The sea that stands lower because water gives

The pressure that only knows depth begins hydrostatics with $p = \rho g h$, and the formula is so good that it is rarely noticed that it assumes something. It assumes $\rho$ is the same at every depth — that each layer of water weighs the same whatever is piled on top of it. For air that is obviously false; why the air thins with height is the story of a gas whose density falls by a factor of $e$ every eight kilometres. For water it is almost true, and the textbooks call water incompressible.

“Almost” deserves a number, because the ocean is deep. At the bottom of the Challenger Deep, eleven kilometres down, every square metre of water is carrying the weight of eleven thousand tonnes of water above it, and water, pressed that hard, gives. The amount it gives is small in any one layer, but a small effect summed over a column eleven kilometres tall is not small at all, and three things follow from it that an incompressible ocean would not have: a higher pressure at the bottom, a lower sea surface, and a channel in which sound can travel for thousands of kilometres.

How much water gives #

The stiffness of a fluid against being squeezed is its bulk modulus $K$: the pressure needed to reduce its volume by a given fraction, per unit of that fraction. Seawater at 0 °C and ordinary salinity has a bulk modulus of about 2.16 gigapascals at the surface, so a pressure of 21.6 megapascals — two thousand metres of water — reduces its volume by about one per cent. Water also stiffens as it is squeezed. The international equation of state for seawater expresses this with a bulk modulus that rises with pressure, by 3.24 pascals of stiffness for every pascal of pressure at 0 °C, and that is the law used here.

The same stiffness shows up wherever water or oil is used as though it could not be squeezed. Force multiplied, and nothing gained treats a hydraulic press as a lever made of liquid, and at the pressures of industrial hydraulics — tens of megapascals — the oil in the lines compresses by a per cent or so, which is why a hydraulic actuator behaves like a stiff spring rather than a rigid rod and why long hydraulic lines make a machine feel soft. Floats that drift through the deep ocean measuring its temperature face the same number from the other side. The depth past which it must sink shows that a body more compressible than the water around it loses buoyancy as it descends and has an unstable depth; the floats are therefore built with hulls whose compressibility is matched to seawater’s or slightly below it, so that a float pushed down is not dragged further, and their designers need seawater’s bulk modulus to within a few per cent to get that right.

With the equation of state the column can be built from the top down: start with the surface density, compute the pressure a thin layer lower from the weight of the layer, compute the density at that pressure, and continue. At no point is the density assumed.

The line leaves the vertical immediately and keeps going. By the ocean’s mean depth of 3.68 kilometres, a litre of water holds 1.67 per cent more mass than a litre at the surface; at the bottom of the deepest trench it holds 4.71 per cent more. The curve is almost exactly straight, which is itself informative: the density grows in proportion to the pressure, and the pressure grows in proportion to the depth, so to first order the density grows in proportion to the depth. The slight flattening that the stiffening introduces is far too small to see at this scale.

The real ocean adds temperature and salinity on top, and they matter as much as pressure at the surface and far less in the deep, where the water is nearly uniformly cold. A column of 0 °C water isolates the one effect this essay is about.

The pressure an incompressible ocean would miss #

A denser column weighs more, so the pressure at the bottom exceeds $\rho_0 g h$.

The excess grows as the square of the depth. To first order it is $\rho_0^2g^2z^2/2K_0$ — the dashed curve — because each layer is compressed in proportion to its depth, and the extra weight of all the layers down to a depth $z$ adds up to half the product of $z$ with the compression at $z$. At the mean depth of the ocean the excess is a third of a megapascal, three atmospheres. At the bottom of the Challenger Deep it is 2.68 megapascals, 2.42 per cent of the whole, which is the weight of 266 metres of water that the squeezing has packed into the same depth. The integrated curve runs slightly below the estimate at great depth, because the stiffening makes the deepest layers compress less than a constant bulk modulus would.

The same squeeze sets a limit the other way. Water can also be put under tension, and the height a siphon cannot pass shows a column held together by cohesion at negative pressure. Under tension the compressibility runs in reverse and the water expands slightly, and the stiffness that resists compression is also what lets a thin column carry a pull of megapascals before it breaks — a property of how tightly water’s molecules are packed that the incompressible idealisation hides in both directions.

Instruments on deep landers measure this directly. A pressure gauge on the floor of a trench reads a depth only if the density of the water above it is known, and converting one to the other with a constant density would put the gauge at the wrong depth by more than two hundred metres. Oceanographers convert pressure to depth with the equation of state and a model of the water’s temperature and salinity, and the depths quoted for the deepest trenches carry uncertainties of a few tens of metres that come mostly from the water above rather than from the gauge.

A sea thirty metres lower #

The compressed column has a second consequence, and it is the one with a map attached. The mass of water in a column is fixed by how much water there is. A compressed column holds that mass in less height than an uncompressed one, so the sea surface stands lower than it would if water could not be squeezed.

For a column at the ocean’s mean depth the answer is 31.0 metres. The dependence is quadratic, because a deeper column is both taller and more compressed at its base: a kilometre of water loses 2.3 metres to squeezing, six kilometres lose 81. Over the ocean as a whole the lost height spreads out as the sea surface finds its level, and because the deep basins count as the square of their depth, the ocean-wide figure is somewhat more than the value at the mean depth — a few tens of metres in all.

If water were incompressible, the sea would stand some thirty metres higher than it does, and the coastlines of the world would be where the thirty-metre contour now runs. That is about half the rise that melting all the ice in Greenland and Antarctica would produce, from a property of water that the textbooks round to zero. It is also why the ocean’s response to warming has a compressibility term: a column that expands when heated expands against the weight above it, and the expansion coefficient of seawater itself changes with pressure, so the deep ocean’s contribution to sea-level rise per degree is not the surface water’s.

The channel compressibility digs for sound #

The third consequence is acoustic, and it is where the effect of squeezing is not a small correction at all.

The speed of sound in a fluid is the square root of its stiffness over its density, $\sqrt{K/\rho}$ in the simplest form. For seawater at 0 °C the equation of state gives 1,449 metres per second at the surface, the value every empirical formula for sound in cold seawater starts from — which is the check that the equation of state is the right one. Squeezing raises both the stiffness and the density, but the stiffness faster, because $K$ rises by 3.24 times the added pressure while the density rises only by the fractional compression. So sound travels faster the deeper it goes.

Temperature pulls the other way. Warm water carries sound faster than cold, and the upper ocean is warm, with a thermocline below it in which the temperature falls to near freezing within a kilometre or so.

Near the surface the cooling wins and the sound speed falls with depth; below the thermocline the temperature has stopped changing and the squeezing takes over, raising the speed by 15.5 metres per second for every kilometre — close to the 16 of the standard empirical formula, computed here from nothing but the bulk modulus. Between the two lies a minimum, at 1,145 metres in this model, and it has a dramatic consequence. A sound wave travelling nearly horizontally from that depth is always turning towards slower water, as the bend at the boundary describes for light in a smoothly varying medium; it bends back towards the minimum from above and from below, and oscillates about it for as long as it lasts. Sound in the channel does not spread into three dimensions but into two, and how a wave thins out shows what that does to its loudness: it falls as one over the distance instead of one over the distance squared.

The channel was found in the 1940s, independently by Maurice Ewing in the United States and by Leonid Brekhovskikh in the Soviet Union, and named the SOFAR channel: small explosive charges detonated at its depth were heard thousands of kilometres away. Whales use it; submarines hide from it and in it; a programme of the 1990s timed sound pulses across whole ocean basins to measure the ocean’s average temperature by the change in the speed of sound. None of it would exist in an incompressible ocean. The deep sound channel is made of the same compressibility that lowers the sea by thirty metres, turned from a correction to the weight of a column into the whole of the reason sound speeds up with depth.

One length that says whether a column can ignore its weight #

All three effects are governed by one comparison: the depth of the column against the length $K/\rho g$ — the height of a column whose weight would, at constant stiffness, compress it by the whole of itself.

For seawater it is 214 kilometres, and the ocean’s mean depth is 1.7 per cent of it, which is exactly the fractional increase in density at that depth. Every number in this essay is some multiple of that ratio: the density change is the ratio, the extra pressure is half its square times $\rho_0 g$ times the depth, and the lost sea level is half the ratio times the depth. A column much shorter than its $K/\rho g$ can be treated as incompressible; one that is a few per cent of it cannot, if the few per cent matter.

The same length for a gas is the most familiar number in the atmosphere. An isothermal gas has a bulk modulus equal to its pressure, because halving its volume doubles its pressure, and $p/\rho g$ is $RT/Mg$ — the 8.4-kilometre scale height of why the air thins with height. The atmosphere’s thinning and the ocean’s squeezing are the same ratio of stiffness to weight, and they differ only in that for air the ratio is set by the temperature and for water by the forces between molecules. Mercury, thirteen times denser than seawater and almost exactly thirteen times stiffer, happens to have the same length; granite and steel are ten times longer. The bottom of the ocean is at 1.7 per cent of its compressibility length; the top of Everest is at a whole scale height of the air’s.

The same ratio decides when a solid planet can ignore its own compression, and for the Earth it cannot: the mantle’s rock, with a length of a couple of thousand kilometres, is squeezed by a planet thousands of kilometres deep, and the density rise with depth that the pull that grows on the way down needs is partly made of exactly this.

Where the squeezed column stops #

One temperature and one salinity. The column is 0 °C seawater of salinity 35 from top to bottom. The real ocean is warm on top and has a salinity that varies by a few parts per thousand, and those change the surface density by more than a kilometre of pressure does. The compressional part computed here is the one that dominates the deep ocean, where the water is uniformly cold.

A truncated equation of state. The bulk modulus is taken as linear in pressure, the first two terms of the international equation. The next term changes the density at the bottom of the deepest trench by about a part in ten thousand.

Static water. Everything is hydrostatic. Currents, tides and the ocean’s own circulation move the surface by a metre or so from where a static calculation would put it — small against thirty metres, but the difference that satellite altimetry actually measures.

A model thermocline. The sound-speed figure uses a single exponential temperature profile and the standard empirical dependence of sound speed on temperature. Real channels sit at depths from a few hundred metres in polar seas, where the surface is cold and the minimum comes to the top, to over a kilometre in the subtropics.

An ocean, not a column #

Each figure is a single vertical column, and none shows the ocean as a surface. The thirty metres of lowered sea level is not uniform: deep basins lose more and shallow seas less, and the surface then flows to a common level, so the actual change at any coast depends on the whole ocean’s shape — a two-dimensional calculation that no column draws. The sound figure plots a speed profile and not the rays it produces; the bundle of paths oscillating about the channel’s axis, some crossing it every fifty kilometres and some every sixty, is the picture that explains why a signal arrives as a series of pulses rather than one. And no figure shows water’s molecular reason for resisting compression, which is the strong short-range repulsion between molecules already packed nearly as tightly as they will go.

Still open: what the deep ocean’s squeezing does to its heat #

Squeezing a fluid adiabatically warms it, and a parcel of water carried from the surface to the bottom of a trench arrives a degree or so warmer than it started, with no heat added. Oceanographers remove this effect by quoting a potential temperature — the temperature a parcel would have if brought back to the surface — and the distinction matters for how heat taken up by the ocean is accounted for. As the ocean absorbs most of the heat added by the rise in greenhouse gases, the deep water below two kilometres has begun to warm measurably, and how much of the resulting sea-level rise comes from that deep layer — where both the expansion per degree and the compressibility differ from the surface — is known with an uncertainty comparable to the effect itself, because the deep ocean is sparsely measured.

The habit worth carrying away is to ask what an approximation is small compared with. A property that is negligible per unit is not negligible per column, and the length that decides it is the ratio of the material’s stiffness to its weight. Water is incompressible to one part in a hundred per kilometre, and the ocean is four kilometres deep and forty thousand kilometres round.