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A numerical study of lateral grain size sorting by an estuarine front
Estuarine, Coastal and Shelf Science 81 (2009) 345–352
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Estuarine, Coastal and Shelf Science journal homepage: www.elsevier.com/locate/ecss
A numerical study of lateral grain size sorting by an estuarine front Simon P. Neill School of Ocean Sciences, Bangor University, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5AB, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 May 2008 Accepted 4 November 2008 Available online 25 November 2008
A two-dimensional non-hydrostatic model of baroclinic flow was applied to an estuarine cross-section. The model was driven by lateral variations in density and produced the classical bilateral cross-sectional recirculation of an axial convergent front. Simulations of the erosion, transport and deposition of sediment by the frontal secondary flows were applied to a range of grain sizes. The model predicted considerable lateral variation in grain size deposition across the frontal recirculation zone, analogous to the lateral grain size sorting which occurs in river meanders. The sorting primarily took place between the frontal surface convergence and the bankward limit of recirculation. A series of sensitivity tests revealed that the contribution of the front to lateral grain size sorting was strongly influenced by the lateral channel slopes and lateral density gradient. The results from this numerical study support previous suggestions (based on observations of near-surface discontinuities in sediment concentration across the frontal interface) that fronts may act as sieves within the estuarine sediment transport system. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: estuary fronts convergence sediment dynamics numerical model Conwy estuary
1. Introduction Convergent fronts are common and persistent features of many estuaries (Simpson and Turrell, 1986; Ferrier and Anderson, 1997). In an aerial survey of UK estuaries, evidence of convergent fronts was found in 15 out of 26 systems (Brown et al., 1991). There are three main categories of estuarine front (e.g. Simpson and James, 1986; Largier, 1992; O’Donnell, 1993): plume and tidal intrusion fronts (Garvine and Monk, 1974; Simpson and Nunes, 1981), axial convergent fronts (Nunes and Simpson, 1985) and longitudinal (shear) fronts (Huzzey and Brubaker, 1988). Secondary flows associated with such fronts can lead to enhanced surface concentrations of larvae (Eggleston et al., 1998) and pollutants (Swift et al., 1996), enhanced suspended sediment transport (Trevethan and Chanson, 2007) and lateral sediment trapping (Woodruff et al., 2001). In situ measurements of estuarine fronts have confirmed secondary flows of magnitude 0.2 m s1 (Brown et al., 1991), i.e. of order 20% of the longitudinal velocity. Due to the interaction of secondary flows on longitudinal sediment transport, it has been suggested that estuarine fronts may act as ‘‘sieves’’ within the estuarine sediment transport process (Reeves and Duck, 2001). Aerial photographs of frontal systems in the Tay Estuary have demonstrated discontinuities in near-surface sediment concentration across the frontal interface, i.e. the lateral transport of suspended sediment is inhibited by the density gradient. Therefore, depending on the magnitude of secondary flow associated with the front and the sediment grain size, estuarine
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fronts, or a series of such fronts, may act as sieves within the estuarine sediment transport system. The secondary recirculation cell of an estuarine front (driven by lateral density gradients) is a similar mechanism to the helicoidal flow pattern occurring in a river meander (driven by the centripetal force). In the latter case, the centripetal force is generated by channel curvature (Kikkawa et al., 1976). Assuming that all grain size fractions in a river bed are fully mobile, sediment in a river meander may become laterally sorted because the helicoidal flow pattern transports fine grains towards the inner bend, while gravity transports coarse grains towards the outer bend (Frings, 2008). This results in a progressive segregation of coarse and fine size fractions throughout the bend (Fig. 1a). Since an estuarine front has two such helicoidal flows, one occurring either side of the surface expression of the front, it is here hypothesised (using the river meander analogy) that coarser sediment will be preferentially deposited on the bed under the frontal surface convergence and that finer sediment will be preferentially deposited on the bed at the bankward limits of frontal crosssectional recirculation (Fig. 1b). If proven to exist, such a sediment sorting mechanism could have important impacts on benthic biogeochemical and biological processes. In this paper, a non-hydrostatic numerical model of an idealised estuarine cross-section (Section 2) is applied to sediment transport processes occurring due to an axial convergent front (Section 3). The validity and robustness of the results (Section 4) are supported by a series of sensitivity tests (Section 5). The aim of the modelling methodology is to determine whether lateral bed grain size sorting occurs in the region of estuarine fronts, analogous to the sorting which occurs across a river meander. This is the first published testing of this hypothesised grain sorting by an estuarine front.
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a A
A
A
A
A coarse sediment
b
foam line A
surface convergence
A
A fine sediment coarse sediment
fine sediment
fine sediment
Fig. 1. Sketch of secondary flows and resulting hypothesised lateral bed grain size distribution. (a) River meander and (b) axial convergent estuarine front.
2. Numerical model The non-hydrostatic model, TEXSM (Tay Estuary CrossSectional Model) (Neill et al., 2004), was applied in this study to an idealised domain. A non-hydrostatic model was used because the vertical velocities generated by estuarine fronts are important. Such models are commonly applied in cases where vertical accelerations are important and the ratio of vertical to horizontal dimension relatively large (e.g. Koçyigita and Falconer, 2004; Xing and Davies, 2007). TEXSM is a two-dimensional (2D) model of an estuarine cross-section with a varying free surface (Hirt et al., 1975), k–epsilon turbulence (Rodi, 1984), salinity transport and baroclinic flow. The model uses the SOLA algorithm (Hirt et al., 1975) which solves explicitly for across-channel velocity v, vertical velocity w, production of turbulent kinetic energy (TKE) k, dissipation of TKE 3, salinity s, and solves for pressure p implicitly using the successive overrelaxation (SOR) algorithm (Press et al., 1986). The governing equations of TEXSM are the continuity equation
vC vC vC v vC v vC 3t 3t þ þ n þ ðw ws Þ ¼ vt vy vz vy vy vz vz
where C is the concentration of sediment, ws is the particle settling velocity (a function of grain shape and size) and 3t is the sediment diffusion coefficient. Since the same mechanism (i.e. eddy mixing) is used for both turbulent transport of momentum and sediment, it was assumed for simplicity that 3t ¼ nt, since the latter is calculated directly by the k–epsilon model. The bed level change was calculated using a one-dimensional sediment budget equation
vz 1 ðD EÞ ¼ 1n vt
(1)
the y (lateral) Reynolds equation
vn vðnnÞ vðnwÞ 1 vp v vn v n þ nt n þ nt þ þ þ ¼ þ2 r vy vt vy vy vz vy vz vw vn 2 vk þ vy vz 3 vy (2)
Ca ¼
vw vðwnÞ vðwwÞ 1 vp v vw vn n þ nt þ þ ¼ þ þ r vz vy vt vy vz vy vz v vw n þ nt þ2 vz vz
2 vk g bs ðs s0 Þ 3 vz
ðw ws ÞC
rs
(6)
where rs is the density of sediment grains. The calculation of D assumes that deposition due to the longitudinal velocity is negligible for application to the idealised channel described in Section 3 (since the baroclinic vertical component of velocity > barotropic vertical component of velocity). The erosion term E (due to the longitudinal velocity) was estimated by calculating an entrainment function e, with units of concentration (Parker, 1978). The entrainment function requires an estimate of the reference concentration Ca at a reference height above the bed za, calculated as (Smith and McLean, 1977)
and the z (vertical) Reynolds equation
(5)
where z is the bed level, n is bed porosity, E is erosion and D is deposition, calculated for only the row of cells adjacent to the bed using
D ¼
vn vw þ ¼ 0 vy vz
(4)
0:00156Ts 1 þ 0:0024Ts
(7)
26:3scr Ts d50 þ rgðs 1Þ 12
(8)
and
za ¼ (3)
where the overbar denotes a mean quantity, n is the kinematic viscosity, nt is the turbulent (eddy) viscosity, bs is the coefficient of saline contraction, s0 is salinity averaged over the cross-section, r is water density and g is acceleration due to gravity. TEXSM has been extensively validated with laboratory data from a diverse set of cases such as flow over a backward facing step (Kim et al., 1980), the lock exchange experiment (Kneller et al., 1999) and vertical jets in confined fluid depths (Jirka and Harleman, 1979). To apply TEXSM to sediment dynamics, a sediment mass continuity equation was added to the model (e.g. McDowell and O’Connor, 1977)
where scr is the threshold bed shear stress, s is the ratio of grain to water density and Ts is the transport parameter
Ts ¼
s0 scr scr
(9)
where s0 is the bed shear stress, related to the drag coefficient CD through the quadratic friction law
s0 ¼ rCD U 2
(10)
where U is the depth-averaged longitudinal velocity. By assuming that the entrainment function e is equal to the reference concentration Ca (Eq. (7)), entrainment of sediment from
S.P. Neill / Estuarine, Coastal and Shelf Science 81 (2009) 345–352
vC ¼ ews ¼ Ca ws vz za
(11)
where the vertical diffusion coefficient Kz is calculated (assuming a parabolic depth distribution) using
z Kz ¼ ku* z 1 h
+
ρ
v=0.20 ms
ws e
(13)
rs
The entrainment function e appears as a source term in Eq. (4), at the reference height za.
−
ρ
ρ
−1
w=0.05 ms −1
0
10
20
30
(12)
where u* is the friction velocity, k is Von Ka´rma´n’s constant and h is water depth. Therefore, the erosion term in Eq. (5) due to the longitudinal component of velocity can be estimated as
E ¼
−
0 1 2 3 4 5 6
40
50
60
70
80
90
100
lateral distance (m)
b v,10×w (m s−1)
Kz
a depth (m)
the bed can be equated with upward mixing due to turbulence at level za (Harris and Wiberg, 2001)
347
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
v 10×w
0
10
20
30
40
50
60
70
80
90
100
80
90
100
lateral distance (m) 3. Model application
ws ¼
n d
10:362 þ 1:049D3*
1=2
10:36
(14)
where d is grain size and D* is the dimensionless grain size, calculated using
c
0.15 0.1
∂v/∂y (s−1)
The model was parameterised from observations of the axial convergent front which occurs in the Conwy Estuary, UK (Nunes and Simpson, 1985). The front persists for up to 2–3 h during the second half of the flood tide and is associated with a strong surface convergence along the main axis of the channel, visible due to the accumulation of organic matter and an intense foam line (Simpson and Turrell, 1986). During the flood phase of the tide, increased advection of denser water at the centre of the channel, where tidal currents are greater (and hence frictional effects less) than at the shallower channel edges, creates a lateral density gradient which drives the two-celled cross-sectional circulation of the axial convergent front. Axial salinity gradients of order 10 km1 have been observed during the flood phase of the tide, in contrast with axial salinity gradients of order 5 km1 during the ebb phase of the tide (Turrell et al., 1996), hence the front develops exclusively on the flood. Typical lateral variations in density of 0.32 kg m3 have been measured in the Conwy Estuary over a distance of 20 m (Nunes and Simpson, 1985). This change in density of 0.32 kg m3 was set as the initial condition for the TEXSM model as sketched in Fig. 2a, applied to an idealised symmetrical geometry with dimensions characteristic of the Conwy Estuary: a Gaussian channel of width 100 m and maximum depth 6 m. The Conwy Estuary is flood-dominant, with flood currents exceeding ebb currents by approximately 50% (Dyer, 1997). Since sediment transport is a function of bed stress, this asymmetry has led to observations of a threefold increase in depth-averaged suspended sediment concentration during the flood, in contrast to the ebb (West and Sangodoyin, 1991). Hence, at the time when the axial convergent front has fully developed, there is a considerable quantity of sediment in suspension. Grain size was parameterised based on survey results collected at the time of the construction of the Conwy Tunnel (Hydraulic Research Station, 1979) and from sampling reported by Knight, (1981). Typical values of the median grain size (d50) ranged from 100 to 800 mm, with coarser sediment generally found in the channel centre and finer sediment towards the channel edges. The grain sizes were translated into settling velocities using (Soulsby, 1997)
0.05 0 −0.05 −0.1 −0.15
0
10
20
30
40
50
60
70
lateral distance (m) Fig. 2. TEXSM modelled lateral (v) and vertical (w) velocities for axial convergent front at peak flood in a Gaussian channel. In (a) the initial density field is shown by vertical dashed lines. (a) Velocity vectors, (b) v and w close to bed and (c) divergence of v close to bed.
D* ¼
gðrs =r 1Þ 1=3
n2
d
(15)
Applying Eq. (14), the range of settling velocities required to represent the in situ median grain sizes was calculated to of order 0.005–0.11 m s1. Therefore, by considering 12 grain sizes, the modelled settling velocities were as listed in Table 1. No assumptions were made on the vertical distribution of sediment concentration, e.g. by the implementation of Rouse profiles. Rather, the 2D concentration field in the estuarine crosssection was calculated (from an initial condition of zero concentration throughout the cross-section) by Eq. (4) (with a flux boundary condition at the bed), applied to the range of 12 modelled grain sizes. The peak (mid-channel) depth-averaged longitudinal
Table 1 Range of modelled settling velocities (ws) and corresponding grain sizes (d). ws (m s1)
d (mm)
0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
94 135 199 257 317 381 452 530 618 716 825 946
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velocity when the front is fully developed was parameterised as U ¼ 1 ms1 and the drag coefficient parameterised as CD ¼ 0.0039 for the Conwy (Hunter, 1982). It was assumed that the depthaveraged longitudinal velocity varied parabolically across the estuarine cross-section. This was confirmed by running a depthaveraged numerical model of a uniform Gaussian channel with a constant value of CD ¼ 0.0039, driven by a velocity of magnitude 1 m s1 normal to the boundary. Since model simulation times for the sediment concentration in the estuarine cross-section to reach steady state were relatively short (of order h/ws ¼ 60–1200 s for the range of grain sizes considered), the assumption that longitudinal velocity was steady over the duration of the model simulations was valid. The model was run both with secondary flows (Run A) and without secondary flows (Run B). Therefore, an examination of the distribution of bed level z for (Run A)–(Run B) for the range of modelled grain sizes demonstrates the contribution of baroclinic processes to lateral sediment transport (and hence lateral grain size sorting). The model assumes that there is no localised gradient in the longitudinal component of velocity, i.e. the modelled cross-section is representative of steady, uniform flow, for example in a straight estuarine reach at mid-flood. In such a channel cross-section with steady barotropic flows, assuming Coriolis forces can be neglected, no mechanism other than diffusion acts to transport sediment laterally. The baroclinic flow, however, drives the two-celled circulation of the axial convergent front (Fig. 2a). Deposition occurred only when w ws < 0 in the row of model grid cells adjacent to the bed, and erosion due to the longitudinal component of velocity was included in the calculation (Eq. (13)). The model has a no-slip boundary condition on the bed and channel edges, and a free-slip at the free surface. 4. Results The model was applied (with dy ¼ 2 m, dz ¼ 0.25 m and a time step dt ¼ 0.1 s) to each of the twelve grain sizes (Table 1) for a duration of 15 min (a time period over which the longitudinal velocity was assumed not to vary significantly and which allowed the cross-sectional concentration fields to attain steady state). Since the grain sizes were modelled independently, there was no interaction between the different classes of grain size. Typical time series of vertical concentration profiles (for d ¼ 135 mm) were plotted at 120 s intervals for four locations across the channel (Fig. 3). For the initial condition at t ¼ 0 (not plotted), concentration was zero throughout the cross-section. Generally, the near-bed concentrations increased rapidly for all four locations as time progressed, and reduced exponentially with height from the bed. However, two main exceptions occurred. Firstly, at y ¼ 50 m (channel centre and hence frontal surface convergence), the nearsurface concentrations were >0 for t 240 s. Near-surface concentrations were continuously highest in the region of the surface convergence due to the combined effects of lateral advection and diffusion of sediment into this region. Secondly, at y ¼ 39 m there was an anomaly in sediment concentration between the surface and a depth of 4 m, with the peak occurring close to the centre of each bilateral recirculation cell (Fig. 2). This increased concentration was mainly due to the occurrence of peak vertical velocity w of order 0.05 m s1 at y ¼ 39 m, considerably greater than the settling velocity for this grain size (0.01 m s1). For the simulation plotted, little change occurred in the concentration profiles after t ¼ 240 s, other than the evolution of the profile at y ¼ 33 m. In general, it was the sediment concentration at the bankward limits of the recirculation zone (y ¼ 33 m) which took the longest time to reach steady state since the sediment had been transported over the greatest lateral distance from the erosional source (channel centre). At steady state, and in the absence of
secondary flows, the concentration profiles across the channel approximate to Rouse profiles. The contribution of secondary flow to the lateral distribution of bed level change after 15 min of simulation for each grain size (i.e. each modelled settling velocity) was plotted (Fig. 4). Note that since the hydrodynamics and resulting sediment transport were symmetrical about the channel centreline, the results have been plotted only for one side (the left-hand-side) of the channel. Clearly, the secondary flow associated with the front had the greatest influence on the lateral distribution of finer grains and the least influence on the lateral distribution of coarser grains. For the finer grain sizes, maximum deposition occurred close to the bankward limit of frontal recirculation (y ¼ 33 m) and there was a minimum close to the centre of the channel. By combining the model results for all grain sizes, the contribution of secondary flow to median grain size sorting was calculated across the channel, assuming that the deposition was well mixed (Fig. 5). The minimum d50 (275 mm) was found at y ¼ 35 m, i.e. close to the bankward limit of frontal recirculation (Fig. 2). The maximum d50 (396 mm) was found at the centre of the channel, as hypothesised in Fig. 1b. Therefore, the presence of the front has resulted in a lateral sorting of sediment grain size of magnitude 121 mm for this benchmark case. 5. Sensitivity tests To confirm the validity of the results, a series of sensitivity runs were performed, the most extensive of which was application of the model to an entire flood cycle (the front does not form on the ebb). This was implemented by assuming that the lateral density gradient was a sinusoidal function, increasing from zero at the beginning of the flood (well-mixed conditions) to a maximum gradient at high water. Hence, the TEXSM model allowed the secondary flows to evolve from zero (no density gradient) to a maximum (as plotted in Fig. 2). The longitudinal velocity was correspondingly varied over this time with a sinusoidal function of amplitude 1 m s1 to represent a spring flood tide. This longitudinal velocity was applied as an across-channel parabolic distribution to simulate spatial and temporal variations in erosion due to varying longitudinal velocity throughout the flood phase of the tide. The modelling methodology described in Section 3 was applied at 15 min intervals using the secondary flows output from TEXSM in conjunction with these longitudinal velocities. The lateral distribution of z for each grain size was recorded over each 15 min interval. A new cross-sectional concentration field was then calculated for the evolving secondary flow field. The lateral distribution of z for each grain size was summed cumulatively over the flood phase of the tidal cycle. The series of simulations was repeated for zero secondary flows. The difference between the two sets of simulations was used to calculate the contribution of the front to the lateral distribution of median grain size sorting over a spring flood cycle. This result was compared to the steady state case (Fig. 6). The distribution was similar to the steady state case (minimum d50 at the bankward limit of recirculation and maximum d50 close to the centre of the channel). However, the magnitude of sorting across the front was reduced by a factor of 2.5, indicating that whereas net lateral grain size sorting due to an axial convergent front will occur over an entire flood tide, the net sorting is likely to be less than the sorting occurring at peak flood by a factor of 2–3. Five additional sensitivity runs were performed (Table 2, Fig. 6). These included simulating a neap longitudinal flow, increasing the lateral density gradient by a factor of 2, decreasing the lateral density gradient by a factor of a half, applying a constant velocity (rather than a parabolic distribution) across the channel, and simulating a rectangular channel case (a constant depth of 6 m across a 100-m wide channel). The latter had the most significant
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t=120s
349
t=240s
0
0 y=33m
y=33m y=39m
1
y=39m
1
y=45m
y=45m
y=50m
y=50m 2
depth (m)
depth (m)
2
3
3
4
4
5
5
6
0
10
20
6
30
0
concentration (kg m−3)
10
20
t=360s
t=480s
0
0 y=33m
y=33m
y=39m
1
y=39m
1
y=45m
y=45m
y=50m
y=50m 2
depth (m)
depth (m)
2
3
3
4
4
5
5
6
0
10
20
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30
0
concentration (kg m−3)
10
20
t=600s
t=720s 0 y=33m
y=33m
y=39m
1
y=39m
1
y=45m
y=45m
y=50m
y=50m
2
2
depth (m)
depth (m)
30
concentration (kg m−3)
0
3
3
4
4
5
5
6
30
concentration (kg m−3)
0
10
concentration (kg
20
m−3)
30
6
0
10
20
30
concentration (kg m−3)
Fig. 3. Time evolution of vertical profiles of sediment concentration at four locations across the channel. These simulations are shown for a settling velocity of 0.01 m s1 (d ¼ 135 mm).
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0.04
0.02
0.01
0
400
350
300
−0.01
−0.02
benchmark entire flood tide neap high ∂ρ/∂y low ∂ρ/∂y constant across−channel U rectangular channel
450
d50 ( m)
0.03
Δ ζ (m)
500
d=94 μm d=135 μm d=199 μm d=257 μm d=317 μm d=381 μm d=452 μm d=530 μm d=618 μm d=716 μm d=825 μm d=946 μm
0
10
20
30
40
50
250
0
10
20
30
40
50
lateral distance (m)
lateral distance (m) Fig. 4. Modelled contribution of secondary flow to bed level change (Dz) for the range of twelve grain sizes after 15 min of simulation for benchmark case.
Fig. 6. Sensitivity tests on contribution of secondary flow to lateral distribution of median grain size (d50).
effect on the resulting lateral distribution of d50. Due to a lack of channel slopes, stronger near-bed lateral flow resulted (Fig. 7), and hence the minimal value of d50 occurred further bankward, compared to the more realistic Gaussian channel (Fig. 6). However, for the rectangular channel case, a fining of grain size deposition occurred at the centre of the channel due to enhanced lateral transport of finer grain sizes by the stronger secondary flows, uninhibited by the presence of lateral channel slopes. Due to this fining, the peak median grain size for the rectangular channel case occurred closer to the centre of each bilateral recirculation cell, rather than at mid-channel. This channel centre fining also occurred for the case with high lateral density gradient, again due to stronger resulting secondary flows.
the location where vv/vy is a minimum (y ¼ 37 m) and the smallest deposition expected where vv/vy is a maximum (y ¼ 50 m, i.e. channel centre). Examining Fig. 4, this was the case for grain sizes 94 and 135 mm. For d > 135 mm, however, peak deposition occurred progressively closer to the centre of the channel with increasing grain size, since secondary flows had less influence on the coarser sediments. The results of the sensitivity tests indicated that the magnitude of the secondary flow (due to variations in the lateral density gradient), although not significantly affecting the lateral location of maximum/minimum d50, did have a significant role in the magnitude of the sorting (Fig. 6). Defined as the difference between the maximum and minimum d50 occurring across the channel (Dd50), this magnitude varied between Dd50 ¼ 69 mm (reduced lateral density gradient) to Dd50 ¼ 151 mm (increased lateral density gradient), in contrast to the benchmark value of Dd50 ¼ 121 mm. Therefore, regardless of the spring/neap character of the tide, in circumstances of an increased axial salinity gradient (e.g. high freshwater input), the contribution of an estuarine front to lateral grain size sorting will be enhanced. For the case of increased lateral density gradient, a slight fining of sediment occurred at the centre of the channel due to considerable erosion of finer sediments (94– 199 mm) from this region. A high concentration of this finer sediment remained in suspension at steady state (due to high values of w), hence less deposition at the outer limits of frontal recirculation. In the case of a river meander, used as an analogy, the inner bank acts as a lateral constraint on the recirculation zone (Fig. 1a). In the estuarine front case, this boundary is diffuse and will vary considerably with tidal range and phase. Therefore, it is difficult to determine where the bankward limits of recirculation will occur and hence exactly what the influence of lateral bed slopes will have on the sediment transport due to secondary flows. However, the sensitivity tests applied over an entire flood cycle demonstrated that the lateral grain size sorting by an estuarine front will be much greater at times of peak longitudinal velocity since suspended sediment concentration will be high. Hence, simulations which use the steady state secondary flow field at times of peak longitudinal currents are useful for examining lateral bed-slope effects, demonstrated to have a large influence on the lateral grain size sorting (Fig. 6). The model simulations demonstrate the potential contribution of estuarine fronts to lateral grain size sorting. In situ observations will also include effects due to longitudinal variations in velocity/ bathymetry and channel curvature. However, a small influence in
6. Discussion The model predicted the largest d50 at the centre of the channel and the smallest d50 at the bankward limits of recirculation (Fig. 5), as hypothesised in Fig. 1b. These locations can be related to the near-bed lateral component of velocity (Fig. 2b), specifically in terms of the near-bed divergence of the lateral velocity (Fig. 2c). For a typical grain size, the largest deposition is generally expected at
400 380
d50 ( m)
360 340 320 300 280 260
0
10
20
30
40
50
lateral distance (m) Fig. 5. Contribution of secondary flow to lateral distribution of median grain size (d50) for benchmark case.
S.P. Neill / Estuarine, Coastal and Shelf Science 81 (2009) 345–352
351
Table 2 Details of sensitivity tests. U is the depth-averaged longitudinal velocity with maximum value U max at the centre of the channel and Dr is the lateral change in density. Sensitivity test
U max ðms1 Þ
Lateral distribution of U
Dr (kg m3)
Lateral bathymetry profile
Benchmark Entire flood tide Neap High lateral density gradient Low lateral density gradient Constant across-channel longitudinal velocity Rectangular channel
1.0 Variousa 0.5 1.0 1.0 1.0 1.0
Parabolic Parabolic Parabolic Parabolic Parabolic Constantb Parabolic
0.32 Variousa 0.32 0.64 0.16 0.32 0.32
Gaussian Gaussian Gaussian Gaussian Gaussian Gaussian Constant d ¼ 6 m
a b
In this unsteady case, velocity and lateral density gradient were varied sinusoidally over time from zero (initial condition) to their maximum values. Constant value of U max applied across the channel.
such a biologically important region (due to accumulated organic matter at estuarine fronts) may lead to a large effect on estuarine benthic biogeochemical and biological processes. For example, estuarine meiofaunal diversity (dominated by nematodes) is closely related to sediment grain size (Hodda, 1990). With predicted increased precipitation occurring during the biologically productive summer season (Beniston et al., 2007) and hence a corresponding increase in lateral estuarine density gradients, the relatively modest contribution of processes such as lateral grain size sorting by estuarine fronts may become more important in the latter half of the 21st century. The model simulations demonstrate how secondary flows associated with an axial convergent front can act as sieves within the estuarine sediment transport system, as suggested by Reeves and Duck (2001). In qualitative agreement with observations in the Tay Estuary of discontinuities in near-surface sediment concentrations across the frontal interface, the model similarly predicted
depth (m)
a
0 1 2 3 4 5 6
ρ−
ρ+
ρ−
v=0.20 m s−1 w=0.05 m s−1 0
10
20
30
40
50
60
70
80
90
100
an order of magnitude change in near-surface sediment concentrations across the front (Fig. 3). Finally, axial convergent fronts are located in the main channel of an estuary. In the case of estuarine fronts which occur at bathymetry breaks (e.g. longitudinal fronts), fine sediment placed into suspension over the shoals may become laterally trapped by the density interface and not freely pass into the main channel. Therefore, further research is required into the role of the individual categories of estuarine front on sediment sieving. 7. Conclusions A numerical model of an estuarine cross-section, parameterised on the axial convergent front occurring in the Conwy Estuary, UK, demonstrated that considerable lateral bed grain size sorting can occur due to secondary frontal flows. The sorting took place between the surface convergence and the bankward limits of recirculation and is sensitive to the lateral density gradient and lateral channel slopes. The magnitude of the sorting is small compared to the effect of lateral shear in the longitudinal velocity, but in such a biologically productive region as an estuarine front, the influence on estuarine benthic biogeochemical and biological processes may be significant. The results from this numerical study support previous suggestions (based on aerial photographs of nearsurface discontinuities in sediment concentration across the frontal interface) that fronts may act as sieves within the estuarine sediment transport system.
lateral distance (m)
v,10×w (m s−1)
b
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
Acknowledgements v 10×w
0
10
20
30
40
50
60
70
80
90
Thanks to Prof. Alan G. Davies at Bangor University and to four anonymous reviewers, whose combined input considerably enhanced the final version of the manuscript. This work was supported by the Higher Education Funding Council for Wales and the Welsh Assembly Government. 100
lateral distance (m)
c
0.15
v/ y (s−1)
0.1 0.05 0 −0.05 −0.1 −0.15
0
10
20
30
40
50
60
70
80
90
100
lateral distance (m) Fig. 7. TEXSM modelled lateral (v) and vertical (w) velocities for axial convergent front at peak flood in a rectangular channel. In (a) the initial density field is shown by vertical dashed lines. (a) Velocity vectors, (b) v and w close to bed and (c) divergence of v close to bed.
References Beniston, M., Stephenson, D.B., Christensen, O.B., Ferro, C.A.T., Frei, C., Goyette, S., Halsnaes, K., Holt, T., Jylha, K., Koffi, B., Palutikof, J., Scholl, R., Semmler, T., Woth, K., 2007. Future extreme events in European climate: an exploration of regional climate model projections. Climatic Change 81 (Suppl. 1), 71–95. Brown, J., Turrell, W.R., Simpson, J.H., 1991. Aerial surveys of axial convergent fronts in UK estuaries and the implications for pollution. Marine Pollution Bulletin 22, 397–400. Dyer, K.R., 1997. Estuaries: A Physical Introduction, third ed. John Wiley and Sons, England. Eggleston, D.B., Armstrong, D.A., Elis, W.E., Patton, W.S., 1998. Estuarine fronts as conduits for larval transport: hydrodynamics and spatial distribution of Dungeness crab postlarvae. Marine Ecology Progress Series 164, 73–82. Ferrier, G., Anderson, J.M., 1997. A multi-disciplinary study of frontal systems in the Tay Estuary, Scotland. Estuarine, Coastal and Shelf Science 45, 317–336. Frings, R.M., 2008. Downstream fining in large sand-bed rivers. Earth-Science Reviews 87, 39–60. Garvine, R.W., Monk, J.D., 1974. Frontal structure of a river plume. Journal of Geophysical Research 79, 2251–2259.
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Harris, C.K., Wiberg, P.L., 2001. A two-dimensional, time-dependent model of suspended sediment transport and bed reworking for continental shelves. Computers & Geosciences 27, 675–690. Hirt, C., Nichols, B., Romero, N., 1975. SOLA – A Numerical Solution Algorithm for Transient Fluid Flows. Tech. Rep. LA-5852. Los Alamos National Laboratory. Hodda, M., 1990. Variation in estuarine littoral nematode populations over 3 spatial scales. Estuarine, Coastal and Shelf Science 30, 325–340. Hunter, J.R., 1982. Modelling the Dynamics of the Conwy Estuary Under High Flow Conditions. Tech. Rep. U82-1. Unit for Coastal and Estuarine Studies, University College of North Wales. Huzzey, L.M., Brubaker, J.M., 1988. The formation of longitudinal fronts in a coastalplain estuary. Journal of Geophysical Research 93, 1329–1334. Hydraulic Research Station, 1979. A55 North Wales Coast Road: Hydrographic and Hydraulic Survey of Conwy Estuary 1978. Tech. Rep. EX869. Hydraulic Research Station. Jirka, G.H., Harleman, D.R.F., 1979. Stability and mixing of a vertical plane buoyant jet in confined depth. Journal of Fluid Mechanics 94, 275–304. Kikkawa, H., Ikeda, S., Kitagawa, A., 1976. Flow and bed topography in curved open channels. Journal of the Hydraulics Division – ASCE 102, 1327–1342. Kim, J., Kline, S.J., Johnston, J.P., 1980. Investigation of a reattaching turbulent shear layer: flow over a backward-facing step. Journal of Fluids Engineering, Transactions of the ASME 102, 302–308. Kneller, B.C., Bennet, S.J., McCaffrey, W.D., 1999. Velocity structure, turbulence and fluid stresses in experimental gravity currents. Journal of Geophysical Research 104, 5381–5391. Knight, D.W., 1981. Some field measurements concerned with the behaviour of resistance coefficients in a tidal channel. Estuarine, Coastal and Shelf Science 12, 303–322. Koçyigita, M.B., Falconer, R.A., 2004. Three-dimensional numerical modelling of wind-driven circulation in a homogeneous lake. Advances in Water Resources 27, 1167–1178. Largier, J.L., 1992. Tidal intrusion fronts. Estuaries 15, 26–39. McDowell, D.M., O’Connor, B.A., 1977. Hydraulic Behaviour of Estuaries. Macmillan Press, London. Neill, S.P., Copeland, G.J.M., Ferrier, G., Folkard, A.M., 2004. Observations and numerical modelling of a non-buoyant front in the Tay Estuary, Scotland. Estuarine, Coastal and Shelf Science 59, 173–184.
Nunes, R.A., Simpson, J.H., 1985. Axial convergence in a well-mixed estuary. Estuarine, Coastal and Shelf Science 20, 637–649. O’Donnell, J., 1993. Surface fronts in estuaries: a review. Estuaries 16, 12–39. Parker, G., 1978. Self-formed straight rivers with equilibrium banks and mobile bed. Part 1. The sand-silt river. Journal of Fluid Mechanics 89, 109–125. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vettering, W.T., 1986. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge. Reeves, A.D., Duck, R.W., 2001. Density fronts: sieves in the estuarine sediment transfer system? Physics and Chemistry of the Earth Part B – Hydrology Oceans and Atmosphere 26, 89–92. Rodi, W., 1984. Turbulence Models and Their Applications in Hydraulics – A State of the Art Review, second ed. IAHR. Simpson, J.H., James, I.D., 1986. Coastal and estuarine fronts. In: Mooers, N. (Ed.), Baroclinic Processes on Continental Shelves. AGU, Washington, pp. 63–93. Simpson, J.H., Nunes, R.A., 1981. The tidal intrusion front – an estuarine convergence zone. Estuarine, Coastal and Shelf Science 13, 257–266. Simpson, J.H., Turrell, W., 1986. Convergent fronts in the circulation of tidal estuaries. In: Wolfe, D. (Ed.), Estuarine Variability. Academic Press, New York, pp. 139–152. Smith, J.D., McLean, S.R., 1977. Spatially averaged flow over a wavy surface. Journal of Geophysical Research 82, 1735–1746. Soulsby, R.L., 1997. Dynamics of Marine Sands. Tech. Rep. SR 466. HR Wallingford. Swift, M.R., Fredriksson, D.W., Celikkol, B., 1996. Structure of an axial convergence zone from acoustic Doppler current profiler measurements. Estuarine, Coastal and Shelf Science 43, 109–122. Trevethan, M., Chanson, H., 2007. Detailed measurements during a transient front in a small subtropical estuary. Estuarine, Coastal and Shelf Science 73, 735–742. Turrell, W.R., Brown, J., Simpson, J.H., 1996. Salt intrusion and secondary flow in a shallow, well-mixed estuary. Estuarine, Coastal and Shelf Science 42, 153–169. West, J.R., Sangodoyin, A.Y.A., 1991. Depth-mean tidal current and sediment concentration relationships in three partially mixed estuaries. Estuarine, Coastal and Shelf Science 32, 141–159. Woodruff, J.D., Geyer, W.R., Sommerfield, C.K., Driscoll, N.W., 2001. Seasonal variation of sediment deposition in the Hudson River estuary. Marine Geology 179, 105–119. Xing, J., Davies, A.M., 2007. On the importance of non-hydrostatic processes in determining tidally induced mixing in sill regions. Continental Shelf Research 27, 2162–2185.
Contents lists available at ScienceDirect
Estuarine, Coastal and Shelf Science journal homepage: www.elsevier.com/locate/ecss
A numerical study of lateral grain size sorting by an estuarine front Simon P. Neill School of Ocean Sciences, Bangor University, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5AB, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 May 2008 Accepted 4 November 2008 Available online 25 November 2008
A two-dimensional non-hydrostatic model of baroclinic flow was applied to an estuarine cross-section. The model was driven by lateral variations in density and produced the classical bilateral cross-sectional recirculation of an axial convergent front. Simulations of the erosion, transport and deposition of sediment by the frontal secondary flows were applied to a range of grain sizes. The model predicted considerable lateral variation in grain size deposition across the frontal recirculation zone, analogous to the lateral grain size sorting which occurs in river meanders. The sorting primarily took place between the frontal surface convergence and the bankward limit of recirculation. A series of sensitivity tests revealed that the contribution of the front to lateral grain size sorting was strongly influenced by the lateral channel slopes and lateral density gradient. The results from this numerical study support previous suggestions (based on observations of near-surface discontinuities in sediment concentration across the frontal interface) that fronts may act as sieves within the estuarine sediment transport system. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: estuary fronts convergence sediment dynamics numerical model Conwy estuary
1. Introduction Convergent fronts are common and persistent features of many estuaries (Simpson and Turrell, 1986; Ferrier and Anderson, 1997). In an aerial survey of UK estuaries, evidence of convergent fronts was found in 15 out of 26 systems (Brown et al., 1991). There are three main categories of estuarine front (e.g. Simpson and James, 1986; Largier, 1992; O’Donnell, 1993): plume and tidal intrusion fronts (Garvine and Monk, 1974; Simpson and Nunes, 1981), axial convergent fronts (Nunes and Simpson, 1985) and longitudinal (shear) fronts (Huzzey and Brubaker, 1988). Secondary flows associated with such fronts can lead to enhanced surface concentrations of larvae (Eggleston et al., 1998) and pollutants (Swift et al., 1996), enhanced suspended sediment transport (Trevethan and Chanson, 2007) and lateral sediment trapping (Woodruff et al., 2001). In situ measurements of estuarine fronts have confirmed secondary flows of magnitude 0.2 m s1 (Brown et al., 1991), i.e. of order 20% of the longitudinal velocity. Due to the interaction of secondary flows on longitudinal sediment transport, it has been suggested that estuarine fronts may act as ‘‘sieves’’ within the estuarine sediment transport process (Reeves and Duck, 2001). Aerial photographs of frontal systems in the Tay Estuary have demonstrated discontinuities in near-surface sediment concentration across the frontal interface, i.e. the lateral transport of suspended sediment is inhibited by the density gradient. Therefore, depending on the magnitude of secondary flow associated with the front and the sediment grain size, estuarine
E-mail address: [email protected] 0272-7714/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ecss.2008.11.009
fronts, or a series of such fronts, may act as sieves within the estuarine sediment transport system. The secondary recirculation cell of an estuarine front (driven by lateral density gradients) is a similar mechanism to the helicoidal flow pattern occurring in a river meander (driven by the centripetal force). In the latter case, the centripetal force is generated by channel curvature (Kikkawa et al., 1976). Assuming that all grain size fractions in a river bed are fully mobile, sediment in a river meander may become laterally sorted because the helicoidal flow pattern transports fine grains towards the inner bend, while gravity transports coarse grains towards the outer bend (Frings, 2008). This results in a progressive segregation of coarse and fine size fractions throughout the bend (Fig. 1a). Since an estuarine front has two such helicoidal flows, one occurring either side of the surface expression of the front, it is here hypothesised (using the river meander analogy) that coarser sediment will be preferentially deposited on the bed under the frontal surface convergence and that finer sediment will be preferentially deposited on the bed at the bankward limits of frontal crosssectional recirculation (Fig. 1b). If proven to exist, such a sediment sorting mechanism could have important impacts on benthic biogeochemical and biological processes. In this paper, a non-hydrostatic numerical model of an idealised estuarine cross-section (Section 2) is applied to sediment transport processes occurring due to an axial convergent front (Section 3). The validity and robustness of the results (Section 4) are supported by a series of sensitivity tests (Section 5). The aim of the modelling methodology is to determine whether lateral bed grain size sorting occurs in the region of estuarine fronts, analogous to the sorting which occurs across a river meander. This is the first published testing of this hypothesised grain sorting by an estuarine front.
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a A
A
A
A
A coarse sediment
b
foam line A
surface convergence
A
A fine sediment coarse sediment
fine sediment
fine sediment
Fig. 1. Sketch of secondary flows and resulting hypothesised lateral bed grain size distribution. (a) River meander and (b) axial convergent estuarine front.
2. Numerical model The non-hydrostatic model, TEXSM (Tay Estuary CrossSectional Model) (Neill et al., 2004), was applied in this study to an idealised domain. A non-hydrostatic model was used because the vertical velocities generated by estuarine fronts are important. Such models are commonly applied in cases where vertical accelerations are important and the ratio of vertical to horizontal dimension relatively large (e.g. Koçyigita and Falconer, 2004; Xing and Davies, 2007). TEXSM is a two-dimensional (2D) model of an estuarine cross-section with a varying free surface (Hirt et al., 1975), k–epsilon turbulence (Rodi, 1984), salinity transport and baroclinic flow. The model uses the SOLA algorithm (Hirt et al., 1975) which solves explicitly for across-channel velocity v, vertical velocity w, production of turbulent kinetic energy (TKE) k, dissipation of TKE 3, salinity s, and solves for pressure p implicitly using the successive overrelaxation (SOR) algorithm (Press et al., 1986). The governing equations of TEXSM are the continuity equation
vC vC vC v vC v vC 3t 3t þ þ n þ ðw ws Þ ¼ vt vy vz vy vy vz vz
where C is the concentration of sediment, ws is the particle settling velocity (a function of grain shape and size) and 3t is the sediment diffusion coefficient. Since the same mechanism (i.e. eddy mixing) is used for both turbulent transport of momentum and sediment, it was assumed for simplicity that 3t ¼ nt, since the latter is calculated directly by the k–epsilon model. The bed level change was calculated using a one-dimensional sediment budget equation
vz 1 ðD EÞ ¼ 1n vt
(1)
the y (lateral) Reynolds equation
vn vðnnÞ vðnwÞ 1 vp v vn v n þ nt n þ nt þ þ þ ¼ þ2 r vy vt vy vy vz vy vz vw vn 2 vk þ vy vz 3 vy (2)
Ca ¼
vw vðwnÞ vðwwÞ 1 vp v vw vn n þ nt þ þ ¼ þ þ r vz vy vt vy vz vy vz v vw n þ nt þ2 vz vz
2 vk g bs ðs s0 Þ 3 vz
ðw ws ÞC
rs
(6)
where rs is the density of sediment grains. The calculation of D assumes that deposition due to the longitudinal velocity is negligible for application to the idealised channel described in Section 3 (since the baroclinic vertical component of velocity > barotropic vertical component of velocity). The erosion term E (due to the longitudinal velocity) was estimated by calculating an entrainment function e, with units of concentration (Parker, 1978). The entrainment function requires an estimate of the reference concentration Ca at a reference height above the bed za, calculated as (Smith and McLean, 1977)
and the z (vertical) Reynolds equation
(5)
where z is the bed level, n is bed porosity, E is erosion and D is deposition, calculated for only the row of cells adjacent to the bed using
D ¼
vn vw þ ¼ 0 vy vz
(4)
0:00156Ts 1 þ 0:0024Ts
(7)
26:3scr Ts d50 þ rgðs 1Þ 12
(8)
and
za ¼ (3)
where the overbar denotes a mean quantity, n is the kinematic viscosity, nt is the turbulent (eddy) viscosity, bs is the coefficient of saline contraction, s0 is salinity averaged over the cross-section, r is water density and g is acceleration due to gravity. TEXSM has been extensively validated with laboratory data from a diverse set of cases such as flow over a backward facing step (Kim et al., 1980), the lock exchange experiment (Kneller et al., 1999) and vertical jets in confined fluid depths (Jirka and Harleman, 1979). To apply TEXSM to sediment dynamics, a sediment mass continuity equation was added to the model (e.g. McDowell and O’Connor, 1977)
where scr is the threshold bed shear stress, s is the ratio of grain to water density and Ts is the transport parameter
Ts ¼
s0 scr scr
(9)
where s0 is the bed shear stress, related to the drag coefficient CD through the quadratic friction law
s0 ¼ rCD U 2
(10)
where U is the depth-averaged longitudinal velocity. By assuming that the entrainment function e is equal to the reference concentration Ca (Eq. (7)), entrainment of sediment from
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vC ¼ ews ¼ Ca ws vz za
(11)
where the vertical diffusion coefficient Kz is calculated (assuming a parabolic depth distribution) using
z Kz ¼ ku* z 1 h
+
ρ
v=0.20 ms
ws e
(13)
rs
The entrainment function e appears as a source term in Eq. (4), at the reference height za.
−
ρ
ρ
−1
w=0.05 ms −1
0
10
20
30
(12)
where u* is the friction velocity, k is Von Ka´rma´n’s constant and h is water depth. Therefore, the erosion term in Eq. (5) due to the longitudinal component of velocity can be estimated as
E ¼
−
0 1 2 3 4 5 6
40
50
60
70
80
90
100
lateral distance (m)
b v,10×w (m s−1)
Kz
a depth (m)
the bed can be equated with upward mixing due to turbulence at level za (Harris and Wiberg, 2001)
347
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
v 10×w
0
10
20
30
40
50
60
70
80
90
100
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lateral distance (m) 3. Model application
ws ¼
n d
10:362 þ 1:049D3*
1=2
10:36
(14)
where d is grain size and D* is the dimensionless grain size, calculated using
c
0.15 0.1
∂v/∂y (s−1)
The model was parameterised from observations of the axial convergent front which occurs in the Conwy Estuary, UK (Nunes and Simpson, 1985). The front persists for up to 2–3 h during the second half of the flood tide and is associated with a strong surface convergence along the main axis of the channel, visible due to the accumulation of organic matter and an intense foam line (Simpson and Turrell, 1986). During the flood phase of the tide, increased advection of denser water at the centre of the channel, where tidal currents are greater (and hence frictional effects less) than at the shallower channel edges, creates a lateral density gradient which drives the two-celled cross-sectional circulation of the axial convergent front. Axial salinity gradients of order 10 km1 have been observed during the flood phase of the tide, in contrast with axial salinity gradients of order 5 km1 during the ebb phase of the tide (Turrell et al., 1996), hence the front develops exclusively on the flood. Typical lateral variations in density of 0.32 kg m3 have been measured in the Conwy Estuary over a distance of 20 m (Nunes and Simpson, 1985). This change in density of 0.32 kg m3 was set as the initial condition for the TEXSM model as sketched in Fig. 2a, applied to an idealised symmetrical geometry with dimensions characteristic of the Conwy Estuary: a Gaussian channel of width 100 m and maximum depth 6 m. The Conwy Estuary is flood-dominant, with flood currents exceeding ebb currents by approximately 50% (Dyer, 1997). Since sediment transport is a function of bed stress, this asymmetry has led to observations of a threefold increase in depth-averaged suspended sediment concentration during the flood, in contrast to the ebb (West and Sangodoyin, 1991). Hence, at the time when the axial convergent front has fully developed, there is a considerable quantity of sediment in suspension. Grain size was parameterised based on survey results collected at the time of the construction of the Conwy Tunnel (Hydraulic Research Station, 1979) and from sampling reported by Knight, (1981). Typical values of the median grain size (d50) ranged from 100 to 800 mm, with coarser sediment generally found in the channel centre and finer sediment towards the channel edges. The grain sizes were translated into settling velocities using (Soulsby, 1997)
0.05 0 −0.05 −0.1 −0.15
0
10
20
30
40
50
60
70
lateral distance (m) Fig. 2. TEXSM modelled lateral (v) and vertical (w) velocities for axial convergent front at peak flood in a Gaussian channel. In (a) the initial density field is shown by vertical dashed lines. (a) Velocity vectors, (b) v and w close to bed and (c) divergence of v close to bed.
D* ¼
gðrs =r 1Þ 1=3
n2
d
(15)
Applying Eq. (14), the range of settling velocities required to represent the in situ median grain sizes was calculated to of order 0.005–0.11 m s1. Therefore, by considering 12 grain sizes, the modelled settling velocities were as listed in Table 1. No assumptions were made on the vertical distribution of sediment concentration, e.g. by the implementation of Rouse profiles. Rather, the 2D concentration field in the estuarine crosssection was calculated (from an initial condition of zero concentration throughout the cross-section) by Eq. (4) (with a flux boundary condition at the bed), applied to the range of 12 modelled grain sizes. The peak (mid-channel) depth-averaged longitudinal
Table 1 Range of modelled settling velocities (ws) and corresponding grain sizes (d). ws (m s1)
d (mm)
0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
94 135 199 257 317 381 452 530 618 716 825 946
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velocity when the front is fully developed was parameterised as U ¼ 1 ms1 and the drag coefficient parameterised as CD ¼ 0.0039 for the Conwy (Hunter, 1982). It was assumed that the depthaveraged longitudinal velocity varied parabolically across the estuarine cross-section. This was confirmed by running a depthaveraged numerical model of a uniform Gaussian channel with a constant value of CD ¼ 0.0039, driven by a velocity of magnitude 1 m s1 normal to the boundary. Since model simulation times for the sediment concentration in the estuarine cross-section to reach steady state were relatively short (of order h/ws ¼ 60–1200 s for the range of grain sizes considered), the assumption that longitudinal velocity was steady over the duration of the model simulations was valid. The model was run both with secondary flows (Run A) and without secondary flows (Run B). Therefore, an examination of the distribution of bed level z for (Run A)–(Run B) for the range of modelled grain sizes demonstrates the contribution of baroclinic processes to lateral sediment transport (and hence lateral grain size sorting). The model assumes that there is no localised gradient in the longitudinal component of velocity, i.e. the modelled cross-section is representative of steady, uniform flow, for example in a straight estuarine reach at mid-flood. In such a channel cross-section with steady barotropic flows, assuming Coriolis forces can be neglected, no mechanism other than diffusion acts to transport sediment laterally. The baroclinic flow, however, drives the two-celled circulation of the axial convergent front (Fig. 2a). Deposition occurred only when w ws < 0 in the row of model grid cells adjacent to the bed, and erosion due to the longitudinal component of velocity was included in the calculation (Eq. (13)). The model has a no-slip boundary condition on the bed and channel edges, and a free-slip at the free surface. 4. Results The model was applied (with dy ¼ 2 m, dz ¼ 0.25 m and a time step dt ¼ 0.1 s) to each of the twelve grain sizes (Table 1) for a duration of 15 min (a time period over which the longitudinal velocity was assumed not to vary significantly and which allowed the cross-sectional concentration fields to attain steady state). Since the grain sizes were modelled independently, there was no interaction between the different classes of grain size. Typical time series of vertical concentration profiles (for d ¼ 135 mm) were plotted at 120 s intervals for four locations across the channel (Fig. 3). For the initial condition at t ¼ 0 (not plotted), concentration was zero throughout the cross-section. Generally, the near-bed concentrations increased rapidly for all four locations as time progressed, and reduced exponentially with height from the bed. However, two main exceptions occurred. Firstly, at y ¼ 50 m (channel centre and hence frontal surface convergence), the nearsurface concentrations were >0 for t 240 s. Near-surface concentrations were continuously highest in the region of the surface convergence due to the combined effects of lateral advection and diffusion of sediment into this region. Secondly, at y ¼ 39 m there was an anomaly in sediment concentration between the surface and a depth of 4 m, with the peak occurring close to the centre of each bilateral recirculation cell (Fig. 2). This increased concentration was mainly due to the occurrence of peak vertical velocity w of order 0.05 m s1 at y ¼ 39 m, considerably greater than the settling velocity for this grain size (0.01 m s1). For the simulation plotted, little change occurred in the concentration profiles after t ¼ 240 s, other than the evolution of the profile at y ¼ 33 m. In general, it was the sediment concentration at the bankward limits of the recirculation zone (y ¼ 33 m) which took the longest time to reach steady state since the sediment had been transported over the greatest lateral distance from the erosional source (channel centre). At steady state, and in the absence of
secondary flows, the concentration profiles across the channel approximate to Rouse profiles. The contribution of secondary flow to the lateral distribution of bed level change after 15 min of simulation for each grain size (i.e. each modelled settling velocity) was plotted (Fig. 4). Note that since the hydrodynamics and resulting sediment transport were symmetrical about the channel centreline, the results have been plotted only for one side (the left-hand-side) of the channel. Clearly, the secondary flow associated with the front had the greatest influence on the lateral distribution of finer grains and the least influence on the lateral distribution of coarser grains. For the finer grain sizes, maximum deposition occurred close to the bankward limit of frontal recirculation (y ¼ 33 m) and there was a minimum close to the centre of the channel. By combining the model results for all grain sizes, the contribution of secondary flow to median grain size sorting was calculated across the channel, assuming that the deposition was well mixed (Fig. 5). The minimum d50 (275 mm) was found at y ¼ 35 m, i.e. close to the bankward limit of frontal recirculation (Fig. 2). The maximum d50 (396 mm) was found at the centre of the channel, as hypothesised in Fig. 1b. Therefore, the presence of the front has resulted in a lateral sorting of sediment grain size of magnitude 121 mm for this benchmark case. 5. Sensitivity tests To confirm the validity of the results, a series of sensitivity runs were performed, the most extensive of which was application of the model to an entire flood cycle (the front does not form on the ebb). This was implemented by assuming that the lateral density gradient was a sinusoidal function, increasing from zero at the beginning of the flood (well-mixed conditions) to a maximum gradient at high water. Hence, the TEXSM model allowed the secondary flows to evolve from zero (no density gradient) to a maximum (as plotted in Fig. 2). The longitudinal velocity was correspondingly varied over this time with a sinusoidal function of amplitude 1 m s1 to represent a spring flood tide. This longitudinal velocity was applied as an across-channel parabolic distribution to simulate spatial and temporal variations in erosion due to varying longitudinal velocity throughout the flood phase of the tide. The modelling methodology described in Section 3 was applied at 15 min intervals using the secondary flows output from TEXSM in conjunction with these longitudinal velocities. The lateral distribution of z for each grain size was recorded over each 15 min interval. A new cross-sectional concentration field was then calculated for the evolving secondary flow field. The lateral distribution of z for each grain size was summed cumulatively over the flood phase of the tidal cycle. The series of simulations was repeated for zero secondary flows. The difference between the two sets of simulations was used to calculate the contribution of the front to the lateral distribution of median grain size sorting over a spring flood cycle. This result was compared to the steady state case (Fig. 6). The distribution was similar to the steady state case (minimum d50 at the bankward limit of recirculation and maximum d50 close to the centre of the channel). However, the magnitude of sorting across the front was reduced by a factor of 2.5, indicating that whereas net lateral grain size sorting due to an axial convergent front will occur over an entire flood tide, the net sorting is likely to be less than the sorting occurring at peak flood by a factor of 2–3. Five additional sensitivity runs were performed (Table 2, Fig. 6). These included simulating a neap longitudinal flow, increasing the lateral density gradient by a factor of 2, decreasing the lateral density gradient by a factor of a half, applying a constant velocity (rather than a parabolic distribution) across the channel, and simulating a rectangular channel case (a constant depth of 6 m across a 100-m wide channel). The latter had the most significant
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t=120s
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t=240s
0
0 y=33m
y=33m y=39m
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y=39m
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y=45m
y=45m
y=50m
y=50m 2
depth (m)
depth (m)
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3
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concentration (kg m−3)
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20
t=360s
t=480s
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y=39m
1
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1
y=45m
y=45m
y=50m
y=50m 2
depth (m)
depth (m)
2
3
3
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concentration (kg m−3)
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1
y=39m
1
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y=50m
2
2
depth (m)
depth (m)
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concentration (kg m−3)
0
3
3
4
4
5
5
6
30
concentration (kg m−3)
0
10
concentration (kg
20
m−3)
30
6
0
10
20
30
concentration (kg m−3)
Fig. 3. Time evolution of vertical profiles of sediment concentration at four locations across the channel. These simulations are shown for a settling velocity of 0.01 m s1 (d ¼ 135 mm).
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0.04
0.02
0.01
0
400
350
300
−0.01
−0.02
benchmark entire flood tide neap high ∂ρ/∂y low ∂ρ/∂y constant across−channel U rectangular channel
450
d50 ( m)
0.03
Δ ζ (m)
500
d=94 μm d=135 μm d=199 μm d=257 μm d=317 μm d=381 μm d=452 μm d=530 μm d=618 μm d=716 μm d=825 μm d=946 μm
0
10
20
30
40
50
250
0
10
20
30
40
50
lateral distance (m)
lateral distance (m) Fig. 4. Modelled contribution of secondary flow to bed level change (Dz) for the range of twelve grain sizes after 15 min of simulation for benchmark case.
Fig. 6. Sensitivity tests on contribution of secondary flow to lateral distribution of median grain size (d50).
effect on the resulting lateral distribution of d50. Due to a lack of channel slopes, stronger near-bed lateral flow resulted (Fig. 7), and hence the minimal value of d50 occurred further bankward, compared to the more realistic Gaussian channel (Fig. 6). However, for the rectangular channel case, a fining of grain size deposition occurred at the centre of the channel due to enhanced lateral transport of finer grain sizes by the stronger secondary flows, uninhibited by the presence of lateral channel slopes. Due to this fining, the peak median grain size for the rectangular channel case occurred closer to the centre of each bilateral recirculation cell, rather than at mid-channel. This channel centre fining also occurred for the case with high lateral density gradient, again due to stronger resulting secondary flows.
the location where vv/vy is a minimum (y ¼ 37 m) and the smallest deposition expected where vv/vy is a maximum (y ¼ 50 m, i.e. channel centre). Examining Fig. 4, this was the case for grain sizes 94 and 135 mm. For d > 135 mm, however, peak deposition occurred progressively closer to the centre of the channel with increasing grain size, since secondary flows had less influence on the coarser sediments. The results of the sensitivity tests indicated that the magnitude of the secondary flow (due to variations in the lateral density gradient), although not significantly affecting the lateral location of maximum/minimum d50, did have a significant role in the magnitude of the sorting (Fig. 6). Defined as the difference between the maximum and minimum d50 occurring across the channel (Dd50), this magnitude varied between Dd50 ¼ 69 mm (reduced lateral density gradient) to Dd50 ¼ 151 mm (increased lateral density gradient), in contrast to the benchmark value of Dd50 ¼ 121 mm. Therefore, regardless of the spring/neap character of the tide, in circumstances of an increased axial salinity gradient (e.g. high freshwater input), the contribution of an estuarine front to lateral grain size sorting will be enhanced. For the case of increased lateral density gradient, a slight fining of sediment occurred at the centre of the channel due to considerable erosion of finer sediments (94– 199 mm) from this region. A high concentration of this finer sediment remained in suspension at steady state (due to high values of w), hence less deposition at the outer limits of frontal recirculation. In the case of a river meander, used as an analogy, the inner bank acts as a lateral constraint on the recirculation zone (Fig. 1a). In the estuarine front case, this boundary is diffuse and will vary considerably with tidal range and phase. Therefore, it is difficult to determine where the bankward limits of recirculation will occur and hence exactly what the influence of lateral bed slopes will have on the sediment transport due to secondary flows. However, the sensitivity tests applied over an entire flood cycle demonstrated that the lateral grain size sorting by an estuarine front will be much greater at times of peak longitudinal velocity since suspended sediment concentration will be high. Hence, simulations which use the steady state secondary flow field at times of peak longitudinal currents are useful for examining lateral bed-slope effects, demonstrated to have a large influence on the lateral grain size sorting (Fig. 6). The model simulations demonstrate the potential contribution of estuarine fronts to lateral grain size sorting. In situ observations will also include effects due to longitudinal variations in velocity/ bathymetry and channel curvature. However, a small influence in
6. Discussion The model predicted the largest d50 at the centre of the channel and the smallest d50 at the bankward limits of recirculation (Fig. 5), as hypothesised in Fig. 1b. These locations can be related to the near-bed lateral component of velocity (Fig. 2b), specifically in terms of the near-bed divergence of the lateral velocity (Fig. 2c). For a typical grain size, the largest deposition is generally expected at
400 380
d50 ( m)
360 340 320 300 280 260
0
10
20
30
40
50
lateral distance (m) Fig. 5. Contribution of secondary flow to lateral distribution of median grain size (d50) for benchmark case.
S.P. Neill / Estuarine, Coastal and Shelf Science 81 (2009) 345–352
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Table 2 Details of sensitivity tests. U is the depth-averaged longitudinal velocity with maximum value U max at the centre of the channel and Dr is the lateral change in density. Sensitivity test
U max ðms1 Þ
Lateral distribution of U
Dr (kg m3)
Lateral bathymetry profile
Benchmark Entire flood tide Neap High lateral density gradient Low lateral density gradient Constant across-channel longitudinal velocity Rectangular channel
1.0 Variousa 0.5 1.0 1.0 1.0 1.0
Parabolic Parabolic Parabolic Parabolic Parabolic Constantb Parabolic
0.32 Variousa 0.32 0.64 0.16 0.32 0.32
Gaussian Gaussian Gaussian Gaussian Gaussian Gaussian Constant d ¼ 6 m
a b
In this unsteady case, velocity and lateral density gradient were varied sinusoidally over time from zero (initial condition) to their maximum values. Constant value of U max applied across the channel.
such a biologically important region (due to accumulated organic matter at estuarine fronts) may lead to a large effect on estuarine benthic biogeochemical and biological processes. For example, estuarine meiofaunal diversity (dominated by nematodes) is closely related to sediment grain size (Hodda, 1990). With predicted increased precipitation occurring during the biologically productive summer season (Beniston et al., 2007) and hence a corresponding increase in lateral estuarine density gradients, the relatively modest contribution of processes such as lateral grain size sorting by estuarine fronts may become more important in the latter half of the 21st century. The model simulations demonstrate how secondary flows associated with an axial convergent front can act as sieves within the estuarine sediment transport system, as suggested by Reeves and Duck (2001). In qualitative agreement with observations in the Tay Estuary of discontinuities in near-surface sediment concentrations across the frontal interface, the model similarly predicted
depth (m)
a
0 1 2 3 4 5 6
ρ−
ρ+
ρ−
v=0.20 m s−1 w=0.05 m s−1 0
10
20
30
40
50
60
70
80
90
100
an order of magnitude change in near-surface sediment concentrations across the front (Fig. 3). Finally, axial convergent fronts are located in the main channel of an estuary. In the case of estuarine fronts which occur at bathymetry breaks (e.g. longitudinal fronts), fine sediment placed into suspension over the shoals may become laterally trapped by the density interface and not freely pass into the main channel. Therefore, further research is required into the role of the individual categories of estuarine front on sediment sieving. 7. Conclusions A numerical model of an estuarine cross-section, parameterised on the axial convergent front occurring in the Conwy Estuary, UK, demonstrated that considerable lateral bed grain size sorting can occur due to secondary frontal flows. The sorting took place between the surface convergence and the bankward limits of recirculation and is sensitive to the lateral density gradient and lateral channel slopes. The magnitude of the sorting is small compared to the effect of lateral shear in the longitudinal velocity, but in such a biologically productive region as an estuarine front, the influence on estuarine benthic biogeochemical and biological processes may be significant. The results from this numerical study support previous suggestions (based on aerial photographs of nearsurface discontinuities in sediment concentration across the frontal interface) that fronts may act as sieves within the estuarine sediment transport system.
lateral distance (m)
v,10×w (m s−1)
b
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2
Acknowledgements v 10×w
0
10
20
30
40
50
60
70
80
90
Thanks to Prof. Alan G. Davies at Bangor University and to four anonymous reviewers, whose combined input considerably enhanced the final version of the manuscript. This work was supported by the Higher Education Funding Council for Wales and the Welsh Assembly Government. 100
lateral distance (m)
c
0.15
v/ y (s−1)
0.1 0.05 0 −0.05 −0.1 −0.15
0
10
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30
40
50
60
70
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100
lateral distance (m) Fig. 7. TEXSM modelled lateral (v) and vertical (w) velocities for axial convergent front at peak flood in a rectangular channel. In (a) the initial density field is shown by vertical dashed lines. (a) Velocity vectors, (b) v and w close to bed and (c) divergence of v close to bed.
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