Model help:AgDegNormalSub

AgDegNormalSub

This program is used to calculate the evolution of upward-concave bed profiles in rivers carrying uniform sediment in subsiding basins.

Model introduction

This module computes the time evolution of a river toward steady state as it flows into a subsiding basin. The subsidence rate s is assumed to be constant in time and space. The sediment is assumed to be uniform with size D. A Manning-Strickler formulation is used for bed resistance. A generic relation of the general form of that due to Meyer-Peter and Muller is used for sediment transport. The flow is computed using the normal flow approximation. The river is assumed to have a constant width.

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

• Exner equation for uniform sediment from a flume-like setting

[math]\displaystyle{ \left ( 1 - \lambda _{p} \right ) {\frac{\partial \eta}{\partial t}} = - {\frac{\partial q_{t}}{\partial x}} }[/math]

(1)

[math]\displaystyle{ \left ( 1 - \lambda _{p} \right ) \left ( {\frac{\partial \eta }{\partial t}} + \delta \right ) = - {\frac{I_{f}}{r_{B}}} \left ( 1 + \Lambda \right ) \Omega {\frac{\partial q_{t}}{\partial x}} }[/math]

(2)

• Ratio of depositional to channel width

[math]\displaystyle{ r_{B} = {\frac{B_{d}}{B_{c}}} }[/math]

(3)

• The maximum possible length of the fluvial reach

[math]\displaystyle{ L_{max} = {\frac{I_{f} \left ( 1 + \Lambda \right ) \Omega }{r_{B}}} {\frac{q_{tf}}{\left ( 1 - \lambda _{p} \right ) \delta}} }[/math]

(4)

• Friction coefficient in Manning-Strickler formulation

[math]\displaystyle{ C_{f} ^ \left ( {\frac{-1}{2}}\right ) = \alpha _{r} \left ( {\frac{H}{k_{c}}} \right ) ^ \left ( {\frac{1}{6}} \right ) }[/math]

(5)

• Total bed material load per unit width

[math]\displaystyle{ {\frac{q_{t}}{\sqrt{R g D}D}} = \alpha_{t} \left ( {\frac{\psi _{s} \tau _{b}}{\rho R g D}} - \tau _{c} ^* \right ) ^ \left ( n_{t} \right ) }[/math]

(6)

Nomenclature

Symbol	Description	Unit
X	Streamwise coordinate	m
ΔX	Spatial step length	m
t	Temporal coordinate	seconds
Qw	Flood discharge	m3/s
If	Flood intermittency	-
Bc	Channel width	m
D	Characteristic grain size	mm
λp	Bed porosity	-
kc	Composite roughness height	mm
L	Length of reach	m
Δt	Time step	year
Ntoprint	number of time steps to printout	-
Nprint	number of printouts	-
M	Number of spatial intervals	-
au	Upwinding coefficient (1 = full upwind, 0.5 = central difference)	-
αr	Coefficient in Manning-Strickler	-
αs	Coefficient in sediment transport relation	-
nt	Exponent in sediment transport relation	-
τc *	Reference Shields number in sediment transport relation	-
φs	Fraction of bed shear stress due to skin friction	-
R	Submerged specific gravity	-
Cz	Non-dimensional Chézy friction coefficient	-
σ	subsidence rate	-
rB	channel width	m
Λ	wash load deposited per unit bed material load	-
Ω	channel sinuosity	-
qt	total sediment transport rate per unit channel width	-
Bd	effective depositional width	-
Lmax	maximum possible length of the fluvial reach	-
SI	initial bed slope	-
qt / qtf	ratio between the sediment transport and feed rate of bed material	-
ρs	density of the sediment	kg / m3
ρ	density of the water	kg / m3
g	acceleration of gravity	m / s2
τb	total boundary shear stress	-
nt	specified parameter	-

Output

Symbol	Description	Unit
η	Bed surface elevation	m
S	Bed slope	-
H	Water depth	m
τb	Total (skin friction + form drag) Shields number	-
qt	total bed material load	m2/s
qtf	sediment input rate of bed material load per unit channel width	m2 / s

Notes

The program computes the approach to mobile-bed equilibrium in a river carrying uniform material and flowing into a subsiding basin. It is a descendant of AgDegNormal. Three relatively minor changes have been implemented as follows: a) The input parameters have been modified to include the following parameters: subsidence rate σ, ratio of depositional width to channel width r_B, ratio of wash load deposited per unit bed material load Λ and channel sinuosity Ω; b) The code has been modified so as to include subsidence in the calculation of mass balance; c) The output has been modified to show the time evolution of not only the profile of bed elevation η, but also the profiles of bed slope S and the ratio q_t/q_tf, where q_tf denotes the volume feed rate of bed material load per unit width.

All sediment transport is assumed to occur in a specified fraction If of time during which the river is in flood. The volume bed material transport rate per unit width during floods is denoted as qt; the upstream feed value is denoted as q_tf.

Sediment is deposited not only on the channel as it aggrades, but across a wider depositional zone as the channel migrates and avulses in response to aggradation. It is assumed that for each unit of bed material load that deposits across the depositional zone, L units of wash load deposit; here L (>= 0)is a user-specified parameter.

Channel sinuosity, denoted as W (>= 1), and the ratio of channel width to depositional width, denoted as r_B (>=1), are also user-specified. The initial condition is specified in terms of a constant initial bed slope S_I.

In performing this calculation, the following parameters must be specified: L = reach length; M = number of spatial intervals, so that the spatial step length = L/M; dt = time step length; Ntoprint = number of time steps to a printout; Nprint = number of printouts in the calculation. The calculation assumes that the bed elevation at the downstream end of the domain is fixed.

• Note on model running

Flow is calculated assuming normal flow approximation

The water depth is calculated using a Chézy formulation, when only the Chézy coefficient is specified in the input text file. The Manning-Strickler formulation is implemented, when only the roughness height, k_c, and the coefficient α_r are given in the input text file. When all the three parameters are present, the program will ask the user which formulation they would like to use.

If the input channel length is longer than the maximum possible length of the fluvial reach, the program cannot perform the calculation.

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

Key papers

Links

• [Model:AgDegNormalSub]