Model help:AgDegNormal
AgDegNormal
This module calculates a) the equilibrium sediment transport rate and b) the morphodynamic evolution of a reach due to a change in sediment input rate.
Model introduction
The module 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. Bed elevation at the downstream end is assumed to be fixed. The model is based on total bed material load. The model is 1D, assumes a rectangular channel and neglects wall effects.
By modifying the sediment feed rate (G_{tf}) at the upstream end, the river can be forced to aggrade or degrade to a new equilibrium. The module computes this evolution.
Model parameters
Uses ports
This component has no uses ports.
Provides ports
- Model: Provides IRF functionality.
Main equations
- Flow in the channel (using Manning-Strickler formulation)
[math]\displaystyle{ C_{z}={\frac{U}{u_{*}}}=\alpha _{r}\left ( \frac{H}{K_{c}} \right )^{\frac{1}{6}} }[/math] (1)
- grain roughness (Used as roughness height when bedforms are absent)
[math]\displaystyle{ k_{s} = n_{k} D }[/math] (2)
- water conservation for a quasi-steady flow
[math]\displaystyle{ Q = q_{w} B = U B H }[/math] (3)
- Boundary shear stress
[math]\displaystyle{ \tau _{b} = \rho u_{*} ^2 = \rho g H S }[/math] (4)
- Shields number (Shields stress)
[math]\displaystyle{ \tau ^* = {\frac{\tau _{b}}{\rho R g D}} = {\frac{H S}{R D}} }[/math] (5)
- Submerged specific gravity
[math]\displaystyle{ R = {\frac{\rho _{s}}{\rho}} - 1 }[/math] (6)
- Water depth
[math]\displaystyle{ H = [{\frac{\left (k_{c} \right ) ^{\frac{1}{3}} Q_{w}^2}{\alpha _{r} g B^2 S}}]^{\frac{3}{10}} }[/math] (7)
- 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] (8)
- Einstein number
[math]\displaystyle{ q_{t} ^* = {\frac {q_{t}}{\sqrt{R g D} D}} }[/math] (9)
- Cumulative time of the river has been in flood
[math]\displaystyle{ t_{f} = I_{f} t }[/math] (10)
- 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] (11)
- Volume sediment transport rate per unit width obtained at the graded state
[math]\displaystyle{ q_{t} = {\frac{G_{tf}}{\rho_{s}Bl_{f} t_{a}}} }[/math] (12)
- 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] (13)
- Exner equation of sediment continuity (average over many floods)
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}}= - {\frac{\partial l_{f} q_{t}}{\partial x}} }[/math] (14)
- Spatial derivative of the total bed material load per unit width
[math]\displaystyle{ \frac{\Delta q_{t,i}}{\Delta X}=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} }[/math] (15)
- Bed slope computed in each node
[math]\displaystyle{ S_{i}=\left\{\begin{matrix} \frac{\eta _{1}-\eta _{2}} {\Delta x} & i=1\\ \frac{\eta _{i-1}- \eta _{i+1}} {2\Delta X} & i=2...M \\ \frac{\eta _{M} - \eta _{M+1}}{\Delta X} & i=M+1 \end{matrix}\right. }[/math] (16)
- Initial bed elevation
[math]\displaystyle{ \eta_{i}=S \left ( L - x_{i} \right ) }[/math] (17)
- non-dimensional total shear stress
[math]\displaystyle{ \tau _{b} ^* = {\frac{\tau _{b}}{\left ( \rho _{s} - \rho \right ) g D}} }[/math] (18)
Symbol | Description | Unit |
---|---|---|
Q | flood discharge | L ^{3} / T |
t | time step | T |
B | river width | L |
D | grain size of the bed sediment | L |
λ_{p} | bed porosity | - |
k_{c} | composite roughness height | L |
G | imposed annual sediment transfer rate from upstream | M / T |
G_{tf} | upstream sediment feed rate | - |
ξ_{d} | downstream water surface elevation | L |
L | length of reach under consideration | L |
q_{w} | water discharge per unit width | L^{2} / T |
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_{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 / T |
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 | - |
u_{*} | shear velocity | L / T |
α_{r} | coefficient in Manning-Strickler resistance relation | - |
τ_{b} ^{*} | non-dimensional total shear stress | - |
Output
Symbol | Description | Unit |
---|---|---|
η | river bed elevation | L |
H | water depth | L |
ξ | water surface elevation | L |
τ_{b} | bed shear stress | - |
S | bed slope | - |
Notes
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 maximum number of computational nodes, M, is 99 (this is the case for all of the AgDeg functions).
The model calculates the water depth with a Chezy formulation, if only the Chézy coefficient is specified in the input file. The code uses a Manning-Strickler formulation, when only the roughness height, k_{c}, and the coefficient α_{r} are given in the input text file. If all these parameters are in the text file, the program will ask the user which formulation he 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., and Angevine, C.L., 1992. The large-scale dynamics of grain-size variation in alluvial basins. 1: Theory. Basin Research, 4, 73-90.
- Meyer-Peter, E., and Müller, R., 1948. Formulas for bed-load transport Proceedings, 2^{nd} Congress International Association for Hydraulic Research, Rotterdam, the Netherlands, 39-64.