005. Estimates of Reynolds stress in a highly energetic shelf sea. Ocean Dynamics 55, 490–498. Stacey, M.T., Monismith, S.G., Burau, J.R., 1999. Measurements of Reynolds stress profiles in unstratified tidal flow. Journal of Geophysical Research 104 (C5), 10933–10949.

31

Taylor, G.I., 1937. The statistical theory of isotropic turbulence. Journal of Aeronautic Science 4, 311–315. Trowbridge, J.H., Elgar, S., 2003. Spatial scales of stress-carrying nearshore turbulence. Journal of Physical Oceanography 33, 1122–1128. Willes, P.J., Rippeth, T.P., Simpson, J.H, Hendricks, P.J., 2006. A novel technique for measuring the rate of turbulent dissipation in the marine environment. Geophysical Research Letters 33, L21608. Williams, E., Simpson, J.H., 2004. Uncertainties in estimates of Reynolds stress and TKE production rate using the ADCP variance method. Journal of Atmospheric and Oceanic Technology 21, 347–357. Whipple, A.C., Luettich, R.A., Seim, H.E., 2006. Measurements of Reynolds stress in a wind-driven lagoonal estuary. Ocean Dynamics 56, 169–185.