A substorm timed by webers and volts

The magnetopause stands where the solar wind’s pressure matches the Earth’s field, about ten Earth radii upstream, and its position barely moves because it depends on the sixth root of the wind’s pressure. The auroral oval sits where field lines from the boundary reach the ground, and a calculation from the pressure balance alone places it within a few degrees of where it is seen. Both results describe the magnetosphere as a cavity. Neither says anything about how energy gets into it, and the answer to that is not a pressure but a circulation.

The solar wind carries the Sun’s magnetic field with it, frozen into the plasma as any field in a good conductor is. When that interplanetary field points southward, it is antiparallel to the Earth’s field at the nose of the magnetopause, and the two reconnect: a field line from the Earth and one from the wind join, and the joined line, with one foot on the Earth and the other in the wind, is dragged over the poles into the tail. In the tail the open lines reconnect again and return, closed, to the dayside. Dungey described that loop in 1961. Its elegance is that every part of it can be counted in one unit — magnetic flux, in webers — and every rate in it is a voltage, because a weber per second is a volt. The whole cycle becomes bookkeeping, and the bookkeeping predicts things that look like weather.

A rate that is a voltage #

The rate at which reconnection opens flux at the dayside magnetopause is the electric field of the solar wind, as seen from the Earth, integrated along the line where the reconnection happens. The wind’s motional electric field is its speed times its magnetic field — for a 450 km/s wind carrying 5 nanotesla, 2.25 millivolts per metre — and across the tens of thousands of kilometres of the magnetopause’s width that becomes tens of kilovolts. Only a fraction of the wind’s field lines that meet the magnetopause reconnect; the rest drape round it. The fraction is the least well determined number in the subject, and it is handled empirically.

The empirical law in the figure, fitted by Milan and colleagues to the rate at which the polar cap was measured to grow, makes the dayside voltage rise as the wind speed to the four-thirds power, in proportion to the transverse field, and as the sine of half the clock angle raised to the power four and a half. The clock angle is the direction of the field in the plane facing the Earth: zero pointing north, parallel to the Earth’s field at the nose, and 180 degrees pointing south, antiparallel. A 5 nanotesla field pointing due south opens flux at 57 kilovolts. The same field turned to within twenty degrees of north opens it at 22 volts — three thousand times less.

That steepness is the whole of the solar wind’s leverage. The wind’s pressure sets the size of the cavity, and the cavity’s size responds as a sixth root. The field’s clock angle sets whether energy enters at all, and it responds as a power of four and a half. A coronal mass ejection — and they come several times a day near the maximum of the solar cycle and a few times a week near its minimum — that arrives fast and dense with its field pointing north does almost nothing; a slower one with its field pointing south can drive a great storm. The clock angle is measured by a spacecraft an hour upstream, which is why geomagnetic forecasts are good for about an hour.

Open flux, and where it can be seen #

The flux that dayside reconnection opens does not vanish into the wind. Each open field line keeps one foot on the Earth, in the polar regions, and the set of those feet is the polar cap: the region of the ionosphere threaded by open field lines, above which the aurora does not form because there is no closed field line to carry particles into it. The polar cap’s area is the open flux divided by the strength of the field at the surface, and its edge is the poleward boundary of the auroral oval.

The relation runs both ways, and that is what makes the budget observable. A quiet magnetosphere holds about 0.4 gigawebers of open flux, and the edge of its polar cap sits near 77 degrees of magnetic latitude. When the open flux has grown to 0.9 GWb the edge has moved to 70 degrees; in a great storm, with more than a gigaweber open, it reaches 66 degrees and below. An ultraviolet camera on a spacecraft looking down at the pole, or a chain of ground magnetometers recording where the auroral currents flow, therefore measures the open flux directly — the central quantity of the Dungey cycle — without any instrument in the solar wind at all. The oval’s expansions and contractions over hours are the cycle’s loading and unloading, drawn on the sky.

Real polar caps are not circles centred on the magnetic pole. They are displaced towards the night side, bulge where the tail’s closure is active, and respond to the east–west component of the interplanetary field by shifting towards dawn or dusk. The circular cap in the figure gets the latitude right to a degree or two, which is the precision with which the oval’s position is a flux measurement.

Loading and unloading #

The polar cap’s area changes as the difference between two rates: flux opened on the dayside, and flux closed in the tail. If they balanced at every instant, the cap would hold a fixed flux and the magnetosphere would convect steadily. They do not balance. Dayside reconnection responds within minutes to the field arriving at the nose; tail reconnection responds to the state of the tail, and it turns on abruptly once the tail has been loaded enough.

The simplest version of that behaviour is a sawtooth. Under steady southward field, dayside reconnection adds flux to the polar cap at a constant 57 kilovolts. The cap grows, the oval moves equatorward, and open flux piles into the tail lobes, stretching the tail’s field and storing magnetic energy. At some threshold — about 0.9 gigawebers in the figure — the stretched tail becomes unstable, reconnection in the near tail switches on explosively, and flux is closed faster than it is opened. That is a substorm: the aurora on the night side brightens suddenly, surges poleward and westward, and currents of a million amperes flow along field lines into the ionosphere. Within an hour the cap has shrunk back to its quiet size, and the loading begins again.

Loading half a gigaweber at 57 kilovolts takes 2.4 hours; in the figure each whole cycle takes 3.9. Substorms under steady southward driving are observed to recur every two to four hours. The timing of the most dramatic event in the near-Earth environment — a release of energy visible across a hemisphere — is, to first order, a flux difference divided by a voltage.

A time constant set by arithmetic #

The same division gives the time to load a substorm at any rate of driving.

At twenty kilovolts — a weakly southward field — loading takes seven hours, longer than the interplanetary field usually keeps one orientation, so the field turns northward before the tail reaches its threshold and the loaded flux leaks away through slow reconnection without a substorm. At fifty kilovolts it takes 2.8 hours, and substorms recur at about that interval. At a hundred and fifty kilovolts it takes under an hour, and the magnetosphere cannot unload in discrete bursts fast enough; it settles either into a state of steady convection, with the tail closing flux as fast as the dayside opens it, or into a storm, in which the ring current of trapped particles grows so large that the whole field at the equator is measurably depressed.

The three regimes of magnetospheric behaviour — quiet, substorms, storms — are thus distinguished by one number, the dayside voltage, against one flux, the capacity of the tail. What sets that capacity — why onset happens near 0.9 gigawebers rather than 0.6 or 1.2 — is less well understood, and the threshold itself varies with the wind’s pressure and the season. But the time constant of the cycle, given the threshold, is not a model. It is conservation of flux.

A potential the ionosphere will not follow #

The voltage that drives the cycle also appears at the ground, and there it runs into a limit. The open field lines move across the polar cap from noon to midnight as the wind drags them tailward, and their motion is an electric field across the cap — the polar cap potential, typically 30 to 100 kilovolts, measured by radars that track the drift of the ionospheric plasma. For weak driving the potential follows the solar wind’s electric field in proportion. For strong driving it does not.

The potential across the polar cap drives currents through the conducting ionosphere — the Pedersen currents — and the currents close along field lines back to the magnetopause. There they make magnetic fields of their own, and those fields weaken the Earth’s field at the nose of the magnetopause where the reconnection happens. The stronger the driving, the larger the current, the more the reconnection site is disturbed, and the less of the solar wind’s electric field reaches the ionosphere. The model in the figure, due to Hill and developed by Siscoe and colleagues, combines a linear response to the wind with a ceiling set by the ionosphere’s conductance: about 200 kilovolts for a conductance of 10 siemens at typical wind pressure. The measured potentials do saturate, at 150 to 250 kilovolts, in the largest storms, even when the solar wind’s electric field is ten times what would be needed to reach that value linearly.

The ceiling depends on the conductance, and the conductance depends on sunlight. The ionosphere’s conductivity is set by how much of it is ionised, and in the summer hemisphere, with the polar cap sunlit around the clock, it is several times higher than in the winter hemisphere in darkness. So the same solar wind imposes a different potential on the two hemispheres, and the seasonal and daily variation of geomagnetic activity carries a contribution from the ionosphere’s illumination as well as from the geometry of the Earth’s tilted dipole against the interplanetary field. A layer a hundred kilometres up, lit or unlit, limits the coupling of the solar wind to a magnetosphere a hundred thousand kilometres across.

Where the energy goes #

The flux budget is also an energy budget. Open flux in the tail lobes is stored magnetic energy — a few times $10^{15}$ joules at substorm onset — and the substorm releases a large part of it in about an hour. Some goes down the field lines into the auroral ionosphere, heating it and lighting the aurora; some goes into accelerating particles and injecting them earthward, where they drift around the Earth and join the trapped population whose three clocks keep them in the belts; some is carried away down the tail in a plasmoid of reconnected flux. The heating of the thermosphere during storms raises its density at satellite altitudes by factors of several, increasing the drag on low-orbiting spacecraft — a direct line from the clock angle of a field measured a million kilometres upstream to the decay of an orbit.

The particles injected during substorms and storms are the source of the energetic electrons that damage spacecraft electronics, and the inner belt, drained over the South Atlantic, and the outer belt, which fills and empties with each storm, are the reservoirs where they collect. The currents that flow along field lines during a substorm induce electric fields in the ground and in long conductors on it, and the largest storms have tripped power grids. All of it is driven by a voltage of tens to hundreds of kilovolts, set at the dayside magnetopause, by the orientation of a few nanotesla of field.

How the voltage is measured #

Each term in the budget is observed by a different instrument, and none of them is a voltmeter. The polar cap’s area comes from images: ultraviolet cameras on spacecraft in high polar orbits photograph the whole auroral oval at once every couple of minutes, and the area inside its poleward edge, multiplied by the polar field, is the open flux. The rate of change of that area is the difference between the dayside and nightside voltages. The polar cap potential comes from radars. A network of high-frequency radars around both poles, each looking across a thousand kilometres of the ionosphere, measures the line-of-sight velocity of plasma irregularities by the Doppler shift of the echoes they return, and the velocities combined give the flow pattern across the whole cap — two convection cells, anticlockwise at dusk and clockwise at dawn — whose electric potential is found by integrating the flow.

The dayside voltage alone cannot be measured directly. It is inferred as the rate of change of the open flux plus the nightside rate, which is estimated separately from the motion of the oval’s poleward edge at midnight. That is the circle the empirical coupling law closes: fitted to how the polar cap grows when the tail is quiet, it gives the dayside voltage as a function of the upstream wind, and it can then be used to forecast the growth of the cap when the tail is not quiet. The consistency of the whole — images, radars and upstream measurements agreeing on one budget — is the evidence that the flux picture is right.

What the budget leaves out #

The Dungey cycle as drawn here has one opening site and one closing site, each switched on or off. The real magnetosphere reconnects at many places at once; the dayside rate responds to the east–west component of the interplanetary field as well as the north–south one, twisting the whole polar cap; and northward field drives reconnection too, poleward of the cusps, which closes open flux from the dayside and creates a much weaker circulation in the opposite sense. The loading–unloading sawtooth is the cleanest of several modes the magnetosphere shows under steady driving: steady convection, sawtooth events with periodic injections every few hours, and isolated substorms are all observed, and what selects among them is not fully understood.

The empirical coupling law is also a fit. It was calibrated on the observed rate of change of the polar cap’s area, and different fits to different data give exponents on the clock angle between two and six. The dependence is steep in all of them, which is the robust result; its exact form is uncertain enough that forecasts of the dayside rate from upstream data carry errors of tens of per cent. And the saturation model’s ceiling depends on a conductance that varies across the polar cap and changes during the storm itself, as auroral precipitation ionises the ionosphere and raises its conductance exactly where the currents flow.

The same circuit elsewhere #

Other planets show how particular the Earth’s case is. Mercury’s magnetosphere is so small that its Dungey cycle turns over in minutes rather than hours, and its substorms have been observed by a spacecraft in orbit to load and unload about every two minutes — the same flux arithmetic with a hundredfold smaller capacity. Jupiter’s magnetosphere is driven mainly by the planet’s own rotation rather than by the wind, and its aurora is powered by the corotation of plasma from the volcanic moon Io rather than by reconnection with the solar wind; the Dungey cycle is present there but minor. The Earth sits between the two, a magnetosphere large enough to store energy for hours and small enough to be driven by the solar wind rather than by itself.

Still open: what sets the threshold #

The loading rate is measured and the unloading is observed; what triggers the switch from one to the other is still argued over. The tail becomes unstable at some combination of stretching and stored flux, and whether the instability begins with reconnection twenty Earth radii down the tail or with a disruption of the current sheet ten radii closer, which then triggers the reconnection, has been debated for thirty years and investigated by a fleet of five spacecraft placed along the tail for the purpose. The answer determines what the threshold flux is and why it varies from one substorm to the next. The rest of the cycle is flux conservation, and it can be checked every night that the aurora moves: the oval’s latitude gives the open flux, the upstream field gives the voltage, and their ratio gives, to within an hour, the time until the next substorm.