An atmosphere that bends light round the Earth

In the winter of 1596–97 a Dutch expedition searching for a north-east passage to Asia was trapped by ice on the coast of Novaya Zemlya, at about 76° north, and spent the polar night in a hut built from driftwood. On 24 January, one of them recorded, the rim of the Sun was seen above the horizon. By their reckoning, and by any astronomical calculation, it should not have appeared for another two weeks. The observation was doubted for centuries — it seemed more likely that the navigators had lost count of the days than that the Sun had risen early — and it is now understood to have been correct.

Ordinary refraction lifts the Sun at the horizon by about 34 arcminutes, a little more than its own diameter, and at high latitude, where the Sun climbs out of the polar night very slowly, that is worth a day or two of early sunrise. Two weeks at 76° north requires the Sun to be seen when it is about five degrees below the horizon — ten times the normal refraction. The atmosphere can do that, and the condition under which it does so can be stated as one number.

Why a ray curves at all #

Light travels more slowly in denser air, and a wavefront crossing a gradient of density turns towards the denser side, the way a line of marchers wheels when its inner end slows down. In the atmosphere the density falls with height, so a horizontal ray turns downward — it curves in the same sense as the Earth’s surface, and that is why the horizon seen through air is slightly more distant than the geometric one and why objects just beyond it are lifted into view.

The ray’s curvature is simply the rate at which the refractive index changes with height. For air the refractive index differs from one by an amount proportional to the pressure divided by the temperature, so its vertical gradient has two parts: the pressure falls with height because of the weight of the air above, and the temperature changes with height by whatever the local conditions dictate. Writing both out gives the ray’s curvature as proportional to

$$0.0342 + \frac{\mathrm{d}T}{\mathrm{d}h}$$

in kelvin per metre, where the first term is the pressure’s contribution and the second the temperature gradient’s. In the standard atmosphere the temperature falls by 6.5 kelvin per kilometre, the bracket is 0.028, and a horizontal ray curves at about a fifth of the Earth’s rate.

The gradient at which light follows the ground #

The bracket depends on the temperature gradient, and the temperature gradient can have either sign.

When the air near the ground is colder than the air above it — an inversion, the same arrangement that in a stellar atmosphere turns a limb bright instead of dark — the density falls faster with height than the pressure alone would make it, and the ray curves more strongly. The curvature rises linearly with the inversion’s strength, and at about ten or eleven kelvin per hundred metres, depending on the temperature and pressure, it equals the curvature of the Earth. A ray launched horizontally at that gradient curves down exactly as fast as the ground curves away from it and stays at the same height all the way round.

Stronger than that, and the ray curves down faster than the ground. It cannot escape the cold layer: a ray starting slightly upward rises, turns and comes back down; reaching the surface, which over ice or calm sea reflects strongly at grazing angles, it is reflected upward and repeats. The layer has become a waveguide, and light within a small range of angles is trapped in it, following the curve of the Earth for as long as the layer continues. That is the opening figure: five of the eight rays drawn remain in the layer, hopping along it, for the whole three hundred kilometres.

Inversions that strong are rare over land in daytime, when the sun warms the ground and the air near it. They are common over sea ice, snow and cold sea in calm weather, especially in the polar winter: the surface radiates heat to the sky, cools the air in contact with it, and the lowest few hundred metres can be many degrees colder than the air above — a gradient of ten degrees per hundred metres is unusual, but not extreme, in the high Arctic.

A threshold, not a gradual effect #

The distinction between an inversion below the threshold and one above it is sharp, and it is worth seeing directly.

Below the threshold the inversion enhances refraction: rays are bent more strongly, the horizon is pushed further out, and objects beyond it are raised into view — looming, which over cold water is familiar enough that sailors have used it to see coastlines that should be hidden. But each ray still escapes eventually, and the extra refraction is limited to a fraction of a degree. Above the threshold the character changes. Trapped rays are guided for as long as the duct continues, and the total bending is no longer set by the gradient at all.

Why the early Sun measures a distance #

That last point is the key to Novaya Zemlya, and it is counter-intuitive enough to be worth drawing.

A ray guided by a duct turns through the same angle as the ground it follows, one degree for every 111 kilometres of the Earth’s surface. Traced backwards from an observer inside the duct, a line of sight that is trapped runs round the curve, and when the duct ends it continues in a straight line from wherever it left — pointing at a part of the sky that is below the observer’s geometric horizon by the angle the duct has turned through. An object in that direction, such as the Sun, is seen at the apparent horizon.

So the refraction produced by a duct is not proportional to the inversion’s strength. Any inversion above the threshold traps light; how far the light is carried round the curve depends only on how far the inversion extends. To see the Sun five degrees below the horizon, as at Novaya Zemlya, the duct had to run for about 560 kilometres between the ship and the point at which the sunlight entered it — a single, continuous inversion over the frozen Barents and Kara Seas. That is a statement about the meteorology of the Arctic winter, not about the physics of refraction, and it is why the phenomenon is rare even where strong inversions are common: they have to be extensive as well as strong.

The Sun in such a sighting does not look like the Sun. The light reaching the observer through the duct comes only through the narrow range of angles the duct traps, so what is seen is a thin, distorted, flickering band — sometimes several bands stacked above one another — rather than a disc. The account from Novaya Zemlya describes only the rim appearing, which is what a duct would deliver.

What two weeks means at 76 degrees north #

The size of the effect in days, rather than degrees, comes from how slowly the Sun climbs back at high latitude, and that is worth spelling out because it is what made the observation so hard to believe.

At 76° north the Sun is below the horizon at noon for about three months in the middle of winter. Its noon altitude as the polar night ends is set by its declination, which changes most slowly near the solstice and is climbing by only about a third of a degree a day in late January. A Sun five degrees below the horizon at noon is therefore about two weeks away from rising, and ordinary refraction, which lifts it by about half a degree, advances the first sunrise by a day or two. Everything beyond that has to come from somewhere else. The noon Sun five degrees down is also the Sun of civil twilight — the sky is already bright and the southern horizon glows — so the expedition was not watching a dark sky suddenly lit; it was watching the glowing horizon produce a sliver of the disc itself, weeks before any almanac allowed.

That arithmetic is also why the effect is almost only ever reported from high latitudes. At the latitude of London, a Sun five degrees below the horizon rises within half an hour. The same duct would advance the sunrise by half an hour and be noticed by nobody. At 76° north the same five degrees is two weeks, because the Sun is barely moving vertically at all.

One object seen several times #

The stacking has a precise origin, and it shows the duct’s effect on images of ordinary objects as well as on the Sun.

Each line of sight from an observer meets a distant object at some height. In a normal atmosphere, higher lines of sight meet higher points, one for one, and the object is seen as it is. Inside a duct, the lines of sight that hop along the layer arrive at heights that rise and fall with the elevation at which they left, and the mapping from direction to height folds. A single point on the object is reached by several lines of sight at different elevations, and so is seen several times, one above another. Between the folds the image is alternately upright and inverted, and at each fold it is stretched without limit, because many directions map onto nearly the same height.

That is the superior mirage. A coastline beyond the horizon appears lifted, stretched vertically into cliffs and towers, and repeated in bands above itself — the Fata Morgana, named for the enchantress believed to raise castles in the air over the Strait of Messina, where warm air over a cold sea produces the same structure.

A stronger inversion curves the trapped rays more tightly, so each hop is shorter and the lines of sight reaching a given distance have made more of them. The image folds more times and is repeated in more, thinner bands. That is the qualitative difference between a gently lifted, stretched coastline and a shattered Fata Morgana, and it is set by the inversion’s strength, while the distance the image can be seen from is set by the duct’s extent.

A green flash that lasts for minutes #

The duct also explains a stranger report from the same kind of place. At sunset the last sliver of the Sun’s upper rim sometimes turns green for a second or two, because the atmosphere disperses sunlight like a prism: the green image of the Sun is lifted slightly more than the red, and for a moment only the green rim is still above the horizon. Normally the flash is brief and tiny, because the separation between the colours is a few arcseconds and the Sun sinks through it in a second.

A mirage magnifies it. Where the image of the Sun’s rim falls near one of the folds in the mapping drawn above, it is stretched vertically without limit, and the thin green edge is stretched with it into a band large enough to see plainly and lasting as long as the rim takes to cross the fold. At high latitudes, where the Sun skims along the horizon at a shallow angle rather than dropping steeply through it, a rim can stay near a fold for a long time. An Antarctic expedition in 1929 reported a green flash visible, on and off, for more than half an hour as the Sun moved along a horizon made irregular by the ice — an observation that, like the one at Novaya Zemlya, sounds impossible until the duct is included.

The same physics at radio wavelengths #

Radio waves are refracted by the atmosphere too — the wet part of the air is one of the media a spacecraft’s signal is corrected for — and far more strongly by water vapour than light is, because water’s molecular dipole can follow a radio field but not an optical one. Ducts for radio waves therefore form wherever humidity falls steeply with height as well as where temperature rises — above all in the evaporation duct over warm seas, a layer a few metres to tens of metres deep in which the air right at the surface is saturated and the air above is drier.

Radio ducting is common and consequential. Radars see ships and coastlines hundreds of kilometres beyond their normal horizon; television and radio stations interfere with others far outside their intended range; and a radar looking through a duct sees returns from the sea surface that can be mistaken for targets. The same guiding that carried the Sun’s light over the Barents Sea carries a radar’s pulse along the sea surface, and the threshold is the same one — a refractivity gradient at which the ray’s curvature matches the Earth’s — expressed in radio refractivity rather than optical. The measurements that probe the atmosphere with radio signals passing through it edgewise have to recognise ducting layers in the lowest kilometre, where the signal can be trapped and the usual inversion from bending angle to refractivity fails.

What the rays leave out #

The figures trace rays through an atmosphere that is horizontally uniform: the same inversion over the whole path. A real duct begins and ends, changes depth, and is broken by open water or a change in the surface, and where it ends the trapped light leaks out. The early Sun at Novaya Zemlya needs the duct to have been continuous over hundreds of kilometres, which is exactly what makes the phenomenon rare, and the figures cannot show how often that happens.

They also treat the inversion as a single layer with a constant gradient and a sharp top. Real inversions have smooth profiles, sometimes several stacked layers, and the detailed shape of the mirage depends on the profile’s shape, not only its maximum gradient. The Fata Morgana’s towers are the signature of a layered profile, and reproducing a particular mirage requires the profile to be known to a precision rarely available.

The observer’s height matters as well, and the figures fix it at five metres. A duct traps only rays that start inside it at shallow enough angles, so an observer standing inside the cold layer sees the guided light and one standing on a hill above it does not. The expedition’s lookout on the ice was in the duct; a mast-head a few tens of metres higher, in warmer air above a shallow inversion, might have seen nothing. The same Sun, on the same morning, could have been visible from the ground and invisible from above it — the reverse of the usual rule that height extends the horizon.

And they ignore turbulence, which in a real duct is small because the stable stratification suppresses it, but not zero. The shimmer of a superior mirage — its images flickering and breaking up — is turbulence acting on light that has been guided for a long way, and it limits how sharply any image through a duct can be seen.

Still open: reading the air from its mirages #

The relation can be run backwards. A mirage’s shape — how many images, where they fold, how they are stretched — is a function of the temperature profile along the path, and an observed mirage is therefore a measurement of that profile, made at no cost by anyone who can photograph the horizon. Mirages of the setting Sun, whose shape is known in advance, have been used to infer the temperature structure of the lowest few hundred metres of the atmosphere over the sea, and old records of sightings like the one at Novaya Zemlya carry, in principle, information about the inversions over polar ice centuries before anyone measured them. Turning a qualitative description in a journal into a quantitative profile needs more than the description usually gives, and whether historical mirages can be read as a climate record — of how often, and how extensively, the polar inversion formed — is an open question about the records rather than about the optics. The optics, as the flattened sunset and the distorted field already showed, is the same refraction throughout; only the gradient changes.