Chapter 5 Coastal Engineering: Equilibrium Breach Profile

Chapter 5 Coastal Engineering: Equilibrium Breach Profile#

Introduction: Equilibrium Beach Profile
Simulation: Dean Profile Simulator
Self-Assessment

1. Introduction#

An equilibrium beach profile refers to a coastal state in which the net sand transport into or out of a littoral cell does not alter the shape of the beach, resulting in a stable shoreline. This condition occurs under consistent wave and water level impacts. However, equilibrium is typically short-lived, as wave energy and water levels are dynamic and continuously changing.

Dean Profile with Bruun Rule and Structural Impact#

This Python script models the equilibrium beach profile using the Dean formulation, and evaluates shoreline retreat due to sea level rise (SLR) using the Bruun Rule. It also incorporates the effect of coastal structures (e.g., seawalls, groins) on retreat magnitude.

Model Description#

The equilibrium beach profile is defined by:

\[ h(x) = A \cdot x^{2/3} \]

Where:

\(( h(x) \)): water depth at distance \(( x \)) offshore
\(( A \)): Dean parameter (m(^{1/3})), related to sediment grain size

The Bruun Rule estimates shoreline retreat \(( R \)) as:

\[ R = \alpha \cdot \frac{S \cdot L}{h + B} \]

Where:

\(( S \)): Sea level rise (m)
\(( L \)): Active profile width (m)
\(( h \)): Depth of closure (m)
\(( B \)): Berm elevation (m)
\(( \alpha \)): Structure impact factor (dimensionless)

###🧮 Interactive Parameters

A: Dean parameter (default = 0.17 m\((^{1/3}\)))
SLR: Sea level rise (default = 0.5 m)
alpha: Structure impact factor (default = 1.0)
- \(( \alpha < 1 \)): structure reduces retreat
- \(( \alpha > 1 \)): structure increases retreat
- \(( \alpha = 1 \)): baseline Bruun Rule

Dean Profile with Bruun Rule and Structure-Induced Beach Narrowing#

This Python model simulates the equilibrium beach profile using the Dean formulation, estimates shoreline retreat due to sea level rise (SLR) using the Bruun Rule, and incorporates the impact of coastal structures that narrow the active beach profile.

Model Overview: Dean Profile#

The equilibrium beach profile is defined by:

\[ h(x) = A \cdot x^{2/3} \]

Where:

\(( h(x) \)): water depth at distance ( x ) offshore
\(( A \)): Dean parameter (m(^{1/3})), related to sediment grain size

🔹 Bruun Rule with Structural Impact#

Shoreline retreat \(( R \)) is estimated as:

\[ R = \alpha \cdot \frac{S \cdot L_{\text{eff}}}{h + B} \]

Where:

\(( S \)): Sea level rise (m)
\(( L_{\text{eff}} \)): Effective active profile width, reduced by structure
\(( h \)): Depth of closure (m)
\(( B \)): Berm elevation (m)
\(( \alpha \)): Structure impact factor
- \(( \alpha < 1 \)): structure reduces retreat
- \(( \alpha > 1 \)): structure increases retreat

🔹 Beach Narrowing#

The narrowing factor represents the fractional reduction in active beach width due to the presence of a structure:

\[ L_{\text{eff}} = L \cdot (1 - \text{narrowing\_factor}) \]

Interactive Parameters#

A: Dean parameter (default = 0.17 m(^{1/3}))
SLR: Sea level rise (default = 0.5 m)
alpha: Structure impact factor (default = 1.0)
narrowing_factor: Fractional narrowing of beach width (default = 0.2)

Structural Impact: Beach Narrowing Factor#

Coastal structures such as seawalls, revetments, and groins often lead to a reduction in active beach width, a phenomenon known as beach narrowing. This effect is quantified in the model using a narrowing factor, which represents the fractional loss of usable beach width due to the presence of a structure.

🔹 Definition#

\[ L_{\text{eff}} = L \cdot (1 - \text{narrowing\_factor}) \]

Where:

\(( L \)): original active profile width
\(( L_{\text{eff}} \)): effective width after narrowing
narrowing_factor: fraction of beach width lost (0 to 1)

Typical Values from Literature#

Scenario	Narrowing Factor (fraction of width lost)
Well-designed seawall	0.10–0.20
Typical armoring (revetment)	0.30–0.40
Severe erosion or poor design	0.50+

These values are based on observed impacts of shoreline armoring on beach morphology.

Armored beaches narrowed by an average of 36% ± 5%
Unarmored beaches narrowed by 16% ± 6%
Complete beach loss occurred in 29% of armored sections

Modeling Implications#

Use narrowing_factor = 0.2 for modest structure impact
Use narrowing_factor = 0.4 for typical long-term armoring
Use narrowing_factor > 0.5 for severe erosion or poorly designed structures

This parameter allows the model to simulate how structures constrain natural beach migration and amplify retreat under sea level rise.

Plot Features#

Original profile (solid teal line)
Retreated profile (dashed orange line)
Vertical arrows showing:
- Total shoreline retreat (red)

Reference#

The equilibrium beach profile (EBP), defined by h = Ay^{2/3}, models how beaches adjust to wave energy and sediment properties [], []. The Bruun Rule predicts that sea level rise causes the entire profile to shift landward and upward, leading to shoreline retreat [Bruun, 1962],[U.S. Army Corps of Engineers, 2011]. This assumption is based on a stable profile shape, no longshore transport, and a sediment budget bounded by the depth of closure. Dean’s refinements link profile steepness to sediment grain size and wave climate. Coastal structures can interrupt this adjustment, often accelerating erosion and preventing natural shoreline migration [U.S. Army Corps of Engineers, 2011].

2. Simulation#

🏖️ Dean Profile Simulator with Beach Type & Bruun Rule Retreat#

This interactive Jupyter tool combines empirical beach profiling with climate and infrastructure-driven change modeling to visualize shoreline evolution.

🧠 What It Does#

Uses the Dean profile equation: ( h = A \cdot x^{2/3} ) to model equilibrium beach shape
Computes shoreline retreat due to Sea Level Rise (SLR) using the Bruun Rule
Adjusts for structural narrowing using a narrowing factor
Classifies beach type based on Dean parameter A
Visualizes both original and retreated profiles, with key annotations

🎛️ User Inputs#

Parameter	Meaning
`A (m¹ᐟ³)`	Dean sediment scale parameter
`SLR (m)`	Sea Level Rise
`α`	Structure impact multiplier
`Narrowing (%)`	Proportion of beach width removed

📊 Outputs#

Two profiles: original and retreated shoreline shape
Retreat distance (R) computed from Bruun Rule adjusted by structure and narrowing
Beach type classification (e.g., fine sand, gravel) based on A
Annotations for SLR, retreat magnitude, narrowing effect
📈 Inverted y-axis plot simulating real bathymetric depth view

🧭 How to Interpret#

Higher SLR or α → greater horizontal retreat
Larger A values → coarser beach types and steeper profiles
Narrowing factor simulates coastal infrastructure impact
Useful for exploring how different conditions affect beach morphology and resilience

This tool bridges sediment profile theory with shoreline management—ideal for planners, educators, and modelers.

import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider

def classify_beach_type(A):
    """
    Return a description of beach type based on Dean parameter A.
    """
    if A < 0.12:
        return "Fine sand beach (e.g., low-energy, muddy shores)"
    elif A < 0.17:
        return "Medium-fine sand beach (e.g., Atlantic barrier islands)"
    elif A < 0.22:
        return "Medium sand beach (e.g., Gulf Coast typical)"
    elif A < 0.30:
        return "Coarse sand beach (e.g., high-energy open coasts)"
    else:
        return "Gravel or cobble beach (e.g., steep reflective profiles)"

def dean_profile_with_beach_type(A, SLR, alpha, narrowing_factor, L=500, h_closure=5, berm_height=2):
    """
    Plot Dean profile with Bruun Rule retreat, structural impact, and beach type annotation.
    """
    # Effective profile width
    L_eff = L * (1 - narrowing_factor)

    # Retreat calculation
    R = alpha * (SLR * L_eff) / (h_closure + berm_height)

    # Profiles
    x = np.linspace(0, L, 500)
    h_original = A * x**(2/3)
    h_retreat = A * (x + R)**(2/3)

    # Beach type classification
    beach_type = classify_beach_type(A)

    # Plot
    plt.figure(figsize=(10, 6))
    plt.plot(x, h_original, label='Original Profile', color='teal')
    plt.plot(x, h_retreat, label='Retreated Profile', color='orange', linestyle='--')

    # Annotate retreat
    plt.axvline(0, color='teal', linestyle=':')
    plt.axvline(R, color='orange', linestyle=':')
    plt.annotate(f'Total Retreat = {R:.1f} m',
                 xy=(R/2, 1),
                 xytext=(R/2, 3),
                 arrowprops=dict(arrowstyle='->', color='red'),
                 color='red',
                 fontsize=10)

    # Annotate SLR
    plt.annotate(f'SLR = {SLR:.2f} m',
                 xy=(L - 50, SLR),
                 xytext=(L - 100, SLR + 2),
                 arrowprops=dict(arrowstyle='->', color='blue'),
                 color='blue',
                 fontsize=10)

    # Annotate narrowing
    plt.annotate(f'Narrowing: {narrowing_factor*100:.0f}%',
                 xy=(L_eff, 0),
                 xytext=(L_eff + 20, 2),
                 arrowprops=dict(arrowstyle='->', color='purple'),
                 color='purple',
                 fontsize=10)

    # Annotate beach type
    plt.text(10, max(h_original)*0.9, f'Beach Type: {beach_type}',
             fontsize=11, color='darkgreen', bbox=dict(facecolor='lightyellow', edgecolor='gray'))

    plt.xlabel('Distance Offshore (m)')
    plt.ylabel('Water Depth (m)')
    plt.title('Dean Profile with Bruun Rule, Structural Impact, and Beach Type')
    plt.gca().invert_yaxis()
    plt.grid(True)
    plt.legend()
    plt.tight_layout()
    plt.show()

# Interactive sliders
interact(
    dean_profile_with_beach_type,
    A=FloatSlider(value=0.17, min=0.10, max=0.35, step=0.01, description='A (m¹ᐟ³)'),
    SLR=FloatSlider(value=0.5, min=0.0, max=2.0, step=0.1, description='SLR (m)'),
    alpha=FloatSlider(value=1.0, min=0.5, max=2.0, step=0.1, description='Structure Impact α'),
    narrowing_factor=FloatSlider(value=0.2, min=0.0, max=0.9, step=0.05, description='Narrowing (%)')
);

3. Self-Assessment#

Conceptual Questions#

What does the Dean parameter ( A ) represent physically, and how does it relate to sediment grain size and wave energy?
How does sea level rise (SLR) influence shoreline retreat according to the Bruun Rule? What assumptions does the Bruun Rule make about sediment transport and profile adjustment?
Why do coastal structures like seawalls and groins lead to beach narrowing? What are the long-term implications of this narrowing for coastal resilience?
How does the narrowing factor affect the active profile width ( L_{\text{eff}} ), and why is this important when modeling retreat?
What are the limitations of using a one-dimensional equilibrium profile model to simulate real-world coastal change?

Reflective Questions#

When adjusting the Dean parameter ( A ), how does the shape of the beach profile change? What does this tell you about the sediment characteristics of different coastlines?
If you increase the structure impact factor ( \alpha ), what happens to the retreat distance? How might this vary between updrift and downdrift sides of a groin?
How would you modify this model to account for sediment supply from rivers or offshore sources?
In your region (e.g., Gulf Coast), what types of structures are most common, and how might they influence the values of ( \alpha ) and the narrowing factor?
What are the trade-offs between using hard structures versus nature-based solutions (e.g., dunes, marshes) to manage shoreline retreat?

Quiz Questions (Click to Reveal Answers)#

Which of the following beach types is most likely to have a Dean parameter ( A ) greater than 0.30?
- A) Fine sand beach
- B) Medium sand beach
- C) Gravel beach
- D) Mudflat
✅ Show Answer
**Answer: C — Gravel beach**

If the narrowing factor is 0.4, what percentage of the active beach profile width is lost?
- A) 20%
- B) 40%
- C) 60%
- D) 80%
✅ Show Answer
**Answer: B — 40%**

The Bruun Rule assumes that sediment is:
- A) Lost offshore
- B) Imported from rivers
- C) Conserved and redistributed landward
- D) Trapped by vegetation
✅ Show Answer
**Answer: C — Conserved and redistributed landward**

Increasing the structure impact factor ( \alpha ) will:
- A) Reduce shoreline retreat
- B) Increase shoreline retreat
- C) Have no effect
- D) Only affect berm height
✅ Show Answer
**Answer: B — Increase shoreline retreat**

Which parameter directly controls the steepness of the beach profile in the Dean model?
- A) Sea level rise
- B) Berm height
- C) Dean parameter ( A )
- D) Depth of closure
✅ Show Answer
**Answer: C — Dean parameter \( A \)**

Chapter 5 Coastal Engineering: Equilibrium Breach Profile

Contents

Chapter 5 Coastal Engineering: Equilibrium Breach Profile#

1. Introduction#

Dean Profile with Bruun Rule and Structural Impact#

Model Description#

Dean Profile with Bruun Rule and Structure-Induced Beach Narrowing#

Model Overview: Dean Profile#

🔹 Bruun Rule with Structural Impact#

🔹 Beach Narrowing#

Interactive Parameters#

Structural Impact: Beach Narrowing Factor#

🔹 Definition#

Typical Values from Literature#

Modeling Implications#

Plot Features#

Reference#

2. Simulation#

🏖️ Dean Profile Simulator with Beach Type & Bruun Rule Retreat#

🧠 What It Does#

🎛️ User Inputs#

📊 Outputs#

🧭 How to Interpret#

3. Self-Assessment#

Conceptual Questions#

Reflective Questions#

Quiz Questions (Click to Reveal Answers)#