Model help:Acronym1D
Acronym1D
This program acts the same way as the Acronym1 do, with the addition of a “flow duration curve;” the program calculates the same values (though taken as a mean annual) as well as the mean annual water discharge, and the characteristics of the flow duration curve.
Model introduction
“Acronym1_D” simply adds a flow duration curve to the algorithm of “Acronym1_R” in order to compute the average volume gravel bedload transport rate per unit width, as well as the average bedload grain size distribution. In addition, it computes the annual mean values of the water discharge, depth, shear velocity and Shields stress based on surface geometric mean size. The values D_{alg}, σ_{alg}, D_{al90}, D_{al70}, D_{al50} and D_{al30} associated with the mean grain size distribution of the bedload are computed along with the corresponding values for the surface material, D_{sg}, σ_{sg}, D_{s90}, D_{s70}, D_{s50} and D_{s30}. Finally, the program computes the volume gravel bedload transport rate per unit width, the water discharge, flow depth, the shear velocity and the Shields stress associated with each range in the flow duration curve, along with the fraction of time that the flow is in that range.
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
- Characteristic grain size for the ith grain size range (spans (D_{b,i}, D_{b,i+1})) (for i=1...N)
<math>D_{i}= \sqrt{ D_{b, i} D_{b, i+1} } </math> (1)
- Fraction in the surface layer F_{i} for the ith grain size range (for i=1...N)
<math>F_{i}= \left ( F_{f, i} - F_{f, i+1} \right ) / 100 </math> (2)
- Grain Size on the base-2 logarithmic scale:
<math>\Psi= LN_{2}\left (D\right) = {\frac{log_{10}\left (D\right)}{log_{10}\left (2\right)}} </math> (3)
- Geometric mean size of the surface material
<math>D_{sg}=2^\left (\bar\Psi_{s} \right ) </math> (4)
<math>\bar\Psi_{s}= \sum\limits_{i=1}^N \Psi_{i} F{i} </math> (5)
- The geometric standard deviations
<math>\sigma_{sg}= 2 ^\sigma </math> (6)
- the arithmetic standard deviations
<math>\sigma ^2= \sum\limits_{i=1}^N \left (\Psi_{i} - \bar\Psi \right )^2 F_{i} </math> (7)
- The transport relation
<math>W_{i}^*= {\frac{Rgq_{bi}}{F_{i}u_{*} ^3}}= 0.00218 G \left (\Phi \right ) </math> (8)
<math>\phi= \omega \phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) </math> (9)
<math> \phi_{sgo}= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} </math> (10)
<math>\tau_{sg} ^*={\frac{u_{*} ^2}{R g D_{sg}}} </math> (11)
<math> G \left ( \phi \right )=\left\{\begin{matrix} 5474 \left ( 1 - {\frac{0.853}{\phi}} \right ) ^ \left (4.5 \right ) & \phi > 1.59 \\ exp[ 14.2\left ( \phi - 1\right ) - 9.28 \left ( \phi - 1 \right )^2 ] & 1 <= \phi <= 1.59 \\ \phi ^\left (14.2 \right ) & \phi < 1 \end{matrix}\right. </math> (12)
<math>\omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \phi_{sgo} \right ) }} [ \omega_{O} \left ( \phi_{sgo} \right ) - 1 ] </math> (13)
- total volume gravel bedload transport rate per unit width summed over all sizes
<math>q_{bT}= \sum\limits_{i=1}^N q_{bi} </math> (14)
- fraction of gravel bedload in the ith grain size range
<math>p_{i}= {\frac{q_{bi}}{q_{bT}}} </math> (15)
- Geometric mean of the bedload
<math>D_{lg}= 2 ^\left (\bar\psi_{l} \right ) </math> (16)
<math>\Psi_{l}= \sum\limits_{i=1}^\left (Np \right ) \Psi_{i} p_{i} </math> (17)
- Geometric standard deviation of the bedload
<math>\delta_{lg}= 2 ^\left ( \delta_{l} \right ) </math> (18)
<math>\delta_{l} ^2= \sum\limits_{i=1}^\left (Np \right ) \left ( \Psi_{i} - \bar\Psi_{l} \right )^2 p_{i} </math> (19)
- Grain sizes in the bedload material
<math>D_{lx}= 2 ^\left (\Psi_{lx} \right ) </math> (20)
<math>\Psi_{lx}= \Psi_{b, i+1} + {\frac{\Psi_{b, j} - \Psi_{b, i+1}}{p_{f, i} - p_{f, i+1}}}\left ( x - p_{f, i+1} \right ) </math> (21)
<math>\Psi_{b, i}= Ln_{2} \left ( D_{b, j} \right ) </math> (22)
- Characteristic flow in the kth range (k=1...M)
<math>Q_{wr,k}= {\frac{1}{2}} \left ( Q_{wd,k} + Q_{wd, k+1} \right ) </math> (23)
- Fraction of time the flow is in the kth range
<math>P_{Q,k}= {\frac{p_{eQ,k+1} - p_{eQ,k}}{100}} </math> (24)
Symbol | Description | Unit |
---|---|---|
D | grain size | L |
D_{i} | characteristic grain size for the ith grain size range (i=1...N) | L |
F_{i} | fraction in surface layer for the ith grain size range(for i =1...N) | - |
τ_{ssrg} ^{*} | equals to 0.0386 | - |
ψ | grain sizes on the base-2 logarithmic ψ scale | |
ρ | density of water | M / L^{3} |
ρ_{s} | density of sediment | M / L^{3} |
R | submerged specific density of sediment, equals to (ρ_{s} /ρ-1) | - |
u | shear velocity of flow | L / T |
g | acceleration of gravity | L / T^{2} |
τ_{b} | boundary shear stress on the bed | M / (L T) |
u_{*} | shear velocity on the bed, equals to sqrt(τ_{b} / ρ ) | L / T |
p_{i} | fraction of gravel bedload in the ith grain size range | L |
ψ_{s} | equals to τ_{bs} / τ_{b} | |
W_{i} ^{*} | dimensionless bedload transport rate for ith grain size | - |
q_{bi} | volume gravel bedload transport per unit width of grains in the ith size range | L ^{2} / T |
ω | straining relation in Parker (1990a,b) bedload relation for mixtures | |
G(Φ) | function in Parker (1990a, b) and Wilcock and Crowe (2003) bedload relation for gravel mixture | |
ω_{0} | function relation in Parker (1990a, b) bedload relation for mixture | - |
Φ | parameter in Parker (1990a, b) bedload relation for mixures | |
Φ_{sgo} | equals to τ_{sg} ^{*} / τ_{ssrg} ^{*} | - |
ψ_{l} | grain sizes on the base-2 logarithmic ψ scale for bedload | - |
ψ_{lx} | grain sizes on the base-2 logarithmic ψ scale for bedload such that x percent of the material is finer | - |
D_{g} | geometric mean size | L |
σ_{g} | geometric standard deviation | - |
D_{x} | diameter such that x% of the distribution is finer | L |
Q_{wd,k} | The kth discharge | L^{3} / T |
Q_{wr,k} | Characteristic flow in the kth range | L^{3} / T |
p_{eQ,k} | the percentage of time the kth flow is exceeded | - |
p_{Q,k} | fraction of time that the flow is in the kth range | - |
Output
Symbol | Description | Unit |
---|---|---|
q_{bT} | total volume gravel bedload transport rate per unit width summed over all sizes | L^{2} / T |
τ_{sg} ^{*} | Shields number based on surface geometric mean size | - |
D_{sg} | geometric mean size of the surface material | L |
σ_{sg} | geometric standard deviations of the surface material | - |
D_{lg} | geometric mean of the bedload material | L |
σ_{lg} | geometric standard deviations of the bedload material | - |
D_{alg} | geometric mean size of the bedload | L |
σ_{alg} | geometric standard deviations of the bedload | - |
σ | arithmetic standard deviations of the surface materials | - |
σ_{lg} | geometric standard deviation of the bedload | - |
D_{sx} | grain size in the surface material, such that x percentage of the material is finer | L |
D_{lx} | grain size in the bedload material, such that x percentage of the material is finer | L |
D_{alx} | mean grain size in the bedload material, such that x percentage of the material is finer | L |
σ_{l} | arithmetic standard deviations of bedload materials | - |
Q_{a} | mean water discharge | L^{3} / T |
H_{a} | mean water depth | L |
u_{*a} | mean shear velocity | L / T |
τ_{ga} | mean Shields stress | - |
Notes
- Note on the program
The flow duration curve is specified in terms of the pairs (Q_{wd,k}, p_{eQ,k}), k = 1..M+1. Here k = 1 corresponds to the highest flow in the curve, with an exceedance percentage p_{eQ} of zero, and k = M+1 corresponds to the lowest flow in the curve, with an exceedance percentage p_{eQ} of 100. The lowest flow on the curve Q_{wd,M} must exceed zero.
Let Y_{k} be any parameter defined for each of the flow ranges k = 1..M. The mean value Y_{a} averaged over the flow duration curve is then given as
Y_{a} = Sum(Y_{k} p_{Q,k}), k=1...M
For example, if the fractions in the bedload in each grain size range within flow range k are given as p_{k,i} then the average fractions of the bedload p_{ai} are given as
p_{ai} = Sum(p_{k,i} p_{Q,k})
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., 1990a. Surface based bedload transport relation for gravel rivers. Journal of Hydraulic Research, 28(4), 417~436.(DOI:10.1080/00221689009499058)
- Parker, G.,1990b. The "ACRONYM" series of Pascal programs for computing bedload transport in gravel rivers. External Memorandum M 220, St. Anthony Falls Hydraulic Laboratory, University of Minnesota.