Morphodynamic Equilibria in Tidal Embayments with Decreasing Cross-Section

1
Morphodynamic Equilibria in Tidal Embayments with
Decreasing Cross-Section

• Henk Schuttelaars
• The Institute for Marine and Atmospheric research
  Utrecht (IMAU)
• Utrecht University

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Contents

• Introduction
• Model Formulation
• Numerical Experiments
• Comparison with Observations
• Conclusions Future Research

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Tidal Embayments
Introduction

• Semi-enclosed bodies of water
• Connected to the open sea
• Driven by tides

Examples

• Frisian Inlet System
• Western Scheldt
• Inlets East Coast of the US

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Marine Part of the Western Scheldt
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Amplitude Hor. Vel.
Phase Hor. Vel.
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Salinity Distribution in the Western Scheldt
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Properties

• Length 160 km
• Depth 20 m - 50 m - decreasing
• Complex Pattern of Channels and Shoals
• Main Channels 3-1
• Dredging
• Cyclic Behaviour
• Fractal Patterns

• Tidal Range Vl 3.2 m Ant 4.0 m
  decreasing
• M2/M4 0.1 (at entrance)
• Phase difference (approx. 1 tidal wave length)

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Research Questions

?

• Do Long-Term Equilibria Exist
• Are They Unique and Stable

PARA- METERS
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Research Questions

?

• Do Long-Term Equilibria Exist
• Are They Unique and Stable
• Can We Explain/Model/Predict

PARA- METERS

• Cyclic Bar behaviour
• Formation of Main Channels
• Splitting of Channels
• Formation of Thresholds

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Model Formulation

• Idealized Models

Water Motion
Short Time Scale
Sediment Transport
Long Time Scale
Bed Evolution
Averaging
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Model Equations and Assumptions

• Depth Averaged Shallow Water Equations
• Only Bed Erodible
• Noncohesive Material
• Suspended Load Transport
• Sediment Balance

Fine Sand
hole
bar
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The Equations
Continuity
Momentum
Concentration
Bed Evolution
The Boundary Conditions
X0
XL
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Sediment Flux
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Geometry
Side View
W
W
Top View
E
Embayment
R
X0
XL
15
Parameter Continuation
Short Embayment

• Analytical Solution
• constantly sloping bed
• spatially uniform hor and vert
  velocities
• spatially uniform bed stress

For other parameters, continue this known solution
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Parameter Continuation
Short Embayment

• Analytical Solution
• constantly sloping bed
• uniform hor and vert velocities
• spatially uniform bed stress
• net sediment import after channel reduction

For other parameters, continue this known solution
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The Experiments

• Reducing WR
• Different Types of Equilibria

Experiment 1

• Comparison with data

Experiment 2
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The Experiments

• Reducing WR
• Local Lorentz Linearization ( L 110 km ltlt
  LMAX)
• Different Types of Equilibria

Experiment 1

• Comparison with data
• Global Lorentz Linearization (L 100 km
  LMAX)

Experiment 2
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Maximum Embayment Length
Figure A1. The dimensionless frictional length
scale Lf Lg of the M2 tide versus the
dimensionless length of the em-bayment. Here Lg
is the frictionless tidal wavelength and Lf has
been computed using (A2) with 06. The water
depths Hf and Hmax are determined by the
characteristics of the morphodynamic equilibrium.
The intersection of these curves with the solid
curve determines the maximum embayment length
Lmax.
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Width Variation
Bed Profile
WR/WE
Sea
Land
110 km
WR/WE 1
WR/WE 0.5

• Bed Very Deep
• Standing Wave
• M4extern amplified

• Bed Weakly Concave
• Travelling Wave
• M4extern negligible

Decreasing WR
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Amplitude of M4extern Hor Vel
WR/WE
110 km
Sea
Land
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Frictionally Dominated Equilibrium
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External Equilibrium
Plate 2. Contour plots of the dominant
contributions to the net sediment flux (defined
in (11)) for the morphodynamic equilibrium that
owes its existence to the presence of external
overtides. Dependent variables are the
dimensionless coordinate x and the length scale
ratio LLg. Here Figure 2a shows us Cs , Figure
2b shows us Cs , Figure 2c shows u 2c C 2c ,
and Figure 2d shows u 2c C 2c . The scaling
for the fluxes is identical to that in Figure 5.
Here 015, other parameter values are as in
Table 3.
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Comparison With Field Data (1)
Width
Width-Averaged Depth
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Comparison With Field Data (2)
Sea
Land
Sea
Land
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Conclusions

• Two Types of Equilibria
• Idealized Model Reproduces Global
  Characteristics Quite Well
• Multiple Equilibria
• Maximum Embayment Length

• externally driven
• friction-related

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Future Research

• Closer Comparison With Process-Based Models
  and Observations
• Introduce Two Dimensional Perturbations

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Multiple Equilibria