1. Introduction
A tidal bore is a moving hydraulic jump that forms at the leading edge of a flood tide in specific macrotidal estuaries. It can be generated by the horizontal convergence of funnel-shaped estuaries with tidal ranges larger than 4–6 m (e.g., Chanson 2010, 2011, 2012; Fan et al. 2014). These tidal bores sweep through estuaries with significant kinetic energy, leading to intense sediment transport and significant changes in geomorphic features (Chen et al. 1990; Chanson et al. 2011; Fan et al. 2014; Furgerot et al. 2016), inducing violent mixing and scalar dispersion (Koch and Chanson 2008; Simpson et al. 2004; Tu and Fan 2017), interacting with coastal infrastructure (Zeng et al. 2017) and playing a unique role in estuarine ecosystems (Donnelly and Chanson 2005).
Given the shape of their leading edges, tidal bores can be categorized as undular bores or breaking bores (Lubin and Chanson 2017). An undular tidal bore front is followed by a train of well-formed, quasiperiodic undulations, and its upstream tidal bore Froude number (Fr1) has a value less than 1.3–1.4 (Lubin and Chanson 2017). A breaking tidal bore has a collapsing front due to the free surface instability associated with a marked roller, and Fr1 is generally larger than 1.4–1.6 (Lubin and Chanson 2017).
Intense turbulent activities and associated significant sediment transport are two fundamental properties of tidal bores. Turbulent velocity has been recorded in laboratory experiments under undular and breaking bores, and substantial velocity changes arise when tidal bores arrive (Koch and Chanson 2008, 2010). Lagrangian particle tracking under undular and breaking bores was used to explore the turbulent mixing of light particles (Chanson and Tan 2010). Their results demonstrated a rapid longitudinal dispersion of lightweight particles as well as some selective dispersions based on the particle’s vertical elevation. Docherty and Chanson (2012) found evidence of transient recirculation in the near-bed layer immediately after the breaking bore roller. Chanson (2010, 2011) and Leng and Chanson (2016) demonstrated that under bore fronts, the Reynolds stress in an upper water column was larger than that in the lower part of the water column. The physical experiments by Leng and Chanson (2015) show that the upstream propagation of the roller toe was highly turbulent and unsteady.
Pan et al. (2007) reproduced the characteristics of the Qiantang bore using a two-dimensional model through numerical experiments. Large eddy simulations of weak breaking tidal bores revealed large velocity fluctuations and flow recirculation structures (Lubin et al. 2010). Furuyama and Chanson (2010) showed the existence of some short-lived flow reversal next to the bed immediately after the bore front passage based on modified large eddy simulation model. Wang and Pan (2018) integrated a new formula of sediment carrying capacity into the finite-volume coastal ocean model and refined the settling velocity accounting for the flocculation of fine sediment. The significant rise in sediment concentration under tidal bores was reproduced accurately. The numerical methods for bore generation were compared for tidal bore simulation, and the effects of mesh size and turbulence models were discussed (Zhang et al. 2022).
Field observations of the turbulence dynamics of tidal bores have gradually emerged in the past two decades. Observations in the Dee River estuary (Simpson et al. 2004) showed that large values of stress (>2 Pa) and shear production (∼5 W m−3) are induced by tidal bores. Turbulent kinetic energy (TKE) is input through the release of energy in the bore, which gives rise to a transient but intense injection of energy at the bore front. Reungoat et al. (2015) conducted observations on the Garonne River and showed that unusual transient turbulence occurred by bore propagation. Their results suggested the advection of large-scale eddies in the wake of the bore front. The observation of undular tidal bores (Sée River, Mont-Saint-Michel Bay, northwest France) by Furgerot et al. (2016) demonstrated that highly sheared flow and pressure variations result in sediment resuspension and that positive vertical velocities transport the high near-bed sediment to the upper water column. Tu and Fan (2017) conducted detailed observations on the tidal bore of the Qiantang Estuary. They demonstrated that the breaking of tidal bores dynamically resembles the breaking waves in a surf zone. The horizontal advection of the dissipation rate most likely results in an imbalance between the generation and dissipation of the local turbulence. Moreover, the breaking process of tidal bores and the associated secondary waves lead to enhanced vertical mixing and vertical distribution of the advected sediments. At the early stage of a tidal bore, the vertical mixing of suspended sediment is more convective than diffusive due to the elevated length scales associated with breaking-bore-generated turbulence (Tu et al. 2021).
Most studies on tidal bores have been carried out in straight estuarine channels without consideration of lateral processes. Detailed studies on the lateral currents of tidal bores in curved channels are rare. Given that a meandering channel is a common estuarial morphology, exploring these lateral dynamics is necessary to deepen the understanding of tidal bores. The classic two-layer helical flow occurs around an estuarine bend, with surface flow pointing to the outer bank and bottom flow directed to the inner bank, when a depth-averaged equilibrium between the cross-channel sea level slope and centrifugal force is achieved (Kalkwijk and Booij 1986; Chant 2002). The pattern of lateral circulation is influenced by vertical stratification (Lerczak and Geyer 2004), different advection of density water (Nunes and Simpson 1985), Ekman veering in the bottom boundary layer (Johnson and Ohlsen 1994; Ott et al. 2002), tidal amplitude, and river discharge (Chant 2002). For tidal bores rushing through curved bends, the unsteady nature of tidal bores may be an important dynamic feature reshaping the vertical structure and temporal variation in lateral currents. The laboratory experiments with flat bottoms and curved channels by Fan et al. (2023) illustrate that the tidal bore height of the inner bank is smaller than the height in straight channels, while the tidal bore height of the outer bank is larger than that in straight channels. The lateral velocity shows a complicated structure and temporal variation.
The major objective of this study is to investigate the mean flow and turbulent dynamics of tidal bores in a curved channel located in the upper Qiantang Estuary, Hangzhou Bay, China. This research explores 1) the characteristics of the vertical structure and temporal variation in lateral currents during the passage of tidal bores, 2) the mechanism that is responsible for the variation in lateral flows, and 3) the mechanism of turbulence generation at the front of the tidal bore. The paper is organized as follows. Section 2 presents the field measurements and data processing method. The main results of the field observations, with an emphasis on lateral currents and turbulence structures, are shown in section 3. In section 4, the mechanism and influence of lateral currents are discussed, and the generation of turbulence at tidal bore fronts is explored. The main conclusions and their implications are summarized in section 5.
2. Field observations
a. Study area and field campaign
In situ observations were carried out in a meandering estuarine channel located in the upper Qiantang Estuary, Hangzhou Bay, China. Hangzhou Bay is a typical funnel-shaped embayment, with a cross-bay width decreasing significantly from approximately 100 km at the bay mouth to approximately 3 km near Da-Que-Kou (DQK). Due to the nonlinear distortion of the tidal wave propagation induced by the convergent estuary and shallowing topography, the mean tidal range increases sharply from ∼3 m at the bay mouth to ∼5.5 m at the bay head, with tidal asymmetry becoming increasingly evident. Tidal bores form at the transect upstream at nearly ∼5 km of GanPu and fully develop at the section near YanGuan (YG). Further upstream of YG, the tidal bore begins to decay and finally disappears upstream of Wen-Yan (WY).
Our observation site was located in a meandering channel just downstream of bend QiBao, which has a curvature to the south. Based on the topography of the observed channel, the radius of the flood thalweg has a value of ∼5200 m and that of the ebb bend is ∼10 900 m.
An upward-looking Signature1000 ADCP was installed on a bottom-mounted quadruped to observe the mean and turbulent velocities of the tidal bores. The observations were conducted from 16 to 18 August 2015 for four semidiurnal tidal cycles during spring tides. The 5-beam ADCP recorded beam velocity with 0.2-m vertical bins. It samples 8 subpings at 1-s intervals for effective data recording. Simultaneously, the 1-Hz pressure signal was recorded by the pressure sensor of the ADCP.
b. Data analysis
1) Velocity coordinates
To show the streamwise and lateral dynamics more clearly, the east–north Cartesian coordinates are transformed to along- and across-channel coordinates. The direction of the tidal channel points 35° west from the north, and the lateral direction (y axis) points perpendicularly to the inner bank of the tidal channel (Fig. 1c). The mean velocity and associated turbulent parameters were transformed to streamwise-lateral coordinates.
Topography of the Hangzhou Bay–Qiantang Estuary system and the observation site in the study area. (a) Location of the Hangzhou Bay–Qiantang Estuary system in the East China Sea (red box). (b) The topography of Hangzhou Bay with black dashes marked with mean tidal ranges in meters (after Pan et al. 2019). The bathymetric data were obtained from field surveys by the Zhejiang Institute of Hydraulic and Estuary in 2014. (c) Location (black dot) of the observation site near the Qibao curved band. The northeast coordinate system is shown as a dotted line, and the x–y (streamwise–lateral) coordinate system is shown as a black line. The angle between the two systems is 35°. The radius of curvature of the flood band is nearly 5200 m, whereas the curvature radius of the ebb bank reaches ∼10 900 m. (d) Photo of the breaking bore when it arrives at the observation site. (e) The bathymetry of the transect across the observed channel. Water levels immediately before and after the bore front are shown with the area used to estimate the tidal bore Froude number.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
2) The valid cells
The velocity recorded by the ADCP in the water layers immediately adjacent to the transducer and the vicinity of the water surface is invalid due to the blank zone and side-lobe interference (Priego-Hernández et al. 2019). In this study, the blank zone is 0.2 m, and the height of the ADCP transducers is 0.58 m above the bed. The data quality is relatively poor within the layer approximately 1 m below the water surface. Thus, the valid cells of the ADCP are in the range of 0.78 m above the bed and 1 m below the surface.
3) Turbulent quantities
The breaking of tidal bores and the associated secondary waves may induce periodic fluctuations in the velocity time series. Several methods have been proposed to separate turbulence and wave-induced fluctuations, such as the paired differencing method (Trowbridge 1998; Shaw and Trowbridge 2001; Feddersen and Williams 2007), coherent spectra method (Benilov and Filyushkin 1970; Bendat and Piesol 2000), and empirical mode decomposition (EMD) method (Qiao et al. 2016; Bian et al. 2018, 2020). Here, the EMD method was used to separate wave signals from the turbulent velocity because the method is especially suitable for nonstationary and nonlinear time series. Detailed information on the EMD method was presented by Huang et al. (1998), who first proposed processing algorithms; later, Qiao et al. (2016) and Bian et al. (2018, 2020) successfully applied this method to analyze oceanic turbulence. The EMD method decomposes observed time series into several intrinsic mode functions (IMFs) and a residual signal. If the peak frequency of an IMF of velocities corresponds to the peak frequency of pressure, the IMF is classified as a wave component and should be removed from the original dataset. The segment of beam 1 velocity at 4 m above the bottom during the first 10 min of the second bore is shown as an example (Fig. 2). The velocity segment is decomposed into 7 IMFs, and the pressure data during the same interval are decomposed into 6 IMFs. Both the first IMFs of the velocity and pressure segments have peaks at the frequency band of 0.12–0.15 Hz. This finding suggests that the first IMF of velocity is most likely affected by the fluctuation of the free surface and should be removed from the original velocities. Additionally, the tail of the spectrum of IMF1 has a gentler slope than the Kolmogorov −5/3 scale, which suggests that IMF1 is contaminated by noise. Removing IMF1 can decrease the influence of both surface waves and noise. Notably, the turbulence signal in IMF1 has also been removed, which may lead to the underestimation of turbulence parameters. However, as the Reynolds shear, production, and TKE are most related to the turbulence macroscale, we expect the removal of IMF1 to have little effect on the turbulence parameters.
The wave-turbulence decomposition procedure of the velocity time series. (a) EMD decomposition of beam-1 velocity at 4 m above the bottom during the first 10 min of the second tidal bore. (b) The power spectra of each IMF of the velocity segment. (c) The power spectra of each IMF of the pressure segment during the same interval.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
In a semidiurnal tidal environment, the time scale of the Reynolds average is usually set to between 5 and 20 min. This averaging period is supposed to be long enough to resolve the largest turbulent eddies but not so long that the turbulent processes cannot be regarded as quasi-stationary (Rippeth et al. 2003). During the passage of the tidal bore, the variation in mean flow occurs drastically fast, and the 5-min averaging period is too long to meet the quasi-stationary assumption. Thus, we reduce the period of the Reynolds average to 1 min following the studies of tidal bores by Simpson et al. (2004) and Tu et al. (2019).
The mean flow variation trend is further detected by cubic polynomial fitting in the 1-min data section. We choose the time series of beam-1 velocity at 4 m above the bottom at the second tidal bore front as an example (Fig. 3). The velocity is highly unsteady in the first 1-min interval. Subtracting the 1-min average velocity directly from the raw data retains the variation in the mean flow, which can lead to an overestimation of the turbulent velocity (Fig. 3b). Cubic polynomial fitting can reasonably estimate the mean velocity variation. Subtracting the raw velocity by cubic polynomial fitting results in a reasonable estimation of the turbulent velocity (Fig. 3c).
(a) The beam-1 velocity at 4 m above the seabed during the first 10 min of the second tidal bore (gray line) with the first 1-min time series highlighted (black line). The mean flow variation by cubic polynomial fitting is also added as a thick black line. (b) The turbulent fluctuation derived by subtracting the mean value from the raw data. (c) The turbulent velocity estimated by subtracting the cubic polynomial fitting from the raw data.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
After Reynolds decomposition of the beam velocity and removal of the influence of waves and noise, the Reynolds stress can be estimated through the variance method (Lu and Lueck 1999a,b; Stacey et al. 1999; Rippeth et al. 2002; Williams and Simpson 2004; Togneri et al. 2017; Guerra and Thomson 2017). The methodology from Guerra and Thomson (2017) extends the variance technique to include expressions for the Reynolds stresses for nonzero tilt. During our measurements, the pitch and roll were never more than 0.5° and 1.2°, respectively, and met the assumption of small-angle approximations for pitch and roll. The Reynolds stresses and turbulent kinetic energy from the 5-beam ADCP are estimated following Guerra and Thomson (2017).
4) Lateral momentum equation
From our observations, the local acceleration, centrifugal acceleration, Coriolis acceleration, and lateral Reynolds stress can be directly estimated based on the velocity recorded by the ADCP. The time difference operation can amplify the noise in the velocity, thereby obscuring major variations. To reduce the influence of noise, 5-point smoothing is conducted on the time series of velocities at each level. For local accelerations, 5-point smoothing is conducted again after acquiring the time difference of the smoothed velocity.
3. Results
a. Tidal bore Froude number and phase speed
The basic shape of the tidal bore front can be predicted by the upstream Froude number (Fr1). According to Chanson (2012), Fr1 in irregular channel bathymetry with parallel walls next to the waterline is defined as Fr1 = [(A2/A1)2/2 + (A2/A1)/2]1/2, where A1 is the channel cross-sectional area before the tidal bore and A2 is the cross-sectional area at the bore front (Fig. 1e). The velocity in the moving frame (u1 + C) can be estimated as
b. Mean flow structure
The water level increased abruptly by ∼1.2 m when tidal bores arrived at the observation site over a few minutes, and the identified tidal ranges during the period of observation were ∼2.5 m (Fig. 4). The time series of the water level shows the highly asymmetric character of the tidal waves. The flood phase refers to the moment when the along-channel velocity (u) is negative, whereas the ebb phase corresponds to the time when u > 0. The duration of the flood tide phase is approximately 2.7 h, while the mean ebb phase continues for almost 9.8 h.
(a) Tidal variations in streamwise velocity at the observation site. (b) Tidal variations in lateral velocity at the observation site.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
The streamwise flow is highly asymmetric, with larger velocities and shorter durations of flood tides (Fig. 4a). Remarkably sharp changes in the streamwise flow direction occur when the tidal bore arrives at the observation site with the bottom velocity being stronger than the surface velocity. When a tidal bore arrives, the along-channel velocity of the whole water depth has approximately the same variation magnitude as shown by velocity profiles 2 and 3 in Fig. 5. This finding suggests that the streamwise barotropic pressure gradient (SBTPG) is the main force driving the variation in velocity, as the SBTPG exerts vertically uniform acceleration on tidal currents. Ignoring the effect of bed shear stress, the SBTPG needs more time to reverse the direction of the faster surface flow, while the direction of slower bottom current responds to the SBTPG more quickly. The formation of the bottom-enhanced velocity profiles demonstrates that the velocity structures of tidal bores are highly influenced by the vertical structure and magnitude of the ebb velocity. The logarithmic velocity profile predicted by classic bottom boundary layer theory may not be suitable in tidal bore fronts, as the velocity is experiencing significant local acceleration and does not reach a state of quasi balance between the SBTPG and the bottom shear stress.
Streamwise velocity profiles near the tidal bore front. Profiles 3, 4, and 5 show the bottom enhanced current at the tidal bore front. Velocity profiles 1 and 2 show the ebb current just before the arrival of the tidal bore. The sampling time of the velocity profiles is labeled with black dots in the time series of sea surface level in the inset.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
The lateral flow is relatively complicated with respect to both the temporal variation and vertical structure (Fig. 4b). During the passage of the tidal bore, the lateral currents of both the surface and bottom layers were directed to the inner bank (Fig. 6a). The velocity direction of the water column is relatively uniform (Fig. 6b) and lasts nearly 10 min. Then, a two-layer structure develops, with a positive surface velocity pointing at the outer bank and a negative bottom velocity directed at the inner bank (Fig. 6a). The two-layer structure resembles the classic two-layer helical flow developed in curved channels (Fig. 6c). During the late flood tide, uniform lateral flow develops, which points at the outer bank (Fig. 6d). The uniform lateral current is significant, with velocities reaching ∼1.2 m s−1 at hours 4, 15, 28, and 39. The same vertical structure and temporal variation in lateral flows appear in every semidiurnal tidal cycle, which implies that a robust mechanism is responsible for the variation in lateral currents.
(a) Temporal variation in lateral velocity in the surface layer (red line) and bottom layer (blue line) with superimposed free-surface elevation (gray line). The vertical dashed lines in each tidal cycle indicate the time corresponding to the periods of the three phases of lateral currents: the negative lateral current at the very beginning of the tidal bore, the helical flow in the middle of the flood tide, and the positive lateral flow at the end of the flood tide. (b)–(d) The characteristic velocity profiles of the lateral currents, with red lines showing lateral currents and blue lines indicating streamwise currents.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
c. Turbulence characteristics
The squared velocity shear (S2) varies like tidal currents (Fig. 7a). At peak flood tide and peak ebb tide, S2 reaches the intratidal extreme value. During flood tides, S2 is approximately 1 s−2 at the near-bed layer and gradually decreases toward 0.1 s−2 at the surface layer.
Tidal variations in (a) magnitude of squared mean shear (S2), (b) shear production, (c) streamwise Reynolds stress, (d) lateral Reynolds stress, and (e) TKE.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
The intratidal variation in shear production is impacted by the breaking of tidal bore fronts as well as velocity shear (Fig. 7b). The shear production peaks at tidal bore fronts suggest the influence of energy released by the collapse of the leading edge of tidal bores. In addition, shear production has the same intratidal fluctuation as S2, with the greatest value appearing at peak flood and ebb tide. The order of shear production reaches nearly O(10−2) in the bottom layer and O(10−3) in the middle and upper layers.
Both streamwise and lateral Reynolds stress show clear peaks at the tidal bore front (Figs. 7c,d). This finding suggests that the Reynolds stresses are also closely related to the TKE injected by bore breaking. The upward decreasing velocity profile causes a negative streamwise Reynolds stress near the tidal bore front. Then, the streamwise Reynolds stress becomes positive as the velocity increases downward. The lateral Reynolds stress at the tidal bore front is positive, corresponding to the inner-bank-toward lateral current. The negative lateral Reynolds stress develops as the two-layer helical and outer-bank-pointing lateral currents provide positive lateral velocity shear. During the peak flood tide, the Reynolds stress at the near-bed layer is ∼1 Pa. At the peak ebb tides, the Reynolds stress value reaches ∼0.8 Pa.
At the tidal bore front, a very pronounced peak in the TKE appears at the arrival of the leading edge of the tidal bores, with TKE densities as high as ∼0.06 J kg−1 (Fig. 7e). In addition to the tidal bore fronts, the TKE has the same variation as S2, where peak TKE values appear at peak ebb and flood tides. Figure 8 shows a zoomed-in view of the TKE during the first half hour of the third tidal bore. As shown by the white cap passage, a breaking bore generally lasts less than 3 min. The considerable TKE of the whole water column emerges within the first 3 min of the tidal bore, which may be attributed to the breaking process. Then, the TKE in the lowest layer begins to decline sharply; however, considerable TKE in the middle and surface layers lasts for another 10 min. Physical experiment results (Rouse et al. 1959; Docherty and Chanson 2012) suggest that breaking-produced turbulence is transient. For example, Rouse et al. (1959) illustrated that the TKE input from a hydraulic jump is mainly distributed in a zone with a horizontal scale twice the depth behind the leading edge. Notably, the Reynolds number is one order of magnitude larger in this field observation (∼5.0 × 106) than in the physical experiment [∼1.0 × 105 by Docherty and Chanson (2012)]. This result suggests that the viscous forces can dissipate the breaking injected turbulence more quickly in the physical experiments.
A zoomed-in view of the turbulence kinetic energy (TKE) during the first half hour of the third flood tide.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
4. Discussion
Our observations reveal two interesting features associated with tidal bores: 1) the complicated lateral currents promoted by tidal bores in the curved channel and 2) elevated turbulent activity lasting ∼10 min above the middle layer after bore breaking. In this section, the two phenomena are discussed in detail.
a. Driving forces of the lateral currents
The depth-averaged terms in the lateral momentum balance [Eq. (3)] are shown in Fig. 9. The lateral barotropic pressure gradient (LBTPG) discussed in this section is estimated as the residual term between local acceleration and the summation of centrifugal acceleration, Coriolis acceleration, and bottom friction.
The depth-averaged lateral momentum of Eq. (3). (a) The temporal variation in local acceleration (red line), centrifugal acceleration (black line), Coriolis force (blue line), and lateral bottom shear stress (gray line). (b) The temporal variation in local acceleration (red line), the summation of centrifugal forcing and Coriolis force (dashed line) and the lateral barotropic pressure gradient (LBTPG). The blue line shows the LBTPG estimated as the residual term of Eq. (3), and the black line indicates the LBTPG deduced from the phase difference between the outer bank and the observation site. The vertical dotted lines indicate the periods corresponding to the three stages of the lateral currents.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
The lateral bottom shear stress plays a minor role in adjusting the magnitude of lateral currents, as it is an order of magnitude smaller than the other terms in lateral momentum budges (Fig. 9a). The depth-averaged local acceleration is negative when lateral currents point to the inner bank at the fronts of the tidal bores. Then, the local acceleration gradually becomes positive when the two-layer helical flows develop during the middle of the flood tides. The local acceleration continuously increases until both the surface and bottom layers of the lateral currents are directed to the outer bank during late flood tides. At the end of the flood tides, the local acceleration becomes negative, indicating that the positive lateral currents gradually decrease and finally cease.
The centrifugal acceleration is positive during whole tidal cycles, as both the flood and ebb bends have the same streamwise direction of curvature (Fig. 9a). Due to the relatively shorter curvature radius of the flood bend and faster flood current, centrifugal acceleration is much more evident during flood tides. The Coriolis acceleration has the same direction as the centrifugal acceleration during flood tides and offsets much of the centrifugal acceleration during ebb tides (Fig. 9a). The combination of Coriolis and centrifugal acceleration shows that it always tends to drive the currents toward the outer bank during the flood tide, and this needs to be balanced by other forces to match the variation in local acceleration (Fig. 9b).
The LBTPG is negative at the arrival of tidal bores (blue line in Fig. 9b). After the passage of tidal bore fronts, the absolute value of the LBTPG begins to decrease and approaches zero in the middle of the flood tides. At the end of the flood tides, the LBTPG decreases again, resulting in negative local accelerations.
The results of the lateral momentum budget clearly illustrate the mechanism controlling the temporal variation and vertical structure of lateral currents. Figure 10 shows the vertical profiles of the centrifugal acceleration, Coriolis acceleration, and LBTPG at the three characteristic times associated with the lateral velocity profiles at the same moment. The substantial negative LBTPG at the tidal bore front overcomes the effects of centrifugal and Coriolis acceleration and exerts negative acceleration on the currents. Thus, the lateral currents of the whole water layer point to the inner bank (Figs. 10a,b). As the LBTPG decreases, the increasing summation of centrifugal and Coriolis acceleration gradually overcomes the LBPTG during the middle of the flood tides. Since the streamwise velocity is faster at the upper layer than the velocity near the bottom layer, the Coriolis and centrifugal forces push the upper layer toward the outer bank more easily, while the bottom layer’s lateral currents remain negative. Thus, two-layer helical flows develop during the middle of the flood tides (Figs. 10c,d). When the LBTPG continuously decreases, the Coriolis and centrifugal acceleration eventually push the whole layer of lateral currents toward the outer bank (Figs. 10e,f).
The vertical structures of lateral currents and lateral momentum budgets at three characteristic times. (a),(b) The lateral currents toward the inner bank and the corresponding lateral momentum terms at the tidal bore front, respectively. (c),(d) The two-layer helical currents and the lateral momentum budgets during the middle of the flood tides. (e),(f) The lateral current toward the outer bank and the associated lateral momentum terms at the end of the tidal bore.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
b. Mechanism of LBTPG variation
The tidal averaged depth of the outer bank is ∼−6.5 m, which is nearly 1 m deeper than the observation site. Assuming that the heights of the tidal bores and ebb currents before bores are the same along the transect of the observation channel, the bore celerity near the outer bank (Co) is estimated to be 7.2, 8.0, 8.4, and 8.2 m s−1 for the four tidal bores. The bore celerity at the observation site (Ci) is 6.7, 7.5, 8.0, and 7.8 m s−1 for the four tidal bores. After the passage of tidal bores (nearly 7 min after the beginning of flood tides), the hypercritical flood currents transition into a subcritical flow. The celerity of tidal waves in subcritical states is simply estimated as
The variation and magnitude of the LBTPGpha are the same as those of the LBTPG calculated as the residual component in lateral momentum budgets (Fig. 9b). At the beginning of the tidal bores, the quicker jump in water level near the outer bank results in an inner bank toward the LBTPG. As the tidal bore passes, the LBTPGpha gradually decreases and is overwhelmed by the summation of centrifugal and Coriolis acceleration. This finding indicates that the celerity difference of tidal waves between the outer and inner banks is an important reason for the variation in the LBTPG. The lateral celerity difference along the transect is not unique. Li et al. (2019) found that the bore celerity varies over the transect at Daquekou of Hangzhou Bay, with the deeper region having a relatively quicker bore celerity. The topography in a curved channel generally has a deeper water depth near the outer bank. Thus, the topography-induced LBTPG should be an essential driving force of lateral currents when tidal bores pass through a meandering channel. Notably, the bore celerity may be faster in the inner bank than in the outer bank if Eq. (5) is not met. This result may occur in shallow water with a significant tidal bore jump and can induce completely different lateral flow patterns during the passage of a tidal bore.
When the tidal bore flows through the downstream curved bend of the observation site (Fig. 1), lateral currents may also develop but in the opposite direction due to the adverse curvature of the downstream channel. Hence, the advection term −u(∂υ/∂x) may always restrain the development of the lateral currents as it transports adverse lateral momentum from downstream to the observation sites. Bottom friction may play a more important role in reducing lateral currents near the inner bank as the water depth shallows toward the inner bank shoal. Thus, the term −υ(∂υ/∂y) is negative as it transports high lateral momentum from the outer bank to the inner bank. This enforces the inner bank toward lateral currents at the tidal bore front. Similarly, during the middle and end of the flood tides, the term −υ(∂υ/∂y) also enforces the lateral currents. From the qualitative analysis, the advection term −u(∂υ/∂x) restrains the development of the lateral current, and the term −υ(∂υ/∂y) redistributes the lateral momentum along the lateral transect. In summary, the advection terms do not directly induce lateral currents, and they merely play a role in controlling their magnitude.
c. Influence of lateral flow on the shear production of turbulence
The depth-averaged lateral velocity shear averaged over the first three tidal bores is shown in Fig. 11. Lateral velocity shear is most evident during flood tide, and it accounts for nearly 40% of the total velocity shear. The tidal-phase-averaged lateral shear stress is also evident during flood tide, especially at the tidal bore front (Fig. 11). Figure 12a illustrates that lateral shear production accounts for approximately 40%–60% of the total shear production during flood tides for the four semidiurnal tidal cycles. Specifically, the contribution of two-layer helical flow during the middle flood tide to shear production is most significant (Fig. 12b). The lateral current generates a significant amount of TKE via shear production, which is then dissipated by molecular viscosity. Thus, lateral currents provide an efficient way to dissipate tidal bore energy.
The depth-averaged lateral shear stresses (orange line) and lateral component velocity shear (blue line) after the tidal phase averaged process. The variation in water level is illustrated by a black dotted line.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
(a) Comparison between total shear production and the lateral component of shear production. (b) The proportion of lateral component shear production to total shear production. The dots show the mean ratio during the three stages of the lateral currents.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
d. Turbulence energy at the tidal bore front
Temporal variations in the significant wave height (Hs) and wave periods (T) of secondary waves. The dashed line shows the variation trend. (b) Temporal variation in the wave-amplitude-based Reynolds number (Rew). All the parameters are in the time interval of the first half hour of the third tidal bore.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
Several studies have confirmed the existence of wave-induced turbulence, such as Babanin (2006), Babanin and Haus (2009), and Dai et al. (2010). If the wave oscillation is strong enough, the orbital motion can be turbulent rather than laminar. The wave energy can continuously feed the turbulence until the orbital motion becomes laminar. Babanin and Haus (2009) proposed the wave-amplitude-based Reynolds number as a criterion of wave-induced turbulence: Rew = α2ω/ν, where α and ω are the wave amplitude and angular frequency (ω = 2π/T), respectively, and ν is the kinematic viscosity of the water. The pressure sampled during the fourth tidal bore is used to illustrate the variation in Hs, T, and Rew (Fig. 13). The critical wave Reynolds number indicating the occurrence of wave-induced turbulence has an approximate value of Recr = 3000. The Rew is larger than 20 000 when the surface TKE is larger than 0.01 J kg−1 at the beginning of tidal bores, indicating development of wave-generated turbulence. The corresponding variation between Rew and TKE supports that wave-induced turbulence can give rise to the long duration of TKE.
The wave-induced turbulence mechanism is further examined by comparing the surface TKE against the mechanical energy contained by surface waves per meter length (Ew = gA2/8). Figure 14 shows the variation in surface TKE at the layer 2 m below the water surface as a function of wave energy during flood tides. The TKE with Rew > 3000 is utilized, and The TKE during the first 3 min of each flood tide is excluded because it is mainly induced by bore breaking. Clearly, there is a positive correlation between the surface TKE and surface wave energy with a correlation coefficient of r ≈ 0.70. The close relationship between surface TKE and surface wave energy supports that the secondary waves generated by bore breaking can promote the development of TKE. Hence, the breaking of the tidal bore can generate turbulent energy through two processes: first, the collapse of the water surface may directly convert the potential energy into turbulent energy, which is the main mechanism leading to the significant TKE at the tidal bore fronts; then, the secondary waves induced by bore breaking may continuously result in turbulence through the wave-generated turbulence mechanism.
The TKE at the layer 2 m below the surface with Rew larger than 3000 as a function of surface wave energy (stars). The black line shows the best linear fit of the two parameters. The logarithmic coordinate is used to better exhibit the two parameters spanning several orders of magnitude.
Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0044.1
5. Conclusions
This study investigated the lateral currents and turbulence dynamics during the passage of tidal bores in a curved channel located upstream of the Qiantang Estuary.
Field observations reveal the complicated variation in lateral currents during the flood tide. When the tidal bore front arrives at the curved channel, the current of almost the whole water layer deflects to the inner bank. During the middle of the flood tides, a two-layer helical flow develops with the surface layer pointing at the outer bank and the bottom layer directed at the inner bank. The vertical structure resembles the typical secondary flow in a curved bend. At the end of the flood tides, the lateral current of almost the whole water column points at the outer bank. The results of lateral momentum budgets demonstrate that the lateral currents are mainly controlled by the balance between the local acceleration, lateral barotropic pressure gradient (LBTPG), centrifugal acceleration, and Coriolis acceleration. Both centrifugal acceleration and Coriolis acceleration tend to push the currents toward the outer bank during flood tide, while the LBTPG restricts the outer-bank-toward acceleration. When the tidal bore arrives at the curved channel, the deeper topography near the outer bank can induce faster propagation of the bore phase than the inner bank, which results in a quicker jump in the water level near the outer bank. Hence, the LBTPG pointing from the outer bank to the inner bank is established at the front of tidal bores and pushes the currents toward the inner bank. During the middle of the flood tides, the rising rate of the water level slows, and the water level difference between the outer bank and inner bank gradually decreases. Then, the weakened LBTPG is overcome by the summation of centrifugal and Coriolis acceleration. As the surface velocity is larger than the bottom velocity, the centrifugal and Coriolis accelerations in the surface layer are stronger than those in the bottom layer. The direction of surface currents is more easily altered by these outer-bank-toward accelerations. Therefore, there will always be a period when the surface flow is directed at the outer bank while bottom currents remain pointed at the inner bank. This is the main process for the development of two-layer helical flow during the middle flood tide. When the LBTPG continues to decrease, the currents of the whole water column are deflected to the outer bank. For classic curved-bend flow, the balance between centrifugal, Coriolis, and LBTPG is quasi-stationary, while the balance between the three terms is unsteady in the case of tidal bores.
A detailed view of the temporal variation and vertical structure of TKE shows that bore breaking is responsible for the significant TKE at the very beginning of the tidal bore. The elevated TKE above the middle layer lasts more than 10 min. The secondary waves stimulated by bore breaking have the features of high frequency and large amplitude. Then, the secondary waves gradually calm down with longer wave periods and smaller wave heights. The correspondence between violent secondary waves and the enhanced TKE implies that wave-induced turbulence is a potential mechanism that promotes turbulence. The wave-based Reynolds number reaches the order of O(104), which is much larger than the criterion of 3000 for wave-generating turbulence. This finding offers evidence that the second wave can be a candidate for continuous feeding turbulence after the tidal bore front.
This study may provide a better understanding of associated phenomena. First, the thalweg swing in the estuarine channel where tidal bores occur is an important dynamical geomorphology problem, as it concerns the safety of navigation and stability of seawalls. Understanding tidal bore lateral currents can help to predict net sediment transport in the lateral direction and the associated geomorphological evolution. Second, seawater occasionally climbs over seawalls during middle and late flood tides near the observation site. The overtopping seawall process may be highly related to the lateral current that points to the outer bank. Third, turbulence generation through bore breaking can induce intratidal asymmetry in eddy viscosity. The tidal bore residual flow has a two-layer structure, comparable to the process of tidal-straining-induced estuarine circulation, with surface currents directing to the sea and bottom currents pointing toward the toward land. Hence, the breaking bore adjusts the material transport and dispersion process by shaping the residual currents.
Acknowledgments.
This work was financially supported by the National Key R&D Program of China (Grant 2023YFC3008100), the Scientific Research Fund of the Second Institute of Oceanography, MNR (Grant JG2308), the National Natural Science Foundation of China (Grants 41906146 and 42276176), the Key R&D Program of Zhejiang Province (Grant 2022C03044); the Zhejiang Provincial Ten Thousand Talents Plan (Grant 2020R52038), and the Zhejiang Provincial Project (Grant 330000210130313013006). We thank our colleagues at the Zhejiang Institute of Hydraulics and Estuaries for organizing and conducting field measurements. Many thanks to two referees for their valuable comments and suggestions on this paper.
Data availability statement.
The data used to analysis the turbulence dynamic and mean flow variation are publicly available at https://figshare.com/s/bf036ac951cb075b78cf.
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