Astronomy & Celestial Mechanics
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Gravitation

The tide that holds up the deep ocean

Cold water sinks near the poles and fills the deep ocean, and something has to stir heat back down into it or the whole ocean below the surface would become one cold pool. The stirring costs about two terawatts. The winds supply some of it. The rest comes from the Moon: about a terawatt of the tide's energy is lost not on continental shelves but in the deep ocean, where the tide flowing over ridges and seamounts makes waves on the ocean's internal layers that break far below the surface.

Assumes Tides and Tidal heating.

15 min read 6 figures Small perturbations, large consequencesScales that defeat intuition

The deep ocean is cold. Below a kilometre or so the water everywhere is within a few degrees of freezing, and it got there by sinking: in the Greenland and Norwegian seas and round the edge of Antarctica, surface water cooled by the polar air becomes dense enough to plunge to the bottom and spread through the ocean basins. That has been going on for millions of years, and it raises a question that is not obvious until it is asked. If cold water keeps arriving at the bottom, why has the whole ocean below the warm surface layer not filled up with it?

The answer is that it rises again, slowly, everywhere, and as it rises it is warmed by heat mixed down from above. The mixing is a turbulent process that needs a supply of mechanical energy, because stirring a stratified fluid lifts dense water and lowers light water, and that costs work against gravity. Walter Munk worked out in 1966 how much mixing the deep ocean needs, and in 1998, with Carl Wunsch, how much power that mixing costs. The answer was about two terawatts, and the search for where those terawatts come from led back to the Moon.

The tide table is the part of the tide that is astronomy: the forcing, its frequencies, and the ocean’s response at a harbour. That essay ended on the part that is not — the internal tides that carry a third of the tidal energy into the deep ocean and break there. This essay follows that energy down.

A heat budget that needs a pump

The deep ocean’s temperature profile is the result of a balance. Cold water upwells at some speed w, carrying cold upward; heat diffuses downward with some effective diffusivity κ; and in a steady state the two balance at every depth.

A temperature profile set by a ratio of mixing to upwelling. Temperature against depth below the warm surface layer for the balance w ∂T/∂z = κ ∂²T/∂z², between 20 °C at the top and 1 °C at four kilometres, for scale heights κ/w of 0.4, 1, 2.5 km. A deep ocean with weak mixing or fast upwelling keeps a thin thermocline over uniform cold water; one with strong mixing or slow upwelling warms a thick layer. The observed profiles of the Pacific have a scale height near a kilometre, which with an upwelling of about 10⁻⁷ m/s — four metres a year — needs κ ≈ 10⁻⁴ m²/s. The profile measures only the ratio; the upwelling has to come from elsewhere, from tracers like radiocarbon.
Fig. 1 Temperature against depth for the balance w ∂T/∂z=κ ∂2T/∂z2w\,\partial T/\partial z = \kappa\,\partial^2 T/\partial z^2, between 20 °C at the top and 1 °C at four kilometres, for scale heights κ/w of 0.4, 1 and 2.5 km. Weak mixing or fast upwelling keeps a thin thermocline over uniform cold water; strong mixing or slow upwelling warms a thick layer.

The solution is exponential, with a scale height equal to κ/w. Weak mixing or fast upwelling gives a thin thermocline sitting on top of uniformly cold water; strong mixing or slow upwelling warms a thick layer. The observed profiles in the Pacific, where the balance is cleanest, have a scale height of about a kilometre. Munk combined that with an upwelling speed estimated from the rate at which the polar seas produce deep water — about four metres a year, spread over the whole ocean — and found that the diffusivity had to be about 10−410^{-4} square metres a second. That number, ten thousand times the molecular diffusivity of heat in water, has framed the subject ever since.

The profile alone measures only the ratio of the two. Radiocarbon, which decays with a half-life of 5,730 years and enters the ocean at the surface, supplies the other: the deep water’s carbon-14 deficit is an age, and it fixes the upwelling speed independently of any model of the mixing.

The price of a diffusivity

Mixing in a stratified fluid is not free. Each parcel of dense water lifted against the stratification gains potential energy, and the turbulence that does the lifting is inefficient: only about a fifth of the energy fed into it ends up as potential energy, the rest dissipating as heat.

The power a diffusivity costs. The power needed to hold the deep ocean's stratification against mixing, P = ρ₀κN²V/Γ, in terawatts, against the vertical diffusivity κ, for buoyancy frequencies of 1 × 10⁻³, 2 × 10⁻³, 3 × 10⁻³ per second, a deep volume of 10¹⁸ cubic metres and a mixing efficiency of 0.2. At Munk's κ = 10⁻⁴ m²/s and N = 2 × 10⁻³ it is 2.1 TW, Munk and Wunsch's figure; at the 10⁻⁵ that tracer releases measure away from rough topography, 0.21. The dashed line is the tide's deep dissipation, about 1 TW: the tides can pay for about half of Munk's diffusivity, and the 10⁻⁵ measured in the open interior needs a tenth of what they supply — so most of the mixing must happen somewhere other than the open interior.
Fig. 2 The power to hold the deep ocean’s stratification against mixing, P=ρ0κN2V/ΓP = \rho_0 \kappa N^2 V/\Gamma, against the diffusivity, for three buoyancy frequencies. At Munk’s κ=10−4 m2 s−1\kappa = 10^{-4}\,\mathrm{m^2\,s^{-1}} and N=2×10−3N = 2\times10^{-3} it is 2.1 TW; at the 10⁻⁵ measured in the open interior, 0.21. Dashed: the tide’s deep dissipation, about 1 TW.

The power needed scales as the diffusivity times the square of the buoyancy frequency — the natural frequency at which a displaced parcel bobs in the stratification — times the volume, divided by the mixing efficiency. With Munk’s diffusivity and a typical deep stratification, it comes out at about two terawatts. That is a large number for a slow process: two thousand gigawatts, comparable with the whole of humanity’s electricity generation, spent continuously on keeping the deep ocean from becoming one cold pool.

Two terawatts has to come from somewhere, and there are only two candidates big enough. The winds work on the ocean surface at a rate of several terawatts, but most of that goes into surface waves and currents and is dissipated within the upper hundred metres. The fraction that reaches the deep ocean, as internal waves radiated downward from storms, is estimated at half a terawatt to a terawatt. The other candidate is the tide.

Where the tide’s energy goes

The total rate at which the Moon and the Sun do work on the Earth through the tides is known precisely, because it is the rate at which the Earth’s rotation slows and the Moon recedes: about 3.7 terawatts. For decades the question was where it was lost.

Where the tide's energy goes, and what the deep ocean needs. Power in terawatts. The Moon and Sun do work on the oceans at about 3.7 TW, measured from the slowing of the Earth's spin and the Moon's recession and, since the 1990s, located by satellite altimetry: about 2.6 TW is lost in shallow seas, about 1 TW in the deep ocean where the tide flows over ridges and seamounts and makes internal waves, and about 0.1 TW in the solid Earth. The open bar is what Munk and Wunsch estimated the deep ocean needs to keep its stratification against the upwelling of cold water, about 2 TW; the winds supply part of it, the tides — on these estimates — about half.
Fig. 3 Power in terawatts: 3.7 TW of tidal dissipation, 2.6 TW in shallow seas, 1 TW in the deep ocean as internal tides, 0.1 TW in the solid Earth; wind work on deep internal waves about 0.5 TW; and the roughly 2 TW Munk and Wunsch estimated the deep ocean needs (open bar).

The traditional answer was the continental shelves. In shallow water the tidal current scrapes against the sea floor and loses energy to bottom friction, and the great shelf seas — the European shelf, the Patagonian shelf, the Yellow Sea, the northwest Australian shelf, the Bay of Fundy — were estimated in the 1970s to account for most of the total. Most, but not all: the budgets never quite closed.

The satellite altimeters launched from 1992 measured the height of the sea surface to a few centimetres every ten days, and the tidal signal in those maps, combined with the equations of motion, located the dissipation directly. Gary Egbert and Richard Ray found in 2000 that about a quarter of it — roughly a terawatt — occurs in the deep ocean, concentrated over the mid-ocean ridges, the island arcs and the chains of seamounts, where nothing rubs against a shallow floor. A terawatt of tidal energy is lost in water three or four kilometres deep, and it is about half of what Munk and Wunsch said the deep ocean needs.

The same energy, counted three ways

The 3.7 terawatts is one of the best-measured numbers in geophysics, because it is measured three ways that share nothing. The first is astronomical: laser pulses returned from the reflectors on the Moon — the same ranging that tests whether the Moon would lean towards the Sun — measure its recession at 3.8 centimetres a year, and the angular momentum the Moon gains is the angular momentum the Earth’s spin loses. The second is geodetic: the slowing of the Earth’s rotation, read from eclipse records two thousand years old and from the modern timekeeping that let the clock leave the Sun, corrected for the changes in the Earth’s shape since the ice ages. The third is oceanographic: the satellite altimetry, which maps the tide’s height and lets its energy flux be followed across the ocean until it disappears. The three agree to a few per cent.

The solid Earth’s share is small because rock is stiff and nearly elastic: how much a world gives to a tidal pull is set by its rigidity, and the Earth’s rocky body flexes by about thirty centimetres and dissipates a few per cent of the energy. A body whose interior is less rigid, or held in an eccentric orbit by its neighbours, can dissipate enormously more — Io’s volcanoes are a moon heated by not being allowed to relax. On the Earth, the water does the work, and the deep water does a quarter of it.

Waves on the layers inside the ocean

The mechanism is a wave that cannot be seen at the surface. The ocean is stratified, so a parcel of water displaced vertically oscillates at the buoyancy frequency, and the stratification carries waves — internal waves — whose restoring force is the difference in density between layers, about a thousand times weaker than the restoring force at the air–sea surface. Internal waves are slow, with heights of tens to a hundred metres in the ocean’s interior and only centimetres at its surface.

When the tidal current, moving back and forth twice a day over the whole depth of the ocean, flows across a ridge, it forces the stratified water up and down the ridge’s flanks at the tidal frequency. That generates internal waves at the tidal frequency — internal tides — which propagate away from the ridge, up and down through the stratification as well as horizontally. The conversion is efficient where the topography is steep and tall: over the Hawaiian Ridge, measured in detail in the early 2000s, some twenty gigawatts of the surface tide is converted into internal tides. Part of the internal tide breaks near where it was made; part travels thousands of kilometres before it does. Where it breaks, it mixes.

That makes the Moon a stirrer of the deep ocean by a route that would be hard to guess. The tide is a difference in the Moon’s pull across the Earth, a few parts in ten million of gravity; it raises a surface bulge of a metre; the bulge’s current, a few centimetres a second in the open ocean, flows over a ridge; and the internal waves it makes, breaking a kilometre down, lift cold water into warm and keep the deep ocean layered.

A latitude beyond which a tide cannot help

Internal waves have a frequency range. They cannot oscillate faster than the buoyancy frequency, and they cannot oscillate slower than the inertial frequency f=2Ωsin⁡ϕf = 2\Omega\sin\phi, below which the Earth’s rotation turns a displaced parcel round in a circle before the stratification can restore it.

The latitudes beyond which a tide cannot make internal waves. The inertial frequency f = 2Ω sin φ against latitude (black), and the frequencies of four tidal constituents, in radians per hour. Internal waves exist only between f and the buoyancy frequency, so a tide can radiate internal waves only where its frequency exceeds f — equatorward of its critical latitude: M₂ 74.5°, S₂ 85.8°, K₁ 30.0°, O₁ 27.6°. The semidiurnal tides can stir the deep ocean almost everywhere; the diurnal tides only within thirty degrees of the equator. Beyond its critical latitude a diurnal tide flowing over a ridge makes trapped motions that stay near the topography rather than waves that carry its energy away.
Fig. 4 The inertial frequency against latitude, and the frequencies of four tidal constituents. A tide radiates internal waves only equatorward of its critical latitude: M2M_2 74.5°, S2S_2 85.8°, K1K_1 30.0°, O1O_1 27.6°.

So each tidal constituent can radiate internal tides only where its frequency exceeds the local inertial frequency — equatorward of a critical latitude. For the principal lunar semidiurnal tide, M2M_2, that is 74.5 degrees; for the principal diurnal constituents, K1K_1 and O1O_1, about thirty. Poleward of thirty degrees a diurnal tide flowing over a ridge cannot make a wave that carries its energy away; it makes motions trapped against the topography instead, which dissipate where they are.

The semidiurnal tides therefore do most of the deep stirring, over most of the ocean. There is a subtler effect near 29 degrees, where half the M2M_2 frequency equals the inertial frequency: there, an internal tide can transfer its energy to waves at half its frequency, which have short vertical wavelengths and break quickly. Measurements across that latitude have found enhanced mixing there, which is a direct demonstration that the latitude-dependent physics of the figure operates in the ocean and not just on paper.

Mixing where the floor is rough

The global budget wants a diffusivity of 10−410^{-4}. The first direct measurement of mixing in the ocean’s interior, made in 1993 by releasing a tracer into the stratified water of the eastern North Atlantic and watching it spread for two years, found 10−510^{-5} — ten times less. That was a problem for the budget, because a diffusivity of 10−510^{-5} everywhere would need only a tenth of the power and would leave the deep ocean’s temperature unexplained.

The resolution came from the floor. Measurements in the Brazil Basin a few years later found mixing weak over the smooth abyssal plains and strong over the rough flanks of the Mid-Atlantic Ridge, increasing towards the bottom and reaching 10−310^{-3} within a few hundred metres of the topography — exactly where internal tides are generated and break. The deep ocean is not mixed uniformly. It is mixed intensely in a small fraction of its volume, over rough topography, and weakly elsewhere, and the average is what Munk’s profile measures. The tracer had been released in a quiet place.

That distribution has consequences beyond the budget. The upwelling that balances the mixing happens where the mixing is, so the deep circulation is not a uniform slow rise but a set of upwelling zones along the flanks of ridges and seamounts, connected by flows along the bottom. The geography of the deep circulation is, in part, the geography of the sea floor that the tide flows over.

How long the deep ocean takes to turn over

The same balance fixes how quickly the deep ocean is renewed.

How long the deep ocean takes to turn over. The time for the deep ocean's volume, 10¹⁸ m³, to be replaced by upwelling through an area of 3.3·10¹⁴ m², with the upwelling set by the mixing, w = κ/h for a scale height h of 1 km, against the diffusivity. At κ = 10⁻⁴ m²/s the deep water is renewed in about 1000 years, close to the radiocarbon ages of the deep Pacific; at the 10⁻⁵ measured in the open interior it would take about 10,000 years. The shaded band is the range of radiocarbon ages of deep water, from a few hundred years in the North Atlantic to about fifteen hundred in the North Pacific.
Fig. 5 The time for the deep ocean to be replaced by upwelling, with w = κ/h for h = 1 km, against the diffusivity. At κ=10−4 m2 s−1\kappa = 10^{-4}\,\mathrm{m^2\,s^{-1}} it is about a thousand years; at 10⁻⁵, about ten thousand. Shaded: the radiocarbon ages of deep water, a few hundred to about fifteen hundred years.

At Munk’s diffusivity the upwelling is about a ten-millionth of a metre a second, and the deep ocean’s volume is replaced in about a thousand years — close to the radiocarbon ages of deep water, which run from a few hundred years in the North Atlantic, near where it sinks, to fifteen hundred in the North Pacific, at the end of its path. With the interior diffusivity the tracer release found, it would take ten thousand. The radiocarbon ages, measured independently, confirm that the mixing averaged over the whole deep ocean is close to Munk’s value, and therefore that the rough-topography mixing is real and global.

The turnover time sets how quickly the deep ocean takes up heat and carbon dioxide from the atmosphere, which is why the question matters well beyond oceanography. The deep ocean holds fifty times as much carbon as the atmosphere, and the rate at which it exchanges carbon with the surface is, at the end of a chain of physical steps, a rate set partly by the Moon.

A nearer Moon

The Moon is receding, by about four centimetres a year, and it was closer in the past.

A deep ocean stirred harder by a nearer Moon. The power in the lunar tide, relative to today's, against the Moon's distance as a fraction of its present one, at fixed ocean response: the tide-raising force goes as a⁻³ and the dissipated power as its square, a⁻⁶. At the 96.5% the tidal rhythmites of 620 million years ago record, the lunar tide carried 24% more power; at 90%, 88% more. If the deep share of that power scaled with it, a nearer Moon would have mixed the deep ocean harder and turned it over faster — at fixed geography, which is never true over that long, since the shelves that take most of today's dissipation come and go with the continents.
Fig. 6 The power in the lunar tide relative to today’s against the Moon’s distance, at fixed ocean response: it scales as a−6a^{-6}. At the 96.5% that tidal rhythmites record for 620 million years ago it was 24% higher; at 90%, 88% higher.

The tide-raising force goes as the inverse cube of the distance, and the power dissipated, which goes as the square of the tide’s amplitude, as the inverse sixth power. Tidal rhythmites — layered sediments deposited in tidal estuaries, whose layers count days and months — and the rhythm of precession preserved in older rock record the Moon’s distance in the past: the rhythmites put it at about 96.5 per cent of its present distance 620 million years ago, which at a fixed ocean response would mean a lunar tide a quarter more powerful. Cyclic sediments 1.4 billion years old put the Moon at about 89 per cent of its present distance, where the lunar tide’s power at fixed response would have been about twice today’s.

Whether that stirred the ancient deep ocean harder is a question about geography as much as about the Moon. Most of today’s dissipation is on shelves whose resonances depend on the shapes of the continents, and the deep conversion depends on the ridges and the stratification, all of which have changed. The figure is drawn at fixed response, which is never true over hundreds of millions of years. What it shows is that the astronomical part of the forcing was larger, by an amount the rhythmites now measure, and that any model of the ancient ocean’s circulation has a lunar term in it.

What the arithmetic leaves out

One diffusivity stands for a distribution. The power, the profile and the turnover time are computed for a single average diffusivity. Mixing concentrated over rough topography gives the same average with a different geography, and the power needed depends on where the mixing happens, because the stratification is stronger in the upper deep ocean than near the bottom.

The mixing efficiency is uncertain. The figure of one fifth is a canonical value with a factor of two either way in different regimes of turbulence, and the power needed scales inversely with it.

The budget’s numbers are estimates. The 1 TW of deep tidal dissipation comes from fitting altimetry to the equations of motion, with an uncertainty of a few tenths of a terawatt; the wind’s contribution to deep internal waves is less certain still; and the two terawatts needed depends on how much of the deep water’s upwelling happens in the Southern Ocean, where wind-driven upwelling along sloping density surfaces can bring water up without much mixing at all.

The Southern Ocean is a second route. The strong westerly winds round Antarctica drive an upwelling along tilted density surfaces that returns deep water towards the surface adiabatically, without mixing across them. How much of the deep return flow goes that way rather than through mixing in the basins has been one of the main revisions to the picture since 1998, and it lowers the power the mixing needs.

Still open: how much of the deep ocean’s stirring is lunar

The broad picture is now settled: about a terawatt of tidal energy is converted into internal tides over rough topography and breaks in the deep ocean, a comparable amount of wind energy reaches it as internal waves, and between them they mix the deep ocean at a rate the radiocarbon ages and temperature profiles require. What is not settled is the split, and where each part goes.

The internal tides are partly predictable — they are periodic at the tidal frequencies — and partly not, because their phase drifts with the stratification they travel through. The wide-swath satellite altimeters now in orbit see their surface signatures, a few centimetres high, in two dimensions for the first time. Mapping where they are generated, how far they travel and where they break would turn the global average into a geography, and would say how much of the deep ocean’s circulation — and of its uptake of the atmosphere’s heat and carbon — depends on the Moon’s distance and the shape of the sea floor it pulls the water across.

Where this goes next

The tide's spectrum, with every frequency set by the sky. The equilibrium tide-raising potential broken into its largest constituents, each a sinusoid whose frequency is a sum of whole-number multiples of six astronomical rates — the Earth's turning relative to the Moon, the Moon's and the Sun's motions round the sky, and the slow turning of the lunar perigee and node — and whose height is its equilibrium amplitude relative to M₂. They fall in three bands: near two cycles a day (M₂ at 1.9323, S₂ at exactly 2, N₂, K₂), near one (K₁, O₁, P₁, Q₁), and near zero (fortnightly, monthly, semi-annual). Every frequency is known to many decimal places from astronomy; what a harbour does to each, its own amplitude and phase, is not, and is the only thing a tide table has to measure. A tide table is astronomy with the harbour fitted Nobody can compute the tide at a harbour from the Moon and the Sun, because the ocean's response depends on coastlines and depths nobody models well enough. Tides are predicted to centimetres years ahead anyway, because the frequencies are pure astronomy and only the response to each one has to be measured. The method needs a record long enough to tell neighbouring frequencies apart, and a correction for a cycle of 18.6 years. Integrating the measured recession back: the Moon reaches the Earth 1.54 Gyr ago. The Earth–Moon separation and the length of the Earth's day, integrated backwards from the measured present recession rate of 3.83 cm per year. Constant-Q tidal friction makes a^(13/2) linear in time, so the history is a single line in a variable nobody plots, and it is calibrated to the laser-ranging measurement rather than to a modelled k₂/Q — the k₂/Q it implies is 0.0257, or Q = 11.6 for the Earth's k₂ of 0.299, which is a startlingly dissipative Earth. Run back at that rate the separation reaches zero 1.54 Gyr ago and crosses the Roche limit at 2.88 Earth radii only 4 years before it, so the drawing is cut off there rather than extrapolated. The Moon is 4.5 Gyr old, so this is a refutation and not a date: the present rate cannot have been the rate, and a mean Q of 34 — drawn dashed, reaching 4.51 Gyr — is the sort of value the age requires. Tidal rhythmites at 620 Myr put the day at 21.9 h and the Moon at 96.5 per cent of its present distance, and this history reads 20.1 h and 92.4 per cent — too fast and too close, which is the same failure the zero crossing is. Day length follows from total angular momentum, 23.93 h today, 9.84 h at half the present lunar distance and 4.97 h at the Roche limit, and depends on the separation alone: it is the same curve whatever Q is. The rate of lengthening the recession requires is 2.10 ms per century, against a tidal total of about 2.3 including the Sun's tide, which slows the Earth without moving the Moon, and an observed 1.75 from ancient eclipses and occultations — the shortfall being the Earth's moment of inertia falling as the mantle rebounds from the last glaciation. A day five hours long The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing. The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one. The tide is a difference, which is why there are two of them The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

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Buoyancy frequencyCritical latitudeEarth moon systemInternal wavesOcean mixingStratificationTidal dissipationTides