Model help:SteadyStateAg
SteadyStateAg
This model is a calculator for approach to equilibrium in recirculating and feed flumes
Model introduction
This program implements the calculation for steady-state aggradation of a sand-bed river in response to sea level rise at a constant rate.
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{ {\frac{\partial \eta_{dev}}{\partial t}} = 0 }[/math] (9)
[math]\displaystyle{ \hat{x} = {\frac{x}{L}} }[/math] (10)
[math]\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L}} }[/math] (11)
[math]\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = {\frac{S}{S_{u}}} = 1 - \beta \hat{x} }[/math] (12)
[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] (13)
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] (14)
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] (15)
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] (16)
- 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] (17)
- 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] (18)
Symbol | Description | Unit |
---|---|---|
Q_{bf} | flood of volume wash load per unit width | L^{3} / T |
Q_{tbf} | total volume bed material load at bankfull flow | L^{3} / T |
Q_{tbf,feed} | upstream bed material sediment feed rate during flood | L^{3} / T |
Λ | units of wash load deposited in the fan per unit bed material load deposited | |
I_{f} | 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 |
B_{f} | floodplain width | L |
Ω | channel sinuosity | - |
λ_{p} | bed porosity | - |
Cz | Chezy resistance coefficient | - |
S_{fbl} | initial fluvial bed slope | - |
dη_{d} / 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 |
H_{bf} | bankfull water depth | L |
B_{bf} | bankfull channel width | L |
η | bed surface elevation | L |
Sl | bed surface slope | - |
q_{bT} | volume bedload transport per unit width | L^{2} / T |
B^ | dimensionless bankfull width | - |
Q^ | dimensionless flow discharge | - |
Q^_{t} | dimensionless total volume bed material load | - |
H^ | dimensionless bankfull depth | - |
C_{f} | bed friction coefficient | - |
α_{EH} | parameter for the Engelund-Hansen relation, equals to 0.05 | - |
τ_{form} ^{*} | Channel-forming Shields number | - |
S | down-channel bed slope | - |
κ_{d} | kinematic sediment diffusivity | - |
ξ_{do} | sea level elevation at time t = 0 | - |
ξ_{d} | aggradating rate of river bed | - |
η_{dev} | downstream elevation deviation | - |
x^ | dimensionless down-channel coordinate | - |
β | user-defined parameter | - |
S_{u} | upstream slope at x = 0 | - |
η^ | dimensionless bed surface elevation | - |
G_{feed} | mean annual rate of bed material load available for deposition (include wash load) | - |
G_{t,feed} | mean annual feed rate of bed material load available for deposition in the reach | - |
L_{V} | valley length | - |
H_{n} | normal depth at given Q_{bf} and Q_{tbf,feed} | L |
S_{n} | normal slope at given Q_{bf} and Q_{tbf,feed} | - |
B_{n} | normal width at given Q<bf> and Q_{tbf,feed} | - |
Fr_{n} | normal Froude number at given Q_{bf} and Q_{tbf,feed} | - |
Notes
Assumptions used in the model:
a) Downchannel reach length L is specified; x = L corresponds to the point where sea level is specified;
b) The river is assumed to have a floodplain width B_{f} that is constant, and is much larger than bankfull width B_{bf}.
c) The river is sand-bed with characteristic size D.
d) All the bed material sediment is transported at rate Q_{tbf} during a period constituting (constant) fraction I_{f} of the year, at which the flow is approximated as at bankfull flow, so that the annual yield = I_{f}Q_{tbf}.
e) Sediment is deposited across the entire width of the floodplain as the channel migrates and avulses. For every mass unit of bed material load deposited, Λ mass units of wash load are deposited in the floodplain. It is assumed that the supply of wash load from upstream is always sufficient for deposition at such a rate. In addition, it is assumed for simplicity 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.
f) Sea level rise is constant at rate ξ_{d}. For example, during the period 5,000 – 17,000 BP the rate of rise can be approximated as 1 cm per year.
g) The river is meandering throughout sea level rise, and has constant sinuosity Ω.
h) The flow can be approximated using the normal-flow assumptions. (But the analysis easily generalizes to a full backwater formulation.)
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 ξ_{d} 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.
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.