# 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

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
Parameter Description Unit
Flood discharge (q) m2 / s
Intermittency (I) flood intermittency -
upstream bed material sediment feed rate during flood (Q) m2 / m
Grain size of bed material (D) mm
Chezy resistance coefficient (C) cofficient in the Chezy relation -
Exponent in load relation (n) -
Critical Shields stress in load relation (T) -
Elevation of top of forest (E) m
Initial elevation of forest bottom (e) m
Initial fluvial bed slope (f) -
Subaqueous basement slope (b) -
initial length of fluvial zone (s) m
Slope of forest face (Sa) -
Submerged specific gravity of sediment (R) -
Bed porosity (L) -
Manning-Strickler coefficient (k) coefficient in the Manning-Strickler relation -
Manning-Strickler coefficient (r) coefficient in the Manning-Strickler relation -
Coefficient in total bed material relation (a) -
Number of fluvial nodes (M) -
Time step (t) days
Number of printouts after initial one -
Iterations per each printout -
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

• Water surface elevation
 $\displaystyle{ \eta = \eta_{f}[s_{s} \left (t\right ), t] - S_{a}[x - s_{s}\left ( t \right )] }$ (1)
• Exner equation for shock condition
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) \int _{s_{s}\left (t\right )} ^ \left ( s_{b} \left (t\right ) \right ){\frac{\partial \eta}{\partial t}} d x = I_{f} \{q_{t}[s_{s}\left (t \right ), t] - q_{t} [s_{b}\left (t\right ),t] \} }$ (2)
 $\displaystyle{ \dot{s_{s}} = {\frac{1}{\left (S_{a} - S_{s} \right )}}[{\frac{I_{f} q_{ts}}{\left ( 1 - \lambda_{p}\right ) \left (s_{b} - s_{s} \right )}} - {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}}] }$ (3)
 $\displaystyle{ \left (S_{a} - S_{b} \right ) \dot{s}_{b} = \left (S_{a} - S_{s}\right ) \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial t}}|_{s_{s}} }$ (4)
• Moving boundary coordinate
 $\displaystyle{ \hat{x} = {\frac{x}{S_{s}\left (t\right )}} }$ (5)
 $\displaystyle{ \hat{t} = t }$ (6)
• Exner equation for moving-boundary coordinate
 $\displaystyle{ \left ( 1 - \lambda_{p} \right ) [\left ({\frac{\partial \eta_{f}}{\partial \hat{t}}} - {\frac{\dot{s}_{s}}{s_{s}}} \hat{x} {\frac{\partial \eta_{f}}{\partial \dot{x}}}\right )] = - {\frac{1}{s_{s}}} I_{f} {\frac{\partial q_{t}}{\partial \dot{x}}} }$ (7)
• Shock condition for moving-boundary coordinate
 $\displaystyle{ \left (s_{b} - s_{s} \right )[{\frac{\partial \eta_{f}}{\partial \hat{t}}}|_{\hat{x} = 1} + S_{a} \dot{s}_{s}] = {\frac{I_{f} q_{t} \left (1, \hat{t}\right )}{\left ( 1 - \lambda_{p}\right )}} }$ (8)
• Continuity condition for moving-boundary coordinate
 $\displaystyle{ \dot{s}_{b} = {\frac{S_{a} \dot{s}_{s} + {\frac{\partial \eta _{f}}{\partial \hat{t}}}|_{\hat{x} = 1}}{\left ( S_{a} - S_{b}\right )}} }$ (9)
• Sediment transport relation

1) Total bed material transport

 $\displaystyle{ q_{t} = \sqrt{R g D} D q_{t} ^* }$ (10)
 $\displaystyle{ q_{t}^* = \alpha_{t}[\tau^* - \tau_{c}^*]^ \left (n_{t}\right ) }$ (11)
• Normal flow approximation
 $\displaystyle{ \tau^* = \left ( {\frac{C_{f} q_{w}^2}{g}}\right )^ \left ({\frac{1}{3}}\right ) {\frac{S^ \left ({\frac{2}{3}}\right )}{R D}} }$ (12)
 $\displaystyle{ C_{f} = Cz^ \left (-2\right ) }$ (13)
 $\displaystyle{ S = - {\frac{1}{s_{s}}} {\frac{\partial \eta _{f}}{\partial \dot{x}}} }$ (14)
• Boundary conditions
 $\displaystyle{ s_{s} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{s} \left (\hat{t}\right ) + \dot{s}_{s} \Delta \hat{t} }$ (15)
 $\displaystyle{ s_{b} \left ( \hat{t} + \Delta \hat{t} \right ) = s_{b} \left (\hat{t}\right ) + \dot{s}_{b} \Delta \hat{t} }$ (16)
 $\displaystyle{ \eta_{b} \equiv \eta [S_{b} \left (\hat{t} \right ), \hat{t}] = \eta_{d} - S_{s} \left ( s_{b} - s_{s}\right ) }$ (17)
• Calculation of derivatives
 $\displaystyle{ {\frac{\partial \eta}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{\eta_{i+1} - \eta_{i}}{\Delta \hat{x}}} & i = 1 \\ {\frac{\eta_{i+1} - \eta_{i-1}}{2 \Delta \hat{x}}} & i = 2...M \\ {\frac{\eta_{i} - \eta_{i-1}}{\Delta \hat{x}}} & i = M+1 \end{matrix}\right. }$ (18)
 $\displaystyle{ {\frac{\partial q_{t}}{\partial \hat{x}}}|_{i} = \left\{\begin{matrix} {\frac{q_{t,j+1} - q_{tf}}{2 \Delta \hat{x}}} & i = 1 \\ {\frac{q_{t,i} - q_{t,i-1}}{2 \Delta \hat{x}}} & 2 \lt = i \lt = M \end{matrix}\right. }$ (19)

## 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: