7.3 Tidal Forces

Ocean tides result from the differential gravitational attraction of the Moon and Sun acting across Earth's diameter. The equilibrium tide theory provides the theoretical framework, while harmonic analysis decomposes observed tides into predictable constituents with remarkable precision.

Gravitational Tidal Force Derivation

The tidal force arises from the difference between the gravitational acceleration at a point on Earth's surface and the acceleration at Earth's center. For a body of mass M at distance R from Earth's center, the tidal force on a unit mass at position r from Earth's center is:

$$F_t = \frac{2GMmr}{R^3}$$

The tidal force scales as the inverse cube of distance, not the inverse square

Derivation: Tidal Force from First Principles

Step 1: Gravitational acceleration at the near side (distance $R - r$ from the Moon):

$$a_{\text{near}} = \frac{GM}{(R-r)^2} = \frac{GM}{R^2}\frac{1}{(1 - r/R)^2} \approx \frac{GM}{R^2}\left(1 + \frac{2r}{R}\right)$$

Step 2: Gravitational acceleration at the far side (distance $R + r$):

$$a_{\text{far}} = \frac{GM}{(R+r)^2} \approx \frac{GM}{R^2}\left(1 - \frac{2r}{R}\right)$$

Step 3: At Earth's center, the acceleration is simply $a_{\text{center}} = GM/R^2$.

Step 4: The differential (tidal) acceleration is the difference between near/far side and center:

$$\Delta a = a_{\text{near}} - a_{\text{center}} \approx \frac{2GMr}{R^3}$$

Step 5: This gives the $R^{-3}$ dependence: tidal forces fall off as the inverse cube, not inverse square, because we are measuring the gradient of gravity across Earth's diameter.

This cubic dependence explains why the Moon (closer but less massive) produces tides about 2.2 times larger than the Sun:

$$\frac{F_{\text{Moon}}}{F_{\text{Sun}}} = \frac{M_{\text{Moon}}}{M_{\text{Sun}}} \left(\frac{R_{\text{Sun}}}{R_{\text{Moon}}}\right)^3 \approx 2.18$$

Derivation: Moon-to-Sun Tidal Force Ratio

Step 1: The tidal force scales as $F_t \propto M/R^3$, so the ratio depends on $M/R^3$, not $M/R^2$:

$$\frac{F_{\text{Moon}}}{F_{\text{Sun}}} = \frac{M_{\text{Moon}}}{M_{\text{Sun}}} \cdot \left(\frac{R_{\text{Sun}}}{R_{\text{Moon}}}\right)^3$$

Step 2: Substitute numerical values: $M_{\text{Moon}}/M_{\text{Sun}} = 7.342 \times 10^{22} / 1.989 \times 10^{30} = 3.69 \times 10^{-8}$.

Step 3: Distance ratio: $R_{\text{Sun}}/R_{\text{Moon}} = 1.496 \times 10^{11} / 3.844 \times 10^{8} = 389.2$.

Step 4: Cube the distance ratio: $(389.2)^3 = 5.90 \times 10^{7}$. Multiply: $3.69 \times 10^{-8} \times 5.90 \times 10^{7} \approx 2.18$.

The Sun is 27 million times more massive but 389 times farther, and $389^3 \gg 27 \times 10^6$, so the Moon wins.

Moon's Parameters

Mass M = 7.342 × 10²² kg, distance R = 384,400 km. Sidereal orbital period: 27.3 days; synodic period: 29.5 days. Orbital eccentricity e = 0.0549, causing perigean spring tides.

Sun's Parameters

Mass M = 1.989 × 10³° kg, distance R = 1.496 × 10&sup8; km. Despite 27 million times the Moon's mass, it is 389 times farther away. Earth's orbital eccentricity (e = 0.017) causes ~3.4% variation in solar tidal force between perihelion and aphelion.

Tidal Potential

The tide-generating potential at a point P on Earth's surface, expressed in terms of the zenith angle θ of the attracting body, is expanded in Legendre polynomials:

$$U = \frac{GM}{R}\sum_{n=2}^{\infty}\left(\frac{r}{R}\right)^n P_n(\cos\theta)$$

The dominant term (n=2) gives the equilibrium tidal shape

Derivation: Tidal Potential and Legendre Expansion

Step 1: The potential at point P on Earth due to a body at distance R is $\Phi = -GM/|\mathbf{R} - \mathbf{r}|$. Expand using the generating function for Legendre polynomials:

$$\frac{1}{|\mathbf{R} - \mathbf{r}|} = \frac{1}{R}\sum_{n=0}^{\infty}\left(\frac{r}{R}\right)^n P_n(\cos\theta)$$

Step 2: The $n=0$ term is constant (no force) and $n=1$ gives uniform acceleration (produces orbital motion, not tides). The tidal potential starts at $n=2$.

Step 3: Since $r/R \approx 1/60$ for the Moon, higher-order terms are negligible. The $n=2$ term with $P_2(\cos\theta) = (3\cos^2\theta - 1)/2$ dominates, giving the familiar double-bulge shape.

Step 4: The equilibrium tide height follows from balancing the tidal potential against gravity:

$$\zeta = -\frac{U_2}{g} = \frac{GMr^2}{2gR^3}(3\cos^2\theta - 1)$$

Maximum at $\theta = 0, \pi$ (sub-lunar and anti-lunar points); minimum at $\theta = \pi/2$ (flanks).

The second-degree potential produces the familiar tidal bulges:

$$U_2 = \frac{GM r^2}{R^3} P_2(\cos\theta) = \frac{GM r^2}{2R^3}(3\cos^2\theta - 1)$$

The equilibrium tide height is $\zeta = -U_2/g$, yielding maximum elevation of approximately 0.36 m for the Moon and 0.16 m for the Sun.

Doodson (1921) decomposed the tidal potential into species based on the cosine power:

Long-period species (n=0): zonal terms independent of Earth's rotation (Mf, Mm, Ssa). Diurnal species (n=1): terms varying once per lunar day from declination effects (K1, O1, P1, Q1). Semidiurnal species (n=2): terms varying twice per lunar day from the equatorial tidal bulge (M2, S2, N2, K2).

Spring and Neap Tides

The Moon and Sun's tidal potentials combine constructively or destructively over the synodic month (~29.5 days):

Spring Tides

At new and full moon, Sun and Moon are aligned (syzygy). Their tidal forces add, producing maximum tidal range. The combined equilibrium amplitude is $\zeta_{\text{spring}} \approx 0.36 + 0.16 = 0.52$ m.

Neap Tides

At first and third quarter moon, Sun and Moon are at 90° (quadrature). Their forces partially cancel, giving minimum range. $\zeta_{\text{neap}} \approx 0.36 - 0.16 = 0.20$ m.

The spring-to-neap ratio is approximately (1 + 0.46)/(1 - 0.46) ≈ 2.7, where 0.46 is the ratio of solar to lunar tidal amplitudes.

Tidal Constituents and Harmonic Analysis

Observed tides are decomposed into harmonic constituents, each with a known frequency determined by astronomical arguments. The tidal height at time t is expressed as:

$$\eta(t) = Z_0 + \sum_{i=1}^{N} A_i f_i \cos(\omega_i t + V_i + u_i - g_i)$$

$A_i$ = amplitude, $\omega_i$ = angular speed, $g_i$ = Greenwich phase lag, $f_i, u_i$ = nodal corrections

M2 (Principal Lunar Semidiurnal)

Period: 12.4206 hr. Speed: 28.984°/hr. Largest constituent globally. Amplitude ~0.24 m (equilibrium).

S2 (Principal Solar Semidiurnal)

Period: 12.0000 hr. Speed: 30.000°/hr. ~46% of M2. The M2-S2 beat produces spring-neap cycle.

K1 (Luni-Solar Diurnal)

Period: 23.9345 hr. Speed: 15.041°/hr. Caused by declination of Moon and Sun. ~58% of M2 amplitude.

O1 (Principal Lunar Diurnal)

Period: 25.8193 hr. Speed: 13.943°/hr. ~41% of M2. Modulated by lunar declination.

N2 (Larger Lunar Elliptic)

Period: 12.6583 hr. Speed: 28.440°/hr. ~19% of M2. Due to ellipticity of Moon's orbit.

K2 (Luni-Solar Semidiurnal)

Period: 11.9672 hr. Speed: 30.082°/hr. ~13% of M2. Declination modulation of semidiurnal tide.

Solid Earth Tides and Ocean Loading

The solid Earth also deforms under tidal forces. The Love numbers $h_2$ and $k_2$ characterize the elastic response:

$$\zeta_{\text{body}} = h_2 \frac{U_2}{g}, \quad \text{where } h_2 \approx 0.609$$

The solid Earth tide has an amplitude of about 20 cm at mid-latitudes. Additionally, the mass of ocean tidal water loading and unloading the crust produces measurable deformation (ocean tidal loading), important for precise GPS positioning and InSAR geodesy.

Love Number Relationships

The potential Love number $k_2 \approx 0.302$ describes how the tidal potential of the solid Earth is modified by deformation. The ocean tide height is $(1 + k_2 - h_2)/g \times U_2$. Precise Love numbers depend on Earth's mantle rheology and are measured from satellite orbit perturbations (LAGEOS, GRACE).

Tidal Triggering

Earth tidal stresses may influence volcanic eruptions, earthquake triggering (particularly at mid-ocean ridges), and ice stream motion. Tidal loading on ice shelves modulates grounding-line ice discharge from Antarctic glaciers at fortnightly and monthly timescales.

Python: Tidal Potential, Harmonic Analysis & Prediction

Python

!/usr/bin/env python3

script.py107 lines

#!/usr/bin/env python3
"""tidal_forces.py - Tidal potential, harmonic analysis, prediction"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import leastsq

G = 6.674e-11
M_moon = 7.342e22; M_sun = 1.989e30; M_earth = 5.972e24
R_earth = 6.371e6; r_moon = 3.844e8; r_sun = 1.496e11
g = 9.81

# --- 1. Tidal potential and equilibrium tide ---
theta = np.linspace(0, 2*np.pi, 500)

def equilibrium_tide(theta, M, R_dist):
    """Equilibrium tide height from Legendre P2."""
    return (3*M*R_earth**2) / (4*M_earth*R_dist) * \
           R_earth / R_dist**2 * (3*np.cos(theta)**2 - 1)

zeta_moon = equilibrium_tide(theta, M_moon, r_moon)
zeta_sun = equilibrium_tide(theta, M_sun, r_sun)

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

# Polar plot of equilibrium tide
ax1 = axes[0, 0]
ax1 = plt.subplot(221, projection='polar')
ax1.plot(theta, zeta_moon*100 + 40, 'b-', lw=2, label='Moon')
ax1.plot(theta, zeta_sun*100 + 40, 'orange', lw=2, label='Sun')
ax1.plot(theta, (zeta_moon+zeta_sun)*100 + 40, 'r--',
         lw=2, label='Combined (spring)')
ax1.set_title('Equilibrium Tidal Bulge (cm)', pad=15)
ax1.legend(fontsize=7, loc='lower right')

# --- 2. Harmonic analysis of synthetic tide gauge data ---
# Generate synthetic tide data with noise
hours = np.arange(0, 30*24, 0.5)  # 30 days, 30-min sampling
# True constituents: (amplitude m, period hr, phase rad)
true_const = [
    (1.2, 12.4206, 0.5),   # M2
    (0.35, 12.0000, 1.2),  # S2
    (0.45, 23.9345, 0.8),  # K1
    (0.30, 25.8193, 2.1),  # O1
]
eta_true = sum(A*np.cos(2*np.pi/T*hours - phi)
               for A, T, phi in true_const)
eta_obs = eta_true + 0.1*np.random.randn(len(hours))

# Least-squares harmonic analysis
periods = [12.4206, 12.0000, 23.9345, 25.8193]
names = ['M2', 'S2', 'K1', 'O1']

def tidal_model(params, t, periods):
    """Model: sum of cosines with amplitude and phase."""
    n = len(periods)
    result = params[0]  # mean level
    for i in range(n):
        A, phi = params[1+2*i], params[2+2*i]
        result += A * np.cos(2*np.pi/periods[i]*t - phi)
    return result

def residuals(params, t, y, periods):
    return y - tidal_model(params, t, periods)

p0 = [0.0] + [1.0, 0.0]*len(periods)  # initial guess
popt, _ = leastsq(residuals, p0, args=(hours, eta_obs, periods))

print('Harmonic Analysis Results:')
print(f'{"Const":>6} {"True A":>8} {"Fit A":>8} {"True phi":>10} {"Fit phi":>10}')
for i, (name, (A_t, T_t, phi_t)) in enumerate(zip(names, true_const)):
    A_fit = abs(popt[1+2*i])
    phi_fit = popt[2+2*i] % (2*np.pi)
    print(f'{name:>6} {A_t:8.3f} {A_fit:8.3f} {phi_t:10.3f} {phi_fit:10.3f}')

# Plot observed vs predicted
ax2 = axes[0, 1]
eta_pred = tidal_model(popt, hours, periods)
ax2.plot(hours/24, eta_obs, 'b-', alpha=0.4, label='Observed')
ax2.plot(hours/24, eta_pred, 'r-', lw=1.5, label='Predicted')
ax2.set_xlabel('Time (days)'); ax2.set_ylabel('Height (m)')
ax2.set_title('Tidal Harmonic Analysis'); ax2.legend()
ax2.set_xlim(0, 7); ax2.grid(True, alpha=0.3)

# --- 3. Spring-neap cycle ---
ax3 = axes[1, 0]
t_long = np.arange(0, 60*24, 1.0)
eta_spring_neap = (1.2*np.cos(2*np.pi/12.4206*t_long) +
                   0.35*np.cos(2*np.pi/12.0*t_long))
ax3.plot(t_long/24, eta_spring_neap, 'b-', lw=0.5)
ax3.set_xlabel('Time (days)'); ax3.set_ylabel('Height (m)')
ax3.set_title('Spring-Neap Cycle (M2 + S2)')
ax3.grid(True, alpha=0.3)

# --- 4. Constituent amplitudes bar chart ---
ax4 = axes[1, 1]
fit_amps = [abs(popt[1+2*i]) for i in range(len(periods))]
true_amps = [tc[0] for tc in true_const]
x = np.arange(len(names))
ax4.bar(x-0.15, true_amps, 0.3, label='True', color='steelblue')
ax4.bar(x+0.15, fit_amps, 0.3, label='Fitted', color='coral')
ax4.set_xticks(x); ax4.set_xticklabels(names)
ax4.set_ylabel('Amplitude (m)')
ax4.set_title('Constituent Amplitudes'); ax4.legend()

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

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Tidal Harmonic Analysis

Fortran

Fit M2, S2, K1, O1 tidal constituents to observed data

program.f90111 lines

program tidal_harmonic_analysis
  ! Fit M2, S2, K1, O1 tidal constituents to observed data
  ! using least-squares normal equations
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.141592653589793_dp
  integer, parameter :: nconst = 4
  integer, parameter :: nobs = 1440   ! 30 days at 30-min intervals
  integer, parameter :: ncol = 2*nconst + 1  ! mean + cos/sin pairs

real(dp) :: periods(nconst), omega(nconst)
  real(dp) :: t(nobs), eta(nobs)
  real(dp) :: A(nobs, ncol), ATA(ncol, ncol), ATb(ncol)
  real(dp) :: x(ncol), amp, phase
  character(len=4) :: names(nconst)
  integer :: i, j, k, ipiv(ncol), info

! Constituent periods (hours)
  periods = (/ 12.4206_dp, 12.0000_dp, 23.9345_dp, 25.8193_dp /)
  names = (/ 'M2  ', 'S2  ', 'K1  ', 'O1  ' /)

do i = 1, nconst
    omega(i) = 2.0_dp * pi / periods(i)
  end do

! Generate synthetic observations (30 days, dt=0.5 hr)
  do i = 1, nobs
    t(i) = real(i-1, dp) * 0.5_dp
    eta(i) = 1.20_dp * cos(omega(1)*t(i) - 0.50_dp) + &
             0.35_dp * cos(omega(2)*t(i) - 1.20_dp) + &
             0.45_dp * cos(omega(3)*t(i) - 0.80_dp) + &
             0.30_dp * cos(omega(4)*t(i) - 2.10_dp)
  end do

! Build design matrix [1, cos(w1*t), sin(w1*t), ...]
  do i = 1, nobs
    A(i, 1) = 1.0_dp  ! mean level
    do j = 1, nconst
      A(i, 2*j)     = cos(omega(j) * t(i))
      A(i, 2*j + 1) = sin(omega(j) * t(i))
    end do
  end do

! Normal equations: ATA * x = ATb
  ATA = 0.0_dp; ATb = 0.0_dp
  do i = 1, nobs
    do j = 1, ncol
      ATb(j) = ATb(j) + A(i,j) * eta(i)
      do k = 1, ncol
        ATA(j,k) = ATA(j,k) + A(i,j) * A(i,k)
      end do
    end do
  end do

! Solve by Gaussian elimination (simple pivoting)
  call solve_linear(ncol, ATA, ATb, x)

! Extract amplitudes and phases
  print *, '=== Tidal Harmonic Analysis Results ==='
  print *, 'Constituent  Amplitude(m)  Phase(deg)'
  do j = 1, nconst
    amp = sqrt(x(2*j)**2 + x(2*j+1)**2)
    phase = atan2(x(2*j+1), x(2*j)) * 180.0_dp / pi
    if (phase < 0.0_dp) phase = phase + 360.0_dp
    print '(A6, F14.4, F12.2)', names(j), amp, phase
  end do

contains

subroutine solve_linear(n, AA, bb, xx)
    integer, intent(in) :: n
    real(dp), intent(inout) :: AA(n,n), bb(n)
    real(dp), intent(out) :: xx(n)
    real(dp) :: factor, maxval_pivot, temp
    integer :: ii, jj, kk, maxrow

do ii = 1, n
      ! Partial pivoting
      maxrow = ii; maxval_pivot = abs(AA(ii,ii))
      do kk = ii+1, n
        if (abs(AA(kk,ii)) > maxval_pivot) then
          maxrow = kk; maxval_pivot = abs(AA(kk,ii))
        end if
      end do
      ! Swap rows
      do jj = 1, n
        temp = AA(ii,jj); AA(ii,jj) = AA(maxrow,jj); AA(maxrow,jj) = temp
      end do
      temp = bb(ii); bb(ii) = bb(maxrow); bb(maxrow) = temp

! Eliminate
      do kk = ii+1, n
        factor = AA(kk,ii) / AA(ii,ii)
        do jj = ii, n
          AA(kk,jj) = AA(kk,jj) - factor * AA(ii,jj)
        end do
        bb(kk) = bb(kk) - factor * bb(ii)
      end do
    end do

! Back substitution
    do ii = n, 1, -1
      xx(ii) = bb(ii)
      do jj = ii+1, n
        xx(ii) = xx(ii) - AA(ii,jj) * xx(jj)
      end do
      xx(ii) = xx(ii) / AA(ii,ii)
    end do
  end subroutine

end program tidal_harmonic_analysis

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Code will be compiled with gfortran and executed on the server

Tidal Prediction and Practical Applications

Once the harmonic constants (amplitude and phase) are determined from observations, tides can be predicted with remarkable accuracy for decades into the future. The prediction is simply the reconstruction of the harmonic series:

$$\eta_{\text{pred}}(t) = Z_0 + \sum_{i=1}^{N} H_i f_i(t) \cos[\omega_i t + V_i(t) + u_i(t) - g_i]$$

Nodal corrections $f_i$ and $u_i$ vary slowly over the 18.61-year lunar nodal cycle

For optimal results, the International Hydrographic Organization recommends analyzing at least 369 days of observations to resolve the closely-spaced S2 and K2 constituents (Rayleigh criterion: ΔT ≥ 1/|ω_1 − ω_2|).

Navigation and Shipping

Precise tidal predictions are essential for under-keel clearance in shallow ports, timing of harbor approaches, and tidal stream atlases for sailing. Published annually in tide tables by national hydrographic offices.

Tidal Energy

Tidal stream turbines extract kinetic energy from tidal currents. Globally, ~100 GW is theoretically extractable. Leading sites include Pentland Firth (Scotland), Bay of Fundy, and Raz Blanchard (France). Tidal barrages capture potential energy from the tidal range.

Geodesy and Satellite Altimetry

Ocean tides are the largest high-frequency signal in satellite altimetry and must be removed to study non-tidal sea level changes. Modern tidal models (FES2014, GOT4.10, TPXO9) achieve sub-centimeter accuracy for M2 in the open ocean.

Biological Rhythms

Many marine organisms synchronize spawning with tidal cycles (e.g., grunion, palolo worms). Intertidal zonation is directly controlled by tidal range. Mangrove and salt marsh ecosystems depend on tidal inundation patterns.

Key Concepts Summary

Tidal Potential

Gravitational potential expanded in Legendre polynomials. Dominant P' term produces the double-bulge equilibrium shape.

Force Scaling

Tidal force &propto; M/R³. The Moon dominates despite lower mass because it is 389 times closer than the Sun.

Harmonic Analysis

Observed tides decomposed into constituents (M2, S2, K1, O1, N2, ...) with known astronomical frequencies. Enables precise prediction.

Spring-Neap Cycle

The beat between M2 and S2 produces a ~14.8 day spring-neap modulation with amplitude ratio ~2.7:1.

← 7.2 Wave Types 7.4 Tidal Patterns →