In river-dominated deltas, bifurcations often develop an asymmetrical morphology; i.e. one of the downstream channels silts up, while the other becomes the dominant one. In tide-influenced systems, bifurcations are thought to be less asymmetric and both downstream channels of the bifurcation remain open. The main aim of this study is to understand how tides influence the morphological development of bifurcations. By using a depth-averaged (2DH, two-dimensional horizontal) morphodynamic model (Delft3D), we simulated the morphological development of tide-influenced bifurcations on millennial timescales. The schematized bifurcation consists of an upstream channel forced by river discharge and two downstream channels forced by tides. Two different cases were examined. In the first case, the downstream channels started with unequal depth or length but had equal tidal forcing, while in the second case the morphology was initially symmetric but the downstream channels were forced with unequal tides. Furthermore, we studied the sensitivity of results to the relative role of river flow and tides. We find that with increasing influence of tides over river, the morphology of the downstream channels becomes less asymmetric. Increasing tidal influence can be achieved by either reduced river flow with respect to the tidal flow or by asymmetrical tidal forcing of the downstream channels. The main reason for this behaviour is that tidal flows tend to be less unequal than river flows when geometry is asymmetric. For increasing tidal influence, this causes less asymmetric sediment mobility and therefore transport in both downstream channels. Furthermore, our results show that bedload tends to divide less asymmetrically compared to suspended load and confirm the stabilizing effect of lateral bed slopes on morphological evolution as was also found in previous studies. We show that the more tide-dominated systems tend to have a larger ratio of bedload-to-suspended-load transport due to periodic low sediment mobility conditions during a transition between ebb and flood. Our results explain why distributary channel networks on deltas with strong tidal influence are more stable than river-dominated ones.
Deltas often consist of distributary channel networks. In these systems, water and sediment are divided at the bifurcations and distributed over the delta. The shape of the delta and the number of active channels depends on many factors like the forcing by rivers, tides, and waves (Galloway, 1975; Rossi et al., 2016; Shaw and Mohrig, 2014); sediment availability, and sediment type (Geleynse et al., 2011). Bifurcations tend to develop differently in river- than in tide-dominated systems, because tides influence the mouth bar formation processes of active river-dominated deltas (Edmonds and Slingerland, 2007; Leonardi et al., 2013; Shaw and Mohrig, 2014). In tidal deltas tides propagate upstream and can induce bidirectional flows. This unique characteristic may lead to a different morphological evolution of the bifurcations than would occur in the river-dominated zone (Frings and Kleinhans, 2008; Hoitink et al., 2017), but this has not been proven yet and the underlying mechanisms have not been studied. The focus of this paper is on the stability and depth asymmetry of bifurcations in tidally influenced deltas. We do not focus on the morphological evolution of the entire delta or the formation process of mouth bars, but we consider a single bifurcation consisting of one upstream and two downstream channels. These are the building blocks of deltas, and the hydro- and morphodynamics of such a system have been studied before by many others (Wang et al., 1995; Bolla Pittaluga et al., 2003; Buschman et al., 2010, 2013; Kleinhans et al., 2008; Sassi et al., 2011).
In river-dominated systems, the morphology of the downstream channels of bifurcations often develops asymmetrically, such that one downstream channel deepens while the other silts up (Kleinhans et al., 2008). In many cases this condition develops into an avulsion. This asymmetric development can be triggered by a small perturbation such as a different bed elevation at the junction (Bolla Pittaluga et al., 2003), by a meandering upstream channel nearby the bifurcation, or by the geometry of the downstream channels such as different lengths of the downstream branches (Kleinhans et al., 2008). The study of this morphological evolution in river-dominated bifurcations was pioneered by Wang et al. (1995). They applied an analytical model to predict the stability of river bifurcations. They found that bifurcations can be stable if any tendency for a downstream branch to become more dominant is counteracted by a relatively large share of the sediment input. Bolla Pittaluga et al. (2003) improved the model in Wang et al. (1995) by taking into account the cross-channel flow that can be induced by an asymmetric cross-sectional profile at the bifurcation. This effect induces a lateral bedload transport, which affects the asymmetric sediment division to the downstream branches. Using this approach, they found that the asymmetry of depth of the two downstream branches depends on the Shields number and on the width-to-depth ratio of the upstream channel at the bifurcation. Bifurcations with high width-to-depth ratio and low Shields number will be unstable and develop asymmetrical depths. Bertoldi and Tubino (2007) confirmed the results by Bolla Pittaluga et al. (2003) using a physical-scale model. Kleinhans et al. (2008) proposed that this asymmetrical depth development is also influenced by meandering of the upstream channel. The meandering bend induces an asymmetrical cross-sectional bed profile and thereby influences the division of sediment at the junction. Bolla Pittaluga et al. (2015) continued the work of Bolla Pittaluga et al. (2003) for a wider range of sediment mobility conditions. They found a range of sediment mobility numbers that result in stable symmetric bifurcations. Meanwhile, bifurcations with sediment mobility higher or lower than this range will grow asymmetrically and avulse. Applying the concept of Bolla Pittaluga et al. (2003), Salter et al. (2017) showed that deposition of sediment at a relatively shallow shelf causes the shorter channel to lengthen and reduce in gradient, thereby balancing the sediment transport division between downstream channels with unequal lengths. Redolfi et al. (2016) eliminated the need for a calibrated parameter in the lateral bedload transport by Bolla Pittaluga et al. (2015), and, by using that approach, Redolfi et al. (2019) showed that stable, symmetric bifurcations can only occur when the width-to-depth ratio of the upstream channel is below the critical limit originally defined in the theory of meandering rivers by Blondeaux and Seminara (1985), where the critical limit value depends on the friction and Shields stress at bifurcation.
In contrast to our knowledge of morphological development of bifurcations in river-dominated systems, our knowledge of this particular area in tide-influenced systems is still limited. Observations suggest that a similar development as in river-dominated systems can occur, as, for example, found in the most upstream bifurcation of the Yangtze Estuary that divides the main channel into the North Branch and South Branch. According to Chen et al. (1982), the North Branch has evolved to be narrower and shallower, while the South Branch has deepened. However, bifurcations in other tide-influenced deltas have downstream channels that seem to have a less asymmetric depth distribution, e.g. the Berau River delta (Buschman et al., 2013) and Kapuas River delta (Kästner et al., 2017). It has been suggested that tidal deltas have more stable distributary channel networks than their river-dominated counterparts (Hoitink et al., 2017), but the underlying mechanisms are unknown. Furthermore, several studies have investigated tidal characteristics at tidal bifurcations. Despite a general understanding on tides and subtidal water division at tidally influenced bifurcations (Buschman et al., 2010, 2013; Sassi et al., 2011; Zhang et al., 2012; Alebregtse and de Swart, 2016), the effect of tides on the morphological evolution of tidal bifurcations has not been fully understood yet. From previous studies it is clear that tides influence the subtidal flow (Buschman et al., 2010; Sassi et al., 2011) and sediment division (Buschman et al., 2013), induce tidal currents that influence the sediment mobility, and can cause cross-channel currents at the junction (Buschman et al., 2013; Kleinhans et al., 2013). In river systems, all these factors are important for the morphological development of the downstream channels and it is expected that this is also the case for tide-influenced systems.
Therefore, the main aim of this paper is to study the effect of tides on the morphological evolution of bifurcations with the focus on how tides contribute to the asymmetrical development. For this purpose, an idealized bifurcating channel was set-up in Delft3D. We simulated the morphological evolution of a system consisting of two downstream channels (branches) forced by tides and an upstream channel forced by river discharge. We consider this system as a building block of each delta system. We studied two cases, i.e. asymmetric geometry of downstream channels and asymmetric tides between the downstream channels. In the former case, the asymmetric downstream geometry was initially prescribed to see how tides affect the asymmetrical development of the downstream channels. The relative effect of tides was investigated by imposing equal tides at downstream boundary of each downstream branch and by using different values for the river discharge in a series of simulations. In the latter case, we imposed unequal tidal forcing at the two downstream boundaries that had a symmetric geometry. In tide-influenced deltas, the asymmetric tides between downstream channels can occur because the downstream channels are connected to other channels with different complexity, which may dissipate the tidal range or slow down the tides unequally before the tides propagate into the downstream channels of the bifurcation.
This paper is organized as follows. The model set-up and methodology are described in Sect. 2. In Sect. 3, the results of the simulated morphological development are presented. Section 4 presents a discussion on the findings. Finally, the conclusions of this study are provided in Sect. 5.
An idealized bifurcating channel was set up and its morphological development was simulated using the depth-averaged version (2DH) of Delft3D. This 2D approach is suitable for long-term large-scale morphodynamic modelling, because it is computationally lighter than a 3D approach. Even though a 3D approach allows for vertical flow patterns (Lane et al., 1999) such as curvature-induced flow, which might be important for the sediment transport process (Daniel et al., 1999), the 2D approach is sufficient for this study since we focus on large-scale morphodynamic evolution and therefore simulating detailed 3D features of flow and morphology is not our goal. Furthermore, the reason to prefer the 2D above the 1D approach is to explicitly simulate cross-channel flow induced by tidal propagation from one branch to another at the junction as observed in Buschman et al. (2010, 2013) and as being identified by Bolla Pittaluga et al. (2003) as an important process for sediment division at the junction.
The model solved the 2DH unsteady shallow water equations using a
semi-implicit alternating direction implicit (ADI) scheme on a staggered
grid (see Lesser et al., 2004).
For bed friction, the Chézy formulation was used with a value of 60 m
The spatial domain consisted of an upstream channel that bifurcates in two
downstream channels. The two downstream channels had a default length of 30 km; although, in one series of simulations the length of one channel was 15 km. The upstream channel had a length of 220 km to ensure that upstream
propagating tides decay smoothly. The downstream channels and the first 20 km of the upstream channel had a convergent width profile, while the
upstream 200 km had a constant width. The channel width was configured by
The spatial domain of the model was discretized in a curvilinear grid and
followed the same method as in Kleinhans et al. (2008) and
Buschman et al. (2010). At the bifurcation
two grid cells had to be removed in the middle of the channel for numerical
reasons (Kleinhans et al., 2008),
as illustrated in Fig. 1. The grid cell length in
the along-channel direction was 80 m. The upstream channel had 12 grid cells
across the channel, whereas in both downstream channels 5 grid cells were
used. Therefore, the grid cell size in the across-channel direction was
spatially varying in order to adapt the funnelling shape of the channel and the additional widening near the bifurcation.
Near the junction this resulted in a typical grid cell width of 62.5 m. Based on
grid size and channel depth, a time step of 6 s was used in all
simulations to have a Courant number smaller than
Because the formation of alternating bars will affect flow and sediment
division at the junction, the channel depth and upstream-prescribed river
discharge were chosen such that the system was in the overdamped bar regime
(Struiksma et al., 1985). To this
end, we conservatively followed the empirical classification proposed by
Kleinhans and van den Berg (2011).
Therefore, the three connected channels had an initial depth of 15 m and a
constant along-channel bed slope of
Depth, width, and length of the downstream channels of bifurcations in deltas
can be unequal. Hence, in Case 1 we started the simulations with an unequal
geometry, either being a difference in depth or length between the two
downstream channels. We simulated the morphological evolution of the
bifurcation until it approximately reached morphodynamic equilibrium
(discussed later on). Note that the length of the branches was fixed in time,
while an initial depth difference does not necessarily result in an
asymmetric equilibrium depth because it can adapt. All simulations belonging
to Case 1 were forced by equal tides from downstream and river discharge
from upstream (settings summarized in Table 1). The depth difference
scenarios were performed in two different ways. First, simulations were
started from a system in which the upstream channel and one downstream
channel were 15 m deep, while the other branch was 7.5 m deep (called
Depth1). The upstream 2 km of the shallow downstream channel was gradually
changed over 2 km to avoid a sudden depth change near the bifurcation. In a
second type of simulation, we started with uniform bathymetry of 15 m depth
and simulated until morphodynamic equilibrium was reached (called Depth2).
Next, one downstream channel was made 0.5 m deeper and the other 0.5 m
shallower. We studied the sensitivity of the results to the relative
magnitude of tides over river discharge by changing the prescribed upstream
discharge. The simulation with largest river discharge (2800 m
Summary of simulations undertaken in the present study and their boundary conditions (river discharge and tidal properties), as well as geometry differences between the downstream channels.
In Case 2 the effect of unequal tidal forcing on morphological development
was studied. In natural systems tides in the two downstream branches can be
unequally forced. For example, when the two branches end in a shelf sea,
amplitude and phase in the two channels can be different because they have a
different position with respect to the amphidromic system in the shelf sea.
Furthermore, in deltas with multiple bifurcations and unequal depths and
channel lengths, tidal amplitude and phase differences will be present in
the channels because propagation speeds and times in the channels are
different. Hence, in Case 2 we started simulations with a symmetric geometry
but with asymmetric tidal forcing, either being a tidal water level
amplitude difference or a tidal phase difference. The corresponding settings
of the simulations can be found in Table 1. The difference in downstream
tidal forcing between the two channels was studied for values between 0
and 0.75 m (
We also performed two control simulations with different discharge, symmetric geometry, and equal tides (see Table 1) to study the equilibrium bed profiles in the absence of any initial asymmetry. The morphology change simulated for Case 1 and Case 2 were caused by the asymmetric forcing/geometry and by the adaptation to the initial conditions. Therefore, the results of the control simulations can be used to better interpret the simulations of Case 1 and Case 2.
The morphological development of the bifurcation was observed by evaluating
for each downstream channel the tidally and spatially averaged depth of the
first 2 km from the bifurcation (Fig. 2, called
The grids in the surroundings of the bifurcation overlaid by the areas where the bed level changes were evaluated (grey boxes) and the grids where the asymmetry indices (red lines) and upstream channel flow (black line) were calculated.
To compare the depth of the two downstream channels, the depth asymmetry
parameter
The sediment mobility was evaluated by calculating the width-averaged value
of the Shields number two grid cells away from the bifurcation, as
illustrated in Fig. 2. The Shields number at each
grid point was calculated as
At the grid locations where we determined the Shields number, we also
determined the tidally averaged (
Results of the two control simulations show that bed levels were initially not in morphodynamic equilibrium. The time-stack diagram of width-averaged depth as a function of space is shown in Fig. 3. The morphology changed over time until an approximate equilibrium was reached, which took about 1200 years. There are two timescales involved. First, there are deposition fronts from the upstream channel that migrate downstream. Second, there is a slower adaptation to the equilibrium condition. The results also show that true morphodynamic equilibrium, in the sense that bed levels are steady, was not achieved after 1200 years. However, bed level changes were small at the end of the simulation. The lowest discharge resulted in the smallest depth for the upstream channel, but the river discharge does not significantly affect the depth of the two downstream channels. This is because both control simulations were imposed by the same tidal forcing, and the morphology of the downstream channels is mainly controlled by the tides. Typical depths are around 8–10 m for the downstream channels and 10–12 m for the upstream one.
Time-stack diagram of width- and tide-averaged depth (colour) of
the upstream channel (
When simulations started with unequal channel depth, a similar evolution as
the control simulations occurred. The morphological evolution was
characterized by three typical timescales. First, there was erosion near
the bifurcation, mainly because of the decrease in the cross-sectional area
directly seaward of the bifurcation. Second, this erosion was followed by
deposition fronts that migrated downstream during the simulation. This
deposition front can be identified by a rapid decrease of the depth in the
downstream channels at the beginning or halfway through the simulation
(Fig. 4). It is similar to the evolution of
Control_Q2800 and Control_Q1596, but this
depositional front was not necessarily similar in the two downstream
channels because of the imposed differences in the initial bed level.
Furthermore, in the lowest discharge simulations (
Same plot as Fig. 3 but for simulations of Depth1. The panels from top to bottom show the results from different simulations (Depth1_Q2800, Depth1_Q1596, Depth1_Q500, respectively), while from left to right panels show the upstream channel, shallow branch, and deep branch, respectively.
The simulations that were based on perturbed equilibrium depth (Depth2) had a different morphological evolution and final equilibrium than the ones that started with 7.5 m depth difference (results not shown). The Depth2 simulation did not show the fast, initial depth response, but was mainly characterized by a slow adaptation to a new equilibrium, because the system was still close to equilibrium at the start of the simulation. It took relatively long to achieve the new equilibrium and total simulation time was 2400 years in this case. Interestingly, although the external forcing for the Depth1 and Depth2 simulations were the same, the final equilibria were different. Because the depth in the channels influences the tidal dynamics (by, for example, the relative importance of friction and by the difference in tidal propagation speed due to the different initial depths), the tide-induced flows were different at the junction and stayed different during the entire simulation. Hence, the equilibrium not only depends on external forcing but also on initial conditions. The initial and final morphology near the bifurcation for all Depth1 and Depth2 simulations can be seen in Appendix A.
The simulations with the
Time-stack diagram of width- and tide-averaged depth as a function
of space for the simulations in
Asymmetric forcing of tides resulted in asymmetric morphological evolution. Because the system started out of equilibrium, the morphological evolution is again characterized by a quick adaptation followed by a slow evolution to the equilibrium. When forced by different tidal amplitude, the downstream branch with the smallest downstream tidal forcing evolved into the shallowest branch (Fig. 6). Interestingly, when tidal amplitude in Branch 1 was decreased from 0.75 to 0.5 m or even 0.25 m the bifurcation evolved into a less asymmetric system. Furthermore, when the two downstream channels were forced by equal amplitudes, but with different phase, this also resulted in the development of an asymmetric morphology of the bifurcation (Fig. 7). In general, the channel with delayed tides developed smallest channel depth, while the channel with earlier tides developed deeper channels. Interestingly, the deposition front in the shallowest branch became stagnant for the largest imposed phase differences, suggesting that the flow magnitude was below the threshold for erosion (static equilibrium). However, the depth around the bifurcation did not become zero and still evolved. The larger the difference in tidal phase at the two downstream boundaries, the shallower the delayed branch became, while the other branch was deeper. The final morphology near the bifurcation for all simulations of this case is provided in Appendix A.
Time-stack diagram of width- and tide-averaged depth as a function
of space for the
Time-stack diagram of width- and tide-averaged depth as a function
of space for the
The results suggest that tides cause less asymmetric bifurcations. To
quantify how tides affect the morphology, the results from all scenarios
were correlated. Figure 8 shows a scatter plot and
linear fit between the final
Relation between depth asymmetry number
According to Eq. (3), in a system with uniform sediment properties and water
density, the sediment mobility in the downstream channels only depends on
the total bed shear stress
The relatively strong river discharge in the simulations performed causes
the ratios of
Comparison between
There are two processes that drive a less asymmetric tidal flow in the more
tide-influenced condition. First, the propagation of tides from the dominant
downstream channel to the other downstream channel balances the tidal flow
in the two downstream channels. This process mainly rules in the tide difference
case. Tidal forcing asymmetry between downstream channels drives tidal
propagation from one downstream channel to the other and results in phase
lags of tidal flow inducing strong cross-channel flow at the junction
(similarly as discussed in Buschman et
al., 2013). This can be seen by a larger cross-channel flow in the upstream
channel near the bifurcations for larger asymmetry between the prescribed
tides in Fig. 10. This cross-channel flow is
dominated by the tides (
Cross-channel flow of
In the theory of Bolla Pittaluga et al. (2003, 2015) the lateral bed slope
causes additional sediment transport into the dominant channel, thereby
having a stabilizing effect on the bifurcation. Here, we used the
van Rijn (1993) sediment transport formulations in
which bed slope only affects the bedload transport and not the suspended
load transport. Based on this, we expected that bedload transport will be
divided less asymmetrically than suspended load transport. To check this
hypothesis, the tidally averaged and width-integrated sediment transport at
the cross sections shown in Fig. 2 were
calculated. We calculated an asymmetry index in a similar way as we did for the
Shields number and depth. The results of the scatter plot of suspended load
asymmetry versus bedload asymmetry index clearly show that suspended load
tends to be divided more asymmetrically at the bifurcation
(Fig. 11a). Only when the system is fully
symmetric or asymmetric is there no difference in asymmetry of bedload and
suspended load transport, because the downstream channels receive an equal
amount of sediment when the downstream channels are symmetric, while only
one downstream channel receives all sediment when an avulsion occurs (both
bedload and suspended load asymmetry are 1). Furthermore, from a scatter
plot of depth asymmetry (
Comparisons of
Defining a different sediment grain size would change the sediment mobility and drive a different ratio of bedload to suspended load transport. These would affect the sediment transport division and therefore the morphological development in the downstream channels. When using finer sediment, this results in a more asymmetric development of the downstream channels, as is shown in Fig. 12. The finer sand induces a larger contribution of suspended load transport to total sediment transport and therefore counteracts the stabilizing effect by the transverse bed-slope effect on the bedload. As a result, the depth asymmetry between downstream channels increases. Similarly, a coarser sediment results in smaller depth asymmetry between the downstream channels.
Initial
The importance of the effect of lateral bed slope to oppose the asymmetrical
morphological development between downstream channels causes the model
results to be sensitive to the parameter
Initial
From the findings presented in this paper, we can predict how tides will influence the morphological evolution of deltas. In the seaward part of tide-influenced deltas, especially those with seaward-widening channels, river flow tends to be small relative to the tidal flows. In these regions we only expect asymmetry in morphology when the branches are unequally forced by tides. The tides tend to keep all the branches open and have similar depths. In the upstream part of deltas, river flows tend to be larger, which can result in large morphological asymmetries. However, the different possible pathways of the tide along the channel networks can generate differences in tidal amplitude and tidal phase between branches, inducing relatively strong tidal currents at the junction. This prevents the closure of one downstream channel and erodes the bed at the junction because of the strong cross-channel flows.
Morphological development of bifurcations occurs on a long timescale and several external causes and internal processes neglected here can affect bifurcation stability (also see the review in Kleinhans et al., 2013), such as of sea level rise (Jerolmack, 2009; van Der Wegen, 2013), changes in upstream discharge or sediment supply (Syvitski and Milliman, 2007), channel bank erosion or growth (Miori et al., 2006), and delta front development that could change the length of a branch (Salter et al., 2017). However, we have provided a basic explanation on how tides can stabilize the morphology of deltas.
In this article, the effect of tides on the morphological development of bifurcations was investigated using a numerical modelling approach in Delft3D. An idealized bifurcation was built by splitting an upstream channel into two downstream branches. The idealized bifurcations were forced by river discharge from upstream and tides from downstream. To identify the effect of tides, two cases were studied, namely geometry difference (length and depth of channels) and tide difference (difference in prescribed tides at the two downstream channels).
The results show that an increased tidal influence compared to river influence results in a less asymmetric morphology of the bifurcation. This increased tidal influence can be achieved either by smaller river discharge or by asymmetric tides from downstream. The main mechanism is that tidal flows tend to be less asymmetric in the two downstream channels than tidally averaged flows. This causes the peak Shields number in the branches to be closer to each other with increasing influence of tides. Furthermore, we have shown that bedload transport tends to be divided less asymmetrically than suspended load due to the influence of lateral bed slopes, which tends to stabilize the system. In our simulations, bifurcations with increased tidal influence had a relatively high ratio of bedload over suspended load transport and therefore developed a less asymmetric morphology than in river-dominated systems. Our results can explain why tides tend to stabilize the bifurcations in deltas.
Initial (left panels) and final (right panels) depth near the
bifurcation for all simulations in Case 1. For the
Final depth for Case 2. For the
Sediment mobility (tide-averaged and maximum), mean flow, and tidal flow amplitude at the cross sections near the bifurcation as shown in Fig. 2 for all simulations. Main channel is the upstream channel, minor branch is the downstream channel that tends to be shallower, and major branch is the deepened downstream channel.
The model set-up for all Delft3D simulations is provided in the Supplement. The results presented were simulated using the Delft3D software package (Delft3D-flow version 4.01.00.rc.04).
The supplement related to this article is available online at:
API, MvdV, and MGK designed the study. API conducted the numerical modelling, performed the output analysis, and interpretation and wrote the article with major input from MvdV and MGK. MvdV conducted part of the output analysis and edited the paper.
The authors declare that they have no conflict of interest.
This research has been supported by the Indonesia Endowment Fund for Education (LPDP) (grant no. 20161222029838).
This paper was edited by Patricia Wiberg and reviewed by John Shaw and one anonymous referee.