Tide and Tidal Datum Explained

This article continues the Elevation & Vertical Datums sub-hub with the tidal-physics complement to MSL. /learn/mean-sea-level-explained covered the 19-year averaging that defines MSL; this article covers what gets averaged — the daily, fortnightly, and annual tide patterns — and the various tidal datums derived from tide-gauge statistics.

What causes tides

The gravitational attraction of the Moon and Sun on Earth's oceans, modulated by Earth's rotation.

Lunar tidal force

The Moon's gravity pulls on Earth's near side more than its far side, creating a tidal bulge on both sides:

• Near-side bulge: ocean water is pulled toward the Moon.
• Far-side bulge: ocean water is pulled less than the rocky Earth, effectively bulging outward relative to it.

The Moon causes the dominant tidal force. Mathematical fact: tidal force scales with mass / distance³. Though the Sun is ~27 million times more massive than the Moon, it's ~400 times further away; the Sun's tidal force is about 46% of the Moon's.

As Earth rotates beneath the (slowly moving) lunar tidal bulges, an observer experiences two high tides and two low tides per day — the M2 (principal lunar semidiurnal) tide.

Solar tidal force

The Sun produces a secondary tidal bulge. When Sun, Earth, and Moon align (new and full moons), the bulges combine to produce extra-large spring tides. When at right angles (first and last quarter moons), they partially cancel, producing smaller neap tides. The spring-neap cycle has a 14.77-day period.

Earth rotation

Earth rotates once per day relative to the Sun (24 hours) and once per ~24.84 hours relative to the Moon (the lunar day is slightly longer because the Moon orbits Earth in the same direction Earth rotates). The M2 period is exactly half the lunar day: 12.42 hours.

Modulating effects

• Lunar declination: the Moon's orbital plane is inclined ~5° to the ecliptic and ~28.5° to Earth's equator. The Moon's position above or below the equator varies; this drives the K1 and O1 diurnal constituents.
• Lunar orbital eccentricity: the Moon's orbit is elliptical, so its distance varies; this drives the N2 constituent.
• Solar declination: similar but smaller solar effects.
• Coastal geometry: continental shelves, bays, and inlets shape and amplify tides.

Tidal constituents

Tides are decomposed into a sum of sinusoidal components (the harmonic constituents), each with a specific frequency derived from astronomical motions. The dominant constituents:

| Constituent | Period (hours) | Description | Typical amplitude | | ----------- | -------------- | ----------- | ----------------- | | M2 | 12.42 | Principal lunar semidiurnal | Largest (dominant) | | S2 | 12.00 | Principal solar semidiurnal | ~46% of M2 | | N2 | 12.66 | Larger lunar elliptical semidiurnal | ~19% of M2 | | K2 | 11.97 | Lunar+solar declinational | ~12% of M2 | | K1 | 23.93 | Lunar+solar diurnal | Variable | | O1 | 25.82 | Principal lunar diurnal | Variable | | P1 | 24.07 | Principal solar diurnal | ~33% of K1 | | Q1 | 26.87 | Larger lunar elliptical diurnal | ~19% of O1 |

These eight constituents account for most of the predictable tide at most locations. NOAA's full prediction uses 396 constituents, including shallow-water harmonics (M4, M6, MS4 — derived from non-linear interaction of M2 and S2 in shallow water).

The Doodson constituent numbering system (developed by Arthur Doodson, 1921) provides a systematic naming and frequency derivation for all constituents. Each constituent has a six-digit “Doodson number” specifying its astronomical origin.

Tide types

Three patterns occur globally, determined by the relative amplitudes of semidiurnal (M2, S2) and diurnal (K1, O1) constituents.

Semidiurnal

Two roughly equal highs and lows per day. Tidal range is similar at all high tides; lows similar.

Common in:

• US East Coast (Boston, New York, Norfolk).
• Most of Western Europe (English Channel, North Sea, Atlantic France).
• Open ocean far from coastlines.

The semidiurnal pattern dominates where M2 strongly exceeds K1+O1.

Diurnal

One high and one low per day. Tidal range typically smaller than in semidiurnal areas.

Common in:

• Gulf of Mexico (Mobile, New Orleans, Galveston).
• Parts of Southeast Asia (Gulf of Thailand, Indonesia).
• Antarctic margin (some locations).

Diurnal patterns occur where K1+O1 dominate over M2+S2 — typically near the antinodes of the semidiurnal tide.

Mixed semidiurnal

Two highs and two lows per day, but distinctly unequal. The higher high water (HHW) is noticeably above the lower high water (LHW); the lower low water (LLW) noticeably below the higher low water (HLW).

Common in:

• US Pacific Coast (San Francisco, Seattle, San Diego).
• Caribbean (parts of).
• Indian Ocean (parts of).

Mixed patterns occur where M2 and K1+O1 are comparable in amplitude.

The amazing tidal ranges

Bay of Fundy — the world record

The Bay of Fundy in Atlantic Canada has the world's highest tidal range:

• Spring tide range at the head of the bay: ~16 m (52 ft).
• Extreme spring tides: ~17+ m.
• Typical neap range: ~12 m.

The mechanism: the bay's geometry has a natural oscillation period close to 12 hours, so M2 tidal forcing produces resonance — like ringing a wine glass at its resonant frequency. The bay is long (270 km), gradually shallowing, and tapering toward its head — geometry that maximizes the resonance amplification.

Visitors to Hopewell Rocks (NB) or Burntcoat Head (NS) can walk on the seabed at low tide and stand under 15+ meters of water 6 hours later.

Other notable ranges

| Location | Approximate spring range | | -------- | ------------------------ | | Ungava Bay (Quebec) | ~16 m | | Bristol Channel / Severn Estuary (UK) | ~15 m | | Mont-Saint-Michel Bay (France) | ~14 m | | Cook Inlet (Alaska) | ~12 m | | Río Gallegos (Argentina) | ~13 m | | English Channel (mid) | ~7-10 m | | Boston, MA | ~3 m | | Open North Atlantic | ~0.5 m | | Mediterranean (most) | ~0.3 m | | Tahiti | ~0.3 m |

Mediterranean and central-Pacific tides are small because the basins don't resonate near tidal frequencies.

Tidal bores

A tidal bore is a wave of water that travels up a shallow river during a rising tide. The bore forms when a large incoming tide is funneled into a narrowing, shallowing river:

• Severn Bore (UK, River Severn): up to 2 m high; surfed by enthusiasts during major spring tides.
• Qiantang Bore (China, Qiantang River): up to 9 m high — the world's largest tidal bore; ancient observation post still standing.
• Pororoca (Amazon River, Brazil): ~4 m high; travels hundreds of km upstream.
• Petitcodiac River (New Brunswick): ~1-2 m, near the Bay of Fundy.

Tidal datums

Tidal datums are vertical reference levels derived from statistical analysis of tide-gauge records over the 19-year National Tidal Datum Epoch (NTDE). The eight common datums, listed top to bottom by elevation:

| Datum | Definition | | ----- | ---------- | | HAT | Highest Astronomical Tide — highest predicted tide from astronomy alone | | MHHW | Mean Higher High Water — average of the higher of the two daily highs | | MHW | Mean High Water — average of all daily high tides | | DTL | Diurnal Tide Level — average of MHHW and MLLW | | MTL | Mean Tide Level — average of MHW and MLW | | MSL | Mean Sea Level — average of all hourly heights | | MLW | Mean Low Water — average of all daily low tides | | MLLW | Mean Lower Low Water — average of the lower of the two daily lows | | LAT | Lowest Astronomical Tide — lowest predicted tide from astronomy alone |

Important properties:

• All are 19-year averages over the NTDE (currently 1983-2001 in the US).
• All are location-specific: each tide gauge has its own values.
• The spacing between datums depends on the local tidal range. At Boston, MHW–MLW is ~3 m. In the Bay of Fundy, MHW–MLW is ~12 m.
• All change over time as MSL changes (each NTDE update revises all datums).

Chart datums

For marine navigation, the chart datum is chosen to be conservative — to understate water depth so ships have at least the charted depth under their keels at most tide stages.

• MLLW is the US Pacific Coast and many other Pacific chart datums.
• MLW is older US Atlantic chart datum (replaced by MLLW in some areas).
• LAT is the IHO international recommendation for new charts since 2007. UK Admiralty charts and many European national charts use LAT.

A chart that lists “5 m depth” relative to LAT means 5 m below the lowest astronomically predicted tide — so at typical tide levels, much more water is actually available.

Vertical datums for engineering

For terrestrial engineering:

• MHW historically used as a boundary between coastal land and tidal waters.
• HAT used for bridge clearance and similar maximum-water-level applications.
• MSL as a general reference (though see /learn/vertical-datums-explained for the modern orthometric and gravimetric alternatives).

Harmonic analysis and prediction

Modern tide prediction uses harmonic analysis of historical tide-gauge records to extract the amplitude and phase of each constituent at a location. The resulting constituent values are then used to predict future tides by summing the sinusoidal components.

The mathematical form:

h(t) = MSL + Σ A_i · cos(ω_i · t + φ_i)

Where for each constituent i:

• A_i is the amplitude.
• ω_i is the angular frequency (from astronomical theory).
• φ_i is the phase (from local observation).

The amplitudes and phases vary by thousands across the world's tide gauges — but the frequencies are universal (astronomical). NOAA maintains the harmonic constants for ~200 US tide stations; UK Admiralty publishes constants for major UK ports; similar agencies exist in most maritime nations.

Prediction accuracy

For deep-ocean and open-coast locations: harmonic prediction is accurate to ~10 cm. The remaining variability is from non-tidal effects: storm surge, atmospheric pressure changes, wind setup, river flow, sea-level anomalies. Total water level prediction combines harmonic tide + storm surge forecasts + sea-level departures.

For complex coastal areas (Bay of Fundy, Severn, estuaries with strong river input): prediction accuracy degrades to ~30-50 cm, requiring local hydrodynamic models (e.g., NOAA OFS, the Operational Forecast System) to refine the harmonic-only prediction.

Historical tide prediction

Before computers, tide prediction was done by mechanical tide prediction machines — gear-and-cam analog computers that summed the constituents mechanically. Lord Kelvin built one in 1873; later versions had 39 constituents and produced year-long tide tables. The US Coast & Geodetic Survey Tide Predicting Machine No. 2 (built 1910) is preserved at the Smithsonian.

Modern tide prediction is purely digital but uses the same harmonic-analysis methodology.

The equilibrium tide vs the dynamic tide

Equilibrium theory (Newton)

The classical idealization: assume the ocean has time to reach gravitational equilibrium with the tidal forces. The result: idealized bulges aligned with the Moon and Sun.

The equilibrium theory predicts the frequencies of the tidal constituents correctly but the amplitudes incorrectly — observed amplitudes differ from equilibrium values by factors of 10-100×.

Dynamic theory (Laplace)

The ocean doesn't have time to reach equilibrium. Ocean wave physics — finite depth, Coriolis force, continental boundaries — control the actual response. Pierre-Simon Laplace developed the dynamic theory in the late 18th century.

The actual tide is a complex standing-wave pattern shaped by ocean basin geometry. Each basin has its own resonant modes; coastal geometry can amplify or suppress specific constituents. The Bay of Fundy resonance is an extreme example.

Modern tide modeling uses numerical models of ocean dynamics (TPXO, FES, OSU TPXO9, GOT4.10) that solve the dynamic equations. These models predict global tides with sub-cm accuracy in the open ocean.

Tides on Earth body

Earth itself flexes under tidal forces. The solid Earth tide has amplitude ~30 cm at the equator — that's right, the ground beneath your feet rises and falls by ~30 cm twice a day from lunar tidal forcing.

Solid Earth tide matters for:

• High-precision GNSS positioning (the receiver itself is moving with the Earth tide).
• Geodynamic studies.
• VLBI (Very Long Baseline Interferometry) measurements.

Solid Earth tide is calculable from theory to ~mm precision; it's removed routinely from geodetic data.

Spring and neap tides

The 14.77-day spring-neap cycle is one of the most visible tidal modulations:

Spring tides

Occur near new and full moons when Sun-Earth-Moon are aligned. Lunar and solar tidal bulges combine, producing:

• Higher high tides than average.
• Lower low tides than average.
• Larger tidal range.

The term “spring tides” has nothing to do with the season — it refers to the ‘springing forward’ of the tide, an older English usage.

Neap tides

Occur near first and last quarter moons when Sun-Earth-Moon form a right angle. Solar tide partially cancels lunar tide:

• Lower high tides than average.
• Higher low tides than average.
• Smaller tidal range (typically ~50% of spring range).

Perigean spring tides ("king tides")

When the Moon is at perigee (closest to Earth in its elliptical orbit) and Sun-Earth-Moon are aligned, the result is a perigean spring tide or king tide. These are the highest tides of the year.

In the US Pacific Northwest and East Coast, king tides regularly cause high-tide flooding in low-lying coastal areas. See /learn/sea-level-rise-explained for how rising MSL makes king-tide flooding more frequent.

Tidal energy

Tidal energy capture is a real but limited renewable resource:

Tidal barrages

Dams across estuaries with turbines that generate power as tides flow in and out.

• La Rance (France, 1966): the world's first large tidal barrage. 240 MW capacity. Still operational.
• Sihwa Lake (South Korea, 2011): 254 MW; currently the world's largest.

Tidal stream turbines

Underwater turbines that capture energy from tidal currents (without dams):

• MeyGen (Scotland, Pentland Firth): one of the largest tidal-stream installations.
• Various pilots in UK, Canada, US, Korea.

Wave energy

Distinct from tides but related: wave energy converters capture wind-generated wave power. Smaller installed capacity than tidal.

The total technically extractable tidal energy globally is ~1-2 TW — meaningful but small compared to wind (~10-100+ TW potential).

Common misconceptions

“Tides are caused by the Moon pulling water up.” Partially. The Moon's gravity pulls water on the near side toward it; the far-side bulge is more subtle — it's the water at the far side being pulled toward the Moon less than the rocky Earth, effectively bulging outward.

“Tides are predictable.” The astronomical component is highly predictable (harmonic analysis at ~10 cm). The total water level depends on storm surge, atmospheric pressure, wind, and other factors that are less predictable.

“Two high tides per day everywhere.” Not everywhere. Diurnal areas (Gulf of Mexico, parts of SE Asia) have one high and one low per day. Mixed-semidiurnal areas have two of each but distinctly unequal.

“Spring tides happen in spring.” No. Spring tides happen at new and full moons (every ~14.77 days), year-round. The word “spring” means “springing forward”, not the season.

“Tidal range is the same everywhere.” No — ranges vary by factor of 50× from ~0.3 m in the Mediterranean to ~16 m in the Bay of Fundy. Local geometry and resonance dominate.

“Chart datums are interchangeable.” No — MLLW, MLW, and LAT differ by location and convention. A chart claiming “5 m depth” relative to MLLW gives different actual water depth than relative to LAT.

“Earth body tides are negligible.” The ~30 cm solid Earth tide is significant for high-precision geodesy. It's removed routinely from GNSS and VLBI measurements.

“The Moon controls all tides equally.” The Moon dominates globally, but the Sun matters (46% of lunar effect). Spring-neap variation demonstrates the Sun's influence.

“Tides happen at the same time every day.” No — the lunar day is 24.84 hours, so M2 tides occur ~50 minutes later each day.

“Harmonic prediction is exact.” It's the astronomical component only. Total water level includes non-astronomical effects (storm surge, pressure) that harmonic prediction misses. Operational tide forecasts combine harmonic + storm surge models.

“Tides are uniform across a port.” No — even within a single harbor, tidal range and timing can vary by minutes and decimeters across distances of kilometers. NOAA publishes tidal constants for many sub-stations within major ports.