Model help:RiverWFRisingBaseLevelNormal
RiverWFRisingBaseLevelNormal
This model is a calculator for disequilibrium aggradation of a sand-bed river in response to rising base level.
Model introduction
This program uses a Chézy formulation and either the Engelund-Hansen relation for bedload in sandy streams or the Parker relation for bedload in gravel bed streams assuming a uniform grain size to solve for the bed evolution, width evolution, and depth evolution in time and space.
Model parameters
Uses ports
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Provides ports
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Main equations
- Exner equation
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}} = - \Omega {\frac{I_{f} \left ( 1 + \Lambda \right )}{B_{f}}} {\frac{\partial Q_{tbf}}{\partial x}} }[/math] (1)
- Relation for sediment transport
1) Dimensionless bankfull width
[math]\displaystyle{ \hat{B} = {\frac{C_{f}}{\alpha_{EH} \left ( \tau_{form}^* \right ) ^ \left (2.5 \right )}} \hat{Q}_{t} }[/math] (2)
2) Down-channel bed slope
[math]\displaystyle{ S = {\frac{R^ \left ({\frac{3}{2}}\right ) C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{\hat{Q}_{t}}{\hat{Q}}} }[/math] (3)
3) Dimensionless bankfull depth
[math]\displaystyle{ \hat{H} = {\frac{\alpha_{EH} \left ( \tau_{form} \right )^2}{\left ( R C_{f} \right ) ^\left ({\frac{1}{2}}\right )}}{\frac{\hat{Q}}{\hat{Q}_{t}}} }[/math] (4)
4) Total volume bed material load at bankfull flow
[math]\displaystyle{ Q_{tbf} = {\frac{\alpha_{EH} \tau_{form}^*}{R C_{f} ^ \left ({\frac{1}{2}}\right )}} Q_{bf} S }[/math] (5)
5) Reduction of the Exner equation
[math]\displaystyle{ {\frac{\partial \eta}{\partial t}} = \kappa_{d}{\frac{\partial ^2 \eta}{\partial x^2}} }[/math] (6)
6) Kinematic sediment diffusivity
[math]\displaystyle{ \kappa_{d} = {\frac{I_{f} \Omega \left ( 1 + \Lambda \right )}{\left ( 1 - \lambda_{p}\right ) B_{f}}} {\frac{\alpha _{EH} \tau_{form}^* Q_{bf}}{R C_{f}^\left ({\frac{1}{2}}\right )}} }[/math] (7)
- Bed elevation
[math]\displaystyle{ \eta = \xi_{do} + \xi_{d} t + \eta_{dev} \left (x, t \right ) }[/math] (8)
- Steady-State aggradation in response to sea level rise condition
[math]\displaystyle{ \hat{x} = {\frac{x}{L}} }[/math] (9)
[math]\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L}} }[/math] (10)
[math]\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = {\frac{S}{S_{u}}} = 1 - \beta \hat{x} }[/math] (11)
[math]\displaystyle{ \beta = {\frac{\left ( 1 - \lambda_{p}\right ) B_{f} \xi_{d} L}{I_{f} \Omega \left ( 1 + \Lambda \right ) Q_{tbf,feed}}} = {\frac{G_{fill}}{G_{feed}}} }[/math] (12)
1) Sediment delivery rate
[math]\displaystyle{ Q_{tbf} = Q_{tbf,feed} - {\frac{\left (1 - \lambda_{p} \right ) B_{f}}{I_{f} \Omega \left ( 1 + \Omega \right )}} \xi_{d} X }[/math] (13)
2)Upstream slope at x = 0
[math]\displaystyle{ S_{u} = {\frac{R C_{f}^\left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }[/math] (14)
3) Elevation profile
[math]\displaystyle{ \hat{\eta} = S_{u} [\left ( 1 - {\frac{1}{2}} \beta \right ) - \hat{x} + {\frac{1}{2}} \beta \hat{x}^2] }[/math] (15)
- Mean annual rate available for deposition
[math]\displaystyle{ G_{feed} = \left ( 1 + \Lambda \right ) G_{t,feed} = \left ( 1 + \Lambda \right ) \left ( R + 1 \right ) Q_{tbf,feed} I_{f} }[/math] (16)
- Amount of sediment required to fill a reach at a uniform aggradation rate
[math]\displaystyle{ G_{fill} = \left ( R + 1 \right ) \left ( 1 - \lambda_{p} \right ) B_{f} L_{v} \xi_{d} }[/math] (17)
- Morphodynamics of the approach to steady-state response to rising sea level
1) Governing partial differential equation
[math]\displaystyle{ {\frac{\partial \eta_{dev}}{\partial t}} + \dot{\xi}_{d} = \kappa_{d} {\frac{\partial ^2 \eta_{dev}}{\partial x^2}} }[/math] (18)
2) Boundary conditions
[math]\displaystyle{ - {\frac{\partial \eta_{dev}}{\partial x}} = S_{u} = {\frac{R C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }[/math] (19)
[math]\displaystyle{ \eta_{dev} \left (L, t \right ) = 0 }[/math] (20)
3) Initial condition
[math]\displaystyle{ \eta_{dev} \left (x,0\right ) = \eta_{dev,I}\left (x\right ) }[/math] (21)
| Symbol | Description | Unit |
|---|---|---|
| Qbf | flood of volume wash load per unit width | L3 / T |
| Qtbf | total volume bed material load at bankfull flow | L3 / T |
| Qtbf,feed | upstream bed material sediment feed rate during flood | L3 / T |
| Λ | units of wash load deposited in the fan per unit bed material load deposited | |
| If | intermittency | - |
| D | grain size of bed material | L |
| R | submerged specific gravity of sediment (e.g. 1.65 for quartz) | - |
| L | reach length (downchannel distance) | L |
| Bf | floodplain width | L |
| Ω | channel sinuosity | - |
| λp | bed porosity | - |
| Cz | Chezy resistance coefficient | - |
| Sfbl | initial fluvial bed slope | - |
| dηd / dt | rate of rise of downstream base level (should be positive) | L / T |
| M | number of intervals | - |
| Δx | spatial step | L |
| Δt | time step | T |
| Mtoprint | number of time steps to printout | - |
| Mprint | number of printouts | - |
| e | rate of downstream base level rise | L / T |
| p | number of prints | - |
| i | number of iterations per print | - |
| t | time step | T |
| y | year the base level change begins | T |
| Y | year the base level change ends | T |
| a | coefficient in the Engelund-Hansen 1967 load relation | - |
| n | exponent in the Engelund-Hansen 1967 load relation | - |
| T | channel-forming Shields number for sand-bed streams | - |
| P | coefficient in the Parker 1979 load relation | - |
| N | exponent in the Parker 1979 load relation | - |
| c | critical Shield number | - |
| G | channel-forming Shields number for gravel-bed streams | - |
| x | downstream coordinate | L |
| Hbf | bankfull water depth | L |
| Bbf | bankfull channel width | L |
| η | bed surface elevation | L |
| Sl | bed surface slope | - |
| qbT | volume bedload transport per unit width | L2 / T |
| B^ | dimensionless bankfull width | - |
| Q^ | dimensionless flow discharge | - |
| Q^t | dimensionless total volume bed material load | - |
| H^ | dimensionless bankfull depth | - |
| Cf | bed friction coefficient | - |
| αEH | parameter for the Engelund-Hansen relation, equals to 0.05 | - |
| τform * | Channel-forming Shields number | - |
| S | down-channel bed slope | - |
| κd | kinematic sediment diffusivity | - |
| ξdo | sea level elevation at time t = 0 | - |
| ξd | aggradating rate of river bed | - |
| ηdev | downstream elevation deviation | - |
| x^ | dimensionless down-channel coordinate | - |
| β | user-defined parameter | - |
| Su | upstream slope at x = 0 | - |
| η^ | dimensionless bed surface elevation | - |
| Gfeed | mean annual rate of bed material load available for deposition (include wash load) | - |
| Gt,feed | mean annual feed rate of bed material load available for deposition in the reach | - |
| LV | valley length | - |
| Gfill | Amount of sediment required to fill a reach at a uniform aggradation rate | - |
| Gfeed | Mean annual rate available for deposition | - |
Output
| Symbol | Description | Unit |
|---|---|---|
| τform * | channel-forming Shields number | - |
| Hn | normal depth at given Qbf and Qtbf,feed | L |
| Sn | normal slope at given Qbf and Qtbf,feed | - |
| Bn | normal width at given Q<bf> and Qtbf,feed | - |
| Frn | normal Froude number at given Qbf and Qtbf,feed | - |
Notes
This program is a companion to the program SteadyStateAg, which computes the steady-state aggradation of a river with a specified base level rise at the downstream end. This program computes the time evolution toward steady-state aggradation.
The calculation assumes a specified, constant Chezy resistance coefficient Cz and floodplain width Bf. 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. The flow is computed using the normal flow approximation. The reach has downchannel length L, and base level is allowed to rise at a specified rate at the downstream end. It is assumed that for every unit of sand deposited in the channel/floodplain system in response to sea level rise, Λ units of wash load are deposited, where Λ is a specified constant that might range from 0 to 3 or higher, and that the supply of wash load from upstream is always sufficient for deposition at such a rate. Besides, it is assumed that the porosity of the floodplain deposits is equal to that of the channel deposits. In fact the floodplain deposits are likely to have a lower porosity.
In the equations, is if β > 1 then the there is not enough sediment feed over the reach to fill the space created by sea level rise, and the sediment transport rate must drop to zero before the shoreline is reached. If β < 1 the excess sediment is delivered to the sea. If for a given rate of sea level rise ξd it is found that β > 1 for reasonable values of reach length L, floodplain width Bf and sediment feed rate Gfeed, no steady state solution exists for that rate of sea level rise.
- Note on model running
For the characteristic grain size, a value of < 2 mm will use the Engelund-Hansen relation, whereas ≥ 2 mm will use the Parker relation.
Due to the evolution that is occurring in time, the GetData function has been included in this function, and works the same way as in the AgDeg programs.
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
- Bard, E., Hamelin, B., and Fairbanks, R.G., 1990, U-Th ages obtained by mass spectrometry in corals from Barbados: sea level during the past 130,000 years, Nature 346, 456-458.Doi [10.1038/346456a0]
- Fisk, H.N., 1944, Geological investigations of the alluvial valley of the lower Mississippi River, Report, U.S. Army Corp of Engineers, Mississippi River Commission, Vicksburg, MS.
- Pirmez, C., 1994, Growth of a Submarine meandering channel-levee system on Amazon Fan, Ph.D. thesis, Columbia University, New York, 587 p.
- Paola, C., P. L. Heller and C. L. Angevine, 1992, The large-scale dynamics of grain-size variation in alluvial basins. I: Theory, Basin Research, 4, 73-90.
- Parker, G., and Y. Cui, 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 1: Simplified analytical solution, Journal of Hydraulic Research, 36(1): 75-100.
- Parker, G., Paola, P., Whipple, K. and Mohrig, D., 1998, Alluvial fans formed by channelized fluvial and sheet flow: theory, Journal of Hydraulic Engineering, 124(10), pp. 1-11. Doi: [10.1061/(ASCE)0733-9429(1998)124:10(985)]
- Sinha, S. K. and Parker, G., 1996, Causes of concavity in longitudinal profiles of rivers, Water Resources Research 32(5),1417-1428. Doi: [10.1029/95WR03819]
- USCOE, 1935., Studies of river bed materials and their movement, with special reference to the lower Mississippi River, Paper 17 of the U.S. Waterways Experiment Station, Vicksburg, MS.
- Yatsu, E., 1955, On the longitudinal profile of the graded river, Transactions, American Geophysical Union, 36: 655-663.
