ESurfEarth Surface DynamicsESurfEarth Surf. Dynam.2196-632XCopernicus PublicationsGöttingen, Germany10.5194/esurf-6-989-2018Morphodynamic model of the lower Yellow River: flux or entrainment form for
sediment mass conservation?Morphodynamic model of the lower Yellow RiverAnChengeanchenge08@163.comhttps://orcid.org/0000-0003-1262-5326MoodieAndrew J.https://orcid.org/0000-0002-6745-036XMaHongboFuXudongxdfu@tsinghua.edu.cnhttps://orcid.org/0000-0003-0744-0546ZhangYuanfengNaitoKensukehttps://orcid.org/0000-0001-8683-0846ParkerGaryhttps://orcid.org/0000-0001-5973-5296Department of Hydraulic Engineering, State Key Laboratory of
Hydroscience and Engineering, Tsinghua University, Beijing, ChinaDepartment of Earth, Environmental and Planetary Sciences, Rice
University, Houston, TX, USAYellow River Institute of Hydraulic Research, Zhengzhou, Henan, ChinaDepartment of Civil and Environmental Engineering, Hydrosystems
Laboratory, University of Illinois, Urbana-Champaign, IL, USADepartment of
Geology, Hydrosystems Laboratory, University of Illinois, Urbana-Champaign,
IL, USAChenge An (anchenge08@163.com) and Xudong Fu (xdfu@tsinghua.edu.cn)6November201864989101012May201812June201817October201822October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://esurf.copernicus.org/articles/6/989/2018/esurf-6-989-2018.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/6/989/2018/esurf-6-989-2018.pdf
Sediment mass conservation is a key factor that constrains river
morphodynamic processes. In most models of river morphodynamics, sediment
mass conservation is described by the Exner equation, which may take various
forms depending on the problem in question. One of the most widely used
forms of the Exner equation is the flux-based formulation, in which the
conservation of bed material is related to the stream-wise gradient of the
sediment transport rate. An alternative form of the Exner equation, however,
is the entrainment-based formulation, in which the conservation of bed
material is related to the difference between the entrainment rate of bed
sediment into suspension and the deposition rate of suspended sediment onto
the bed. Here we represent the flux form in terms of the local capacity
sediment transport rate and the entrainment form in terms of the local
capacity entrainment rate. In the flux form, sediment transport is a
function of local hydraulic conditions. However, the entrainment form does
not require this constraint: only the rate of entrainment into suspension is
in local equilibrium with hydraulic conditions, and the sediment transport
rate itself may lag in space and time behind the changing flow conditions.
In modeling the fine-grained lower Yellow River, it is usual to treat
sediment conservation in terms of an entrainment (nonequilibrium) form
rather than a flux (equilibrium) form, in consideration of the condition
that fine-grained sediment may be entrained at one place but deposited only at
some distant location downstream. However, the differences in prediction
between the two formulations have not been comprehensively studied to date.
Here we study this problem by comparing the results predicted by both the
flux form and the entrainment form of the Exner equation under conditions
simplified from the lower Yellow River (i.e., a significant reduction of
sediment supply after the closure of the Xiaolangdi Dam). We use a
one-dimensional morphodynamic model and sediment transport equations
specifically adapted for the lower Yellow River. We find that in a treatment
of a 200 km reach using a single characteristic bed sediment size, there is
little difference between the two forms since the corresponding adaptation
length is relatively small. However, a consideration of sediment mixtures
shows that the two forms give very different patterns of grain sorting:
clear kinematic waves occur in the flux form but are diffused out in the
entrainment form. Both numerical simulation and mathematical analysis show
that the morphodynamic processes predicted by the entrainment form are
sensitive to sediment fall velocity. We suggest that the entrainment form of
the Exner equation might be required when the sorting process of
fine-grained sediment is studied, especially when considering relatively
short timescales.
Introduction
Models of river morphodynamics often consist of three elements: (1) a
treatment of flow hydraulics; (2) a formulation relating sediment transport
to flow hydraulics; and (3) a description of sediment conservation. In the
case of unidirectional river flow, the Exner equation of sediment
conservation has usually been described in terms of a flux-based form in
which temporal bed elevation change is related to the stream-wise gradient of
the sediment transport rate. That is, bed elevation change is related to
∂qs/∂x, where qs is the total volumetric sediment
transport rate per unit width and x is the stream-wise coordinate (Exner, 1920;
Parker et al., 2004). This formulation is also referred to as the
equilibrium formulation, since it considers sediment transport to be at
local equilibrium; i.e., qs equals its sediment transport capacity
qse, as defined by the sediment transport rate associated with local
hydraulic conditions (e.g., bed shear stress, flow velocity, stream power,
etc.) regardless of the variation in flow conditions. Under this
assumption, sediment transport relations developed under equilibrium flow
conditions (e.g., Meyer-Peter and Müller, 1948; Engelund and Hansen,
1967; Brownlie, 1981) can be incorporated directly in such a formulation to
calculate qs, which is related to one or more flow parameters such as
bed shear stress.
An alternative formulation, however, is available in terms of an
entrainment-based form of the Exner equation, in which bed elevation
variation is related to the difference between the entrainment rate of bed
sediment into the flow and the deposition rate of sediment on the bed
(Parker, 2004). The basic idea of the entrainment formulation can be traced
back to Einstein's (1937) pioneering work on bed load transport and has been
developed since then by numerous researchers so as to treat either bed load
or suspended load (Tsujimoto, 1978; Armanini and Di Silvio, 1988; Parker et
al., 2000; Wu and Wang, 2008; Guan et al., 2015). Such a formulation differs
from the flux formulation in that the flux formulation is based on the local
capacity sediment transport rate, whereas the entrainment formulation is
based on the local capacity entrainment rate into suspension. In the
entrainment form, the difference between the local entrainment rate from the
bed and the local deposition rate onto the bed determines the rate of bed
aggradation–degradation and concomitantly the rate of loss–gain of sediment
in motion in the water column. Therefore, the sediment transport rate is no
longer assumed to be in an equilibrium transport state, but may exhibit lags
in space and time after changing flow conditions. The entrainment
formulation is also referred to as the nonequilibrium formulation (Armanini
and Di Silvio, 1988; Wu and Wang, 2008; Zhang et al., 2013).
To describe the lag effects between sediment transport and flow conditions,
the concept of an adaptation length–time is widely applied. This length–time
characterizes the distance–time for sediment transport to reach its
equilibrium state (i.e., transport capacity). Using the concept of the
adaptation length, the entrainment form of the Exner equation can be recast
into a first-order “reaction” equation, in which the deformation term is
related to the difference between the actual and equilibrium sediment
transport rates, as mediated by an adaptation length (which can also be
recast as an adaptation time) (Bell and Sutherland, 1983; Armanini and Di
Silvio, 1988; Wu and Wang, 2008; Minh Duc and Rodi, 2008; El kadi
Abderrezzak and Paquier, 2009). The adaptation length is thus an important
parameter for bed evolution under nonequilibrium sediment transport
conditions, and various estimates have been proposed. For suspended load,
the adaptation length is typically calculated as a function of flow depth,
flow velocity, and sediment fall velocity (Armanini and Di Silvio, 1988; Wu
et al., 2004; Wu and Wang, 2008; Dorrell and Hogg, 2012; Zhang et al.,
2013). The adaptation length of bed load, on the other hand, has been related
to a wide range of parameters, including the sediment grain size (Armanini
and Di Silvio, 1988), the saltation step length (Phillips and Sutherland,
1989), the dimensions of particle diffusivity (Bohorquez and Ancey, 2016),
the length of dunes (Wu et al., 2004), and the magnitude of a scour hole
formed downstream of an inerodible reach (Bell and Sutherland, 1983). For
simplicity, the adaptation length can also be specified as a calibration
parameter in river morphodynamic models (El kadi Abderrezzak and Paquier,
2009; Zhang and Duan, 2011). Nonetheless, no comprehensive definition of
adaptation length exists.
In this paper we apply the two forms of the Exner equation mentioned above
to the lower Yellow River (LYR) in China. The LYR describes the river
section between Tiexie and the river mouth and has a total length of about
800 km. Figure 1a shows a sketch of the LYR along with six major gauging
stations and the Xiaolangdi Dam, which is 26 km upstream of Tiexie. The LYR
has an exceptionally high sediment concentration (Ma et al., 2017),
historically exporting more than 1 Gt of sediment per year with only 49 billion tons of water, leading to a sediment concentration an order of
magnitude higher than most other large lowland rivers worldwide (Milliman
and Meade, 1983; Ma et al., 2017; Naito et al., 2018). However, the LYR has seen a substantial reduction in its sediment
load in recent decades, especially since the operation of the Xiaolangdi Dam beginning in
1999 (Fig. 1b), because most of its sediment load is derived from the
Loess Plateau, which is upstream of the reservoir (Wang et al., 2016; Naito
et al., 2018). Finally, the bed surface material of
the LYR is very fine, as low as 15 µm. This is much finer than
the conventional cutoff of wash load (62.5 µm) employed for sediment
transport in most sand-bed rivers (National Research Council, 2007; Ma et
al., 2017).
(a) Sketch of the lower Yellow River showing six major gauging stations
and the Xiaolangdi Dam. (b) Annual sediment load of the LYR measured at
three
gauging stations since 1950. (c) Grain size distributions of both bed
surface material and suspended load measured at six gauging stations of the
LYR.
When modeling the high-concentration and fine-grained LYR, it is common to
treat sediment conservation in terms of an entrainment-based rather than a
flux-based formulation. This is because many Chinese researchers view the
entrainment formulation as more physically based, as it is capable of
describing the behavior of fine-grained sediment, which when entrained at
one place may be deposited at some distant location downstream (Zhang et
al., 2001; Ni et al., 2004; Cao et al., 2006; He et al., 2012; Guo et al.,
2008). However, the entrainment formulation is more computationally
expensive and more complex to implement. Because the differences in
prediction between the two formulations do not appear to have been studied
in a systematic way, here we pose our central questions. Under what
conditions is it valid to use the entrainment form of the Exner equation,
and under what conditions may the flux form be used? Or more specifically,
which form of the Exner equation is most suitable for the LYR?
Here we study this problem by comparing the results of flux-based and
entrainment-based morphodynamics under conditions typical of the LYR. The
organization of this paper is as follows. The numerical model is described
in Sect. 2. In Sect. 3, the model is implemented to predict the
morphodynamics of the LYR with a sudden reduction of sediment supply, which
serves to mimic the effect of the Xiaolangdi Dam. We find that the two forms of
the Exner equation give similar predictions in the case of uniform sediment,
but show different sorting patterns in the case of sediment mixtures. In
Sect. 4, we conduct a mathematical analysis to explain the results in
Sect. 3; more specifically, we quantify the effects of varied sediment
fall velocity in the simulations. Finally, we summarize our conclusions in
Sect. 5.
Model formulation
In this paper, we present a one-dimensional morphodynamic model for the
lower Yellow River. The fully unsteady Saint–Venant equations are
implemented for the hydraulic calculation. Both the flux form and the
entrainment form of the Exner equation are implemented in the model for
sediment mass conservation. For each form of the Exner equation, we consider
both the cases of uniform sediment (bed material characterized by a single
grain size) and sediment mixtures. Since the sediment is very fine in the
LYR, the component of the load that is bed load is likely negligible (e.g., Ma
et al., 2017), so we consider only the transport of suspended load.
Considering the fact that most accepted sediment transport relations (e.g.,
the Engelund and Hansen, 1967, relation) underpredict the sediment transport
rate of the LYR by an order of magnitude or more (Ma et al., 2017), in our
model we implement two recently developed generalized versions of the
Engelund–Hansen relation, which are based on data from the LYR. These are the
version of Ma et al. (2017) for uniform sediment and the version of Naito
et al. (2018) for sediment mixtures. In cases
considering sediment mixtures, we also implement the method of Viparelli et al. (2010) to store and access bed stratigraphy as the bed aggrades and
degrades.
Since the aim of this paper is to compare the two formulations of the Exner
equation in the context of the LYR rather than reproduce site-specific
morphodynamic processes of the LYR, some additional simplifications are
introduced to the model to facilitate comparison. The channel is simplified
to be a constant-width rectangular channel, and bank (sidewall) effects and
floodplain interactions are not considered. The channel bed is assumed to be
an infinitely deep supplier of erodible sediment with no exposed bedrock,
which is justifiable because the LYR is fully alluvial and has been
aggrading for thousands of years, as copiously documented in Chinese
history. Finally, water and sediment (of each grain size range) are fed into
the upstream boundary at a specified rate, and at the downstream end of the
channel we specify a fixed bed elevation along with a normal flow depth.
These restrictions could be easily relaxed so as to incorporate
site-specific complexities of the Yellow River. Because of the severe
aggradation of the LYR developed before the Xiaolangdi Dam operation, the
LYR is famous for its hanging bed (i.e., bed elevated well above the
floodplain) and no major tributaries need be considered in the simulation.
Flow hydraulics
Flow hydraulics in a rectangular channel are described by the following
1-D
Saint–Venant equations, which consider fluid mass and momentum conservation,
1If∂h∂t+∂qw∂x=01If∂qw∂t+∂∂xqw2h+12gh2=ghS-Cfu2Cf=Cz-2
where t is time, h is water depth, qw is flow discharge per unit width,
g is gravitational acceleration, S is bed slope, u is depth-averaged flow
velocity, Cf is dimensionless bed resistance coefficient, and Cz is
the dimensionless Chézy resistance coefficient. In our model, the fully
unsteady 1-D Saint–Venant equations are solved using a Godunov-type scheme
with the HLL (Harten–Lax–van Leer) approximate Riemann solver (Harten et
al., 1983; Toro, 2001), which can effectively capture discontinuities in
unsteady and nonuniform open channel flows.
In this paper, the full flood hydrograph of the LYR is replaced by a flood
intermittency factor If (Paola et al., 1992; Parker, 2004). According to
this definition, the river is assumed to be at low flow and not transporting
significant amounts of sediment for time fraction 1-If and is in
flood at constant discharge and active morphodynamically for time fraction
If. In the long term, the relation between the flood timescale tf
and the actual timescale t is tf=Ift. With
the consideration that a river is in flood only for a fraction of time, here
we introduce If into the time derivative of all governing equations so
that the flood timescale tf is implemented in the simulation. This
notwithstanding, the results we exhibit later in this paper are all cast in
terms of actual timescale t. Full hydrographs are considered in the
Supplement.
Flux form of the Exner equation
When dealing with uniform sediment, the flux form of the Exner equation can
be written as
1If1-λp∂zb∂t=-∂qs∂x,
where λp is the porosity of the bed deposit, and zb is bed
elevation. Sediment transport is regarded to be in a quasi-equilibrium
state so that the sediment transport rate per unit width qs equals the
equilibrium (capacity) sediment transport rate per unit width qse.
When considering sediment mixtures, an active layer formulation (Hirano,
1971; Parker, 2004) is incorporated in the flux-based Exner equation so
that the evolution of both bed elevation and surface grain size distribution
can be considered. In this formulation, the riverbed is divided into a
well-mixed upper active layer and a lower substrate with vertical
stratigraphic variations. The upper active layer therefore represents the
volume of sediment that interacts directly with suspended load transport
and also exchanges with the substrate as the bed aggrades and degrades.
Discretizing the grain size distribution into n ranges, the mass conservation
relation for each grain size range can be written as
1If1-λpfIi∂∂tzb-La+∂∂tFiLa=-∂qsi∂x,
where qsi is volumetric sediment transport rate per unit width of the
ith grain size range (taken to be equal to its equilibrium value
qsei in the flux formulation), Fi is the volumetric fraction of
surface material in the ith grain size range, fIi is the volumetric fraction
of material in the ith grain size range exchanged across the
surface–substrate interface as the bed aggrades or degrades, and La is
the thickness of the active layer. For bedform-dominated sand-bed rivers,
La is often related to the height of dunes (Blom, 2008) so that the
vertical sorting processes due to bedform migration can be considered. In
this paper, a constant value of La is implemented in the simulation.
Summing Eq. (5) over all grain size ranges, one can find that the governing
equation for bed elevation in the case of sediment mixtures is the same as Eq. (4) upon replacing
qs with qsT=Σqsi, where
qsT denotes the total sediment transport rate per unit width summed over
all size ranges. Reducing Eq. (5) with Eq. (4), we get
1If1-λpLa∂Fi∂t+Fi-fIi∂La∂t=fIi∂qsT∂x-∂qsi∂x.
Therefore, the flux formulation Eqs. (4) and (6) are implemented as
governing equations for sediment mixtures, with Eq. (4) describing the
evolution of bed elevation and Eq. (6) describing the evolution of surface
grain size distribution. The exchange fractions fIi between the active
layer and the substrate are calculated using the following closure relation.
fIi=fi|zb-La∂zb-La∂t<0αFi+1-αpsi∂zb-La∂t>0
That is, the substrate is transferred into the active layer during
degradation, and a mixture of suspended load and active layer material is
transferred into substrate during aggradation. In Eq. (7), fizb-La is the volumetric fraction of substrate material
just beneath the interface, psi=qsi/qsT is the fraction of
bed material load in the ith grain size range, and α is a specified
parameter between 0 and 1. The formulation is adapted from Hoey and Ferguson (1994) and Toro-Escobar et al. (1996), who originally used it for bed load.
In this paper, a value of 0.5 is specified for α.
The method of Viparelli et al. (2010) is applied in our model to store
substrate stratigraphy and provide information for fizb-La (i.e., the topmost sublayer in Viparelli et al.,
2010). The reader can refer to the original reference of Viparelli et al. (2010) for more details or refer to An et al. (2017) for a concise
description of how to implement this method in a morphodynamic model.
When solving the flux form of the Exner equation, a first-order upwind
scheme is implemented to discretize the spatial derivatives, and a
first-order explicit scheme is implemented to discretize the temporal
derivatives.
Entrainment form of the Exner equation
The entrainment-based Exner equation for uniform sediment is
1If1-λp∂zb∂t=-vsE-r0C.
In Eq. (8), vs is the fall velocity of sediment particles; E is the
dimensionless entrainment rate of sediment normalized by sediment fall
velocity; C is the depth-flux-averaged volume sediment concentration; and
ro=cb/C is the recovery coefficient of suspended load, which
denotes the ratio between the near-bed sediment concentration cb and the
flux-averaged sediment concentration C. By definition, r0 is related to
the concentration profile of suspended load and is expected to be no less
than unity in cases appropriate for a depth-averaged shallow-water treatment
of flow and morphodynamics. Therefore, the first term on the right-hand side
of Eq. (8), i.e., vs⋅E, denotes the sediment entrainment rate per
unit area; the second term on the right-hand side of Eq. (8), i.e.,
vs⋅r0⋅C, denotes the sediment deposition rate per unit
area.
For the sediment fall velocity vs, we compare two widely used relations:
the relation of Dietrich (1982) and the relation of Ferguson and Church (2004). Results show that these two relations give almost the same fall
velocity for the bed material load of the LYR, whose grain sizes typically fall
in the range of 15 to 500 µm. Therefore, only the relation of
Dietrich (1982) is implemented in our simulations in this paper. Readers can
refer to Sect. S1 of the Supplement for more details.
In the entrainment formulation the sediment transport rate qs is not
necessarily in its equilibrium state, but the dimensionless entrainment rate
E is taken to be at capacity. The sediment transport rate qs is
calculated according to the following continuity relation.
qs=huC
For the dimensionless entrainment rate E, we assume that sediment transport
reaches its equilibrium state (qs=qse) when the sediment
deposition rate and the sediment entrainment rate balance each other
(r0C=E). Therefore, E can be back-calculated from qse as
E=r0qseqw.
For the depth-flux-averaged sediment concentration C, another equation is
implemented describing the conservation of suspended sediment in the water
column.
1If∂hC∂t+∂huC∂x=vsE-r0C
The entrainment-form Exner equation for sediment mixtures also uses the
active layer formulation described in Sect. 2.2. Mass conservation of each
grain size range can be written as
1If1-λpfIi∂∂tzb-La+∂∂tFiLa=-vsiEi-r0iCi,Ei=r0iqseiqw,
where the subscript i denotes the ith size range of sediment grain size.
Summing Eq. (12) over all grain size ranges, we get the governing equation
for bed elevation.
1If1-λp∂zb∂t=-∑j=1nvsjEj-r0jCj
Reducing Eq. (12) with Eq. (14) we get the governing equation for surface
fraction Fi.
1If1-λpLa∂Fi∂t+Fi-fIi∂La∂t=fIi∑j=1nvsjEj-r0jCj-vsiEi-r0iCi
The governing equation for the sediment concentration of each grain size
Ci can be written as
1If∂hCi∂t+∂huCi∂x=vsiEi-r0Ci,
and the sediment transport rate per unit width for the ith size range
qsi obeys the following continuity relation:
qsi=huCi.
In the entrainment formulation, the closure relation for fIi is the same
as that used in the flux formulation (i.e., Eq. 7), and the substrate
stratigraphy is also stored and accessed using the method of Viparelli et al. (2010). When discretizing the entrainment form of the Exner equation, a
first-order upwind scheme is implemented for the spatial derivatives, and
a first-order explicit scheme is implemented for the temporal derivatives.
Sediment transport relationUniform sediment
To close the Exner equations described in Sect. 2.2 and 2.3, equations
for equilibrium sediment transport rate qse (qsei) are still needed.
For the simulations using uniform sediment, we implement the generalized
Engelund–Hansen relation proposed by Ma et al. (2017). This equation is
based on data from the LYR and can be written in the following dimensionless
form:
qs∗=αsCfτ∗ns,
where qs∗ is the dimensionless sediment transport rate per unit
width (i.e., the Einstein number), and τ∗ is dimensionless
shear stress (i.e., the Shields number). They are defined as
qs∗=qseRgDD,τ∗=τbρRgD,τb=ρCfu2,
where D is the characteristic grain size of the bed sediment (here
approximated as uniform); τb is bed shear stress; and R is the
submerged specific gravity of sediment defined as (ρs-ρ)/ρ, in which ρs is the density of sediment, and ρ is
the density of water. The sediment submerged specific gravity R is specified as
1.65 in this paper, which is an appropriate estimate for natural rivers and
corresponds to quartz.
In the relation of Ma et al. (2017), the dimensionless coefficient αs=0.9
and the dimensionless exponent ns=1.68. These values
are quite different from the original relation of Engelund and Hansen (1967), in which αs=0.05 and
ns=2.5. Ma et al. (2017) demonstrated that such differences imply that the riverbed of the LYR
is dominated by low-amplitude bedform features (dunes) approaching
the upper-regime plane bed. According to this finding, form drag is then
neglected in our modeling, and all of the bed shear stress is used for
sediment transport.
Sediment mixtures
We implement the relation of Naito et al. (2018) to
calculate the equilibrium sediment transport rate of size mixtures. Using
field data from the LYR, Naito et al. (2018)
extended the Engelund and Hansen (1967) relation to a surface-based
grain-size-specific form, in which the suspended load transport rate of the
ith size range is tied to the availability of this size range on the bed
surface:
qsei=Ni∗Fiu∗3RgCf,
where Ni∗ is the dimensionless sediment transport rate in the
ith size range, and u∗ is shear velocity calculated from the bed
shear stress τb.
u∗=τbρ
The transport relation itself takes the form
Ni∗=Aiτg∗DsgDiBi,
in which Di is the characteristic grain size for sediment in the ith
size range, Dsg is the geometric mean grain size in the active layer,
and τg∗ is the dimensionless bed shear stress associated
with Dsg. The parameters τg∗, coefficient Ai,
and exponent Bi are calculated as follows.
τg∗=τbρRgDsgAi=0.46DiDsg-0.84Bi=0.35DiDsg-1.16
If Ai and Bi are specified as constant values in Eq. (24), then the
sediment transport rate for each size range depends only on the flow shear
stress and the characteristic grain size of this size range, without being
affected by other size ranges. But according to Eqs. (26) and (27), the
coarser the sediment the smaller the values of Ai and Bi will be,
thus leading to reduced mobility for coarse sediment (and increased mobility
for fine sediment) due to the presence of grains of other sizes. Thus the
relations in Eqs. (26) and (27) serve as a hiding function that allows for grain
sorting.
We note that a form of the Engelund–Hansen equation for mixtures was
introduced by Van der Scheer et al. (2002) and implemented by Blom et al. (2016). Blom et al. (2017) further extended this relation to a more general
framework capable of including hiding effects. These forms,
however, have not been calibrated to the LYR data and are thus not suitable
for the LYR.
Numerical modeling of the LYR using the two forms of the Exner
equation
In this section, we conduct numerical simulations using both the flux form
and the entrainment form of the Exner equation, with the aim of studying under
what circumstances the two forms give different predictions. Numerical
simulations are conducted in the setting of the LYR. We specify a 200 km
long channel reach for our simulations, along with a constant channel width
of 300 m and an initial longitudinal slope of 0.0001. Bed porosity
λp is specified as 0.4. Based on field measurements of the LYR
available to us, we implemented a dimensionless Chézy resistance coefficient
Cz of 30, which corresponds to a dimensionless bed resistance
coefficient Cf of 0.0011. For the entrainment form of the Exner equation, we
specify the ratio of near-bed sediment concentration to flux-averaged
sediment concentration r0(r0i)=1. Such a value of r0
(r0i) corresponds to a vertically uniform profile of sediment
concentration and will thus give a maximum difference between the
prediction of the entrainment form and the prediction of the flux form. More
discussion about the effects of r0 is presented in Sect. 4.3.
A constant flow discharge of 2000 m3 s-1 (corresponding to a flow discharge per unit width qw
of 6.67 m2 s-1) is introduced at the inlet of the channel with
the flood intermittency factor If estimated as 0.14 (Naito et
al., 2018). The downstream end is specified far from the river mouth to
neglect the effects of backwater. Therefore, the bed elevation is held
constant and the water depth is specified as the normal flow depth at the
downstream end of the calculational domain. The above flow discharge per unit
width qw combined with the bed slope S as well as the bed
resistance coefficient Cf leads to a normal flow depth of 3.69 m. In
our simulation, we use the height of bedforms in the LYR to determine the
thickness of the active layer (Blom, 2008). According to the field survey of
Ma et al. (2017), the characteristic height of bedforms in the LYR is about
20 % of the normal flow depth, which can fall in the range suggested by
the data analysis of Bradley and Venditti (2017). This eventually leads to an
estimate of active layer thickness of La=0.738 m. The sublayer
in the substrate to store the vertical stratigraphy is specified with a
thickness of 0.5 m.
Two cases are considered here. In the first case, the sediment grain size
distribution of the LYR is simplified to a uniform grain size of 65 µm.
This is based on the measured grain size distribution of bed material at the
Lijin gauging station, which has a median grain size of D50=66.6µm, a geometric mean grain size of Dg=65.5µm, and a
geometric standard deviation σg=2.0, as shown in Fig. 1c.
In the second case, we consider the effects of sediment mixtures. The grain
size distribution of the initial bed is based on the bed material at the
Lijin gauging station, as shown in Fig. 1c, but we renormalize the
measured grain size distribution with a cutoff for wash load at 15 µm as
suggested by Ma et al. (2017). The renormalized grain size distribution for
the initial bed as implemented in the case of sediment mixtures is shown in
Fig. 2, with a total number of grain size fractions of 5. In both
cases, simulations start with an equilibrium state in which sediment supply
rate, sediment transport rate, and equilibrium sediment transport rate are
the same so that the initial state of the channel is in equilibrium. Then
we cut the sediment supply rate (of each size range) to only 10 % of the
equilibrium sediment transport rate and keep this sediment supply rate. This
is to mimic the reduction of sediment load in the LYR in recent years, as
shown in Fig. 1b. The grain size distribution of sediment supply in the
case of sediment mixtures is shown in Fig. 2.
The 200 km channel reach is discretized into 401 cells, with cell size
Δx of 500 m. In the case of uniform sediment, we specify a time step
for morphologic calculation Δtm=10-4 years and a time
step for hydraulic calculation Δth=10-6 years. In the
case of sediment mixtures, we specify a time step for morphologic
calculation Δtm=10-5 years and a time step for
hydraulic calculation Δth=10-6 years. Computational
conditions are briefly summarized in Table 1. The computational conditions
we implement are much simpler than the rather complicated conditions of the
actual LYR. But it should be noted that the aim of this paper is not to
reproduce specific aspects of the morphodynamic processes of the LYR, but to
compare the flux form and entrainment form of the Exner equation in the context
of conditions typical of the LYR.
Grain size distributions of both the initial bed and the sediment
supply in the case of sediment mixtures. For the initial bed, the surface
and substrate grain size distributions are the same. The grain size
distribution of the initial bed is renormalized based on the field data at
the Lijin gauging station. The grain size distribution of the sediment
supply equals the grain size distribution of bed material load at
equilibrium. Grain sizes in the range of wash load have been removed from
both distributions.
Summary of computational conditions for numerical modeling of the
LYR.
ParameterValueChannel length L200 kmChannel width B300 mInitial slope SI0.0001Dimensionless Chézy resistance coefficient Cz30Flow discharge per unit width qw6.67 m2 s-1Flood intermittency factor If0.14Ratio of near-bed concentration to average concentration r0(r0i)1Characteristic grain size in the case of uniform sediment65 µmSubmerged specific gravity of sediment R1.65Porosity of bed deposits λp0.4Cell size Δx500 mTime step for morphologic calculation Δtm10-4 years (uniform sediment) 10-5 years (sediment mixtures)Time step for hydraulic calculation Δth10-6 yearsCase of uniform sediment
In this case, we implement a uniform grain size of 65 µm for both the
bed material and sediment supply. Such a grain size is nearly equal to the
observed median grain size (or geometric mean grain size) of bed material at
the Lijin gauging station. The relation of Ma et al. (2017) is implemented to
calculate the transport rate of bed material suspended load. This relation
provides an equilibrium sediment transport rate per unit width qse of
0.0136 m2 s-1 under the given flow discharge, bed slope, and sediment
grain size. With a flood intermittency factor If of 0.14, this further
gives a mean annual bed material load of 47.8 Mt a-1. Adding in wash load
according to the estimate of Naito et al. (2018),
the total mean annual load is 86.9 Mt a-1, a value that is of the same order of
magnitude as averages over the period 2000–2016 (89–126 Mt a-1 depending on
the site), i.e., since the operation of the Xiaolangdi Dam beginning in 1999 (Fig. 1b). The
sediment supply rate qsf we specify at the upstream end of the channel
is only 10 % of the equilibrium sediment transport rate (i.e., the sediment
supply rate is cut by 90 % from the equilibrium state) such that
qsf=0.00136 m2 s-1.
Figure 3 shows the modeling results using the flux form of the Exner
equation. As we can see in the figure, the bed degrades and the sediment
load decreases in response to the cutoff of sediment supply. Such
adjustments start from the upstream end of the channel and gradually migrate
downstream. Figure 4 shows the modeling results using the entrainment form
of the Exner equation. A comparison between Figs. 4 and 3 shows that the
entrainment form and the flux form give very similar predictions in this
case. The entrainment form provides a somewhat slower degradation (at the
upstream end the flux form predicts a 3 m degradation, whereas the
entrainment form predicts a 2.3 m degradation) and a more diffusive sediment
load reduction. Such more diffusive predictions of sediment load variation
can be ascribed to the condition of nonequilibrium transport that is
embedded in the entrainment form. This issue will be studied analytically in
Sect. 4. Here we present the results for only 0.2 years after the cutoff of
sediment supply, since the differences between the predictions of the two
forms tend to be the most evident shortly after the disruption but gradually
diminish as the river approaches the new equilibrium (El kadi Abderrezzak
and Paquier, 2009). Modeling results over a longer timescale will be
discussed in Sect. 4.3.
The 0.2-year results for the case of uniform sediment using the flux
form of the Exner equation: time variation of (a) bed elevation zb and water
surface (WS), (b) sediment load per unit width qs of the LYR in response
to the cutoff of sediment supply. The inset shows detailed results near the
upstream end.
The 0.2-year results for the case of uniform sediment using the
entrainment form of the Exner equation: time variation of (a) bed elevation
zb and water surface (WS), (b) sediment load per unit width qs of
the LYR in response to the cutoff of sediment supply. The inset shows
detailed results near the upstream end.
To further quantify the differences between the predictions of the two
forms, we propose the following normalized parameter:
δy=yE-yFyF×100%,
where y denotes an arbitrary variable calculated by the morphodynamic model,
and subscripts F and E denote results using the flux form and the entrainment
form, respectively. Therefore, δ (y) denotes the difference between
the prediction of the two forms yF and yE normalized by the prediction
of the flux form yF.
Table 2 gives a summary of the maximum values of δ along the channel
at different times in the case of uniform sediment. The values of δ
for both zb and qs are presented. As we can see from the table, the
maximum value of δ(zb) along the calculational domain stays
within 4 % in the first 0.2 years after the cutoff of sediment supply. This
indicates that the flux form and the entrainment form can indeed give almost
the same prediction in terms of bed elevation in this case. But in the case
of the sediment load per unit width qs, the maximum value of δ(qs) can be as high as 20 %, indicating that even though the two
forms give qualitatively similar patterns of evolution in terms of sediment
load as shown in Figs. 3 and 4, a quantitative difference is clearly evident
due to the more diffusive nature of the predictions of the entrainment form.
The value of δ(qs) is largest at the beginning of the
simulation and is then gradually reduced with time. It should be noted that
the values of δ(zb) depend on the choice of elevation datum. In
this paper bed elevation at the downstream end is fixed as 0 m, which serves
as the elevation datum. In the simulation of this paper, the maximum value
of δ(zb) almost always occurs at the upstream end where bed
elevation does not deviate far from the initial value of 20 m.
Quantification of the difference between predictions of the flux
form and the entrainment form in the case of uniform sediment. The maximum
values of δ(zb) and δ(qs) in the calculational
domain are presented every 0.04 years.
The above results show that the flux form and the entrainment form can
provide similar predictions of the LYR when the bed sediment grain size
distribution is simplified to a uniform value of 65 µm. To understand
under what conditions the two forms will lead to more different results, we
conduct an idealized run using the entrainment form in which the sediment
fall velocity vs is arbitrarily multiplied by a factor of 0.05. That is
to say, we keep the sediment grain size at 65 µm in the computation of
the Shields number, but let the sediment fall velocity in Eqs. (8) and (10)
equal only 1/20 of the value calculated by the relation of Dietrich (1982)
from this grain size. With a much smaller, and indeed intentionally
unrealistic, sediment fall velocity the entrainment form predicts very
different results as shown in Fig. 5. The adjustment of the sediment load
becomes even more diffusive in space: it takes almost the entire 200 km reach
for the sediment load to adjust from the upstream disruption to the
equilibrium transport rate. Meanwhile, there is barely any bed degradation
at the upstream end after 0.2 years, in correspondence with the fact that the
spatial gradient of qs becomes quite small. In Table 2 we also exhibit
the δ values for this idealized run. It is no surprise that both
δ(zb) and δ(qs) are high, as the entrainment form
and flux form predict very different patterns with such an arbitrarily
reduced sediment fall velocity.
The 0.2-year results for the case of uniform sediment using the
entrainment form of the Exner equation: time variation of (a) bed elevation
zb and water surface (WS), (b) sediment load per unit width qs of
the LYR in response to the cutoff of sediment supply. Sediment fall velocity
vs is arbitrarily multiplied by a factor of 0.05 while holding bed grain
size constant in this run. The inset shows detailed results near the
upstream end.
In Sect. S2, we also conduct numerical simulations with
hydrographs. Results indicate that our conclusions based on constant flow
discharge also hold when hydrographs are considered: the flux form and the
entrainment form (with the sediment fall velocity not adjusted) of the Exner
equation give very similar predictions using a characteristic grain size of
65 µm.
Case of sediment mixtures
In this section we consider the morphodynamics of sediment mixtures rather
than the case of a uniform bed grain size implemented in Sect. 3.1. The
grain size distribution of the initial bed is based on field data at the
Lijin gauging station and is shown in Fig. 2. Using the sediment transport
relation of Naito et al. (2018) for mixtures, such a
grain size distribution combined with the given bed slope and flow discharge
leads to a total equilibrium sediment transport rate per unit width
qseT of 0.0272 m2 s-1. With a flood intermittency factor If of
0.14, this further gives a mean annual bed material load of 95.5 Mt a-1.
Adding in wash load according to the estimate of Naito et al. (2018), the total mean annual load is 173.7 Mt a-1, a value that is of
the same order of magnitude as averages over the period 2000–2016 (89–126 Mt a-1 depending on the site), i.e.,
since the operation of the
Xiaolangdi Dam beginning in 1999
(Fig. 1b). The sediment supply rate of each grain size range is set at
10 % of its equilibrium sediment transport rate. This results in a total
sediment supply rate of qsf=0.00272 m2 s-1 and a grain size
distribution of the sediment supply (shown in Fig. 2) that is identical to
the grain size distribution of the equilibrium sediment load before the
cutoff. That is, the grain size distribution of sediment supply does not
change; only the total sediment supply is reduced by 90 %. Again we
exhibit simulation results for only 0.2 years here, a value that is enough to
show the differences between the two forms, flux and entrainment, as applied
to mixtures. Modeling results over a longer timescale are presented in
Sect. 4.3.
Figure 6 shows the simulation results using the flux form of the Exner
equation. As a result of the reduced sediment supply at the inlet, bed
degradation occurs first at the upstream end and then gradually migrates
downstream. The total sediment transport rate per unit width qsT is also
reduced as a response to the cutoff of sediment supply. More specifically,
the evolution of qsT shows marked evidence of advection, with at least
two kinematic waves being observed within 0.2 years. Actually, as illustrated
by Stecca et al. (2014, 2016), each grain size fraction should induce a
migrating wave. As shown in Fig. 6b, the fastest kinematic wave migrates
beyond the 200 km reach within 0.06 years, and the second fastest kinematic
wave migrates for a distance of about 60 km in 0.2 years. Figure 6c and d show the results for the surface geometric mean grain size Dsg and
geometric mean grain size of suspended load Dlg, respectively. As can be
seen therein, both the bed surface and the suspended load coarsen as a
result of the cutoff of sediment supply. This represents armoring mediated
by the hiding functions of Eqs. (26) and (27). Such coarsening is not
evident near the upstream end, possibly due to the inverse slope visible in
Fig. 6a. Similarly to the variation of qsT, the patterns of time
variation of both Dsg and Dlg also exhibit very clear kinematic
waves, with migration rates about the same as those of qsT.
The 0.2-year results for the case of sediment mixtures using the flux
form of the Exner equation: time variation of (a) bed elevation zb and water
surface (WS), (b) total sediment load qsT, (c) surface geometric mean
grain size Dsg, and (d) geometric mean grain size of sediment load of the
LYR in response to the cutoff of sediment supply. The inset shows detailed
results near the upstream end.
Figure 7 shows the simulation results obtained using the entrainment form of
the Exner equation. In general, the patterns of variation predicted by the
entrainment form have similar trends and magnitudes to those predicted by
the flux form: the bed degrades near the upstream end, the suspended load
transport rate is reduced in time, and both the bed surface and the suspended
load coarsen as a result of the cutoff of sediment supply. But the results
based on the two forms exhibit very evident differences when multiple grain
sizes are included. That is, the results predicted by the entrainment form
are sufficiently diffusive so that the variations of qsT, Dsg, and
Dlg (Fig. 7b, c, and d) do not show the advective character seen
in Fig. 6. Figure 7c, however, shows the same armoring as in the case of
calculations with the flux form. No clear kinematic waves can be observed in
Fig. 7. Table 3 gives a summary of the values of δ in the case of
sediment mixtures. The prediction of bed elevation is not affected much when
multiple grain sizes are considered, with δ(zb) being no more than 3.5 % within 0.2 years.
The δ values of qsT, Dsg, and Dlg are, however,
relatively large since the two forms predict quite different patterns of
variations, as shown in Figs. 6 and 7.
The 0.2-year results for the case of sediment mixtures using the
entrainment form of the Exner equation: time variation of (a) bed elevation
zb and water surface (WS), (b) total sediment load qsT, (c) surface
geometric mean grain size Dsg, and (d) geometric mean grain size of
sediment load of the LYR in response to the cutoff of sediment supply. The
inset shows detailed results near the upstream end.
Quantification of the difference between predictions of the flux
form and the entrainment form in the case of sediment mixtures. The maximum
values of δ in the calculational domain are presented at different
times.
The results shown in Fig. 8 have also been calculated using the entrainment
form of the Exner equation, but here the sediment fall velocities vsi
used in Eqs. (14)–(16) are arbitrarily multiplied by a factor of 20. That
is, we still apply the grain size distribution in Fig. 2, but the sediment
fall velocities implemented in the simulation are 20 times the corresponding
fall velocities calculated by the relation of Dietrich (1982). In the case
of uniform sediment in Sect. 3.1, we arbitrarily reduce the sediment fall
velocity to force a difference between the predictions from the entrainment
form and those from the flux form. Here we arbitrarily increase the sediment
fall velocity with the aim of determining under what conditions the sorting
patterns predicted by the two forms converge. As we can see in Fig. 8, with
such a larger and intentionally unrealistic sediment fall velocity, the
general trend of variations predicted by the entrainment form does not
change, but the results show a notably less diffusive pattern. The
variations of qsT, Dsg, and Dlg show more advection compared
with Fig. 7, and at least two kinematic waves appear within 0.2 years. It
should be noted that even though these kinematic waves appear after we
arbitrarily increase the sediment fall velocity, they are more diffusive
than those obtained from the flux formulation and also migrate with a slower
celerity compared with those predicted by the flux form, especially for
the fastest kinematic wave in the modeling results.
Table 3 summarizes the δ values for this run. The values of
δ(zb) become smaller with arbitrarily increased sediment fall
velocities except for t=0.06 years. A relatively large value of δ(zb) at t=0.06 years occurs near the downstream end of the channel,
where the entrainment form predicts some slight degradation. Also, δ(qsT) is quite large at t=0.01 years and 0.03 years, even though the
results for the case of increased fall velocities become qualitatively more
similar to the prediction of the flux form. This is because the flux form
and the entrainment form with arbitrarily increased sediment fall velocities
predict different celerities for the fastest kinematic wave. The error
δ(qsT) becomes smaller from t=0.06 years as the fastest
kinematic wave migrates beyond the channel reach. The error δ(Dlg) behaves similarly to δ(qsT), with δ(Dlg) being quite large at t=0.01 years and 0.03 years near the
fastest kinematic wave, but gradually becoming smaller as time passes. The
error δ(Dsg) stays low within the whole 0.2-year period,
possibly because the fastest kinematic wave of Dsg has a small
magnitude, as shown in Fig. 8c.
The 0.2-year results for the case of sediment mixtures using the
entrainment form of the Exner equation: time variation of (a) bed elevation
zb and water surface (WS), (b) total sediment load qsT, (c) surface
geometric mean grain size Dsg, and (d) geometric mean grain size of
sediment load of the LYR in response to the cutoff of sediment supply.
Sediment fall velocities vsi are arbitrarily multiplied by a factor of
20 in this run while keeping the grain sizes invariant. The inset shows
detailed results near the upstream end.
In Sect. S3, we present additional numerical cases that
are similar to the cases in this section, except that hydrographs are
implemented instead of constant discharge. Results indicate that our
conclusions based on constant flow discharge also hold when hydrographs are
considered. The flux form and the entrainment form (with the sediment fall
velocity not adjusted) of the Exner equation predict quite different
patterns of grain sorting, with the flux form exhibiting a more advective
character than the entrainment form.
DiscussionAdjustment of sediment load and the adaptation length
In Sect. 3.1, our simulation shows that in the case of uniform sediment,
the flux form and the entrainment form of the Exner equation give very
similar predictions for a given sediment size of 65 µm. However, if we
arbitrarily reduce the sediment fall velocity by a multiplicative factor of
0.05, the prediction given by the entrainment form will become much more
diffusive in terms of both zb and qs. The diffusive nature of the
entrainment form and the important role played by the sediment fall
velocity can be explained in terms of the governing equation.
In the entrainment form, the equation governing suspended sediment
concentration is
1If∂hC∂t+∂huC∂x=vsE-r0C,
i.e., the same as Eq. (11). The sediment transport rate per unit width
qs=huC=qwC, and the dimensionless entrainment rate E=r0qse/qw. In order to simplify the mathematical analysis, here
we consider only the adjustment of sediment concentration in space and
neglect the temporal derivative in Eq. (29) so that we get
∂qs∂x=vsE-r0C=1Ladqse-qs,Lad=qwvsr0,
where Lad can be identified as the adaptation length for suspended
sediment to reach equilibrium. This definition of adaptation length is
similar to those in Wu and Wang (2008) and Ganti et al. (2014).
If we consider the spatial adjustment of sediment load shortly after the
cutoff of sediment supply, we can further neglect the nonuniformity of the
capacity (equilibrium) transport rate qse along the channel, and Eq. (30) can be solved with a given upstream boundary condition. That is, with
the boundary condition
qsx=0=qsf,
Eq. (30) can be solved to yield
qs=qse+qsf-qsee-xLad.
Here qsf is the sediment supply rate per unit width at the upstream end.
According to Eq. (33), qs adjusts exponentially in space from qsf to
qse, which also coincides with our simulation results in Sect. 3.1, as
shown in Figs. 3–6. The adaptation length Lad is the key parameter that
controls the distance for qs to approach the equilibrium sediment
transport rate qse. More specifically, qs attains 1-1/e
(i.e.,
63.2 %) of its adjustment from qsf to qse over a distance
Lad. Therefore, the larger the adaptation length, the slower qs
adjusts in space so that the more evident lag effects and diffusivity are
exhibited in the entrainment form. In the flux form, however, the sediment
load responds simultaneously with the flow conditions so that Lad=0
and qs=qse along the entire channel reach.
For the case of uniform sediment in Sect. 3.1, qw=6.67 m2 s-1
and ro is specified as unity. Therefore, the value of Lad is
determined only by the sediment fall velocity vs. Figure 9 shows the
value of the adaptation length Lad for various sediment grain sizes,
with the sediment fall velocity vs calculated by the relation of
Dietrich (1982). From the figure we can see that Lad decreases sharply
with the increase in grain size, indicating that the lag effects between
sediment transport and flow conditions are evident for very fine sediment
but gradually disappear when sediment is sufficiently coarse. For the
sediment grain size of 65 µm implemented in Sect. 3.1, the
corresponding Lad=1.88 km, which is much smaller than the 200 km
reach of the computational domain. In this case and in general, the
predictions of the flux form and the entrainment form show little difference
when Lad/L≪1, where L is domain length. However, if we
arbitrarily multiply the sediment fall velocity by a factor of 0.05, then
Lad becomes 37.60 km. With such a large adaptation length, it is no
surprise that the entrainment form gives very different predictions from the
flux form.
The evolution of bed elevation zb can also be affected by the value of
Lad. For example, in the case of uniform sediment in Sect. 3.1, the
flux form corresponds to an adaption length of zero. As a result, the flux
form yields a spatial derivative of qs near the upstream end that is
relatively large, thus leading to fast degradation from the upstream end. In
the case of the entrainment form, however, the spatial derivative of
qs is small with a large Lad, thus leading to a slower and more
diffusive bed degradation. This is especially evident when we arbitrarily
reduce the sediment fall velocity by a factor of 0.05 while keeping grain
size invariant.
The above analysis also holds for sediment mixtures, except that each grain
size range will have its own adaptation length. Here we neglect the temporal
derivative in Eq. (29) and analyze only the spatial adjustment of sediment
load. If we neglect the spatial derivative in Eq. (29) and conduct a similar
analysis for sediment concentration, we would find that the temporal
adjustment of sediment concentration is also described by an exponential
function of time, in analogy to Eq. (33).
Patterns of grain sorting: advection vs. diffusion
In Sect. 3.2 we find that the flux form and entrainment form of the Exner
equation provide very different patterns of grain sorting for sediment
mixtures: kinematic sorting waves are evident in the flux form but are
diffused out in the entrainment form. The diffusivity of grain sorting
becomes smaller and the kinematic waves appear, however, if we arbitrarily
increase the sediment fall velocity by a factor of 20. In this section, we
explain this behavior by analyzing the governing equations.
First we rewrite the sediment transport relation of Naito et al. (2018) in the following form.
qsei=Fiqriqri=u∗3RgCfAiτg∗DgDiBi
Substituting Eq. (34) into Eq. (6), which is the governing equation for
surface fraction Fi in the flux form, we get
1If1-λpLa∂Fi∂t+Fi-fIi∂La∂t=fIi∂∑j=1nFjqrj∂x-∂Fiqri∂x.
Equation (36) can be written in the form of a kinematic wave equation with
source terms as follows.
∂Fi∂t+cFi∂Fi∂x=SFicFi=Ifqri1-λpLa1-fIiSFi=-IfFi1-fIi1-λpLa∂qri∂x+IffIi1-λpLa∂∑j=1n,j≠iFjqrj∂x-Fi-fIi1-λp∂La∂t
Here cFi is the ith celerity of a kinematic wave and SFi denotes
source terms. Since the surface geometric mean grain size Dsg, the total
sediment load per unit width qsT (which equals the equilibrium sediment
transport rate qseT), and the geometric mean grain size of sediment load
Dlg are all closely related to the surface grain size fractions
Fi, the evolution of these three parameters shows marked advective
behavior when simulated by the flux form of the Exner equation. However, the
evolution of bed elevation zb is related to ∂qsT/∂x, which is dominated by diffusion if qsT is
predominantly slope dependent (as is the case here). The advection–diffusion
character of the flux form of the Exner equation for sediment mixtures has been
documented thoroughly in a series of papers (e.g., Stecca et al., 2014, 2016; An et al., 2017). The reader can reference these papers
for more details.
Relation between adaptation length Lad and grain size D. The
values of flow discharge per unit width qw and recovery coefficient
r0 are the same as those in Sect. 3.1. The relation of Dietrich (1982)
is implemented for sediment fall velocity.
Now we turn to the entrainment form of the Exner equation. Combined with the
sediment transport rate per unit width qsi=huCi=qwCi and the dimensionless entrainment rate Ei=r0iqsei/qw, Eqs. (16) and (15) can be written as
1If∂qsiu∂t+∂qsi∂x=vsiroiqwqsei-qsi,1If1-λpLa∂Fi∂t+Fi-fIi∂La∂t=fIi∑j=1nvsjr0jqwqsej-qsj-vsir0iqwqsei-qsi,
where Eq. (40) denotes the conservation of suspended sediment and Eq. (41)
denotes the conservation of bed material. If we rewrite Eq. (40) in the
form
qsi=qsei-qwvsiroi1If∂qsiu∂t+∂qsi∂x,
then qsi can be solved iteratively. With an initial guess of qsi=qsei and neglecting the temporal derivatives, we obtain the second-order
solution of qsi as
qsi=qsei-qwvsiroi∂∂xqsei-qwvsiroi∂qsei∂x.
Details on the iteration scheme are given in Sect. S4.
Substituting Eqs. (43) and (34) into Eq. (41), we find that
1If1-λpLa∂Fi∂t+Fi-fIi∂La∂t=fIi∑j=1n∂∂xFjqrj-qwvsjroj∂Fjqrj∂x-∂∂xFiqri-qwvsiroi∂Fiqri∂x.
Expanding out the last two terms in Eq. (44) using the chain rule, after
some work the relation for the conservation of bed material can be expressed
as follows.
∂Fi∂t+cEi∂Fi∂x-νi∂2Fi∂x2=SEicEi=1-fIiIf1-λpLaqri-2qwvsir0i∂qri∂xνi=1-fIiIfqwqri1-λpLavsir0iSEi=IffIi1-λpLa∑j=1n,j≠i∂∂xFjqrj-qwvsjroj∂Fjqrj∂x-1-fIiIf1-λpLaFi∂qri∂x-qwvsiroiFi∂2qri∂x2-Fi-fIiLa∂La∂t
Here cEi is the celerity of a kinematic wave, νi is the
diffusivity coefficient, and SEi denotes source terms.
From Eq. (45) we can see that the governing equation for Fi in the
entrainment form is an advection–diffusion equation rather than the
kinematic wave equation of the flux form. The surface geometric mean grain
size Dsg is governed by Eq. (45), which describes the variation of the
surface fractions Fi from which it is computed. The equilibrium sediment
transport rate qsei is governed by Eq. (45) because we implement a
surface-based sediment transport relation as shown in Eq. (34). According to
Eq. (43), the total sediment load per unit width qsT and the geometric
mean grain size of sediment load Dlg must also be closely related to the
surface grain size fractions Fi. Therefore, the diffusion terms in Eq. (45) can lead to dissipation of the kinematic waves in
Fig. 7b, c, and d.
From Eq. (47), we can also see that the diffusivity coefficient νi is related to the sediment fall velocity vsi: the larger the
sediment fall velocity, the smaller the diffusivity coefficient. Thus when
we increase the sediment fall velocity arbitrarily by a factor of 20 in
Sect. 3.2, the kinematic waves become more evident as a result of the
reduction of diffusivity.
Moreover, if we compare the celerity of kinematic waves in both the flux form
and the entrainment form, we have
cEicFi=1-rci,rci=2Ladiqri∂qri∂x,
where Ladi is the adaptation length for the ith size range as defined by
Eq. (31). More specifically, the value of rci depends on ∂qri/∂x. For our numerical simulation in Sect. 3.2,
∂qri/∂x > 0 as a result of bed degradation
progressing from the upstream end, thus leading to a positive value of
rci and an entrainment celerity cEi that is smaller than the
corresponding flux celerity cFi. This is consistent with our numerical
results: the kinematic waves in Fig. 8 predicted by the entrainment form are
somewhat smaller than the kinematic waves in Fig. 6 predicted by the flux
form.
Modeling implications and limitations
In Sect. 3, two numerical cases are presented to compare the flux form and
the entrainment form of the Exner equation, but only within 0.2 years after
the cutoff of sediment supply. Here we run both numerical cases for a longer
time (5 years). Table 4 shows the results of the case of uniform sediment
(as described in Sect. 3.1) within 5 years, and Table 5 shows the results
of the case of sediment mixtures (as described in Sect. 3.2) within 5 years. For both cases, the δ values corresponding to relative
deviation between the flux and entrainment forms become quite small after 1 year, thus validating our assumption that the predictions of the two forms
tend to be most evident shortly after disruption, but gradually diminish
over a longer timescale. Moreover, if the water and sediment supply are
kept constant for a sufficiently long time, the flux form and entrainment
form of the Exner equation predict exactly the same equilibrium in terms of
both the channel slope and the bed surface texture. Under such conditions,
the sediment transport rate (of each size range) equals the equilibrium
sediment transport rate (of each size range) and also equals the
sediment supply rate (of each size range).
Quantification of the difference between predictions of the flux
form and the entrainment form in the case of uniform sediment. The maximum
δ values in the calculational domain are presented for 5 years.
Quantification of the difference between predictions of the flux
form and the entrainment form in the case of sediment mixtures. The maximum
δ values in the calculational domain are presented for 5 years.
Based on the numerical modeling and mathematical analysis in this paper, we
suggest that the entrainment form of the Exner equation be used when
studying the river morphodynamics of fine-grained sediment (or, more
specifically, sediment with small fall velocity). This is because the
adaptation length La and the diffusivity coefficient νi are
large for fine sediment, but the flux form of the Exner equation does not
account for lag effects or the diffusivity of individual size fractions, thus
leading to unrealistic simulation results. Such unrealistic simulation
results can include an overestimation of advection as sediment sorts (as
shown in the case of sediment mixtures) and an overestimation of the
aggradation–degradation rate (as shown in the case of uniform sediment) when
sufficiently small grain sizes (or sediment fall velocities) are considered.
Field surveys of the LYR observe no clear sorting waves: the grain size
distribution adjusts smoothly both in space and in time, thus indicating
that the more physically based entrainment form is more applicable in terms
of the sorting processes of the LYR. It should be noted, however, that the
difference in the predictions of the two forms of the Exner equation tends to be
large shortly after disruption, but gradually diminishes over time. The flux
form of the Exner equation, on the other hand, is particularly applicable
for coarse sediment or when the sediment transport is dominated by bed load
(e.g., gravel-bed rivers). The above results could have practical
implications in regard to a wide range of issues including dam construction,
water and sediment regulation, flood management, and ecological restoration
schemes. The results can also be used as a reference for other fine-grained
fluvial systems similar to the LYR, such as the Pilcomayo River in
Paraguay and Argentina, South America (Martín-Vide et al., 2014).
It should be noted that in the morphodynamic models of this paper, we
implement the mass and momentum conservation equations for clear water
(i.e., Eqs. 1 and 2) to calculate flow hydraulics instead of the
mass and momentum equations for water–sediment mixture as suggested by Cao
et al. (2004, 2006). More specifically, Cui et al. (2005)
have pointed out that when sediment concentration in the water is
sufficiently small, bed elevation can be taken to be unchanging over
characteristic hydraulic timescales, and the effects of flow–bed exchange
on flow hydraulics can be neglected. For the two simulation cases in this
paper, the volume of sediment concentration C drops from about 2×10-3 to about 2×10-4 in the case of uniform sediment
and from about 4×10-3 to about 4×10-4 in the
case of sediment mixtures due to the cutoff of sediment supply at the
upstream end. These dilute concentrations validate our implementation of
mass and momentum conservation equations for clear water. Our assumption is
not necessarily correct for the entire Yellow River. Upstream of our study
reach, especially upstream of Sanmenxia Dam, the flow is often
hyperconcentrated (Xu, 1999).
Considering the fact that in our numerical simulations a constant inflow
discharge (along with a flood intermittency factor) is implemented, and also
considering that the morphodynamic timescale is much larger than the
hydraulic timescale in our case, the quasi-steady approximation or even the
normal flow approximation can be introduced to further save computational
efforts (Parker, 2004). But one thing that should be noted is that in our
simulation results in Sect. 3, the bed exhibits an inverse slope near the
upstream end. The normal flow assumption becomes invalid under such
circumstances, thus requiring a full unsteady shallow water model.
By definition, the recovery coefficient ro is the ratio of the near-bed
to the flux-depth-averaged concentration of suspended load and is thus
related to the concentration profile. In our simulation r0 is specified
as unity. That is, density stratification effects of suspended sediment are
neglected, and the vertical profile of sediment concentration is regarded as
uniform. However, in natural rivers, the value of r0 can vary
significantly under different circumstances (Cao et al., 2004; Duan and
Nanda, 2006; Zhang and Duan, 2011; Zhang et al., 2013). In general, the
value of r0 is no less than unity and can be as large as 12 (Zhang and
Duan, 2011). Therefore, according to our mathematical analysis in Sect. 4.1 and 4.2, r0=1
corresponds to a maximum adaptation length Lad, a
maximum diffusivity coefficient νi, and a minimum ratio of
celerities cEi/cFi, thus leading to the largest difference between
the flux form and the entrainment form. When sediment concentration is
sufficiently high, hindered settling effects reduce the sediment fall
velocity. Considering the fact that the sediment concentrations considered
in our simulation are fairly small, hindered settling effects are not likely
significant. More study on stratification and hindered settling effects
would be useful in the case of the LYR.
In this paper, a one-dimensional morphodynamic model with several
simplifications is implemented to compare the flux-based Exner equation and
the entrainment-based Exner equation in the context of the LYR. However, a
site-specific model of the morphodynamics of the LYR without these
simplifications would be much more complex. For example, in our 1-D
simulation we observe bed degradation after the closure of the Xiaolangdi
Dam, but we cannot resolve its structure in the lateral direction. In
natural rivers, bed degradation is generally not uniform across the channel
width, but may be concentrated in the thalweg. Moreover, the spatial
variation of channel width and initial slope, which are not considered in
this paper, are also important when considering applied problems. The
abovementioned issues, even though not the aim of this paper, merit future
research (e.g., He et al., 2012). Chavarrias et al. (2018) have
reported that morphodynamic models considering mixed grain sizes may be
subject to instabilities that result from complex eigenvalues of the system
of equations. No such instabilities were encountered in the present work.
Conclusions
In this paper, we compare two formulations for sediment mass conservation in
the context of the lower Yellow River, i.e., the flux form of the Exner equation
and the entrainment form of the Exner equation. We represent the flux form
in terms of the local capacity sediment transport rate and the entrainment
form in terms of the local capacity entrainment rate. In the flux form of
the Exner equation, the conservation of bed material is related to the
stream-wise gradient of sediment transport rate, which is in turn computed
based on the quasi-equilibrium assumption according to which the local
sediment transport rate equals the capacity rate. In the entrainment form of
the Exner equation, on the other hand, the conservation of bed material is
related to the difference between the entrainment rate of sediment from the
bed into the flow and the deposition rate of sediment from the flow onto the
bed. A nonequilibrium sediment transport formulation is applied here so
that the sediment transport rate can lag in space and time behind changing
flow conditions. Despite the fact that the entrainment form is usually
recommended for the morphodynamic modeling of the LYR due to its
fine-grained sediment, there has been little discussion of the differences
in predictions between the two forms.
Here we implement a 1-D morphodynamic model for this problem. The fully
unsteady Saint–Venant equations are implemented for the hydraulic
calculation. Both the flux form and the entrainment form of the Exner equation
are implemented for sediment conservation. For each formulation, we include
the options of both uniform sediment and sediment mixtures. Two generalized
versions of the Engelund–Hansen relation specifically designed for the LYR
are implemented to calculate the quasi-equilibrium sediment transport rate
(i.e., sediment transport capacity). They are the version of Ma et al. (2017) for uniform sediment and the version of Naito et al. (2018) for sediment mixtures. The method of Viparelli et al. (2010) is implemented to store and access bed stratigraphy as the bed
aggrades and degrades. We apply the morphodynamic model to two cases with
conditions typical of the LYR.
In the first case, a uniform bed material grain size of 65 µm is
implemented. We study the effect of the cutoff of sediment supply, as occurred
after the operation of the Xiaolangdi Dam beginning in 1999. We find that the flux form
and the entrainment form give very similar predictions for this case.
Through quantification of the difference between the two forms with a
normalized measure of relative difference, we find that the difference in the
prediction of bed elevation is quite small (< 4 %), but the difference
in the prediction of sediment load can be relatively large (about 20 %)
shortly after the cutoff of sediment supply.
The results for the case of uniform sediment can be explained by analyzing
the governing equation of sediment load qs. In the flux form, the volume
sediment transport rate per unit width qs equals the local
equilibrium (capacity) value qse. In the entrainment form, however,
we find that the difference between qs and qse decays exponentially
in space. The adaptation length Lad=qw/(vsr0) is the
key parameter that controls the distance for qs to approach its
equilibrium value qse. The larger the adaptation length, the more
different the predictions of the two forms will be. For computational
conditions in this case, the adaption length is relatively small (Lad=1.88 km).
In the second case the bed material consists of mixtures ranging from 15 to 500 µm. We find that the flux form and the entrainment form
give very different patterns of grain sorting. Evident kinematic waves occur
at various timescales in the flux form, but no evident kinematic waves can
be observed in the entrainment form. The different sorting patterns are
reflected in the evolution of surface geometric mean grain size Dsg,
total sediment load qsT, and geometric mean grain size of sediment load
Dlg, but are not reflected in the evolution of bed elevation zb.
The different sorting patterns exhibited in the case of sediment mixtures
can be explained by analyzing the governing equation for bed surface
fractions Fi, i.e., the grain-size-specific conservation of bed material.
We find that in the flux form, the governing equation for Fi can be
written in the form of a kinematic wave equation. In the entrainment form,
however, the governing equation for Fi is an advection–diffusion
equation. It is the diffusion term that leads to the dissipation of
kinematic waves. Moreover, in the advection–diffusion equation arising from
the entrainment form, the coefficient of diffusivity is inversely
proportional to the sediment fall velocity. In addition, under the condition
of bed degradation the wave celerity is smaller than that arising from the
flux form.
Overall, our results indicate that the more complex entrainment form of the
Exner equation might be required when the sorting processes of fine-grained
sediment (or sediment with small fall velocity) is studied, especially at
relatively short timescales. Under such circumstances, the flux form of the
Exner equation might overestimate advection in sorting processes and
the aggradation–degradation rate due to the fact that it cannot account for
the relatively large adaptation length or diffusivity of fine particles.
Notations
Cdepth-flux-averaged sediment concentrationCfdimensionless bed resistance coefficientCzdimensionless Chézy resistance coefficientcbnear-bed sediment concentrationcEcelerity of the kinematic wave corresponding to Fi in the entrainment formcFicelerity of the kinematic wave corresponding to Fi in the flux formDsediment grain sizeEdimensionless entrainment rate of sedimentFivolumetric fraction of surface material in the ith size rangefIivolumetric fraction of sediment in the ith size range exchangedacross the surface–substrate interfaceggravitational accelerationhwater depthIfflood intermittency factorLathickness of active layerLadadaptation length of suspended loadpsivolumetric fraction of bed material load in the ith size rangeqrinormalized sediment transport rate per unit width for the ith size range, defined by Eq. (34)qsvolumetric sediment transport rate per unit widthqseequilibrium volumetric sediment transport rate (capacity) per unit widthqsfsediment supply rate per unit widthqwflow discharge per unit widthRsubmerged specific gravity of sedimentr0user-specified parameter denoting the ratio between the near-bedsediment concentration and the flux-averaged sediment concentrationSbed slopettimeudepth-averaged flow velocityu∗shear velocityvssediment fall velocityxstream-wise coordinatezbbed elevationαcoefficient in Eq. (6) for interfacial exchange fractionsΔthtime step for hydraulic calculationΔtmtime step for morphologic calculationΔxspatial step lengthδnormalized parameter quantifying the fraction differencebetween the entrainment form and the flux formλpporosity of bed depositνidiffusivity coefficient corresponding to Fi in the entrainment formρdensity of waterρsdensity of sedimentτbbed shear stressτ∗dimensionless shear stress (Shields number)
All data needed to evaluate the conclusions in the paper are present in the paper and/or
the Supplement. Additional data related to this paper may be requested from the authors.
The Supplement related to this article is available online at: https://doi.org/10.5194/esurf-6-989-2018-supplement
CA, XF, and GP designed the study.
CA performed the simulation with help from HM and KN. CA wrote the paper.
AJM, XF, and YZ provided substantial editorial feedback. CA, AJM, and GP performed the analysis.
HM, YZ, and KN collected data from the LYR for the simulation.
The authors declare that they have no conflict of
interest.
Acknowledgements
The participation of Chenge An and Xudong Fu was made possible in part by
grants from the National Natural Science Foundation of China (grants
51525901 and 91747207) and the Ministry of Science and Technology of China
(grant 2016YFC0402406). The participation of Andrew J. Moodie, Hongbo Ma,
Kensuke Naito, and Gary Parker was made possible in part by grants from
the National Science Foundation (grant EAR-1427262). The participation of
Yuanfeng Zhang was made possible in part by a grant from the National Natural
Science Foundation of China (grant 51379087). Part of this research was
accomplished during Chenge An's visit to the University of Illinois at
Urbana-Champaign, which was supported by the China Scholarship Council (file
no. 201506210320). The participation of Andrew J. Moodie was also supported
by a National Science Foundation Graduate Research Fellowship (grant
145068). We thank the morphodynamics class of 2016 at the University of
Illinois at Urbana-Champaign for their participation in preliminary modeling
efforts. We thank Astrid Blom and two other anonymous reviewers for their
constructive comments, which helped us greatly improve the paper.
Edited by: Jens Turowski
Reviewed by: Astrid Blom and two anonymous referees
ReferencesAn, C., Fu, X., Wang, G., and Parker, G.: Effect of grain sorting on gravel
bed river evolution subject to cycled hydrographs: Bed load sheets and
breakdown of the hydrograph boundary layer, J. Geophys. Res.-Earth, 122, 1513–1533, 10.1002/2016JF003994,
2017.Armanini, A. and Di Silvio, G.: A one-dimensional model for the transport of
a sediment mixture in non-equilibrium conditions, J. Hydraul. Res., 26, 275–292, 10.1080/00221688809499212, 1988.
Bell, R. G. and Sutherland, A. J.: Nonequilibrium bedload transport by
steady flows, J. Hydraul. Eng., 109, 351–367, 1983.Blom, A.: Different approaches to handling vertical and streamwise sorting
in modeling river morphodynamics, Water Resour. Res., 44, W03415,
10.1029/2006WR005474, 2008.Blom, A., Viparelli, E., and Chavarrías, V.: The graded alluvial river:
profile concavity and downstream fining, Geophys. Res. Lett., 43,
1–9, 10.1002/2016GL068898, 2016.Blom, A., Arkesteijn, L., Chavarrías, V., and Viparelli, E.: The
equilibrium alluvial river under variable flow and its channel-forming
discharge, J. Geophys. Res.-Earth, 122, 1924–1948,
10.1002/2017JF004213, 2017.Bohorquez, P. and Ancey, C.: Particle diffusion in non-equilibrium bedload
transport simulations, Appl. Math. Model., 40, 7474–7492,
10.1016/j.apm.2016.03.044, 2016.
Bradley, B. R. and Venditti J G.: Reevaluating dune scaling relations,
Earth-Sci. Rev., 165, 356–376, 2017.
Brownlie, W. R.: Prediction of flow depth and sediment discharge in open
channels, W. M. Keck Laboratory of Hydraulics and Water Resources,
California Institute of Technology, Pasadena, USA, Rep. KH-R-43A, 232 pp.,
1981.
Cao, Z., Pender, G., Wallis, S., and Carling, P.: Computational dam-break
hydraulics over erodible sediment bed, J. Hydraul. Eng.,
130, 689–703, 2004.Cao, Z., Pender, G., and Carling, P.: Shallow water hydrodynamic models for
hyperconcentrated sediment-laden floods over erodible bed, Adv. Water Resour., 29, 546–557, 10.1016/j.advwatres.2005.06.011, 2006.Chavarrías, V., Stecca, G., and Blom, A.: Ill-posedness in modeling
mixed sediment river morphodynamics, Adv. Water Resour., 114,
219–235, 10.1016/j.advwatres.2018.02.011, 2018.Cui, Y., Parker, G., Lisle T. E., Pizzuto, J. E., and Dodd, A. M.: More on
the evolution of bed material waves in alluvial rivers, Earth Surf. Proc. Land., 30, 107–114, 10.1002/esp.1156, 2005.Dietrich, E. W.: Settling velocity of natural particles, Water Resour. Res., 18, 1626–1982, 10.1029/WR018i006p01615, 1982.Dorrell, R. M. and Hogg, A. J.: Length and time scales of response of
sediment suspensions to changing flow conditions, J. Hydraul. Eng., 138, 430–439, 10.1061/(ASCE)HY.1943-7900.0000532, 2012.
Duan, J. G. and Nanda, S. K.: Two-dimensional depth-averaged model
simulation of suspended sediment concentration distribution in a groyne
field, J. Hydrol., 324, 426–437, 2006.
Einstein, H. A.: Bedload transport as a probability problem, PhD thesis,
Mitt. Versuchsanst. Wasserbau Eidg. Tech. Hochsch, Zurich, Switzerland,
1937.El kadi Abderrezzak, K. and Paquier, A.: One-dimensional numerical modeling
of sediment transport and bed deformation in open channels, Water Resour. Res., 45, W05404, 10.1029/2008WR007134, 2009.
Engelund, F. and Hansen, E.: A monograph on sediment transport in alluvial
streams, Technisk Vorlag, Copenhagen, Denmark, 1967.
Exner, F. M.: Uber die Wechselwirkung zwischen Wasser und Geschiebe in
Flussen, Sitzber. Akad. Wiss Wien, 134, 169–204, 1920 (in German).Ferguson, R. I. and Church, M.: A simple universal equation for grain
settling velocity, J. Sediment. Res., 74, 933–937,
10.1306/051204740933, 2004.Ganti, V., Lamb, M. P., and McElroy, B.: Quantitative bounds on
morphodynamics and implications for reading the sedimentary record, Nat. Commun., 5, 3298, 10.1038/ncomms4298, 2014.Guan, M., Wright, N. G., and Sleigh, P. A.: Multimode morphodynamic model
for sediment-laden flows and geomorphic impacts, J. Hydraul. Eng., 141, 04015006,
10.1061/(ASCE)HY.1943-7900.0000997, 2015.Guo, Q., Hu, C., and Takeuchi, K.: Numerical modeling of hyper-concentrated
sediment transport in the lower Yellow River, J. Hydraul. Res.,
46, 659–667, 10.3826/jhr.2008.3009, 2008.
Harten, A., Lax, P. D., and van Leer, B.: On upstream differencing and
Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25,
35–61, 1983.He, L., Duan, J. G., Wang G., and Fu, X.: Numerical simulation of unsteady
hyperconcentrated sediment-laden flow in the Yellow River, J. Hydraul. Eng., 138, 958–969,
10.1061/(ASCE)HY.1943-7900.0000599, 2012.
Hirano, M.: On riverbed variation with armoring, Proc. Jpn. Soc. Civ. Eng.,
195, 55–65, 1971, (in Japanese).Hoey, T. B. and Ferguson, R.: Numerical simulation of downstream fining by
selective transport in gravel bed rivers: Model development and
illustration, Water Resour. Res., 30, 2251–2260, 10.1029/94WR00556,
1994.Ma, H., Nittrouer, J. A., Naito, K., Fu, X., Zhang, Y., Moodie A. J., Wang,
Y., Wu, B., and Parker, G.: The exceptional sediment load of fine-grained
dispersal systems: Example of the Yellow River, China, Sci. Adv.,
3, e1603114, 10.1126/sciadv.1603114, 2017.
Martín-Vide, J. P., Amarilla, M., and Zárate, F. J.: Collapse of
the Pilcomayo River, Geomorphology, 205, 155–163, 2014.
Meyer-Peter, E. and Müller, R.: Formulas for bed-load transport, in
Proceeding of the 2nd IAHR Meeting, International Association for Hydraulic
Research, 7–9 June 1948, Stockholm, Sweden, 39–64, 1948.
Milliman, J. D. and Meade, R. H.: World-wide delivery of river sediment to
the oceans, J. Geol., 91, 1–21, 1983.Minh Duc, B. and Rodi, W.: Numerical simulation of contraction scour in an
open laboratory channel, J. Hydraul. Eng., 134, 367–377,
10.1061/(ASCE)0733-9429(2008)134:4(367), 2008.
Naito, K., Ma, H., Nittrouer, J. A., Zhang, Y., Wu, B., Wang, Y., Fu, X., and Parker,
G.: Extended Engelund-Hansen type sediment transport relation for mixtures
based on the sand-silt-bed Lower Yellow River, China, J. Hydraul. Res., accepted,
2018.National Research Council: River Science at the U.S. Geological Survey,
Washington, DC: The National Academies Press,
10.17226/11773, 2007.
Ni, J. R., Zhang, H. W., Xue, A., Wieprecht, S., and Borthwick, A. G. L.:
Modeling of hyperconcentrated sediment-laden floods in Lower Yellow River,
J. Hydraul. Eng., 130, 1025–1032, 2004.
Paola, C., Heller, P. L., and Angevine, C. L.: The large-scale dynamics of
grain-size variation in alluvial basins, I: Theory, Basin Res., 4,
73–90, 1992.Parker, G.: 1D Sediment Transport Morphodynamics with Applications to Rivers
and Turbidity Currents, available at:
http://hydrolab.illinois.edu/people/parkerg//morphodynamics_e-book.htm
(last access: 31 October 2018), 2004.
Parker, G., Paola, C., and Leclair, S.: Probabilistic Exner sediment
continuity equation for mixtures with no active layer, J. Hydraul. Eng., 126, 818–826, 2000.Phillips, B. C. and Sutherland A. J.: Spatial lag effects in bedload
sediment transport, J. Hydraul. Res., 27, 115–133,
10.1080/00221688909499247, 1989.Stecca, G., Siviglia, A., and Blom, A.: Mathematical analysis of the
Saint-Venant-Hirano model for mixed-sediment morphodynamics, Water Resour. Res., 50, 7563–7589, 10.1002/2014WR015251, 2014.Stecca, G., Siviglia, A., and Blom, A.: An accurate numerical solution to
the Saint-Venant-Hirano model for mixed-sediment morphodynamics in rivers,
Adv. Water Resour., 93, 39–61,
10.1016/j.advwatres.2015.05.022, 2016.
Toro, E. F.: Shock-capturing methods for free-surface shallow flows, John
Wiley, 2001.Toro-Escobar, C. M., Parker, G., and Paola, C.: Transfer function for the
deposition of poorly sorted gravel in response to streambed aggradation,
J. Hydraul. Res., 34, 35–53, 10.1080/00221689609498763,
1996.
Tsujimoto, T.: A probabilistic model of sediment transport processes and its
application for erodible-bed problems, PhD thesis, Kyoto University,
Kyoto, Japan, 1978 (in Japanese).
Van der Scheer, P., Ribberink, J. S., and Blom, A.: Transport formulas for
graded sediment; Behaviour of transport formulas and verification with data.
Research Report 2002R-002, Civil Engineering, University of Twente,
Netherlands, 2002.Viparelli, E., Sequeiros, O. E., Cantelli, A., Wilcock, P. R., and Parker,
G.: River morphodynamics with creation/consumption of grain size
stratigraphy 2: numerical model, J. Hydraul. Res., 48,
727–741, 10.1080/00221686.2010.526759, 2010.Wang, S., Fu, B., Piao, S., Lü, Y., Ciais, P., Feng, X., and Wang, Y.:
Reduced sediment transport in the Yellow River due to anthropogenic changes,
Nat. Geosci., 9, 38–41, 10.1038/ngeo2602, 2016.
Wu, W. and Wang, S. S. Y.: One-dimensional modeling of dam-break flow over
movable beds, J. Hydraul. Eng., 133, 48–58, 2007.
Wu, W. and Wang, S. S. Y.: One-dimensional explicit finite-volume model for
sediment transport, J. Hydraul. Res., 46, 87–98, 2008.Wu, W., Vieira, D. A., and Wang, S. S. Y.: A 1-D numerical model for
nonuniform sediment transport under unsteady flows in channel networks,
J. Hydraul. Eng., 130, 914–923,
10.1061/(ASCE)0733-9429(2004)130:9(914), 2004.
Xu, J.: Erosion caused by hyperconcentrated flow on the Loess Plateau of
China, Catena, 36, 1–19, 1999.
Zhang, H., Huang, Y., and Zhao, L: A mathematical model for unsteady
sediment transport in the Lower Yellow River, Int. J.
Sediment Res., 16, 150–158, 2001.Zhang, S. and Duan, J. G.: 1D finite volume model of unsteady flow over
mobile bed, J. Hydrol., 405, 57–68,
10.1016/j.jhydrol.2011.05.010, 2011.
Zhang, S., Duan, J. G., and Strelkoff T. S.: Grain-scale nonequilibrium
sediment-transport model for unsteady flow, J. Hydraul. Eng., 139, 22–36, 10.1061/(ASCE)HY.1943-7900.0000645, 2013.