Modeling tidal current and asymmetry in partially mixed ... - CiteSeerX

Modeling tidal current and asymmetry in partially mixed estuary using an unstructured grid model Wei-Bo Chen1, Wen-Cheng Liu2, Chin Wu3, Nobuaki Kimura2, Adam Bechle3 and Ming-Hsi Hsu1 1

Department of Bienvironemntal Systems Engineering, National Taiwan University, Taipei 10617, Taiwan 2 Department of Civil and Disaster Prevention Engineering, National United University, Miao-Li 36003, Taiwan. 3 Departemnt of Civil and Environmental Engineering, University of Wisconsin-Madison, USA ABSTRACT A three-dimensional, unstructured grid, hydrodynamic model is implemented and applied to the Danshuei River estuarine system and its adjacent cosatal sea in northern Taiwan. The model is calibrated and verified with available data of vertical velocity profile using ADP (Acoustic Doppler Profilers), horizontal velocity profile and water level using an Automated River Video Imaging System (ARVIS). The validated model is then used to investigate the tidal asymmetry in the Danshuei Rive estuary. Vertical profiles of salinity, velocity and eddy diffusivity show a distinct asymmetry between the flood and ebb tides and the tidal mixing at the bottom is stronger on flood than on ebb. The velocity profile shows a subsurface maximum on flood but a nearly linear distribution on ebb due to the non-tidal pressure gradient. The phenomena are consistent with the ADP measured results. The vertical profiles of tidally averaged currents show stronger at surface layer in the spring than neap tide, but weaker at bottom layer in the spring than neap tide. During the neap tide, there is complex horizontal spatial velocity pattern, consistent with the observations by ARVIS.

Keywords: Three-dimensional hydrodynamic model, Acoustic Doppler Profilers, tidal current, tidal asymmetry, eddy diffusivity, Danshuei River estuary.

Introduction Tidal currents provide a major mechanism for generating turbulent mixing in estuaries and continental shelves. Recent observations have revealed significant asymmetry in turbulent mixing over a tidal cycle. Jay and Smith (1990) analyzed data collected from the Columbia River estuary and found a flood–ebb asymmetry: enhanced shear and stratification during ebb tides but stronger mixing and weaker stratification during flood tides. Nepf and Geyer (1996) investigated intratidal variations in boundary layer structure in a straight section of the Hudson River estuary. They found that active mixing with a gradient Richardson number less than 1⁄4 is confined to a well-mixed near bed layer on floods but occurs throughout regions of significant stratification on ebbs. More recently, Geyer et al. (2000) estimated eddy viscosity in the Hudson River and found that flood values exceeded ebb values by a factor of 2. Other estuarine field studies have also documented this tidal asymmetry and suggested that the asymmetric mixing needs to be considered when calculating tidally averaged vertical fluxes (e.g., Stacey et al. 2001). In addition to the flood–ebb asymmetry, turbulent mixing exhibits large fluctuations over the spring–neap cycle. For example, Peters (1999) found that all flood tides have substantial mixing but that spring ebbs have the strongest mixing extending throughout the water column. In weakly stratified flows found in regions of the continental shelf influenced by lateral freshwater inputs (e.g., the Rhine outflow area of the North Sea and the Liverpool Bay), Simpson et al. (1990, hereafter S90) independently discovered the same phenomenon and described the switching between the stratified and mixed states over a single tidal cycle as strain-induced periodic stratification (SIPS). In the Liverpool Bay, Simpson et al. (2002) and Rippeth et al. (2001) found pronounced asymmetry in energy dissipation between the flood and ebb tidal regimes. During the ebb, the water column stratifies and strong dissipation is confined to the lower half of the water column. By contrast, during the flood, stratification is eroded with complete vertical mixing occurring at high water and higher values of dissipation extending throughout the water column. In this paper we use a realistic three-dimensional (3D) numerical model of Danshuei River to examine the variations of flow and salinity fields over flood–ebb and spring–neap tidal cycles. Our goal is to better understand the relationship between turbulent mixing and estuarine circulation in Danshuei River and gain new insights into the dynamics of estuarine circulation. SELFE is a new unstructured-grid model designed for the effective simulation of 3D baroclinic circulation across river-to-ocean scales (Zhang and Baptista 2008). We hope that the modeling results reported in this paper will motivate future observational studies into the tidal variability in Danshuei River.

Study site The Danshuei River estuary (Fig. 1a and 1b) is the largest estuarine system in Taiwan. Its drainage basin includes the capital city of Taipei. Its tidal influence spans a total length of about 82 km, encompassing the entire length of the Danshuei River and the downstream reaches of its three major tributaries, namely,

the Tahan Stream, the Hsintien Stream, and the Keelung River. Its drainage area encompasses 2728 km2 with a total channel length of 327.6 km. Except for occasional storm surges induced by hurricanes, the major forces that cause barotropic flows in this area are the astronomical tide at the river mouth and river discharges upriver. The astronomical tide may extend as far upriver as the Cheng-Ling Bridge on the Tahan Stream, the Hsiu-Lang Bridge on the Hsintien Stream and the Chiang-Pei Bridge on the Keelung River (Fig. 1a). Tidal propagation is the dominant mechanism controlling the elevation of the water’s surface and its flooding profiles. The M 2 tide is the primary tidal constituent at the river mouth with a tidal range of 2.17 m at mean tide and up to 3 m at spring tide. Due to cross-sectional contraction and wave reflection, the mean tidal range may reach a maximum of 2.39 m within the system. The phase relationship between the tidal elevation and tidal flow exhibits certain characteristics of standing waves (Hsu et al. 1999). Sea water intrudes upriver as a result of both tidal advection and classical two-layer estuarine circulation. The salinity varies according to an intra-tidal time scale in response to the ebb and flow of floods, and it varies on longer time scales in response to freshwater inflow. The limit of salt intrusion may extend beyond 25 km in the Tahan Stream from the river mouth during the low-flow period. The baroclinic pressure gradient due to the salinity distribution is sufficiently large to push the denser salt water upriver along the bottom layer of the estuary, resulting in classical two-layer circulation involving net upriver flow in the bottom layer and net downriver flow in the upper layer (Liu et al. 2004; Liu 2006). The resulting estuarine circulation strengthens as the river flow decreases. Six million people, over a quarter of Taiwan’s entire population, live in the catchment area of the Danshuei River system. The Danshuei River estuary receives untreated domestic discharges and both treated and untreated industrial discharges from its tributaries; the river is heavily polluted by trace metals and organic materials (Jeng and Han 1994; Fang and Lin 2002; Hung et al. 2007). Approximately 20.72 m3/s of mostly untreated domestic sewage enters the Danshuei River system (Wu et al. 1997), in addition to an unknown quantity of waste effluent from industries in the upper estuary (Fang 2000). Doong et al. (2002) reported that a variety of organochlorine pesticide residues still persist in the fluvial sediments of the Tahan Stream, suggesting that these river sediments have been contaminated for decades. Understanding the transport of particulate pollutants is of critical importance for environmental management in the estuary and coastal ocean area.

Model description Governing equations and boundary conditions The numerical modeling of ocean circulation at scales ranging from estuaries to ocean basins is gaining maturity. A plethora of codes are available, and many of these are open-source. Most modern oceanic and

estuarine circulation codes solve some form of the three-dimensional Navier Stokes equations, complemented with conservation equations for the water volume and salt content. Common codes use either structured (POM, Blumberg and Mellor 1987; TRIM, Casulli and Cheng 1992; ROMS, Song and Haidovgel 1994; HEM-3D, Hamrick 1996) or unstructured (ADCIRC, Luettich et al. 1991; UnTRIM, Casulli and Walters 2000; FVCOM, Chen and Liu 2003; ELCIRC, Zhang et al. 2004) grids, and they are typically based on finite difference, finite element, or hybrid approaches involving finite volumes. In this paper, we use a three-dimensional semi-implicit Eulerian-Lagrangian finite element model (SELFE, Zhang and Baptista 2008) to simulate the Danshuei River estuarine system and its adjacent coastal ocean region. The SELFE program solves the Reynolds stress averaged Navier-Stokes equations, including conservation laws for the water’s mass, momentum and salt content under the hydrostatic and the Boussinesq approximations. These equations yield the free-surface elevation, three-dimensional water velocity and salinity:

G ∂W ∇ ⋅u + =0 ∂z η G ∂η + ∇ ⋅ ∫ udz = 0 −h ∂t G G Du JG ∂ ∂u ) = F − g∇η + (ν Dt ∂z ∂z ∧ JG G G g η 1 F = − f k × u + α g∇ψ − ∇PA − ∫ ∇ρ d ζ + ∇ ⋅ ( μ∇u )

(1)

DS ∂ ∂S = (κ ) Dt ∂z ∂z

(4)

ρ = ρ 0 ( p, S )

(5)

ρ0

ρ0

(2)

(3)

z

where ( x, y ) : the horizontal Cartesian coordinates; z : the vertical coordinate, positive in the upward ⎛ ∂ G ∂ G⎞ direction; ∇ = ⎜ i, j ⎟ ； t : time; η ( x, y, t ) : free-surface elevation; h( x, y ) : bathymetric depth; ⎝ ∂x ∂y ⎠ G u ( x, y, z , t ) : horizontal velocity with the Cartesian components (U ,V ) ; W : vertical velocity; f : ∧

Coriolis factor; g : acceleration due to gravity; ψ (φ , λ ) : earth tidal potential; α : effective Earth elasticity factor; ρ ( x, t ) : water density with the reference value ρ 0 = 1025 kg / m3 ; PA (x,y,t): free surface atmospheric pressure; p : pressure; S : salinity; ν : vertical eddy viscosity; μ : horizontal eddy viscosity and κ : vertical eddy diffusivity for salt. The vertical boundary conditions for the momentum equation are as follows. At the water’s surface, the balance between the internal Reynolds stress and the applied shear stress is: G (6) ∂u JJG = τ w at z = η ν ∂z where the specific stress τ w can be parameterized using either the approach developed by Zeng et al. (1998) or that proposed by Pond and Pickard (1998).

The bottom surface boundary condition plays an important role in the SELFE formulation as it involves the unknown velocity. Specifically, at the bottom surface, the no-slip condition (U = V = 0) is usually replaced by a balance between the internal Reynolds stress and the bottom frictional stress: G (7) ∂u = τ b at z = −h ν ∂z JJG JJG where the bottom stress is τ b = CD ub ub . The velocity profile along the bottom boundary layer obeys the logarithmic law: G ln[( z + h) / z ] JJG 0 u= ub , ( z0 − h ≤ z ≤ δ b − h) , ln(δ b / z0 )

(8)

which is smoothly matched to the exterior flow. In Eq. (8), δ b is the thickness of the bottom computational cell, z0 is the roughness of the bottom surface, and ub is the velocity along the bottom surface as measured at the top of the lowermost computational cell. The Reynolds stress inside the boundary layer is derived from Eq. (8) as: G (9) JJG ∂u ν = ub . ν ∂z ( z + h) ln(δ b / z0 ) According to the turbulence closure theory (Umlauf and Buchard 2003), the eddy viscosity can be expressed as: 1 (10) ν = 2s K 2l m

where the stability function, the turbulent kinetic energy, and the macroscale mixing length are respectively given by: sm = g 2 1 23 B1 CD ub 2 l = κ 0 ( z + h) K=

2

(11)

1

In Eq. (11), κ 0 =0.4 is von Karman’s constant, and g 2 and B1 are constants where g 2 B1 3 = 1 . Tennekes et al. (1973) reported that the Reynolds stress is not constant within the boundary layer. Hanert et al. (2007) developed and compared numerical discretizations that explicitly take into account the logarithmic behavior of the velocity within the ocean’s bottom boundary layer. This is achieved by discretizing the governing equations using the finite element method and either enriching or modifying the set of shape functions used to approximate the velocity field. Hanert and colleagues concluded that their proposed method can capture the velocity field in the bottom boundary layer. However, eq. (12) was implemented in a straightforward way in our finite element model. In this study, a constant value was adopted for simplicity, yielding: G JJG 1 κ0 ∂u CD 2 ub ub , ( z0 − h ≤ z ≤ δ b − h) ν = ∂z ln(δ b / z0 )

(12)

where the drag coefficient, calculated from Eqs. (6), (7), and (12), is given by: 1 δ CD = ( ln b ) −2 . κ 0 z0

(13)

Eq. (12) suggests that the vertical viscosity term in the momentum equation vanishes inside the boundary layer, which is logically to be expected for the case studied here. Due to the relatively large time steps in the hydrodynamic model, we used multiple sub-steps to track particles consistent with the Courant-Friedrichs-Lewy (CFL) condition. During every sub-step of time, the movement of the particle due to advection was calculated using the method proposed by Blanton (1993). Combined with a quick search algorithm embedded in SELFE to determine the destination cell of each particle, the velocity of the particle at every position was obtained by linear interpolation if the particle was in a triangular cell or by bilinear interpolation if it was in a quadrilateral cell. Geological structure of the model

Bottom topography data for the coastal sea and Danshuei River estuarine system were obtained from the National Center for Ocean Research and from the Water Resources Agency, Taiwan. The deepest point within our study area extends to 110 m below mean sea level near the northeast corner of the model, in the coastal sea (Fig. 1c). Our mesh model of the Danshuei River estuarine system and its adjacent coastal sea consists of 5119 polygons (Fig. 2). Higher resolution grids are used in the Danshuei River estuary, and coarser grids are used in the coastal sea area. For the hybrid “SZ” vertical grid, we used 10 z-level and 10 evenly-spaced S-levels. A large time step was used to perform simulations with this grid when there were no signs of numerical instability.

Model verification To determine the accuracy of our model in practical applications, we used a large set of observational data to validate the model and to verify its ability to predict the water surface elevation, tidal current, and salinity. Water surface elevation and tidal current

We used data from 1999 for daily freshwater discharges upriver from the Tahan Stream, Hsintien Stream, and Keelung River. Jan et al. (2001) reported that a tide model with five variables ( M 2 , S 2 , N 2 , K 1 , and O1 ) can be used to calculate all relevant tidal components in the Taiwan Strait to a sufficient level of accuracy. Therefore, a tide with five constituents was adopted here as a forcing function at the coastal sea boundaries. Table 1 lists the amplitudes and phases used in our simulation. These conditions allow us to investigate the model’s response to the interactions between tidal forcing and various discharges entering the river. The freshwater discharge inputs from three tributaries (e.g., the Tahan Stream, Hsintien Stream, and Keelung River) during the two periods are presented in Figures 3a and 4a. Figures 3b-3f and 4b-4f depict the computed surface elevations together with the measured data from five stations (Danshuei River mouth, Taipei Bridge, Hsin-Hai Bridge, Chung-Cheng Bridge, and Ta-Chih Bridge). In general, the model accurately reproduces the variations in the water level. Table 2 presents statistics of relevance to

the calculated water surface elevation. The mean absolute errors for the difference between the measured hourly surface elevations and the computed values during the period from December 1st to December 15th, 1999 for the Danshuei River mouth, Taipei Bridge, Hsin-Hai Bridge, Chung-Cheng Bridge, and Ta-Chih Bridge are 0.280 m, 0.246 m, 0.324 m 0.248 m, and 0.299 m, respectively. The corresponding root-mean-square errors are 0.327 m, 0.291 m, 0.363 m, 0.287 m, and 0.316 m. To evaluate the model’s ability to calculate tidal currents, an intensive survey of the current speed at five transects was conducted on April 16th, 1999. The current speed was measured by trained technicians on boats every half hour for a period of 13 daylight hours. The velocity data were acquired using handheld current meters that measured the current’s magnitude but not its direction. The data were recorded by hand, and flow was visually determined in the “ebb” or “flood” direction. We encountered some uncertainty in determining the current’s direction during the period around slack tides. To compare the measured and computed velocities, the computed velocity in the horizontal plane was converted to the velocity along the channel. Figure 5a presents the water surface elevation at the Danshuei River mouth by survey date. Figures 5b-5f compare the time series data for the velocity along the channel (i.e., axial velocity) on April 16th, 1999. Our model satisfactorily predicted the velocity along the channel. We conclude that the flood tidal current is weaker and has a shorter duration than that of the ebb tidal current in the estuary. This result can be explained in terms of the tidal asymmetry caused by the interaction between the incoming tide and the river flow. The mean absolute errors and root-mean-square errors for the differences between the computed and measured data are listed in Table 3. Salinity

The salinity distribution reflects the combined result of several river processes. This parameter controls the density circulation and modifies mixing processes. In the present study, salinity data from the five stations were used to validate the model. The salinities of the open boundaries of the coastal sea area were set to 35 ppt . The upstream boundaries of the salinity at the three tributaries were set to zero due to the freshwater discharges that enter at these boundaries. Figure 6 compares the time series salinities for the computed and measured data at the Kuan-Du Bridge, Taipei Bridge, Hsin-Hai Bridge, Chung-Cheng Bridge, and Pa-Ling Bridge on April 16th, 1999. Table 4 also shows the mean absolute errors and root-mean-square errors for the differences between the computed and measured salinities. Overall, the SELFE model reflected the large variations in the salinity over one tidal cycle.

Flood–ebb asymmetry First, we analyse the flood–ebb asymmetry during a spring-tide. Tidal currents are strong in the shallow lower Bay. We select Kuan-Du Bridge for a detailed analysis of the flood–ebb cycle. As shown in Fig. 7a, the depth-averaged current oscillates at theM2 frequency. To get a complete picture of the flood–ebb asymmetry, we plot the time-depth distributions of current, salinity, and vertical diffusivity and energy dissipation rate over several tidal cycles (Fig.8). The along-channel current shows alternating flooding and ebbing currents. Most of the flooding currents have a subsurface maximum while the ebbing currents always reach the maximum at the surface. This asymmetry in the vertical salinity profile appears to be related to the tidal straining effect but the small stratification difference observed between the flood and ebb tides suggests that horizontal advection could also play a role. Fig.9 is shown along-channel distribution of ((a), (b)) current, ((c), (d)) salinity, ((e), (f)) logarithm of eddy diffusivity and ((g), (h)) logarithm of turbulence kinetic energy at the ebb ((a), (c), (e), (g)) or flood ((b), (d), (f), (h)). when the tide is in the ebb phase in the lower and part of the middle river channel but in the flood phase in the upper river channel. If we compare the instantaneous diffusivity distributions with the tidally averaged diffusivity distribution, we find marked variations over the flood–ebb tidal cycle.

Spring–neap cycle Fig. 10 is along-channel distribution of tidally averaged ((a), (b)) current, ((c), (d)) salinity, ((e), (f)) logarithm of eddy diffusivity and ((g), (h)) logarithm of turbulence kinetic energy at the spring ((a), (c), (e), (g)) or neap ((b), (d), (f), (h)). As shown in Fig. 10 the circulation strength is stronger during the neap tides. The landward return flow increases by 50%. The strength of residual estuarine circulation depends on the competition between the estuary’s head-to-mouth density gradient and vertical momentum fluxes. In more turbulent conditions such as during spring tides, more momentum imparted to the circulation by virtue of the horizontal density gradient is diffused vertically, resulting in slower circulation. Strong mixing/vertical momentum exchange is confined to the upper and lower river channel during the spring tide. In contrast, the mixing intensity and stratification in the mid-river channel show relatively moderate changes between the spring and neap tides.

Conclusion We have used a numerical model to investigate how flood–ebb and spring–neap tidal cycles affect turbulent mixing, stratification and residual circulation in Danshuei River Vertical profiles of salinity, velocity, eddy diffusivity and energy dissipation rates show a marked asymmetry between the flood and ebb tides. The asymmetric tidal mixing causes significant variation of salinity distribution over the flood–ebb tidal cycle. However, there are only minor changes in the subtidal currents within a tidal cycle. Therefore, it appears that the flood–ebb mixing asymmetry does not significantly affect the residual

gravitational circulation in Danshuei River. Results presented in this study are based on the outputs from a numerical model. Although this model have been validated against a variety of observational data, there are to our knowledge no existing data with adequate temporal resolution to resolve the flood–ebb and spring–neap tidal cycles in Danshuei River. Our model results which reveal significant variability in salinity distribution and residual circulation over the spring–neap cycle should motivate future observations of tidal variability in Danshuei River.

References 1.

Brown, A. R., S. H. Derbyshire, and P. J. Mason, 1994: Large eddy simulation of stable atmospheric boundary layers with a revised stochastic sub-grid model. Quart. J. Roy. Meteor. Soc., 120, 1485–1512.

2.

Burchard, H., and H. Baumart, 1998: The formation of estuarine turbidity maxima due to density effects in the salt wedge. A hydrodynamic process study. J. Phys. Oceanogr., 28, 309–321.

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Geyer, W. R., J. H. Trowbridge, and M. M. Bowen, 2000: The dynamics of a partially mixed estuary.

J. Phys. Oceanogr., 30, 2035–2048. 4.

Jay, D. A., and J. D. Smith, 1990: Residual circulation in shallow estuaries. 2. Weakly stratified and partially mixed, narrow estuaries. J. Geophys. Res., 95, 733–748.

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Mason, P. J., and S. H. Derbyshire, 1990: Large-eddy simulation of the stably-stratified atmospheric boundary layer. Bound.-Layer Meteor., 53, 117–162.

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Metais, O., and M. Lesieur, 1992: Spectral large-eddy simulations of isotropic and stably-stratified turbulence. J. Fluid Mech., 239, 157–194.

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Moeng, C.-H., 1984: A large-eddy simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci., 41, 2052–2062.

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Nepf, H. M., and W. R. Geyer, 1996: Intratidal variations in stratification and mixing in the Hudson estuary. J. Geophys. Res., 101, 12 079–12 086.

10. Simpson, J. H., and A. J. Souza, 1995: Semidiurnal switching of stratification in the region of freshwater influence of the Rhine. J. Geophys. Res., 100, 7037–7044. 11. Zhang, Y. L., & Baptista, A. M. (2008). SELFE: A semiimplicitEulerian–Lagrangian finite-element model for cross-scale ocean circulation. Ocean Modelling, 21(3-4), 71-96.

Fig. 1 a The map of the Danshuei River estuarine system and its adjacent coastal sea, b 3D view of the Danshuei River estuarine system and adjacent coastal sea, and c the contour of Danshuei River estuarine system and adjacent coastal sea

Fig. 2 An unstructured grid representing the modeling domain

Fig. 3 a Freshwater discharge inputs from three tributaries during the period of August 1–16, 1999, and the comparison of water surface elevation at: b Danshuei River c Taipei Bridge, d Hsin-Hai Bridge, e Chung-Cheng Bridge, and f Ta-Chih Bridge

Fig. 3 a Freshwater discharge inputs from three tributaries during the period of August 1–16, 1999, and the comparison of water surface elevation at: b Danshuei River mouth, c Taipei Bridge, d Hsin-Hai Bridge, e Chung-Cheng Bridge, and f Ta-Chih Bridge

Fig. 4 a Freshwater discharge inputs from three tributaries during the period of December 1–16, 1999, and the comparison of water surface elevation at: b Danshuei River mouth, c Taipei Bridge, d Hsin-Hai Bridge, e Chung-Cheng Bridge, and f Ta-Chih Bridge

Fig. 5 a Water surface elevation at the Danshuei River mouth and the comparison of computed longitudinal velocity with time series data at: b Kuan-Du Bridge, c Taipei Bridge, d Hsin-Hai Bridge, e Chung-Cheng Bridge, and f Pa-Ling Bridge

Fig. 6 The comparison of simulated and measured salinity on April 16, 1999, at a Kuan-Du Bridge, b Taipei Bridge, c Hsin-Hai Bridge, d Chung-Cheng Bridge, and e Pa-Ling Bridge

07/01 1.5

07/02

07/03

07/04

07/05

07/06

(a)

1.2 Ebb

Velocity ( m / s )

0.9 0.6 0.3 0 -0.3 -0.6 -0.9 Flood -1.2 -1.5 0

12

24

36

48

60

72

84

96

108

120

Time ( hr ) 07/01 1.2

07/02

07/03

07/04

07/05

07/06

(b)

Bed stress ( N / m-2 )

0.9 0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 0

12

24

36

48

60

72

84

96

108

Time ( hr ) Fig. 7. Time series of (a) tidal current and (b) bed stress at Kuan-Du Bridge

120

Fig. 8. Time-depth distributions of (a) current (ms-1), (b) salinity (ppt), (c) eddy diffusivity (m2s-1), (d) density (kgm-3) and (e) turbulence kinetic energy (m2s-2) Kuan-Du Bridge

Fig. 9. Along-channel distribution of ((a), (b)) current, ((c), (d)) salinity, ((e), (f)) logarithm of eddy diffusivity and ((g), (h)) logarithm of turbulence kinetic energy at the ebb ((a), (c), (e), (g)) or flood ((b), (d), (f), (h))

Fig. 10. Along-channel distribution of tidally averaged ((a), (b)) current, ((c), (d)) salinity, ((e), (f)) logarithm of eddy diffusivity and ((g), (h)) logarithm of turbulence kinetic energy at the spring ((a), (c), (e), (g)) or neap ((b), (d), (f), (h))