Braided Magnetic Flux Ropes Are Found at Both Human and Light Year Scales

Investigating solar corona structures has led Paul Bellan, Caltech professor of applied physics, and his former graduate student Yang Zhang (PhD '24) to discover a new equilibrium state of the magnetic field and its associated plasma. The solar corona, the outermost part of the Sun's atmosphere, is much less dense than the Sun's surface but is a million times hotter. The corona is composed of strong magnetic fields confining plasma, a gaseous soup of charged particles (electrons and ions). The new equilibrium, called a double helix, applies not only to the solar corona but also to much larger astrophysical configurations such as the Double Helix Nebula located near the center of the Milky Way galaxy.

Solar corona structures such as flares often have the form of magnetic flux ropes: twisted tubes of plasma-containing magnetic fields. Such a rope can be visualized as a plasma-filled garden hose with a stripe wrapped around it in a helical pattern. An electric current flows along the length of the hose, and the helical stripe corresponds to the twisted magnetic field. Because it is charged, plasma conducts electric currents and is attached, or "frozen," into magnetic fields.

Magnetic flux ropes occur in a variety of situations ranging from the human scale—say, a laboratory experiment—to the absolutely huge: solar flares that are few hundred thousand kilometers long. Astrophysical structures with magnetic flux ropes can also span hundreds or even thousands of light-years.

In a large laboratory vacuum chamber, Bellan and Zhang (now a NASA Jack Eddy postdoctoral fellow at Princeton) produced solar flare replicas measuring between 10 and 50 centimeters long. "We have two electrodes inside the vacuum chamber, which has coils producing a magnetic field spanning the electrodes. Then we apply high voltage across the electrodes to ionize initially neutral gas to form a plasma," Yang explains. "The resulting magnetized plasma configuration automatically forms a braided structure."

This braided structure consists of two flux ropes that wrap around one another to form a double helix structure. In the experiments, this double helix was observed to be in a stable equilibrium—in other words, it holds its structure without tending to twist tighter or untwist. In a new paper, Zhang and Bellan demonstrate that the stable equilibrium of these double-helix flux ropes can be understood, analyzed, and predicted accurately in mathematical terms.

Though the properties of single flux ropes are well known, braided flux ropes were not well understood—especially those configurations in which the electric currents flow in the same direction along both of the braided strands. Scientists have modeled the other possible situation—where currents flow in one direction in one flux rope and in the opposite direction in the other—but this scenario is thought to be unlikely in nature.

The same-current configuration is especially important because it would be susceptible to kinking and expansion driven by hoop forces—phenomena observed both in braided solar structures and in laboratory experiments. Such kinking and expansion should not occur when current flows in opposite directions in the braided strands (a "no-net-current" state).

Previously, scientists assumed that braided flux ropes where the strands have current flowing in the same direction would always merge, because parallel currents magnetically attract each another. However, in 2010, researchers at Los Alamos National Laboratory found that such flux ropes instead bounce off one another as they come closer together.

"There was clearly something more complicated going on when the flux ropes are braided, and now we have shown what that is," Bellan says. "If you have electrical currents flowing along two helical wires that wrap around each other to form a braided structure, as seen in our lab, the components of the two currents flowing along the length of the two wires are parallel and attract, but the components of the two currents flowing in the wrapping direction are anti-parallel and repel. This combination of both attractive and repulsive forces means there will be a critical helical angle at which these opposing forces balance, producing an equilibrium. If the helical flux ropes twist tighter, there will be too much magnetic repulsion; if they twist more loosely, there will be too much magnetic attraction. At the critical angle of twist, the helical structure arrives at its lowest energy state, or equilibrium."

The next task was to create a mathematical model of this behavior—something not previously done. Using what Bellan describes as "brute force mathematics," Zhang created a set of equations that could apply to multiple flux tubes in various configurations, including braided ropes, and showed there is indeed a state at which the attractive and repulsive forces balance each other, creating an equilibrium. "And as an unexpected bonus, Yang can calculate the magnetic fields inside and outside the flux ropes, and the current and pressure inside them," Bellan says, "giving us a full picture of the behavior of these braided structures."

Zhang tested his mathematical model against the Double Helix Nebula, an astrophysical plasma formation located 25,000 light-years from Earth that covers a 70 light-year swath of space, to see if the equations could describe a large model as well as it did the structures he and Bellan created in the lab. "What was rather amazing about this calculation is that Yang didn't really need to know much about the nebula," Bellan says. "Just knowing the diameter of the strands and the periodicity of the twist, numbers that can be observed astronomically, Yang was able to predict the angle of twist that yielded an equilibrium structure, and that was consistent with observations of this nebula. One of the most exciting aspects of this research is that magnetohydrodynamics, the theory of magnetized plasmas, turns out to be fantastically scalable. When I first started looking into this, I thought the phenomena of magnetic structures at different scales were qualitatively similar, but because their sizes are so different, they couldn't be described by the same equations. It turns out that this is not so. What we see in lab experiments and in solar and astrophysical observations are governed by the same equations."

The paper, titled "Magnetic Double Helix," was published in Physical Review Letters. The work was funded by the National Science Foundation.