A storm weighed by a compass needle

A magnetometer at a low-latitude observatory records the horizontal component of the Earth’s field, about thirty thousand nanotesla, and on a quiet day it varies by a few tens of nanotesla with the daily heating of the upper atmosphere. On some days the record does something else. The field rises a little and abruptly, then over a few hours falls by fifty, a hundred, occasionally several hundred nanotesla below its quiet level, and then recovers over a day or several. That is a geomagnetic storm, and the fall is the signature of a current of charged particles circling the Earth several Earth radii up.

The current is made by the particles trapped on the closed field lines of the inner magnetosphere. They gyrate around the field, bounce between mirror points in the two hemispheres, and drift slowly around the planet — ions westward, electrons eastward — because the field is stronger on the side of each gyration nearer the Earth. Opposite charges drifting in opposite directions make a net westward current, a ring around the Earth, and its field opposes the Earth’s at the surface near the equator. During a storm the solar wind’s electric field drives fresh plasma from the tail into this region — the region where electrons are also carried inward and energised — the trapped population grows by orders of magnitude, and the ring current grows with it.

The remarkable thing is how directly the ground record measures the population. Michael Dessler and Eugene Parker in 1959, and Bengt Sckopke in 1966, showed that the field depression at the Earth’s centre is proportional to the total kinetic energy of the trapped particles and depends on nothing else about them.

Why only the energy matters #

The theorem is a statement about magnetised plasma in a dipole field, and the reason it ignores so much is worth tracing.

A trapped particle contributes to the current in three ways: its gradient drift, its curvature drift and its gyration, which in a population with a gradient of density produces a net magnetisation current. Each depends on the particle’s energy, its pitch angle and where it is. But all three can be written in terms of the plasma’s pressure — the parallel and perpendicular components — and the total magnetic moment of a current distribution in a dipole field, integrated over all space, comes out proportional to the integral of the pressure, which is the total kinetic energy. The field at the Earth’s centre from that current is then fixed by the dipole geometry. The distribution of particles in radius, in energy and in pitch angle cancels out.

So a hundred keV protons at four Earth radii and ten keV electrons at six, carrying the same total energy, depress the field at the centre by the same amount. The ground magnetometer cannot tell them apart, and does not need to: it measures the population’s energy in joules.

The theorem holds strictly at the Earth’s centre for a population symmetric about the axis in an undistorted dipole. A ground station is at the surface, a real ring current is lopsided — stronger on the dusk side during a storm’s main phase, because fresh particles enter there — and the outer field is not a dipole. The practical index averages several low-latitude stations around the globe to reduce the asymmetry, and the theorem is used as a calibration good to tens of per cent rather than as an identity. That is still a remarkable thing: a measurement of the energy content of a volume of space a hundred thousand kilometres across, made by compass needles at four observatories.

A deflection noticed before anyone knew what made it #

The effect was seen long before it was understood. In 1741 Anders Celsius and his assistant Olof Hiorter in Uppsala noticed that a compass needle they were watching for other reasons swung erratically on nights of bright aurora, and that colleagues in London saw it swing at the same time — the first evidence that whatever disturbed the needle was large, not local. In 1859 Richard Carrington watched a white-light flare on the Sun, and seventeen hours later the magnetometers at Kew recorded the largest disturbance they had ever seen, telegraph lines sparked and ran without batteries, and aurorae were seen in the Caribbean. The connection between the Sun and the needle was established in one event — and storms were soon found to crowd towards the maximum of the sunspot cycle — and its mechanism took another century.

The mechanism had to wait for two things: the discovery that the space around the Earth is filled with trapped particles, made by the first satellite Geiger counters in 1958, and the theory of how charged particles drift in a dipole field. The theorem followed within a year of the first. What makes the history striking is that the measurement it explains is the oldest continuous record of the space environment there is. Magnetic observatories have recorded the horizontal field every hour for more than a century and a half, so the energy of the ring current — a population nobody could sample directly until the space age — has a record going back to before the Carrington storm.

An index built from four observatories #

The storm index in use today is computed from four observatories at low latitude, spread round the globe so that between them they sample the ring current’s field from every side: Hermanus in South Africa, Kakioka in Japan, Honolulu, and San Juan in Puerto Rico. Each station’s record has its quiet-day variation removed — the daily cycle driven by solar heating of the ionosphere — and its slow drift with the Earth’s own changing field. What remains at each station is scaled by the cosine of its magnetic latitude, to refer it to the equator, and the four are averaged hour by hour.

The averaging is the practical answer to the asymmetry the theorem assumes away. During a storm’s main phase the depression is stronger on the dusk side of the Earth, where freshly injected ions drift; a single station would see the storm grow and shrink as the Earth turned it under the asymmetry. Four stations spaced in longitude average most of that out, leaving an approximation to the symmetric part the theorem describes. Higher-resolution indices from more stations, at one-minute cadence, show the asymmetric part separately, and it is often as large as the symmetric one while the storm is building.

The particles themselves #

The theorem says the ground record measures the trapped particles’ energy, and since the 1990s that energy has been measured directly. Spacecraft in elliptical orbits through the inner magnetosphere carry instruments that count ions and electrons by energy, species and direction; integrating the measured pressure over the volume of the ring current gives its energy content, and comparing it with the ground index tests the theorem.

The comparisons show what the theorem’s assumptions predict. The bulk of the storm-time ring current is ions with energies between ten and a few hundred kiloelectronvolts, at three to six Earth radii, carrying total energies of $10^{15}$ to $10^{16}$ joules in large storms — the range the theorem reads off the index. The agreement is within tens of per cent for moderate storms and degrades in the largest, where the tail current and the distortion of the field contribute to the ground index and the theorem’s dipole is a poor approximation. The electrons of the outer radiation belt, far more energetic individually, carry little of the energy: there are far fewer of them, and the ring current is an ion phenomenon.

Those spacecraft have also measured where the inner edge of the ring current sits, and how close to the Earth its particles reach during the largest storms — close enough, in extreme cases, to approach the altitudes where the inner belt dips lowest over the South Atlantic and satellites pass through it every day.

A storm built and drained #

A storm index rises and falls as energy flows into and out of the ring current, and the simplest description of that budget, written down by Burton, McPherron and Russell in 1975, is a single equation: the rate of change of the depression is an injection term, driven by the solar wind’s dawn–dusk electric field, minus a loss term proportional to the depression itself.

The injection is proportional to the dawn–dusk electric field of the solar wind, the product of its speed and the southward component of its magnetic field. That is the same quantity that sets the rate at which the dayside opens flux and drives the whole circulation of the magnetosphere; a storm is what happens when that circulation runs hard for hours, and the plasma it carries inward from the tail is energised by being compressed into the stronger field near the Earth. The loss term is a single lifetime, a crude summary of several processes, and the recovery it predicts is a simple exponential.

The model is crude and it works well enough to be used for forecasting. Given the solar wind measured by a spacecraft upstream of the Earth, it predicts the storm index an hour ahead to within a few tens of nanotesla for most storms. Its failures are informative: the largest storms recover in two stages rather than one, and that is the subject of a later section.

A storm the size of the worst on record #

The same equation with a stronger and longer drive shows how the largest storms arise.

Nothing exotic is needed. A coronal mass ejection arriving at a thousand kilometres a second with a field of twenty or thirty nanotesla pointing south for most of a day drives the injection term ten times harder than a typical storm, and the ring current grows until its losses, which grow with it, balance the drive. The size of the largest storms is therefore set by the most extreme solar wind the Sun produces, multiplied by how long its field points the wrong way. The Carrington event of 1859 — estimated from the few magnetometer records that survive at between eight hundred and well over a thousand nanotesla — needed a solar wind of that kind sustained for longer.

The energy in such a storm, from the theorem, is a few times $10^{16}$ joules: an hour of the solar wind’s kinetic power flowing past the magnetosphere’s cross-section. Most of the energy that enters the magnetosphere in a storm goes into heating the upper atmosphere at high latitudes and into the auroral particles; the ring current holds a fraction, and its loss over the following day is part of where the rest goes.

The boundary current, which the theorem does not count #

The storm index contains a second current that has nothing to do with the trapped particles. The magnetopause, where the Earth’s field stands off the solar wind, carries a current of its own — the Chapman–Ferraro current that confines the field — and its field at the Earth points the same way as the Earth’s, raising the field at the ground.

When a pressure pulse in the solar wind reaches the Earth, the boundary is pushed inward, its current strengthens, and the ground field jumps upward within minutes — the sudden commencement that so often begins a storm. The empirical correction removed from the storm index before it is read as ring-current energy goes as the square root of the pressure, with a coefficient of about sixteen nanotesla per root nanopascal. That square root is not an accident of fitting. It is the sixth-root standoff cubed, read at the ground.

The two currents therefore have to be separated before the theorem can be applied, and the separation uses the measured solar wind pressure. During the main phase of a storm, when the pressure is high and the ring current is growing, the boundary current can hide tens of nanotesla of the ring current’s depression; a storm index read without the correction understates the trapped energy.

A recovery in two steps #

The single lifetime in Burton’s equation fails most clearly in the recovery of large storms. The index recovers quickly for the first several hours, then slowly over the following days, and the two-step shape is a measurement of what the ring current is made of.

The main loss process is charge exchange. A trapped ion passing through the Earth’s extended hydrogen exosphere can take an electron from a hydrogen atom and become a fast neutral atom, which is no longer trapped by the field and leaves on a straight line. The rate depends on the ion’s species and energy through the charge-exchange cross-section, and for the ring current’s typical energies oxygen ions are lost several times faster than protons.

Oxygen in the ring current comes from the Earth’s own ionosphere, pulled up and accelerated along the field lines during storms. In quiet times the ring current is mostly protons from the solar wind and the ionosphere; in large storms oxygen can carry half its energy or more. The two-step recovery is then the oxygen being lost first and the protons afterwards, and the time at which the recovery slows says how much oxygen there was. The fast neutral atoms leaving by charge exchange can themselves be imaged from orbit, which turns the loss process into a way of seeing the ring current whole: a picture of the trapped population taken in the atoms it sheds.

What the theorem leaves out #

The theorem assumes a dipole and an axially symmetric ring current, and both fail during a storm’s main phase. The ring current is then strongly asymmetric, with most of its current closing through the ionosphere on the dusk side rather than circling the Earth, and its own field distorts the inner magnetosphere enough that the dipole is a poor description of the field it sits in. The tail current, flowing across the night side, also depresses the ground field, and for some storms it contributes as much to the index as the ring current. Each of these breaks the one-to-one mapping between index and energy, and modern estimates of the ring current’s energy combine the index with models of the field and with direct measurements of the particles by spacecraft that pass through the region.

There is also a self-consistency the simple picture misses. As the ring current grows, its field weakens the Earth’s field in the inner magnetosphere, which changes the drift paths of the particles, which changes the current. In the largest storms the depression is a substantial fraction of the field at the ring current’s location, and the particles and the field have to be solved together.

The same measurement elsewhere #

The theorem is not specific to the Earth. Any planet with a dipole field and a trapped plasma population has a ring current whose field at the planet’s centre measures its total energy, and the same arithmetic applies with the planet’s own field and radius. At Jupiter and Saturn the trapped plasma is dominated by material from moons rather than from the solar wind, and its energy content inflates their magnetospheres into discs; the ring currents there are the main reason those magnetospheres are larger than their dipole fields alone would make them.

On the Earth the measurement has a practical value beyond physics. The depression of the ground field during a storm is accompanied by rapid changes in the field, which induce electric fields in the ground and currents in long conductors: power lines, pipelines, submarine cables. The storm index, read as an energy, is a measure of how much the magnetosphere has stored and will release, and the rate of change of the field at the ground is what threatens grids. The storm of March 1989 failed a power system in nine seconds of rapidly changing field.

Still open: what the largest possible storm is #

The Carrington storm is the largest in the record, and its size is uncertain by a factor of two because the record is so thin. Whether larger storms are possible depends on how strong a southward field a coronal mass ejection can carry, how fast it can arrive, and whether the ring current’s growth saturates — whether its losses and its own distortion of the field cap the depression before the drive runs out. Burton’s equation has no saturation in it; physical models do, through the inflation of the inner magnetosphere and the shortening of particle lifetimes as the ring current moves inward. Estimates of a once-in-a-century storm range from the Carrington level to two or three times it, and the answer bears directly on how robust the grids that depend on the ground field’s quiet need to be.