The Sun reading open

Coronal Loop Cycles

A loop of hot plasma above the Sun, heated near its feet, that fills, overcools, rains back down and fills again, hour after hour.

The bright arches above the Sun are magnetic field lines lit up by plasma at a million degrees and more. The plasma can only move along its loop. Heat a loop mostly near its two feet, and it has no steady state. Heated plasma boils up from the surface and fills the loop. The fuller the loop, the faster it radiates, until the top loses heat faster than heat can reach it. The cooling runs away, the plasma condenses into cool, dense knots, and they slide back down a leg as coronal rain. The drained loop heats up, and the cycle starts again. With nowhere steady to settle, the loop cycles. Solar physicists call this thermal non-equilibrium, and telescopes see it as loops that brighten and dim over hours.

One Loop, Seen From the Side

100 Mm of loop, 32 Mm tall · color is temperature · blue dots are coronal rain

The Same Loop in State Space

temperature against density at the top of the loop · the last 8 hours, newest darkest

temperature at the top of the loop (MK)—

density at the top (10¹⁵ particles per m³)—

coolest plasma more than 10 Mm up, and coronal rain—

hours between rain events—

time (hours)—

starting

A real calculation along one loop, far simpler than the corona. It solves the one-dimensional equations of a hot, ionized gas along a fixed magnetic loop 100 Mm long, on 880 cells 125 km wide, with gravity, heat conduction along the field and radiation. The loop’s shape and width are fixed, and the magnetic field does nothing else. The heating is prescribed, because what really heats the corona is still an open question. The left foot is heated 1.3 times more than the right. The thin layer between the surface and the corona is too steep for this grid, so the model uses a standard correction that broadens it (transition-region adaptive conduction). The bottom 2 Mm of each leg is held near its starting state, so the surface acts as a deep reservoir. At the default setting, a grid twice as fine gives rain every 0.98 hours instead of 1.12, so the cycle here is about 14% slow. This short loop cycles faster than the long loops seen to pulse over several hours. Time is the solver’s clock, at most 15 model minutes each second.

Two Kinds of Loop

what the picture shows, and where the attractor-like shape is

The arch in the top picture is a loop in space: a magnetic field line that rises out of one patch of the surface and back into another, made visible by the hot plasma trapped along it. Its shape comes from the field, not from any cycle, and a steady loop has the same arch.

The curve below it is a loop in state space: the temperature and the density at the top of the loop, plotted against each other as they change. When the loop cycles, the point comes back around nearly the same closed curve every time, a limit cycle. That closed curve, not the arch, is the attractor-like shape.

The Lorenz butterfly is also a curve in state space, drawn from three numbers of a convecting fluid, and the fluid looks nothing like a butterfly. Lorenz’s curve never closes and never repeats; this one closes. The arch is a picture of neither.

Why the Reading Is Open

an authored reading, not a measurement

Three things, classified separately. The program that runs the panel is an attractlet model under § 7.4, as on the Lorenz page. The plasma in the modeled loop is a second thing. The Sun’s real loops are a third.

Name the boundary. Take X to be the plasma in one loop, with the heating and the magnetic loop that holds it supplied from outside X.

The case for attractlet. Same test as the Weather page: the cycle carries heat delivered to it and releases none of its own. Every hour of it runs on heating the loop does not produce. Press Stop the heating: within about 45 minutes the top falls from 1.2 MK to 45,000 K and the loop empties to under 1% of its density. That is the fifth test of § 7.4, a configuration that “relaxes to an inert or passive state when uncoupled.”

What is left open. The flows, the runaway cooling and the rain are the plasma’s own doing, not something the heating arranges, which is the same shape as the convection cells: the machinery inside the boundary, the supply outside it. The site has not settled which of those two facts governs. And the heating itself is thought to come from the magnetic field, braided and shaken by motions at the surface; widen X to include those and the reading has to be redone, which a model that prescribes the heating cannot do.

The Shape of the Basin

measured on this model

Two settings, two outcomes. With the heating concentrated near the feet, the loop cycles and rains: at a heating scale length of 5 Mm the rain comes every 1.12 hours, at 7 Mm every 1.86, at 9 Mm every 2.89. Spread the heating out and the rain stops: from 12 Mm up the loop settles into a nearly steady glow, its top at 1.9 MK at 12 Mm and 2.4 MK at 30 Mm, wandering by a few percent.

In between, a cycle with no rain. At 10 Mm the top swings between 1.16 and 1.40 MK every 4.4 hours and never gets cold enough to condense. Run into that setting from a cycling loop or from a steady one and the model ends in the same state, the two runs agreeing to within 1%.

The same cycle, whatever the start. Starting the corona at 3 MK instead of 1 MK gives the same rain interval, 1.12 hours, over the same range of temperature and density. Press Drain the loop to strip 70% of the plasma: rain returns about 1.4 hours later, and the interval is back to 1.12 hours within about ten hours. On the steady setting, 20 Mm, the same draining is undone in about an hour, the density back within a few percent of where it was.

Uncouple it and it stops. Press Stop the heating: the top cools from 1.2 MK to 45,000 K within 45 minutes, the loop empties to under 1% of its density, and nothing starts it again.

What Telescopes See

the real loops this model stands in for

Long-period pulsations lasting several hours are common in coronal loops, and they have been traced to cycles of evaporation and condensation driven by steady heating near the feet (Froment et al. 2015). Coronal rain, cool knots falling along loops, is seen in many active regions.

In models, the cycles need heating concentrated low in the legs, with the heating at the top of the loop less than about a tenth of the heating at the feet, and not too lopsided between the two feet. Cycle periods run from about 2 to 15 hours, depending on the heating and the loop’s shape, and the cycles are generally limit cycles (Klimchuk 2019). This loop is shorter than most of those studied, and its cycle is faster.

Froment, C., et al. (2015). Evidence for evaporation-incomplete condensation cycles in warm solar coronal loops. The Astrophysical Journal, 807, 158.
Klimchuk, J. A. (2019). The distinction between thermal nonequilibrium and thermal instability. Solar Physics, 294, 173.
Klimchuk, J. A., Patsourakos, S., & Cargill, P. J. (2008). Highly efficient modeling of dynamic coronal loops. The Astrophysical Journal, 682, 1351.
Johnston, C. D., et al. (2020). Modelling the solar transition region using an adaptive conduction method. Astronomy & Astrophysics, 635, A168.

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