Model help:BedrockAlluvialTransition
BedrockAlluvialTransition
This is used to calculate aggradation and degradation with a migrating bedrock-alluvial transition at the upstream end.
Model introduction
This program calculates the bed surface evolution at predefined nodes relative to moving boundary nodes for a transition from bedrock to allvium.
Model parameters
Uses ports
This will be something that the CSDMS facility will add
Provides ports
This will be something that the CSDMS facility will add
Main equations
- Continuity condition at the bedrock-alluvial transition
[math]\displaystyle{ {\frac{\partial \eta}{\partial t}} | _{s_{ba}} - S | _{s_{ba}} \dot{s}_{ba} = - S_{b} | _{s_{ba}} \dot{s}_{ba} }[/math] (1)
- coordinate transformation
[math]\displaystyle{ \bar{x} = {\frac{x - s_{ba} \left ( t \right )}{s_{d} - s_{ba} \left ( t \right )}} }[/math] (2)
[math]\displaystyle{ \bar{t} = t }[/math] (3)
- Bed slope
[math]\displaystyle{ S = - {\frac{\partial \eta}{\partial x}} = - {\frac{1}{s_{d} - s_{ba}}} {\frac{\partial \eta}{\partial \bar{x}}} }[/math] (4)
- Exner equation
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}} = - I_{f} {\frac{\partial q_{t}}{\partial x}} }[/math] (4)
- Speed of migration of the bedrock-alluvial transition
[math]\displaystyle{ \dot{s}_{ba} = - {\frac{1}{S_{b} |_{\bar{x} = 0}}} {\frac{\partial \eta}{\partial \bar{t}}} | _{\bar{x} = 0} }[/math] (5)
| Symbol | Description | Unit |
|---|---|---|
| x | downstream coordinate (0 = initial bedrock - alluvial transition) | L |
| Sl | bed slope | - |
| qb | volume bedload transport per unit width | L2 / T |
| H | water depth | L |
| qw | water discharge per unit width during floods | L2 / T |
| If | flood intermittency (0 < I <= 1) | - |
| Q | volume sediment feed rate per width at upstream and during flood | L2 / T |
| D | grain size of alluvium | L |
| Cz | coefficient in Chezy relation, Cf | - |
| b | slope of bedrock basement | - |
| Sinit | initial slope of alluvial region | - |
| L | bed porosity | - |
| k | coefficient in Manning-Strickler relation | - |
| a | coefficient in Manning-Strickler relation | - |
| d | position of the downstream end of the reach | L |
| M | number of spatial intervals | - |
| t | time step | T |
| i | number of interations per print | - |
| p | number of prints | - |
| sd | position of the downstream end of the reach, equals to initial length of alluvial region | - |
| qtf | volume sediment feed rate per width at upstream end during floods | L2 / T |
| Sb | slope of bedrock basement | - |
| dt | time step | T |
| Mtoprint | number of steps to printout | - |
| Mprint | number of printouts | - |
| sba | position of the bedrock-alluvial transition, change with time | - |
| S | alluvial bed slope | - |
| dot{sba} | speed of migration of the bedrock-alluvial transition | - |
| Sb | slope of the bedrock channel | - |
| bar{x} | dimensionless coordinate | - |
| bar{t} | dimensionless time | - |
| x | downstream coordinate | L |
| η | bed surface elevation | L |
| λp | porosity of sediment | - |
Notes
This program computes fluvial aggradation/degradation with a bedrock-alluvial transition. The bedrock-alluvial transition is located at a point sba(t) which is free to change in time. A bedrock basement channel with slope Sb is exposed from x = 0 to sba(t); it is covered with alluvium from x = sba(t) to x = sd, where Sd is fixed. Initially sba = 0. The bedrock basement channel is assumed to undergo no incision on the time scales at which the alluvial reach responds to change. In computing bed level change on the alluvial reach, the normal (steady, uniform) flow approximation is used. Base level is maintained at x = sd, where bed elevation h = 0. The Engelund-Hansen relation is used to compute sediment transport rate, so the analysis is appropriate for sand-bed streams. Resistance is specified in terms of a constant Chezy coefficient Cz.
From the continuity equation, it could be derived that If the bed aggrades, the transition moves upstream; if the bed degrades the transition moves downstream.
- Note on model running
The water depth is calculated using a Chézy formulation, when only the Chézy coefficient is present in the inputted file, and with the Manning-Strickler formulation, when only the roughness height, kc, value is present. When both 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
Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams, Technisk Vorlag, Copenhagen, Denmark.
Parker, G. and Muto, T., 2003, 1D numerical model of delta response to rising sea level, Proc. 3rd IAHR Symposium, River, Coastal and Estuarine Morphodynamics, Barcelona, Spain, 1-5 September.
Sklar, L., and W. E. Dietrich, 1998, River longitudinal profiles and bedrock incision models: Stream power and the influence of sediment supply, in Rivers Over Rock: Fluvial Processes in Bedrock Channels, Geophys. Monogr. Ser., vol. 107, edited by K. J. Tinkler and E. E. Wohl, pp. 237–260, AGU, Washington, D. C.
Whipple, K. X., G. S. Hancock, and R. S. Anderson, 2000, River incision into bedrock: Mechanics and relative efficacy of plucking, abrasion, and cavitation, Geol. Soc. Am. Bull., 112, 490–503.
Wong, M. and Parker, G., submitted, The bedload transport relation of Meyer-Peter and Müller overpredicts by a factor of two, Journal of Hydraulic Engineering
