Model help:Acronym1D

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

First tab header

Second tab header

Etc. tab header

Parameter	Description	Unit
First parameter	Description parameter	[Units]

Parameter	Description	Unit
First parameter	Description parameter	[Units]

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]\displaystyle{ 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]\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)

Nomenclature

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 q_bTa, as well as the average bedload grain size distribution (D_b,i, p_af,i), i = 1..N+1. In addition, it computes the values Q_a, H_a, u_∗a and τ_ga∗ corresponding to 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 q_bT, the water discharge Q_w, flow depth H, the shear velocity u_∗ and the Shields stress τ_g∗ associated with each range in the flow duration curve, along with the fraction of time p_Q that the flow is in that range.

The flow duration curve is specified in terms of the pairs (Q_wd,k, p_eQ,k), k = 1..M+1, where Q_wd,k denotes the kth discharge and p_eQ,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 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.

• Note on equations

In order to implement the equations 9~15, it is necessary to specify a) the surface grain size distribution (D_f,i, F_f,i) and b) the shear velocity u_∗. This results in a predicted values of q_bi.

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 q_bi are known the total volume bedload transport rate per unit width q_bT and the fractions pi in the bedload can be calculated as equations 16, 17.

The results are presented in terms of q_bT 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_lg and σ_lg, respectively, from the relations of equations 18, 19, 20, 21.

The percent finer in the bedload p_f,i for the grain size D_f,i is obtained from the fractions p_i as follows: p_f,1 = 100 p_f,i = p_f,i-1 - 100 p_i-1 (i=2~N+1)

Let D_sx and D_lx 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_l50 denote the median sizes of the surface and bedload material, respectively. Once F_f,i is specified (p_f,i is computed) the value D_sx (D_lx) 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_f,i ≤ x ≤ p_f,i+1. Then we got equation 22, 23, using equation 24.

The characteristic flow Q_wr,k in each range and fraction of time the flow is in that range p_Q,k are computed with equation 25, 26, here k is ranged from 1 to M.

Let Y_k 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 Y _{= /Sigma 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 = /Sigma 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)

Gary Parker

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.

Links

• [Model:Acronym1]