# 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

Parameter Description Unit
Input directory path to input files
Site prefix Site prefix for Input/Output files
Case prefix Case prefix for Input/Output files
Parameter Description Unit
Chezy Or Manning, Chezy-1 or Manning-2
Bedload relation, Parker or Wilock, Parker-1 or Wilock-2
Parameter Description Unit
Flood discharge m3 / s
gravel input m2 / s
Intermittency -
base level m
initial bed slope -
reach length m
Time step days
no. of intervals(100 or less) -
Number of printouts -
Iterations per each printout -
factor by which Ds90 is multiplied for roughness height -
factor by which Ds90 is multiplied for active layer thickness -
Manning-Strickler coefficient r
Submerged specific gravity of sediment
bed porosity, gravel
upwinding coefficient for load spatial derivatives in Exner equation (> 0.5 suggested)
coefficient for material transferred to substrate as bed aggrades
channel sinuosity
ratio of depositional width to channel width
Chezy resistance coefficient -
Parameter Description Unit
Model name name of the model -
Author name name of the model author -

## 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

• Total bedload transport over all grain sizes
 $\displaystyle{ q_{bT} = \sum\limits_{i=1}^N q_{bi} }$ (1)
• Fraction of bedload in the ith grain size range
 $\displaystyle{ p_{bi} = {\frac{q_{bi}}{q_{bT}}} }$ (2)
• Exner equation describing the evolution of grain size distribution of the active layer
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) [L_{a}{\frac{\partial F_{i}}{\partial t}} + \left (F_{i} - f_{li}\right ) {\frac{\partial L_{a}}{\partial t}}] = - I_{f} {\frac{\partial q_{bT} p_{bi}}{\partial x}} + I_{f} f_{li} {\frac{\partial q_{bT}}{\partial x}} }$ (3)
• Fraction in the ith grain size range of materials exchanged between the surface and substrate as the bed aggrades or degrades
 $\displaystyle{ f_{li} = \left\{\begin{matrix} f_{i}|_{Z = \eta - L_{a}} & {\frac{\partial \eta}{\partial t}} \lt 0 \\ \lambda F_{i} + \left ( 1 - \lambda \right ) p_{bi} & {\frac{\partial \eta}{\partial t}} \gt 0 \end{matrix}\right. }$ (4)
• Surface-based bedload transport formulation for mixtures

1) Thickness of the active (surface) layer of the bed

 $\displaystyle{ L_{a} = n_{a} D_{s90} }$ (5)

2) Dimensionless grain size specific Shields number

 $\displaystyle{ \tau_{i}^* \equiv {\frac{\tau_{b}}{\rho R g D_{i}}} = {\frac{u_{*}^2}{R g D_{i}}} }$ (6)

3) Grain size specific Einstein number

 $\displaystyle{ q_{bi}^* = {\frac{q_{bi}}{\sqrt{R g D_{i}}D_{i} F_{i}}} }$ (7)

4) Dimensionless grain size specific bedload transport rate

 $\displaystyle{ W_{i}^* \equiv {\frac{q_{bi}^*}{\left ( \tau_{i}^* \right )^ \left ({\frac{3}{2}}\right )}} = {\frac{R g q_{bi}}{\left (u_{*}\right )^3 F_{i} }} }$ (8)
• Bedload relation for mixtures due to Parker (1990a, b)
 $\displaystyle{ W_{i}^* = 0.00218 G\left (\phi_{i} \right ) }$ (9)
 $\displaystyle{ \phi_{i}= \omega \phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) }$ (10)
 $\displaystyle{ \phi_{sgo}= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} }$ (11)
 $\displaystyle{ \tau_{sg} ^*={\frac{u_{*} ^2}{R g D_{sg}}} }$ (12)
 $\displaystyle{ G \left ( \phi \right )= \left\{\begin{matrix} 5474 \left ( 1 - {\frac{0.853}{\phi}} \right ) ^ \left (4.5 \right ) & \phi \gt 1.59 \\ exp[ 14.2\left ( \phi - 1\right ) - 9.28 \left ( \phi - 1 \right )^2 ] & 1 \lt = \phi \lt = 1.59 \\ \phi ^\left (14.2 \right ) & \phi \lt 1 \end{matrix}\right. }$ (13)
 $\displaystyle{ \omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \phi_{sgo} \right ) }} [ \omega_{O} \left ( \phi_{sgo} \right ) - 1 ] }$ (14)
 $\displaystyle{ D_{sg}=2^\left (\bar\Psi_{s} \right ) }$ (15)
 $\displaystyle{ \bar\Psi_{s}= \sum\limits_{i=1}^N \Psi_{i} F{i} }$ (16)
 $\displaystyle{ \delta_{s} ^2 = \sum\limits_{i=1}^N \left (\Psi_{i} - \bar{\Psi}_{s} \right )^2 F_{i} }$ (17)
• Bedload relation for mixture due to Wilcock and Crowe (2003)
 $\displaystyle{ W_{i}^* = G \left (\phi_{i}\right ) }$ (18)
 $\displaystyle{ G = \left\{\begin{matrix} 0.002 \phi^ \left (7.5 \right ) & for \phi \lt 1.35 \\ 14 \left (1 - {\frac{0.894}{\phi_{0.5}}}\right )^ \left (4.5 \right ) & \phi \gt = 1.35 \end{matrix}\right. }$ (19)
 $\displaystyle{ \phi_{i} = {\frac{\tau_{sg}^*}{\tau_{ssrg}^*}} \left ( {\frac{D_{i}}{D_{sg}}}\right )^ \left (-b \right ) }$ (20)
 $\displaystyle{ \tau_{sg}^* = {\frac{u_{*}^2}{R g D_{sg}}} }$ (21)
 $\displaystyle{ \tau_{ssrg}^* = 0.021 + 0.015 exp \left (-14 F_{s}\right ) }$ (22)
 $\displaystyle{ b = {\frac{0.69}{1 + exp \left (1.5 - {\frac{D_{i}}{D_{sg}}}\right )}} }$ (23)
• Roughness hight
 $\displaystyle{ k_{s} = n_{k} D_{s90} }$ (24)
• Boundary shear stress
 $\displaystyle{ \tau_{b,k} = \rho u_{*}^2 = \left ({\frac{k_{s,k}^\left ({\frac{1}{3}} \right ) q_{w}^*}{\alpha_{r}^2}} \right )^ \left ({\frac{3}{10}}\right ) g^\left ({\frac{7}{10}}\right ) S_{k}^ \left ({\frac{7}{10}}\right ) }$ (25)
• Bed slope
 $\displaystyle{ S_{k} = \left\{\begin{matrix} {\frac{\eta_{1} - \eta_{2}}{\Delta x}} & k = 1 \\ {\frac{\eta_{k-1} - \eta_{k+1}}{2 \Delta x}} & k = 2...M \end{matrix}\right. }$ (26)
• Shields number based on the geometric mean size
 $\displaystyle{ \tau_{sg}^* = \left ({\frac{k_{s}^ \left ({\frac{1}{3}}\right ) q_{w}^2}{\alpha_{r}^2 g}}\right ) ^ \left ({\frac{3}{10}}\right ) {\frac{S^ \left ({\frac{7}{10}}\right )}{R D_{sg}}} }$ (27)
• Shear velocity based on the surface geometric mean size
 $\displaystyle{ u_{*} = \left ({\frac{k_{s}^ \left ({\frac{1}{3}}\right ) q_{w}^2}{\alpha_{r}^2}}\right )^ \left ({\frac{3}{20}}\right ) g^ \left ({\frac{7}{20}}\right ) S^ \left ({\frac{7}{20}}\right ) }$ (28)
• Volume bedload transport rate per unit width in the ith grain size
 $\displaystyle{ q_{bi} = F_{i}{\frac{u_{*}^3}{R g}}W_{i}^* }$ (29)

## 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 ηd.

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 DbU and DbL such that ffU = 100 (1.00) and ffL = 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, kc, 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: