Acronym1D
“Acronym1_D” combines the scheme of “Acronym1_R” with a flow duration curve. The bedload transport rate and bedload grain size distribution are computed for each flow of the curve, and then averaged to yield a mean bedload transport rate and a mean bedload grain size distribution.
Model introduction
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 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 (Db,i, Db,i+1)) (for i=1...N)
[math]\displaystyle{ D_{i}= \sqrt{ D_{b, i} D_{b, i+1} } }[/math] (1)
- Fraction in the surface layer Fi for the ith grain size range (for i=1...N)
[math]\displaystyle{ F_{i}= \left ( F_{f, i} - F_{f, i+1} \right ) / 100 }[/math] (2)
- Grain Size on the base-2 logarithmic scale:
[math]\displaystyle{ \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]\displaystyle{ D_{sg}=2^\left (\bar\Psi_{s} \right ) }[/math] (4)
[math]\displaystyle{ \bar\Psi_{s}= \sum\limits_{i=1}^N \Psi_{i} F{i} }[/math] (5)
- The geometric standard deviations
[math]\displaystyle{ \sigma_{sg}= 2 ^\sigma }[/math] (6)
- the arithmetic standard deviations
[math]\displaystyle{ \sigma ^2= \sum\limits_{i=1}^N \left (\Psi_{i} - \bar\Psi \right )^2 F_{i} }[/math] (7)
- The transport relation
[math]\displaystyle{ W_{i}^*= {\frac{Rgq_{bi}}{F_{i}u_{*} ^3}}= 0.00218 G \left (\Phi \right ) }[/math] (8)
[math]\displaystyle{ \phi= \omega \Phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) }[/math] (9)
[math]\displaystyle{ \Phi_{sgo}= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} }[/math] (10)
[math]\displaystyle{ \tau_{sg} ^*={\frac{u_{*} ^2}{R g D_{sg}}} }[/math] (11)
- φ> 1.59
[math]\displaystyle{ G \left ( \Phi \right )= 5474 \left ( 1 - {\frac{0.853}{\Phi}} \right ) ^ \left (4.5 \right ) }[/math] (12)
- 1<=φ<=1.59
[math]\displaystyle{ G\left (\Phi \right )= exp[ 14.2\left ( \Phi - 1\right ) - 9.28 \left ( \Phi - 1 \right )^2 ] }[/math] (13)
- φ< 1
[math]\displaystyle{ G\left (\Phi \right )= \Phi ^\left (14.2 \right ) }[/math] (14)
[math]\displaystyle{ \omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \Phi_{sgo} \right ) }} [ \omega_{O} \left ( \Phi_{sgo} \right ) - 1 ] }[/math] (15)
- total volume gravel bedload transport rate per unit width summed over all sizes
[math]\displaystyle{ q_{bT}= \sum\limits_{i=1}^N q_{bi} }[/math] (16)
- fraction of gravel bedload in the ith grain size range
[math]\displaystyle{ p_{i}= {\frac{q_{bi}}{q_{bT}}} }[/math] (17)
- Geometric mean of the bedload
[math]\displaystyle{ D_{lg}= 2 ^\left (\bar\psi_{l} \right ) }[/math] (18)
[math]\displaystyle{ \Psi_{l}= \sum\limits_{i=1}^\left (Np \right ) \Psi_{i} p_{i} }[/math] (19)
- Geometric standard deviation of the bedload
[math]\displaystyle{ \delta_{lg}= 2 ^\left ( \delta_{l} \right ) }[/math] (20)
[math]\displaystyle{ \delta_{l} ^2= \sum\limits_{i=1}^\left (Np \right ) \left ( \Psi_{i} - \bar\Psi_{l} \right )^2 p_{i} }[/math] (21)
- Grain sizes in the bedload material
[math]\displaystyle{ D_{lx}= 2 ^\left (\Psi_{lx} \right ) }[/math] (22)
[math]\displaystyle{ \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] (23)
[math]\displaystyle{ \Psi_{b, i}= Ln_{2} \left ( D_{b, j} \right ) }[/math] (24)
- The kth discharge
[math]\displaystyle{ Q_{wr,k}= {/frac{1}{2}} \left ( Q_{wd,k} + Q_{wd, k+1} \right ) }[/math] (25)
- The percentage of time the kth flow is exceeded
[math]\displaystyle{ P_{Q,k}= {\frac{p_{eQ,k+1} - p_{eQ,k}}{100}} }[/math] (26)
| Symbol | Description | Unit |
|---|---|---|
| D | grain size | mm |
| Di | characteristic grain size for the ith grain size range (i=1...N) | mm |
| Fi | fraction in surface layer for the ith grain size range(for i =1~N) | - |
| τssrg * | equals to 0.0386 | - |
| ψ | grain sizes on the base-2 logarithmic ψscale | |
| Dsg | geometric mean size of the surface material | mm |
| σsg | geometric standard deviations of the surface materials | - |
| σ | arithmetic standard deviations of the surface materials | - |
| ρ | density of water | kg/m3 |
| ρs | density of sediment | kg/m3 |
| R | submerged specific density of sediment, equals to (ρs /ρ-1) | - |
| u | shear velocity of flow | m / s |
| g | acceleration of gravity | m / s2 |
| τb | boundary shear stress on the bed | kg / (m s) |
| u* | shear velocity on the bed, equals to | m / s |
| qbi | volume gravel bedload transport per unit width of grains in the ith size range | m 2 |
| pi | fraction of gravel bedload in the ith grain size range | mm |
| Dlg | geometric mean of the bedload | mm |
| σlg | geometric standard deviation of the bedload | - |
| Dsx | grain size in the surface material, such that x percentage of the material is finer | mm |
| Dlx | grain size in the bedload material, such that x percentage of the material is finer | - |
| ψs | equals to τbs / τb | |
| Wi * | dimensionless bedload transport rate for ith grain size | - |
| ω | straining relation in Parker (1990a,b) bedload relation for mixtures | |
| τsg * | Shields number based on surface geometric mean size | - |
| G(Φ) | function in Parker (1990a, b) and Wilcock and Crowe (2003) bedload relation for gravel mixture; the function is different in each case | |
| ω0 | function relation in Parker (1990a, b) bedload relation for mixture | - |
| Φ | ||
| Φsgo | - | |
| τsgo | ||
| ψl | - | |
| σl | ||
| ψlx | - |
Output
| Symbol | Description | Unit |
|---|---|---|
| qbT | total volume gravel bedload transport rate per unit width summed over all sizes | - |
| Dg | geometric mean | mm |
| τg * | shields number based on surface geometric mean size | kg / (m s) |
| σg | geometric standard deviation | - |
| Dx | diameter such that x% of the distribution is finer | mm |
Notes
- Note on the program
“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 qbTa, as well as the average bedload grain size distribution (Db,i, paf,i), i = 1..N+1. In addition, it computes the values Qa, Ha, u∗a and τga∗ corresponding to annual mean values of the water discharge, depth, shear velocity and Shields stress based on surface geometric mean size. The values Dalg, σalg, Dal90, Dal70, Dal50 and Dal30 associated with the mean grain size distribution of the bedload are computed along with the corresponding values for the surface material, Dsg, σsg, Ds90, Ds70, Ds50 and Ds30. Finally, the program computes the volume gravel bedload transport rate per unit width qbT, the water discharge Qw, flow depth H, the shear velocity u∗ and the Shields stress τg∗ associated with each range in the flow duration curve, along with the fraction of time pQ that the flow is in that range.
The flow duration curve is specified in terms of the pairs (Qwd,k, peQ,k), k = 1..M+1, where Qwd,k denotes the kth discharge and peQ,k denotes the percentage of time this flow is exceeded. Here k = 1 corresponds to the highest flow in the curve, with an exceedance percentage peQ of zero, and k = M+1 corresponds to the lowest flow in the curve, with an exceedance percentage peQ of 100. The lowest flow on the curve Qwd,M must exceed zero.
- Note on equations
In order to implement the equations 9~15, it is necessary to specify a) the surface grain size distribution (Df,i, Ff,i) and b) the shear velocity u∗. This results in a predicted values of qbi.
If the boundary shear stress at the bed includes a component of form drag, the component must be removed before computing u∗.
Once the parameters qbi are known the total volume bedload transport rate per unit width qbT and the fractions pi in the bedload can be calculated as equations 16, 17.
The results are presented in terms of qbT and the grain size distribution of the bedload, which is computed from the values of pi. These same fractions pi are used to compute the geometric mean and geometric standard deviation of the bedload Dlg and σlg, respectively, from the relations of equations 18, 19, 20, 21.
The percent finer in the bedload pf,i for the grain size Df,i is obtained from the fractions pi as follows: pf,1 = 100 pf,i = pf,i-1 - 100 pi-1 (i=2~N+1)
Let Dsx and Dlx denote sizes in the surface and bedload material, respectively, such that x percent of the material is finer. For example, if x = 50 then Ds50 and Dl50 denote the median sizes of the surface and bedload material, respectively. Once Ff,i is specified (pf,i is computed) the value Dsx (Dlx) can be computed by interpolation. The interpolation should be done using a logarithmic scale for grain size. For example, consider the computation of Dlx where pf,i ≤ x ≤ pf,i+1. Then we got equation 22, 23, using equation 24.
The characteristic flow Qwr,k in each range and fraction of time the flow is in that range pQ,k are computed with equation 25, 26, here k is ranged from 1 to M.
Let Yk be any parameter defined for each of the flow ranges k = 1..M. The mean value Ya averaged over the flow duration curve is then given as Ya = /sum/limits_{k=1}^M Yk pQ,k For example, if the fractions in the bedload in each grain size range within flow range k are given as pk,i then the average fractions of the bedload pai are given as pai = /sum/limits_{k=1}^M pk,i pQ,k
- Note on model running
Notice in the Key for Inputs: the GSD and the flow duration curve are separated by a row of zeros. This row is very important, as it acts as a trigger to assign the next values to the hydrograph.
If the friction velocity is too high or low for this formulation to work, the program alerts the user and exits.
The depth is calculated iteratively, and if the depth calculation does not converge, the program will alert the user that the calculation bombed and will automatically exit.
Apart from these additions, the same notes that applied to Acronym1 functions apply here as well.
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: http://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.
- 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.
