Model help:AgDegNormalSub: Difference between revisions

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==Model introduction==
==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.  
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.  


<div id=CMT_MODEL_PARAMETERS>
<div id=CMT_MODEL_PARAMETERS>
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==Main equations==
==Main equations==
* Exner equation for uniform sediment from a flume-like setting
* Exner equation for uniform sediment from a river
::::{|
::::{|
|width=500px|<math> \left ( 1 - \lambda _{p} \right ) {\frac{\partial \eta}{\partial t}} = - {\frac{\partial q_{t}}{\partial x}} </math>
|width=500px|<math> \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>
|width=50px align="right"|(1)
|width=50px align="right"|(1)
|}
::::{|
|width=500px|<math> \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>
|width=50px align="right"|(2)
|}
|}
* Ratio of depositional to channel width
* Ratio of depositional to channel width
::::{|
::::{|
|width=500px|<math> r_{B} = {\frac{B_{d}}{B_{c}}} </math>
|width=500px|<math> r_{B} = {\frac{B_{d}}{B_{c}}} </math>
|width=50px align="right"|(3)
|width=50px align="right"|(2)
|}
|}
* The maximum possible length of the fluvial reach
* The maximum possible length of the fluvial reach
::::{|
::::{|
|width=500px|<math> L_{max} = {\frac{I_{f} \left ( 1 + \Lambda \right ) \Omega }{r_{B}}} {\frac{q_{tf}}{\left ( 1 - \lambda _{p} \right ) \delta}} </math>
|width=500px|<math> L_{max} = {\frac{I_{f} \left ( 1 + \Lambda \right ) \Omega }{r_{B}}} {\frac{q_{tf}}{\left ( 1 - \lambda _{p} \right ) \delta}} </math>
|width=50px align="right"|(4)
|width=50px align="right"|(3)
|}
* Friction coefficient in Manning-Strickler formulation
::::{|
|width=500px|<math> C_{f} ^ \left ( {\frac{-1}{2}}\right ) = \alpha _{r} \left ( {\frac{H}{k_{c}}} \right ) ^ \left ( {\frac{1}{6}} \right ) </math>
|width=50px align="right"|(5)
|}
* Total bed material load per unit width
::::{|
|width=500px|<math> {\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>
|width=50px align="right"|(6)
|}
|}


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!Symbol!!Description!!Unit
!Symbol!!Description!!Unit
|-
|-
| X
| Q
| Streamwise coordinate
| flood discharge
| m <sup>3</sup> / s
|-
| x
| streamwise coordinate
| m
| m
|-
|-
| ΔX
| η
| Spatial step length
| river bed elevation
| m
| m
|-
|-
| t
| t
| Temporal coordinate
| time step
| seconds
| year
|-
| Q<sub>w</sub>
| Flood discharge
| m<sup>3</sup>/s
|-
| I<sub>f</sub>
| Flood intermittency
| -
|-
|-
| B<sub>c</sub>
| B<sub>c</sub>
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|-
|-
| D
| D
| Characteristic grain size
| grain size of the bed sediment
| mm
| mm
|-  
|-
| σ
| subsidence rate
| -
|-
| r<sub>B</sub>
| the ratio of depositional width to channel width
| -
|-
| Ω
| channel sinuosity
| -
|-
| Λ
| units of wash load deposited in the system per unit of bed material load
| -
|-
| λ<sub>p</sub>
| λ<sub>p</sub>
| Bed porosity
| bed porosity
| -
| -
|-
| R
| submerged specific gravity
| -
|-
| ξ<sub>d</sub>
| downstream water surface elevation
| m
|-
| q<sub>w</sub>
| water discharge per unit width
| m<sup>2</sup> / s
|-
|-
| k<sub>c</sub>
| k<sub>c</sub>
| Composite roughness height
| composite roughness height
| mm
| m
|-
| G
| imposed annual sediment transfer rate from upstream
| tons / annum
|-
| G<sub>tf</sub>
| upstream sediment feed rate
| -
|-
| ξ<sub>d</sub>
| downstream water surface elevation
| m
|-
|-
| L
| L
| Length of reach
| length of reach under consideration
| m
| m
|-
|-
| Δt
| L<sub>max</sub>
| Time step
| maximum length of basin that the sediment supply can fill
| year
| m
|-
| q<sub>w</sub>
| water discharge per unit width
| m<sup>2</sup> / s
|-
|-
| N<sub>toprint</sub>
| i
| number of time steps to printout
| number of time steps per printout
| -
| -
|-
|-
| N<sub>print</sub>
| p
| number of printouts
| number of printouts desired
| -
| -
|-
|-
| M
| M
| Number of spatial intervals
| number of spatial intervals
| -
| -
|-
|-
| a<sub>u</sub>
| R
| Upwinding coefficient (1 = full upwind, 0.5 = central difference)
| submerged specific gravity of sediment
| -
| -
|-
|-
| α<sub>r</sub>
| S<sub>f</sub>
| Coefficient in Manning-Strickler
| friction slope
| -
| -
|-
|-
| α<sub>s</sub>
| F<sub>r</sub>
| Coefficient in sediment transport relation
| Froude number
| -
| -
|-
|-
| n<sub>t</sub>
| U
| Exponent in sediment transport relation
| flow velocity
| m / s
|-
| C<sub>f</sub>
| bed friction coefficient
| -
| -
|-
|-
| τ<sub>c</sub> <sub>*</sub>
| g
| Reference Shields number in sediment transport relation
| acceleration of gravity
| m/ s^2
|-
| α<sub>r</sub>
| coefficient in Manning-Stricker, dimensionless coefficient between 8 and 9
| -
| -
|-
|-
| φ<sub>s</sub>
| k<sub>s</sub>
| Fraction of bed shear stress due to skin friction
| grain roughness
| m
|- 
| n<sub>k</sub>
| dimensionless coefficient typically between 2 and 5
| -
|- 
| τ<sup>*</sup>
| Shield number
| -
| -
|-
|-
| R
| ρ
| Submerged specific gravity
| fluid density
| kg / m<sup>3</sup>
|-
| ρ<sub>s</sub>
| sediment density
| kg / m<sup>3</sup>
|-
| τ<sub>c</sub>
| critical Shields number for the onset of sediment motion
| -
| -
|-
|-
| Cz
| ψ<sub>s</sub>
| Non-dimensional Chézy friction coefficient
| the fraction of bed shear stress
| -
| -
|-
|-
| σ
| q<sub>t</sub> <sup>*</sup>
| subsidence rate
| Einstein number
| -
| -
|-
|-
| r<sub>B</sub>
| q<sub>t</sub>
| channel width
| volume sediment transport rate per unit width
| m
| -
|-
|-
| Λ
| I<sub>f</sub>
| wash load deposited per unit bed material load
| flood intermittency
| -
| -
|-
|-
| Ω
| t<sub>f</sub>
| channel sinuosity
| cumulative time the river has been in flood
| s
|-
| G<sub>t</sub>
| the annual sediment yield
| tone/yr
|-
| t<sub>a</sub>
| the number of seconds in a year
| -
| -
|-
|-
| q<sub>t</sub>
| Q<sub>f</sub>
| total sediment transport rate per unit channel width
| sediment transport rate during flood discharge
|-
| α<sub>t</sub>
| dimensionless coefficient in the sediment transport equation, equals to 8
| -
| -
|-
|-
| B<sub>d</sub>
| n<sub>t</sub>
| effective depositional width
| exponent in sediment transport relation, equals to 1.5
| -
| -
|-
|-
| L<sub>max</sub>
| τ<sub>c</sub> <sup>*</sup>
| maximum possible length of the fluvial reach
| reference Shields number in sediment transport relation, equals to 0.047
|-
| C<sub>f</sub>
| bed friction coefficient, equals to τ<sub>b</sub> / (ρ U<sup>2</sup> )
| -
| -
|-
|-
| SI
| C<sub>Z</sub>
| initial bed slope
| dimensionless Chezy resistance coefficient.
|-
| S<sub>l</sub>
| initial bed slope of the river
| -
| -
|-
|-
| q<sub>t</sub> / q<sub>tf</sub>
| η<sub>i</sub>
| ratio between the sediment transport and feed rate of bed material
| initial bed elevation
| -
| -
|-
|-
| ρ<sub>s</sub>
| x
| density of the sediment
| downstream coordinate
| kg / m<sup>3</sup>
| m
|-
| τ
| shear stress on bed surface
| N / m<sup>2</sup>
|-
| q<sub>b</sub>
| bed material load
| tons / year
|-
| Δx
| spatial step length, equals to L / M
| m
|-
| Q<sub>w</sub>
| flood discharge
| m<sup>3</sup> / s
|-
| Δt
| time step
| year
|-
|-
| ρ
| Ntoprint
| density of the water
| number of time steps to printout
| kg / m<sup>3</sup>
| -
|-
|-
| g
| Nprint
| acceleration of gravity
| number of printouts
| m / s<sup>2</sup>
| -
|-
|-
| τ<sub>b</sub>
| a<sub>U</sub>
| total boundary shear stress
| upwinding coefficient (1=full upwind, 0.5=central difference)
| -
| -
|-
|-
| n<sub>t</sub>
| α<sub>s</sub>
| specified parameter
| coefficient in sediment transport relation
| -
| -
|-
|-
|}
|}


'''Output'''
'''Output'''
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|-
|-
| η
| η
| Bed surface elevation
| bed surface elevatioon
| m
|-
| H
| water depth
| m
| m
|-
|-
| S
| ξ
| Bed slope
| water surface elevation
| -
| m
|-
|-
| H
| L<sub>max</sub>
| Water depth
| maximum length of basin the the sediment supply can fill
| m
| m
|-
|-
| τ<sub>b</sub>
| τ<sub>b</sub>
| Total (skin friction + form drag) Shields number
| bed shear stress
| kg / (s^2 m)
|-
| S
| bed slope
| -
| -
|-
|-
| q<sub>t</sub>
| q<sub>t</sub>
| total bed material load
| total bed material load
| m<sup>2</sup>/s
|-
| q<sub>tf</sub>
| sediment input rate of bed material load per unit channel width
| m<sup>2</sup> / s
| m<sup>2</sup> / s
|-
|-
|}
|}
   </div>
   </div>
</div>
</div>


==Notes==
==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:
some assumptions:
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.
 
 
The program 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<sub>B</sub>, ratio of wash load deposited per unit bed material load Λ and channel sinuosity Ω;
a) The input parameters have been modified to include the following parameters: subsidence rate σ, ratio of depositional width to channel width r<sub>B</sub>, 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;
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<sub>t</sub>/q<sub>tf</sub>, where q<sub>tf</sub> denotes the volume feed rate of bed material load per unit width.
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<sub>t</sub>/q<sub>tf</sub>, where q<sub>tf</sub> 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<sub>tf</sub>.
The ratio of depositional to channel width, r<sub>B</sub>, has been introduced to model the fact that in an aggrading river sediment deposits not only in the channel itself, but also in a much wider belt (e.g. the floodplain or basin width, due to overbank deposition, channel migration and avulsion). Here channel width is denoted as B<sub>c</sub> (which can be taken to be synonymous with bankfull width) and effective depositional width is denoted as B<sub>d</sub>.


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.
The parameter Λ that represents the units of wash load deposited per unit of bed material is introduced to consider that in the 1D formulation implemented in this model it is assumed that deposition occurs not only in the channel but on a much wider area (e.g. the floodplain).  Sediment deposited in the channel is mostly made of bed material but sediment deposited around the channel contains a significant amount of wash load.  A precise mass balance for wash load is beyond the scope of this model. For simplicity it is assumed that for every unit of sand deposited in the system, Λ units of wash load are deposited.  It is also assumed that the supply of wash load from upstream is always sufficient for deposition at such a rate.  This is not likely to be strictly true, but should serve as a useful starting assumption.


Channel sinuosity, denoted as W (>= 1), and the ratio of channel width to depositional width, denoted as r<sub>B</sub> (>=1), are also user-specified. The initial condition is specified in terms of a constant initial bed slope S<sub>I</sub>.
The parameter Ω has been introduced to consider that channels may be sinuous. Here it is assumed that the channel has a sinuosity, Ω, but that the depositional surface across which it wanders is rectangular. In the present formulation the sinuosity is defined as the ratio of downchannel distance per unit of downvalley distance.


In performing this calculation, the following parameters must be specified:
Boundary and initial conditions are equal to that implemented for the ancestor model AgDegNormal.  
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
* Note on model running
Flow is calculated assuming normal flow approximation
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<sub>c</sub>, and the coefficient α<sub>r</sub> 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. The maximum possible length of the fluvial reach, Lmax, is defined as the maximum length of basin that the sediment supply can fill; at this length the sediment transport rate out of the basin drops precisely to zero.  
 
If the input channel length is longer than the maximum possible length of the fluvial reach, the program cannot perform the calculation.


==Examples==
==Examples==

Revision as of 14:10, 10 May 2011

The CSDMS Help System

AgDegNormalSub

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

Model introduction

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.

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 -
Grain size of bed material mm
I -
Manning-Strickler coefficient, k -
Slope of forest face -
upstream bed material sediment feed rate during floods m2 / s
L -
Time step days
Iterations per each printout
Number of printout m
Number of fluvial nodes
u
Manning-Strickler coefficient, r
Coefficient in total bed material relation
Exponent in load relation
Critical Shield stress
p
Submerged specific gravity of sediment
initial length of fluvial zone m
B -
O -
Y -
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 for uniform sediment from a river
[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] (1)
  • Ratio of depositional to channel width
[math]\displaystyle{ r_{B} = {\frac{B_{d}}{B_{c}}} }[/math] (2)
  • 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] (3)

Notes

some assumptions: 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.


The program 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 rB, 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 qt/qtf, where qtf denotes the volume feed rate of bed material load per unit width.

The ratio of depositional to channel width, rB, has been introduced to model the fact that in an aggrading river sediment deposits not only in the channel itself, but also in a much wider belt (e.g. the floodplain or basin width, due to overbank deposition, channel migration and avulsion). Here channel width is denoted as Bc (which can be taken to be synonymous with bankfull width) and effective depositional width is denoted as Bd.

The parameter Λ that represents the units of wash load deposited per unit of bed material is introduced to consider that in the 1D formulation implemented in this model it is assumed that deposition occurs not only in the channel but on a much wider area (e.g. the floodplain). Sediment deposited in the channel is mostly made of bed material but sediment deposited around the channel contains a significant amount of wash load. A precise mass balance for wash load is beyond the scope of this model. For simplicity it is assumed that for every unit of sand deposited in the system, Λ units of wash load are deposited. It is also assumed that the supply of wash load from upstream is always sufficient for deposition at such a rate. This is not likely to be strictly true, but should serve as a useful starting assumption.

The parameter Ω has been introduced to consider that channels may be sinuous. Here it is assumed that the channel has a sinuosity, Ω, but that the depositional surface across which it wanders is rectangular. In the present formulation the sinuosity is defined as the ratio of downchannel distance per unit of downvalley distance.

Boundary and initial conditions are equal to that implemented for the ancestor model AgDegNormal.

  • Note on model running

Flow is calculated assuming normal flow approximation

If the input channel length is longer than the maximum possible length of the fluvial reach, the program cannot perform the calculation. The maximum possible length of the fluvial reach, Lmax, is defined as the maximum length of basin that the sediment supply can fill; at this length the sediment transport rate out of the basin drops precisely to zero.

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:

See also: Help:Images or Help:Movies

Developer(s)

Gary Parker

References

Key papers

Links

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