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

Parameter	Description	Unit
Input directory	Determine if you want to use the "GUI" interface to provide input parameter values or use a text file with the input parameters by providing the location of the file on the server.	[-]
Site prefix	Part of the input and output file name e.g. the name of the geographic location, or project	[-]
Case prefix	Part of the input and output file name that provides you the opportunity to do different scenario simulations for e.g. the same location, or project	[-]

Parameter	Description	Unit
Flood discharge		[m²/s]
Intermittency		[-]
Channel width		[m]
Grain size		[mm]
Bed porosity		[-]
Roughness height		[mm]
Ambient Bed Slope		[-]
Imposed sediment transport rate		[tons/year]
Length of reach		[m]
Time step		[days]
Iterations per each printout		[-]
Number of printouts		[-]
Number of fluvial nodes		[-]
Upwinding coefficient		[-]
Manning-Strickler coefficient α_r		[-]
Coefficient in sediment transport relation		[-]
Exponent in sediment tranpsort relation		[-]
Critical Shields stress		[-]
Fraction bed shear stress	Fraction of bed shear stress due to skin friction	[-]
Submerged specific gravity		[-]

Parameter	Description	Unit
Model name	The name of the model	[-]
Author name	The developer of the model	[-]

Uses ports

This component has no uses ports.

Provides ports

Model: Provides IRF functionality.

Main equations

Manning-Strickler formulation

[math]\displaystyle{ C_{f}=\alpha _{r}\left ( \frac{H}{K_{c}} \right )^{\frac{1}{6}} }[/math]

(1)

Total bed material load per unit width

[math]\displaystyle{ \frac{q_{t}}{{\sqrt{RgDD}}}=\alpha_{t}\left ( \frac{\varphi_{s}\tau _{b}}{\rho RgD} -\tau_{c}^* \right )^{n_{t}} }[/math]

(2)

Exner equation

[math]\displaystyle{ \left ( 1-\lambda_{p} \right )\frac{\partial \eta }{\partial t}=-I_{f}\frac{\partial q_{t}}{\partial X} }[/math]

(3)

Discretized Exner equation

[math]\displaystyle{ \eta \left| _{i,t+\Delta t}=\eta \right |_{i,t}-\frac{I_{f}}{1-\lambda _{p}}\frac{\Delta q_{t,i}}{\Delta X}\Delta t }[/math]

(4)

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]

(5)

Bed slope is computed in each node

[math]\displaystyle{ S=\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]

(6)

Initial profile

[math]\displaystyle{ \eta \left ( x,t \right ) | _{t=0}=\eta _{Id}+S_{I}\left ( L-x \right ) }[/math]

(7)

Nomenclature

Symbol	Description	Unit
X	Streamwise coordinate	m
ΔX	Spatial step length	m
t	Temporal coordinate	seconds
C_f	Non-dimensional friction coefficient	-
Q_w	Flood discharge	m³/s
I_f	Flood intermittency	-
B_c	Channel width	m
D	Characteristic grain size	mm
λ_p	Bed porosity	-
K_c	Composite roughness height	mm
S_I	Ambient bed slope	-
G_tf	Imposed annual sediment transport rate	tons/annum
L	Length of reach	m
Δt	Time step	year
M	Number of spatial intervals	-
A_u	Upwinding coefficient (1 = full upwind, 0.5 = central difference)	-
α_r	Coefficient in Manning-Strickler	-
α_s	Coefficient in sediment transport relation	-
η_t	Exponent in sediment transport relation	-
τ^*_c	Reference Shields number in sediment transport relation	-
φ_s	Fraction of bed shear stress due to skin friction	-
R	Submerged specific gravity	-
Cz	Non-dimensional Chézy friction coefficient	-

Output

Symbol	Description	Unit
η	Bed surface elevation	m
S	Bed slope	-
H	Water depth	m
τ_b	Total (skin friction + form drag) Shields number	-
a_t	Bed material load per unit channel width	m²/s

Notes

The maximum number of computational nodes, M, is 99;
The model calculates the water depth with a Chezy formulation, if only the Chézy coefficient is specified in the input text 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.
The model prompts user whether he would like to append some of the characteristic values for the initial and final equilibrium state to the output file, or write them in a separate file.

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: http://csdms.colorado.edu/wiki/Special:Upload
Create link to the file on your page: [[Image:<file name>]].

Developer(s)

Gary Parker

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.

Links

Model:AgDegNormal
http://vtchl.uiuc.edu/people/parkerg/word_files.htm (Chapter 14, "1D aggradation and degradation of rivers: normal flow assumption" of the e-book).