Tides

Why local tides are hard

Goal: understand why a physically correct model gives a practically useless answer, and how real tide prediction actually works.

The equilibrium model's hidden assumptions

Everything so far assumed Earth is covered by a deep, uniform ocean that responds instantly. Every part of that is false.

Continents are in the way. The bulge cannot travel freely westward; it hits Africa.

Water cannot move fast enough. The bulge would need to travel at about 1,600 km/h at the equator. A shallow-water wave in a 4 km deep ocean travels at about 700 km/h. The ocean physically cannot keep up, so the tide is not a bulge riding under the Moon; it is a wave sloshing around basins.

Basins resonate. Each ocean basin has a natural period set by its size and depth. When that is near 12.4 hours, the tide is amplified enormously, like pushing a child on a swing at the right rhythm. The Bay of Fundy resonates almost perfectly and gets 16-metre tides.

Shallow water distorts. The crest travels faster than the trough, so rise and fall take unequal times.

Friction lags everything. Which is where the lunitidal interval comes from.

Weather. ~1 cm of sea level per millibar of pressure. Wind piles water up.

Consequences

• Most coasts: two highs a day (semidiurnal)
• Some coasts: one (diurnal), parts of the Gulf of Mexico, Fremantle
• Some: two unequal (mixed), much of the Pacific coast
• Mediterranean: almost no tide, because it is nearly enclosed
• Bay of Fundy: 16 m

How real prediction works

Not from this geometry at all. It works by harmonic analysis:

1. Measure water level at a station, every few minutes, for a year or more
2. Decompose the record into ~40 sine waves at known astronomical frequencies

(M2, S2, N2, K1, O1...)

1. Measure the amplitude and phase of each one at that station
2. Predict by adding the waves back up

The astronomy supplies the frequencies, those come from orbital mechanics and are the same everywhere. Only measurement can supply the amplitudes and phases, those are properties of the coastline.

That is the deep lesson of this module:

A good model often needs both theory and local data. Knowing which parts must come from measurement is most of the skill.

Checkpoint

• I can explain why the ocean cannot keep up with the bulge
• I know what basin resonance is and can name an example
• I can explain why some coasts have one high a day
• I know what harmonic analysis is
• I can say which parts of a tide prediction come from theory and which

from measurement

Questions to think about

• Harmonic analysis needs a year of data per station. What about uninstrumented

coasts?

• Could you predict tides for a coastline that does not exist yet, a planned

reclamation, say?

• What other everyday predictions are theory plus local calibration?

Next: Module 05, Your local sky.