Model help:AgDegNormGravMixSubPW
AgDegNormGravMixSubPW
It is the calculator for evolution of upward-concave bed profiles in rivers carrying sediment mixtures in subsiding basins.
Model introduction
This program calculates the bed surface evolution for a river of constant width with a mixture of gravel sizes with a load computed either by the Parker relation or the Wilcock-Crowe relation, as in the case of AgDegNormGravMixPW, but this program also takes into effect the subsidence.
Model parameters
Uses ports
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Provides ports
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Main equations
- Total bedload transport over all grain sizes
[math]\displaystyle{ q_{bT} = \sum\limits_{i=1}^N q_{bi} }[/math] (1)
- Fraction of bedload in the ith grain size range
[math]\displaystyle{ p_{bi} = {\frac{q_{bi}}{q_{bT}}} }[/math] (2)
- Exner equation describing the evolution of grain size distribution of the active layer
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) [L_{a}{\frac{\partial F_{i}}{\partial t}} + \left (F_{i} - f_{li}\right ) {\frac{\partial L_{a}}{\partial t}}] = - I_{f} {\frac{\partial q_{bT} p_{bi}}{\partial x}} + I_{f} f_{li} {\frac{\partial q_{bT}}{\partial x}} }[/math] (3)
- Fraction in the ith grain size range of materials exchanged between the surface and substrate as the bed aggrades or degrades
[math]\displaystyle{ f_{li} = \left\{\begin{matrix} f_{i}|_{Z = \eta - L_{a}} & {\frac{\partial \eta}{\partial t}} \lt 0 \\ \lambda F_{i} + \left ( 1 - \lambda \right ) p_{bi} & {\frac{\partial \eta}{\partial t}} \gt 0 \end{matrix}\right. }[/math] (4)
- Surface-based bedload transport formulation for mixtures
1) Thickness of the active (surface) layer of the bed
[math]\displaystyle{ L_{a} = n_{a} D_{s90} }[/math] (5)
2) Dimensionless grain size specific Shields number
[math]\displaystyle{ \tau_{i}^* \equiv {\frac{\tau_{b}}{\rho R g D_{i}}} = {\frac{u_{*}^2}{R g D_{i}}} }[/math] (6)
3) Grain size specific Einstein number
[math]\displaystyle{ q_{bi}^* = {\frac{q_{bi}}{\sqrt{R g D_{i}}D_{i} F_{i}}} }[/math] (7)
4) Dimensionless grain size specific bedload transport rate
[math]\displaystyle{ W_{i}^* \equiv {\frac{q_{bi}^*}{\left ( \tau_{i}^* \right )^ \left ({\frac{3}{2}}\right )}} = {\frac{R g q_{bi}}{\left (u_{*}\right )^3 F_{i} }} }[/math] (8)
- Bedload relation for mixtures due to Parker (1990a, b)
[math]\displaystyle{ W_{i}^* = 0.00218 G\left (\phi_{i} \right ) }[/math] (9)
[math]\displaystyle{ \phi_{i}= \omega \phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) }[/math] (10)
[math]\displaystyle{ \phi_{sgo}= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} }[/math] (11)
[math]\displaystyle{ \tau_{sg} ^*={\frac{u_{*} ^2}{R g D_{sg}}} }[/math] (12)
[math]\displaystyle{ G \left ( \phi \right )= \left\{\begin{matrix} 5474 \left ( 1 - {\frac{0.853}{\phi}} \right ) ^ \left (4.5 \right ) & \phi \gt 1.59 \\ exp[ 14.2\left ( \phi - 1\right ) - 9.28 \left ( \phi - 1 \right )^2 ] & 1 \lt = \phi \lt = 1.59 \\ \phi ^\left (14.2 \right ) & \phi \lt 1 \end{matrix}\right. }[/math] (13)
[math]\displaystyle{ \omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \phi_{sgo} \right ) }} [ \omega_{O} \left ( \phi_{sgo} \right ) - 1 ] }[/math] (14)
[math]\displaystyle{ D_{sg}=2^\left (\bar\Psi_{s} \right ) }[/math] (15)
[math]\displaystyle{ \bar\Psi_{s}= \sum\limits_{i=1}^N \Psi_{i} F{i} }[/math] (16)
[math]\displaystyle{ \delta_{s} ^2 = \sum\limits_{i=1}^N \left (\Psi_{i} - \bar{\Psi}_{s} \right )^2 F_{i} }[/math] (17)
- Bedload relation for mixture due to Wilcock and Crowe (2003)
[math]\displaystyle{ W_{i}^* = G \left (\phi_{i}\right ) }[/math] (18)
[math]\displaystyle{ G = \left\{\begin{matrix} 0.002 \phi^ \left (7.5 \right ) & for \phi \lt 1.35 \\ 14 \left (1 - {\frac{0.894}{\phi_{0.5}}}\right )^ \left (4.5 \right ) & \phi \gt = 1.35 \end{matrix}\right. }[/math] (19)
[math]\displaystyle{ \phi_{i} = {\frac{\tau_{sg}^*}{\tau_{ssrg}^*}} \left ( {\frac{D_{i}}{D_{sg}}}\right )^ \left (-b \right ) }[/math] (20)
[math]\displaystyle{ \tau_{sg}^* = {\frac{u_{*}^2}{R g D_{sg}}} }[/math] (21)
[math]\displaystyle{ \tau_{ssrg}^* = 0.021 + 0.015 exp \left (-14 F_{s}\right ) }[/math] (22)
[math]\displaystyle{ b = {\frac{0.69}{1 + exp \left (1.5 - {\frac{D_{i}}{D_{sg}}}\right )}} }[/math] (23)
- Roughness hight
[math]\displaystyle{ k_{s} = n_{k} D_{s90} }[/math] (24)
- Boundary shear stress
[math]\displaystyle{ \tau_{b,k} = \rho u_{*}^2 = \left ({\frac{k_{s,k}^\left ({\frac{1}{3}} \right ) q_{w}^*}{\alpha_{r}^2}} \right )^ \left ({\frac{3}{10}}\right ) g^\left ({\frac{7}{10}}\right ) S_{k}^ \left ({\frac{7}{10}}\right ) }[/math] (25)
- Bed slope
[math]\displaystyle{ S_{k} = \left\{\begin{matrix} {\frac{\eta_{1} - \eta_{2}}{\Delta x}} & k = 1 \\ {\frac{\eta_{k-1} - \eta_{k+1}}{2 \Delta x}} & k = 2...M \end{matrix}\right. }[/math] (26)
- Shields number based on the geometric mean size
[math]\displaystyle{ \tau_{sg}^* = \left ({\frac{k_{s}^ \left ({\frac{1}{3}}\right ) q_{w}^2}{\alpha_{r}^2 g}}\right ) ^ \left ({\frac{3}{10}}\right ) {\frac{S^ \left ({\frac{7}{10}}\right )}{R D_{sg}}} }[/math] (27)
- Shear velocity based on the surface geometric mean size
[math]\displaystyle{ u_{*} = \left ({\frac{k_{s}^ \left ({\frac{1}{3}}\right ) q_{w}^2}{\alpha_{r}^2}}\right )^ \left ({\frac{3}{20}}\right ) g^ \left ({\frac{7}{20}}\right ) S^ \left ({\frac{7}{20}}\right ) }[/math] (28)
- Volume bedload transport rate per unit width in the ith grain size
[math]\displaystyle{ q_{bi} = F_{i}{\frac{u_{*}^3}{R g}}W_{i}^* }[/math] (29)
Symbol | Description | Unit |
---|---|---|
x | streamwise coordinate | L |
t | time step | T |
Z | boundary-attached upward normal coordinate | L |
D_{bi} | ith bounding size specifying percents or fractions finer of grain size distribution | L |
λ_{p} | bed porosity | - |
α | the parameter that governs the grain size distribution of the sediment at the active layer-substrate interface during bed aggredation (0 <= α <= 1) | - |
F_{fi} | grain size distribution of the active layer for initial condition | - |
F_{i} | fraction of material in the surface layer in the ith grain size range, i = 1...N | - |
D_{i} | characteristic size of the ith grain size range | L |
f_{subfi} | fraction of sediment in the ith grain size range in the substrate layer for initial condition | - |
F_{fli} | percent finer than ith grain size range for the bed surface for initial condition | - |
F_{subfli} | percent finer than ith grain size range for the substrate layer for initial condition | - |
f_{li} | fraction of sediment in the ith grain size range in the active-layer substrate interface | - |
p_{i} | fraction of sediment in the ith grain size range in the bedload | - |
F_{sub,i} | fraction of substrate material in the ith size range | - |
D_{s90} | the diameter of the bed surface such that the 90% of the sediment is finer | L |
n_{a} | user specified order-one non dimensional constant | - |
p_{ffi} | the percent that is finer than the ith size range for upstream boundary conditon | - |
η_{d} | fixed bed elevation at the downstream end of the modeled reach | L |
k_{c} | composite roughness height | L |
G | function in Parker (1990a, b) and Wilcock and Crowe (2003) bedload relation for gravel mixture | |
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 | - |
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_{i} ^{*} | dimensionless grain size specific Shields number | - |
τ_{b} | bed shear stress | - |
u_{*} | shear velocity | L / T |
q_{bi} ^{*} | Einstein bedload number for ith grain size | L / T |
W_{i} ^{*} | dimensionless grain size specific bedload transport rate | - |
φ_{i} | parameter in Parker (1990a,b) bedload relation for mixtures | - |
ω | straining parameter in Parker (1990a,b) bedload relation for
mixtures |
- |
φ_{sg} ^{*} | Shields number based on surface geometric mean size | - |
φ_{ssrg} ^{*} | reference Shields number close to (but above) the threshold of motion, equals to 0.0386 for Parker (1990a,b) | - |
ω_{O} | function relation in Parker (1990a, b) bedload relation for mixture | - |
δ_{O} | function relation in Parker (1990a, b) bedload relation for mixture | - |
D_{sg} | geometric mean size of surface layer sediment | L |
δ_{s} | arithmetic standard deviation of surface size distribution | - |
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 |
α_{t} | dimensionless coefficient in the sediment transport equation, equals to 8 | - |
b | reference distance above the bed where sediment entrainment is specified | - |
n_{k} | an order-one dimensionless number | - |
F_{s} | mass fraction of the surface sediment that is sand | - |
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 | - |
D_{sub50} | median size of the substrate layer | L |
D_{subg} | geometric mean size of the substrate layer | L |
L_{a} | thickness of the active layer | L |
σ | subsidence rate | L / T |
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 | - |
τ | shear stress on bed surface | - |
q_{bi} | volume bedload transport rate per unit width of material in the ith grain size range | L^{2} / T |
q_{bT} | total volume bedload transport rate per unit width | L^{2} / 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 | - |
φ | transverse angle of inclination of bed | - |
λ | wavelength of bedform | - |
φ_{sgo} | equals to τ_{sg} ^{*} / τ_{ssrg} ^{*} | - |
Output
Symbol | Description | Unit |
---|---|---|
η | bed surface elevatioon | L |
H | water depth | L |
ξ | water surface elevation | L |
τ_{b} | bed shear stress | M / (T^{2} L) |
S | bed slope | - |
L_{max} | maximum length of basin that the sediment supply can fill | L |
Notes
The river is assumed to be morphologically active for If fraction of time, during which the flow is approximated as constant. Otherwise, the river is assumed to be morphologically dead.
The river flows into a basin that is subsiding with rate σ. The basin has constant width. For each unit of bedload deposited, L units of washload (typically sand transported in suspension) is deposited across the depositional basin.
If run for a sufficient length of time, the river profile approaches a steady-state balance between subsidence. At this steady state the profile displays both an upward-concave elevation profile and downstream fining of the surface material.
The upstream point, at which sediment is fed, is fixed in the horizontal to be at x = 0. The vertical elevation of the upstream point may change freely as the bed aggrades or degrades.
The reach has constant length L, so that the downstream point is fixed in the horizontal at x = L. This downstream point has a user-specified initial elevation η_{d}.
Gravel bedload transport of mixtures is computed with a user-specified selection of the Parker (1990), or Wilcock-Crowe (2003) surface-based formulations for gravel transport.Sand and finer material must first be excluded from the grain size distributions, which then must be renormalized for gravel content only, in the case of the Parker (1990) relation. In the case of the Wilcock-Crowe (2003) relation, the sand is retained in the computation.
The grain size distributions of the sediment feed, initial surface material and substrate material must be specified. It is assumed that the grain size distribution of the sediment feed rate does not change in time, the initial grain size distribution of the surface material is the same at every node, the grain size distribution of the substrate is the same at every node and does not vary in the vertical.
The program does not store the vertical and streamwise structure of the new substrate created as the bed aggrades. As a result, is cannot capture the case of aggradation followed by degradation. Again, the constraint is easy to relax, but at the price of increased memory requirements for storing the newly-created substrate.
The flow is calculated using the normal flow (local equilibrium) approximation.
- Note on model running
In the case of the load relation due to Parker (1990), the grain size distributions are automatically re-normalized because the relation is for the transport of gravel only in the case of the load relation due to Wilcock-Crowe (2003), the sand and the fine sediment are retained for the computation.
The input grain size distributions may be on a 0-100% or a 0.00-1.00 scale, and the program will automatically scale.
The input grain size distributions must have bounds at 0% and 100% (1.00) to properly perform the calculation. If the user does not input the bounds the program will automatically interpolate upper and lower bounds D_{bU} and D_{bL} such that f_{fU} = 100 (1.00) and f_{fL} = 0
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, k_{c}, 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
- Parker, G., 1990, Surface-based bedload transport relation for gravel rivers, Journal of Hydraulic Research, 28(4): 417-436.
- Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment, Journal of Hydraulic Engineering, 129(2), 120-128.