Dynamics of Turbulence in Low-Speed Oscillating Bottom-Boundary ...

Environmental Fluid Mechanics 2: 291–313, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Dynamics of Turbulence in Low-Speed Oscillating Bottom-Boundary Layers of Stratified Basins ANDREAS LORKEa, LARS UMLAUFb, TOBIAS JONASa and ALFRED WÜESTa a Applied Aquatic Ecology (APEC); Limnological Research Center, EAWAG, Kastanienbaum, Switzerland; b Departement de Génie Civil (DGC), Ecole Polytechnique Fédéral de Lausanne (EPFL), Lausanne, Switzerland

Received 11 December 2001; accepted in revised form 7 June 2002 Abstract. This paper focuses on the impact of an oscillating low-speed current on the structure and dynamics of the bottom-boundary layer (BBL) in a small stratified basin. A set of high-resolution current profile measurements in combination with temperature-microstructure measurements were collected during a complete cycle of the internal oscillation (‘seiching’) in the BBL of Lake Alpnach, Switzerland. It was found that even a relatively long seiching period of 24 hours significantly changed the form of the near-bottom current profiles as well as the dynamics of the turbulent dissipation rate compared to the steady-state law-of-the-wall. A logarithmic fit to the measured current profiles starting at a distance of 0.5 m above the sediment led to inconsistent estimates of both friction velocity and roughness length. Moreover, a phase lag between the current and the turbulent dissipation of 1.5 hours and a persistent maximum in the current profile at a height of 2.5 to 3 m above the sediment were observed. The experimental findings were compared to the results of a k-ε turbulence model and showed good agreement in general. Specifically, the inconsistent logarithmic fitting results and the observed phase lag were reproduced well by the model. Key words: bottom-boundary layer, dissipation rate, k-ε model, lake, phase lag, temperature microstructure, turbulence

1. Introduction A number of recent publications emphasized the importance of the bottom boundary layer (BBL) in lakes, fjords and oceanic sub-basins for the overall mixing efficiency (e.g., [1–6]). Wüest et al. [5] have shown in a case study for Lake Alpnach that 90% of the energy that enters the stratified part of the lake via internal seiching is dissipated by friction within the BBL. This boundary mixing in combination with lateral advection contributes significantly to the basin-wide diffusivity, estimated for instance by tracer experiments [2]. The importance of the BBL for sediment-water exchange processes has also been acknowledged in a number of oceanic investigations. An early study of the structure in the BBL on the continental shelf by Caldwell and Chriss [7] revealed the possibility of a viscous sublayer several millimeters thick below a logarithmic layer. Velocity profile measurements within this logarithmic boundary layer ∗ Corresponding author, E-mail: [email protected]

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were widely used to estimate the bottom stress τ in terms of the friction velocity √ u∗ = τ/ρ and the bottom roughness in terms of the roughness length scale z0 or the drag coefficient C1m by fitting the measured velocity profiles to the lawof-the-wall (e.g., [8–11]). Additionally, recent acoustic profiling techniques also permitted the direct evaluation of profiles of the turbulent kinetic energy and the vertical Reynolds stress and thus provided a direct verification of the law-of-thewall relations in high-speed tidal flows [12, 13]. In this paper, the effect of the non-stationarity of the boundary layer due to oscillatory currents driven by internal seiching is explored. Using simultaneous measurements of high-resolution velocity profiles and turbulent dissipation rates obtained from temperature microstructure measurements in the seiche-induced oscillating BBL of a lake, the dynamic behavior of currents and turbulence in nonstationary BBLs will be investigated. Compared to tidal flows, bottom currents in lakes are about an order of magnitude smaller and, most remarkably, the pressuregradient term in the momentum equation is not predominately balanced by the Reynolds stress, but by the rate of change term. It will be illustrated that this fact has considerable consequences for the validity of the frequently applied law-ofthe-wall relations. After a short description of the measurements, we demonstrate in Section 3 that a logarithmic fit to the measured velocity profiles starting at 0.5 m above the sediment surface can lead to significant over-estimations of the bottom stress and the roughness length. In the discussion the reasons behind this discrepancy will be explored by comparing the temporal dynamics of the observed oscillating BBL with a numerical turbulence-closure (k − ε) model. Particularly the reasons behind the failure of the logarithmic fitting procedure for providing appropriate estimates of the bottom stress and roughness length as well as the phase lag between the current and the turbulence are discussed. In the final section we will show an interesting feature of the oscillatory BBL currents: The vertical velocity profile exhibiting a distinct maximum at a height of 2.5 m above the bottom leads to unstable stratification and thus to additional convective mixing within the BBL. The results of the measurements and model calculations will be summarized in the conclusion section. This paper focuses on the impact of the periodic forcing on the structure of the vertical profiles of the current velocity and the turbulence in the BBL. The comprehensive analysis of seiching and BBL dynamics in this lake, which have been published at several instances [14, 15], will not be repeated in this article. 2. Measurements 2.1. STUDY SITE Measurements were carried out for 22 hours on May 16–17, 2000 in Lake Alpnach, a small (4.76 km2 ) sub-basin of Lake Lucerne in Central Switzerland. It has an elliptical shape with axes lengths of 5 and 1.5 km, respectively, and a maximum

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Figure 1. Geometry and bathymetry of Lake Alpnach (Switzerland). ‘P’ denotes the approximate position of the measurements, where the depth is close to the maximum depth of 34 m.

depth of 34 m. The measurements were carried out close to the center of the lake at a depth of 32 m (Figure 1). Because of the mountainous topography of the surroundings there is a diurnal wind forcing (lake breeze) along the main axis of the lake [14]. This periodicity is in resonance with the second vertical/first horizontal mode of the internal basin-scale seiching described by Münnich et al. [14]. 2.2. CURRENT PROFILES A Nortek ADCP (Acoustic Doppler Current Profiler) was used for near-bottom current profiling. Deployed on the bottom facing upward, the ADCP was operated in high-resolution mode, which is a ping-to-ping coherent mode offering high accuracy at potentially small depth cell sizes. The system overview and settings are summarized in Table 1. The profiling range was 0.44 to 4.8 m above the sediment. Using the built-in compass, the single-ping current data were transformed to earth coordinates and stored on-line. A RDI Workhorse ADCP (1.2 MHz) was used in parallel and 50 m apart from the Nortek ADCP at the start of the experiment. The RDI was operated selfcontained with comparable settings as the Nortek ADCP (Table 1). Unfortunately, the RDI ran out of power after 6 hours of measurements. However, the obtained data were sufficient for a comparison of both ADCPs and to check the reproducibility and reliability of the current profiles. The result of this comparison showed an excellent agreement with the form of the measured velocity profiles as well as with the measured velocities.

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Table I. System and deployment settings of the ADCP. System frequency Beam configuration Ping frequency Height of 1st depth cell above sediment Depth cell size Number of depth cells

1500 kHz 3-beam, convex, 25◦ 0.3 Hz 0.44 m 0.07 m 64

2.3. TEMPERATURE MICROSTRUCTURE Temperature microstructure profiles were measured near the ADCP location using a modified SeaBird SBE-9 microstructure CTD, which was equipped with a FP-07 fast response thermistor and a conductivity microstructure cell [16]. The profiler was operated freely sinking with a speed of approximately 10 cm/s and a sampling rate of 96 Hz. This corresponds to a vertical resolution of ≈1 mm. Profiles from 15 m depth down to 15 cm above the local lake’s bottom were taken every 15 min. Dissipation rates of turbulent kinetic energy ε were calculated from the temperature microstructure by fitting the calculated spectra of temperature fluctuations to the theoretical Batchelor spectra [17]. Partially, artificial noise in the form of heavy spikes contaminated the temperature profiles and therefore, as a conservative measure, 20 to 30% of the profile segments had to be removed from further analysis. These spikes were most probably produced by a grounding problem. To ensure no contamination of the dissipation estimates we did not try to remove the spikes from the signal. 3. Results 3.1. TEMPERATURE AND STRATIFICATION Figure 2 summarizes the dynamics of temperature and stratification between 15 m depth and 15 cm above the bottom, as obtained from the temperature microstructure profiles. The overall vertical temperature change in this depth range was less than 3 ◦ C. The seiche motion with isotherm displacements of up to 5 m is clearly visible in the isotherm plot in Figure 2. Depending on the phase of the seiching, the water column was continuously stratified almost down to the bottom or a well-mixed BBL of up to 5 m in height was established. 3.2. CURRENT STRUCTURE The current velocities were transformed into a longitudinal and a transverse component ulong and utrans by minimizing the transverse current component for the entire data set while rotating the coordinate system in the horizontal plane. The

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Figure 2. The lower panel shows a contour plot of isotherm depth vs. time as obtained from the temperature measurements with the microstructure profiler from 15 m depth down to the bottom. Isotherm spacing is 0.2 ◦ C. The two upper panels show sample profiles at the indicated times (arrows).

resulting direction of the longitudinal current was 25◦ . This is in good agreement with the orientation of the main axes of the lake (Figure 1), coinciding with the main direction of the wind. The time series of the three current components measured at 4 m height above the bottom are shown in Figure 3. Most of the time, the current magnitude was dominated by the longitudinal direction for which a cosine fit gave a period of 23.5 h (Figure 3). The transverse current shows some minor oscillations with higher frequencies that can be attributed to transverse seiching of the lake. The transverse seiching can dominate the current magnitude during the low-speed periods of the longitudinal current component. Since the low-frequency part of the current is of greater importance for our further analysis, we will focus on the longitudinal current component only. To remove Doppler noise and high-frequency fluctuations of the current structure, the data needed to be averaged over time. Spectra of the original currentvelocity time series showed a clear peak at a period of 9.3 min. This period is almost identical to the theoretical period of surface seiching (1st barotropic basin mode) of Lake Alpnach with T = 9.6 min. We chose this period for time-averaging of the current data and hence averaged 150 single-ping profiles to a set of 146 profiles in total. Figure 4 illustrates a sample of consecutive profiles at the time of the second flow-reversal of the longitudinal current velocity. This plot shows

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Figure 3. Time series of the longitudinal, transverse and vertical velocities as measured by the ADCP at a height of 4 m above the bottom. A cosine fit to the longitudinal velocity with a period of 23.5 h is indicated.

Figure 4. Sample profiles of the along-lake velocity ulong from the second half of the observed seiching period. The profiles are offset and all plotted with the given velocity scale. The measurement times are given above or below each profile.

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Figure 5. Measured profiles of the along-lake velocity ulong during high (12:34) and low current speed (16:10). The logarithmic law-of-the-wall according to Equation (1) was fitted to the high-speed profile. The inset plot shows the same profile and fit with a semi-logarithmic scaling to visualize the excellent agreement of the data to the logarithmic fit up to the characteristic velocity maximum at z ≈ 2.75 m.

clearly the development and breakdown of the BBL. The velocity profiles were fitted to the law-of-the-wall,
 z u∗ ln , (1) ulong = κ z0 √ with the friction velocity u∗ = τ/ρ (τ is the shear stress and ρ is the density) and the roughness length z0 as the free parameters. κ = 0.41 is the von Kármán constant and z the distance from the bottom. As long as the velocity exceeded a threshold value of about 1 cm/s as a maximum value within the range of measurments, most of the profiles where in excellent agreement with the logarithmic profile. For velocities smaller than this threshold, the logarithmic profile broke down and no characteristic features of a BBL could be identified. Figure 5 shows examples of profiles above and below the threshold velocity including the logarithmic fit. A special feature of all velocity profiles exceeding the threshold was a distinct velocity maximum above the logarithmic part of the profile at a height between 2.5 and 3.5 m. The friction velocity u∗ obtained from fitting all profiles to the law-of-the-wall is shown in Figure 6. This figure demonstrates that ulong scales well with u∗ . Due to the nature of the fitting, the estimates of the roughness length z0 are much more erroneous because of its logarithmic dependence and the huge extrapolation from the measured range of z to z0 . Figure 7 shows the histogram of the estimates.

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Figure 6. Time series of the friction velocity u∗ as obtained from the law-of-the-wall and the along-lake current velocity ulong measured at a height z of 1 m above the sediment.

Figure 7. Histogram of the roughness length z0 estimates from the law-of-the-wall scaling on a logarithmic scale. The overall logarithmic mean of z0 = 1.4 cm is indicated.

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Most of the values were found to lie within a lognormal distribution. However, the distribution is skewed towards smaller values and an overall logarithmic mean gives z0 = 1.4 cm. Another parameter for describing bottom friction is the drag coefficient C1m defined by u2∗ = C1m · u21m

(2)

(u1m is the velocity measured at a height of 1 m above the bottom). By combining Equation (1) and Equation (2), the drag coefficient can be directly computed from z0 by  2 κ (3) C1m =     . ln 1zm 0 This results in a value of C1m = 9 × 10−3 for the logarithmic mean of z0 = 1.4 cm. This fits well with the scaling of u∗ and u1m according to Equation (2) as shown in Figure 6. Assuming a rough flow regime, the roughness length z0 should be related to the topographical bottom structure at the sediment boundary only and hence should be constant with time since the observed bottom stress is too weak to alter the sediment topography. A smooth flow regime on the other hand would result in a reciprocal relationship between z0 and u∗ [18] which cannot be seen in our results. The huge scatter of the z0 estimates as shown in Figure 7 suggests that (i) the uncertainty of the method as mentioned above is very high or (ii) that the assumption of a law-of-the-wall according to Equation (1) was not valid. The logarithmic mean values of z0 and the corresponding drag coefficients C1m are almost five times higher as the common literature value estimated by Elliott [19] to C1m = 1.6 × 10−3 . Increased drag coefficients (C1m = 3.2 × 10−3 to 8.8 × 10−3 ) were found by Chriss and Caldwell [20] and were attributed to an increased form drag introduced by larger scale roughness elements at the sediment surface. However, Dimai et al. [10] found drag coefficients of C1m = 4.9 × 10−3 and 6.9 × 10−3 , respectively, by using different methods during a former study in Lake Alpnach, while scuba divers found no evidence for additional roughness elements at the sediment surface around the study site. 3.3. DISSIPATION The friction velocity u∗ can be used to estimate the rate of turbulent kinetic energy dissipation ε using ε=

u3∗ κ ·z

(4)

[7]. These estimates were plotted together with the dissipation rate estimates from the temperature microstructure at a height of 1 m above the bottom in Figure 8.

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Figure 8. Turbulent kinetic energy dissipation rates ε at 1 m above the sediment as obtained from the temperature microstructure measurements (
) and from the law-of the-wall fit to the current profiles according to Equation (4) (
).

Although both estimates show the same temporal pattern, there are strong discrepancies between both. The dissipation rate estimated from u∗ is on average one and sometimes more than two orders of magnitude greater than the microstructure based estimates. In addition, microstructure-based estimates are lagging behind the values obtained from the law-of-the-wall. This becomes more obvious looking at the correlation between both estimates in Figure 9. The phase lag can be estimated from that figure as ϕ ≈ 1.5 h. 4. Discussion Although the measured current profiles in the BBL of Lake Alpnach show an excellent agreement with the logarithmic interpolation, the obtained results for u∗ and z0 using the law-of-the-wall scaling are doubtful and both seem to be overestimated. Both, the distinct maximum found in the current profiles as well as the observed phase lag between the microstructure-based dissipation estimates and the current velocity, indicate that the temporal dynamics is an important factor. A phase lag between the current and turbulent kinetic energy and its dissipation rate in (oscillating) tidal flows was modeled by Baumert and Radach [21] and observed by Simpson et al. [22]. Deviations from the law-of-the-wall and a distinct velocity maximum at a certain height above the bottom were found in near-bed

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velocity profiles by Soulsby and Dyer [23] and Foster et al. [24] under different forcing periods. In the following discussion we investigate the effect of the seicheinduced oscillatory current on the form and dynamics of near-bottom velocity and turbulence profiles. 4.1. STOKES ’ OSCILLATORY BBL An academic case, which corresponds to the seiche-induced BBL in Lake Alpnach, is Stokes’ solution for an oscillatory BBL [25]. Figure 10 shows a schematic sketch of the problem. A water column is forced by a horizontally oscillating bottom, with a velocity UB (t) = UB0 cos(ωt), where t denotes time and ω is the forcing frequency. The Navier–Stokes equations in combination with the continuity equation and a no-slip boundary condition can be solved analytically for this problem and yield a velocity profile u(z, ˜ t) in the water column as a function of height above the bottom z of: 
√ ω 0 − ω/2νz z + ωt , (5) cos − u(z, ˜ t) = UB e 2ν with ν denoting turbulent viscosity. The above equation describes an exponentially damped vertical oscillation in the current profile u(z, ˜ t) with the vertical ‘wave√ −1 number’ ω/2ν [m ] as illustrated in Figure 10. In order to describe the inverse forcing, an oscillating water column above a fixed bottom, the bottom velocity U 0B can be subtracted from the above equation, yielding: u(z, t) = u(z, ˜ t) − UB0 (t).

(6)

The height of the first maximum from the bottom depends on the forcing frequency ω and the viscosity ν. The larger ω becomes the closer this velocity maximum moves towards the bottom. To obtain the observed velocity-maximum height of 2 to 3 m with a 24 h forcing period, we have to choose a turbulent viscosity of ν ≈ 4 × 10−4 m2 /s, which seems to be a reasonable estimate within the BBL [26]. However, the turbulent viscosity is constant with depth and time in this simple model which might not be an appropriate approximation in our case. Encouraged by the good correspondence of this very simple model approach, we continued with the numerical modeling of the system. 4.2. K -ε MODELING OF THE OSCILLATING BBL A number of applications of differential turbulence closure models to oscillating oceanic boundary layers have been reported. Weatherly and Martin [27] and Simpson et al. [22] applied a zero-equation (level 2) model of Mellor and Yamada [28] to oceanic turbulent boundary layers and investigated the dynamic properties of turbulence. However, zero-equation models do not predict a phase lag between the velocity shear and turbulent quantities, which is assumed to be of minor importance

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in high-speed tidal flows, but may also be significant in low-speed boundary layers of small stratified basins as illustrated below. In addition to this, these models may compute unstable profiles of the turbulent diffusivity as illustrated by Burchard and Baumert [29]. Vager and Kagan [30] conducted a theoretical investigation of unstratified rotating tidal flows with a one-equation model. Applications of one- and two-dimensional two-equation models with simple Algebraic Stress Models (ASMs) to laboratory and stratified real-world tidal flows have been reported by Smith and Takhar [31, 32]. A complete second-order closure was used by Richards [33] for his theoretical investigation of the stratified, rotating and oscillating boundary layer. No comparison to measurements was included, though. Only a few comparisons of measured and modeled tidal currents were conducted, e.g., by Baumert and Radach [21] and Burchard et al. [34]. These authors used a k − ε model and were able to explain the essential physics of tidal flows, most remarkably the characteristic tidal time-lag between the current and the turbulent kinetic energy or its rate of dissipation, respectively. It can be summarized that differential closure schemes are a powerful tool for the description of dynamic boundary layers in oscillating tidal flows. However, in spite of their great success in oscillating tidal flows, similar investigations in oscillating bottom boundary layers in lakes have, to our knowledge, never been reported. This is a surprising fact as it is evident that such turbulence models, if implemented in three-dimensional circulation models for lakes (see, e.g., [35, 36]) determine the important turbulent fluxes of dissolved solids and momentum through the boundary layer. Since the turbulent momentum flux predicted by such models determines, to a large extent, the decay of internal seiches, it is crucial for the dynamics of the most characteristic motion in stratified lakes. It is, however, not obvious that the encouraging results obtained with two-equation models in oscillating tidal flows also do apply immediately to oscillating boundary layers in lakes. There, bottom currents are at least one order of magnitude lower and turbulence occurs in general at much lower Reynolds numbers, throwing into question the applicability of the simple but popular high-Reynolds number two-equation models. Besides this, the seiching motions in lakes cover a much wider range of periods (e.g., 10 h for Baldeggersee, 100 h for Lake Constance, see [37, 38]) compared to the ocean, where the main contribution in most cases comes from the approximately semi-diurnal M2 tide. It is therefore unclear, to what extent the phase shift relations between turbulent and mean quantities obtained for the ocean (see [21, 22]) are valid also in stratified lakes and reservoirs. To investigate these questions, we modeled the seiche-induced boundary layer as a horizontally infinite, oscillating boundary layer over a rough plate. Turbulence was modeled with a standard k − ε model as described by Rodi [39]. Stratification effects were ignored, since the temperature gradient in the well-mixed boundary layer was dynamically insignificant [40] and chemical stratification by dissolved solids and sediments could also be excluded [26].

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To capture the essential physics of the problem, we forced the model with a periodic, cosine-shaped pressure-gradient with the observed period T = 24 h, and for comparison also with the shorter periods of T = 6 h and T = 12 h. The amplitude of the pressure-gradient was adjusted ‘by hand’ to yield the observed maximum current speed of 3–4 cm/s in each case (see Figure 3). The numerical results displayed below represent a quasi steady-state, where the influence of the initial disturbances had died away and the phase and the amplitude of the oscillating fields were approximately constant. Note that, as demonstrated by Umlauf (2001), the turbulent parameters in the boundary layer are quite sensitive with respect to the value of the roughness length z0 . He suggested z0 = 0.001 m, a value that is considerably smaller than the doubtful value inferred from the logarithmic fits, z0 = 0.014 m, but much more consistent with the structure of the sediment near the measuring site. Displayed in Figure 11 are modeled time series of the velocity at different heights above the sediment for the period of T = 24 h. It is evident that the velocity closely follows a sinusoidal shape, indicating that the main balance in the momentum budget is between the inertial term and the pressure-gradient (which follows a cosine). Frictional effects amount only to a very small phase shift at different heights, hardly visible on the scale of the plot and certainly undetectable with any confidence from the measurements. Even though for the smaller periods of T = 6 h and T = 12 h (not shown), the phase shift at different levels is somewhat clearer, it appears also unlikely that it could be recovered from measurements. Note also, that the main balance in the momentum equation is at no time between the pressuregradient and the friction terms (see [40]). This is very distinct from tidally induced flows, where clear phase shifts of the velocity at different levels above the sediment have been reported, and the main balance is, at least during periods of high speeds, between the pressure-gradient and the rate terms [12, 13, 21]. In contrast to the computed velocity records, a clear phase shift in the turbulent quantities at different levels above the sediment is evident as illustrated in Figure 12 for the 12 h period and in Figure 13 for the observed 24 h period. In both cases, the phase shift is more pronounced for the turbulent kinetic energy compared to the rate of dissipation. Also in both cases, the phase shift increases with increasing distance from the sediment. A closer examination of the time series of the velocity and the rate of dissipation revealed a phase shift between these two quantities of approximately 1.0 h at 1 m distance from the sediment for the period T = 24 h. Taking the deviation of the measured velocities from a sinusoidal shape into account, this value is in reasonable agreement with the measured phase lag of 1.5 h (see Figure 9). Measured time series of the turbulent kinetic energy were not available. It is evident that measurements in oscillating boundary layers in lakes with a smaller seiching period could reveal the relative phase lag much clearer and could be used to investigate the properties of two-equation models in a unique way. Note in this context, that in the low-speed flows encountered here it is not the lag of the velocity shear dictating the phase lag of the turbulent quantities as assumed in

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Figure 9. Normalized correlation between the two estimates of the turbulent kinetic energy dissipation rates ε from Figure 8 versus lag time. A positive lag means that the estimate from the law-of-the-wall scaling is leading the microstructure-based estimate and vice versa. The correlation maximum can be found at a lag time of about +1.5 h. The correlation values were normalized to this peak value.

the theory of Simpson et al. [22], who computed the phase lag based on (5). With ν ≈ 4 × 10−4 m2 /s (see above), this equation predicts a phase difference of 1.73 h between the velocity at 1m and 2.5 m height, a result that is clearly contradicted by Figure 11. For smaller periods (not shown) this discrepancy becomes even more evident. Thus, the dynamics of turbulence in low-speed boundary layers of lakes must be considered significantly different from those in oceanic tidal flows. Profiles of the velocity at the time t = T /4 of maximum current speed for T = 12 h and T = 24 h, displayed in Figure 14, reveal that the law-of-the-wall (also indicated in these figures) is valid for the lowest 0.5 m, at most. This is in apparent contradiction to the ‘perfect’ logarithmic fit to the experimental data in Figure 5. Figure 15, in which the velocity profiles for T = 6 h and T = 12 h are displayed, but now displayed only above a height of 0.5 m above the ground (as in the measurements), reveal why this is the case: Ignoring the lowest 0.5 m, i.e., precisely that region in which the law-of the-wall is valid, both profiles can be closely approximated by a spurious logarithmic ‘law-of-the-wall’. In both cases, however, the inferred values of the roughness length, z0 , and the friction velocity, u∗ , are overestimated by factors of 5 and 1.5, respectively. This is the most significant result of this section, since it illustrates clearly why, in spite of the good logarithmic fit of the measured velocity profiles, u∗ and hence via Equation (4) also ε are overestimated so significantly (see Figure 8). The rate of dissipation

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Figure 10. Schematic illustration of the velocity profiles in Stokes’s oscillatory BBL. The bottom is a flat plate, horizontally oscillating with the velocity U (t) = U0 cos(ωt) and the angular frequency ω. Current velocity profiles at representative times t as fractions of the forcing period T = 2π/ω according to Equation (5) are shown as a function of height above the bottom.

Figure 11. Velocity, u, in a seiche-induced BBL as calculated by the k − ε model at different heights above the sediment. The seiching period was 24 h.

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Figure 12. Time series of the turbulent kinetic energy k (left panel) and its rate of dissipation ε (right panel) for one complete seiche cycle (of T = 12 h duration) at different heights above the sediment.

Figure 13. Same as Figure 12, but now for a seiche period of T = 24 h.

computed by the k − ε model, however, yields maximum values of ε ≈ 10−8 W/kg (see Figure 13), which are in reasonable agreement with the measured values (see Figure 8). The law-of-the-wall relations, strictly valid only for stationary turbulent boundary layers, are thus not appropriate for low-speed oscillating boundary layers driven by a pressure gradient, even if the period is as large as T = 24 h. 4.3. STRATIFICATION IN THE OSCILLATING BBL The general dynamics of temperature and stratification was discussed in the beginning (Figure 2). The two sample temperature profiles shown in Figure 2 exhibit completely different stratification, from continuously stratified almost down to the bottom to a well-mixed BBL of 5 m height. Since both profiles were measured

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Figure 14. Calculated velocity profile at the time t = T /4 for a 12 h (left panel) and 24 h (right panel) seiche period T. The corresponding logarithmic law-of-the-wall velocity profile based on the calculated friction velocity u∗ are plotted as well.

Figure 15. Calculated velocity profiles at the time t = T /4 for a 6-h and a 12-h seiche period T. Also plotted are fits of these profiles to a logarithmic law-of-the-wall starting at 0.5 m above the bottom.

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during the zero crossing of the longitudinal current velocity, the difference cannot be attributed to the actual mixing conditions at the sampling site. The different stratification pattern was produced by seiche-driven horizontal advection. Away from the nodal point of the internal seiching the BBL is strained and compressed by the inclination of the isotherms. The effect seems surprisingly strong for the position of the sampling site, close to the center of the lake (Figure 1) and is an important feature, which should be kept in mind while determining the thickness of the BBL from single-point measurements. The seiching creates horizontal temperature gradients also close to the bottom and since the seiche-driven currents are horizontal, the vertical structure of the near-bottom current profile is influencing the vertical temperature stratification. This will be explained for two different seiche phases. Figure 16 shows the near-bottom temperature profiles and the current velocity profile at around 14:30 (May 16, 2000). As it can be deduced from Figure 3, the seiche-induced current has just passed the peak-current velocity and the flow starts to decelerate. From the temperature-contour plot (Figure 2) it can be seen that the well-mixed BBL is compressed and warm water is moving to the sampling site. The same features are obvious in the 4 subsequent temperature profiles shown in Figure 16. The characteristic maximum in the current velocity is well established at this time and its impact can be seen in the two intermediate temperature profiles. The warm water in the maximum-speed horizon is transported faster than in the horizons above and beneath. However, stratification above was too strong for the establishment of temperature inversions and hence for the production of convective mixing. But the situation changes at the opposite phase of the seiching when the stratification is weak as shown in Figure 17. This figure shows two temperature profiles and a corresponding current velocity profile around 02:30 (May 17, 2001). At this time the seiche induced current velocity was just passing its negative peak-current and again was decelerating (Figure 3). Cold water was transported to the sampling site and the well-mixed BBL was strained (Figure 2). As shown in Figure 15 the characteristic maximum in the current-velocity profile produces inversions in the temperature profiles because the cold water was transported faster at this depth horizon than in the layer below. The BBL becomes unstable since the contribution of salinity to the near-bottom density stratification is negligible in Lake Alpnach [26]. Although the imposed vertical temperature gradient is weak, this effect produces additional mixing within the BBL by forced convection. Unfortunately the quality of the temperature microstructure measurements did not allow to measure this effect in the dissipation of kinetic energy. Very similar processes were observed in the coastal oceans in regions of freshwater inflow (ROFI). Tidal straining of the horizontal isopycnals produces periodical density inversions and hence convective mixing [41]. However the scales and the impact of these processes are much larger there. The additional mixing in terms of the dissipation rate and the turbulent kinetic energy produced by tidal straining was described recently by Rippeth et al. [42].

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Figure 16. Temperature and current velocity profiles from the BBL. The measurement times are indicated.

5. Conclusions Coherent acoustic Doppler profiler enable the resolution of the very low current velocities found in the BBL of lakes with very high spatial and temporal resolution. Measurements obtained in the seiche-induced BBL of Lake Alpnach showed a perfect logarithmic velocity profile between the lowest resolved depth cell (z ≥ 0.5 m) and a persistent velocity maximum at 2 to 3 m above the bottom. The seiching period was 24 h. A logarithmic fit of the profiles to the law-of-the-wall was applied to infer the turbulent friction velocity u∗ and the roughness length scale z0 . Dissipation rates of turbulent kinetic energy were estimated from u∗ using the law-of-the-wall relations. Although the profile range where the law-of-the-wall was applied (z > 0.5 m) showed an almost perfect logarithmic form, turbulent quantities do not satisfactorily obey the law-of-the-wall. The behavior is strongly influenced by the oscillatory dynamics of the forcing and hence both quantities, u∗ and z0 were overestimated considerably. Moreover, a phase lag of 1.5 h between the temperature microstructure based dissipation rate estimates and the current velocity was observed. Using a simple analytical model (Stokes’ oscillatory BBL) and a numerical turbulence-closure (k − ε) model, it was shown that at least partly the observed characteristic velocity maximum as well as the deviations from the law-of-the-wall

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Figure 17. Same as Figure 16 at different times of measurements.

can be attributed to the oscillatory dynamics of the system. Also the numerical model reproduced the phase lag between turbulent kinetic energy dissipation and velocity. The results showed that a classical law-of-the-wall layer exists for all forcing periods, but it is gradually confined to a thin near-sediment region for decreasing periods. Even for the relatively long 24-hour forcing period, this layer was shallower than 0.5 m and hence not resolved by the measurements. Recent theories for tidal flow [22] assuming the mean flow shear to be the key control on the timing of the dissipation phase lag were shown to fail in the low-speed boundary layers investigated here, where the rate and turbulent transport terms in the budgets of k and ε contribute significantly. Since the observed 24-h period is a natural period, which can be observed easily in many natural systems, one should carefully check the validity of the law-of thewall-scaling before its application. For higher-frequency oscillations the described effects are even more pronounced and for further studies it would be interesting to investigate the effect for shorter forcing periods. In contrast to the well-investigated tidal boundary layers in the ocean, in which the dominant current component is the M2 tide, typical periods in lakes cover a much wider range. The investigation of phase lags in a variety of such boundary layers could provide a serious test on the two-equation models applied to such situations.

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Further it was shown that the differential horizontal transport in the BBL can produce unstable stratification and hence additional turbulent kinetic energy for mixing. A follow-up question is how much turbulent kinetic energy can be produced by this 2-dimensional mechanism and under which circumstances it can be important for the overall budget of turbulent kinetic energy of a lake? Acknowledgements We thank M. Schurter, B. Müller, M. Märki, C. Dinkel and J. Little for their great help in the field. D. McGinnis kindly improved the English. This work was financially supported by the Swiss National Science Foundation 20-50761.97 and 2000-063723.00. References 1. 2. 3.

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