7.4 Tidal Patterns

The dynamic theory of tides explains how ocean basin geometry, resonance, and the Coriolis effect transform the simple equilibrium tide into the complex tidal patterns observed globally. Amphidromic systems, tidal mixing fronts, and internal tides govern coastal and deep-ocean dynamics.

Amphidromic Points and Co-tidal Charts

In the dynamic theory, the Coriolis effect causes tidal waves to rotate around points of zero amplitude called amphidromic points. Co-tidal lines radiate outward showing the progression of the tidal crest, while co-range lines connect points of equal tidal amplitude:

$$\eta(r,\theta,t) = A(r)\cos(\omega t - \theta + \phi_0)$$

Tidal wave rotates around the amphidromic point; amplitude increases with distance from it

Derivation: Amphidromic Point Formation

Step 1: A Kelvin wave propagates along a coast with the boundary on its right (Northern Hemisphere). Its amplitude decays exponentially away from the coast over the Rossby radius $R_d = \sqrt{gH}/f$:

$$\eta_1 = A\,e^{-y/R_d}\cos(kx - \omega t)$$

Step 2: In a closed or semi-enclosed basin, the wave reflects off the far boundary, producing a second Kelvin wave traveling in the opposite direction:

$$\eta_2 = B\,e^{-(L_y - y)/R_d}\cos(kx + \omega t)$$

Step 3: Superposition $\eta = \eta_1 + \eta_2$ creates a pattern where the two waves cancel at specific points. Setting $A(r) = 0$, the amplitude vanishes where the two wave amplitudes are equal and phases differ by $\pi$ -- these are amphidromic points.

Step 4: The Coriolis effect ensures the combined pattern rotates around these nodes. Co-tidal lines radiate outward like clock hands, completing one full rotation per tidal period.

The global M2 tide is organized into about a dozen major amphidromic systems. In the Northern Hemisphere, tides rotate counterclockwise (cyclonic) around amphidromic points; in the Southern Hemisphere, clockwise. The North Sea M2 tide is a classic example with three amphidromic points in the basin.

Degenerate amphidromic points occur when one of the opposing Kelvin waves is much weaker than the other; the amphidromic point moves to the coast and becomes a virtual amphidrome on land.

Co-tidal Lines

Lines of simultaneous high water radiating from amphidromic points like spokes of a wheel. Each line is labeled with the hour of high water relative to the Moon's transit.

Co-range Lines

Contours of equal tidal range (amplitude). Zero at the amphidromic point, increasing outward. Maximum range occurs at the basin boundaries.

Laplace Tidal Equations

The dynamic theory of tides is governed by the Laplace tidal equations (LTE), which are the linearized shallow water equations forced by the tide-generating potential:

$$\frac{\partial u}{\partial t} - fv = -g\frac{\partial \eta}{\partial x} + F_x$$

$$\frac{\partial v}{\partial t} + fu = -g\frac{\partial \eta}{\partial y} + F_y$$

$$\frac{\partial \eta}{\partial t} + \frac{\partial (Hu)}{\partial x} + \frac{\partial (Hv)}{\partial y} = 0$$

where $F_x, F_y$ include the tidal forcing and bottom friction terms

Derivation: Laplace Tidal Equations from Shallow Water Equations

Step 1: Start from the full 3D momentum equations. Apply the hydrostatic approximation ($\partial p/\partial z = -\rho g$), valid because horizontal scales $\gg$ ocean depth.

Step 2: Vertically integrate to get the nonlinear shallow water equations:

$$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} + f\hat{k} \times \mathbf{u} = -g\nabla\eta + \frac{\boldsymbol{\tau}_s - \boldsymbol{\tau}_b}{\rho H}$$

Step 3: Linearize about a rest state ($|\mathbf{u}| \ll 1$, $|\eta| \ll H$), dropping the advective term $(\mathbf{u} \cdot \nabla)\mathbf{u}$:

$$\frac{\partial u}{\partial t} - fv = -g\frac{\partial \eta}{\partial x}, \quad \frac{\partial v}{\partial t} + fu = -g\frac{\partial \eta}{\partial y}$$

Step 4: Add the tide-generating potential gradient $F_x = -g\,\partial(\eta_{\text{eq}})/\partial x$ as forcing, where $\eta_{\text{eq}}$ is the equilibrium tide. The linearized continuity equation is:

$$\frac{\partial \eta}{\partial t} + \frac{\partial(Hu)}{\partial x} + \frac{\partial(Hv)}{\partial y} = 0$$

These three equations are the Laplace Tidal Equations -- linearized, forced shallow water equations on a rotating Earth.

Resonance occurs when the natural period of a basin matches a tidal constituent period. The Bay of Fundy has a natural period of ~13 hours, close to the M2 period of 12.42 hours, resulting in the world's largest tides (>16 m range).

Tidal Currents and Tidal Ellipses

Tidal currents describe the horizontal water motion driven by the tidal wave. In shallow coastal areas, currents reverse direction (reversing currents), while in deeper water the Coriolis effect causes the current vector to trace an ellipse over one tidal period:

$$u(t) = U_{\max}\cos(\omega t - \phi_u), \quad v(t) = V_{\max}\cos(\omega t - \phi_v)$$

The tidal ellipse is characterized by its semi-major axis, semi-minor axis, inclination, and sense of rotation

Reversing Currents

In narrow channels and coastal straits, currents flow in one direction during flood and reverse during ebb. The tidal ellipse degenerates to a line. Typical speeds: 0.5–3 m/s.

Rotary Currents

In open water, the current vector rotates over a tidal cycle, tracing an ellipse. The rotation sense depends on latitude and basin geometry. Often clockwise in the Northern Hemisphere.

Form Factor and Tidal Classification

The form factor classifies the character of the local tide based on the ratio of principal diurnal to semidiurnal constituents:

$$F = \frac{A_{K1} + A_{O1}}{A_{M2} + A_{S2}}$$

Derivation: Form Factor Physical Meaning

Step 1: The diurnal constituents $K_1$ and $O_1$ arise from the Moon's and Sun's declination relative to the equatorial plane. They produce one high and one low per day. Their sum $A_{K1} + A_{O1}$ measures the total diurnal tidal energy at a location.

Step 2: The semidiurnal constituents $M_2$ and $S_2$ arise from the equatorial component of the tidal bulge. They produce two highs and two lows per day. Their sum $A_{M2} + A_{S2}$ measures total semidiurnal energy.

Step 3: The form factor is their ratio:

$$F = \frac{A_{K1} + A_{O1}}{A_{M2} + A_{S2}} = \frac{\text{diurnal signal strength}}{\text{semidiurnal signal strength}}$$

Step 4: Thresholds: $F < 0.25$ means semidiurnal energy dominates by 4:1 (pure semidiurnal); $F > 3.0$ means diurnal energy dominates by 3:1 (pure diurnal). Intermediate values produce mixed tides with diurnal inequality -- unequal successive highs or lows.

F < 0.25: Semidiurnal

Two nearly equal highs and lows per day. Atlantic coast of North America and Europe.

0.25 < F < 1.5: Mixed (mainly semidiurnal)

Two unequal highs and lows. US Pacific coast. Diurnal inequality prominent.

1.5 < F < 3.0: Mixed (mainly diurnal)

Occasionally only one high per day. Parts of Southeast Asia.

F > 3.0: Diurnal

One high and one low per day. Gulf of Mexico, parts of South China Sea.

Tidal Mixing Fronts (Simpson-Hunter Criterion)

In shelf seas, the competition between tidal stirring (which mixes the water column) and solar heating (which stratifies it) determines whether waters are well-mixed or stratified. The Simpson-Hunter parameter defines the transition:

$$\log_{10}\left(\frac{h}{u^3}\right) \approx 2.7 \text{ (critical value)}$$

h = water depth (m), u = tidal current amplitude (m/s)

h/u³ < 500

Strong tidal mixing dominates. Waters remain vertically well-mixed year-round. Low stratification.

h/u³ > 500

Stratification wins. Seasonal thermocline develops in summer. Nutrients trapped below pycnocline.

Tidal mixing fronts (the boundary between stratified and mixed waters) are biologically important: they concentrate nutrients and phytoplankton, supporting enhanced productivity and fisheries.

Tidal Mixing and Biological Productivity

The Celtic Sea, North Sea, and Georges Bank are classic examples where tidal mixing fronts sustain rich fisheries. The nutrient-rich mixed side provides a continuous supply of nutrients, while the stratified side allows phytoplankton to remain in the euphotic zone. Frontal convergence concentrates plankton, attracting higher trophic levels (fish, seabirds, marine mammals).

Internal Tides: Barotropic to Baroclinic Conversion

When the barotropic (surface) tide flows over abrupt topography (ridges, shelf breaks, seamounts) in a stratified ocean, it generates internal tides — internal waves at tidal frequencies:

$$P_{\text{conversion}} \sim \frac{1}{2}\rho_0 U_0^2 N h_{\text{topo}}^2 \frac{\omega}{\sqrt{\omega^2 - f^2}}$$

where $U_0$ is barotropic tidal velocity, N is buoyancy frequency, $h_{\text{topo}}$ is topographic height

The global conversion rate from barotropic to baroclinic tides is estimated at ~1 TW, roughly half of the total tidal dissipation. Internal tides propagate thousands of kilometers from generation sites (e.g., Hawaiian Ridge, Luzon Strait) and eventually break, providing ~50% of the deep ocean mixing required to maintain the abyssal stratification.

Tidal Prism

The tidal prism is the volume of water exchanged between an estuary and the ocean during one tidal cycle: $V_p = A \cdot \Delta h$, where A is the estuary surface area and Δh is the tidal range. This controls flushing time, salinity intrusion, and pollutant dilution.

Python: Amphidromic System, Tidal Ellipse & Simpson-Hunter

Python

!/usr/bin/env python3

script.py82 lines

#!/usr/bin/env python3
"""tidal_patterns.py - Amphidromic systems, tidal ellipses, mixing fronts"""
import numpy as np
import matplotlib.pyplot as plt

# --- 1. Amphidromic system (Kelvin wave rotating in a basin) ---
Lx, Ly = 500e3, 400e3      # basin dimensions (m)
g, f = 9.81, 1.0e-4         # gravity, Coriolis
H = 50.0                    # depth (m)
c = np.sqrt(g * H)          # wave speed
Rd = c / abs(f)              # Rossby radius
omega_M2 = 2*np.pi / (12.42*3600)  # M2 frequency

nx, ny = 200, 160
x = np.linspace(0, Lx, nx)
y = np.linspace(0, Ly, ny)
X, Y = np.meshgrid(x, y)

# Superpose two Kelvin waves to create amphidromic point
k = omega_M2 / c
phase_t = 0.0  # snapshot at t=0
# Northward propagating wave along west coast
eta1 = np.exp(-X/Rd) * np.cos(k*Y - phase_t)
# Southward propagating (reflected) wave
eta2 = 0.8 * np.exp(-(Lx-X)/Rd) * np.cos(k*Y + phase_t + np.pi/3)
eta = eta1 + eta2

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

ax1 = axes[0, 0]
# Amplitude (co-range)
amp = np.sqrt(eta1**2 + eta2**2)
cf = ax1.contourf(X/1e3, Y/1e3, amp, levels=15, cmap='YlOrRd')
plt.colorbar(cf, ax=ax1, label='Amplitude')
ax1.set_xlabel('x (km)'); ax1.set_ylabel('y (km)')
ax1.set_title('Co-range (Amplitude) Pattern')

ax2 = axes[0, 1]
# Phase (co-tidal) - using angle of complex amplitude
eta_complex = eta1 * np.exp(1j*0) + eta2 * np.exp(1j*np.pi/3)
phase = np.angle(eta_complex)
cf2 = ax2.contourf(X/1e3, Y/1e3, np.degrees(phase),
                    levels=12, cmap='hsv')
plt.colorbar(cf2, ax=ax2, label='Phase (degrees)')
ax2.set_xlabel('x (km)'); ax2.set_ylabel('y (km)')
ax2.set_title('Co-tidal (Phase) Pattern')

# --- 2. Tidal current ellipse ---
ax3 = axes[1, 0]
t = np.linspace(0, 2*np.pi, 200)
U_maj, U_min = 1.0, 0.3     # m/s semi-axes
incl = np.radians(30)        # inclination
u_ell = U_maj * np.cos(t) * np.cos(incl) - U_min * np.sin(t) * np.sin(incl)
v_ell = U_maj * np.cos(t) * np.sin(incl) + U_min * np.sin(t) * np.cos(incl)

ax3.plot(u_ell, v_ell, 'c-', lw=2)
ax3.plot(u_ell[0], v_ell[0], 'ro', ms=8, label='t=0')
ax3.arrow(u_ell[50], v_ell[50],
          u_ell[51]-u_ell[50], v_ell[51]-v_ell[50],
          head_width=0.05, color='yellow')
ax3.set_xlabel('u (m/s)'); ax3.set_ylabel('v (m/s)')
ax3.set_title('Tidal Current Ellipse')
ax3.set_aspect('equal'); ax3.grid(True, alpha=0.3)
ax3.legend()

# --- 3. Simpson-Hunter parameter ---
ax4 = axes[1, 1]
h_range = np.linspace(5, 200, 100)      # depth (m)
u_range = np.linspace(0.1, 2.0, 100)    # tidal velocity (m/s)
H_grid, U_grid = np.meshgrid(h_range, u_range)
SH = np.log10(H_grid / U_grid**3)

cf3 = ax4.contourf(H_grid, U_grid, SH, levels=20, cmap='RdYlBu_r')
ax4.contour(H_grid, U_grid, SH, levels=[2.7],
            colors='black', linewidths=2)
plt.colorbar(cf3, ax=ax4, label='log10(h/u^3)')
ax4.set_xlabel('Depth h (m)'); ax4.set_ylabel('Tidal velocity u (m/s)')
ax4.set_title('Simpson-Hunter Parameter (front at 2.7)')

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

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Laplace Tidal Equations Solver

Fortran

Solve linearized shallow water (Laplace tidal) equations

program.f9080 lines

program laplace_tidal_equations
  ! Solve linearized shallow water (Laplace tidal) equations
  ! on a rectangular basin to find tidal eigenmodes
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.141592653589793_dp
  real(dp), parameter :: g = 9.81_dp
  integer, parameter :: nx = 100, ny = 80
  real(dp) :: Lx, Ly, dx, dy, dt, f0, H0
  real(dp) :: u(nx,ny), v(nx,ny), eta(nx,ny)
  real(dp) :: u_new(nx,ny), v_new(nx,ny), eta_new(nx,ny)
  real(dp) :: omega_force, A_force, t_current
  real(dp) :: drag_coeff, max_eta
  integer :: i, j, step, nsteps

Lx = 500.0e3_dp; Ly = 400.0e3_dp  ! basin size (m)
  dx = Lx / real(nx-1, dp)
  dy = Ly / real(ny-1, dp)
  H0 = 50.0_dp          ! uniform depth (m)
  f0 = 1.0e-4_dp         ! Coriolis parameter
  drag_coeff = 1.0e-3_dp ! linear bottom drag

! M2 tidal forcing
  omega_force = 2.0_dp * pi / (12.42_dp * 3600.0_dp)
  A_force = 0.5_dp       ! forcing amplitude (m)

! Time stepping (CFL condition)
  dt = 0.4_dp * min(dx, dy) / sqrt(g * H0)
  nsteps = nint(5.0_dp * 12.42_dp * 3600.0_dp / dt)  ! 5 M2 periods

! Initialize
  u = 0.0_dp; v = 0.0_dp; eta = 0.0_dp

print *, 'Laplace Tidal Equations Solver'
  print '(A,F8.1,A,F8.1,A)', ' Basin: ', Lx/1e3, ' x ', Ly/1e3, ' km'
  print '(A,F6.1,A)', ' Depth: ', H0, ' m'
  print '(A,F6.1,A)', ' dt = ', dt, ' s'
  print '(A,I8)', ' Total steps: ', nsteps

do step = 1, nsteps
    t_current = real(step, dp) * dt

! Update momentum (leapfrog-like with forward Euler)
    do j = 2, ny-1
      do i = 2, nx-1
        u_new(i,j) = u(i,j) + dt * ( &
          f0 * v(i,j) - g * (eta(i+1,j) - eta(i-1,j))/(2.0_dp*dx) &
          - drag_coeff * u(i,j) / H0 &
          + A_force * omega_force * cos(omega_force * t_current) &
            * sin(pi * real(j-1,dp) / real(ny-1,dp)) )
        v_new(i,j) = v(i,j) + dt * ( &
          -f0 * u(i,j) - g * (eta(i,j+1) - eta(i,j-1))/(2.0_dp*dy) &
          - drag_coeff * v(i,j) / H0 )
      end do
    end do

! Update surface elevation (continuity)
    do j = 2, ny-1
      do i = 2, nx-1
        eta_new(i,j) = eta(i,j) - dt * H0 * ( &
          (u_new(i+1,j) - u_new(i-1,j))/(2.0_dp*dx) + &
          (v_new(i,j+1) - v_new(i,j-1))/(2.0_dp*dy) )
      end do
    end do

! Boundary conditions: no normal flow, zero eta gradient
    u_new(1,:) = 0.0_dp; u_new(nx,:) = 0.0_dp
    v_new(:,1) = 0.0_dp; v_new(:,ny) = 0.0_dp

u = u_new; v = v_new; eta = eta_new

if (mod(step, nsteps/10) == 0) then
      max_eta = maxval(abs(eta))
      print '(A,I8,A,F10.4,A)', ' Step ', step, &
            '  max|eta| = ', max_eta, ' m'
    end if
  end do

print *, 'Simulation complete.'
end program laplace_tidal_equations

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Tidal Resonance in Ocean Basins

Resonance occurs when the natural oscillation period of a basin matches a tidal constituent period. The natural period for a rectangular basin is given by Merian's formula:

$$T_n = \frac{2L}{n\sqrt{gh}}, \quad Q = \frac{T_n}{|T_n - T_{\text{tide}}|}$$

Q is the quality factor; higher Q means stronger resonant amplification

Bay of Fundy

Natural period ~13 hr (close to M2 = 12.42 hr). Resonance amplifies M2 tide from ~1 m in the open Atlantic to >16 m. The funnel shape further concentrates energy. World's largest tidal range.

Bristol Channel

Natural period ~12.5 hr. Near-perfect M2 resonance. Tidal range 14 m at Avonmouth. Proposed site for tidal barrage energy generation (Severn Barrage project).

Gulf of Mexico

Near-resonant for diurnal constituents (K1, O1). Produces diurnal tides despite M2 dominance in the open ocean. Form factor F > 3.0 in many areas.

Mediterranean

Near an M2 amphidromic point. Very small tides (<0.5 m typically). Natural period far from semidiurnal. Strait of Messina produces local tidal currents.

Tidal Energy Dissipation

The global tidal dissipation rate is approximately 3.7 TW, primarily from the M2 constituent. This energy is split between two main sinks:

$$\dot{E}_{\text{total}} \approx 3.7 \text{ TW} = \underbrace{2.5 \text{ TW}}_{\text{shelf friction}} + \underbrace{1.2 \text{ TW}}_{\text{internal tide conversion}}$$

Bottom friction in shallow shelf seas (particularly in the North Atlantic and Indonesian archipelago) accounts for roughly two-thirds. The remaining one-third is converted to internal tides at deep-ocean topographic features, providing the energy for abyssal mixing that maintains the global overturning circulation. This dissipation also causes a secular deceleration of Earth's rotation (day lengthening by ~2.3 ms/century) and recession of the Moon (~3.8 cm/yr).

Tidal Datums and Reference Levels

Tidal datums are standard elevation references defined by tidal statistics computed over the National Tidal Datum Epoch (currently 1983–2001 in the US). They are essential for navigation, coastal engineering, and legal boundaries:

Chart Datum (LAT/MLLW)

Lowest Astronomical Tide (LAT, international) or Mean Lower Low Water (MLLW, US). Depth on nautical charts is measured below this datum to ensure charted depths are conservative for navigation safety.

Mean Sea Level (MSL)

Average water level over the datum epoch. Used as the zero reference for land elevation (topographic maps). MSL varies geographically by up to 2 m due to ocean dynamic topography.

Mean High Water (MHW/MHHW)

Average of all high water levels (MHW) or higher high waters (MHHW). Used for coastal flood mapping, building setback lines, and defining the legal boundary of tidal waters in many jurisdictions.

Highest Astronomical Tide (HAT)

Maximum predicted tidal level under all astronomical conditions. Occurs once per 18.6-year nodal cycle. Used for defining flood-free elevations and infrastructure clearance requirements (bridge heights, port facilities).

Tidal Bores and Estuarine Tides

In funnel-shaped estuaries with large tidal ranges, the incoming flood tide can steepen into a tidal bore — a wall of water propagating upstream against the river current. The bore speed is given by:

$$c_{\text{bore}} = \sqrt{gh_2 \cdot \frac{h_2 + h_1}{2h_1}}$$

where $h_1$ is the pre-bore depth and $h_2$ is the post-bore depth; Froude number $Fr = c/\sqrt{gh_1} > 1$

Notable tidal bores include the Qiantang River bore (China, up to 9 m), the Severn bore (UK, 2 m), the Amazon pororoca (Brazil, 4 m), and the Ganges bore (Bangladesh). Conditions favoring bore formation include:

Geometric Requirements

Large tidal range (>4 m), shallow estuary, funnel-shaped convergence, low river flow. The rate of tidal rise must exceed the rate at which the estuary can accommodate the incoming volume.

Tidal Asymmetry

Shallow-water nonlinearity transfers energy to overtides (M4 = 2×M2 frequency). When flood duration < ebb duration, flood currents are stronger, enhancing sediment import and estuary infilling. The M4/M2 amplitude ratio quantifies asymmetry.

Key Concepts Summary

Amphidromic Systems

Points of zero amplitude around which the tidal wave rotates. Co-tidal lines show phase; co-range lines show amplitude. The global M2 tide has ~12 major amphidromic points.

Form Factor

F = (K1+O1)/(M2+S2) classifies tides as semidiurnal (F<0.25), mixed, or diurnal (F>3.0). Determined by local basin geometry and resonance.

Tidal Mixing Fronts

Simpson-Hunter criterion log'(h/u³) ≈ 2.7 separates stratified from well-mixed shelf seas. Biologically productive boundaries.

Internal Tides

Barotropic-to-baroclinic conversion at topography generates ~1 TW of internal tidal energy. Provides half of deep ocean mixing.

← 7.3 Tidal Forces 7.5 Tsunamis →