# Model help:AgDegBW

## AgDegBW

It is the Calculator for aggradation and degradation of a river reach using a backwater formulation. This program computes 1D bed variation in rivers due to differential sediment transport. The sediment is assumed to be uniform with size D. All sediment transport is assumed to occur in a specified fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler relation is used for bed resistance. A generic Meyer-Peter Muller relation is used for sediment transport. The flow is computed using a backwater formulation for gradually varied flow.

## Model introduction

The model calculates a) an ambient mobile-bed equilibrium, and b)the response of a river reach to either 1) changed sediment input rate at the upstream end of the reach starting from t = 0 or 2) changed downstream water surface elevation at the downstream end of the reach starting from t = 0, where t is the temporal coordinate. The code is very similar to AgDegNorm. The main difference between the two codes is in the procedure to compute the water depth. In AgDegNorm the flow is assumed normal (i.e. steady and uniform), while in AgDegBW the flow is assumed steady and it is computed solving the backwater equation. The case of Froude-subcritical flow, for which Fr < 1, is considered herein. This implies that integration of the backwater equation must proceed upstream from x = L, with x streamwise coordinate and L length of the modeled reach. Both a Chezy and a Manning-Striclker formulation can be used to compute the flow.

## 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
Flood discharge m3 / s
Intermittency -
Channel Width m
Grain size mm
Bed Porosity -
Roughness height mm
Ambient Bed Slope
Imposed Annual Sediment Transfer Rate from Upstream tons / annum
Imposed water surface elevation m
Intervals
Length of reach m
Number of Time Steps per Printout
Number of printout
Upwinding coefficient (1 = full upwind, 0.5 = central difference)
Coefficient in Manning-Strickler Resistance Relation
Coefficient in Sediment Transport Relation
Exponent in Sediment Transport Relation
Critical Shield stress
Fraction of bed shear stress that is skin friction
Submerged specific gravity of sediment
Time step year
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

• Computation of the flow

The backwater equation

 $\displaystyle{ {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - F_{r} ^2}} }$ (1)
• Friction slope
 $\displaystyle{ S_{f} = C_{f} F_{r} ^2 }$ (2)
• Froude number
 $\displaystyle{ F_{r} ^2 = {\frac{U^2}{g H}} = {\frac{q_{w} ^2}{g H^3}} }$ (3)
• Flow velocity
 $\displaystyle{ U = {\frac{q_{w}}{H}} }$ (4)
• The bed friction coefficient ( assumed to obey a Manninbg-Strickler resistance )
 $\displaystyle{ C_f ^ \left ( {\frac{-1}{2}} \right ) = C_{z} = \alpha_{r} \left ( {\frac{H}{k_{c}}} \right ) ^{\frac{1}{6}} }$ (5)
• grain roughness (Used as roughness height when bedforms are absent)
 $\displaystyle{ k_{s} = n_{k} D }$ (6)
• The relation between bed slope S and bed elevation η (Froude-subcritical flow (Fr < 1))
 $\displaystyle{ S = -{\frac{\partial \eta}{\partial x}} }$ (7)
• Water surface elevation (Froude-subcritical flow (Fr < 1))
 $\displaystyle{ \epsilon = \eta + H }$ (8)
• Shields number
 $\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} = {\frac{C_{f} U^2}{R g D}} = {\frac{C_{f} {\frac{q_{w} ^2}{H^2}}}{R g D}} }$ (9)
• Bed shear stress
 $\displaystyle{ \tau_{b} = \rho C_{f} U^2 }$ (10)
• Submerged specific gravity
 $\displaystyle{ R = {\frac{\rho_{s}}{\rho}} - 1 }$ (11)
• Computation of the sediment transport (Meyer-Peter and Muller equation )
 $\displaystyle{ q_{t} ^* = \left\{\begin{matrix} \alpha_{t} \left ( \varphi_{s} \tau ^2 - \tau_{c} ^* \right ) ^ \left ( n_{t} \right ) & \tau^* \gt \tau_{c}^* \\ 0 & \tau^* \lt = \tau_{c}^* \end{matrix}\right. }$ (12)
• Einstein number
 $\displaystyle{ q_{t} ^* = {\frac {q_{t}}{\sqrt{R g D} D}} }$ (13)
• Cumulative time of the river has been in flood
 $\displaystyle{ t_{f} = I_{f} t }$ (14)
• Annual sediment yield with a graded state at this slope
 $\displaystyle{ G_{t} = \rho_{s} q_{t} Bl_{f} t_{a} }$ (15)
• Volume sediment transport rate per unit width obtained at the graded state
 $\displaystyle{ q_{t} = {\frac{G_{tf}}{\rho_{s}B I_{f} t_{a}}} }$ (16)
• Computation of bed variation
• Exner equation of sediment continuity (assume that qt is zero for most of the time)
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial\eta}{\partial t}} = - {\frac{\partial q_{t}}{\partial x}} }$ (17)
• Exner equation of sediment continuity (average over many floods)
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}}= - {\frac{\partial I_{f} q_{t}}{\partial x}} }$ (18)
• Numerical scheme
• Initial bed elevation
 $\displaystyle{ \eta_{i}=S \left ( L - x_{i} \right ) }$ (19)
• Computation of the depth of upstream node
 $\displaystyle{ {\frac{H_{i+1} - H_{i}}{\Delta x}} = F_{back} \left ( H \right ) = {\frac{{\frac{\eta _{i+1} - \eta _{i}}{\Delta x} - {\frac{1}{\alpha _{r} ^2}}\left ({\frac{H}{k_{c}}}\right )^\left ({\frac{-1}{3}} \right ){\frac{q_{w}^2}{g H^3}}}}{\frac{q_{w}^2}{g H^3}}} }$ (20)
• Boundary condition of depth
 $\displaystyle{ H_{M + 1} = \xi _{d} - \eta _{M+1} }$ (21)
• A predictor-corrector scheme used to solve for H
 $\displaystyle{ H_{pred}=H_{i+1} - F_{back}\left (H_{i+1}\right ) \Delta x }$ (22)
 $\displaystyle{ H_{i} = H_{i+1} - {\frac{1}{2}}[F_{back} \left ( H_{pred} \right )+ F_{back} \left (H_{i+1} \right )] \Delta x }$ (23)
• Spatial derivative of the total bed material load per unit width
 $\displaystyle{ \frac{\Delta q_{t,i}}{\Delta X}=\left\{\begin{matrix} a_{u}\frac{q_{t,i}-q_{t,i-1}}{\Delta X}+\left ( 1-a_{u} \right )\frac{q_{t,i+1}-q_{t,i}}{\Delta X} & i=1...M \\ {\frac{q_{t,i} - q_{t,i-1}}{\Delta x}} & i=M + 1 \end{matrix}\right. }$ (24)

## Notes

The model computes variation in river bed level η(x, t), where x denotes a streamwise coordinate and t denotes time, in a river with constant width B. The bed sediment is characterized in terms of a single grain size D and submerged specific gravity R. The reach under consideration has length L. Water surface elevation at the downstream end is prescribed. The model is based on a calculation of total bed material load. The model is 1D, assumes a rectangular channel and neglects wall or bank effects.

By modifying the upstream sediment feed rate Gtf and/or the downstream water surface elevation ξd, the river can be forced to aggrade or degrade to a new equilibrium. The program computes this evolution.

Actual rivers tend to be morphologically active only during floods. That is, most of the time they are not doing much to modify their morphology. The simplest way to take this into account is to assume an intermittency If such that the river is in flood a fraction I of the time, during which Q = Qf and qt is the sediment transport rate at this discharge (Paola et al., 1992). For the other (1 – If) fraction of time the river is assumed not to be moving sediment.

Output is controlled by the parameters Ntoprint and Nprint. The code will implement Ntoprint time steps. In addition to output pertaining to the initial state, the code implements Nprint outputs, to that the total number of time steps executed is equal to Nprint x Ntoprint.

• Note on model running

The downstream water surface elevation must exceed the Hc, critical depth, which is equal to (Qw 2/(Bc 2g))1/3, otherwise the user is alerted, and the program exits.

The water depth is calculated using a Chézy formulation, and the Manning-Strickler formulation is implemented, when only the roughness height, kc, and the coefficient αr are given in input file. When all the three parameters 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: