Tides: from astronomical forcing to wave response

Tides are the rhythmic rise and fall of the sea – the most predictable ocean motion of all, and one of the oldest studied. In this final lecture, Serena Illig explains what drives them, and how the ocean actually responds. The key idea that ties the week together: tides are forced by astronomy – the pull of the Moon and Sun – but they respond as the shallow-water waves you met earlier in the week.

The Earth and Moon dance together

The Moon doesn’t simply circle the Earth’s centre. The Earth and Moon orbit their common centre of mass – a point that, because the Earth is 81 times heavier, lies inside the Earth, about 4,700 km from its centre. As the Moon goes around, the Earth swings in a small circuit of its own around that shared point.

The tide-producing force

Two forces are in play. The centrifugal force of this orbital motion points away from the Moon and is the same everywhere on Earth. The Moon’s gravitational pull, though, is stronger on the near side and weaker on the far side.

The tide-producing force is the difference between these two. On the side facing the Moon, gravity wins, and the net force points toward the Moon. On the far side, the centrifugal force wins, and the net force points away. The result is a pair of bulges on opposite sides of the Earth – the ocean stretched into a gentle ellipsoid, the equilibrium tide.

This difference is subtle but crucial: while gravity and centrifugal force fall off with distance squared, the tidal force falls off with distance cubed. That’s why the Sun, despite its overwhelming mass, produces a tidal effect only about half the Moon’s – it is simply too far away.

Why two tides a day

The Earth spins once a day beneath these two bulges. So any fixed point passes through two highs and two lows every day – the semi-diurnal tide. The distance between the bulges – half the Earth’s circumference, about 20,000 km at the equator – is the wavelength of the tidal wave.

And here the week’s physics returns. The ocean is only a few kilometres deep, so this vast tidal wave is a shallow-water wave. That has a striking consequence: the tidal wave’s natural speed is around 200 m/s, but to stay directly under the Moon it would need to travel at over 400 m/s. It can’t keep up. So real tides are never truly in equilibrium – they lag and reshape, as we’ll see.

The lunar day and the Moon’s tilt

Two complications make lunar tides richer. First, the Moon moves: by the time the Earth has spun once, the Moon has advanced along its orbit, so the Earth must turn about 50 minutes more to face it again. This is why high tides arrive about 50 minutes later each day, and why successive high tides are 12 hours 25 minutes apart.

Second, the Moon’s orbit is tilted relative to the equator, so the bulges are tilted too. Away from the equator, this gives one big high tide and one small one each day – the “diurnal inequality.” Lunar tides are therefore a blend of a semi-diurnal part (M2) and a diurnal part (M1).

Solar tides, spring and neap

The Sun does the same, more weakly – a semi-diurnal part (S2) and a diurnal part (S1), with a solar semi-diurnal period of exactly 12 hours.

Now combine Sun and Moon. Their semi-diurnal tides have almost the same period, so – exactly like the interfering waves from the previous lecture – they beat against each other over a fortnight:

• When Sun, Earth, and Moon line up (new and full Moon), the two forcings reinforce, giving the strong spring tides
• A week later, when Sun and Moon are at right angles, they partly cancel, giving the weak neap tides

This spring-neap cycle repeats roughly every two weeks – the rhythm familiar to anyone who lives by the sea.

From the ideal tide to the real ocean

The equilibrium picture is only the forcing. The real ocean response is shaped by continents, basin geometry, and rotation. Tidal waves travel along coastlines as the Kelvin waves from the last lecture – coast on their right in the northern hemisphere – and they reflect and interfere in complex ways.

This is why tidal range varies so dramatically from place to place. Across the English Channel, the tide is only a few metres on the English side but exceeds 11 metres near Saint-Malo on the French coast – enough to power a tidal energy station. Around the ocean basins, the tide rotates about special no-tide points called amphidromic systems, wheeling around them like the hand of a clock.

Tides in ocean models

In practice, tides are not one wave but many – the dominant M2, S2, M1, and S1, plus more than 390 smaller constituents. Models introduce them either through the astronomical forcing directly or at their open boundaries. And once generated, tides do far more than raise and lower the sea: they drive strong tidal currents, generate internal waves, and are one of the ocean’s great engines of mixing.

That mixing role brings the week full circle – from the forces that move the ocean, through balances, waves, and tides, back to the stirring that shapes the sea.

As always, focus on the physical pictures rather than the mathematics.