Model help:AgDegBW
AgDegBW
It is the Calculator for aggradation and degradation of a river reach using a backwater formulation. This program computes 1D bed variation in rivers due to differential sediment transport. The sediment is assumed to be uniform with size D. All sediment transport is assumed to occur in a specified fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler relation is used for bed resistance. A generic Meyer-Peter Muller relation is used for sediment transport. The flow is computed using a backwater formulation for gradually varied flow.
Model introduction
The model calculates a) an ambient mobile-bed equilibrium, and b)the response of a river reach to either 1) changed sediment input rate at the upstream end of the reach starting from t = 0 or 2) changed downstream water surface elevation at the downstream end of the reach starting from t = 0, where t is the temporal coordinate. The code is very similar to AgDegNorm. The main difference between the two codes is in the procedure to compute the water depth. In AgDegNorm the flow is assumed normal (i.e. steady and uniform), while in AgDegBW the flow is assumed steady and it is computed solving the backwater equation. The case of Froude-subcritical flow, for which Fr < 1, is considered herein. This implies that integration of the backwater equation must proceed upstream from x = L, with x streamwise coordinate and L length of the modeled reach. Both a Chezy and a Manning-Striclker formulation can be used to compute the flow.
Model parameters
Uses ports
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Provides ports
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Main equations
- Computation of the flow
The backwater equation
[math]\displaystyle{ {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - F_{r} ^2}} }[/math] (1)
- Friction slope
[math]\displaystyle{ S_{f} = C_{f} F_{r} ^2 }[/math] (2)
- Froude number
[math]\displaystyle{ F_{r} ^2 = {\frac{U^2}{g H}} = {\frac{q_{w} ^2}{g H^3}} }[/math] (3)
- Flow velocity
[math]\displaystyle{ U = {\frac{q_{w}}{H}} }[/math] (4)
- The bed friction coefficient ( assumed to obey a Manninbg-Strickler resistance )
[math]\displaystyle{ C_f ^ \left ( {\frac{-1}{2}} \right ) = C_{z} = \alpha_{r} \left ( {\frac{H}{k_{c}}} \right ) ^{\frac{1}{6}} }[/math] (5)
- grain roughness (Used as roughness height when bedforms are absent)
[math]\displaystyle{ k_{s} = n_{k} D }[/math] (6)
- The relation between bed slope S and bed elevation η (Froude-subcritical flow (Fr < 1))
[math]\displaystyle{ S = -{\frac{\partial \eta}{\partial x}} }[/math] (7)
- Water surface elevation (Froude-subcritical flow (Fr < 1))
[math]\displaystyle{ \epsilon = \eta + H }[/math] (8)
- Shields number
[math]\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} = {\frac{C_{f} U^2}{R g D}} = {\frac{C_{f} {\frac{q_{w} ^2}{H^2}}}{R g D}} }[/math] (9)
- Bed shear stress
[math]\displaystyle{ \tau_{b} = \rho C_{f} U^2 }[/math] (10)
- Submerged specific gravity
[math]\displaystyle{ R = {\frac{\rho_{s}}{\rho}} - 1 }[/math] (11)
- Computation of the sediment transport (Meyer-Peter and Muller equation )
[math]\displaystyle{ q_{t} ^* = \left\{\begin{matrix} \alpha_{t} \left ( \varphi_{s} \tau ^2 - \tau_{c} ^* \right ) ^ \left ( n_{t} \right ) & \tau^* \gt \tau_{c}^* \\ 0 & \tau^* \lt = \tau_{c}^* \end{matrix}\right. }[/math] (12)
- Einstein number
[math]\displaystyle{ q_{t} ^* = {\frac {q_{t}}{\sqrt{R g D} D}} }[/math] (13)
- Cumulative time of the river has been in flood
[math]\displaystyle{ t_{f} = I_{f} t }[/math] (14)
- Equilibrium (graded) states
- Annual sediment yield with a graded state at this slope
[math]\displaystyle{ G_{t} = \rho_{s} q_{t} Bl_{f} t_{a} }[/math] (15)
- Volume sediment transport rate per unit width obtained at the graded state
[math]\displaystyle{ q_{t} = {\frac{G_{tf}}{\rho_{s}B I_{f} t_{a}}} }[/math] (16)
- Computation of bed variation
- Exner equation of sediment continuity (assume that q_{t} is zero for most of the time)
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial\eta}{\partial t}} = - {\frac{\partial q_{t}}{\partial x}} }[/math] (17)
- Exner equation of sediment continuity (average over many floods)
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}}= - {\frac{\partial I_{f} q_{t}}{\partial x}} }[/math] (18)
- Numerical scheme
- Initial bed elevation
[math]\displaystyle{ \eta_{i}=S \left ( L - x_{i} \right ) }[/math] (19)
- Computation of the depth of upstream node
[math]\displaystyle{ {\frac{H_{i+1} - H_{i}}{\Delta x}} = F_{back} \left ( H \right ) = {\frac{{\frac{\eta _{i+1} - \eta _{i}}{\Delta x} - {\frac{1}{\alpha _{r} ^2}}\left ({\frac{H}{k_{c}}}\right )^\left ({\frac{-1}{3}} \right ){\frac{q_{w}^2}{g H^3}}}}{\frac{q_{w}^2}{g H^3}}} }[/math] (20)
- Boundary condition of depth
[math]\displaystyle{ H_{M + 1} = \xi _{d} - \eta _{M+1} }[/math] (21)
- A predictor-corrector scheme used to solve for H
[math]\displaystyle{ H_{pred}=H_{i+1} - F_{back}\left (H_{i+1}\right ) \Delta x }[/math] (22)
[math]\displaystyle{ H_{i} = H_{i+1} - {\frac{1}{2}}[F_{back} \left ( H_{pred} \right )+ F_{back} \left (H_{i+1} \right )] \Delta x }[/math] (23)
- Spatial derivative of the total bed material load per unit width
[math]\displaystyle{ \frac{\Delta q_{t,i}}{\Delta X}=\left\{\begin{matrix} a_{u}\frac{q_{t,i}-q_{t,i-1}}{\Delta X}+\left ( 1-a_{u} \right )\frac{q_{t,i+1}-q_{t,i}}{\Delta X} & i=1...M \\ {\frac{q_{t,i} - q_{t,i-1}}{\Delta x}} & i=M + 1 \end{matrix}\right. }[/math] (24)
Symbol | Description | Unit |
---|---|---|
Q | flood discharge | L ^{3} / T |
x | streamwise coordinate | L |
t | time step | T |
B | river width | L |
D | grain size of the bed sediment | L |
λ_{p} | bed porosity | - |
ξ_{d} | downstream water surface elevation | L |
q_{w} | water discharge per unit width | L^{2} / T |
k_{c} | composite roughness height | L |
G | imposed annual sediment transfer rate from upstream | M / T |
G_{tf} | upstream sediment feed rate | - |
L | length of reach under consideration | L |
i | number of time steps per printout | - |
p | number of printouts desired | - |
M | number of spatial intervals | - |
R | submerged specific gravity of sediment | - |
S_{f} | friction slope | - |
F_{r} | Froude number | - |
U | flow velocity | L / T |
C_{f} | bed friction coefficient | - |
g | acceleration of gravity | L / T^{2} |
α_{r} | coefficient in Manning-Stricker, dimensionless coefficient between 8 and 9 | - |
k_{s} | grain roughness | L |
n_{k} | dimensionless coefficient typically between 2 and 5 | - |
τ^{*} | Shield number | - |
ρ | fluid density | M / L^{3} |
ρ_{s} | sediment density | M / L^{3} |
τ_{c} | critical Shields number for the onset of sediment motion | - |
ψ_{s} | the fraction of bed shear stress | - |
q_{t} ^{*} | Einstein number | - |
q_{t} | volume sediment transport rate per unit width | L^{2} / T |
I_{f} | flood intermittency | - |
t_{f} | cumulative time the river has been in flood | T |
G_{t} | the annual sediment yield | M / T |
t_{a} | the number of seconds in a year | - |
Q_{f} | sediment transport rate during flood discharge | L^{2} / T |
α_{t} | dimensionless coefficient in the sediment transport equation, equals to 8 | - |
n_{t} | exponent in sediment transport relation, equals to 1.5 | - |
τ_{c} ^{*} | reference Shields number in sediment transport relation, equals to 0.047 | |
C_{f} | bed friction coefficient, equals to τ_{b} / (ρ U^{2} ) | - |
C_{Z} | dimensionless Chezy resistance coefficient. | |
S_{l} | initial bed slope of the river | - |
η_{i} | initial bed elevation | L |
τ | shear stress on bed surface | - |
q_{b} | bed material load | M / L |
Δx | spatial step length, equals to L / M | L |
Q_{w} | flood discharge | L^{3} / T |
Δt | time step | T |
Ntoprint | number of time steps to printout | - |
Nprint | number of printouts | - |
a_{U} | upwinding coefficient (1=full upwind, 0.5=central difference) | - |
α_{s} | coefficient in sediment transport relation | - |
n_{k} | parameter such that k_{s} = n_{k} D_{s90} | - |
Output
Symbol | Description | Unit |
---|---|---|
η | bed surface elevatioon | T |
H | water depth | T |
ξ | water surface elevation | T |
τ_{b} | bed shear stress | M / (T^{2} L) |
S | bed slope | - |
q_{t} | total bed material load | L^{2} / T |
Notes
The model computes variation in river bed level η(x, t), where x denotes a streamwise coordinate and t denotes time, in a river with constant width B. The bed sediment is characterized in terms of a single grain size D and submerged specific gravity R. The reach under consideration has length L. Water surface elevation at the downstream end is prescribed. The model is based on a calculation of total bed material load. The model is 1D, assumes a rectangular channel and neglects wall or bank effects.
By modifying the upstream sediment feed rate G_{tf} and/or the downstream water surface elevation ξ_{d}, the river can be forced to aggrade or degrade to a new equilibrium. The program computes this evolution.
Actual rivers tend to be morphologically active only during floods. That is, most of the time they are not doing much to modify their morphology. The simplest way to take this into account is to assume an intermittency I_{f} such that the river is in flood a fraction I of the time, during which Q = Q_{f} and q_{t} is the sediment transport rate at this discharge (Paola et al., 1992). For the other (1 – I_{f}) fraction of time the river is assumed not to be moving sediment.
Output is controlled by the parameters Ntoprint and Nprint. The code will implement Ntoprint time steps. In addition to output pertaining to the initial state, the code implements Nprint outputs, to that the total number of time steps executed is equal to Nprint x Ntoprint.
- Note on model running
The downstream water surface elevation must exceed the H_{c}, critical depth, which is equal to (Q_{w} ^{2}/(B_{c} ^{2}g))^{1/3}, otherwise the user is alerted, and the program exits.
The water depth is calculated using a Chézy formulation, and the Manning-Strickler formulation is implemented, when only the roughness height, k_{c}, and the coefficient α_{r} are given in input file. When all the three parameters are present, the program will ask the user which formulation they would like to use.
Examples
An example run with input parameters, BLD files, as well as a figure / movie of the output
Follow the next steps to include images / movies of simulations:
- Upload file: https://csdms.colorado.edu/wiki/Special:Upload
- Create link to the file on your page: [[Image:<file name>]].
See also: Help:Images or Help:Movies
Developer(s)
References
- Paola, C., Heller, P. L. & Angevine, C. L. 1992 The large-scale dynamics of grain-size variation in alluvial basins. I: Theory. Basin Research, 4, 73-90.
- Meyer-Peter, E., and Müller, R. 1948 Formulas for bed-load transport. Proceedings, 2nd Congress International Association for Hydraulic Research, Rotterdam, the Netherlands, 39-64.