Characterization of Alluvial Formation by Stochastic Modelling of Paleo- fluvial Processes: the Concept and Method
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Jiang, Zhenjiao, Mariethoz, Gregoire, Farrell, Troy, Schrank, Christoph,& Cox, Malcolm (2015) Characterization of alluvial formation by stochastic modelling of paleo- fluvial processes: The concept and method. Journal of Hydrology, 524, pp. 367-377.
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Notice: Please note that this document may not be the Version of Record (i.e. published version) of the work. Author manuscript versions (as Sub- mitted for peer review or as Accepted for publication after peer review) can be identified by an absence of publisher branding and/or typeset appear- ance. If there is any doubt, please refer to the published source. https://doi.org/10.1016/j.jhydrol.2015.03.007 1 A stochastic formulation of sediment accumulation and transport to characterize
2 alluvial formations
3 Zhenjiao Jiang a,*, Gregoire Mariethoz b, Troy Farrell c, Christoph Schrank a, Malcolm Cox a
4
5 a School of Earth, Environmental & Biological Sciences, Queensland University of
6 Technology, Brisbane, QLD, 4001, Australia.
7
8 b School of Civil and Environmental Engineering, University of New South Wales, Sydney,
9 NSW, 2052, Australia.
10
11 c School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD,
12 4001, Australia.
13
14 Zhenjiao Jiang *
15 Email: [email protected]
16
17 Abstract
18 Modelling fluvial processes is an effective way to reproduce basin evolution and recreate
19 riverbed morphology. One-dimensional (1D) fluvial process-based models are widely used
20 because of their computational efficiency and flexible parameterization of hydraulic and
21 sediment properties. However, currently used 1D models have the limitation that spatial and
22 temporal variations of the flow velocity are not fully considered. To address this, we derive a
23 stochastic fluvial process model (SFPM) on the basis of the Exner equation. Stochastic
24 velocity function is applied in the model that accounts for influences of riverbed and channel
25 evolution. The riverbed slope evolves with the sediment accumulation, and the velocity
1
26 changes due to this are accounted for dynamically in the SFPM. The effect of channel
27 evolution is accounted for in the stochastic velocity function in term of the probability of the
28 channel occurring at the position of interest, and this probability is estimated from the river
29 discharge and the width of fluvial trace. In order to couple the stochastic velocity function
30 into the fluvial process model, the Exner equation is developed as two separate equations
31 namely, a “mean equation”, which yields the mean sedimentation thickness, and a
32 “perturbation equation”, which yields the variance of sedimentation thickness. SFPM is
33 applied in two synthetic cases, and the results suggest that SFPM can be used for stochastic
34 analysis of fluvial processes at the basin scale.
35 Key words: process-based model, fluvial, perturbation theory, spectral approach, numerical
36 simulation.
37 1. Introduction
38 Rivers are one of the most dynamic external forces interacting with and modifying the
39 Earth’s surface. Sediment erosion and deposition in rivers (fluvial processes) affect the
40 geomorphic evolution of land surfaces and basin stratigraphy. Various models have been
41 developed over the past decades to quantitatively describe fluvial processes, including
42 geostatistical models that statistically mimic the final results of fluvial processes, and
43 process-based models that quantify the physics of fluvial processes (e.g. Koltermann and
44 Gorelick, 1996; Paola, 2000; Van De Wiel et al., 2011). Geostatistical methods interpolate
45 the data values based on probability rules inferred from the data measurements. These
46 methods can be conditioned to the measured information, but their applicability can be
47 limited by a lack of data. In contrast, process-based models describe the mechanics of fluvial
48 processes, and can simulate the lithology distribution in the absence of data measurements (Li
49 et al., 2004; Tetzlaff, 1990).
2
50 A classical process-based model describing fluvial processes is the Exner equation, which
51 is established on the basis of the mass balance of sediment transport in rivers and sediment
52 accumulation on the riverbed (Exner, 1925; Leliavsky, 1955). A general Exner equation was
53 derived by Paola and Voller (2005) that considers the influence of tectonic uplift and
54 subsidence, soil formation and creep, compaction and chemical precipitation and dissolution.
55 The mass balance equation for a wide range of specific problems, such as short- or long- term
56 riverbed evolution, can be extracted from the general Exner equation by combining and
57 dropping negligible or undetermined terms.
58 The models extracted from the general Exner equation and widely used nowadays include
59 for example: the convective model (Davy and Lague, 2009; Paola and Voller, 2005), where
60 the sediment flux and accumulation at the position of interest is assumed to be controlled by
61 the upstream landscape features and sediment input; the diffusion model (Paola et al., 1992;
62 Paola and Voller, 2005), which simulates influences of both upstream and downstream
63 situations on the target positions; and the fractional model (Voller et al., 2012), which
64 accounts for non-local upstream and downstream influences.
65 In these fluvial process models (FPM), the flow velocity, which represents the stream
66 energy, is the key input parameter. The velocity can be resolved by a fluid dynamics model
67 (FDM) based on the Navier-Stokes equations (e.g. Gonzalez-Juez et al., 2009; Necker et al.,
68 2005). Approaches that couple FPM and FDM can yield a detailed description of the fluvial
69 processes and the channel evolution, however these are mostly limited to controlled
70 laboratory settings. At the catchment scale, a coupled FPM-FDM has been applied in two-
71 dimensional planes, where the vertical velocity variation is neglected (Koltermann and
72 Gorelick, 1992). Fully-coupled modelling of the fluvial processes and fluid dynamics,
73 however, is still a challenge, partly because applying the FDM requires precise knowledge of
74 the initial and boundary flow conditions, which are generally not available, and partly
3
75 because extensive computational time is required (e.g. Koltermann and Gorelick, 1992;
76 Lesshafft et al., 2011; Simpson and Castelltort, 2006).
77 Due to the geological and hydrogeological complexity, the flow velocity and fluvial
78 processes are difficult to simulate deterministically. Subsequently, stochastic fluvial process
79 models have been developed, for example, to account for the probability distribution of
80 sediments sizes (Parker et al., 2000), the stochastisity of river discharge (e.g. Lague, 2014;
81 Molnar et al., 2006; Tucker and Bras, 2000), and the stochasticity of particle motion (e.g.
82 Furbish et al., 2012; Roseberry et al., 2012).
83 In this study, we derive a stochastic fluvial-processes model to account for the influences
84 of those geological and hydrogeological factors that affect the velocity and can be
85 represented by the statistics of flow velocity. These factors include riverbed and channel
86 evolution, river discharge, and riverbed and bank friction within the river channel.
87 A perturbation approach is employed to develop a stochastic model based on the
88 convective fluvial process model (Davy and Lague, 2009). The velocity in our model is
89 characterized by a stochastic description consisting of an ensemble mean component and a
90 variance (or perturbation component). We mainly derive the analytical solutions for the
91 statistics (mean and variance) of sediment load in the river and sedimentation thickness on
92 the riverbed on a short time-scale, assuming that velocity changes within this short timescale
93 are negligible. A numerical scheme is used to advance these short timescale analytical
94 solutions over the entire simulation time, and the velocity is updated according to the changes
95 in the river discharge and/or variation in the riverbed slope when the simulation time
96 increases by a short timescale.
97 This study is organized as follows: Section 2 introduces the convective fluvial process
98 model, defines the ensemble velocity and then develops the model as a stochastic model by
99 employing perturbation theory. Section 3 derives the analytical solutions for the sediment
4
100 load and sedimentation thickness. The algorithm of implementing the solutions is
101 summarized in Section 4. Finally, Section 5 applies the stochastic model in two synthetic
102 cases.
103 However, this current study mainly presents the derivation of a stochastical fluvial
104 processes-based model (SFPM) and the resultant solutions are applied in synthetic cases. The
105 application of SFPM in reproducing the lithology architecture of the Late Permian Betts
106 Creek Beds in the Galilee Basin, Australia will be perused in our companion study.
107 2 Governing equations
108 2.1 Mass balance equation
109 The mass balance equation describing fluvial processes is expressed as two separate
110 equations (Davy and Lague, 2009), one describing the sediment transport in the river:
∂η(x,t) ∂v(x,t)η(x,t) + − E(x,t) + D(x,t) = 0 , ∂t ∂x (1)
111 and another describing the sediment accumulation on the riverbed:
∂z(x,t) 1 = [D(x,t) − E(x,t)] , ∂t 1−ϕ (2)
112 where chemical precipitation and dissolution, and the abrasion of the sediment particles are
113 not considered. Here η is the sediment load in the river (L3/L2), which represents the volume
114 of sediments in the water column of a unit bottom area, v is the stream velocity (L/T), x is the
115 distance along the stream from its origin (L), t is time (T), z is the sedimentation thickness (L),
116 ϕ is the porosity of deposited sediment, E is the erosion rate of sediment (L/T), and D is the
117 deposition rate of sediment (L/T). Expressions for E and D are given in Appendix A.
118 If we consider the fluvial processes on an arbitrary position in the fluvial trace, η and z at
119 this position will present uncertainties, firstly due to potential river channel evolution that the
120 river may not be flowing through this position (Fig. 1a), and secondly due to the inherent
121 non-uniform distribution of v in the river channel (relating to such as frictions of river bank
5
122 and riverbed) when the river is flowing through this position. The uncertainty in v induces the
123 perturbation of E and D, and further leads to the uncertainty in η and z (Fig. 1b).
124
125 Figure 1. (a) depicts a fluvial trace schematic consisting of the current river channel (solid
126 line) and the historical channel positions (dashed lines), (b) the uncertainty in v induces the
127 perturbation of E and D, and further leads to the uncertainty in η and z, where solid arrows
128 represent the forward influences and dashed arrow represents the backward influences, and
129 prime represents the perturbation of the quantities. “River channel” in this current study
130 indicates the range covered by the water flow at a fixed time and “fluvial trace” indicates the
131 range where the sediments were deposited by the river flow in the past.
132 Without the dynamic fluid simulation based on the Stokes-Navier equation, Eqs. (1) and (2)
133 can only solve η and z in the fixed river channel, assuming that river channel always pass the
134 position of interest and v in the river channel is commonly estimated by Manning formula
135 (Section 2.2). As an alternative, this current study derives the statistics (mean and variance)
136 of η and z, where the variance quantifies the uncertainties induced by both channel evolution
137 and inherent velocity perturbation in the river channel. For this purpose, v in Eqs. (1) and (2)
138 should be first redefined.
139 2.2 Velocity revisited
140 2.2.1 Manning velocity
141 In the absence of the Navier Stokes equation and for a fixed river channel, the Manning
142 formula is often used to estimate v (e.g. Lague, 2010; Le Méhauté, 1976) :
6
2 1 c f Q(x,t) v (x,t) = R(x,t) 3 S(x,t) 2 = . M n(x,t) A(x,t) (3)
143 Here vM is the average stream velocity on the cross-section of the river channel perpendicular
144 to the river flow direction (hereafter referred to as the Manning velocity), cf is a conversion
145 factor (L1/3/T), n is the Manning coefficient, S is the riverbed slope, R is the hydraulic radius
146 of river channel (L), Q is the river discharge (L3/T) and A is the wetted area (L2). The channel
147 cross-section, perpendicular to the river flow, is assumed to have a quadrangular shape,
148 therefore,
W (x,t)H (x,t) R(x,t) = c c , Wc (x,t) + 2H c (x,t) (4)
A(x,t) = Wc (x,t)H c (x,t) ,
149 where Wc is the width of the river channel and Hc is the depth of the river flow.
150 In Eq. (3), Q and initial value of S are the major user-specific parameters. Eq. (3) and (4)
151 can then be used to determine Wc, Hc and vM , given an additional relationship either between
152 Q and Wc (Viparelli et al., 2011) or between Wc and Hc (Parker et al., 1998),
W (x,t) Q (x,t) W (x,t) c = [ c ]a or c = a . Wref Qref H c (x,t) (5)
153 Here a is a constant, Qref and Wref are a reference river discharge and a reference river
154 channel width corresponding to Qref, respectively.
155 2.2.2 Inherent velocity perturbation in the river channel
156 The manning velocity, vM , represents the averaged stream velocity on the cross-section of
157 river channel perpendicular to the river flow. However, v on an arbitrary cross section is
158 unevenly distributed. To express this, v in the river channel ( vc ) is expressed as the sum of a
159 mean velocity ( vc ) and a perturbation ( vc′ ):
v (x,t) = v (x,t) + v′ (x,t) , c c c (6)
7
σ 2 (x,t) = v′ (x,t)v′ (x,t) , vc c c
160 where σ 2 is the variance of v in the river channel. vc c
161 Assuming that vc on each cross section follows the Gaussian distribution (Leopold and
162 Wolman, 1957), then applying the definition of a confidence interval for Gaussian distributed
163 quantities (Cox and Hinkley, 1979), we obtain that
v (x,t) = v (x,t) , c M (7) 164 and
σ 2 (x,t) = α ⋅ v 2 (x,t) . vc M (8)
165 Here α is a Gaussian coefficient equal to 0.26, which is adjustable if vc does not strictly
166 follow a normal distribution. Specifically, α decreases if the vc distribution is negatively
167 skewed and increases if the vc distribution is positively skewed.
168 It is noted that thus far Eqs. (3), (7) and (8) account for the non-uniform distribution of
169 velocity within the river channel, relating to river discharge (Q) and river morphology (S and
170 A) on v. We now consider a further modification of v due to the channel evolution.
171 2.2.3 Define channel evolution in the statistics of the velocity
172 Channel evolution includes: (1) Wc changes relating to Q changes, and (2) the river channel
173 movement induced by bank erosion and deposition. Wc variation is explicitly accounted for in
174 the 1D fluvial process model, via Wc -Q relationship inferred by Eqs. (3)-(5) when calculating
175 v. However, the channel movement and its feedback on the fluvial processes cannot be
176 simulated dynamically within 1D fluvial process model, because the vertical and lateral
2 177 variation of v are not simulated in detail, but are merely inferred as the perturbation term,σ v ,
178 in Eq. (8).
179 Here we regard the effects of channel movement as the uncertainty of river channel
180 locations (Fig. 1a). To quantify the influences of channel movement on η and z, we define
8
181 the uncertainty of channel location as the probability of channel occurrence at a position of
182 interest in the fluvial trace. Assuming that the channel has an equal probability of passing at
183 each position on the cross-section of fluvial trace perpendicular to the river flow, we obtain
184 that:
Wc (x,t) p(x,t) = , (9) W f (x)
185 where p is the probability of channel occurrence at position x and time t, Wf is the width of
186 fluvial trace, which can be surveyed by geophysical tools such as seismic survey, airborne
187 electromagnetic surveys and remote sensing (e.g. Jin et al., 2011; Montgomery and Morrison,
188 1999; Vrbancich, 2009), and Wc can be solved from Eqs. (3)-(5) once Q is input. If the river
189 channel does not move and the channel evolution is only induced by Q changes, p=1
190 The probability of channel occurrence, p, is then defined in the statistics of v by
191 considering that when the channel does not pass through the position of interest, then v is
192 zero at this position, however, when the channel passes through this position, then inherent v
193 changes follow a Gaussian distribution and the statistics of v in the river channel has been
194 defined in Eqs. (7) and (8). In addition, v changes due to channel movement occur slowly,
195 and on a different time-scale to inherent v variation in the river channel relating to such as
196 riverbed and bank frictions. However, these two time-scale v variations can be defined in the
197 ensemble statistics of v (Furbish et al., 2012; Gibbs, 2010), because ensemble statistics are
198 calibrated at a fixed time over all possible values of v.
199 As a consequence, the terms of flow velocity in Eqs. (7) and (8) are revised to consider the
200 channel movement (derivations in Appendix B):
v(x,t) = p(x,t) ⋅ v (x,t) , M (10)
σ 2 (x,t) = [[1− p(x,t)][2 − p(x,t)] + αp(x,t)]v2 (x,t) , v M (11) 201
9
2 202 where v and σ v are the ensemble mean and variance of velocity used in this study.
203 The situation that channel is away for the position of interest is quantified as the
204 probability of zero velocity (Eq. B2 in Appendix B). The erosion can occur only when the
205 velocity becomes sufficiently large to assure the shear stress larger than the critical shear
206 stress (Eqs. A1 and A3 in Appendix A). Therefore, using zero velocity to indicate the channel
207 location can assure that the erosion does not occur at the position of interest when the channel
208 does not pass at this position.
209 However, the deposition and erosion can occur simultaneously and D is expressed by an
210 equation which is unrelated to the flow velocity Eq. (A6) (Appendix A). Effects of channel
211 movement on D have not yet been considered if using zero velocity as an indicator of channel
212 location. With regard to that D is zero when the river channel does not pass at the position of
213 interest, coefficient in D expression (Eq. A6) is multiplied by p to represent the effects of
214 channel movement.
215 Both channel evolution and inherent velocity perturbation in the river channel are now
2 2 216 included in v and σ v (Eqs. 10 and 11). To infer v and σ v , the major user-specified
217 parameters include Q, initial values of S and Wf. Wf is the only additional parameter required
2 218 in the calculation of v and σ v , when compared to using merely vc in the traditional
219 deterministic model (e.g. Viparelli et al., 2011).
220 Following this, we derive the solution for the statistics of η and z as the functions of v
2 221 and σ v (in Section 3). For this purpose, v and v′ should be first coupled in the governing
222 equations (1) and (2), on the basis of perturbation theory.
223 2.3 Mass balance equation revisited
224 Eqs. (1) and (2) are nonlinear partial different equation, which should be solved numerically
225 according to the following simplifications (Lanzoni and Seminara, 2002): (1) the fluvial trace,
10
226 river channel and riverbed are discretized into n segments, and v , E and D are held constant
227 within each segment, (2) η and z are calculated at short timescale (time step), and then are
228 advanced to results at the entire simulation period.
229 The time steps are selected according to the variation of v , which depends on S and Q
230 (Eqs. 3 and 9). S evolves due to the sediment accumulation, whist Q is changed due to the
231 rainfall. v changes induced by sediment accumulation are much slower than the variation
232 induced by Q. As these two events alter the velocity at different frequencies, two (internal
233 and external) time steps are proposed (Fig. 2). The internal (small) time step is selected
234 according to the time series of Q, while the external (large) time step is selected according to
235 the time that significant changes of S are induced by sediment accumulation and that the
236 accumulated sediment approximately stabilizes on the riverbed.
237
238 Figure 2. Flow chart of modelling sediment load and sedimentation thickness. The velocity is
239 assumed to be a constant within one internal time step and one river segment, and is updated
240 according to the river discharge when the computation time increases by one internal time
241 step, and is updated according to the riverbed slope when the computation time increases by
242 one external time step. The analytical solutions of sediment load and sedimentation thickness
243 are derived in Section 3.
11
244 Within one river segment and one internal time step, Eqs. (1) and (2) can be rewritten as:
∂η(x ,t) ∂η(x ,t) k + v k − E + D = 0 , k k k (12) ∂t ∂xk
245 and
∂zk (t) 1 = [D − E ] , k k (13) ∂t 1−ϕk
246 where k indicates the k-th segment (k=1, 2, … n), xk is the coordinates in k-th segment. The
247 spatial and temporal variables in Eqs. (12) and (13) are simplified as follows: (1) independent
248 variables vk, Ek and Dk are assumed to be constant within k-th segment, (2) dependent
249 variable zk does not change spatially within one segment due to the constant Ek and Dk, but zk
250 changes with the time due to erosion and deposition processes, (3) η(xk ,t) is still a spatial and
251 temporal variable, because deposition and erosion keep occurring, and the sediment volume
252 entrapped in the river changes.
253 Perturbation method is used to the statistics of vk in Eq. (12), which considers the
254 quantities in the equation as a sum of the ensemble mean and a perturbation surrounding the
255 mean (Holmes, 2013):
v = v + v′ , v′ = 0 ,σ 2 = v′v′ , k k k k vk k k
Dk = Dk + Dk′ , Dk′ = 0 ,
E = E + E′ , E′ = 0 , k k k k (14)
2 η(xk ,t) =η (xk ,t) +η′(xk ,t) , η′(xk ,t) = 0 ,ση (xk ,t) =η′(xk ,t)η′(xk ,t) ,
= + ′ , z′ (t) = 0 , σ 2 (t) = z′ (t)z′ (t) , zk (t) zk (t) zk (t) k zk k k
256 where the bar indicates ensemble mean of these quantities and the prime indicates a zero-
257 mean perturbation.
258 Substituting Eq. (14) in (12) gives:
12
∂[η (x ,t) +η′(x ,t)] ∂[η (x ,t) +η′(x ,t)] k k + (v + v′ ) k k − E − E′ + D + D′ = 0 . k k k k k k (15) ∂t ∂xk
259 Taking the ensemble mean on both sides of Eq. (15) to remove the first-order perturbation
260 terms, leaves a mean equation:
∂η (x ,t) ∂η (x ,t) ∂η′(x ,t) k + v k + v′ k − E + D = 0 . k k k k (16) ∂t ∂xk ∂xk
261 Subtracting Eq. (16) from (15) yields the perturbation equation:
∂η′(x ,t) ∂η′(x ,t) ∂η (x ,t) k + v k + v′ k − E′ + D′ = 0 . k k k k (17) ∂t ∂xk ∂xk
262 Eq. (17) can be rewritten as:
∂η′(x ,t) ∂η′(x ,t) k + v k − w v′ = 0 , k uk k (18) ∂t ∂xk
263 where wuk = wsk + wdk + wek represents the influence of the velocity changes on the sediment
∂η (xk ,t) 264 load via sediment flux ( wsk = − ), deposition (wdk) and erosion (wek). wek satisfies ∂xk
265 (Appendix A):
w = k c ρ (3v 2 + σ 2 ) , ek ek dk f k vk (19)
2.5 2 1.5 266 where kek is the erosion efficiency coefficient (L ∙T /M ), cdk is the dimensionless drag
3 267 coefficient , ρ f is the density of water (M/L ).
268 wdk represents the influence of vk′ on ηk′ via altering Dk. Recalling Fig. 1b, vk′ does not
269 directly induce Dk changes (Winterwerp and Van Kesteren, 2004), therefore, wdk = 0 in Eq.
270 (18).
271 Making use of Eq. (14) in the sediment accumulation model (Eq. 13) yields a mean
272 equation:
13
∂zk (t) 1 = [D − E ], k k (20) ∂t 1−ϕk
273 and a perturbation equation:
∂z′k (t) 1 = [D′ − E′ ], k k (21) ∂t 1−ϕk
274 Eqs. (16), (18), (20) and (21) form the basic partial different equations of the stochastical
275 fluvial process-based model (SFPM). Their semi-analytical solutions are then derived in
276 Section 3. The governing equations (16) and (20) can be solved for η and z , given certain
277 boundary conditions. However, rather than finding a relationship between η′ , z′ and v′ from
2 2 2 2 278 Eqs. (18) and (21), results are expressed by functions as ση ~ fη (σ v ) and σ z ~ f z (σ v ) .
2 2 279 Section 3 derives solutions for η , ση , z and σ z within one internal time step. As shown
2 280 in Fig. 2, when the simulation time increases by one internal time step, v and σ v are updated
281 due to Q changes, and when the simulation period increases by one external time step, v and
2 282 σ v are updated due to S variation induced by sediment accumulation (using Eq. 3 in Eqs. 10
2 2 2 283 and 11). Making use of these updated v and σ v in solutions of η , ση , z and σ z , these
284 quantities can be solved out after loops of internal and external time steps.
285 3. Semi-analytical solutions
286 3.1 Solution for the variance of sediment load
2 287 This section derives the analytical solution of ση from Eq. (18), using the nonstationary
288 spectral method on the basis of Fourier transform.
289 The Fourier-Stieltjes representation of the random process η′ is (Gelhar and Axness, 1983;
290 Li and McLaughlin, 1995; Ni et al., 2010):
∞ η′(x ,t) = φ(x ,t,κ)dZ (κ) , k ∫−∞ k v (22)
291 and v′ is represented by:
14
∞ κ v′ = ei xk dZ (κ) , k ∫−∞ v (23)
292 where dZv (κ) is the complex Fourier amplitude of flow velocity, i = −1 , κ is the
293 magnitude of the wave number vector, and φ is a transfer function between v′ and η′ .
294 Substituting Eqs. (22) and (23) in Eq. (18) gives:
∂φ κ ∂φ κ (xk ,t, ) (xk ,t, ) iκxk + vk − wuk e = 0 . ∂t ∂xk (24)
295 At the source of the river, η′ is determined by the natural boundary condition, which is
296 independent to the flow velocity. Therefore,
η′(0,t) = 0 . (25) 297 The Fourier representation of Eq. (25) implies:
φ(0,t,κ) = 0. (26) 298 Solving Eq. (24) subject to Eq. (26) results in a general solution for φ(x,t,κ) at each node
299 m (details in Appendix C):
m 1 κ∆ 1 w φ(xˆ ,t,κ) = ( ei x − ) ( uk ) , m ∑ (27) iκ iκ k=1 vk
300 where m is the index of node (m= 0, 1, 2,…,n), ∆x is the length of discretized segment (L),
301 xˆm is the coordinates of node m.
302 Once φ is expressed analytically, the variance of η can be calculated as:
∞ η′η′ = φ(x,t,κ) ⋅φ * (x,κ)s (κ)dκ , ∫−∞ vv
∞ η′v′ = eiκx ⋅φ * (x,t,κ)s (κ)dκ , ∫−∞ vv (28)
∞ * ∂η′ κ ∂φ (x,t,κ) v′ = ei x s (κ)dκ , ∂x ∫−∞ ∂x vv
15
* 303 where φ represents the conjunction of φ , and svv (κ) is the spectral density function of the
304 ensemble velocity. A commonly used spectral density function for many natural quantities is
305 the exponential model (Dagan, 1989; Zhu and Satish, 1999) :
2κ 2σ 2λ3 s (κ) = v , vv π (1+ κ 2λ2 )2 (29)
306 where λ is the correlation scale of the flow velocity (L). The upstream flow velocity affects
307 the downstream velocity, and hence, λ at different locations on the river is the flow distance
308 ( λ = x ).
309 Substituting Eqs. (27) and (29) in (28) yields the closed-form expressions for the variance
2 310 of sediment load (ση ) and covariance between sediment load and velocity (σηv ) at m-th
311 node:
∆x 2 − m 2 λ + ∆x λ w σ 2 =η′η′ = 2λ (1− m e m ) ( uk ) σ 2 , ηm m m m ∑ vm λm k=1 vk (30) xˆ xˆ − m − m−1 m λ λ w σ =η′ v′ = (x e m − x e m ) ( uk ) ⋅σ 2 . ηmvm m m m m−1 ∑ vm k=1 vk
312 where λm is the correlation scale of the flow velocity at m-th node.
2 313 Eq. (30) suggests that the impact of v′ on the η′ (or ση ) is accumulated from upstream to
m w 314 downstream, as both λ and the coefficient term ∑( uk ) increase with the distance from the k =1 vk
315 river source.
316 In addition, the second-order perturbation term in Eq. (16) becomes:
x − m ∂η′ w xˆ λ v′ m = um (1− m )e m σ 2 , m vm (31) ∂x vm λm
317 which is zero because λm = xˆm .
16
318 3.2 Solution for the mean sediment load
319 Solving the mean equation (16) is a traditional Cauchy problem, subject to the boundary
320 condition (Hadamard, 2003):
η(0,t) = q (t) , s (32)
321 where qs(t) is the first-kind boundary condition at the river source (L), which expresses the
322 sediment input from the river source.
323 As a result, the sediment load at each node m can be written as:
m ∆x m ∆x m ∆x η = E k + q (t − k ) − D k . m ∑ k s ∑ ∑ k (33) k=1 vk k=1 vk k=1 vk
324 Eq. (33) suggests that η at xm is attributed to qs from the river source and the erosion over
m m ∆xk ∆xk 325 a distance xm. The term, ∑ Ek + g(t − ∑ ) , is defined as the potential load. k =1 vk k =1 vk
326 Simultaneously, the potential load decreases to η due to the deposition along the flow path.
327 3.3 Solution for the mean and variance of sedimentation thickness
328 Sediment is accumulated on the riverbed due to deposition and removed from the riverbed
329 due to erosion. After an internal time step ∆t , the mean and perturbation of sedimentation
330 thickness at node m (Eqs. 20 and 21) can be rewritten as:
1 zm = [Dm − Em ]∆t , 1−ϕm (34)
331 and
1 z′ = [D′ − E′ ]∆t . m m m (35) 1−ϕm
332 While the distribution of ηm is simulated by Eq. (33), zm is solved from Eq. (34).
333 Furthermore, using expressions of E′ and D′ (Eq. A5 and A10 in Appendix A) in the
334 perturbation equation (35) leads to the variance of sedimentation thickness:
17
1 β σ 2 = z′ z′ = [ m η′ − w v′ ]2∆t 2 , zm m m 2 m em m (36) (1−ϕm ) H cm
335 which can be further expressed as:
1 β β σ 2 = [( m )2σ 2 − 2w m σ + w2 σ 2 ]∆t 2 , zm 2 ηm em ηvm em vm (1−ϕm ) H cm H cm (37)
336 where βm is the deposition coefficient (L/T) (Appendix A).
337 It is noted that although the basic sediment transport and accumulation equations are given
2 338 in one segment of the river (Eqs. 12 and 13), the solutions of η , ση (Eqs. 30 and 33), z and
2 339 σ z (Eqs. 34 and 37) are written as general forms over the entire longitude profile of the river
340 system.
341 In order to simulate the spatial distribution of different lithologies, the transport and
342 accumulation of sediments of multiple grain sizes can be modelled based on the exhausted
343 deposition assumption, involving that fine-grain sediments can only be deposited after the
344 coarse-grain sediments are exhausted (Paola et al., 1992). By the end of each time step, the
345 deposited lithology is recorded together with z .
346 4. Algorithm
347 Based on derivations above, the algorithm of implementing the stochastic model is
348 summarized as follow:
349 (1) Select a cross-section parallel with the general river flow direction. This cross-section is
350 not necessarily on the river channel, but sediment at different positions along this cross
351 section should be created by the same river channel (for meandering river) or the same
352 sets of river channels (for the braided river).
353 (2) Discretize the selected cross-section into n segments.
354 (3) Choose the time steps according to sedimentation rate and frequency of Q changes.
355 (4) Input S in Eq. (3).
18
356 (5) Input Q at each segment and calculate v , σ 2 , p, W and H (k=1, 2,…n) of the river k vk ck ck
357 channel according to Eqs. (3-5), (9-11)
358 (6) Assess Ek (Eq. A4) and we in Eq. (A5);
x 359 (7) Calculate the time interval ( k ) for river flow through each segment; vk
360 (8) Calculate the potential ηm at each node m (m=0, 1, … n) as the sum of erosion and
361 sediment input (Eq. 33);
m xk 362 (9) Calculate the real ηm in Eq. (33) after deposition of ∑ Dk for node m. Because ηm k=1 vk
363 affects Dm (Eq. A9), the real ηm and Dm should be calculated simultaneously. At the
364 river source (m=0), the potential ηm=0 is merely the sediment input. The deposition rate
365 ( D1 ) at the first segment is assumed to be constant, which is calculated corresponding to
366 η0 . For the first node, the potential load is the sum of the erosion over the length of the
367 first segment plus sediment input. However, the real η1 for this node should remove the
368 deposition volume over the first segment (deposition rate is D1 and deposition period is
x 369 1 ). Subsequently, for the m-th node, the potential load is the sum of erosion over a v1
m 370 distance of ∑ xk plus the sediment input from river source, and the real load needs the k=1
m xk 371 removal of a deposition volume ∑ Dk . k =1 vk
372 (10) zm is recorded simultaneously with the calibration of Dm and Em in steps (8) and (9)
373 according to the mean equation (34).
2 2 374 (11) ση is estimated by Eq. (30), whilst σ z is calculated by Eq. (37);
19
375 (13) When the computation time increase by one internal time step, input new Q, and when
376 the computation time increase by external time step, recalculate the regional S induced by
377 the sediment accumulation. Go back to step (5) and repeat the steps until the total
378 computation time is larger than the target simulation time.
379 The algorithm is implemented in Matlab and applied to two synthetic cases.
380 5. Application in synthetic cases
381 The major advance of the proposed stochastic fluvial model is that the model can yield the
382 most likely sedimentation thickness ( z ) and also the potential variation (σ z ) induced by
383 uncertainty of the flow velocity. User-specific parameters in the model mainly include initial
384 riverbed slope (S), river discharge (Q), width of fluvial traces (Wf) (which determine the
385 ensemble velocity in Section 2.2) and the sediment input from the river source (qs in Eq. 32).
386 Other parameters for the sediment and fluid properties are assumed to be fixed in this current
387 study (Table 1).
388 To demonstrate the effects of proposed model, two synthetic examples are presented
389 mainly to discuss the dependence of z and σ z on S and Q, with the following simplifications:
390 (1) qs from the river source is assumed to be unrelated with Q, and their influences on the
391 sediment accumulation are discussed separately. The interaction between qs and Q is
392 beyond the scope of this study.
393 (2) Streamwise variation of Q is neglected assuming that within the stream section of
394 interests, there are no major tributaries, and no significant water loss induced by
395 infiltration and evaporation.
396 (3) Single kind of sediment (sands) transport and accumulation are discussed.
397 (4) The deposited sand can be eroded, but the original bedrock is assumed to be non-erodible.
398
399
20
400 Table 1. Default parameters used in fluvial process modelling
Coefficients value 1/3 Conversion factor, cf , m /s 1.0
-5 Concentration coefficient, kd , - 3×10
3 Density of sediment, ρs , kg/m 2700
3 Density of water, ρ f , kg/m 1000
Drag coefficient, cd , - 0.005
1.5 3 1.5 -6 Erosion efficiency, ke, m/year (m s /kg ) 3×10 Gaussian coefficient, α , - 0.26 Gravitational acceleration, g, m/s2 9.8 Kinematic viscosity of the water, ν , m2/s 10-6
Magnitude of external time step, Tex, years 100 Magnitude of internal time step, ∆t , days 15 Manning coefficient, n, - 0.034 Porosity, ϕ , - 0.2 Particle size, d, m 0.01 Ratio between width and depth, Wc/H, - 20
2 2 Threshold shear stress, τ c , kg∙m /s 4.3 401
402 5.1 Synthetic example-1
403 This example describes the sediment aggregation in an empty basin under a constant
3 404 sediment input from the river source (qs=0.0001 m) and a constant Q (50 m /s). The initial
405 riverbed is assumed to have an exponential-shape (Fig. 3a), which represents the sedimentary
406 environment varying from hill to coastal plain. Sedimentary process is simulated over a
407 period of 100 years. Because the sedimentation rate is low (less than 0.025 m/year in Fig. 3a
408 and 3b), sediment accumulation on the riverbed does not significantly change the flow
409 velocity within 100 years. Therefore, the simulation is conducted within one external time
410 step (Fig. 2). In addition, the internal time step is selected according to the variation
21
411 frequency of Q. In this synthetic case, Q is assumed to be a constant, the magnitude of
412 internal time step is arbitrarily selected as the default value in the model (15 days in Table 1).
413
414 Figure 3. (a) sands accumulation on the riverbed due to the sediment input from the river
415 source. (b) sedimentation thickness and potential variation and (c) highly nonlinear
416 distribution of the standard deviation of sedimentation thickness, relating to (d) the pattern of
417 the river channel and (e) the probability of channel occurrence within the range of fluvial
418 trace, which are quantitatively described by (f) the statistics of ensemble flow velocity and (g)
419 the perturbation of erosion and deposition rates or (h) thickness perturbation under unit
420 velocity perturbation.
421
422 Fig. 3b illustrates z and σ z of the sedimentation thickness. The nonlinear pattern of mean
423 sedimentation thickness can compare qualitatively with the results from laboratory by Cui et
424 al. (2003) and Sklar et al. (2009). This pattern is induced by the coupled possesses of
425 deposition and erosion.
22
426 D reduces along the river flow direction because deposition keeps occurring and η
2 427 decreases. E reduces along the river flow direction due to the decreases of v and σ v (Fig.
428 3f). However, the decreasing D and E play reverse roles on z , as deposition increases z ,
429 whist erosion decreases z . Therefore, a nonlinear pattern of z is produced in Fig. 3b that z
430 increases with the distance to the river source at the upstream part of the river. After z peaks
431 to 2.2 m at the position of about 2000 m, z decreases along the river flow at the downstream
432 part of the river.
433 The potential variation of z is inferred from the standard deviation of sedimentation
434 thickness (σ z ), which is high-nonlinearly distributed parallel with the river flow direction
435 (Fig. 3c), due to the contrast between Wc and Wf (Fig. 3d1), and exponential-shape riverbed
436 (Fig. 3a).
437 As shown in Fig. 3d, Wc increases with the streamwise distance, given a spatially constant
438 Q (50 m3/d). The wider river channel at the downstream part indicates that the river channel
439 has a larger possibility to pass at one position within the fluvial traces on the downstream part
440 than that on the upstream part (as Wf is fixed as 50 m). In other words, p in Eq. (9) increases
441 along river flow direction (Fig. 3e). In addition, as S decreases with the streamwise distance
2 442 (Fig. 3a), vM resulting from Eq. (3) decreases. Therefore, σ v reduces streamwise (Fig. 3f) due
443 to the increasing p and decreasing vM (according to Eq. 11). Because the influence of p on the
444 v is weaker than influence of vM in this synthetic case, v also presents a decreasing trend
445 along the river flow similar to the trend of vM (Fig. 3f).
446 The geometry of the channel, fluvial traces and riverbed determine the spatial distribution
2 447 of v and σ v , which then affects the sedimentary process via the deposition and erosion of
2 ε S 448 sediment. σ z is composed of deposition perturbation ( D′ = η′ ) and erosion perturbation H c
23
449 ( E′ = wemvm′ ) (Eq. 36). The contrast between deposition perturbation and erosion perturbation
D′ 450 is illustrated in Fig. 3g, where the deposition factor is calculated as f = , whist the D D′ + E′
E′ 451 erosion factor is f = . As a result, the erosion factor reduces along the river flow E D′ + E′
2 452 direction due to the decreases of v and σ v (Fig. 3f and Eq. A4), but the deposition factor
2 453 increases with the streamwise distance because η′ (or ση ) is accumulated along the river
454 flow (Eq. 30).
455 The coupled processes of deposition and erosion in Fig. 3g contribute to a nonlinear trend
σ σ 456 of z (Fig. 3h and Eq. 37), which decreases at the upstream part of the river. After z σ v σ v
457 reaches the lowest value at a position around 1500 m, it increases along the river flow at the
σ z 458 downstream part of the river. Furthermore, because trends of and σ v are reverse at the σ v
459 downstream part (when distance to the river source become larger than 1500 m), the resultant
460 σ z in Fig. 3c decreases again with the streamwise distance at further downstream part after
461 peaking at a position of about 6000 m.
462 5.2 Synthetic example-2
463 This illustrated case investigates the incision processes (over a period of 100 years) of the
464 riverbed formed in previous example, under different Q (10, 50 and 100 m3/d). In this
465 example, sediment input (qs) from the river source is assumed to be zero, and the processes
466 inducing riverbed evolution are the sediment erosion and redeposition.
24
467
468 Figure 4. (a) Incision rate increases with river discharge and (b) the stronger perturbation of
469 remaining sediment thickness induced by the larger river discharge rate. (c) Erosion rate is far
470 larger than the redeposition rate.
471 Fig. 4a shows that significant incision occurs at the upstream part of the river due to the
2 472 strong erosion induced by large σ v and v (considering the velocity trend in Fig. 4.4f under
473 Q=50 m3/d). As indicated by Fig. 4c, the erosion rate herein is far larger than the redeposition
474 rate, and hence the eroded sediments from the upstream part are largely transported out of the
475 system rather than being redeposited at the downstream part.
476 In Fig. 4b, perturbation of sediment thickness (standard deviation, σ z ) is mainly induced
477 by the erosion process, as the redeposition process is too weak. Under a constant Q, σ z
478 decreases along the river flow direction, which is different with trend in Fig. 3c, because the
2 479 trend in Fig. 4b is mainly induced by the decreasing σ v and E′ (recalling Fig. 3f and 3g), but
480 weakly affected by the redeposition process.
481 Moreover, increases of Q enlarge Wc and p within the fluvial trace. As a result, v
2 482 increases with Q due to the increases of both p and Manning velocity (vc). However, σ v can
483 either be enlarged or reduced relating to S, Wf and Q, as the proportion part
484 (1− p)(2 − p) + αp of Eq. (11) (which represents the velocity perturbation induced by the
485 probability of channel occurrence) decreases with the increasing Q, while the Manning
2 486 velocity part ( vM , which is able to represent the velocity perturbation in the river channel in
25
2 487 Eq. 8) increases. In this illustrated case, increases of Q enlarge both the σ v and v , and
488 enhance E . As a result, more sediment on the riverbed is incised at Q of 100 m3/d, and a
489 larger perturbation of remaining sediment thickness can also be induced.
490 6. Conclusion
491 Riverbed and channel evolution, and variation of river discharge affect fluvial processes,
492 essentially via flow velocity changes. The major contribution of this study is in coupling
493 these factors which can induce flow velocity changes in a 1D stochastic fluvial process model
494 (SFPM), by introducing the ensemble statistics (mean and perturbation) of velocity as key
495 parameters in the model instead of the solely stream velocity. The ensemble statistics of
496 velocity can be determined as the function of riverbed slope, width of fluvial traces and river
497 discharge. In addition, a wide range of events inducing velocity changes, such as evaporation
498 and infiltration, can be considered in the model by adjusting the input river discharge.
499 SFPM is developed based on perturbation theory and the non-stationary spectral approach.
500 Output variables of SFPM are the mean and variance of sediment load in the river and
501 sedimentation thickness on the riverbed. SFPM is then applied to two synthetic cases to
502 demonstrate the effects of the model. As a result, both variance and mean values of
503 sedimentation thickness are nonlinearly distributed over space, induced by the nonlinear
504 pattern of riverbed and of river channel, and also of the uncertainty of channel location in the
505 fluvial trace due to the channel movement.
506 SFPM is developed to simulate the longitude profile of sediment within one fluvial trace,
507 which are formed by the single river channel (for the meandering river) or the single sets of
508 river channels (for such as the braided river and alluvial fan). Within one cross section of
509 fluvial trace perpendicular to the river flow direction, the river channel is assumed to have the
510 same possibility to pass at the any positions on this cross section. Limited by this assumption,
511 SFPM cannot simulate the sediment distribution perpendicular to the river flow. Instead, this
26
512 potential lateral distribution is expressed as most likely sediment thickness and the potential
513 variation.
514 SFPM is potentially combined with cellular models (e.g. Sun et al., 2002) to simulate the
515 sedimentary processes within multiple channelized fluvial traces. In addition, the river
516 discharge is assumed to be a deterministic user-specific variable in the model. It may be of
517 interest to couple the probability distribution of river discharge into the ensemble statistics of
518 flow velocity in the future.
519 Appendix A
520 The velocity is a key parameter which affects the sediment transport and accumulation via
521 altering the erosion and deposition rates. This section incorporates ensemble velocity in the
522 expressions of erosion and deposition rates.
523 1. Erosion rate
524 A widely used expression for the erosion rate is given as (e.g. Lague et al., 2005; Tucker,
525 2004):
E = k (τ 3/ 2 −τ 3/ 2 ) , τ > τ , e c c A1
2.5 2 1.5 2 2 526 where ke is the erosion efficiency (L ∙T /M ), τ is the shear stress (ML /T ) and τ c is the
2 2 527 critical shear stress above which the erosion starts (ML /T ). τ c depends on river bed
528 lithology which is calculated as (Cornelis et al., 2004):
τ = k (ρ − ρ )gd , c t s f A2
3 529 where kt is shear parameter, ρs is the dry density of sediments (M/L ), and ρ f is the density
530 of water (M/L3) , d is the diameter of sediments (L).
531 As this study derives the stochastic fluvial process-based model by introducing the
532 statistics of velocity, Eq. (A1) is converted to a function of velocity.
533 The shear stress and stream velocity satisfy (Dade and Friend, 1998):
27
τ = c ρ v2 , d f A3
534 where cd is the dimensionless drag coefficient.
535 Substituting Eq. (A3) in Eq. (A1) and making use of perturbation theory (Eq. 14), the
536 erosion rate is separated as a mean:
E = k [(c ρ )1.5 (v 2 + 3σ 2 )v −τ ], e d f v c A4 537 and a perturbation:
E′ = w v′ , e A5
1.5 2 2 538 where we = ke (cd ρ f ) (3v + σ v ) .
539 2. Deposition rate
540 Assume that erosion and deposition can occur simultaneously, the deposition rate can be
541 simply written as (e.g. Davy and Lague, 2009; Winterwerp and Van Kesteren, 2004):
η D = β , A6 H c
η 542 where is depth-averaged concentration (L3/L3) and β is the deposition coefficient, H c
543 relating to the settling velocity (ε s ), vertical sediment concentration distribution and in this
544 study also relating to the probability of channel occurrence (p):
β = k ε p , d s A7 545 In this current study, the vertical sediment concentration distribution is not simulated, its
546 influence on D are represent by a concentration coefficient kd .
547 ε S is the settling velocity of the particles (L/T), which relates to the fluid and sediments
548 properties (Dade and Friend, 1998):
2 gd ρs − ρ f ε = for d<1 mm, S A8 18ν ρ f
28
ρs − ρ f ε S = gd for d ≥1mm, ρ f
549 where ν is the kinematic viscosity of water (L2/T).
550 Making use of Eq. (14) in Eq. (A6) results in the mean of deposition rate:
β D = η , H c A9
551 and the perturbation
β D′ = η′ , A10 H c
552 It is noted that D′ in Eq. (17) indicates the influence of v′ on η′ via altering the D. As D is
553 unrelated to the stream velocity directly (Eq. A6 and Fig. 1), D′ = 0 in Eq. (17). In contrast,
554 η′ feedbacks on D, which then affects z in Eqs. (36). Therefore, D′ is nonzero in estimates of
2 555 σ z , and is given by Eq. (A10).
556 Appendix B
557 The possible velocity on an arbitrary cross-section of the river channel is given as the vector:
Vc = [v ,v ,v ,...v ] , 1 2 3 n1 B1
558 where n1 the size of the vector, vi is the possible values of stream velocity (i=1, 2,… n1).
559 The ensemble velocity vector is composed by the velocity in the river channel (river
560 flowing through the position of interest) and zero (river channel does not pass at the position):
V = [v ,v ,v ,...v ,0,0..0]. 1 2 3 n1 B2
561 The size of V is n1+n2 and n2 is the number of zero velocity, and the ratio between n1 and
562 n1+n2 is the probability of channel occurrence, which can be expressed as the ratio between
563 the width of river channel and of fluvial trace:
n W p = 1 = c . n1 + n2 W f B3
29
564 where Wc is the channel width and W f is the width of fluvial trace.
565 Assuming that the flow velocity in the river channel follows Gaussian distribution. The
566 mean of Vc is manning velocity and the variance is given in Eq. (8). The statistics (mean and
567 variance) of flow velocity in the river channel can also be written as the matrix pattern:
n1 1 vc = ∑vi , n1 i=1 B4 1 n1 σ 2 = (v − v )2 . vc ∑ i c n1 i=1
568 Moreover, the mean and variance of the ensemble velocity in Eq. (B2) is written as:
+ 1 n1 n1 n2 v = ( v + 0) = p ⋅ v , + ∑ i ∑ c B5 n1 n2 i=1 i=n1+1
+ 1 n1 n1 n2 σ 2 = (v − pv )2 + ( pv )2 , v + ∑ i c ∑ c B6 n1 n2 i=1 i=n1+1
569 Eq. (B6) is further developed as follows:
+ 1 n1 n1 n2 σ 2 = [v − v + (1− p)v ]2 + ( pv )2 v + ∑ i c c ∑ c n1 n2 i=1 i=n1+1 B7 + + + 1 n1 n2 n1 n2 n1 n1 n2 = n σ 2 + 2(1− p)v [ v′ − (0 − v )] + (1− p)2 v 2 + ( pv )2 . + 1 vc c ∑ i ∑ c ∑ c ∑ c n1 n2 i=1 i=n1+1 i=1 i=n1+1
570 Considering that
+ + 1 n1 n2 1 n1 n2 E(v′) = (v − v ) = v′ = 0 , ∑ i c ∑ i B8 n1 i=1 n1 i=1
571 Eq. (B7) can then be written as:
1 σ 2 = [n σ 2 + 2(1− p)v ⋅ n v + n (1− p)2 v 2 + n ( pv )2 ] v 1 vc c 2 c 1 c 2 c n1 + n2 B9 2 = [αp + (1− p)(2 − p) +]vc .
30
572 p =1(Wf = Wc) indicates that the river channel does not evolve within the simulation period.
573 In this situation, v uncertainty only relates to the inherent v variation in the river channel, and
574 Eq. (B9) becomes same as Eq. (8). In contrast, p=0 (Wc= 0) indicates that there is not river
2 575 channel occurrence, therefore, σ v are zero.
576 Appendix C
577 The first order partial differential equation (24) subject to Eq. (26) can be solved as the
578 Cauchy problem, which converts the Eq. (24) as (Hadamard, 2003):
dt dx dφ(x ,t,κ) = k = k . iκxk 1 vk wuk e C1
579 φ(xk ,t,κ) in Eq. (C1) is solved subject to spatial boundary condition (Eq. 24) rather than
580 a temporal initial condition. At the end of the first segment,
wu1 1 iκxˆ1 wu1 1 φ(xˆ1,t,κ) = e − . v1 iκ v1 iκ C2
581 where xˆi indicate the distance to the river source at the end of i-th segement (i=0, 1, 2… n). 582 The first-order partial differential equation (Eq. 24) represents the single-direction
583 influence (Jost, 2012). This means that in a given river segment, the sediment load are only
584 affected by river flow from upstream segments, but are unrelated to the downstream state of
585 the system. For each segment, Eq. (C1) is solved subject to the boundary condition at the
586 beginning of this segment, that is, φ(xˆ1,t,κ) at the end of first segment can be considered as
587 the boundary condition for sediment transport in the second segment, and so on. Therefore,
588 φ(x,t,κ) at the end of the second segment can be expressed as:
wu2 1 iκ (xˆ2 − xˆ1 ) wu1 1 iκxˆ1 wu1 1 wu2 1 φ(xˆ2 ,t,κ) = e + e − ( + ) . v2 iκ v1 iκ v1 iκ v2 iκ C3
589 Similarly, we can write the φ(x,t,κ) at the end of the m-th segment as:
31
m−1 m w 1 κ ˆ − ˆ 1 w κ ˆ − ˆ 1 w φ(xˆ ,t,κ) = um ei (xm xm−1 ) + ( uk )ei (xk xk−1 ) − ( uk ) . m ∑ ∑ C4 vm iκ iκ k =1 vk iκ k =1 vk
590 The entire river is discretized into n segments with equal separation distance so that
591 ∆xk = xˆk − xˆk −1 is a constant larger than zero. Therefore, we can write a general solution for
592 φ(x,t,κ) at each node m as:
m 1 κ∆ 1 w φ(xˆ ,t,κ) = ( ei x − ) ( uk ) , m ∑ C5 iκ iκ k=1 vk
593 where m is the index of node, m= 0, 1, 2,…,n.
594 Acknowledgment
595 Funding support for this study came from China Scholarship Council, and financial support
596 from Exoma Energy Ltd is also gratefully acknowledged. Professor Chris Fielding is thanked
597 for his constructive suggestions on conceptualizing the sedimentology environment. Daniel
598 Owen and Matthias Raiber are thanked for their suggestions on the manuscript.
599 References
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