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)
 $\displaystyle{ {\frac{\partial \eta_{dev}}{\partial t}} = 0 }$ (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)

## Notes

Assumptions used in the model:

a) Downchannel reach length L is specified; x = L corresponds to the point where sea level is specified;

b) The river is assumed to have a floodplain width Bf that is constant, and is much larger than bankfull width Bbf.

c) The river is sand-bed with characteristic size D.

d) All the bed material sediment is transported at rate Qtbf during a period constituting (constant) fraction If of the year, at which the flow is approximated as at bankfull flow, so that the annual yield = IfQtbf.

e) Sediment is deposited across the entire width of the floodplain as the channel migrates and avulses. For every mass unit of bed material load deposited, Λ mass units of wash load are deposited in the floodplain. It is assumed that the supply of wash load from upstream is always sufficient for deposition at such a rate. In addition, it is assumed for simplicity that the porosity of the floodplain deposits is equal to that of the channel deposits. In fact the floodplain deposits are likely to have a lower porosity.

f) Sea level rise is constant at rate ξd. For example, during the period 5,000 – 17,000 BP the rate of rise can be approximated as 1 cm per year.

g) The river is meandering throughout sea level rise, and has constant sinuosity Ω.

h) The flow can be approximated using the normal-flow assumptions. (But the analysis easily generalizes to a full backwater formulation.)

In the equations, is if β > 1 then the there is not enough sediment feed over the reach to fill the space created by sea level rise, and the sediment transport rate must drop to zero before the shoreline is reached. If β < 1 the excess sediment is delivered to the sea. If for a given rate of sea level rise ξd it is found that β > 1 for reasonable values of reach length L, floodplain width Bf and sediment feed rate Gfeed, no steady state solution exists for that rate of sea level rise.

## 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:

## References

• Bard, E., Hamelin, B., and Fairbanks, R.G., 1990, U-Th ages obtained by mass spectrometry in corals from Barbados: sea level during the past 130,000 years, Nature 346, 456-458.Doi [10.1038/346456a0]
• Fisk, H.N., 1944, Geological investigations of the alluvial valley of the lower Mississippi River, Report, U.S. Army Corp of Engineers, Mississippi River Commission, Vicksburg, MS.
• Pirmez, C., 1994, Growth of a Submarine meandering channel-levee system on Amazon Fan, Ph.D. thesis, Columbia University, New York, 587 p.
• Paola, C., P. L. Heller and C. L. Angevine, 1992, The large-scale dynamics of grain-size variation in alluvial basins. I: Theory, Basin Research, 4, 73-90.
• Parker, G., and Y. Cui, 1998, The arrested gravel front: stable gravel-sand transitions in rivers. Part 1: Simplified analytical solution, Journal of Hydraulic Research, 36(1): 75-100.
• Parker, G., Paola, P., Whipple, K. and Mohrig, D., 1998, Alluvial fans formed by channelized fluvial and sheet flow: theory, Journal of Hydraulic Engineering, 124(10), pp. 1-11. Doi: [10.1061/(ASCE)0733-9429(1998)124:10(985)]
• Sinha, S. K. and Parker, G., 1996, Causes of concavity in longitudinal profiles of rivers, Water Resources Research 32(5),1417-1428. Doi: [10.1029/95WR03819]
• USCOE, 1935., Studies of river bed materials and their movement, with special reference to the lower Mississippi River, Paper 17 of the U.S. Waterways Experiment Station, Vicksburg, MS.
• Yatsu, E., 1955, On the longitudinal profile of the graded river, Transactions, American Geophysical Union, 36: 655-663.