The cold reservoir overhead

The ceiling on every engine found that a heat engine needs two reservoirs, a hot one to draw heat from and a cold one to reject it to, and that its efficiency can never exceed one minus the ratio of their temperatures. On the Earth’s surface the cold reservoir is almost always the surroundings: the air, a river, the ground. All are close to the same temperature, about 15 °C on average, which is why every engine and every refrigerator on the planet is built around that number.

There is another cold reservoir, available over every square metre of the planet on every clear night, and it is reached not by conduction or convection but by radiation. It is the sky. The ground and everything on it radiate heat upward continuously, as all warm bodies do, and the sky radiates heat back down; but a clear sky radiates back much less than the ground sends up, because part of what it would return comes from the cold upper atmosphere, and part of what the ground sends goes straight through the air to space. A surface that can see a clear sky is in radiative contact with something much colder than the air beside it.

A sky colder than the air #

Every surface radiates in proportion to the fourth power of its absolute temperature, $\varepsilon\sigma T^4$, where $\sigma$ is the Stefan–Boltzmann constant and $\varepsilon$ the surface’s emissivity — about 0.95 for paint, wood, skin, soil and most non-metals, and much lower for polished metal, as the glow that says nothing about the surface found. The sky radiates too, from the water vapour and carbon dioxide in the air above. Its downward radiation is commonly written as if the air at ground level were radiating with an effective emissivity $\varepsilon_{\text{sky}}$, which for a clear, dry sky is about 0.72, for a clear humid one about 0.82, and under low cloud close to one.

That effective emissivity below one has two sources. The air’s emitting gases are colder higher up, and the height a planet is seen from found that radiation escaping to space comes from that colder upper level; looking up, the ground sees the same cold layer. And in a band of wavelengths around ten micrometres the clear air is nearly transparent: the spectrum that is a thermometer at every height found satellites looking down through that window at the ground itself. Looking up through it, the ground sees space at three kelvin.

Why the upper air is colder at all is the business of the cloud that cools more slowly than the air: air carried upward expands and cools, and the atmosphere settles at a temperature that falls about six and a half degrees with every kilometre of height. A sky made of that air, seen from below, is a radiator whose effective temperature is set less by the air at the surface than by the air a kilometre or two above it, where most of the downward radiation in the absorbing bands comes from. Thinner and drier air moves that level higher and colder, which is why the clear desert sky and the sky over a mountain are colder reservoirs than the sky over a humid coast.

The result is that a surface at the air temperature, facing a clear dry sky, sends up about a hundred watts per square metre more than it receives. That is a substantial cooling power — a tenth of full sunshine — and it acts all night. The sky’s radiation balances the surface’s only if the surface is about twenty-three degrees colder than air at 15 °C: that is the sky’s effective radiative temperature. Under cloud the balance point is barely below the air, because the cloud base is nearly as warm as the ground and radiates almost as a black body.

The hundred watts is easy to check by hand. Air at 15 °C is 288 K, and a black body at that temperature radiates $\sigma T^4 \approx 390$ W/m². A surface of emissivity 0.95 sends up 95 per cent of that, about 371 W/m². The clear dry sky sends down 72 per cent of the black-body figure, about 281 W/m², and the surface absorbs 95 per cent of that, about 267 W/m². The difference, 104 W/m², is the net loss at the air’s own temperature — and it does not depend on the surface being hot, only on its seeing a sky that is radiatively colder than itself.

How far below the air a surface falls #

The sky does not get to cool a surface all the way to its radiative temperature, because the air keeps warming it by convection, at a rate proportional to the temperature difference times a coefficient that depends on the wind. The surface settles where its net radiative loss equals the heat the air delivers. In still air, where the coefficient is a few watts per square metre per kelvin, an insulated surface — a car roof, a leaf, a sheet of glass lifted off the ground — sits ten degrees or more below the air under a clear dry sky. A breeze mixes warm air down to it and the drop shrinks; under cloud it is a degree or so whatever the wind.

That is why clear, still nights are the cold ones, and why gardeners protect seedlings by covering them: a sheet of fabric overhead hides the sky and replaces it with a radiator at nearly the air’s temperature. It is also why frost forms first on surfaces that are poorly connected to the ground and see much of the sky — car roofs and windscreens, the tops of hedges, the leaves of grass — and last under trees.

The cooling does not stop at the surface. Ground and grass chilled by the sky chill the air in contact with them, and that cold, dense air lies in a shallow layer under warmer air above — a nocturnal temperature inversion, often a few metres deep, in which a thermometer at head height reads several degrees warmer than one at the grass tips. On sloping ground the cold layer drains downhill like water and pools in hollows and valley floors, which is why gardeners speak of frost pockets, and why an orchard planted halfway up a slope survives a spring night that kills blossom on the valley floor a few hundred metres away.

What the surface is made of matters as well as what it sees. The metal that feels colder than the wood found that a surface’s grip on its own temperature depends on how much heat it can draw from inside itself, and the same quantity governs how fast the sky can pull it down. A thin sheet of steel on a car roof has almost nothing beneath it to draw on and follows the sky’s pull within minutes; a thick wooden deck or a slab of concrete feeds its surface with heat stored during the day and stays nearer the air for hours. Frost appears on the car before the path for the same reason the railing feels colder than the bench.

Frost with the air above freezing #

So frost does not need freezing air. It needs a surface below freezing and air moist enough to deposit ice on it, and a clear sky supplies the first with the air well above freezing. The idealised insulated surface of the figure would reach freezing in still air under a dry sky even at 13 °C. Real surfaces do less well, for two reasons the figure leaves out: as soon as the surface cools below the air’s dew point, water condenses on it and releases its latent heat, which slows further cooling; and grass and soil draw heat up from the ground beneath. With both included, ground frost on a clear calm night typically appears with the air at three to five degrees — the weather forecast’s “ground frost with air temperatures above freezing”.

The same physics once made ice in deserts. In India and Iran, shallow trays of water set on beds of straw in pits open to the sky, insulated from the ground and sheltered from wind, froze on clear winter nights with the air above freezing, and ice was harvested this way for centuries before refrigeration. The Persian yakhchal, with its tall shading wall and deep storage pit, combined radiative ice-making with insulation to keep ice into the summer.

The ice on the windscreen should not be confused with the other way cold nights lift a garden. The frost that lifts by drinking is about ice growing inside the soil, fed by water drawn upward toward the freezing front, and it can take weeks of cold to develop. Radiative frost is a surface event of a single night: it needs only a clear sky, still air, and a surface poorly coupled to anything warmer than itself.

An engine whose cold side is the sky #

If the sky is a cold reservoir, it can in principle run an engine. Take heat from the air at 15 °C, pass it through a reversible engine, and reject the waste heat through a radiator facing the sky. The question is how much work that arrangement could deliver, and the answer is set by a trade-off like the one the engine that has to finish found for any engine that must deliver power rather than only efficiency.

A radiator at the air’s temperature sheds the most heat, a hundred watts per square metre, but an engine between two reservoirs at the same temperature does no work. A radiator at the sky’s radiative temperature gives the engine a large temperature difference but sheds nothing, because it is in radiative balance with the sky. In between, the work is the heat shed times the Carnot factor, and it peaks at about two watts per square metre, with the radiator about eleven degrees below the air. That is tiny compared with sunlight — a solar panel delivers about two hundred watts per square metre in full sun — but it is available at night, when sunlight is not.

The two-watt figure follows from a short argument. Near the air temperature the heat a radiator sheds falls roughly in a straight line with how far it sits below the air, from about 104 W/m² at zero difference to nothing at the balance point, about 23 degrees below. The Carnot factor rises in proportion to the same difference divided by the air’s absolute temperature. The product of a rising line and a falling one peaks halfway between, at about eleven degrees below the air, where the heat shed is half its maximum and the Carnot factor is eleven parts in 288. Half of 104 times eleven over 288 is about two watts per square metre. The number is small because the sky is only twenty-odd degrees colder than the air in radiative terms, and a reservoir that close in temperature can lend very little work however much heat it accepts.

What the sky does have is size. The work left in two buckets of water found that two finite bodies drift together in temperature as an engine runs between them, so the work available is less than Carnot’s ceiling with fixed temperatures promises. The sky is not finite in that sense: heat radiated through the window goes to space and is never returned, and the upper air that radiates the rest is replaced by weather faster than any radiator could warm it. Between the air and the sky the reservoirs stay fixed all night, which is exactly the idealisation Carnot’s argument assumes. And because a real radiator warms as heat is pushed through it, the engine actually rejects its heat across a range of temperatures, the complication the temperature an engine really takes its heat at deals with on the hot side; the ceiling drawn here is for a radiator held at a single temperature, the best case.

Devices have been built to try. A thermoelectric generator with one face warmed by the air and the other attached to a sky-facing radiator, demonstrated in 2019, produced about 25 milliwatts per square metre, enough to light a small LED; later designs reached a few hundred milliwatts. An engine with one number in it found that a thermoelectric couple reaches only a fraction of the Carnot efficiency set by its figure of merit, which is why the demonstrations fall well below the ideal two watts. The ceiling, set by the sky’s temperature and the Stefan–Boltzmann law, is the one drawn here.

Cooling below the air in sunshine #

The sky keeps radiating cold in the daytime too. A surface in sunshine also absorbs whatever sunlight it fails to reflect, and that usually swamps the radiative loss: a dark roof in summer sun runs tens of degrees above the air. But if the surface reflects almost all the sunlight while still radiating strongly in the infrared, the radiative loss can win. Under a clear dry sky the figure needs about 86 per cent of the sunlight reflected just to stay at the air temperature, and more to fall below it. Ordinary white paint, reflecting about 85 per cent, does not quite manage. The threshold is again simple arithmetic. In air at 30 °C, 303 K, a surface of emissivity 0.95 under a clear dry sky has a net radiative loss of about 127 W/m²; full sunshine of 900 W/m² brings in that much if the surface absorbs only 14 per cent of it. Every percentage point of extra reflectance beyond that is nine more watts per square metre the sky can spend pulling the surface below the air. In 2014 a multilayer coating designed by Aaswath Raman, Shanhui Fan and their colleagues — reflecting 97 per cent of sunlight and emitting strongly in the atmospheric window — stayed five degrees below the air in direct sun, and paints and films that do the same have since become commercial products.

The model here is simpler than those coatings, treating the surface’s emission and the sky’s as broadband. A real daytime radiative cooler gains by being selective: emitting strongly only in the window near ten micrometres, where the sky sends little back, and reflecting at other infrared wavelengths, where the sky is warm. That lets it reach temperatures further below the air than a surface that radiates at every wavelength, which shares in the sky’s warm radiation outside the window as well as its cold radiation through it.

What the reservoir depends on #

Humidity. Water vapour is the main absorber in the atmospheric window’s edges, so a humid sky is warmer and the cooling weaker. Radiative cooling works best in deserts and at altitude, and worst in the humid tropics, where the need is greatest.

The view. A surface sees the sky only over the part of its hemisphere not blocked by buildings, trees or hills, and the sky near the horizon is warmer than the zenith, because there the line of sight passes through much more air. A courtyard or a street canyon sees a narrower and warmer sky than an open field.

Clouds and pollution. Low cloud closes the window almost completely; high thin cloud partly. Aerosols that absorb in the infrared warm the sky’s radiation. The cold reservoir is a property of clean, clear air.

The ceiling. No surface can radiate more than a black body at its own temperature, and the sky always returns part of it, so the net cooling a sky-facing surface can draw at the air’s temperature is about a hundred to a hundred and fifty watts per square metre under ordinary clear skies, and well-made coolers measure somewhat less. The heat that crosses a gap too narrow for light found that this ceiling can be broken between two surfaces separated by less than a wavelength, but that loophole needs a second surface close by. The sky is as far away as anything can be, and a radiator facing it is held strictly to Stefan and Boltzmann’s limit.

Still open: how much cooling the sky can supply #

Radiative cooling is being developed for buildings, where a sky-facing roof that stays below the air temperature could reduce or replace air conditioning, and for power stations, whose cooling water could be chilled by radiators facing the night sky instead of evaporated in towers. The physics is not in doubt. The open questions are about scale and durability: whether coatings that reflect 97 per cent of sunlight stay that clean and white for decades outdoors, how much the cooling falls in humid or polluted air, and whether large areas of sky-facing surface, reflecting more sunlight and radiating more heat to space than the ground they cover, would measurably change local climate.

There is also an older and stranger question underneath the arithmetic. The three-kelvin space seen through the atmospheric window is, in thermodynamic terms, an almost limitless cold reservoir, and the Earth uses it all the time: every watt of sunlight the planet absorbs is eventually radiated to space, much of it through that window, and that flow from a hot Sun to cold space is what drives the weather and every living thing. A surface facing the clear sky taps the same flow in the other direction. The sky over every field is a heat sink colder than anything nearby, reached by light rather than by contact, and the frost on a car roof on a clear morning is the smallest everyday proof of it.