Shear Instabilities and Stratified Turbulence in an Estuarine Fluid Mud

Junbiao Tu aState Key Laboratory of Marine Geology, Tongji University, Shanghai, China

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Daidu Fan aState Key Laboratory of Marine Geology, Tongji University, Shanghai, China

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Feixiang Sun aState Key Laboratory of Marine Geology, Tongji University, Shanghai, China

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Alexis Kaminski bDepartment of Mechanical Engineering, University of California, Berkeley, Berkeley, California

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William Smyth cCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Abstract

This study presents field observations of fluid mud and the flow instabilities that result from the interaction between mud-induced density stratification and current shear. Data collected by shipborne and bottom-mounted instruments in a hyperturbid estuarine tidal channel reveal the details of turbulent sheared layers in the fluid mud that persist throughout the tidal cycle. Shear instabilities form during periods of intense shear and strong mud-induced stratification, particularly with gradient Richardson number smaller than or fluctuating around the critical value of 0.25. Turbulent mixing plays a significant role in the vertical entrainment of fine sediment over the tidal cycle. The vertical extent of the billows identified seen in the acoustic images is the basis for two useful parameterizations. First, the aspect ratio (billow height/wavelength) is indicative of the initial Richardson number that characterizes the shear flow from which the billows grew. Second, we describe a scaling for the turbulent dissipation rate ε that holds for both observed and simulated Kelvin–Helmholtz billows. Estimates for the present observations imply, however, that billows growing on a lutocline obey an altered scaling whose origin remains to be explained.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: D. Fan, ddfan@tongji.edu.cn; W. Smyth, bill.smyth@oregonstate.edu

Abstract

This study presents field observations of fluid mud and the flow instabilities that result from the interaction between mud-induced density stratification and current shear. Data collected by shipborne and bottom-mounted instruments in a hyperturbid estuarine tidal channel reveal the details of turbulent sheared layers in the fluid mud that persist throughout the tidal cycle. Shear instabilities form during periods of intense shear and strong mud-induced stratification, particularly with gradient Richardson number smaller than or fluctuating around the critical value of 0.25. Turbulent mixing plays a significant role in the vertical entrainment of fine sediment over the tidal cycle. The vertical extent of the billows identified seen in the acoustic images is the basis for two useful parameterizations. First, the aspect ratio (billow height/wavelength) is indicative of the initial Richardson number that characterizes the shear flow from which the billows grew. Second, we describe a scaling for the turbulent dissipation rate ε that holds for both observed and simulated Kelvin–Helmholtz billows. Estimates for the present observations imply, however, that billows growing on a lutocline obey an altered scaling whose origin remains to be explained.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: D. Fan, ddfan@tongji.edu.cn; W. Smyth, bill.smyth@oregonstate.edu

1. Introduction

Turbidity maxima with high concentrations of suspended sediment are ubiquitous in estuaries due to a combination of hydrodynamic and sediment dynamic processes (Burchard et al. 2018, and references therein). In hyperturbid estuaries, the near-bottom layer of high-concentration, fine (particle size < 63 μm) suspended sediment is generally referred to as fluid mud (FM; McAnally et al. 2007). The motion of estuarine FM is modulated by the tidal cycle (Bruens et al. 2012), with resuspension and entrainment during phases of high flow velocity and settling in slower flow. Current shear, induced by bottom friction, provides an energy source for shear instability, while stratification induced by concentrated sediment (e.g., Becker et al. 2018; Tu et al. 2019) tends to stabilize the flow. Although this process has been reproduced in the laboratory (Scarlatos and Mehta 1993), field observations of shear instability in FM remain scarce. In the turbid Ems estuary, Held et al. (2013, 2019) observed cusped waves were suggestive of asymmetric Holmboe instability (Carpenter et al. 2007). Tu et al. (2020, hereafter T20) observed symmetric Kelvin–Helmholtz (KH) billows on an estuarine lutocline for the first time using echosounder images. They showed that the instabilities might play an important role in sediment entrainment and mixing across the lutocline. However, detailed measurements of FM and/or lutoclines linking shear instabilities and turbulent mixing are scarce due to technical challenges (Becker et al. 2018; Sottolichio et al. 2011). In this paper we analyze measurements, including the turbulent kinetic energy dissipation rate, from the Changjiang estuary through which the Yangtze River empties into the East China Sea.

KH billows have been well observed in the laboratory (e.g., Thorpe 1973). In the ocean, observations were successfully carried out photographically (Woods 1969), acoustically (e.g., Moum et al. 2003) and using finely spaced temperature sensor arrays (e.g., Hebert et al. 1992; Van Haren and Gostiaux 2010). Echosounder images have been used to identify shear instabilities in salt-wedge estuaries by Geyer et al. (2010) and Tedford et al. (2009). KH instability is also observed throughout Earth’s atmosphere (Lee 1997; Fukao et al. 2011; Fritts et al. 2014), where they often form banded clouds.

These observations have shown that billows can be interpreted using the gradient Richardson number Ri = N2/S2, where N2 = (−g/ρ)(∂ρ/∂z) is the squared buoyancy frequency and S2 = (∂u/∂z)2 + (∂υ/∂z)2 is the squared vertical shear of the mean horizontal current. The coordinate z points vertically upward, opposite to the gravitational acceleration g; ρ is the density (determined in this case by salinity, temperature, and suspended sediment concentration), and u and υ are horizontal velocity components to be specified later.

In the limit of inviscid, nondiffusive, steady flow, KH instability may arise if the shear is sufficiently strong, relative to stratification, that the minimum value of Ri < 0.25 (Miles 1961; Howard 1961). Observations of fully turbulent KH billows also suggest a critical Ri of 0.25 (e.g., Chang et al. 2016). However, underresolution might cause the Ri estimates to be too high compared to the canonical value of 0.25 (e.g., Moum et al. 2003). In forced shear flows, Ri can fluctuate around 0.25 due to the repetitive growing, breaking, and decaying of the billows (e.g., T20), a state often referred to as marginal instability (e.g., Smyth et al. 2019; Smyth 2020).

Shear instability is a major mechanism of turbulent mixing in oceanic stratified flow (Smyth and Moum 2012). Observations in the ocean interior (Chang et al. 2016; Moum et al. 2003; Seim and Gregg 1994) and in estuarine flows (Geyer et al. 2010) reveal elevated turbulence levels coinciding with the billows that result from the instability. Estimation of turbulent quantities such as the turbulent kinetic energy dissipation rate is challenging due to the need to measure fluctuating quantities accurately on fine spatiotemporal scales (Caulfield 2021), often from a moving platform. Thus, it is useful to parameterize mixing using readily measured quantities (e.g., Klymak and Legg 2010). Here we discuss a parameterization based on the billow height as registered by an echosounder.

We present observations of shear instabilities within an estuarine FM as it interacts with the oscillating tidal flow. We use measurements from a shipboard echosounder and CTD–OBS casts (conductivity–temperature–depth and optical backscatter sensor), as well as water samples. A bottom-mounted tripod system was added in order to measure the tidal flow using an ADCP (acoustic Doppler current profiler) and near-bed turbulence via high-resolution acoustic pulse-coherent Doppler velocity profiles. Acoustic imagery of the billows is presented, together with analyses of their properties and formation mechanisms. The billows’ aspect ratio can be used to infer the initial flow condition prior to billows’ formation. The echograms are further used to quantify the turbulent mixing, which may explain the exchange of the FM with the overlying, less turbid water. We discuss the possibility of a turbulent mixing parameterization based on readily obtained quantities: the density profile and the vertical scale of the billows as shown in the acoustic images.

This study extends the previous work of T20. In that work, we analyzed observations from the Jiaojiang estuary, examining hydrographic variations (tidal current, salinity, SSC, etc.) during a 25-h tidal cycle, echosounder images showing internal wave and instabilities, and velocity and density profiles synchronous with typical billow trains. Here, using new observations made in the Changjiang estuary, we repeat those analyses and in addition measure the turbulence associated with the shear instabilities. We also combine previous and present observations with DNS results and propose parameterizations for the TKE dissipation rate and the initial Richardson number based on the vertical extent of the observed billows.

In section 2 we describe observational details and analysis methods. Results are described and discussed in the context of existing knowledge and research questions in sections 36. We focus in turn on variations over the tidal cycle (section 3), on the physical mechanisms for individual instability events (section 4), on the turbulent mixing that results (section 5), and on the applicability of ε parameterization to the lutocline case (section 6). While shear instability is the underlying mechanism, we find significant differences between these observations and instances of shear instability in other geophysical regimes. In section 7 we summarize the major findings and suggest future work.

2. Data and methods

The North Branch of the Changjiang estuary (the lowest reach of the Yangtze River, Fig. 1) in southeast China is a shallow, tidally dominated distributary with mean water depth up to ∼8 m (Dai et al. 2016). The sea bed is mainly mud with mean grain size ∼8 μm. Approximately 25 h of data were measured starting from 1000 local time (LT) 2 October 2018 in the main tidal channel of the North Branch. The mean water depth was ∼6 m during the observational period.

Fig. 1.
Fig. 1.

Study area, satellite image courtesy of USGS. The red circle shows the location of observational site.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

A dual-frequency echosounder (24/200 kHz) was deployed on the vessel side at 0.6 m below the sea surface to measure water depth and collect acoustic images. Vertical profiles of salinity, temperature, and turbidity were measured every 0.5 h using a CTD with a built-in OBS 3+, sampling at 2 Hz. The OBS outputs were converted to suspended sediment concentration (SSC) using the filtered water samples. The water–sediment mixture density was determined combining SSC, salinity, and temperature. The buoyancy frequency was calculated using this density, which was dominated by high SSC. The determination of SSC and density is described in Text S1 in the online supplemental material. The salinity and temperature measured by CTD are used to calculate seawater density ρsw. The density of the water–sediment mixture is then estimated using the SSC via ρ = ρsw[1 − (SSC/ρs)] + SSC (Wright et al. 1986), where ρs = 2650 kg m−3 is the sediment density.

Concurrent near-bed observations were carried out using a bottom-mounted tripod. A downward-looking 2-MHz Aquadopp HR profiler (referred to as ADP hereafter) was set to burst mode, collecting 1024 profiles at 300-s intervals with a sampling rate of 4 Hz, so that the burst duration was 256 s. The ADP collected velocity profiles between 0.04 and 0.92 mab (meters above consolidated bed) with a vertical bin size of 0.04 m. An uplooking ADCP was deployed at 1.9 m mab sampling at 0.5 Hz and recording an average over 180 s with a vertical resolution of 0.5 m. A 2D electromagnetic current meter (EMCM) was deployed at 1 mab sampling at 1 Hz and recording horizontal velocity components averaged over 30 s. More details of the instrumentation are summarized in Table 1.

Table 1

Summary of instrumentation.

Table 1

Observations by ADP with a pulse-to-pulse correlation c < 50 and/or backscatter amplitude a < 30 are excluded. If the number of excluded values exceeds 50% of the total data points in a segment, the segment is rejected. The ADP data can suffer from signal attenuation under high sediment concentration due to its high operating frequency and is therefore quality controlled prior to further analysis. More details of quality control of the ADP data are given in Text S2 in the supplemental material. The quality-controlled ADP along beam velocities were used to determine the turbulent kinetic energy dissipation rate, ε, in the bottom layer 0.04–0.92 mab, using the structure function method (see appendix A for details). The velocity records are rotated into streamwise (u, positive upstream) and spanwise (υ, positive toward the south bank) velocities. The shear production P = −〈uw′〉∂u/∂z, where u′ and w′ are streamwise and vertical velocity perturbations, was calculated independently for comparison. Estimates of P and ε roughly agree (see appendix A), and differences are consistent with expected buoyancy effects, suggesting that the structure function method provides reliable dissipation estimates.

Interpretation of these data was aided by three-dimensional DNS (direct numerical simulations) of turbulent KH billows. The simulations focus on a single wavelength of an infinite KH wave train as it grows, breaks down into turbulence and ultimately dissipates. From the DNS output we estimate echo strength, billow height, and various turbulence statistics. Details are given in appendix B.

3. Intratidal variations in hydrographic structure and fluid mud distribution

As indicated by velocities measured at mid-water column (3.4 mab, Fig. 2d), the durations and magnitude of flood and ebb currents are comparable. A persistent presence of FM is evident throughout the entire ∼25-h tidal cycle (Fig. 2b). The thickness of the FM layer (defined by SSC > 10 g L−1, black contour, Fig. 2b) varies between 1 and 2.5 m depending on the tidal phase. Another lutocline was observed between 0 and 1 mab (red contour, Fig. 2b) with SSC greater than 100 g L−1, suggesting that the viscosity of FM may be elevated (Mehta 2013). However, this effect is expected to be minor as we focus on the layers with internal waves and instabilities where SSCs are mainly O(10) kg m−3 (see Figs. 3a2–f2 and appendix C for further discussion).

Fig. 2.
Fig. 2.

Depth–time variations in (a) salinity and (b) SSC, and time series of (c) SSC and (d) streamwise velocities at different elevations. Near-bed structure of (e) streamwise velocities measured by ADP, (f) buoyancy frequency squared, (g) shear squared, and (h) gradient Richardson number are also presented. Shadowed areas represent the approximate periods wherein the internal waves and billows are identified from the echosounder images.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

Fig. 3.
Fig. 3.

(a)–(f) Echosounder images and profiles of water–sediment mixture density, streamwise velocity, shear, and buoyancy frequency. (a1)–(f1) Zoomed-in views for typical structures. For disturbances interpreted as KH billows, braids are indicated as white dashed curves and symbol “B” and cores represented as “C.” Shaded bands show the depth range of the acoustic image at left. Gray dots indicate cubic spline interpolations using the lowest two data points from ADCP and the three uppermost data points from ADP. Densities of water–sediment mixture are represented by σt = ρ − 1000. SSC values > 100 g L−1 (the upper limit of the measurements) were indicated by red dots. Black and red dotted lines in (a3)–(f3) represent S2/4 and N2, respectively. Note that S2/4 > N2, is equivalent to Ri < 1/4. Gray dots in (a3)–(f3) indicate S2/4 based on interpolated velocities. On the frames at the right, horizontal gray bands indicate the depth range shown in the images at left.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

It appears that, during high-flow periods (hours 11–16 and 22–28), the FM layer thickens and serves as a sediment source for the overlying water (Fig. 2c). Note that during periods of flow transition (hours 16–19 and 29–36), the upper-layer water slows down and changes from flood to ebb, and velocity within FM is close to zero, suggesting that the FM was not mixing with the overlying water. Thus, the fresher water is trapped in the lower FM layer while the salty upper layer continues to flow upstream. As a consequence, reduced salinity was observed in lower water column compared to the water immediately above, especially within the FM (e.g., hour 16–20s, Fig. 2a), a phenomenon frequently observed in estuarine FM environments [e.g., the Huanghe (Wang and Wang 2010), the Gironde (Sottolichio et al. 2011), or the Ems (Becker et al. 2018)]. The decreased salinity might be artificial as the CTD conductivity cell could be affected by high sediment concentration (Kineke and Sternberg 1995). However, experimental studies indicate this effect is negligible at low salinity (<10 psu) (Dai et al. 2011; Fig. S1 in the supplemental material), as is the case in this study. Thus, we believe that the salinity reduction in the near bed region is associated with separate flow regimes within and out of FM as discussed above.

Flow regimes within the FM layer were investigated using the near-bed ADP data. Although the nominal range of ADP was 0.04–0.96 mab, the available velocity profile range is reduced due to acoustical signal attenuation by FM (Fig. 2e). The buoyancy frequency N (Fig. 2f) varies approximately between 0.3 and 1.7 s−1, indicating strong sediment stratification. Stratification is usually stronger in estuaries than in the open ocean, but these values are well above a typical value (N = 0.3 s−1) for salt-wedge estuaries (Geyer et al. 2010) as well as that at a lutocline in hypertidal estuary (tidally averaged value of ∼0.4 s−1 and maximum value of 0.7 s−1, Becker et al. 2018). The squared shear S2 ranges from 104 to 2.5 s−2 (Fig. 2g). Low values of Ri generally coincide with observations of instability and turbulence. The greater variation in shear compared to stratification indicates that the temporal variations in Ri (Fig. 2h) are dominated by shear, i.e., low Ri values correspond primarily to strong shear, and vice versa.

Around peak flood and peak ebb, large velocity differences were observed between 1 and 3.4 mab (Fig. 2d). Remarkable SSC differences occur between 1 and 2 mab, indicating the presence of a sharp density interface at this region (Fig. 2c). The interaction between current shear and mud-induced stratification might lead to the formation of internal waves and/or billows (T20). Such is the case in this study as well-defined billows were identified from the echosounder image at several periods covering both flood and ebb phase (hours 11.5–12.5; hours 14–17; hours 20–23; hours 24–25.5, gray shading areas in Fig. 2). These periods were also characterized by Ri below or fluctuating around 0.25 at regions below 1 mab (gray shaded areas in Fig. 2h), a condition favoring shear instability. Periods with extremely high Ri values suggest reduced flow (e.g., hours 17–19; hours 30–32) or turbulence dampening by high SSC (e.g., hours 32–35 when current is not weak).

4. Internal waves, instability, and billows

Echosounder images often reveal wavelike structures suggestive of shear instability (Fig. 3, shaded bands) in the near-bed region. No surface waves (Trowbridge and Traykovski 2015) or bed forms (Tedford et al. 2009) were present during the observational period.

a. Stratification, shear, and instability

We next describe six examples of wavelike disturbances in the echogram as illustrated in Fig. 3. Interpretation of these results is subject to three important caveats. First, the time dependence of the echogram represents an unknown combination of true time evolution and advection by the horizontal mean flow. Second, the incomplete velocity measurements are often interpolated through regions of interest. Third, neither density nor velocity profiles represent the initial state prior to instability growth. The latter is a challenge because the initial state is used to define the particular shear instability mechanism in theoretical, numerical and laboratory studies (e.g., Smyth and Carpenter 2019). We return to this issue in section 4b.

  • At hour 11.5 (Fig. 3a), the acoustic signal was enhanced, indicating the growth of wavelike disturbances and small-scale structure centered at ∼0.7 mab. Following the growth phase, braid-core structures are visible (Fig. 3a1). Stable stratification was localized in the layer 0.4–1.2 mab, a region with significant shear as well, resulting in Ri < 0.25 at some depths (Fig. 2h; Fig. 3a3). In the later part of the measurement, a wavelike signal emerged. The asymmetric form suggests wave steepening and resembles the distinctive S-shape identified by Geyer et al. (2010) and others in observations of KH billows. We therefore interpret this event as resulting from shear instability.

  • At hour 12, billows appeared at a greater height, 1.2–2 mab (Fig. 3b). Steepening is evident, with sharp cusps on both crests and troughs. Although no direct velocity measurements are available in this region, large shear is implied by the interpolated velocities in Fig. 3b2.

  • The most beautiful, well-organized billows with clear, symmetric braid-core structures (KH category) occur at hour 16 at 0.8–1.6 mab (Fig. 3c). This vertical range corresponds approximately to the stratified layer. Note that, although the sign of the shear is reversed in this case (Fig. 3c3), the polarity of the billow structures remains the same, with the braids (the brightest red regions) descending in time. This is because the mean flow direction reverses with the tidal phase, i.e., the product of u and du/dz has the same sign in all cases.

  • At hour 20.5, the acoustic image suggests internal wavelike structures with no sign of steepening that would indicate instability (Fig. 3d). The Ri values below 0.9 mab appear to change from a stable (Ri > 0.25) to an unstable regime (Ri < 0.25) (Fig. 2h). It is interesting that the Ri values tend to decrease to values < 0.25 at the top and base of the layer containing the billows (Fig. 2d4). Note that the lowest S2/4 data point in Fig. 2d4 is likely unreliable as the flow velocities inside this particular FM layer decay sharply toward the bottom and remains low, hence the vertical gradient vanishes. This is also true for other times with extremely large Ri values caused by small S2 at the lowest heights approaching the (hydrodynamic) bed (i.e., u ∼ 0 m s−1).

  • At hour 21 (Fig. 3e), the wavelike features resemble KH billows, with discernible roll-up structure. The interpolated velocity profile suggests that the shear increased while the density gradient remained comparable to hour 20.5, resulting in decreased Ri (Fig. 3e4).

  • At hour 25 (Fig. 3f), the echosounder signal reveals a combination of overturning billows and small-scale density variations that suggest turbulent breakdown of the billows. The shear layer appears to coincide with the density interface, resulting in shear instabilities occurring at 0.6–1.2 mab.

Fig. 4.
Fig. 4.

Aspect ratio vs initial minimum Richardson number Ri0. Symbols represent results from laboratory experiments and DNS as indicated in the legend. The straight line shows the least squares fit (1). Shaded bars indicate in situ observations of aspect ratio, from which we infer Ri0 using (1).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

In summary, the six examples shown in Fig. 3 exhibit shear instabilities, (breaking) internal waves, and small-scale structure. The Ri values (estimated over 256 s) in the regions with billows are well below 0.25 at hour 11.5 and hour 25. For other periods, instability coincides with near-bed Ri below or fluctuating around 0.25. These periods were also characterized by large (interpolated) current shear in the mid-upper water column. The approximated Ri values are found to fluctuate around or slightly above 0.25 during these periods.

b. Aspect ratio and the initial Ri

The ratio of maximum billow height to wavelength has been called the aspect ratio (e.g., T20), and is equivalent to the steepness as defined in the original laboratory experiments of Thorpe (1973). The aspect ratio is known to be a strong function of Ri0, the minimum gradient Richardson number that existed when the instability first began to grow (Thorpe 1973). Ri0 is difficult to define in nature but is crucial for connecting observations with the theory of idealized linear instabilities (e.g., Miles 1961; Smyth and Carpenter 2019), with laboratory experiments (e.g., Thorpe 1973) and with numerical simulations (e.g., Mashayek et al. 2017; Kaminski and Smyth 2019). The aspect ratio is defined as hes/λ. In this ratio, hes is the billow height, measured graphically from the echosounder image. The wavelength λ is the product of the wave period, measured from the echosounder image, and the mean velocity at the height of the billows. In this subsection we explore the variability of the aspect ratio and see how it can be used to infer Ri0.

The dimensions of observed oceanic billows are highly site specific, with wavelength varying over two decades: O(1) m on estuarine lutoclines (Held et al. 2019; T20), several meters to over 100 m in estuarine pycnoclines (Geyer et al. 2017; Tedford et al. 2009), and 75–700 m in the deep ocean (Chang et al. 2016; Van Haren and Gostiaux 2010). Atmospheric KH billows can be even larger, with wavelengths of several kilometers (Fukao et al. 2011). In contrast to this wide variation, the aspect ratio is relatively constant. T20 summarized 10 oceanic cases with thermohaline stratification and found that the aspect ratio ranges between 0.08 and 0.31 with mean 0.15 and standard deviation 0.08. Cases with a near-bed lutocline have yielded larger values [0.28–0.62 (Jiang and Wolanski 1998), 0.34–0.53 (Held et al. 2019), and 0.14–0.58 (T20)].

A first estimate of hes/λ may be obtained by assuming that the billow height equals the thickness of the shear layer. DNS and laboratory experiments show that this is approximately true, though mature billows can be larger by factors of 2–3 (e.g., Smyth and Moum 2000). For a KH billow growing on an idealized shear layer, the wavelength is ∼7 times the shear layer thickness prior to the onset of the instabilities (Moum et al. 2011; Smyth and Carpenter 2019), suggesting an aspect ratio of 1/7 = 0.14. The average value 0.15 found by T20 for the oceanic cases is close to this, but the lutocline aspect ratios are significantly larger.

Echosounder images (e.g., Fig. 3) permit straightforward graphical measurement of the height and the period of a train of billows. The wavelength of the billows is approximated by multiplying the period with an estimate of the velocity at the height of maximum shear (Smyth and Carpenter 2019; T20). The billows observed here have periods of 5–8 s. A large range of horizontal propagation velocities leads to wavelength estimates varying between 0.7 and 3.6 m. Aspect ratios estimated from the six cases described in Fig. 3 vary between 0.2 and 0.71, with mean value of 0.4 and standard deviation of 0.2. Thus, both previous and present observations agree that the billow aspect ratio on near-bed lutoclines is elevated compared to that of the oceanic thermohaline cases.

A possible explanation for the increased aspect ratio of the lutocline billows is reduced Ri0 associated with flow over the boundary. Compiling published results from previous laboratory (Thorpe 1973) and DNS studies (Fritts et al. 2011), as well as the present DNS (see details in section 5 and appendix B), we obtain an empirical approximation
Ri0=0.250.39hes/λ,
as shown in Fig. 4.

Based on (1), the range of oceanic values compiled by T20, hes/λ = 0.15 ± 0.08, suggests that those billows correspond to laboratory or DNS models with Ri0 between 0.15 and 0.20 (thick shading on Fig. 4). In the lutocline case, the well-developed KH billows with clear braid-core structures have hes/λ= 0.58, 0.41, 0.69, 0.24, and 0.54 (Fig. 3a4 in T20; Figs. 3a,c,e,f in this study). These values are uncertain by ∼50% due to our limited knowledge of the mean flow velocity, but they illustrate the use of (1), suggesting Ri0 = 0.02, 0.09, 0, 0.15, and 0.04, respectively. These Ri0 values are well below the Miles–Howard threshold for inviscid shear instability (Ri0 = 1/4; Miles 1961; Howard 1961) and are consistent with the proximity of the benthic boundary, where shear is produced directly via bed friction and the fluid is typically well mixed. In a Monin–Obukhov boundary layer, for example, Ri drops linearly to zero at the boundary (e.g., Grachev et al. 2015; Scotti and White 2016).

Jiang and Mehta (2002) made a similar comparison based on a “global Richardson number,” measured directly in flows where waves already existed (whereas our parameter Ri0 is the initial value and is measurable only in laboratory and DNS experiments). Jiang and Mehta found that the wave height decreases with increasing global Ri while the wavelength increases, i.e., the aspect ratio decreases. Insofar as these two variants of the Richardson number are related, the present results support those of Jiang and Mehta. Our objective here is to use the aspect ratio is to infer Ri0, a parameter that is otherwise unmeasurable in flows where billows are already present.

5. Turbulent mixing

a. Effects of shear and stratification

To investigate the turbulent mixing associated with shear instabilities, we focus on hour 11.5, when well-defined billows are identified and profiles of density and high-frequency velocity cover the billows’ vertical extent (Fig. 3a). Figure 5b shows the time dependence of the echogram from which the wavelike billows’ heights were approximated (Fig. 5a). Although this evolution represents an unknown combination of temporal and spatial variability, it is broadly consistent with the growth, breaking and decay of a long train of KH billows advected past an observer (e.g., Smyth et al. 2001; Mashayek et al. 2017). From 0 to 25 s, the acoustic signal is enhanced and wavelike features appear. From 25 to 50 s the billows grow. From 50 to 100 s clear braid-core structures are identified (also in Fig. 5c), indicative of classic KH instability; after 100 s the billows appears to break down into small-scale turbulence.

Fig. 5.
Fig. 5.

Echosounder images and profiles of water–sediment mixture density and streamwise velocity for 150 s near hour 11.5 (Fig. 3a). (a) Billow heights as approximated by taking the vertical extent of each billow as identified from the echosounder image in (b). (c) A close-up of (b) showing the well-defined braid-core structures as signatures of KH billows during 50–100 s. (d) Time series of high-frequency, vertically averaged shear squared based on ADP velocity profiles and an assumed constant buoyancy frequency squared based on the density profiling data with (e) the resulting Ri shown. Vertical variations in shear and (f) N2, (g) Ri, (h) ε, and (i) Reb are shown. These are time averages over a 256-s burst of the ADP.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

The squared shear, averaged vertically over ∼0.7–1 mab, appears to correspond to the apparent billows’ variability, especially an increase at ∼100 s as wavelike motions become KH billows (Fig. 5d, black curve). Because the density profiles were obtained only every 0.5 h, we are obliged to assume a constant N2 (i.e., the observed vertically averaged value) during the 150-s-long period described here. The resulting Ri values fluctuated around 0.25 at 0–50 s, identified as a prebillow period, and drop below 0.25 after 50 s when fully developed billows emerged, suggesting enhanced turbulent mixing (Fig. 5e). The time-averaged S2 appears to decrease toward the bed, whereas the N2 increases toward bed (Fig. 5f), resulting in Ri close to 0.25 at the midelevation (∼0.8 mab, Fig. 5g), which is approximately the central elevation of the observed billows.

In a standard (e.g., Monin–Obukhov) bottom boundary layer, the shear is expected to reach its maximum at the bed. However, in our case the shear decreases toward bed. This, as observed in other FM environments (e.g., Jaramillo et al. 2009), is likely because the velocity inside the FM layer decays abruptly and remains low, hence the vertical gradient of current velocity is low within that layer.

b. Energy dissipation and the turbulent mass flux

The ADP measurements, analyzed using the structure function method (appendix A), allow us to estimate values of the turbulent kinetic energy dissipation rate ε in the bottom meter of the water column in 256-s averages. This interval encompasses the passage of a train of a train of about 20 billows, together with calmer periods before and after (Fig. 5b). The dissipation rate varies between 7 × 10−6 and 2.5 × 10−5 m2 s−3 (Fig. 5g, appendix A) and is balanced mainly by shear production (appendix A, Figs. A1c,d).

It is useful to examine some diagnostics that originated in the analysis of unsheared stratified turbulence. The Ozmidov scale LO = (ε/N3)1/2 is an estimate of the largest scale at which eddies can remain isotropic against the flattening effect of buoyancy. The buoyancy Reynolds number Reb = ε/(νN2), where ν is the kinematic viscosity, is the four-thirds power of the ratio of LO to the Kolmogorov scale, and thus measures the extent of the inertial subrange. Reb must exceed 20–30 for turbulence to be maintained (Stillinger et al. 1983), which is the case for the present observations (30 < Reb <150, Fig. 5h).

The turbulent vertical buoyancy flux can be estimated as Γε, where Γ is the flux coefficient (e.g., Moum 1996). The flux coefficient can be approximated by the constant value 0.2 in the intermediate mixing regime with 7 < Reb < 100 (Osborn 1980; Shih et al. 2005; Smyth 2020), which pertains to this study. Thus, the turbulent mass flux is Jρ = Γε(ρ/g). Inserting Γ = 0.2, ε = 2 × 10−5 m2 s−3, ρ ≈ 1015 kg m−3, and g = 9.8 m s−2, we obtain a mass flux per unit area Jρ = 4 × 10−4 kg s−1 m−2. If this flux acts at the height of 1 mab, the resulting mass loss in the bottom meter is 4 × 10−4 kg s−1 m−3. As can be seen from Fig. 3a2, a typical SSC in the bottom meter is 15 kg m−3. With our estimated mass flux, this would take ∼10.4 h to erase completely. This is close to the semidiurnal tidal period, i.e., the duration of the flood and the ebb phase. As the instabilities have been identified during both flood and ebb phases, they are expected to play a significant role in sediment transport/exchange.

c. Estimating the dissipation rate from echosounder imagery

While echosounder imagery can provide a convenient and comprehensive view of a billow train, extracting quantitative information can be challenging. Lavery et al. (2010) use measurements of high-frequency broadband (160–590 kHz) acoustic backscattering spectra to estimate the dissipation rate associated with KH instabilities by fitting the observed spectra to established models. Here we explore an alternative approach that allows us to estimate the turbulent kinetic energy dissipation rate ε based on an echosounder image.

We assume that, when billows are clearly visible in an echosounder image (e.g., Fig. 3c), their height is related to LO. We therefore approximate the ratio hes/LO by a constant C, and rearrange the definition of the Ozmidov scale, LO = (ε/N3)1/2, to give
ε=C2hes2N3.

But is the assumption hes/LO = constant valid? And if so what is its value? We address these questions using both observational and numerical data. In the Connecticut River estuary, Geyer et al. (2010) observed KH billows with hes = 2 m, ε = 2.4 × 10−4 m2 s−3, and N = 0.19 s−1, yielding LO = 0.19 m and therefore hes/LO = 11. [The estimate of LO is sensitive to the calculation method. Here we averaged values of ε, and of N, from nine locations in a billow train as given in Geyer et al. (2010, their Table 1), then combined the averages to get LO. If the values are combined first, then the resulting LO averaged, the result is 50% higher.] In the same estuary, Holleman et al. (2016) found hes/LO = 3 m/0.24 m = 12.5. In the Kuroshio, Chang et al. (2016) observed large-scale KH instabilities with hes ∼ 100 m. Combining estimates of N2 = 10−4 s−2 and ε = 5 × 10−5 m2 s−3, we obtain LO = 7 m and hes/LO = 14. Averaging these three estimates gives C = hes/LO ≈ 12.5, or C−2 ≈ 0.0064 in (2).

We now explore the relationship between billow height and Ozmidov scale further using a suite of 18 DNS experiments, each covering the growth, breaking, and decay of a KH billow train. These experiments are described in appendix B and in more detail in Kaminski and Smyth (2019). Figure 6 shows the growth and breakdown of the billows as observed by a “virtual echosounder” (appendix B), as well as the temporal variations in LO and hes. The billow height hes increases initially then levels off at a time that is close to t2D (the time at which the kinetic energy in two-dimensional motions is a maximum, or roughly the time of maximum amplitude for the primary KH billow; see appendix B). Beyond this stage, hes increases but only slightly. On the other hand, LO is small during the initial growth of hes but then increases rapidly to a maximum near the time t3D (when 3D motions are most energetic), then decreases rapidly back to zero. Because that maximum is the only nonarbitrary value of LO that can be defined for a KH event, our goal in this DNS analysis is to predict its value, and ultimately that of the corresponding ε. In the observational analyses discussed above, LO is a typical value for the region containing the instability, defined subjectively. The distinction between this and the maximum value of LO is secondary and cannot be made with the available data. In the example shown in Fig. 6, hes and 10 LO are nearly equal for t ∼ t3D, suggesting hes/LO 10, very similar to the observational estimates.

Fig. 6.
Fig. 6.

Results from example DNS with Re = 2000, Ri0 = 0.12, Pr = 1 (from Kaminski and Smyth 2019). Details of the DNS are given in appendix B. (a) Proxy echosounder image extracted from DNS. The nondimensional squared vertical density gradient is plotted. (b) Nondimensional length scales 10 LO and hes vs nondimensional time. Vertical lines show t2D and t3D, the times of maximum kinetic energy in 2D and 3D motions (Smyth et al. 2005).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

The ratio hes/LO in a DNS also depends on other parameters of the initial state: the initial Richardson and Reynolds numbers and the amplitude of the initial noise field. Combining results for various values of these parameters (Fig. 7) we find that the ratio is generally ∼O(10) but can vary significantly. We conclude that (2), with C = 12.5 as suggested previously (or C2 = 0.0064) gives a useful first estimate of the maximum dissipation rate attained by a breaking KH billow.

Fig. 7.
Fig. 7.

Ozmidov scale LO at t3D vs billow height hes at t = t2D, from 18 DNS runs as described by Kaminski and Smyth (2019). An example is shown in Fig. 6. Re and Ri are as given in the legend. Each symbol type represents three simulations with initial turbulence amplitude A = 0.0025, 0.01, and 0.05.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

6. Discussion

The present observations lie in an extreme parameter regime and therefore offer an interesting test of (2). Unfortunately, only one of the examples shown in Fig. 3 has billows located in the lowermost meter of the flow, where ADP measurements are available (Fig. 3a). Moreover, the structure function estimate of ε is available only as an average over the 256-s ADP burst, in which billows are present only part of the time. The Ozmidov scale, based on segment-averaged value of ε, and N, is LO = 0.01 m, an order of magnitude smaller even than the other estuarine cases (Geyer et al. 2010; Holleman et al. 2016). The disturbance height ranges between 0.5 m (when billows are present) and 0.2 m (when they are not). Using the value measured when billows are clearly present, we find hes/LO ∼ 50. Phrased differently, the structure function estimate of ε is smaller by an order of magnitude than the value predicted by the scaling (2) with C = 12.5.

In summary, previous observations supported by the present DNS give consistent values of hes/LO near 12.5. This suggests that (2), with C = 12.5, is a useful approximation for ε. In contrast, our lutocline observation at hour 11.5 gives a much larger value hes/LO ∼ 50. This distinction may be interpreted in three ways:

  1. Uncertainty in the structure function estimate of ε: The structure function method (appendix A) may underestimate ε. The method is based on Kolmogorov’s theory of the inertial subrange, and thus neglects the potentially important effects of stratification and boundary proximity. The required measurements of the fluctuating velocity are missing from some parts of the water column. But the deficit in ε needed to account for the discrepancy in hes/LO is an order of magnitude, whereas the turbulent kinetic energy balance used to test the method suggests at most a factor-of-2 deficit (appendix A).

  2. Time dependence of billow length scales: Our hour 11.5 observation may represent a time interval not characteristic of turbulent billows. As shown by the DNS (Fig. 6), young billows are tall and clearly visible in echosounder images, precisely because they are not yet fully turbulent (Smyth et al. 2001). Graphical estimates of hes are often made at this stage. In contrast, our estimate of ε for the lutocline case is an average over a 256-s period that includes young billows, mature billows in which ε is larger, and quiescent periods in which ε is again small.

  3. Differences in near-lutocline flows: There may be a genuine physical difference in the lutocline case. This possibility is consistent with the tendency of the aspect ratio hes/λ to be relatively large in lutocline cases (section 4b), which appears to be real. As one explanation, one may think of hes as a measure of the potential energy stored in a young billow. More specifically, the potential energy is ∼ hes2N2 (Dillon and Park 1987). When the billow breaks, that potential energy is converted to turbulent kinetic energy and, ultimately, to internal energy via ε. So in the lutocline case, ε is smaller than would be expected given the amount of potential energy available. It could be that the lutocline billows lose energy to some other mechanism besides viscous dissipation, such as radiation of gravity waves into the surrounding sediment-stratified fluid.

7. Summary and future work

After a shipboard observational campaign in a hyperturbid estuarine tidal channel, we analyzed echosounder images, vertical profiles of velocity and suspended sediment-dominated density, as well as the dissipation rate of turbulent kinetic energy ε as derived using the structure function method. The main findings are summarized as follows:

  • Waves, instability, and turbulence are seen within FM layers at both flood and ebb tide.

  • Echosounder images, together with mean velocity profiles, allow estimation of the aspect ratio and therefore of the initial Richardson number characterizing the flow prior to instability growth.

  • The aspect ratio is larger for billows on estuarine lutoclines than those observed in oceanic interior. This may be related to the smaller initial minimum Richardson number.

  • The turbulent mass flux can be significant in suspending sediment over a tidal cycle.

  • Echosounder images, together with density profiles, allow estimates of the turbulent dissipation rate ε. While valid for a broad range of oceanic conditions, this estimate appears to be too small when applied to the lutocline observations. The difference may reflect insufficient measurements, or it may indicate a physical difference in lutocline billows.

These results, including details of the interactions between velocity shear and mud-induced stratification, periodic occurrence and collapse of shear instability, and turbulent mixing, have broadened our understanding of fluid dynamics in a hyperturbid boundary layer. Our results may also suggest realistic modeling scenarios and allow for better predictions of sediment entrainment, mixing, and dispersion of FM in hyperturbid estuarine channels.

Three caveats suggest lines of future investigation:

  • Though rare, our observations include regions where SSC values approach 100 g L−1, suggesting the possibility of variable viscosity (appendix C).

  • In interpreting the observations in terms of shear instability and the gradient Richardson number, we implicitly assume that these sediment-stratified flows respect the Boussinesq approximation. In our observations, that density typically changes by no more than 8% over the lowest meter of the water column (Fig. 3). While this change is small compared with previous cases where the Boussinesq approximation has been assumed (e.g., Daly and Pracht 1968; Schatzmann and Policastro 1984), and in particular where the Boussinesq value 1/4 for the critical Richardson number has been applied (Trowbridge and Kineke 1994), it is possible that this and other parameter values are affected by non-Boussinesq effects. Future research should address this question.

  • While the parameterization (2) is useful, we must understand its boundaries of validity, beginning with why it requires adjustment in the lutocline case. This will require more extensive DNS explorations as well as more targeted observations. The present turbulent mixing estimates are time-averaged over a few minutes thus are not able to resolve the details of individual billows (period ∼5–8 s). Future work should focus on resolving the fine temporal and spatial scales of motion within the braid and core areas, and velocity measurements should cover the water column more completely.

Acknowledgments.

The contribution of J. Tu was with the support of the National Natural Science Foundation of China (NSFC-41906052). J. Tu, F. Sun, and D. Fan were supported by NSFC-41776052 and the Innovation Program of Shanghai Municipal Education Commission (2021-01-07-00-07-E00093). A. Kaminski was supported by the U.S. National Science Foundation under Grants OCE-1537173 and OCE-1657676. W. Smyth was supported by the U.S. National Science Foundation under Grant OCE-1830071. We acknowledge high-performance computing support on Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the U.S. National Science Foundation.

Data availability statement.

The observational and DNS simulations data used in this study are publicly available at https://zenodo.org/record/5558794#.YWGusBpByUk.

APPENDIX A

TKE Dissipation Rate Estimates Using Structure Function

Following Wiles et al. (2006), the TKE dissipation rate εi was estimated along beam i using the second-order structure function,
D(z,r)=[b(z)b(zr)]2,
where b′(z) is the along beam velocity fluctuation at a height z above the bed, r is the along-beam distance between velocity measurements and the overbar denotes a segment time average (256 s).
According to the Kolmogorov’s inertial subrange theory, the structure function can be expressed as
D(z,r)=Cυ2ε2/3r2/3,
where Cυ2=2.1 is an empirical constant (Pope 2000). To obtain estimates of ε, the second-order structure function D(z, r) is fitted to a linear equation using MATLAB’s robust fit algorithm,
D(z,r)=Ar2/3+n.
The slope of the regression, A, is related to the dissipation by
A=Cυ2ε2/3,
and n is an offset related to the Aquadopp noise variance, which is assumed to be independent of r. TKE dissipation rates were estimated from the 256-s segment detrended velocity records (data points = 1024) using Eqs. (A2)(A4). The separation distance r was limited to the distance to the boundary (Mullarney and Henderson 2012). Following Thomson (2012), n is obtained as a free parameter in the fit and further used for quality control by accepting only n<2σu2 and nAr2/3, where σu = 0.0175 m s−1 is a nominal velocity uncertainty (see Text S2 in the supplemental material). The goodness of the fit is assessed by adjusted R squared, Radj2 =1[(1R2)(m1)/(mk1)], and normalized standard error by the mean value (NSE = SE/MEAN). Here R2 is the ordinary square of correlation coefficient between observed and fitted structure functions, m is the total sample size, k = 1 is the number of variables. Only estimates with Radj2 > 0.7 and NSE < 0.3 were retained for further analysis. Though somewhat subjective, visual comparison of the observed and fitted data indicates that the threshold values defined above assure good data quality. Dissipation estimates from three beams were log-averaged to provide a single dissipation estimate for each elevation for a given segment. An example of the estimation of dissipation using structure function is shown in Figs. A1a and A1b. As the available velocity records of ADP were limited by signal attenuation by the FM. Therefore, available dissipation estimates were even less given that at least three velocity points were required to obtain a dissipation estimate, and some of the obtained dissipation rates were further excluded if they failed to pass the quality control described earlier.
The shear production (P = −〈uw′〉∂u/∂z) was calculated as an independent parameter for comparison. The estimate of current shear ∂u/∂z is straightforward using the segment averaged streamwise velocities at different elevations. However, the calculation of Reynolds stress ⟨uw′⟩ can be complicated by nonturbulent motions, countergradient flux, and noise floor (Scully et al. 2011; Walter et al. 2011). To address these issues, we fit the observed cospectra to the model proposed by Kaimal et al. (1972)
kSuw(k)uw=0.88(kk0)1+1.5(kk0)5/3,
where Suw is the cospectra of each ADP segment estimated using the Welch method (Welch 1967) with 50% overlap and applying a Hamming window. The frequency f is then converted to wavenumber k via k = 2πf/U, making Taylor’s frozen turbulence assumption. A two-parameter least squares fitting can be used with the observed Suw(k), to obtain estimates of Reynolds stress ⟨uw′⟩ and the runoff wavenumber k0. We refer the reader to Tu et al. (2019) for a detailed description of the Kaimal fitting method. As can be seen from Figs. A1c and A1d, the dissipation rate derived from structure function and the shear production obtained from independent Kaimal fit method roughly agree with each other, suggesting that the structure function method provides reliable dissipation estimates.
Figure A1d shows that the shear production is somewhat larger than dissipation rate with a mean value of P/ε being 1.7. Given that the fluid flow is strongly stratified by the suspended sediment, it is likely that the imbalance between P and ε is attributable to buoyancy destruction by sediment. Assuming stationary, homogeneous turbulence, the energy balance is given by P = B+ ε. Jones and Monismith (2008) found that P/ε = 3.3 and argued that the sediment concentration gradient near bed might account for the disparity between P and ε. If the sediment settling flux and the vertical turbulent flux of sediment are balanced (consistent with the assumption of stationarity), the turbulent buoyancy flux B can be written as minus the settling flux (Green and McCave 1995),
B=g(ρsρw)ρsρwSSC×ws,
where g is gravity acceleration, ρs is sediment density, ρw is the fluid density, and ws is the sediment settling velocity. Using a ws = 4.6 × 10−4 m s−1, ρs = 1320 kg m−3 for the flocculated sediment in Changjiang estuary (Tang 2007), and SSC = 20 kg m−3 (Fig. 2a1) we obtain an estimate of B ∼ 2 × 10−5 m2 s−3. This value is at the same magnitude as P (4.9 × 10−5 m2 s−3) and ε (2.8 × 10−5 m2 s−3). Although just an estimate of order of magnitude, the buoyancy flux by mixing of suspended sediment explains the discrepancy between shear production and dissipation. This suggests that the structure function estimate of ε is reliable for the present observations.
Fig. A1.
Fig. A1.

(a) Velocity measurements of Aquadopp HR on 2 Oct at 1120 LT [segment 35, indicated by dashed line in (c)]. (b) Structure function and associated fits (lines) to Eq. (4) for a single beam from Aquadopp HR at the same segment as (a). Here, zb represents distance below the Aquadopp HR transducer (m). (c),(d) Comparisons between the estimated dissipation and production are also shown.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0230.1

APPENDIX B

DNS Methods

The direct numerical simulations are based on the Boussinesq approximation to the Navier–Stokes equations. The computation domain is rectilinear. Boundary conditions are periodic in both horizontal directions, free-slip, permeable, and constant buoyancy at the top and bottom. The streamwise periodicity length matches the wavelength of the fastest-growing shear instability.

The initial state is a stratified shear layer with mean streamwise velocity and buoyancy defined by
UΔU=BΔB=tanhzh.

For the example shown in Fig. 6, the constants ΔU, ΔB, and h are chosen such that the initial Reynolds number Re = hΔU/ν = 2000, the Prandtl number Pr = ν/k = 1 and the initial Richardson number Ri = hBU2) = 0.12. To this initial state is added a small-amplitude, quasi-random noise field to trigger the instability. Further details may be found in Kaminski and Smyth (2019), where the simulation shown in Fig. 6 is entry 10 in their Table 1.

Echosounder response is approximated by the squared buoyancy gradient (∂b/∂z)2. To approximate the response of a fixed echosounder to instabilities advecting downstream at speed ΔU, we compute a profile of the squared buoyancy gradient at a point that moves upstream at a constant speed ΔU, namely, (b/z)2]x=ΔUt. If the observation point x  =  −ΔUt encounters a boundary, it is moved a distance Lx to the right in accordance with the periodic boundary condition.

At any given time, the billow height hes is computed as the vertical distance between the upper and lower maxima in the buoyancy gradient (∂b/∂z)2. The dissipation rate ε is calculated from the strain rate tensor at each point in space and averaged both horizontally and over the vertical extent of the turbulent shear layer. The corresponding squared buoyancy frequency N2 is calculated from the mean buoyancy profile and similarly averaged over the shear layer. The latter two quantities are combined to obtain the Ozmidov scale.

Another useful diagnostic is the specific kinetic energies (one-half the squared velocity) contained in various components of the motion (Smyth et al. 2005). The mean kinetic energy is the volume average of U2/2, where U(z, t) is the streamwise velocity averaged over the horizontal directions x and y. Subtracting U from the total velocity field and averaging over y gives u2D(x, z, t), the 2D component of the motion that is dominated by the KH billow. Subtracting U and u2D, from the total velocity field isolates the 3D motions associated with secondary instabilities and turbulence. At the beginning of an instability event, the mean flow is dominant. The 2D component associated with the primary KH billows grows exponentially then saturates, reaching its maximum amplitude at time t2D. As secondary instabilities sap the energy of the billows, the 3D component of the motion grows, and its kinetic energy reaches a maximum at the later time t3D. In the end, the disturbances dissipate and the energy is again contained in the mean flow.

APPENDIX C

The Impact of SSC on Viscosity

Although numerous empirical formulas express the relationship between SSC and viscosity, there is no consensus on a general expression (Mehta 2013). Thomas (1963) proposed a formula for viscosity that can be applied to SSC exceeding several hundreds of grams per liter with particles in the size range between 0.1 and 20 μm (close to mean grain size of 8 μm in this study), η=ηw{1+2.5Фvf+10.05Фvf2+0.062exp[1.875Фvf/(11.595Фvf)]}, where ηw is the dynamic viscosity of water, Фvf=SSC(ρsρw)/[ρs(ρfρw)] is the floc volume fraction (Mehta 2013). Here, ρs = 2650 g L−1 is the sediment particle density, ρw is the water density, and ρf is the floc density. In the same estuary, Guo et al. (2017) found ρf ∼ 1300 g L−1.

As can be seen in Fig. 3, the SSC at the billows elevation generally varies between 10 and 100 g L−1, this gives a viscosity between 1.1ηw and 2ηw. Chen (2021) conducted rheological experiments using mud from the same estuary and found η/ηw ∼ 1.3 at SSC ∼ 100 g L−1. Similarly, Fei (1982) compiled experimental results using fine sediment with similar size (d50 = 6–9 μm) as present study. They found η/ηw values varying between 1.02 and 1.65 given SSC of 10–100 g L−1. Regarding the DNS, Harang et al. (2014) show that, at large Reynolds number (Re > ∼1000), the mud viscosity has no influence on the development of the primary instability, which is used to derived key parameters (e.g., wave height) for the inference of turbulence. The relevant Reynolds number is Re = /ν, where u, δ, and ν are half-velocity difference across the layer, half-thickness of the shear layer, and fluid kinematic viscosity. Given typical values of u ∼ 0.25 m s−1, δ ∼ 0.5 m (see the six cases in Fig. 3), and ν < 2νw = 2 × 10−6 estimated above (where νw is the kinematic viscosity of clear water), we arrive at Re > 6 × 104, indicating turbulent flow.

From an observational perspective, researchers may use SSC and velocity profiles to infer the depth-dependent viscosity on basis of momentum equation (e.g., Vinzon and Mehta 2000; Traykovski et al. 2015). The viscosity derived using this method, however, is termed as a total viscosity including turbulent eddy viscosity (νt) and fluid-mud viscosity (νm); thus, in turbulent flows, νt dominates while in laminar flows νm dominates (Traykovski et al. 2015). The Reynolds number Re > 6 × 104 estimated above indicates that the periods with shear instabilities and internal waves are generally turbulent, hence eddy viscosity plays a key role. Therefore, it is likely that the suspended sediment has minor impact on the total viscosity, and thus on flow dynamics, except very few periods with extremely high SSC approaching 100 g L−1.

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Supplementary Materials

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