Model help:DeltaNorm
DeltaNorm
This is a calculator for evolution of long profile of a river ending in a 1D migrating delta, using the normal flow approximation.
Model introduction
This program calculates the bed surface evolution for a narrowly channelized 1D fan-delta prograding into standing water, as well as calculating the initial and final amounts of sediment in the system.
Model parameters
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
- Water surface elevation
[math]\displaystyle{ \eta = \eta_{f}[s_{s} \left (t\right ), t] - S_{a}[x - s_{s}\left ( t \right )] }[/math] (1)
- Exner equation for shock condition
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) \int _{s_{s}\left (t\right )} ^ \left ( s_{b} \left (t\right ) \right ){\frac{\partial \eta}{\partial t}} d x = I_{f} \{q_{t}[s_{s}\left (t \right ), t] - q_{t} [s_{b}\left (t\right ),t] \} }[/math] (2)
[math]\displaystyle{ \dot{s_{s}} = {\frac{1}{\left (S_{a} - S_{s} \right )}}[{\frac{I_{f} q_{ts}}{\left ( 1 - \lambda_{p}\right ) \left (s_{b} - s_{s} \right )}} - {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}}] }[/math] (3)
[math]\displaystyle{ \left (S_{a} - S_{b} \right ) \dot{s}_{b} = \left (S_{a} - S_{s}\right ) \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}} }[/math] (4)
- Moving boundary coordinate
[math]\displaystyle{ \hat{x} = {\frac{x}{S_{s}\left (t\right )}} }[/math] (5)
[math]\displaystyle{ \hat{t} = t }[/math] (6)
- Exner equation for moving-boundary coordinate
[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) [\left ({\frac{\partial \eta_{f}}{\partial \hat{t}}} - {\frac{\dot{s}_{s}}{s_{s}}} \hat{x} {\frac{\partial \eta_{f}}{\partial \dot{x}}}\right )] = - {\frac{1}{s_{s}}} I_{f} {\frac{\partial q_{t}}{\partial \dot{x}}} }[/math] (7)
- Shock condition for moving-boundary coordinate
[math]\displaystyle{ \left (s_{b} - s_{s} \right )[{\frac{\partial \eta_{f}}{\partial \hat{t}}}|_{\hat{x} = 1} + S_{a} \dot{s}_{s}] = {\frac{I_{f} q_{t} \left (1, \hat{t}\right )}{\left ( 1 - \lambda_{p}\right )}} }[/math] (8)
- Continuity condition for moving-boundary coordinate
[math]\displaystyle{ \dot{s}_{b} = {\frac{S_{a} \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial \hat{t}}}|_{\hat{x} = 1}}{\left ( S_{a} - S_{b}\right )}} }[/math] (9)
- Sediment transport relation
1) Total bed material transport
[math]\displaystyle{ q_{t} = \sqrt{R g D} D q_{t} ^* }[/math] (10)
[math]\displaystyle{ q_{t}^* = \alpha_{t}[\tau^* - \tau_{c}^*]^ \left (n_{t}\right ) }[/math] (11)
- Normal flow approximation
[math]\displaystyle{ \tau^* = \left ( {\frac{C_{f} q_{w}^2}{g}}\right )^ \left ({\frac{1}{3}}\right ) {\frac{S^ \left ({\frac{2}{3}}\right )}{R D}} }[/math] (12)
[math]\displaystyle{ C_{f} = Cz^ \left (-2\right ) }[/math] (13)
[math]\displaystyle{ S = - {\frac{1}{s_{s}}} {\frac{\partial \eta _{f}}{\partial \dot{x}}} }[/math] (14)
- Boundary conditions
[math]\displaystyle{ s_{s} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{s} \left (\hat{t}\right ) + \dot{s}_{s} \Delta \hat{t} }[/math] (15)
[math]\displaystyle{ s_{b} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{b} \left (\hat{t}\right ) + \dot{s}_{b} \Delta \hat{t} }[/math] (16)
[math]\displaystyle{ \eta_{b} \equiv \eta [S_{b} \left (\hat{t} \right ), \hat{t}] = \eta_{d} - S_{s} \left ( s_{b} - s_{s}\right ) }[/math] (17)
- Calculation of derivatives
[math]\displaystyle{ {\frac{\partial \eta}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{\eta_{i+1} - \eta_{i}}{\Delta \hat{x}}} & i = 1 \\ {\frac{\eta_{i+1} - \eta_{i-1}}{2 \Delta \hat{x}}} & i = 2...M \\ {\frac{\eta_{i} - \eta_{i-1}}{\Delta \hat{x}}} & i = M+1 \end{matrix}\right. }[/math] (18)
[math]\displaystyle{ {\frac{\partial q_{t}}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{q_{t,j+1} - q_{tf}}{2 \Delta \hat{x}}} & i = 1 \\ {\frac{q_{t,i} - q_{t,i-1}}{2 \Delta \hat{x}}} & 2 \lt = i \lt = M \end{matrix}\right. }[/math] (19)
| Symbol | Description | Unit |
|---|---|---|
| qw | water discharge / width | L2 / T |
| Cz | dimensionless Chezy resistance coefficient | - |
| R | submerged specific gravity of sediment | - |
| λp | bed porosity | - |
| nt | exponent in load relation | - |
| ηd | water surface elevation of the lake | L |
| Sfl | initial fluvial bed slope | - |
| sfl | initial length of fluvial zone | L |
| Sa | slope of foreset face | - |
| Δt | time step | T |
| M | number of fluvial nodes | - |
| Mtoprint | number of steps until a printout is made | - |
| Mprint | number of printouts after the initial one | - |
| x | downstream coordinate | L |
| η | bed surface elevation | L |
| qb | volume bedload transport per unit width | L2 / T |
| H | water depth | L |
| τ | shear stress | - |
| sU | location of the upstream coordinate | L |
| sbb | reach of the alluvium bottom | L |
| sss | reach of the alluvium top | L |
| ηup | upstream bed surface elevation | L |
| ηtop | bed surface elevation of the top of the forest | L |
| ηbot | bed surface elevation of the bottom of the forest | L |
| q | flood discharge | L2 / T |
| If | flood intermittency | - |
| Q | upstream bed material sediment feed rate during floods | L2 / T |
| D | grain diameter | L |
| E | elevation of the top of the forest | L |
| e | initial elevation of the bottom of the forest | L |
| f | initial fluvial bedslope | - |
| b | subaqueous basement slope | - |
| s | initial length of the fluvial zone | L |
| L | bed porosity | - |
| k | coefficient in the Manning-Strickler relation | - |
| a | coefficient in the total bed material load relation | - |
| r | coefficient in the Manning-Strickler relation | - |
| t | time step | T |
| p | number of prints | - |
| i | number of iterations per print | - |
| ηs(t) | bed elevation at the topset-foreset break | L |
| ηb(t) | bed elevation at the foreset-bottomset break | L |
| ss(t) | coordinate corresponds to the topset-foreset break | L |
| sb(t) | coordinate corresponds to the foreset-bottomset break | L |
| qt | volume total bed material transport rate per unit
width |
L2 / T |
| qtf | upstream sediment feed rate | L2 / T |
| ηf(x,t) | bed elevation on the fluvial region | L |
| dot{s}s | prograding rate of the delta at the topset-foreset break | L / T |
| dot{s}b | prograding rate of the delta at the foreset-bottomset break | L / T |
| Ss | bed slope of the fluvial region at the topset-foreset break | - |
| x^ | dimensionless downstream coordinate | - |
| t^ | dimensionless time | - |
| qt * | Einstein number for total bed material | - |
| αt | coefficient in generic relation for total bed material load | - |
| τ* | Shields number | - |
| τc * | critical Shields number at the threshold of motion | - |
| Cf | bed friction coefficient | - |
| S | down-channel bed slope | - |
| g | acceleration due to gravity | L / T2 |
Notes
This model is used to calculate for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width. The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs the normal flow approximation rather than a full backwater calculation.
In the normal flow formulation, for any given time t = t^:
a) Specify the downstream bed elevation ηd
b) Calculate the bed slope S everywhere, and use this to find H everywhere.
c) Use this to evaluate qt everywhere, including qts at x^ = 1.
d) Implement the shock condition to find dot{s}s. This shock condition requires knowledge of the term d ηf / d t^ |x^ = 1 = d ηd / d t^, which can be directly computed in terms of the imposed function ηd \left (t^\right ).
e) Solve Exner everywhere except the last node (where bed elevation is specified) to find new bed elevations at time Δt^ later.
f) Use continuity condition to find dot{s}b.
- Note on model running
This model assumes a uniform grain size.
The fan builds outward by forming a prograding delta front with an assigned foreset slope.
The water depth is calculated using a Chézy formulation, when only the Chézy coefficient is present in the inputted text 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:
- Upload file: https://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)
References
- Grover, N.C., and Howard, C.L., 1937, The passage of turbid water through Lake Mead, Transactions, American Society of Civil Engineers, 103, 720-732.
- Kostic, S. and Parker, G., 2003a, Progradational sand-mud deltas in lakes and reservoirs. Part 1. Theory and numerical modeling, Journal of Hydraulic Research, 41(2), 127-140.
- Kostic, S. and Parker, G., 2003b, Progradational sand-mud deltas in lakes and reservoirs. Part 2. Experiment and numerical simulation, Journal of Hydraulic Research, 41(2), 141-152
- Swenson, J. B., Voller, V. R., Paola, C., Parker G. and Marr J., 2000, Fluvio-deltaic sedimentation: a generalized Stefan problem, European Journal of Applied Math., 11, 433-452.
