Mixing In Hydraulically Controlled Exchange Flows Biology Essay

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Laboratory experiments are used to determine the bulk mixing efficiency in hydraulically-controlled exchange flows past a topographic constriction that include a lateral contraction and a bottom sill. The flow processes generated in the laboratory are relevant to natural density-driven exchange flows through straits or over sills, commonly found in the oceans. The total amount of turbulent mixing is measured and reported as a bulk mixing efficiency. For flows through simple constrictions or over a vertical weir inserted in the constriction, the bulk efficiency of mixing asymptotes to a constant value of 11% at large Reynolds numbers, independent of flow conditions and constriction geometry, and in good agreement with a scaling analysis. For flow over a sloping sill within the same constriction, the measured efficiency is reduced to 8%. We conclude that the average efficiency in the oceans may be one half of that commonly assumed in general ocean circulation models.

1. Introduction

Vertical mixing plays a key role in maintaining the abyssal density stratification of the oceans in the context of the global overturning circulation of ocean waters [1]. The amount of available energy required to sustain the mixing, the input of turbulent kinetic energy, and the fraction of the input used for the mixing are poorly known [2]. Thus, an understanding of the rates of turbulent mixing in stably stratified turbulence remains important. This study has therefore examined a specific class of flows of fundamental interest in which turbulent mixing occurs, that is buoyancy-driven exchange flows, and in which accurate and precise measurements of the amount of irreversible mixing, hence mixing efficiency are possible. In the laboratory, hydraulic controls that are imposed by geometry or topography lead to a steady exchange flow and shear-induced mixing. This mixing can be quantified by measuring the change in the density structure of the water in the reservoirs. The observed shear instability in flows of this type may also provide insights into the dynamics of mixing relevant to other forms of unstable shear flows. For example, similar buoyancy-driven exchange flows are commonly found in the oceans, where straits or sills control the exchange of water of differing densities between marginal seas and the oceans [3], between water masses in the deep oceans [4, 5] and within estuaries [6].

An important quantity characterising turbulent mixing in stratified shear flows is a mixing efficiency, defined as the fraction of the kinetic energy supplied to turbulence, which is in turn irreversibly converted into potential energy of the density field, while the remainder of the kinetic energy is dissipated by viscosity. Numerical simulations of the full evolution of a Kelvin-Helmholtz (KH) billow [7, 8] have demonstrated a detailed sequence of mixing events, where a `cumulative' mixing efficiency is calculated as a measure of the time-averaged efficiency. The cumulative mixing efficiency is found to approach a value of 0.15, somewhat less than the proposed value of approximately 0.2 for stratified turbulence [9].

Theoretical studies [10] have investigated a turbulent patch generated in an initially linearly stratified fluid and found that even smaller efficiencies of order 0.1 are possible. As a result of a turbulent mixing event, mixed water is locally produced but eventually spreads out laterally until the fluid reaches an equilibrium state. In this state, the potential energy increase attributable to mixing referred to as the large-scale flux Richardson number, or equivalently the bulk mixing efficiency, is half of that calculated for the localised patch.

Here laboratory experiments are used to determine the bulk (space and time averaged) mixing efficiency in hydraulically controlled, steady exchange flows past a topographic constriction. The experimental methods are detailed in §2. The results for all cases are presented in §3. All the results are discussed in §4 and conclusions are given in §5.

2. Experimental methods

2.1. Apparatus and procedure

All experiments were conducted in a tank of length L = 5.26m and width B = 0.2m. Figure 1 shows the apparatus used in the experiments in which one of four symmetric constrictions was placed at the centre of the channel. Three of the constrictions had curved walls (Figure 1a), referred to as `short constrictions', and the fourth with the same curved walls but included a straight central section (Figure 1b), referred to as the `long constriction', having parallel-sided walls 0.06m apart and 0.5m long (the total constriction length Lc was 1.0m for this geometry). The minimum width bo of the short constrictions was such that the contraction aspect ratio bo/B was 0.1, 0.3, or 0.5.

Figure 1. Experimental apparatus showing a plan view of (a) a short constriction; (b) a long constriction; and a channel with (c) a vertical weir; (d) a sloping sill in the long constriction.

In each experiment, the tank was filled with freshwater to a depth of H and a removable barrier was inserted at the central point of the constriction. A density difference across the barrier was created by dissolving a measured quantity of salt into the right reservoir. Both the dense and light fluids were dyed with different colours for flow visualisation. The reservoirs were then stirred thoroughly to ensure homogeneity in density.

The free surface heights of the left and right reservoirs were adjusted to give equal hydrostatic pressures at mid-depth so as to achieve a purely baroclinic exchange. The exchange flow was initiated by the smooth and rapid removal of the barrier over a period of approximately 1 s in each experiment. The barrier was rapidly reinserted into the channel at a time when gravity currents had nearly reached the endwalls of the tank.

Sill-exchange experiments were carried out with one of two different bottom sills being placed at the mid-point of the long constriction (Figure 1c, d). The sills were a vertical weir (0.02m thick and 0.1m high) and a triangular sill with sloping sides (0.5m long and 0.1m high), each spanning the constriction width. In these experiments, the barrier was inserted at the centre of the constant width section of the long constriction so as to make contact with the top of the sill.

2.2. Measurement techniques

Before an experiment was run, the initial densities of the fresh and saltwater reservoirs were measured in an Anton Paar digital density meter. The initial free surface heights in the reservoirs were determined to within ±0.01mm using a digital micrometer gauge with an attached upward-pointing needle immersed in the water.

After the exchange flow was stopped, the water was left to stand for at least two hours, long enough for all long waves to die away but too short for diffusion of salt to affect the density distribution in the reservoirs on length scales greater than 3mm. During this time, the tank was sealed to minimise evaporation. After each experiment, densities from both reservoirs were sampled and measured using the digital density meter to a precision of 10-3 kgm-3. The final heights were measured as required in the calculations of the final potential energies.

2.3. Experimental conditions

The relevant variables included a fractional density difference ∆ρ/ρ2 = (ρ2−ρ1)/ρ2 (across the range 0.001 ≤ ∆ρ/ρ2 ≤ 0.096), where ρ1 and ρ2 are the initial densities of the fresh and saltwater reservoirs, respectively, the reduced gravity g' = g∆ρ/ρ2, where g is acceleration due to gravity, the total water depth H and the kinematic viscosity ν. Flow conditions were examined by varying the flow aspect ratio H/Lc, the contraction aspect ratio bo/B and a Reynolds number.

The most appropriate Reynolds number is the one based on the constriction length Lc, as this is the lengthscale for flow acceleration [11]. This Reynolds number is referred to as the horizontal Reynolds number Re, defined as


lying in the range 1.2x104 ≤ Re ≤ 2.1x105 for all runs.

In the case of flows through simple constrictions, we carried out thirty runs with the short constrictions and nine runs with the long constriction. For the short constrictions, the water depth H was set to 0.1m, 0.2m, or 0.3m. The results for the short constrictions were compared with those for the long constriction, in which the water depth was fixed at 0.2m. These were aimed to examine the sensitivity of the results to the constriction geometry and water depth.

In the case of flows over sills, we conducted six runs with the vertical weir and nine runs with the sloping sill. These were designed to examine the roles of bottom topography in altering mixing processes near the sill. The flow speed scaled with (g'H)1/2 as the water depth over the sill crest was only H/2. Hence, for the sill case horizontal Reynolds number is defined as Re = 0.5 (g'H/2)1/2 Lc/ν. The contraction aspect ratio in this case was fixed at 0.3 and the only variable changed was the density difference ∆ρ.

2.4. Analysis

Following [11], we describe a method of calculating the efficiency of mixing in a buoyancy-driven exchange flow. For the initial state (Figure 2a) with two basins of homogeneous water of different density, the potential energy Pi is given by

, (2)

where h1 and h2 are the initial free surface heights of the fresh and saltwater reservoirs, respectively, and A is the horizontal cross-section of each reservoir, which may be either uniform or dependent on height.

In the final state after the exchange flow and mixing have occurred and altered the properties of water on both sides of the constriction (Figure 2b), the potential energy Pf is determined by the measured density distribution, ρ´1 and ρ´2, in both reservoirs,

, (3)

where h´1 and h´2 are the final free surface heights of the fresh and saltwater, reservoirs, respectively, and again A is the cross-sectional area in each reservoir.

We examine the effects of mixing by defining the hypothetical minimum state that would be present if mixing did not occur (but the same mass were transported). This hypothetical state is obtained by notionally redistributing the measured total amount of salt within each reservoir into fresh and saltwater layers of densities ρ1 and ρ2 (Figure 2c). We write the required thicknesses of the fresh and saltwater layers in each of the reservoirs for this state as hfi (i=1,2) and hsi (i=1,2), respectively. The required redistribution of salt must satisfy conservation of mass such that the depths hs1 and hs2 of the hypothetical interfaces dividing the fresh and saltwater layers in each reservoir are given by


We assume that a linear equation of state is satisfied such that h´i = hsi + hfi, which gives constant volume. The hypothetical minimum potential energy Ph is then given by


Figure 2. (a) The initial state. (b) The final state indicating the density profiles measured after the experiment. (c) The hypothetical minimum state following partial redistribution of salt within each reservoir with no mixing.

The increase in potential energy owing to mixing, or the amount of mixing, Pm is given by the difference in potential energies between the final state and hypothetical state with no mixing, Pm = Pf − Ph. The available potential energy Pa released to the flow is the difference in potential energies between the initial and hypothetical (non-mixing) states, Pa = Pi − Ph. We define the mixing efficiency as the fraction of the available potential energy released to kinetic energy of the flow that is transformed into an irreversible increase in the potential energy of the density distribution above the hypothetical state. The mixing efficiency η for a two-layer exchange flow can then be written as

, (6)

where the denominator Pa is the total mechanical energy supplied, whether as kinetic energy of the mean flow or as turbulent kinetic energy. Thus, η in (6) represents the overall efficiency with which mixing draws on the available energy in this type of flow.

3. Results

3.1. Qualitative observations

In all cases, after the barrier was removed the subsequent density-driven exchange flow led to shear instability and extensive mixing, particularly in the vicinity of the constriction (Figure 3). Hydraulic jumps occurred near the exits of the constriction and gravity currents formed in each reservoir to accommodate the exchange. In experiments with a simple contraction, the flow was characterised by the growth and collapse of KH billows on the strongly sheared interface within the contraction, generating overturns and vertical mixing. These billows, along with the resulting mixed fluid, were carried away from the centre of the constriction and were not present beyond the hydraulic jumps. There were no persistent billows along the stratified region above the gravity currents, apart from those on the current heads. The exchange was effectively stopped as the barrier was rapidly reinserted into the channel at a time when the gravity current noses had nearly reached the endwalls. Thus, throughout the exchange the mean flow in the constriction was in a steady state and the distant endwalls had no influence on the flow.

After the barrier was reinserted, the currents were reflected from the far ends of the tank to form large-amplitude waves, which propagated towards the constriction and were eventually reflected back from the replaced barrier, after which a complex pattern of wave-wave interactions was observed in each reservoir. All of these processes produced a relatively minor amount of mixing.

There were no qualitative differences in the flow for experiments with large Reynolds numbers. Continuous overturning at the sheared interface within the constriction was the dominant feature in these experiments. In experiments with low Reynolds numbers, the shear instability at the interface appeared to be relatively weak and the billows were only intermittent, with consequently weak mixing between the layers.

Figure 3. Intense mixing in the vicinity of the constriction from three cases having bo=60mm, H=0.2m, and with (a) the long constriction for Re=15.5x104 and ∆ρ/ρ2=4.8%; (b) a vertical weir for Re=5.4x104 and ∆ρ/ρ2=4.8\%; (c) a sloping sill for Re=6.8x104 and ∆ρ/ρ2=1.9\%. The flows shown in the frames are in steady states.

Similar features were observed when bottom topography was introduced to the long constriction (Figure 3a), with the less dense fluid moved along the upper surface into the right reservoir and the dense fluid either plunged down (Figure 3b) the vertical side of a vertical weir or ran down a slope along the lee side of a sloping sill (Figure 3c). When the dense overflow reached the bottom of the sill, it began to fill up the left reservoir, after which the dense current was generated and advanced along the base of that reservoir. The interface between the layers upstream of the sill was almost flat, stretched to the exit region of the contraction in the right reservoir, where the billows appeared to roll-up with relatively small amplitudes compared with those in the left reservoir.

3.2. Measurements

Here we provide examples of normalised final density profiles measured from both the left and right reservoirs after each experiment from cases with the long constriction and with bottom topography within the same constriction. In each case, the depth is normalised by the full water depth H and the density is normalised by the initial difference in the densities of both reservoirs. The normalised density sets to zero at the water surface and to unity at the channel base.

Examples of the normalised density profiles from the long constriction case are plotted in Figure 4 for four experiments having different fractional density differences, but the same constriction minimum width bo = 60mm and water depth H = 0.2m. The normalised profiles for the left (Figure 4a) and right reservoirs (Figure 4b) are indistinguishable from each other, suggesting that mixing processes are dynamically similar in both reservoirs.

Figure 4. Final profiles of normalised density (ρ(z)−ρ1)/(ρ2−ρ1), plotted against normalised height z/H in experiments with the long constriction. Frames (a) and (b) are the left and right profiles measured from experiments with H=0.2m, bo=60mm and four different fractional density differences: ∆ρ/ρ2=0.7% (dotted line), ∆ρ/ρ2=1.9% (solid line), ∆ρ/ρ2=3.0% (dashed line) and ∆ρ/ρ2=4.8% (dashed-dotted line), respectively.

Examples of the normalised density profiles from both reservoirs measured after each exchange with a bottom sill are plotted in Figure 5. Frames (a) and (b) show the profiles from experiments with the vertical weir, while frames (c) and (d) from those with the sloping sill. Both cases were run with the same water depth H = 0.2m and three different fractional density differences. There are differences in the left profiles between the vertical weir (Figure 5a) and the sloping sill (Figure 5c) in the regions close to the bottom boundary. The differences suggest that mixing is enhanced in the left reservoir for the sloping sill case, leading to a flow asymmetry in this case.

We also provide an overview of all experiments conducted by plotting in Figure 6 the results for all cases (see §4 for detailed discussion). The resulting mixing efficiencies for both flow over the weir with vertical sides and flows through the constrictions with no bottom topography approach an asymptotic value of η = 0.11 at large Re (shown as the solid line in Figure 6). However, the slightly smaller efficiency of η = 0.08 is measured for the case of flow over the sill with sloping sides (shown as the dashed-dotted line in Figure 6) for again large Re.

Control experiments were carried out to examine the effects of the non-linear equation of state of salt solutions, particularly at large density differences, where mixing tends to decrease the total volume and increase the internal energy, thus reducing the increase in potential energy owing to mixing. However, the influence of these effects on the calculated efficiency was found to be 0.005, smaller than the uncertainty of the measurements. Hence, we have assumed in all calculations that the density depends linearly upon salinity and that non-linear mixing has a negligible effect on the measured efficiencies.

Figure 5. Final profiles of normalised density (ρ(z)−ρ1)/(ρ2−ρ1), plotted against normalised height z/H in experiments with bottom sills. Frames (a) and (b) are the left and right profiles measured from the weir case with H=0.2m and three different fractional density differences: ∆ρ/ρ2=0.7% (dashed line), ∆ρ/ρ2=3.0% (dashed-dotted line) and ∆ρ/ρ2= 9.4% (solid line), respectively. Frames (c) and (d) are the left and right profiles measured from the sill case with H=0.2m and three different fractional density differences: ∆ρ/ρ2=0.7% (dashed line), ∆ρ/ρ2=2.9% (dashed-dotted line) and ∆ρ/ρ2=9.5\% (solid line), respectively.

We also conducted experiments with various run times to include runs in which the barrier was replaced into the channel after the gravity current heads had reflected from the far endwalls. These runs, along with those with the barrier was reinserted before the current noses had reached the endwalls, were designed to examine time-dependent mixing (not detailed here).

Figure 6. The measured mixing efficiencies η as a function of the horizontal Reynolds number Re (1) for all cases. Symbols indicate bo=20mm (circles), 60mm (triangles) and 100mm (squares) for the short constrictions, bo=60mm (diamonds) for the the long constriction, with the weir (bullets) and the sill (asterisks). The dashed line shows the theoretical value (η=0.125) for the efficiency of mixing, the solid line describes the mean (η=0.108) of the measured efficiencies for flows through the constrictions or over the weir with Re > 5x104 and the dashed-dotted line describes the mean (η=0.084) of the measured efficiencies for flows over the sill with Re > 4x104.

The results confirm that the rate of mixing was constant for the steady exchange flow and that mixing associated with the starting and ending of the exchange flow by lifting and replacing the barrier had no significant contributions to the overall amount of mixing. It was also found that the total available energy has largely depleted when the gravity currents reflected from the endwalls for the first time. With only a small portion of the energy released was present as kinetic energy of the mean flow at the moment the barrier was replaced, the mean kinetic energy of the flow could not lead to further significant additional mixing after the exchange flow was effectively stopped.

3.3. Theoretical prediction of mixing efficiency

A scaling analysis (not detailed here) for the mixing efficiency in terms of the energy budgets in the interfacial mixed layer was proposed by [11]. For a linear dependence of the mean density and velocity on depth, the change in potential and kinetic energies of the interfacial mixed layer was calculated. It was found that the irreversible increase in potential energy in the mixed layer relative to the two-layer state with no mixing is ∆P=∆ρgδ2/24, where δ is the thickness of the mixed layer. This potential energy increase is indicative of the amount of mixing, and corresponds to the difference in the kinetic energies between the two-layer (non-mixing) and actual states, which is indicative of the kinetic energy transferred from the mean flow to turbulence in the mixed layer. This kinetic energy difference is given by ∆K=ρo(∆U)2δ/12 where ρo is the mean density and ∆U=(g'H)1/2 is the velocity difference across the mixed layer. Thus, we can estimate the mixing efficiency as


If the time-averaged gradient Richardson number Rig tends to its critical value of 0.25, then the mixing efficiency is predicted to be η = 0.125.

The normalised theoretical estimate of the mixed region thickness δ/H used in (7) is compared with the profiles of dye concentration (Figure 7), where the profiles were obtained by averaging over 400 individual frames from the videos for six experiments. The measured profiles show that the mixed region thickness lies in the range δ/H = 0.23-0.25. For each of these experiments we find only a small variation in the mixed region thickness with position along the channel. The theoretical estimate for the efficiency of mixing η = 0.125 in (7) is in fact predicted as an upper bound for the measured efficiencies (shown as the dashed line in Figure 6).

Figure 7. Measured profiles of normalised mean dye concentration for four different experiments using a short constriction with bo=20mm, and Re=3.5x104 (solid line), Re=4.6x104 (dashed line), Re=9.4x104 (dotted line) and Re=11.6x104 (dashed-dotted line). Normalised dye concentrations are defined such that the unmodified upper and lower layers correspond to values of zero and one, respectively. The dark-solid line denotes the piecewise linear approximation to a normalised concentration profile with δ/H ≈ 0.25.

4. Discussion

We have performed laboratory experiments to determine the bulk mixing efficiency for a range of flow geometries that include lateral contractions and bottom sills. The most obvious effect of mixing in all cases is the production of the substantial volume of mixed water in the reservoirs after the experiment. The methods developed do not allow measurements of the mixing attributable to individual mechanisms or to different localities within the flow. Rather, the methods provide a measure of the cumulative, irreversible mixing over time and length scales large compared with the turbulent events, which leads to calculations of the bulk mixing efficiency.

We have tested the sensitivity of the results to the constriction geometry and water depth in the experiments with simple constrictions, and found only small effects of these parameters on the results. The measured bulk efficiency of mixing is η = 0.11 ± 0.01, provided that the horizontal Reynolds number defined in (1), Re >5x104. Among other sources of the mixing, the observed primary mechanisms responsible for mixing in the exchange flows through the constrictions are shear instability and hydraulic jumps that are associated with the constriction. Thorough and careful observations indicate that the most intense mixing in a controlled exchange flow takes place within the constriction and near the exits of the constriction. Hence, we suggest that the efficiency measured here will characterise mixing in exchange flows. Moreover, the measured bulk efficiency may also characterise mixing in a broader range of two-layer stratified shear flows, where KH billows are the dominant mechanism causing mixing. In experiments with low Reynolds numbers, we find that the shear instability is less efficient at mixing, owing to the absence of continuous small-scale turbulence at the interface.

In the case of flow over a vertical weir, the measured efficiency is found to asymptote to a value of approximately η = 0.11 for large Re. However, for flow over a sill with sloping sides a smaller asymptotic efficiency of η = 0.08 has been measured for large Re. These results are consistent with the scaling analysis that predicts an upper bound on the mixing efficiency of η = 0.125.

5. Conclusions

The aim of this study is to determine the overall mixing efficiency in hydraulically-controlled, two-layer exchange flows past a topographic constriction. Such a constriction include a lateral contraction and a bottom sill. We have found that the steady shear at the interface between the layers leads to the generation of KH billows and persistent, small-scale turbulence within the constriction.

The dependence of mixing efficiency on the details of constriction geometry and flow conditions, or topography, has been examined in detail. For flows through the constrictions, the overall (bulk) efficiency asymptotes to a constant value of 0.11 (±0.01) at large Re, independent of the external parameters. The same value is obtained for flow over the vertical weir. In the case of flow over the sloping sill, the bulk efficiency is measured to be 0.08 at large Re, again independent of the parameters. These results are consistent with the theoretical upper bound prediction for the mixing efficiency of 0.125 for the steady exchange flows.

The results are expected to be applicable to the large-scale oceanic flows as these flows are in the limit of large Re. Hence, the mixing efficiency in such flows will approach the theoretical upper bound of the efficiency, 0.125, fairly close to 0.15 obtained from numerical simulations [7, 8]. The results also provide insights into a broader range of density-stratified shear flows. While the results are supported by theoretical work [10], there remains discrepancies between the current results and those of previous studies [12, 13]. These studies have found that the mixing efficiency asymptotes to the widely proposed value of approximately 0.2, the mixing efficiency for steady, stratified turbulence [9], commonly used in the general ocean circulation model. It has been assumed that 20% of the turbulent kinetic energy in the abyssal ocean is irreversibly converted into potential energy of the density field [1, 2]. However, the smaller asymptotic mixing efficiency raises the possibility that the average mixing efficiency in the oceans may not be as large as widely assumed and that calculations of the global energy balance of the oceans need re-visiting.