The mixture heavier than either water

The layer a parcel cannot leave found that a fluid whose density decreases upward resists vertical motion: a displaced parcel is pushed back and oscillates at the buoyancy frequency. The wave that picks an angle, the reflection that changes the wavelength and the latitude past which a tide cannot split followed the waves such a fluid carries, and the wave that holds a ship back found them draining a ship’s energy. The salt that sinks through a stable sea found the first surprise in stratified fluids: heat and salt diffuse at different rates, so a column that is stable overall can drive salt downward in fingers.

That essay treated density as a straight-line function of temperature and salt. It is not, and the curvature produces a second surprise, independent of diffusion: two waters of exactly equal density, placed side by side, are stable; mixed together, they make water denser than either, which sinks. Oceanographers call this cabbeling, from an old word for the way a mixture becomes heavier. This essay follows where the extra weight comes from, how large it is, and how the same curvature, at its most extreme in fresh water near 4 °C, sets the seasonal life of every temperate lake — and why the ocean does something different.

A straight line against a curve #

The density of seawater depends on its temperature, its salt content and its pressure. At the surface, over the range of temperatures the ocean spans, a simplified equation of state of the kind used in ocean models captures it well: density falls as temperature rises, rises with salinity in proportion, and the rate at which it falls with temperature is itself larger for warm water than for cold. Warm water expands more per degree than cold water does.

The figure is the diagram oceanographers use for this, temperature against salinity, with lines of equal density drawn across it. Because density’s response to temperature changes with temperature, the lines of equal density are curves. Two waters on the same curve — here, one at 4 °C and 33.93 grams of salt per kilogram, the other at 16 °C and 36.53 — have exactly the same density, 1026 kilograms per cubic metre, and side by side they have no reason to move.

Mixing is linear. Temperature and salt are both conserved quantities that simply average when two waters combine, so every mixture of the two parents lies on the straight line between them in the diagram. The straight line and the curved density line meet only at the parents. Everywhere between, the straight line lies on the dense side of the curve, because the curve bows away from it. An even mixture, at 10 °C and 35.23 grams per kilogram, is 0.182 kilograms per cubic metre denser than either parent. It will sink below both.

In symbols the reason is a single inequality. Write the density’s dependence on temperature, at fixed salinity, as a straight line plus a small downward-curving term proportional to the square of the temperature’s departure from a reference. Along a line of equal density the straight part is balanced by salt. The curved part is not: averaging two temperatures and then squaring gives less than squaring each and averaging, and because the curved term subtracts from density, the mixture loses less density to it than its parents did. The excess is the curvature times the square of half the temperature difference — exactly the variance of the two parents’ temperatures about their mean.

This is the whole mechanism. No heat has gone anywhere and no salt has been added; the density excess appears because density is a curved function of quantities that mix in a straight line. The same argument applies to any mixing of any substance whose density depends nonlinearly on its composition — two liquids that shrink when mixed, as alcohol and water famously do, are the familiar case.

The same effect in a measuring jug #

The effect is easiest to believe in a case where the curvature is large. Pour fifty millilitres of ethanol into fifty millilitres of water and the mixture occupies about ninety-six millilitres, not a hundred: the small water molecules fit into the gaps between the larger alcohol ones and both hydrogen-bond to each other more tightly than either does alone. The mass is unchanged, the volume shrinks, and the mixture is denser than the average of its ingredients. Its density is a curved function of its composition, while its composition mixes in a straight line — exactly the situation of the two seawaters, with composition standing in for temperature.

The seawater case is subtler only because its curvature is small and needs a careful diagram to see. What makes it matter is where it acts. An ethanol–water mixture sitting in a jug is merely denser; seawater mixed at a front is denser than the ocean around it, in a fluid where a density difference of a tenth of a kilogram per cubic metre decides whether water sinks a hundred metres or stays at the surface.

How much heavier #

The excess grows as the square of the temperature difference between the parents, since it is the second-order term in the curve that produces it. For waters two degrees apart it is five grams per cubic metre, a small number. For waters ten degrees apart it is an eighth of a kilogram per cubic metre, and for twenty, half a kilogram. Those numbers are not small in oceanographic terms. The entire density difference across the seasonal thermocline — the layer that separates the warm surface water from the cold water beneath and makes the upper ocean one of the most stable fluids on Earth — is typically a few tenths of a kilogram per cubic metre. A mixture made where two water masses ten degrees apart meet can therefore be as much denser than its surroundings as the thermocline is strong.

Such meetings happen at fronts: the boundaries between warm subtropical and cold subpolar water, the edges of major currents such as the Gulf Stream, and the Antarctic Circumpolar Current’s fronts, where waters of very different temperature lie side by side on the same density surface. Mixing along those surfaces, by eddies stirring them together, produces denser water continually, and the dense water sinks. Estimates suggest that cabbeling along density surfaces is a significant source of the downward flow that forms intermediate water masses in the Southern Ocean, and that it transforms a substantial fraction of the water that passes through those fronts. It is a quiet process with no visible sinking plume, and it happens wherever eddies stir warm and cold water of the same density together.

Fresh water, whose curve has a top #

Seawater’s curvature is modest. Fresh water’s, near 4 °C, is the whole story.

Fresh water is densest at 3.98 °C and gets lighter on cooling below that as well as on warming above it. Its density curve has a top. On either side of the top, waters of equal density sit at temperatures symmetric about 3.98 °C, and any mixture of water from opposite sides lands nearer the top and is denser than either parent: water at half a degree and water at seven and a half, equal in density, mix to make the densest fresh water there is. Near the top, density hardly depends on temperature at all to first order, and every mixture across it is a cabbeling mixture.

The peak exists because liquid water is partly a loose, open network of hydrogen-bonded molecules, with the structure of ice in fragments, and cooling towards freezing makes more of it. The open network takes up more room, and below 4 °C its growth outweighs the ordinary contraction of cooling. The melting curve that leans the wrong way found the same open structure making ice less dense than water, and the water that warms on the way down found pressure moving the temperature of maximum density downward in deep lakes. Here the peak is taken as fixed, at the surface, and its consequences follow.

A lake that turns over twice a year #

In summer a temperate lake is strongly stratified: sunlight warms a surface layer a few metres deep, the warm water is light, and it lies on a cold, dense bottom layer that may stay near 4 °C all year. In autumn the surface cools. Cooled surface water is denser than the water beneath it, sinks, and is replaced; the mixing penetrates deeper as the surface layer cools towards the temperature of the water below, until the whole lake approaches 4 °C and mixes from top to bottom. That is the autumn overturn, which carries oxygen down to the bottom and nutrients up to the surface.

Then the curve’s top changes everything. Cooling the surface below 4 °C makes it lighter, so the coldest water stays at the surface and stops mixing downward. The surface can cool to freezing while the bottom stays at 4 °C, and ice forms on top, insulating the water beneath. In spring the ice melts and the surface warms back through 4 °C; as it approaches 4 °C it becomes denser than the water below, sinks, and the lake overturns again before the summer layer forms. Lakes that do this are called dimictic — mixing twice — and the two overturns are the defining events of their year. Both follow from a single maximum in a single curve.

It is also why fish survive winter in lakes, and why the bottom of a deep temperate lake stays near 4 °C through every season: the densest water ends up at the bottom and stays there, and the lake can freeze only from the top.

Lakes that never turn over #

Not every lake gets its two overturns. The mechanism needs the lake’s density to be set by temperature alone, and if something else makes the bottom water denser — dissolved salt, or dissolved gas — the seasonal cooling may never be enough to overcome it. Lakes whose deep water never mixes with their surface are called meromictic, and their bottom layers can remain isolated for centuries, without oxygen, accumulating whatever the sediments release.

The most notorious is Lake Nyos in Cameroon, a crater lake over a volcanic vent that feeds carbon dioxide into its deep water. The dissolved gas made the bottom water slightly denser and kept it in place, and over years the deep water became loaded with carbon dioxide at a pressure held down only by the weight of the water above. In August 1986 something — a landslide, a storm, a cold rain cooling the surface — disturbed the stratification, deep water rose, the pressure on it fell, and the dissolved gas came out of solution in a single eruption, as a bottle of fizzy water does when opened. A cloud of carbon dioxide, denser than air, flowed down the valleys around the lake and killed about 1,700 people. The lake has since been fitted with pipes that bring deep water to the surface continuously, degassing it gently, so that a stratification the lake cannot remove by itself is removed by hand. It is the same physics as the autumn overturn — dense water held down until a small change lets it move — in a lake where the density that holds it is not temperature.

Why the ocean does not freeze like a lake #

Salt changes the picture, and the figure shows how. Dissolved salt lowers the freezing point, by about 0.054 degrees per gram per kilogram, and it lowers the temperature of maximum density much faster, by about 0.22 degrees per gram per kilogram. For fresh water the maximum is four degrees above freezing, so there is a range of temperatures in which cooling makes water lighter, and a lake uses it. As salt is added the maximum falls towards the freezing point, and at 24.6 grams per kilogram and −1.33 °C they meet. Above that salinity the maximum lies below the freezing point, which means it never occurs: water saltier than 24.6 g/kg gets denser all the way down to freezing.

The open ocean, at about 35 g/kg, is on the far side of that crossing. When its surface cools, the cooled water always sinks, however cold it gets, and convection continues until the whole mixed layer — tens to hundreds of metres deep — has cooled to the freezing point, near −1.9 °C. Only then can ice form, and it forms from a column of water all at freezing rather than from a thin cold skin. That is why sea ice forms much later in the season than lake ice at the same latitude, why it grows from a slush of ice crystals in freezing water, and why the formation of sea ice drives deep convection: ice rejects most of its salt as it freezes — and releases the latent heat that changes no temperature — leaving the water beneath even saltier and denser, which sinks and helps to form the densest water in the world ocean, near Antarctica.

Brackish seas sit in between. The Baltic’s surface water, at 5 to 10 g/kg over much of its area, is below the crossing, and it freezes more like a lake than an ocean.

Where the equation of state stops #

The figures use simplified equations of state: a quadratic fit for fresh water near its maximum and a seawater law that captures the curvature in temperature but not all of its dependence on salinity and pressure. The real equation of state of seawater, defined internationally as a function with dozens of terms, also curves with salinity, and its thermal expansion changes with pressure — a second nonlinearity, called thermobaricity, which makes cold water relatively denser at depth and can trigger deep convection in polar oceans. The linear approximations for the temperature of maximum density and the freezing point are good to a few tenths of a degree over the range drawn. None of this changes the argument, which needs only that density is a curved function of quantities that mix linearly.

The lake profiles are idealised, not simulated. A real lake’s seasonal cycle depends on its depth, its exposure to wind, the clarity of its water and the flows entering and leaving it; deep lakes may not overturn every year, shallow ones may mix throughout the summer, and lakes in cold climates may overturn only once. The four profiles show the sequence the density maximum makes possible, not the one any particular lake follows.

What the pictures cannot show #

The mixing figures show the end state of mixing, not the process: in the ocean two water masses are stirred together by eddies over weeks, their interface drawn out into ever finer filaments until molecular diffusion completes the job, and the dense product appears gradually along the whole interface rather than all at once. The figures cannot show where the sinking water goes. In the ocean it slides down along surfaces of slightly greater density until it finds its level, spreading as a thin layer; in a lake it falls to the bottom. And they cannot show that the extra density, though real, releases very little energy compared with the energy spent stirring the waters together: cabbeling converts some of the eddies’ energy into the potential energy of a denser layer that then sinks, and that conversion is part of how the ocean’s large-scale overturning is powered.

Still open: how much of the ocean’s deep water is made this way #

The ocean’s deep and intermediate waters are formed in a few places — the North Atlantic, the Southern Ocean, a few marginal seas — by cooling, freezing and mixing, and how much each process contributes is still being measured. Cabbeling along fronts is invisible to direct observation, because it happens as a slow transformation of water along density surfaces rather than as sinking columns, and it is estimated by combining measured temperature and salinity fields with models of eddy mixing. Those estimates find it significant in the Southern Ocean and near the edges of the Gulf Stream and Kuroshio, but how large it is globally, and how it will change as the ocean warms — which increases the curvature, since warm water’s density responds more strongly to temperature — are open questions for the modelling of the ocean’s circulation.

The habit worth carrying away is to ask whether a property averages the way its ingredients do. Temperature and salt mix in a straight line and density does not, so two waters of equal density make a denser mixture — by a tenth of a kilogram per cubic metre or more at a front — and at water’s density maximum the same curvature makes a lake overturn twice a year and freeze from the top, which saltier water, with no maximum above freezing, cannot do. The ocean and the lake obey one curve; salt decides which side of its peak they live on.