A tide table is astronomy with the harbour fitted

The tide is a difference: the Moon’s pull on the near side of the Earth minus its pull on the centre, and the centre’s minus the far side’s, which together stretch the ocean into two bulges. That explains why there are two high tides a day and why they come about fifty minutes later each day. It does not explain the times or the heights in any tide table, and it cannot, because the real ocean does not form the bulges the equilibrium calculation describes. The water is shallow compared with the width of its basins, it is blocked by continents, and it is being driven at periods close to its own natural ones. What a harbour actually sees is the response of a resonant, irregular ocean to a periodic force, and that response has never been computed from first principles to the precision a tide table quotes.

Tide tables are nonetheless accurate to centimetres and minutes, years in advance. The reason is that the force, though the ocean’s response to it is intractable, has a structure that makes the response unnecessary to compute. The tide-raising potential is a sum of sinusoids at frequencies fixed by the motions of the Moon and the Sun, and a linear system driven by a sum of sinusoids responds at exactly those frequencies, with an amplitude and a phase for each. The frequencies are astronomy, known to many decimal places. Only the amplitudes and phases depend on the harbour, and they can be measured from a record of the sea’s own height.

That is harmonic tidal prediction, and its whole content is in three questions. What are the frequencies? How long must a record be to measure the response to each? And how do the slow astronomical cycles that the frequencies leave out get put back?

Frequencies made of six rates #

The Moon’s tidal pull on a point of the Earth’s surface depends on three things: where on the Earth’s turning surface the point is relative to the Moon, how far the Moon is, and how far the Moon is from the equator. Each changes periodically, and so does the corresponding quantity for the Sun.

The fastest is the turning. Relative to the Moon, the Earth turns once in a mean lunar day of 24 hours and 50 minutes, and because the tidal potential has two bulges, the dominant term repeats twice in that time: the principal lunar semidiurnal constituent, $M_{2}$, at 1.9323 cycles per mean solar day, a period of 12 hours 25 minutes. The Sun produces the same thing on a solar day: $S_{2}$, at exactly two cycles per day.

The other periodicities modulate those. The Moon’s distance varies over the anomalistic month as it moves round its elliptical orbit — one of several months the Moon keeps, none of them a whole number of days — and a modulated sinusoid is a sum of sinusoids at the carrier frequency plus and minus the modulation frequency — which is why $M_{2}$ is accompanied by $N_{2}$, offset by one cycle per anomalistic month. The Moon’s declination swings north and south of the equator every tropical month, and that swing turns the two equal daily bulges into one large and one small, which puts power at once a day: the diurnal constituents $K_{1}$ and $O_{1}$, separated by twice the Moon’s rate round the sky. The Sun does the same over a year, giving $P_{1}$ and the semi-annual Ssa.

Arthur Doodson organised all of this in 1921. Every constituent has a frequency that is a sum of integer multiples of six astronomical rates — the lunar day, the Moon’s mean longitude, the Sun’s, the longitude of the lunar perigee, the longitude of the Moon’s node, and the solar perigee — and the six integers, Doodson’s numbers, label it uniquely. His expansion of the potential has nearly four hundred terms. A few dozen carry essentially all of the energy, and the eleven largest are drawn here.

The figure’s content is the frequencies, and they are not measured but computed. $S_{2}$ is exactly two cycles per day because a mean solar day is a day — the mean Sun, not the real one, which keeps imperfect time over the year and contributes that imperfection to the tide as further small constituents. $M_{2}$ is two cycles per lunar day, which is the solar day lengthened by the Moon’s daily motion round the sky. $K_{1}$ is one cycle per sidereal day, the Earth’s rotation measured against the stars, because the declination effects of both bodies combine into a term locked to the stars rather than to either body. Every line in the tidal spectrum is a statement about orbital mechanics, and those statements have been checked to the precision of the ephemerides themselves.

Springs and neaps are a beat #

The tide’s best-known rhythm is not a constituent at all. It is what two constituents do together.

$M_{2}$ completes 1.9323 cycles a day and $S_{2}$ completes two. They drift relative to each other by 0.0677 cycles a day, and so come back into step every 14.765 days — half the synodic month, the time from new Moon to full. When the two are in step their heights add, the range is the sum of the two amplitudes doubled, and the tide is a spring tide. A week later they are out of step, the heights subtract, and the tide is a neap. The name has nothing to do with the season; it is from the Old English for a rising.

At the harbour drawn, with $M_{2}$ of 1.2 metres and $S_{2}$ of 0.42, the spring range is 3.24 metres and the neap range 1.56 — a factor of two between them, which is typical. Where $S_{2}$ is unusually strong relative to $M_{2}$, as in some basins whose resonances favour the solar period, springs and neaps differ more; where it is weak, less.

The beat picture also explains why spring tides lag the full and new Moon by a day or two in most places. The equilibrium theory puts the springs exactly at syzygy. But the ocean’s response to $M_{2}$ and to $S_{2}$ has different phase lags, set by how far each frequency is from the basin’s natural periods, and the beat’s maximum comes when the two responses, not the two forcings, are in step. The difference in phase lag between $S_{2}$ and $M_{2}$, divided by the rate at which they drift apart, is the delay — called the age of the tide since the eighteenth century, and measured in hours to days at different ports.

The same arithmetic makes every other rhythm in a tide record. $N_{2}$ beats with $M_{2}$ every anomalistic month, so alternate springs are larger, the larger ones coming near lunar perigee. $K_{1}$ beats with $O_{1}$ every half tropical month. A year’s record contains dozens of beats nested inside one another, and each is a pair of lines.

One tide a day or two #

The relative strength of the two bands differs enormously from place to place, and the difference is the clearest demonstration that the harbour, not the sky, sets the shape of the tide.

The ratio is called the form factor: the sum of the $K_{1}$ and $O_{1}$ amplitudes divided by the sum of $M_{2}$ and $S_{2}$. On most of the Atlantic coasts of Europe it is below a quarter, and the tide is semidiurnal: two highs a day of nearly equal height. On the Pacific coast of North America it is around one, and the tide is mixed: two highs a day, one much larger than the other, with the smaller nearly vanishing when the Moon is far north or south of the equator. In parts of the Gulf of Mexico and the South China Sea it exceeds three, and the tide is diurnal: one high water a day.

The astronomical forcing is nearly the same at all of them; the diurnal potential is not a weak term in the equilibrium expansion, about half the semidiurnal one. What differs is the basins. A basin whose natural period is close to twelve hours amplifies the semidiurnal band and suppresses nothing in particular; one whose geometry favours a period near a day does the opposite; and the Gulf of Mexico, connected to the Atlantic through two narrow straits, responds to the diurnal forcing much more strongly than to the semidiurnal. The tide at each place is a filter applied to the same astronomical signal, and the filter is the ocean.

How long a record has to be #

Measuring a harbour’s response is a least-squares fit: the observed sea level as a sum of cosines and sines at the known frequencies, with an amplitude and a phase to be found for each. The fit is linear and solved in one step, and the only question is whether the record is long enough to tell the constituents apart.

Two sinusoids whose frequencies differ by $\Delta f$ are separable only from a record longer than about $1/\Delta f$. Over a shorter record they do not drift through a full cycle relative to each other, and a fit cannot apportion the signal between them: any split of amplitude between the two fits equally well. That is Rayleigh’s criterion, the same limit on resolving two lines in a spectrum, and in tidal analysis it sets a schedule.

$M_{2}$ and $S_{2}$ need 14.8 days; $M_{2}$ and $N_{2}$, 27.6; $K_{1}$ and $O_{1}$, 13.7; $K_{1}$ and $P_{1}$, 183. A practical analysis therefore includes a constituent only if the record resolves it from every larger neighbour, and treats the rest as unknown. With three days of data only $M_{2}$ and $K_{1}$ can be fitted, and the prediction is wrong by tens of centimetres, because everything else is error. With a fortnight $S_{2}$ and $O_{1}$ are added and the springs and neaps appear; with a month $N_{2}$ is added; with six months $K_{1}$ and $P_{1}$ separate. In the synthetic harbour drawn, the error falls from seventy centimetres with three days to sixteen with a month and under one with a year.

What a year leaves is weather. Storm surges, driven by wind and air pressure, raise and lower the sea by tens of centimetres over days, and they are not periodic, so they cannot be predicted harmonically. They also leak into the fitted amplitudes, slightly, because a surge that happens to coincide with a particular constituent’s phase looks like a contribution to it, and that leakage shrinks as the record lengthens. A tide table predicts the tide; the sea level a harbour sees is the tide plus whatever the weather adds.

The cycle of 18.6 years #

A year resolves every constituent in the main bands, and yet the standard analyses are corrected using a cycle nineteen times longer.

The Moon’s orbit is tilted by five degrees to the ecliptic, and its line of nodes — where the orbit crosses the ecliptic — turns backwards once every 18.61 years, under the Sun’s pull. When the ascending node coincides with the vernal equinox, the Moon’s inclination adds to the Earth’s obliquity and the Moon swings as far as 28.6 degrees from the equator each month. Nine years later the two subtract, and it reaches only 18.3. The same nodal regression is what moves eclipses round the calendar.

The diurnal tides are produced by the Moon’s declination, so they swell when it reaches furthest from the equator and shrink when it does not: $O_{1}$ varies by nineteen per cent either way and $K_{1}$ by twelve. The semidiurnal tides are strongest when the Moon is over the equator, so $M_{2}$ varies in the opposite sense, by under four per cent.

In Doodson’s expansion the node’s motion appears as further constituents, sidebands at the main frequencies plus and minus one cycle in 18.61 years. Resolving them would need a record of nearly nineteen years, and few harbours have one. They are instead absorbed into nodal factors: each main constituent’s amplitude and phase are multiplied by a factor computed from the longitude of the node, using series derived from the equilibrium theory, before fitting and before predicting. The fitted amplitudes are therefore mean values, and the predictions apply the factor for the year being predicted. The method assumes the ocean’s response to a sideband is the same as its response to the main line a few parts in ten thousand away in frequency, which is true wherever the response does not change sharply with frequency.

The cycle also appears directly in sea level. Mean high water at a semidiurnal port rises and falls by a few centimetres over 18.6 years, and the effect matters for coastal flooding: a high nodal phase on top of a rising mean sea level brings a decade of more frequent nuisance floods. The current maximum of the nodal cycle for semidiurnal tides falls in the early 2030s.

What was actually measured #

Tide gauges. The longest instrumental records go back to the eighteenth century — Amsterdam from 1700, Brest from 1711, Liverpool from the 1760s — and today several thousand gauges report sea level continuously. From each, harmonic analysis extracts dozens of constituents, and the constants are published and used for every prediction at that place.

The machines. Before computers, predictions were made by mechanical analogues. William Thomson’s tide-predicting machine of 1872 summed ten constituents with a system of pulleys, each turning at a constituent’s frequency and each displaced by its amplitude, with a wire over all of them whose end traced the tide. Later machines summed forty. They ran into the 1960s, and during the Second World War they predicted the tides for the Normandy landings, with the harbour constants for the beaches estimated from short records at nearby ports.

Satellite altimetry. Since 1992, radar altimeters have measured the height of the open ocean to a few centimetres along repeating ground tracks, and the same harmonic analysis applied to their records has produced maps of every major constituent across every ocean. The maps show the amphidromic systems directly — points of zero tide round which the high water rotates — and they constrain where the tides lose energy. The global dissipation measured this way, about 3.7 terawatts, is the same number obtained from the Moon’s recession by lunar laser ranging, which is what sets the slowing of the Earth’s rotation.

Aliasing in orbit. A satellite on a repeating ground track samples each point of the ocean only once every ten days or so, which is far too coarse to see a twelve-hour tide directly. Each constituent therefore appears aliased to a long period set by the repeat cycle, and the orbits were chosen so that the aliases of the major constituents fall at well-separated periods. It is Rayleigh’s criterion again, applied to the aliased frequencies, and it is one of the reasons the altimetry orbits are inclined as they are.

Where the method stops #

Linear response. Harmonic analysis assumes the ocean responds linearly, so that each forcing frequency produces a response at that frequency alone. In shallow water the response is not linear: the wave’s crest travels faster than its trough, the tide steepens, and power appears at sums and differences of the forcing frequencies — overtides like $M_{4}$ at twice $M_{2}$'s frequency, and compound tides like $MS_{4}$. These are added as further constituents with no astronomical forcing of their own, and in estuaries they can be as large as the primary ones.

Weather and seasons. Surges are not tides and cannot be predicted harmonically. Seasonal changes in the ocean’s density and circulation produce annual and semi-annual sea-level changes that a harmonic analysis absorbs into Sa and Ssa, which are therefore largely meteorological at most ports.

A changing ocean. The constants are not permanent. Dredging a channel, building a barrier or raising mean sea level changes the basin, and with it the response. At some ports $M_{2}$ has changed by several per cent over a century, and every analysis assumes the harbour of the record is the harbour of the prediction.

A rigid Earth. The sea is measured against land that also moves: the solid Earth rises and falls by tens of centimetres under the same forcing, by an amount set by its Love numbers, and its own deformation changes the potential the ocean feels. Tide-gauge constants include all of this, since they are measured relative to the gauge; the equilibrium amplitudes do not.

Equilibrium amplitudes. The opening figure’s heights are the equilibrium-tide coefficients, the astronomy’s own weights. A harbour’s amplitudes differ from these, constituent by constituent, by factors that are the whole point of measuring them.

The sky’s frequencies, the sea’s response #

The general shape is a separation that works because one half of a problem is easy and the other is hard. The forcing of the tides is known completely; the ocean’s response is not computable to useful precision. But the response of any linear system to a sum of sinusoids is a sum of sinusoids at the same frequencies, so the hard half reduces to measuring two numbers per frequency, and the easy half supplies the frequencies themselves.

When the input is known exactly and the system is linear, the system does not have to be understood to be predicted — only measured, once per frequency, with a record long enough to resolve the frequencies from each other. The same separation runs through every periodogram of a star, every calibration of an instrument with known input, and every acoustic measurement of a room.

The other reading is about time scales. Every rhythm a tide table encodes is astronomical: the day, the month in three different senses, the year, and the 18.6-year cycle of the node. A harbour’s record is a clock that keeps all of them at once, which is why a long one can be read backwards as easily as forwards — and why the oldest gauges are now as valuable for what they say about the sea’s mean level as for the tide they were installed to measure.

Still open: the tides that are not in the potential #

The harmonic method describes the ocean’s response to the astronomical forcing. It does not describe the internal tides — waves on the density layers inside the ocean, generated where the barotropic tide flows over seamounts and ridges, which carry about a third of the tidal energy into the deep ocean and break there, mixing it. They are periodic at the tidal frequencies but their phase wanders with the ocean’s stratification, so they are only partly predictable, and they show at the surface as height variations of a few centimetres that the newest wide-swath altimeters can see. How much of the deep ocean’s mixing they supply, and therefore how the tides help set the ocean’s overturning circulation, is a question about the same constituents and the same energy budget, followed below the surface.