The water that warms on the way down

The sea that stands lower because water gives took the most familiar approximation in fluid mechanics, that water is incompressible, and found what it hides: the bottom of the ocean is 4.7 per cent denser than the top, the whole sea stands about thirty metres lower than it would if water did not give, and the speed of sound rises with depth enough to dig a channel for it. That essay ended on a question it did not answer. Squeezing a fluid does not only make it denser. Done without letting heat in or out, it makes it warmer, and the deep ocean is squeezed hard.

The effect is small for water and large for the questions it touches: how to read a thermometer at the bottom of the sea, how to decide whether a column of water is stable, how to account for the heat the ocean is taking up. And water, alone among common fluids, has a temperature at which the effect vanishes and below which it runs backwards. That reversal makes a deep freshwater lake stable in a way that can suddenly fail.

A thermometer that lies about the deep #

The drawing constructs a column of seawater the way the deep ocean is built. Its upper few hundred metres are warm and its deep water, formed near Antarctica and in the far North Atlantic and spread slowly through the basins, is uniformly cold. The dashed line is the temperature each parcel of water would have if it were brought up to the surface without exchanging heat — its potential temperature. In the deep, that is 1.00 °C from four kilometres to the bottom, and the pressure on every parcel is set only by the depth, a thousand atmospheres for each ten kilometres.

A thermometer lowered through the same column reads something else. Near the surface the two agree. With depth the thermometer reads progressively warmer than the potential temperature, and below about four kilometres the deep water’s reading actually rises with depth: from a minimum of 1.38 °C to 2.35 °C at almost eleven kilometres, in water that is, by construction, all the same. The rise is compression. A parcel carried from the surface to the bottom of the trench is squeezed by more than a thousand atmospheres, and the work done on it raises its temperature by more than a degree.

This is not a subtlety of the model. It is what is measured. The deepest parts of the ocean’s trenches show exactly this pattern, a temperature minimum at around four kilometres and warmer water below it, and it misled early interpreters into looking for a heat source at the bottom. The geothermal heat coming through the sea floor is real but tiny — it would take centuries to warm a deep trench’s water by a tenth of a degree, far longer than the water spends there — and the rising temperature is the same cold water, squeezed.

Potential temperature, and why the deep ocean needs it #

The practical consequence is that the thermometer reading is the wrong number for almost every question an oceanographer asks. Whether a deep water mass is the same one that sank in the Weddell Sea, whether a column is stable, how much heat the ocean has taken up — all of these need a temperature that does not change when water is merely moved up or down. That is the potential temperature, and it is what ocean data are reported in. A modern refinement, conservative temperature, goes one step further and tracks the heat content rather than the temperature, but the idea is the same: remove the part of a reading that is only pressure.

The same step was taken in the atmosphere first, where it matters far more. The layer a parcel cannot leave decided whether a column of air overturns by comparing its temperature profile with the rate at which a rising parcel cools by expanding, and the potential temperature of air is the device that makes that comparison automatic: a column is stable if its potential temperature rises with height. In water the correction is fifty to a hundred times smaller, and for the upper ocean it can be ignored. For the deep ocean, where temperature differences between water masses are a few tenths of a degree, the adiabatic correction is as large as the signal.

The gradient, and what it is made of #

The warming comes from one thermodynamic identity. For a parcel compressed with no heat exchanged, its temperature rises with pressure at the rate

$$\left(\frac{\partial T}{\partial p}\right)_S = \frac{\alpha T}{\rho c_p},$$

where $\alpha$ is the thermal expansion coefficient, $T$ the absolute temperature, and $\rho c_p$ the heat capacity per unit volume. The logic runs through the expansion coefficient. A material that expands when heated must, by a reciprocity that is one of the Maxwell relations, heat up when compressed without heat exchange, and by an amount proportional to how much it expands. The argument is short. Squeezing a material at constant temperature changes its entropy by an amount set by how much its volume responds to temperature; if no heat may flow, that entropy change has to be undone by a temperature change instead, and the heat capacity says how large. A material that does not expand on heating does not warm on compression.

For seawater $\alpha$ is small and depends strongly on the state of the water. It grows with temperature — warm water expands far more per degree than cold — and, unusually, it grows with pressure. The drawing uses a simplified equation of state for seawater that captures both, and the gradient that follows ranges over a factor of five: 45 millikelvin per kilometre for water at freezing near the surface, 232 for tropical surface water, 126 for cold water under six kilometres of ocean. Deep, cold water warms more per kilometre than the same water would near the surface, because its expansion coefficient has been increased by the pressure it is under.

The same identity explains a small fact about the speed of sound. Newton’s calculation used the isothermal stiffness of air and came out fifteen per cent low; Laplace’s correction was that sound compresses air adiabatically, and adiabatic compression, which warms the air, stiffens it. The size of that stiffening is set by the same $\alpha$. For air it is large; for water it is less than one per cent, and at four degrees, in fresh water, it is zero.

Water that cools when squeezed #

Fresh water is densest near 3.98 °C. Below that temperature it expands as it cools, so its expansion coefficient is negative, and the identity above says that compressing it without heat exchange must cool it. The drawing shows the adiabatic gradient of fresh water passing through zero at the temperature of maximum density and changing sign: water at 10 °C carried down a kilometre warms by 64 millikelvin, water at 1 °C cools by 31.

The temperature of maximum density is itself not fixed. Pressure pushes it down, by about two hundredths of a degree for every bar, so at the bottom of a lake a kilometre deep water is densest near two degrees rather than four. Each curve in the drawing crosses zero at a lower temperature than the one above it. Seawater, whose salt lowers its temperature of maximum density below its freezing point, never reaches this regime; its expansion coefficient stays positive at every temperature it can have as a liquid. Only fresh and brackish water can be squeezed colder.

A coefficient whose sign is set by a temperature #

A thermodynamic effect that changes sign at a particular temperature is not peculiar to water. Push a gas through a porous plug and its temperature changes although no heat goes anywhere — it cools if it starts inside a dome in the plane of pressure and temperature and warms if it starts outside, and hydrogen at room temperature warms. The Joule–Thomson coefficient that decides this is built from the same thermal expansion coefficient, and it passes through zero where $\alpha T$ equals one, the inversion temperature. In both cases a quantity everyone expects to have one sign has the other, because it is really a statement about how a material’s volume responds to heat, and that response can vanish.

What makes water’s version unusual is where it sits. The inversion temperature of nitrogen is hundreds of degrees above anything in ordinary life; the zero of water’s adiabatic gradient is at four degrees, in the middle of the range of temperatures of lakes, rivers and polar surface water. It is a curiosity of the liquid’s hydrogen-bonded structure — the open, ice-like arrangement of molecules that persists in cold water and occupies more volume than the collapsed arrangement that warmer water can afford — and it puts the sign change where it governs how every temperate lake turns over in spring and autumn.

A lake that is stable until it is not #

The moving density maximum makes the stability of a deep freshwater lake a question with a depth in it. Consider a lake whose deep water sits at 3.4 °C, typical of the deepest lakes, and a parcel of colder surface water, at 2.8 °C, pushed down by wind from 250 metres. At 250 metres the temperature of maximum density is close to 3.5 °C, the deep water is nearly at it and so nearly as dense as fresh water can be, and the colder parcel is further from it and lighter. Released, it would float back up. The column looks stable.

Push the parcel deeper and the balance changes. The temperature of maximum density falls with depth, and once it has fallen below the average of the two temperatures — about 3.1 °C here, at a depth of roughly 450 metres — the parcel, being colder, is now nearer to the density maximum than the deep water is, and denser. Past that depth it sinks without any further push, all the way to the bottom. The drawing follows the parcel’s own adiabatic temperature change as it descends, which is small, and finds it heavier than its surroundings by about twenty parts per million at the floor — a small difference, and enough to drive it down through a kilometre of water.

This thermobaric instability, as it is called, is how the deepest lakes renew their bottom water. Lake Baikal, more than 1.6 kilometres deep, has well-oxygenated water at its bottom, which requires that surface water reach it regularly. It gets there in episodes, when winds along the shore push colder water down past the compensation depth, and the parcels then cascade to the floor under their own weight. A water column that is stable to small disturbances and unstable to large ones is an unusual thing, and in fresh water it is a consequence of nothing more exotic than a density maximum that pressure moves.

Stability read in the wrong variable #

The lake parcel is a case of a more general point about stability, and the deep ocean has the mirror image of it. A column is stable if a parcel displaced up or down finds itself heavier or lighter than its new surroundings in the direction that sends it back. That is a comparison made at the parcel’s new pressure, not at its old one, and it is not the same as asking whether the water below is denser than the water above, each at its own depth.

In the trench drawn earlier the in-situ temperature rises with depth below four kilometres, so a naive reading of the thermometer would say the bottom water is warmer, hence lighter, than the water above it, and the column should overturn. It does not, because a parcel moved down from above would warm by exactly the amount the thermometer shows, and arrive at the same temperature as its new neighbours. The column is neutral, with uniform potential temperature. Read in the right variable there is nothing to explain.

The lake is the opposite case: a column that looks stable by any local test and hides a depth below which a displaced parcel no longer comes back. It has the same shape as a body carrying a pocket of gas, which floats at one depth and sinks without limit if pushed below it, because it compresses faster than the water around it. There the body’s gas is more compressible than water; here the parcel’s density responds to pressure differently from its surroundings because it sits on the other side of the density maximum. In both, the question is not how heavy something is but how its weight changes when it is moved, and the answer can change sign at a depth.

The ocean has a subtler version, because the thermal expansion of seawater itself changes with pressure. Two water masses that have the same density at the surface can have different densities at four kilometres, since the colder one’s expansion coefficient grows faster with pressure. Oceanographers therefore compare densities at a reference pressure near the depth of interest, or use a construction called neutral density that follows the local reference everywhere, rather than reducing every parcel to the surface. The same thermobaric effect that drives a lake’s renewal quietly redirects where water masses spread in the deep ocean.

One formula across four materials #

Written per kilometre of descent in a column held up by its own weight, the adiabatic gradient becomes $g\alpha T/c_p$, and it is one of the few formulas that can be applied without change to the atmosphere, the ocean and the rock beneath both. For an ideal gas $\alpha T$ is exactly one, since a gas at constant pressure expands in proportion to its absolute temperature, and the gradient is $g/c_p$ — the 9.8 kelvin per kilometre of dry air, which needs no equation of state at all. For seawater $\alpha T$ is a few hundredths, and the gradient is a hundredth of the air’s.

The Earth’s mantle sits between them. Rock’s expansion coefficient is a few times $10^{-5}$ per kelvin and its temperature near two thousand, and the gradient comes out near half a kelvin per kilometre. Over the 2,900 kilometres of the convecting mantle that adds up to about 1,300 kelvin, a large part of the temperature at the base of the mantle — which means that much of the heat at the bottom of the mantle, like the warmth at the bottom of the trench, is the same material squeezed. Geophysicists separate a mantle’s actual temperature profile from its adiabat for the same reason oceanographers use potential temperature: only the departure from the adiabat says anything about heat flowing.

Where the adiabatic picture stops #

Parcels that exchange heat. The whole construction assumes a parcel moves faster than it can exchange heat with its surroundings. For ocean water masses sinking over years this is excellent, because the ocean’s diffusion of heat is slow. For thin layers or slow motions it is not, and where heat diffuses faster than salt — the salt that sinks through a stable sea — the difference between the two diffusions produces instabilities of its own.

A simplified equation of state. Seawater’s density depends on temperature, salinity and pressure in a way that takes a long polynomial to fit to measurement accuracy. The simplified form used here gets the expansion coefficient to within tens of per cent over the ocean’s range, which is enough to show the minimum and its size; the international standard gives the exact numbers, and in-situ temperatures in the deepest trenches agree with it to a few thousandths of a degree.

Salinity. The drawing holds the salinity fixed. Real deep water masses differ in salinity as well as temperature, and it is the combination that decides density; a column can be stable in temperature and unstable in salinity, or the reverse.

What the profile does not show #

The profile is a snapshot of an ocean at rest. The real deep ocean is moving, slowly, and the water in a trench is renewed over decades to centuries by flows that come over the sills at the trench’s edge. The drawing cannot show whether the water at the bottom arrived recently or long ago; the potential temperature can, when compared with that of the water masses that could have supplied it, which is how the path of deep water through the basins has been traced. And the drawing cannot show the small departures from the adiabat — a few thousandths of a degree — that record the geothermal heat and the mixing, which are exactly the quantities someone measuring the deep ocean’s heat budget is looking for.

Still open: the heat the deep ocean is taking up #

Since the 1990s repeated measurements along the same ship tracks have found the water below two kilometres warming — by a few thousandths to a few hundredths of a degree per decade, a small change in temperature and a large amount of heat, because the deep ocean is so vast. Because the expansion coefficient of cold, deep water is larger than that of the same water near the surface, that heat raises sea level by more than an equal amount of heat would near the surface, and because the deep ocean is sparsely sampled, how much of the ocean’s heat uptake and of sea-level rise comes from below two kilometres is known with an uncertainty comparable to the effect itself. Autonomous floats that dive to six kilometres are beginning to fill the gap, and the answer depends on separating a signal of thousandths of a degree from the adiabatic warming a float’s own dive produces in its thermometer reading.

The habit worth carrying away is to ask whether a reading is of a state or of a place. A temperature measured at depth is a property of the water and of the pressure it is under, and only the part that survives bringing the water to a reference pressure says anything about where it came from. The water at the bottom of a trench is warm the way a bicycle pump is warm, and a lake’s cold water can sink through warmer water because the depth it is carried to has changed which of the two is denser.