Twice-daily tides

Why does the sea rise twice a day when there is only one Moon pulling on it?

One Moon, on one side of the Earth, pulling one way. The obvious prediction is one bulge of water facing it, and one high tide as your coast rotates beneath it. Instead most coasts get two, about twelve hours and twenty-five minutes apart. There is water heaped up on the side facing away from the Moon, where its pull is weakest. That is the awkward fact, and any account that does not explain it has not explained tides.

Begin with the one thing gravity certainly does: it weakens with distance. The Earth is nearly thirteen thousand kilometres wide, so the near face sits about three per cent closer to the Moon than the far face. Small, but the Moon does not pull equally on the two.

Now ask a question that sounds like a technicality and is actually the whole answer: what is the Earth doing? It is not anchored. It is in free fall toward the Moon, exactly as the Moon is toward it. And in free fall, a uniform pull is not felt at all — an astronaut in orbit is being pulled hard by the Earth and floats regardless, because everything around her is falling at the same rate.

So what can be felt? Only the difference from the average. If a piece of ocean is pulled harder than the Earth as a whole is being accelerated, it goes ahead of the Earth. If it is pulled less, the Earth goes ahead of it. Do you see where that leads?

Take the near side. It is closer, so the Moon pulls it harder than average — it moves moonward relative to Earth's centre. That bulge was expected.

Now take the far side. It is further away, so it is pulled less than average. The solid Earth accelerates moonward and leaves it behind. Relative to the Earth's centre it drifts away from the Moon — which is to say, outward, away from the ground. A second bulge, produced by weakness rather than by pull.

That is the entire mechanism, and notice it needed no spin. The near side is pulled more than the centre, the far side less, and both differences point outward from the centre. Squeeze in at the sides, stretch out at the ends: the Earth is being pulled taut, not tugged.

How strong is it? At each end, the residual works out at a little over one micrometre per second squared — roughly one part in nine million of ordinary surface gravity. Absurdly feeble to lift a rock, but the ocean is a fluid spread across a hemisphere, and a whisper applied over that area, twice a day for ages, is enough to move it.

Then the Earth turns. Rotate any coast through this fixed double-ended pattern and it passes two bulges per rotation. Why twelve hours and twenty-five minutes rather than twelve exactly? Because while you rotated, the Moon moved on in its month-long orbit, so it takes about fifty extra minutes to come back overhead.

That last clause matters enough to spend the rest of this section on, along with a correction.

The correction first. You will often read that the far bulge comes from centrifugal force as the Earth swings about the Earth-Moon barycentre. Something can be salvaged from that account in a rotating frame if it is done carefully, but as a primary explanation it is misleading, and it teaches the wrong thing. The far bulge does not require the pair to be rotating at all: two bodies simply released and falling straight toward each other would show the same stretch. The cause is the gradient of the Moon's gravity across the Earth's width, full stop.

Now the honest limitation of everything above. The picture just built is the "equilibrium tide", and it gets the pattern right and the reality wrong. It predicts a rise of about half a metre, and it predicts high water when the Moon is overhead. Neither is what tide tables show.

The reason is that the ocean cannot instantly settle into that shape. Continents are in the way, and the bulge would have to travel westward faster than a shallow-water wave can propagate. So what actually happens is that the tidal force acts as a rhythmic driver on the ocean basins, and each basin responds according to its own size, depth, and natural period — in effect, resonance. Water rotates around fixed nodes called amphidromic points, pushed by the Coriolis effect. The Bay of Fundy, whose geometry happens to ring in near-tune with the semidiurnal rhythm, sees a range of some sixteen metres. Much of the Mediterranean barely notices the tide. And in parts of the Gulf of Mexico and Southeast Asia, the basin responds better to the once-daily component of the forcing than the twice-daily one, so those coasts get a single tide a day — which is precisely why "twice daily" is a description of the driving force rather than a law about coastlines.

One more piece: the Sun does the same thing, at about 46 per cent of the Moon's strength. When the two line up we get the larger spring tides, and when they act at right angles the smaller neaps.

• How tidal friction is slowing the Earth's rotation and pushing the Moon steadily further away.
• Why the same differential force, taken to an extreme, tears bodies apart near a massive object.

Every term the collection defines is gathered in the glossary.