Model help:AgDegNormGravMixSubPW: Difference between revisions

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==Main equations==
==Main equations==
* Exner equation of conservation of channel bed sediment
 
::::{|
|width=530px|<math> \left ( 1 - \lambda _{p} \right ) {\frac{\partial \eta}{\partial t}} = - I_{f} {\frac{\partial q_{bT}}{\partial x}} </math>
|width=50p=x align="right"|(1)
|}
* Characteristic diameter
::::{|
|width=530px|<math> D_{i} = \left ( D_{b,i} \ast D_{b,i+1} \right ) ^ \left ( {\frac{1}{2}} \right ) </math>
|width=50p=x align="right"|(2)
|}
::::{|
|width=530px|<math> f_{i} = f_{f,i+1} - f_{f,i} </math>
|width=50p=x align="right"|(3)
|}
* The conservation of sediment in each grain size range
::::{|
|width=530px|<math> \left ( 1 - \lambda _{p} \right ) [L_{a} {\frac{\partial F_{i}}{\partial t}} + \left ( F_{i} - f_{li} \right ) {\frac{\partial L_{a}}{\partial t}}] = -I_{f} {\frac{\partial q_{bT} p_{i}}{\partial x}} + I_{f} f_{li} {\frac{\partial q_{bT}}{\partial x}} </math>
|width=50p=x align="right"|(4)
|}
* Fraction of sediment in the ith grain size range at the active-layer substrate interface (if ∂η / ∂t > 0)
::::{|
|width=530px|<math> f_{li} = \alpha F_{i} + \left ( 1 - \alpha \right ) p_{i}  </math>
|width=50p=x align="right"|(5)
|}
* Fraction of sediment in the ith grain size range at the active-layer substrate interface (if ∂η / ∂t < 0)
::::{|
|width=530px|<math> f_{li} = f_{sub,i}  </math>
|width=50p=x align="right"|(6)
|}
* Roughness height due to skin friction
::::{|
|width=530px|<math> k_{s} = n_{k} D_{s90}  </math>
|width=50p=x align="right"|(7)
|}
::::{|
|width=530px|<math> L_{a} = n_{a} D_{s90}  </math>
|width=50p=x align="right"|(8)
|}
* Non-dimensional bed shear stress
::::{|
|width=530px|<math> \tau _{sg} ^* = {\frac{\tau_{b}}{\left ( \rho _{s} - \rho \right ) g D_{sg}}}  </math>
|width=50p=x align="right"|(9)
|}


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!Symbol!!Description!!Unit
!Symbol!!Description!!Unit
|-
|-
| q
| Q
| water discharge / width
| flood discharge
| m<sup>2</sup> / s
| m <sup>3</sup> / s
|-
| x
| streamwise coordinate
| m
|-
| η
| river bed elevation
| m
|-
| t
| time step
| year
|-
| B
| river width
| m
|-
| D
| grain size of the bed sediment
| mm
|-
| D<sub>bi</sub>
| bound diameter
| mm
|-
| λ<sub>p</sub>
| bed porosity
| -
|-
| α
| the parameter that governs the grain size distribution of the sediment at the active layer-substrate interface during bed aggredation
| -
|-
| F<sub>fi</sub>
| grain size distribution of the active layer for initial condition
| -
|-
| F<sub>i</sub>
| fraction of sediment in the ith grain size range in the active layer
| -
|-
| f<sub>subfi</sub>
| fraction of sediment in the ith grain size range in the substrate layer for initial condition
| -
|-
| F<sub>fli</sub>
| percent finer than ith grain size range for the bed surface for initial condition
| -
|-
| F<sub>subfli</sub>
| percent finer than ith grain size range for the substrate layer for initial condition
| -
|-
| f<sub>li</sub>
| fraction of sediment in the ith grain size range in the active-layer substrate interface
| -
|-
| p<sub>i</sub>
| fraction of sediment in the ith grain size range in the bedload
| -
|-
| F<sub>sub,i</sub>
| fraction of substrate material in the ith size range
| -
|-
| D<sub>s90</sub>
| the diameter of the bed surface such that the 90% of the sediment is finer
| -
|-
| n<sub>a</sub>
| user specified order-one non dimensional constant
| -
|-
| p<sub>ffi</sub>
| the percent that is finer than the ith size range for upstream boundary conditon
| -
|-
|-
| T
| η<sub>d</sub>
| gravel input
| fixed bed elevation at the downstream end of the modeled reach
| m<sup>2</sup> / s
| m
|-
|-
| I
| R
| intermittency
| submerged specific gravity
| -
| -
|-
|-
| e
| ξ<sub>d</sub>
| base level
| downstream water surface elevation
| m
| m
|-
|-
| S
| q<sub>w</sub>
| initial bed slope
| water discharge per unit width
| m<sup>2</sup> / s
|-
| k<sub>c</sub>
| composite roughness height
| 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
| reach length
| length of reach under consideration
| m
| m
|-
|-
| t
| q<sub>w</sub>
| time step
| water discharge per unit width
| days
| m<sup>2</sup> / s
|-
|-
| M
| i
| no. of intervals
| number of time steps per printout
| -
| -
|-
|-
| p
| p
| no. of prints
| number of printouts desired
| -
| -
|-
|-
| i
| M
| no. of iterations per print
| number of spatial intervals
| -
| -
|-
|-
| k
| R
| factor by which Ds90 is multiplied for roughness height
| submerged specific gravity of sediment
| -
| -
|-
|-
| n
| S<sub>f</sub>
| factor by which Ds90 is multiplied for active layer thickness
| friction slope
| -
| -
|-
|-
| r
| F<sub>r</sub>
| coefficient in Manning-Strickler relation
| Froude number
| -
| -
|-
|-
| R
| U
| submerged specific gravity of gravel
| flow velocity
| m / s
|-
| C<sub>f</sub>
| bed friction coefficient
| -
| -
|-
|-
| l
| g
| bed porosity, gravel
| acceleration of gravity
| m/ s^2
|-
| α<sub>r</sub>
| coefficient in Manning-Stricker, dimensionless coefficient between 8 and 9
| -
| -
|-
|-
| u
| k<sub>s</sub>
| upwinding coefficient for load spatial deviations in Exner equation (> 0.5 suggested)
| grain roughness
| m
|- 
| n<sub>k</sub>
| dimensionless coefficient typically between 2 and 5
| -
| -
|-   
|-   
| a
| τ<sup>*</sup>
| coefficient for material transferred to substrate as bed aggrades
| Shield number
| -
|-
| ρ
| 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
| -
|-
| ψ<sub>s</sub>
| the fraction of bed shear stress
| -
|-
| q<sub>t</sub> <sup>*</sup>
| Einstein number
| -
| -
|-
|-
| C
| q<sub>t</sub>
| coefficient, Cf, in the Chezy formulation
| volume sediment transport rate per unit width
| -
| -
|-
|-
| O
| I<sub>f</sub>
| channel sinuosity
| flood intermittency
| -
| -
|-
|-
| t<sub>f</sub>
| cumulative time the river has been in flood
| s
| s
| subsidence rate
|-
| mm/yr
| G<sub>t</sub>
|-
| the annual sediment yield
| B
| tone/yr
| ratio of wash load deposited per unit bed material load deposited
|-
| t<sub>a</sub>
| the number of seconds in a year
| -
| -
|-
|-
| V
| Q<sub>f</sub>
| ratio of wash load deposited per unit bed material load deposited
| sediment transport rate during flood discharge
|-
| α<sub>t</sub>
| dimensionless coefficient in the sediment transport equation, equals to 8
| -
| -
|-
|-
| F<sub>up</sub>
| n<sub>t</sub>
| GSD at the upstream end
| exponent in sediment transport relation, equals to 1.5
| -
| -
|-
|-
| F<sub>down</sub>
| τ<sub>c</sub> <sup>*</sup>
| GSD at the downstream end
| reference Shields number in sediment transport relation, equals to 0.047
| -
|-
|
| C<sub>f</sub>
| L<sub>max</sub>
| bed friction coefficient, equals to τ<sub>b</sub> / (ρ U<sup>2</sup> )
| maximum reach length
| -
| -
|-
| C<sub>Z</sub>
| dimensionless Chezy resistance coefficient.
|-
|-
| S<sub>l</sub>
| S<sub>l</sub>
| the initial bed slope
| initial bed slope of the river
| -
| -
|-
|-
| D<sub>i</sub>
| η<sub>i</sub>
| diameter
| initial bed elevation
| -
| -
|-
|-
| x
| D<sub>sub50</sub>
| downstream coordinate
| median size of the substrate layer
| m
| m
|-
|-
| Sl
| D<sub>subg</sub>
| slope of the bed surface
| geometric mean size of the substrate layer
| -
| m
|-
|-
| p<sub>feed</sub>
| L<sub>a</sub>
| GSD of the feed
| thickness of the active layer
| tons / year
| m
|-
|-
| F<sub>fs</sub>
| σ
| GSD of the substrate
| subsidence rate
| -
| -
|-
|-
| F<sub>f</sub>
| r<sub>B</sub>
| GSD of the final surface
| the ratio of depositional width to channel width
| -
| -
|-
|-
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| channel sinuosity
| channel sinuosity
| -
| -
|-
| r<sub>B</sub>
| ratio of depositional width to channel width
|
|-
|-
| Λ
| Λ
| ratio of wash load depositional per unit bed material load dpeosited
| units of wash load deposited in the system per unit of bed material load
| -
| -
|-
|-
| σ
| τ
| subsidence rate
| shear stress on bed surface
| mm/year
| N / m<sup>2</sup>
|-  
|-
| q<sub>w</sub>
| q<sub>b</sub>
| water discharge/width
| bed material load
| m<sup>2</sup> / s
| tons / year
|-
|-
| η<sub>d</sub>
| Δx
| base level
| spatial step length, equals to L / M
| m
| m
|-
|-
| I<sub>f</sub>
| Q<sub>w</sub>
| flood intermittency
| flood discharge
| -
| m<sup>3</sup> / s
|-
|-
| ∆<sub>t</sub>
| Δt
| time step
| time step
| day
| year
|-
|-
| n<sub>k</sub>
| Ntoprint
| factor by which surface D<sub>s90</sub> is multiplied to obtain roughness height k<sub>s</sub>
| number of time steps to printout
|  
| -
|-
|-
| n<sub>a</sub>
| Nprint
| factor by which surface D<sub>s90</sub> is multiplied to obtain active layer thickness L<sub>a</sub>
| number of printouts
| -
| -
|-
|-
| α<sub>r</sub>
| a<sub>U</sub>
| coefficient in Manning-Strickler resistance relation
| upwinding coefficient (1=full upwind, 0.5=central difference)
| -
| -
|-  
|-
| R<sub>r</sub>
| α<sub>s</sub>
| submerged specific gravity of gravel
| coefficient in sediment transport relation
| -
| -
|-
|-
| λ<sub>p</sub>
| p<sub>feed</sub>
| bed porosity, gravel
| GSD of the feed
| m
| tons / year
|-
|-
| I<sub>f</sub>
| F<sub>fs</sub>
| flood intermittency
| GSD of the substrate
| -
| -
|-
|-
| a<sub>U</sub>
| F<sub>f</sub>
| upwinding coefficient for load spatial derivatives in Exner equation (value > 0.5 suggested)
| GSD of the final surface
| -
| -
|-
|-
| α
| F<sub>up</sub>
| coefficient for material transferred to substrate as bed aggrades
| GSD at the upstream end
| -
| -
|-
|-
| k<sub>v</sub>
| F<sub>down</sub>
| roughness height
| GSD at the downstream end
| -
| -
|-
|-
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{| {{Prettytable}} class="wikitable sortable"
{| {{Prettytable}} class="wikitable sortable"
!Symbol!!Description!!Unit
!Symbol!!Description!!Unit
|-
| η
| bed surface elevatioon
| m
|-
|-
| H
| H
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| m
| m
|-
|-
| τ<sub>sg</sub>
| ξ
| shear stress on the bed surface
| water surface elevation
| N / m<sup>2</sup>
|-
| η
| bed surface elevation
| m
| m
|-
|-
| q<sub>bT</sub>
| τ<sub>b</sub>
| bedload transport rate
| bed shear stress
| m<sup>2</sup> / s
| kg / (s^2 m)
|-
|-
| q<sub>bTf</sub>
| S
| upstream feed rate
| bed slope
| tons / year
| -
|-
|-
| D<sub>sg</sub>
| q<sub>t</sub>
| geometric mean grain size on the bed surface
| total bed material load
| mm
| m<sup>2</sup> / s
|-
|-
| D<sub>s90</sub>
| L<sub>max</sub>
| the diameter such that 90% of the bed surface is finer
| maximum length of basin that the sediment supply can fill
| mm
| m
|-
|-
|}
|}

Revision as of 16:46, 10 May 2011

The CSDMS Help System

AgDegNormGravMixSubPW

It is the calculator for evolution of upward-concave bed profiles in rivers carrying sediment mixtures in subsiding basins.

Model introduction

This program calculates the bed surface evolution for a river of constant width with a mixture of gravel sizes with a load computed either by the Parker relation or the Wilcock-Crowe relation, as in the case of AgDegNormGravMixPW, but this program also takes into effect the subsidence.

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
Chezy Or Manning, Chezy-1 or Manning-2
Bedload relation, Parker or Wilock, Parker-1 or Wilock-2
Parameter Description Unit
Flood discharge m3 / s
gravel input m2 / s
Intermittency -
base level m
initial bed slope -
reach length m
Time step days
no. of intervals(100 or less) -
Number of printouts -
Iterations per each printout -
factor by which Ds90 is multiplied for roughness height -
factor by which Ds90 is multiplied for active layer thickness -
Manning-Strickler coefficient r
Submerged specific gravity of sediment
bed porosity, gravel
upwinding coefficient for load spatial derivatives in Exner equation (> 0.5 suggested)
coefficient for material transferred to substrate as bed aggrades
channel sinuosity
ratio of depositional width to channel width
ratio of wash load deposited per unit bed material load deposited
Chezy resistance coefficient -
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


Notes

The river is assumed to be morphologically active for If fraction of time, during which the flow is approximated as constant. Otherwise, the river is assumed to be morphologically dead.

The river flows into a basin that is subsiding with rate s. The basin has constant width; the ratio of basin width to river width is rB. The river has sinuosity W. For each unit of bedload deposited, L units of washload (typically sand transported in suspension) is deposited across the depositional basin.

In particular, the program computes the time evolution of the spatial profiles of bed elevation, bed slope, total bedload transport rate and grain size distribution of the surface (active) layer of the bed.

If run for a sufficient length of time, the river profile approaches a steady-state balance between subsidence. At this steady state the profile displays both an upward-concave elevation profile and downstream fining of the surface material.

The upstream point, at which sediment is fed, is fixed in the horizontal to be at x = 0. The vertical elevation of the upstream point may change freely as the bed aggrades or degrades.

The reach has constant length L, so that the downstream point is fixed in the horizontal at x = L. This downstream point has a user-specified initial elevation ηd.

Gravel bedload transport of mixtures is computed with a user-specified selection of the Parker (1990), or Wilcock-Crowe (2003) surface-based formulations for gravel transport.Sand and finer material must first be excluded from the grain size distributions, which then must be renormalized for gravel content only, in the case of the Parker (1990) relation. In the case of the Wilcock-Crowe (2003) relation, the sand is retained in the computation.

The grain size distributions of the sediment feed, initial surface material and substrate material must be specified. It is assumed that the grain size distribution of the sediment feed rate does not change in time, the initial grain size distribution of the surface material is the same at every node, the grain size distribution of the substrate is the same at every node and does not vary in the vertical. These constraints are easy to relax.

The program does not store the vertical and streamwise structure of the new substrate created as the bed aggrades. As a result, is cannot capture the case of aggradation followed by degradation. Again, the constraint is easy to relax, but at the price of increased memory requirements for storing the newly-created substrate.

The flow is calculated using the normal flow (local equilibrium) approximation.

In performing the calculation, the following control parameters must be specified: 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.

  • Note on model running

In the case of the load relation due to Parker (1990), the grain size distributions are automatically renormalized because the relation is for the transport of gravel only in the case of the load relation due to Wilcock-Crowe (2003), the sand and the fine sediment are retained for the computation

The user will be prompted by the program as to which bedload relation he would like to use.

The input grain size distributions may be on a 0-100% or a 0.00-1.00 scale, and the program will automatically scale.

The input grain size distributions must have bounds at 0% and 100% (1.00) to properly perform the calculation. If the user does not input the bounds the program will automatically interpolate upper and lower bounds DbU and DbL such that ffU = 100 (1.00) and ffL = 0

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 coefficients αr and nk are given in the inputfile. When all the three parameters are present, the program will ask the user which formulation they would like to use.

There is a maximum reach length equal to: Lmax = I * (1+Λ) * σ * Ω * qbTf / (rB * (1 – λ)), if the L value exceeds this critical value the program will exit and alert the user

The water depth is calculated using a Chézy formulation, when only the Chézy coefficient is present in the inputted file, and with the Manning-Strickler formulation, when only the roughness height, kc, value is present. When both 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:

See also: Help:Images or Help:Movies

Developer(s)

Gary Parker

References

Key papers

Links