Model help:AgDegNormalSub
AgDegNormalSub
This program is used to calculate the evolution of upward-concave bed profiles in rivers carrying uniform sediment in subsiding basins.
Model introduction
The program computes the approach to mobile-bed equilibrium in a river carrying uniform material and flowing into a subsiding basin. It is a descendant of AgDegNormal.
Model parameters
Uses ports
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Provides ports
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Main equations
- Exner equation for uniform sediment from a river
<math> \left ( 1 - \lambda _{p} \right ) \left ( {\frac{\partial \eta }{\partial t}} + \delta \right ) = - {\frac{I_{f}}{r_{B}}} \left ( 1 + \Lambda \right ) \Omega {\frac{\partial q_{t}}{\partial x}} </math> (1)
- Ratio of depositional to channel width
<math> r_{B} = {\frac{B_{d}}{B_{c}}} </math> (2)
- The maximum possible length of the fluvial reach
<math> L_{max} = {\frac{I_{f} \left ( 1 + \Lambda \right ) \Omega }{r_{B}}} {\frac{q_{tf}}{\left ( 1 - \lambda _{p} \right ) \delta}} </math> (3)
Symbol | Description | Unit |
---|---|---|
Q | flood discharge | L ^{3} / T |
x | streamwise coordinate | L |
η | river bed elevation | L |
t | time step | T |
B_{c} | Channel width | L |
D | grain size of the bed sediment | L |
σ | subsidence rate | - |
r_{B} | the ratio of depositional width to channel width | - |
Ω | channel sinuosity | - |
Λ | units of wash load deposited in the system per unit of bed material load | - |
λ_{p} | bed porosity | - |
q_{w} | water discharge per unit width | L^{2} / T |
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 |
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 |
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 | - |
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_{f} | bed friction coefficient, equals to τ_{b} / (ρ U^{2} ) | - |
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 | - |
B_{d} | deposition width | - |
q_{tf} | volume feed rate per unit width of total bed material load | L^{2} / T |
Output
Symbol | Description | Unit |
---|---|---|
H | water depth | L |
ξ | water surface elevation | L |
L_{max} | maximum length of basin the the sediment supply can fill | L |
τ_{b} | bed shear stress | M / (T^{2} L) |
S | bed slope | - |
q_{t} | total bed material load | L^{2} / T |
Notes
some assumptions: The subsidence rate s is assumed to be constant in time and space. The sediment is assumed to be uniform with size D. A Manning-Strickler formulation is used for bed resistance. A generic relation of the general form of that due to Meyer-Peter and Muller is used for sediment transport. The flow is computed using the normal flow approximation. The river is assumed to have a constant width.
The program is a descendant of AgDegNormal. Three relatively minor changes have been implemented as follows:
a) The input parameters have been modified to include the following parameters: subsidence rate σ, ratio of depositional width to channel width r_{B}, ratio of wash load deposited per unit bed material load Λ and channel sinuosity Ω;
b) The code has been modified so as to include subsidence in the calculation of mass balance;
c) The output has been modified to show the time evolution of not only the profile of bed elevation η, but also the profiles of bed slope S and the ratio q_{t}/q_{tf}, where q_{tf} denotes the volume feed rate of bed material load per unit width.
The ratio of depositional to channel width, r_{B}, has been introduced to model the fact that in an aggrading river sediment deposits not only in the channel itself, but also in a much wider belt (e.g. the floodplain or basin width, due to overbank deposition, channel migration and avulsion). Here channel width is denoted as B_{c} (which can be taken to be synonymous with bankfull width) and effective depositional width is denoted as B_{d}.
The parameter Λ that represents the units of wash load deposited per unit of bed material is introduced to consider that in the 1D formulation implemented in this model it is assumed that deposition occurs not only in the channel but on a much wider area (e.g. the floodplain). Sediment deposited in the channel is mostly made of bed material but sediment deposited around the channel contains a significant amount of wash load. A precise mass balance for wash load is beyond the scope of this model. For simplicity it is assumed that for every unit of sand deposited in the system, Λ units of wash load are deposited. It is also assumed that the supply of wash load from upstream is always sufficient for deposition at such a rate. This is not likely to be strictly true, but should serve as a useful starting assumption.
The parameter Ω has been introduced to consider that channels may be sinuous. Here it is assumed that the channel has a sinuosity, Ω, but that the depositional surface across which it wanders is rectangular. In the present formulation the sinuosity is defined as the ratio of downchannel distance per unit of downvalley distance.
Boundary and initial conditions are equal to that implemented for the ancestor model AgDegNormal.
- Note on model running
Flow is calculated assuming normal flow approximation
If the input channel length is longer than the maximum possible length of the fluvial reach, the program cannot perform the calculation. The maximum possible length of the fluvial reach, Lmax, is defined as the maximum length of basin that the sediment supply can fill; at this length the sediment transport rate out of the basin drops precisely to zero.
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
Key papers