This model is a calculator for approach to equilibrium in recirculating and feed flumes

## Model introduction

This program implements the calculation for steady-state aggradation of a sand-bed river in response to sea level rise at a constant rate.

## 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
Mean annual bed material sediment feed rate Mt / yr
Reach length (downchannel distance) km
Rate of sea level rise mm / yr
Bankfull water discharge m3 / s
Floodplain width km
Grain size grain diameter mm
Fraction wash load deposited per unit bed material deposited -
Channel-forming Shield number -
Coefficient in Engelund-Hansen bed material load relation -
Bed porosity -
Flood intermittence -
Chezy resistance coefficient -
Channel sinuosity -
Submerged specific gravity of seidment -
Initial water surface elevation m
Time interval for plotting yr
Number of prints desired -
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

• Exner equation
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}} = - \Omega {\frac{I_{f} \left ( 1 + \Lambda \right )}{B_{f}}} {\frac{\partial Q_{tbf}}{\partial x}} }$ (1)
• Relation for sediment transport

1) Dimensionless bankfull width

 $\displaystyle{ \hat{B} = {\frac{C_{f}}{\alpha_{EH} \left ( \tau_{form}^* \right ) ^ \left (2.5 \right )}} \hat{Q}_{t} }$ (2)

2) Down-channel bed slope

 $\displaystyle{ S = {\frac{R^ \left ({\frac{3}{2}}\right ) C_{f}^ \left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{\hat{Q}_{t}}{\hat{Q}}} }$ (3)

3) Dimensionless bankfull depth

 $\displaystyle{ \hat{H} = {\frac{\alpha_{EH} \left ( \tau_{form} \right )^2}{\left ( R C_{f} \right ) ^\left ({\frac{1}{2}}\right )}}{\frac{\hat{Q}}{\hat{Q}_{t}}} }$ (4)

4) Total volume bed material load at bankfull flow

 $\displaystyle{ Q_{tbf} = {\frac{\alpha_{EH} \tau_{form}^*}{R C_{f} ^ \left ({\frac{1}{2}}\right )}} Q_{bf} S }$ (5)

5) Reduction of the Exner equation

 $\displaystyle{ {\frac{\partial \eta}{\partial t}} = \kappa_{d}{\frac{\partial ^2 \eta}{\partial x^2}} }$ (6)

6) Kinematic sediment diffusivity

 $\displaystyle{ \kappa_{d} = {\frac{I_{f} \Omega \left ( 1 + \Lambda \right )}{\left ( 1 - \lambda_{p}\right ) B_{f}}} {\frac{\alpha _{EH} \tau_{form}^* Q_{bf}}{R C_{f}^\left ({\frac{1}{2}}\right )}} }$ (7)
• Bed elevation
 $\displaystyle{ \eta = \xi_{do} + \xi_{d} t + \eta_{dev} \left (x, t \right ) }$ (8)
• Steady-State aggradation in response to sea level rise condition
 $\displaystyle{ {\frac{\partial \eta_{dev}}{\partial t}} = 0 }$ (9)
 $\displaystyle{ \hat{x} = {\frac{x}{L}} }$ (10)
 $\displaystyle{ \hat{\eta} = {\frac{\eta_{dev}}{L}} }$ (11)
 $\displaystyle{ {\frac{Q_{tbf}}{Q_{tbf,feed}}} = {\frac{S}{S_{u}}} = 1 - \beta \hat{x} }$ (12)
 $\displaystyle{ \beta = {\frac{\left ( 1 - \lambda_{p}\right ) B_{f} \xi_{d} L}{I_{f} \Omega \left ( 1 + \Lambda \right ) Q_{tbf,feed}}} = {\frac{G_{fill}}{G_{feed}}} }$ (13)

1) Sediment delivery rate

 $\displaystyle{ Q_{tbf} = Q_{tbf,feed} - {\frac{\left (1 - \lambda_{p} \right ) B_{f}}{I_{f} \Omega \left ( 1 + \Omega \right )}} \xi_{d} X }$ (14)

2)Upstream slope at x = 0

 $\displaystyle{ S_{u} = {\frac{R C_{f}^\left ({\frac{1}{2}}\right )}{\alpha_{EH} \tau_{form}^*}} {\frac{Q_{tbf,feed}}{Q_{bf}}} }$ (15)

3) Elevation profile

 $\displaystyle{ \hat{\eta} = S_{u} [\left ( 1 - {\frac{1}{2}} \beta \right ) - \hat{x} + {\frac{1}{2}} \beta \hat{x}^2] }$ (16)
• Mean annual rate available for deposition
 $\displaystyle{ G_{feed} = \left ( 1 + \Lambda \right ) G_{t,feed} = \left ( 1 + \Lambda \right ) \left ( R + 1 \right ) Q_{tbf,feed} I_{f} }$ (17)
• Amount of sediment required to fill a reach at a uniform aggradation rate
 $\displaystyle{ G_{fill} = \left ( R + 1 \right ) \left ( 1 - \lambda_{p} \right ) B_{f} L_{v} \xi_{d} }$ (18)