The rise a joule of heat buys

The pressure that only knows depth found that the pressure at a point in still water is the weight of the water above it, per unit area, and nothing else. The sea that stands lower because water gives followed what that weight does to the water underneath: it squeezes it by a few per cent at the bottom of the ocean, and the whole sea stands thirty metres lower for it. The water that warms on the way down found that squeezing warms it, and that the warming per kilometre depends on how much the water swells when heated.

That last quantity, the thermal expansion coefficient, is the subject here, read the other way round. Not what compression does to temperature, but what heating does to height. When the oceans take up heat — and they take up nine-tenths of what the Earth has gained in recent decades — the water expands, and since it cannot expand sideways or downward it expands upward. The sea rises. The surprise is how much the rise depends on where the heat is put, and why a pressure gauge on the sea floor sees none of it.

Taller and exactly as heavy #

Warm a column of water a kilometre deep by a fraction of a degree and its density falls by α times the warming, where α is the expansion coefficient; its height rises by the same fraction. The mass in the column is unchanged — no water has been added — so the weight per square metre on the floor is unchanged too. The column is taller and lighter per cubic metre, and the two changes cancel exactly in the pressure at the bottom.

This is the hydrostatic law doing what it always does, counting weight and not height, and it has a practical consequence that sounds paradoxical. The sea rises for two reasons: water added from melting ice and from the land, and water expanding as it warms. Satellites measuring the height of the sea surface see both. A pressure sensor on the sea floor, or a satellite measuring the pull of the water’s mass, sees only the first. Comparing the two kinds of measurement is how the rise is split into its parts — and the split works precisely because thermal expansion is invisible to anything that weighs.

The rise from a given heat follows directly. Heat Q per square metre warms a layer of thickness H by $Q/\rho c_p H$, and the layer expands by α times that warming times H, so the thickness cancels:

$$\Delta h = \frac{\alpha\,Q}{\rho\,c_p}.$$

Where the heat is spread through the column does not matter, except through α. The figures count heat in watt-years per square metre — what a flux of one watt per square metre delivers in a year, about thirty-two million joules — because the Earth’s present energy imbalance is a little under a watt per square metre, and a watt-year is a year’s worth.

Why the heat is in the sea #

The ocean takes almost all of the heat for a reason that is a matter of counting. The heat capacity of the whole atmosphere, from the ground to its top, equals that of the top three metres or so of the ocean, because a square metre of air column weighs about ten tonnes and a square metre of water three metres deep weighs three, and water stores four times as much heat per kilogram. The ocean is four kilometres deep on average. Whatever imbalance there is between the sunlight the Earth absorbs and the infrared it sends back — the balance the height a planet is seen from sets up — is absorbed, over years, by the one part of the climate system able to hold it without changing temperature much.

That is also why thermal expansion is a slow and long-lived contribution to sea level. Heat enters at the surface and is carried down by mixing and by the sinking of cold water in a few places, and the time for it to reach the abyss is centuries. Like the summer that reaches the cellar in December, the deep ocean’s temperature lags the surface’s by a time set by the depth and the speed of the carrying. Heat taken up now goes on penetrating, and the sea goes on expanding, long after the surface has come into balance — and since the deep yields almost as much rise per joule as the surface, the slow tail is not a small one.

A coefficient that changes by a factor of six #

For most liquids the expansion coefficient is roughly constant, and the rise per joule would be too. Water is not most liquids. Fresh water has its densest point at four degrees and does not expand at all there; below it, it contracts when warmed. Salt pushes the density maximum below the freezing point, so seawater expands at every temperature it reaches, but near freezing it does so only barely, as though it still remembered the anomaly. Warm seawater expands about five times as much per degree as water near freezing.

The reason is the same one why heating a perfect spring changes nothing gave for solids, with a twist. Expansion comes from the asymmetry of the forces between molecules: pushed together they resist harder than pulled apart, so a molecule jostled more by heat sits further out on average. In cold water a second effect opposes it. The molecules are partly arranged in an open, hydrogen-bonded network like that of ice, more spacious than the close-packed liquid, and warming breaks some of that network and lets the liquid collapse. The two effects nearly cancel in cold water and the first wins easily in warm.

Pressure tips the balance as well. Squeezing cold water collapses the open network the way warming does, so less of it is left for heating to collapse, and the ordinary expansion shows through. At two degrees, forty megapascals — four kilometres down — nearly doubles the expansion coefficient. That pressure dependence is the thermobaric effect, which the mixture heavier than either water found making a parcel that sinks far enough keep sinking.

The rise per watt-year, depth by depth #

Down a subtropical column, warm at the top and cold below, the rise a watt-year buys falls steeply through the thermocline as the water cools, reaches a minimum near two kilometres, where the water is as cold as it will get but the pressure is still moderate, and then recovers with depth as pressure raises the coefficient. Down a polar column, cold from top to bottom, there is no thermocline to descend, and the yield simply rises with depth: almost twice as much at four kilometres as at the surface.

That polar curve is the counter-intuitive one. Heat reaching cold water is usually thought of as heat that does little to sea level, and at the surface that is right: the Southern Ocean and the North Atlantic, where surface water is cold, take up a large share of the ocean’s heat at a low exchange rate. But the water that sinks there carries the heat down, and at depth the same water at the same temperature expands much more per joule. Heat does not lose its power to raise the sea by being buried. Below about two kilometres it gains some of it back.

Six places to put the same heat #

The spread is a factor of six. The global ocean’s thermal expansion therefore has no fixed exchange rate. The commonly quoted figure, about a tenth of a metre of rise for every 10²⁴ joules of heat, is a weighted average over where the heat happened to go, and it is a property of the present pattern of ocean circulation rather than of seawater. If future warming shifts more of the uptake to the cold high latitudes, each joule will buy less rise; if more of it reaches the warm upper ocean of the tropics, more. Climate models disagree among themselves about sea-level rise partly for this reason: two models that take up the same heat can store it differently.

The rise is not uniform either. Heat stored in one basin raises the sea surface above it, and the slope this makes in the surface drives currents that move water elsewhere, so the regional pattern of thermal rise is a matter for ocean dynamics, not for the column-by-column arithmetic here. What the arithmetic gives is the global total, because the total volume added is the sum of every column’s expansion wherever it ends up.

A surface with hills in it #

The rise from expansion is not only a global total; it is also a map, and the map drives the ocean. Wherever a column is warmer through its depth, it stands taller, and the sea surface above a warm pool is a gentle hill. Across the Gulf Stream the surface drops by about a metre over a hundred kilometres, from the warm Sargasso Sea to the colder water inshore, and almost all of that metre is thermal expansion of the upper ocean. The pressure at a fixed depth under the hill is higher than under the valley beside it, so water is pushed from hill to valley, and on a spinning planet a push of that kind does not produce a flow down the slope but a flow along it, at right angles, with the hill on the right in the northern hemisphere. The Gulf Stream is that flow.

So oceanographers have measured the expansion of seawater column by column for a century, as dynamic height, to calculate currents from temperature and salinity profiles. The same arithmetic, summed over the whole ocean instead of compared between two columns, is the thermal part of sea-level rise. A change in where the heat is stored changes the hills, the hills change the currents, and the currents change where the next heat is stored.

Checking the exchange rate against the sea #

The pieces can be put together and compared with what is measured. Since the early 1990s, satellite altimeters have tracked the height of the sea surface across the globe to a few centimetres per measurement and a fraction of a millimetre a year in the global average, and they find the sea rising at a little over three millimetres a year on average, faster in the most recent decade. Over the same period the ocean has been gaining heat at roughly ten zettajoules a year, which is an imbalance of about six-tenths of a watt per square metre over the Earth’s whole surface. At a tenth of a metre per 10²⁴ joules, that heat should raise the sea by about a millimetre a year.

Measured directly, from the temperature profiles of thousands of floats, the thermal expansion of the upper two kilometres accounts for a little more than a millimetre a year, roughly a third of the total rise, with the rest from melting glaciers and ice sheets. The two routes agree to within their uncertainties, and the disagreement that remains — a few tenths of a millimetre a year — is about the size of the deep ocean’s contribution, which is exactly the part the floats did not reach.

Tide gauges on coasts, which have recorded sea level for more than a century in some ports, measure something subtly different: the sea relative to the land the gauge is bolted to, which may itself be rising after the weight of ice-age glaciers was removed, or sinking as groundwater is pumped out. Separating the motion of the land from the motion of the sea needs satellite positioning at the gauge, and the global record from gauges is the longer and the harder to read.

Heat that goes deep, and the rise it brings #

The deep ocean below two kilometres holds about half the ocean’s volume and is sampled far more thinly than the upper half. Until the 2010s almost all the systematic measurements of ocean temperature came from floats that dived to two kilometres and no further, and the deeper water was measured only from ships, repeating the same lines across a basin once a decade. Those repeated sections found the deep and bottom water warming, especially water formed round Antarctica, at rates of a few thousandths of a degree per decade.

The figure shows why that matters to sea level more than its smallness suggests. Averaged through the cold, compressed water between two and six kilometres, a watt-year buys 1.22 millimetres of rise, close to the 1.35 averaged through the warm upper two kilometres. Heat in the deep counts almost at par. Whatever fraction of the ocean’s uptake is going below two kilometres, very nearly that fraction of the thermal rise is coming from there, and an uncertainty in the deep’s heat is an uncertainty in the rise of the same size.

A thousandth of a degree, measured #

The arithmetic is unforgiving. A thousandth of a degree, spread through the four kilometres of water below two kilometres, is 0.62 millimetres of sea level and sixteen million joules per square metre — half a watt-year, half a year’s worth of the planet’s present imbalance concentrated into a layer whose temperature has barely moved. The signals reported from the deep basins are a few thousandths of a degree per decade, which puts the whole of the deep’s contribution at a few tenths of a millimetre a year.

To measure that, a thermometer has to be accurate to about a thousandth of a degree and stay so for years, at six hundred atmospheres, in an instrument that dives and surfaces on its own. Floats built for that, part of a deep extension of the global float array, began to be deployed in the late 2010s with thermometers calibrated to a thousandth of a degree and pressure sensors corrected for their own drift. Their task is complicated by the effect the water that warms on the way down described: the temperature a thermometer reads at depth includes the warming of compression, a fraction of a degree, which must be removed by converting to potential temperature before a thousandth of a degree of real change can be seen. A small error in the pressure, or in the equation of state used for the conversion, is a false trend.

Where the arithmetic stops #

The coefficient used here comes from a simplified equation of state, linear in temperature and in pressure, which reproduces the measured expansion of seawater to roughly ten per cent over the ocean’s range and misses its curvature; the full international equation of state does better and is what is used for real heat budgets. Salinity is held fixed, though freshening of the ocean, from melting ice and changing rainfall, expands it too, by an amount that largely cancels in the global total because salt is conserved. The profiles are idealised: real columns have mixed layers, intermediate water masses and fronts. And the rise is computed column by column, which gives the right global total and the wrong regional pattern.

What the hydrostatic part of the argument cannot be wrong about is the invisibility of thermal expansion to the pressure on the floor. That rests on the column’s mass being unchanged, which holds as long as no water has been added or removed above the sensor. Currents that shift mass from one region to another change the bottom pressure without heating anything, and separating that signal from the others is part of the work of the satellites that weigh the oceans.

Still open: how much of the rise comes from the deep #

The deep floats have been in the water for only a few years, the repeated ship sections are a decade apart, and the deep ocean warms unevenly, faster in some basins than others. How much heat is going below two kilometres, and so how much of the present sea-level rise is coming from there, is known with an uncertainty comparable to the deep contribution itself. A related question is whether the circulation that carries heat into the deep is changing: the water sinking round Antarctica has been freshening and its volume may be shrinking, which would change both where the heat goes and how much rise each joule buys. Closing the global sea-level budget — the sum of expansion, ice melt and land water against the rise measured from space — to a tenth of a millimetre a year depends on answering that.

The habit worth carrying away is to ask what an exchange rate depends on before quoting it as a constant. The rise from heat stored in the sea is $\alpha Q/\rho c_p$, and α ranges from 0.40 to 2.60 mm per watt-year per square metre across the places the heat can go — six times — while a pressure gauge on the floor sees none of it, because the column has grown taller without growing heavier. A thousandth of a degree in the deep is worth more than half a millimetre of sea level, and nobody could measure it until recently.