Model help:Acronym1: Difference between revisions
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==Main equations== | ==Main equations== | ||
* | * Characteristic grain size for the ith grain size range (spans (D<sub>b,i</sub>, D<sub>b,i+1</sub>)) (for i=1...N) | ||
::::{| | ::::{| | ||
|width= | |width=530px|<math>D_{i}= \sqrt{ D_{b, i} D_{b, i+1} } </math> | ||
|width= | |width=50p=x align="right"|(1) | ||
|} | |} | ||
* Fraction in the surface layer F<sub>i</sub> for the ith grain size range (for i=1...N) | |||
::::{| | ::::{| | ||
|width= | |width=530px|<math>F_{i}= \left ( F_{f, i} - F_{f, i+1} \right ) / 100 </math> | ||
|width= | |width=50p=x align="right"|(2) | ||
|} | |} | ||
* Grain Size on the base-2 logarithmic scale: | |||
::::{| | ::::{| | ||
|width= | |width=500px|<math>\Psi= LN_{2}\left (D\right) = {\frac{log_{10}\left (D\right)}{log_{10}\left (2\right)}} </math> | ||
|width= | |width=50px align="right"|(3) | ||
|} | |} | ||
* Geometric mean size of the surface material | |||
::::{| | ::::{| | ||
|width= | |width=500px|<math>D_{sg}=2^\left (\bar\Psi_{s} \right ) </math> | ||
|width= | |width=50px align="right"|(4) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=500px|<math> | |width=500px|<math>\bar\Psi_{s}= \sum\limits_{i=1}^N \Psi_{i} F{i} </math> | ||
|width=50px align="right"|(5) | |width=50px align="right"|(5) | ||
|} | |} | ||
* The geometric standard deviations | |||
::::{| | ::::{| | ||
|width=500px|<math>\ | |width=500px|<math>\sigma_{sg}= 2 ^\sigma </math> | ||
|width=50px align="right"|(6) | |width=50px align="right"|(6) | ||
|} | |} | ||
* the arithmetic standard deviations | |||
::::{| | ::::{| | ||
|width=500px|<math>\ | |width=500px|<math>\sigma ^2= \sum\limits_{i=1}^N \left (\Psi_{i} - \bar\Psi \right )^2 F_{i} </math> | ||
|width=50px align="right"|(7) | |width=50px align="right"|(7) | ||
|} | |} | ||
* The transport relation | |||
::::{| | ::::{| | ||
|width= | |width=530px|<math>W_{i}^*= {\frac{Rgq_{bi}}{F_{i}u_{*} ^3}}= 0.00218 G \left (\Phi \right ) </math> | ||
|width= | |width=50p=x align="right"|(8) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>\phi= \omega \Phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) </math> | ||
|width=50p=x align="right"|(9) | |width=50p=x align="right"|(9) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math>\ | |width=530px|<math> \Phi_{sgo}= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} </math> | ||
|width=50p=x align="right"|(10) | |width=50p=x align="right"|(10) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> \ | |width=530px|<math>\tau_{sg} ^*={\frac{u_{*} ^2}{R g D_{sg}}} </math> | ||
|width=50p=x align="right"|(11) | |width=50p=x align="right"|(11) | ||
|} | |} | ||
* φ> 1.59 | |||
::::{| | ::::{| | ||
|width=530px|<math>\ | |width=530px|<math> G \left ( \Phi \right )= 5474 \left ( 1 - {\frac{0.853}{\Phi}} \right ) ^ \left (4.5 \right ) </math> | ||
|width=50p=x align="right"|(12) | |width=50p=x align="right"|(12) | ||
|} | |} | ||
* φ | * 1<=φ<=1.59 | ||
::::{| | ::::{| | ||
|width=530px|<math> G \left ( \Phi \right )= | |width=530px|<math> G\left (\Phi \right )= exp[ 14.2\left ( \Phi - 1\right ) - 9.28 \left ( \Phi - 1 \right )^2 ] </math> | ||
|width=50p=x align="right"|(13) | |width=50p=x align="right"|(13) | ||
|} | |} | ||
* | * φ< 1 | ||
::::{| | ::::{| | ||
|width=530px|<math> G\left (\Phi \right )= | |width=530px|<math> G\left (\Phi \right )= \Phi ^\left (14.2 \right ) </math> | ||
|width=50p=x align="right"|(14) | |width=50p=x align="right"|(14) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>\omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \Phi_{sgo} \right ) }} [ \omega_{O} \left ( \Phi_{sgo} \right ) - 1 ] </math> | ||
|width=50p=x align="right"|(15) | |width=50p=x align="right"|(15) | ||
|} | |} | ||
* total volume gravel bedload transport rate per unit width summed over all sizes | |||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>q_{bT}= \sum\limits_{i=1}^N q_{bi} </math> | ||
|width=50p=x align="right"|(16) | |width=50p=x align="right"|(16) | ||
|} | |} | ||
* fraction of gravel bedload in the ith grain size range | |||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>p_{i}= {\frac{q_{bi}}{q_{bT}}} </math> | ||
|width=50p=x align="right"|(17) | |width=50p=x align="right"|(17) | ||
|} | |} | ||
* Geometric mean of the bedload | |||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>D_{lg}= 2 ^\left (\bar\psi_{l} \right ) </math> | ||
|width=50p=x align="right"|(18) | |width=50p=x align="right"|(18) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>\Psi_{l}= \sum\limits_{i=1}^\left (Np \right ) \Psi_{i} p_{i} </math> | ||
|width=50p=x align="right"|(19) | |width=50p=x align="right"|(19) | ||
|} | |} | ||
* Geometric standard deviation of the bedload | |||
::::{| | ::::{| | ||
|width=530px|<math>\ | |width=530px|<math>\delta_{lg}= 2 ^\left ( \delta_{l} \right ) </math> | ||
|width=50p=x align="right"|(20) | |width=50p=x align="right"|(20) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math>\delta_{ | |width=530px|<math>\delta_{l} ^2= \sum\limits_{i=1}^\left (Np \right ) \left ( \Psi_{i} - \bar\Psi_{l} \right )^2 p_{i} </math> | ||
|width=50p=x align="right"|(21) | |width=50p=x align="right"|(21) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>D_{lx}= 2 ^\left (\Psi_{lx} \right ) </math> | ||
|width=50p=x align="right"|(22) | |width=50p=x align="right"|(22) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<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> | ||
|width=50p=x align="right"|(23) | |width=50p=x align="right"|(23) | ||
|} | |} | ||
::::{| | ::::{| | ||
|width=530px|<math> | |width=530px|<math>\Psi_{b, i}= Ln_{2} \left ( D_{b, j} \right ) </math> | ||
|width=50p=x align="right"|(24) | |width=50p=x align="right"|(24) | ||
|} | |} | ||
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{| {{Prettytable}} class="wikitable sortable" | {| {{Prettytable}} class="wikitable sortable" | ||
!Symbol!!Description!!Unit | !Symbol!!Description!!Unit | ||
|- | |||
| D | |||
| grain size | |||
| mm | |||
|- | |- | ||
| D<sub>i</sub> | | D<sub>i</sub> | ||
| characteristic grain size ( | | characteristic grain size for the ith grain size range (i=1...N) | ||
| mm | | mm | ||
|- | |- | ||
| F<sub>i</sub> | | F<sub>i</sub> | ||
| surface layer (for i =1~N) | | fraction in surface layer for the ith grain size range(for i =1~N) | ||
| - | |||
|- | |||
| τ<sub>ssrg</sub> <sup>*</sup> | |||
| equals to 0.0386 | |||
| - | | - | ||
|- | |- | ||
| ψ | | ψ | ||
| grain sizes | | grain sizes on the base-2 logarithmic ψscale | ||
| | | | ||
|- | |- | ||
| D<sub>sg</sub> | | D<sub>sg</sub> | ||
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|- | |- | ||
| σ<sub>sg</sub> | | σ<sub>sg</sub> | ||
| geometric standard deviations | | geometric standard deviations of the surface materials | ||
| - | | - | ||
|- | |- | ||
| σ | | σ | ||
| arithmetic standard deviations | | arithmetic standard deviations of the surface materials | ||
| - | | - | ||
|- | |- | ||
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|- | |- | ||
| R | | R | ||
| submerged specific | | submerged specific density of sediment, equals to (ρ<sub>s</sub> /ρ-1) | ||
| - | | - | ||
|- | |- | ||
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| kg / (m s) | | kg / (m s) | ||
|- | |- | ||
| u | | u<sub>*</sub> | ||
| shear velocity | | shear velocity on the bed, equals to | ||
| m / s | | m / s | ||
|- | |- | ||
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| m <sup>2</sup> | | m <sup>2</sup> | ||
|- | |- | ||
| | | p<sub>i</sub> | ||
| | | fraction of gravel bedload in the ith grain size range | ||
| | | mm | ||
| D<sub>lg</sub> | | D<sub>lg</sub> | ||
| geometric mean of the bedload | | geometric mean of the bedload | ||
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|- | |- | ||
| D<sub>sx</sub> | | D<sub>sx</sub> | ||
| size in the surface material, such that x percentage of the material is finer | | grain size in the surface material, such that x percentage of the material is finer | ||
| mm | | mm | ||
|- | |- | ||
| D<sub>lx</sub> | | D<sub>lx</sub> | ||
| size in the bedload material, such that x percentage of the material is finer | | grain size in the bedload material, such that x percentage of the material is finer | ||
| - | |||
|- | |||
|- | |- | ||
|} | |} | ||
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| mm | | mm | ||
|- | |- | ||
| τ<sub>g</sub> | | τ<sub>g</sub> <sup>*</sup> | ||
| | | shields number based on surface geometric mean size | ||
| kg / (m s) | | kg / (m s) | ||
|- | |- | ||
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==Notes== | ==Notes== | ||
* Note on equations | * Note on equations | ||
In order to implement the equations | In order to implement the equations 9~15, it is necessary to specify a) the surface grain size distribution (D<sub>f</sub>,i, F<sub>f</sub>,i) and b) the shear velocity u<sub>∗</sub>. This results in a predicted values of q<sub>bi</sub>. | ||
If the boundary shear stress at the bed includes a component of form drag, the component must be removed before computing u<sub>∗</sub>. | If the boundary shear stress at the bed includes a component of form drag, the component must be removed before computing u<sub>∗</sub>. | ||
Once the parameters q<sub>bi</sub> are known the total volume bedload transport rate per unit width q<sub>bT</sub> and the fractions pi in the bedload can be calculated as equations 17 | Once the parameters q<sub>bi</sub> are known the total volume bedload transport rate per unit width q<sub>bT</sub> and the fractions pi in the bedload can be calculated as equations 16, 17. | ||
The results are presented in terms of q<sub>bT</sub> 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 D<sub>lg</sub> and σ<sub>lg</sub>, respectively, from the relations of equations 19, 20, 21 | The results are presented in terms of q<sub>bT</sub> 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 D<sub>lg</sub> and σ<sub>lg</sub>, respectively, from the relations of equations 18, 19, 20, 21. | ||
The percent finer in the bedload p<sub>f,i</sub> for the grain size D<sub>f,i</sub> is obtained from the fractions p<sub>i</sub> as follows: | The percent finer in the bedload p<sub>f,i</sub> for the grain size D<sub>f,i</sub> is obtained from the fractions p<sub>i</sub> as follows: | ||
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p<sub>f,i</sub> = p<sub>f,i-1</sub> - 100 p<sub>i-1</sub> (i=2~N+1) | p<sub>f,i</sub> = p<sub>f,i-1</sub> - 100 p<sub>i-1</sub> (i=2~N+1) | ||
Let D<sub>sx</sub> and D<sub>lx</sub> 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 D<sub>l50</sub> denote the median sizes of the surface and bedload material, respectively. Once F<sub>f,i</sub> is specified (p<sub>f</sub>,i is computed) the value D<sub>sx</sub> (D<sub>lx</sub>) 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 p<sub>f,i</sub> ≤ x ≤ p<sub>f,i+1</sub>. Then we got equation 23 | Let D<sub>sx</sub> and D<sub>lx</sub> 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 D<sub>l50</sub> denote the median sizes of the surface and bedload material, respectively. Once F<sub>f,i</sub> is specified (p<sub>f</sub>,i is computed) the value D<sub>sx</sub> (D<sub>lx</sub>) 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 p<sub>f,i</sub> ≤ x ≤ p<sub>f,i+1</sub>. Then we got equation 22, 23, using equation 24. | ||
==Examples== | ==Examples== |
Revision as of 21:46, 19 April 2011
Acronym1
The Acronym1 programs implement the Parker (1990a) surface-based bedload transport relation in order to compute gravel bedload transport rates.
Model introduction
The gravel is divided into N grain size ranges bounded by N+1 sizes Db,i, i = 1 to N+1. The grain size distribution of the surface (active) layer of the bed is specified in terms of the N+1 pairs (Db,i, Ff,i), i = 1..N+1, where Ff,i denotes the percent finer in the surface layer. Here Db,1 must be the coarsest size, such that Ff,1 = 100, and Db,N+1 must be the finest size, such that Ff,N+1 = 0.
The finest size must equal or exceed 2 mm. That is, the sand must be removed from the surface size distribution, and the fractions appropriately renormalized, in determining the surface grain size distribution to be input into Acronym1.
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)
[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)
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) | - | |||
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 | - |
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 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.
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.