5.5 Coastal Geology

Coasts are dynamic environments shaped by waves, tides, currents, and sea-level change. Beach morphology, sediment transport, and coastal landforms respond to forcing on timescales from individual storms to millennia of tectonic uplift and glacial cycles.

Beach Morphology

A beach profile from land to sea includes the backshore (above normal high tide, including dunes and berm crest), the foreshore (intertidal zone, the sloping face of the beach), and the nearshore (extending to the outer edge of the surf zone). Offshore bars (longshore bars) form where breaking waves deposit sediment. The Dean equilibrium profile describes the idealized cross-shore shape:

$$h(x) = A\,x^{2/3}$$

where $h$ is water depth, $x$ is distance offshore, and $A$ is a shape parameter that depends on sediment grain size ($A \approx 0.1\;\text{m}^{1/3}$ for medium sand).

Berm

Flat terrace built by wave swash at the top of the foreshore. Summer berms are wider; storm berms are higher.

Longshore Bar

Submerged sand ridge parallel to shore where waves break. Multiple bars can form in dissipative beaches.

Trough

Depression between the bar and the beach. Longshore currents flow through this channel during storms.

Sediment Transport

Sediment movement begins when the shear stress exerted by waves or currents exceeds a critical threshold. The Shields parameter characterises the onset of motion:

$$\theta = \frac{\tau_b}{(\rho_s - \rho)\,g\,d}$$

where $\tau_b$ is bed shear stress, $\rho_s$ is sediment density, $\rho$ is water density, and $d$ is grain diameter. Motion begins when $\theta > \theta_{cr} \approx 0.05$.

Bedload Transport

Grains roll, slide, or saltate along the bed. The Meyer-Peter & Mueller formula gives the transport rate:$q_b = 8(\theta - \theta_{cr})^{3/2}\sqrt{(\rho_s/\rho - 1)g d^3}$.

Suspended Load

Fine sediment lifted into the water column by turbulence. Concentration decreases exponentially with height above the bed following the Rouse profile: $C(z)/C_a = [(h-z)/z \cdot a/(h-a)]^{w_s/(\kappa u_*)}$.

Littoral Drift (Longshore Transport)

Waves approaching the coast at an angle drive a net longshore current and sediment transport. The CERC formula (Coastal Engineering Research Center) estimates the annual longshore transport rate:

$$Q_l = K\,\frac{\rho g}{16(\rho_s - \rho)(1-p)}\,H_b^{5/2}\,\sin(2\alpha_b)$$

where $H_b$ is breaking wave height, $\alpha_b$ is the breaker angle, $p$ is sediment porosity, and $K \approx 0.77$.

Bruun Rule for Beach Erosion

The Bruun Rule (1962) relates shoreline retreat to sea-level rise, assuming the beach profile translates landward and upward to maintain its equilibrium shape:

$$R = \frac{S \cdot W^*}{B + h^*}$$

where $R$ is shoreline retreat, $S$ is sea-level rise, $W^*$ is the active profile width, $B$ is berm height, and $h^*$ is the closure depth. For typical values ($W^* = 500$ m, $B + h^* = 12$ m), a 1 m sea-level rise produces ~42 m of retreat.

Coastal Classification & Landforms

Barrier Islands

Long, narrow sand islands parallel to the mainland coast, separated by lagoons. Found on trailing-edge (passive) margins with gentle slopes and abundant sediment (e.g., U.S. East and Gulf Coasts). Migrate landward during sea-level rise through overwash and inlet migration.

Deltas

Classified by dominant process: river-dominated (Mississippi, bird's foot), tide-dominated (Ganges, funnel-shaped), and wave-dominated (Nile, arcuate). Delta morphology reflects the balance between sediment supply and marine reworking.

Estuaries

Semi-enclosed water bodies where freshwater and saltwater mix. Types: salt-wedge(strongly stratified, large river, e.g., Mississippi), partially mixed(moderate river/tidal forcing, e.g., Chesapeake Bay), and well-mixed(strong tidal currents, e.g., Delaware Bay). The salinity intrusion length scales as:

$$L \propto \frac{K_H \cdot A}{Q_r}$$

where $K_H$ is horizontal diffusivity, $A$ is cross-sectional area, and $Q_r$ is river discharge.

Tidal Flats

Extensive low-gradient surfaces alternately flooded and exposed. Dominated by fine sediment and biological activity.

Mangroves

Tropical/subtropical forests at the land–sea interface. Trap sediment, protect coasts, and serve as nursery habitat.

Coral Reefs

Biogenic structures in warm, clear, shallow waters. Fringing, barrier, and atoll types reflect tectonic subsidence history (Darwin).

Python: Bruun Rule, Longshore Transport & Estuary Salinity

Python

!/usr/bin/env python3

script.py57 lines

#!/usr/bin/env python3
"""coastal_geology.py - Bruun rule, longshore transport, estuary salinity."""
import numpy as np
import matplotlib.pyplot as plt

# ── 1. Bruun Rule: Sea-Level Rise Erosion ──
def bruun_retreat(S, W_star=500, B=2, h_star=10):
    """Shoreline retreat (m) for sea-level rise S (m)."""
    return S * W_star / (B + h_star)

slr = np.linspace(0, 2, 100)   # sea-level rise (m)
fig, axes = plt.subplots(1, 3, figsize=(18, 5))

ax = axes[0]
for W in [300, 500, 800]:
    retreat = bruun_retreat(slr, W_star=W)
    ax.plot(slr * 100, retreat, lw=2, label=f'W* = {W} m')
ax.set_xlabel('Sea Level Rise (cm)')
ax.set_ylabel('Shoreline Retreat (m)')
ax.set_title('Bruun Rule: Retreat vs SLR')
ax.legend(); ax.grid(True, alpha=0.3)

# ── 2. Longshore Sediment Transport (CERC formula) ──
def cerc_transport(Hb, alpha_b_deg, K=0.77, rho=1025, rho_s=2650, g=9.81, p=0.4):
    """Annual longshore transport rate (m^3/yr)."""
    alpha_b = np.radians(alpha_b_deg)
    return K * rho * g * Hb**2.5 * np.sin(2 * alpha_b) / (
        16 * (rho_s - rho) * (1 - p))

Hb_range = np.linspace(0.5, 3.0, 100)
ax = axes[1]
for angle in [5, 10, 15, 20]:
    Ql = cerc_transport(Hb_range, angle)
    ax.plot(Hb_range, Ql, lw=2, label=f'alpha = {angle} deg')
ax.set_xlabel('Breaking Wave Height Hb (m)')
ax.set_ylabel('Transport Rate (m3/yr per m)')
ax.set_title('Longshore Sediment Transport (CERC)')
ax.legend(); ax.grid(True, alpha=0.3)

# ── 3. Estuary Salinity Profile ──
def estuary_salinity(x, S_ocean=35, L=50, x0=30, steepness=0.2):
    """Hyperbolic tangent salinity profile along estuary."""
    return S_ocean / 2 * (1 + np.tanh(steepness * (x - x0)))

x_km = np.linspace(0, 80, 200)   # distance from river mouth
ax = axes[2]
for Qr_label, x0_val in [('Low river', 45), ('Med river', 30), ('High river', 15)]:
    S = estuary_salinity(x_km, x0=x0_val)
    ax.plot(x_km, S, lw=2, label=Qr_label)
ax.set_xlabel('Distance from River Mouth (km)')
ax.set_ylabel('Salinity (PSU)')
ax.set_title('Estuary Salinity Profile')
ax.legend(); ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('coastal_geology.png', dpi=150)
plt.show()

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: 1D Morphodynamic Model for Beach Profile Evolution

This program evolves a cross-shore beach profile under wave forcing using a simple energetics-based sediment transport model. The bed level $z_b$ evolves according to the Exner equation:

$$\frac{\partial z_b}{\partial t} = -\frac{1}{1-p}\frac{\partial q_s}{\partial x}$$

Fortran: 1D Morphodynamic Model for Beach Profile Evolution

Fortran

============================================================

program.f9073 lines

program beach_profile_1d
  ! ============================================================
  ! 1D morphodynamic model for cross-shore beach evolution
  ! Uses Exner equation with energetics-based transport
  ! ============================================================
  implicit none
  integer, parameter :: dp = selected_real_kind(15, 307)
  integer, parameter :: nx = 200, nt = 50000
  real(dp) :: zb(0:nx)              ! bed elevation (m, positive up)
  real(dp) :: zb_new(0:nx)
  real(dp) :: qs(0:nx)              ! sediment transport rate (m^2/s)
  real(dp) :: dx, dt, x_max
  real(dp) :: A_dean, H0, T_wave, gamma_b
  real(dp) :: h, Hb, slope, x
  real(dp) :: porosity
  integer  :: i, n

! Parameters
  x_max   = 1000.0_dp             ! cross-shore domain (m)
  dx      = x_max / real(nx, dp)
  dt      = 0.5_dp                 ! time step (s)
  A_dean  = 0.1_dp                 ! Dean profile shape parameter
  H0      = 1.5_dp                 ! offshore wave height (m)
  T_wave  = 8.0_dp                 ! wave period (s)
  gamma_b = 0.78_dp                ! breaker ratio H/h
  porosity = 0.4_dp

! Initial bed: Dean equilibrium profile
  do i = 0, nx
    x = real(i, dp) * dx
    if (x > 10.0_dp) then
      zb(i) = -A_dean * x**(2.0_dp / 3.0_dp)
    else
      zb(i) = 1.0_dp - 0.1_dp * x  ! simple beach face
    end if
  end do

! Time evolution
  do n = 1, nt
    ! Compute transport rate at each point
    do i = 1, nx - 1
      h = max(0.01_dp, -zb(i))     ! local water depth
      Hb = gamma_b * h              ! local breaking height (saturated)
      Hb = min(Hb, H0)             ! cannot exceed offshore height
      slope = (zb(i+1) - zb(i-1)) / (2.0_dp * dx)

! Simplified energetics transport (Bailard 1981 concept)
      qs(i) = 1.0d-4 * Hb**3 * slope / h
    end do
    qs(0)  = 0.0_dp                 ! no transport at shore
    qs(nx) = 0.0_dp                 ! no transport at offshore boundary

! Exner equation: update bed
    do i = 1, nx - 1
      zb_new(i) = zb(i) - dt / ((1.0_dp - porosity) * dx) * (qs(i) - qs(i-1))
    end do
    zb_new(0)  = zb(0)
    zb_new(nx) = zb(nx)
    zb = zb_new
  end do

! Output final profile
  open(unit=10, file='beach_profile.dat', status='replace')
  write(10,'(A)') '# x(m)  zb(m)'
  do i = 0, nx
    write(10,'(F10.2, 1X, F10.4)') real(i,dp)*dx, zb(i)
  end do
  close(10)

write(*,'(A)') 'Beach profile evolution complete.'
  write(*,'(A)') 'Output: beach_profile.dat'

end program beach_profile_1d

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Wave Refraction & Longshore Currents

As waves approach the shore, they slow down in shallow water (since phase speed $c = \sqrt{gd}$ in shallow water). Portions of the wave crest in deeper water travel faster, causing the wave crest to bend (refract) toward alignment with the bottom contours. Snell's law for wave refraction gives:

$$\frac{\sin\alpha_1}{c_1} = \frac{\sin\alpha_2}{c_2}$$

where $\alpha$ is the angle between the wave crest and the depth contour, and $c$ is the local phase speed. Refraction concentrates wave energy on headlands (erosion) and disperses it in bays (deposition).

Longshore Current Velocity

Waves breaking at an angle to the shore drive a longshore current within the surf zone. The Longuet-Higgins (1970) formula gives the mean longshore current velocity:

$$V_l = \frac{5\pi}{16}\frac{\gamma_b g^{1/2}}{c_f}H_b^{1/2}\sin\alpha_b\cos\alpha_b$$

where $\gamma_b$ is the breaker ratio, $c_f$ is a friction coefficient, and $H_b$ is the breaking wave height. Typical longshore currents are 0.3–1 m/s.

Coastal Classification Systems

Emergent vs Submergent

Emergent coasts are rising relative to sea level (tectonic uplift or glacio-isostatic rebound), producing raised marine terraces, exposed wave-cut platforms, and steep rocky shorelines (e.g., U.S. West Coast, Scandinavia). Submergent coasts are sinking or experiencing relative sea-level rise, producing drowned river valleys (rias), flooded glacial valleys (fjords), and wide coastal plains (e.g., U.S. East Coast, Southeast Asia).

Primary vs Secondary

Primary (youthful) coasts are shaped mainly by non-marine processes (tectonics, volcanism, glaciation, river deposition)—their morphology reflects recent geological history. Secondary (mature) coasts have been significantly modified by marine processes (wave erosion, longshore transport, biological construction). Most real coasts exhibit both primary and secondary characteristics.

Key Equations

Closure depth (Hallermeier, 1978):$h^* = 2.28\,H_s - 68.5\,\frac{H_s^2}{gT_s^2}$ where $H_s$ and $T_s$ are the significant wave height and period. Beyond $h^*$, the profile does not change significantly over years.

Iribarren number (surf similarity):$\xi = \tan\beta / \sqrt{H_0/L_0}$ classifies breaker type: spilling ($\xi < 0.5$), plunging ($0.5 < \xi < 3.3$), surging ($\xi > 3.3$).

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