The tone the whole Earth rings at

The pipe that will not carry a low note found that a wave squeezed sideways acquires a lowest frequency, below which nothing travels however long the pipe. The loss that falls as the frequency rises found one mode in a metal pipe that escapes the usual rule about resistance. Both were about guides built on purpose, a few centimetres across, carrying microwaves. This essay is about a guide nobody built: the space between the ground and the ionosphere, a shell about eighty kilometres thick wrapped round the whole planet. It carries radio waves longer than the Earth is round, it has no lowest frequency at all, and because it closes on itself it rings, at about eight hertz, continuously, struck by lightning.

Two walls carry everything #

Seawater and wet ground conduct electricity well enough to reflect radio waves of low frequency almost perfectly. So does the lower ionosphere, the layer of air ionised by sunlight and cosmic rays that begins about sixty kilometres up in daytime and about eighty-five at night. Below its own plasma frequency, as the frequency below which nothing gets in showed, a plasma reflects a radio wave instead of carrying it. Between the two conducting surfaces is air, which barely absorbs at all. That is the recipe for a waveguide, and the radio pioneers who found long waves following the curve of the Earth for thousands of kilometres were using it before anyone had the name.

A pipe has a cutoff because its wall closes on itself. A wave inside has to fit across the pipe in every direction, and below a certain frequency it cannot; the field becomes evanescent and dies within a diameter. A guide made of two parallel plates is different in kind. Between two plates a wave with its electric field running straight from one plate to the other and its magnetic field parallel to both can travel at any frequency, with nothing to fit across the gap: the field is the same everywhere between the plates. That mode, the transverse electromagnetic mode, has no cutoff. The plates can be as close together as one likes compared with the wavelength, and the wave goes on regardless.

The Earth–ionosphere guide is two plates, curved into concentric spheres. At ten hertz the wavelength is thirty thousand kilometres, nearly four hundred times the gap between the walls and three-quarters of the way round the planet, and the guide carries it with a vertical electric field and a horizontal magnetic field exactly as two plates would. Nor does the curvature let it out. The mode that will not turn a corner found that a bent fibre leaks from its outer side, because the part of the field farthest from the centre of the bend would have to travel faster than light in the surrounding medium to keep up, and instead it radiates away. The Earth–ionosphere guide is bent round the whole planet, but its outer side is a conductor, not a cladding: there is no surrounding medium for the field to escape into, and the wave is held to the curve. The channel with no walls bound a wave with nothing but a slower layer and found that one mode always survives; here the binding is by two mirrors, and the surviving mode is the one that needs no room at all.

The wave travels on until it has gone all the way round the world and met itself.

Closing the guide on itself #

A guide that closes on itself is a resonator, because a wave that comes back to its starting point adds to what is still arriving there, and only frequencies at which the return is in step build up. For a ring of the Earth’s circumference, forty thousand kilometres, the first such frequency is the speed of light divided by the circumference, 7.49 hertz, and the rest are its multiples. That is the first guess, and it is wrong in an instructive way, because the Earth is not a ring.

On a sphere a wave does not go round a loop. Radio waves spreading from a lightning stroke at one place travel outward in every direction across the surface, spread to a widest circle at the equator relative to the stroke, and then converge again toward the point exactly opposite — the antipode — where they all arrive together and pass through each other. The standing patterns of a spherical shell are therefore not sines around a circle but the zonal spherical harmonics, the Legendre polynomials of the cosine of the angle from the source, and the condition for resonance becomes $n(n+1) = (2\pi R f/c)^2$ rather than $n^2 = (2\pi R f/c)^2$. Winfried Otto Schumann worked this out in 1952:

$$f_n = \frac{c}{2\pi R}\sqrt{n(n+1)} .$$

The first tone is $\sqrt{2}$ times the ring’s, 10.6 hertz, the second 18.4, the third 25.9. The ratios are not whole numbers, for the same reason that the drum that has no harmonics found its overtones at 1.593 and 2.136 times its fundamental: in two dimensions the geometry decides the allowed patterns, and a string’s integers belong to one dimension only.

The patterns on the globe #

Each pattern is a standing wave laid over the whole planet with the lightning at one pole. The first has the field rising at the stroke and falling at the antipode, with a single node along the great circle a quarter of the way round. The second has the field the same sign at both poles and reversed in a band round the middle, with nodes at 54.7 and 125.3 degrees. The third has three nodes. All of them are largest at the stroke and at its antipode, which is the convergence of the waves made visible: a storm over the Congo puts its strongest signal not only nearby but also in the Pacific, on the far side of the world, where the waves meet again.

Patterns with an odd number have a node exactly 90 degrees from the source. A receiver a quarter of the way round the world from a storm sits on that node, and the storm’s odd tones are invisible to it. This is not an effect of the leaky ceiling or of anything else imperfect; it is the shape of the standing waves on a sphere, and it matters for what any single station can learn.

A leaky ceiling #

The tones measured on the real Earth are 7.8, 14.3, 20.8 and 27.3 hertz, continuing upward in steps of about six and a half. Every one is well below Schumann’s perfect-wall prediction: the first at 74 per cent of it and the fourth at 82. The upper wall is to blame. The ionosphere is not a sheet of metal but a gas whose ionisation, and so whose conductivity, rises gradually over tens of kilometres. The wave’s electric field is reflected near the bottom of that region, its magnetic field penetrates much higher, and the energy the wave carries is spread through a thicker and partly conducting layer. The effect is to slow the wave to about three-quarters of the speed of light and to damp it, so the resonant frequencies come down in proportion and the peaks broaden.

It is a coincidence, then, that the first measured tone, 7.8 hertz, lies close to the naive ring value of 7.49. The sphere raises the frequency by $\sqrt{2}$, and the leaky ceiling lowers it by almost as much. Two errors in opposite directions are easy to mistake for a correct first guess, and many popular accounts of the resonance make exactly that mistake.

The damping and the slowing are two faces of one fact. A reflector that lets part of the field in is a reflector that loses part of the energy, and the node that is not standing still found the same thing for a string tied to a lossy end: the standing wave’s minima are no longer zeros, energy flows steadily through them into the load, and the depth of the pattern measures how much is lost. The Earth’s cavity is that string wrapped round a sphere, with the ionosphere as its imperfect end. By night, when the lower ionosphere thins and rises, the ceiling is higher and sharper, the cavity loses less, and the peaks narrow slightly and move; by day the reverse. The resonance parameters therefore carry a record of the lower ionosphere itself, a region too high for balloons and too low for satellites, which is one of the least directly measured parts of the atmosphere.

The resonance was predicted long before it was seen, because seeing it is hard. The fields are tiny, about a picotesla in the magnetic field — a fifty-millionth of the Earth’s steady field — and buried under the electrical noise of power lines at 50 and 60 hertz. Schumann’s prediction in 1952 was confirmed clearly only in 1960, when Martin Balser and Charles Wagner measured spectra at a quiet site in the United States and found peaks near 8, 14 and 20 hertz.

Fifty strokes a second #

The cavity is struck constantly. Lightning flashes about fifty times a second over the whole Earth, mostly over the tropical landmasses of Africa, the Americas and Southeast Asia, and every stroke is a vertical current of tens of thousands of amperes lasting less than a millisecond, which drives every mode of the cavity at once.

The cavity is a poor resonator. Its quality factor — the number of cycles over which the stored energy falls by a factor of $e^{2\pi}$, or roughly, how many cycles a ring lasts — is between about four and six, compared with tens of thousands for a quartz crystal. After a single stroke the first tone’s amplitude falls by $e$ in about 160 milliseconds, little more than one cycle. As the noise a high Q moves out of the way argued from the other end of the scale, a low quality factor means a broad resonance and a quick forgetting, and the Earth’s cavity forgets almost as soon as it is struck.

Strokes arrive faster than the rings die. The record a magnetometer sees is a continuous, irregular hiss, with no individual stroke standing out from the others and no tone visible on the screen. The resonances are there only as a statistical fact: average the power spectrum over minutes and the peaks emerge, each about a fifth as wide as its own frequency. Only an unusually large stroke, a few times an hour, rises far enough above the background to show its own ring, and these events, called Q-bursts, are what allowed the cavity’s damping to be measured stroke by stroke.

The figure computes the spectrum a station would see from a single region of storms, using the measured frequencies and widths as inputs and the Legendre patterns for the geometry. Twenty degrees from the storms, every tone appears, broad and overlapping. Ninety degrees away the odd tones have gone, because the station sits on their node, and the spectrum is built from the second, fourth and sixth alone. The real Earth has storms in several places at once and their activity moves round the world with the afternoon, so at any station the heights of the peaks rise and fall through the day in a way that encodes where the lightning is.

A thermometer for the tropics #

That encoding made the resonance useful. Because the strokes that drive it are spread over the whole planet and the cavity integrates them all, the intensity of the Schumann resonances measures the global rate of lightning — something no network of local detectors could do until satellites were able to count flashes from orbit. In 1992 Earle Williams showed that the monthly intensity of the first tone tracked the temperature of the tropics, rising by a large fraction for each degree, because the rate of lightning in tropical thunderstorms is extraordinarily sensitive to the instability of the air that feeds them. A handful of magnetometers at quiet sites thus became a thermometer for the whole tropical belt.

The geometry helps further. Several stations at known positions, each seeing a different mixture of each storm region’s tones, can triangulate where the strokes were, and the ratio of the tones’ intensities at one station shifts as the main region of storms moves from Southeast Asia in the morning, by universal time, to Africa and then to the Americas.

The note that hooks at the cutoff #

The cavity’s tones belong to the guide’s lowest mode, the one without a cutoff. Above about two kilohertz the guide carries more. Higher modes, with the field reversing once or more across the gap between ground and ionosphere, each have a cutoff exactly like a pipe’s, at a frequency where the half-wavelength fits the height: $f = mc/2h$. For the night-time ionosphere at about 85 kilometres the first is 1.76 kilohertz.

A lightning stroke a few thousand kilometres away at night arrives on a receiver tuned to these frequencies as a click followed by a short descending whistle, a tweek. The click is the high frequencies, which travel almost at the speed of light. The whistle is the low end of the first higher mode, whose group velocity, $c\sqrt{1 - (f_c/f)^2}$, falls to nothing as the frequency approaches cutoff, so those frequencies arrive later and later and the sound hooks down to a steady tone at the cutoff frequency itself. Reading off where the hook flattens measures the height of the reflecting layer, at a time of night when it is hard to measure by any other means: the figure shows how ten kilometres of height moves the hook by nearly two hundred hertz. The group velocity collapsing at cutoff is exactly the behaviour of the pipe in the pipe that will not carry a low note, here spread out over three thousand kilometres so that it can be heard.

The same lightning produces a third sound, and the comparison is the surprising connection. The whistle that arrives sorted followed the part of a stroke’s radio pulse that escapes upward through the ionosphere, travels along the Earth’s magnetic field into the other hemisphere and comes back as a whistler, a descending note lasting a second. A whistler descends because the magnetised plasma it crosses is dispersive. A tweek descends because the guide it travels in is dispersive, with nothing in the air dispersive at all. Two notes from the same flash, gliding downward for two entirely different reasons: one because of what the medium is, the other because of the shape of the space it fills.

Where the picture stops #

The figures treat the cavity as a uniform spherical shell. The real one is not. The ionosphere is lower on the day side than the night side, so the cavity is lopsided, and its upper boundary is a gradual change of conductivity with two effective heights, one for the electric field and one for the magnetic, which is what modern models use to compute the slowing and the damping that the figures take from measurement. The Earth’s magnetic field makes the upper wall anisotropic and splits each resonance slightly. Ground conductivity varies between seawater, rock and ice. The spectra are computed from the measured frequencies and quality factors as inputs and from a single storm region as the source, when real stations receive the incoherent sum of storms spread over three continents; the absolute heights of the peaks are not meant to match any particular measurement. The tweek figure treats the higher mode as that of two perfect parallel plates, ignoring the curvature of the Earth and the ionosphere’s absorption, which is strong by day and is why tweeks are heard only at night.

The domain of the cavity’s lowest mode is roughly three to fifty hertz, from the frequency at which a wave is too long to fit even once round the world up to the range where the higher modes appear and the guide stops behaving as a resonator and becomes a carrier. That carrier is used: navigation systems such as Omega worked between ten and fourteen kilohertz, and submarine communications use frequencies from tens of hertz to tens of kilohertz, because they penetrate seawater, which shorter waves cannot.

Still open: other planets’ cavities #

Any planet with a conducting surface or interior and an ionosphere has a cavity of its own, and its tones measure both. The Huygens probe, descending through the atmosphere of Saturn’s moon Titan in 2005, recorded a signal near 36 hertz that has been interpreted as a Schumann-like resonance of Titan’s cavity — remarkable because Titan appears to have no lightning, which would require some other source to drive it, and because the lower wall may be not the surface but a conducting ocean under the ice. Whether the signal was a resonance at all, or an artefact of the probe’s own motion, is still argued. Mars, Venus and the giant planets would each ring at frequencies set by their size and the height and leakiness of their ionospheres, and measuring them would locate lightning, or its absence, from a single instrument.

On Earth, the open questions are about the ceiling and the climate. How the resonance parameters respond to changes in the ionosphere over the eleven-year solar cycle is measured but not fully modelled, and whether the global lightning rate the resonances record will rise as the tropics warm, by how much, and whether long series of Schumann data can show it before satellite records are long enough, is an active question.

The habit worth carrying away is to look for the guide that nobody built. Two conducting shells eighty kilometres apart carry a wave of any length, and closed round a sphere of radius R they resonate at $(c/2\pi R)\sqrt{n(n+1)}$ — 10.6 hertz for perfect walls, 7.8 as measured, because the leaky ionosphere slows the wave and gives it a quality factor of about five. Fifty strokes of lightning a second keep the planet ringing, the tones appear only in averaged spectra, and above two kilohertz the same guide’s higher modes have cutoffs that every night-time tweek hooks down to.