Paul M Bellan. American Scientist. Volume 88, Issue 2. Mar/Apr 2000.
When the shadow of the moon swept across Europe, the Middle East and India last August 11th, for a few minutes millions of people were able to gaze up in the sky and observe the surface of the sun while it was in a state of high excitement. It was, of course, mere happenstance that this solar eclipse took place just as the sun was nearing a climax in its 11-year cycle of activity. (The action is expected to peak this summer.) But this coincidence made the play of red and white lights behind the blackened moon appear especially spectacular. No wonder that many American astronomy enthusiasts traveled across the Atlantic just to view this sight. They were, in a sense, maintaining a long-standing scientific tradition.
Indeed, the astronomical literature of the 19th century is replete with tales of expeditions to observe solar eclipses in exotic locales-Peru in 1858, Malacca in 1868 and Japan in 1887, to name a few far-flung destinations. One of the most striking and unexpected results of these forays was the discovery of a phenomenon that proved quite apparent this past August: huge, luminous arches jutting from the solar surface. Eventual
Paul M. Bellan received his bachelor’s degree from the University of Manitoba in 1970. He continued graduate studies in physics at Princeton University, where he received a doctorate in 1976. The following year Bellan joined the Department of Applied Physics at the California Institute of Technology. Apart from simulating solar prominences, his research interests include mapping of the velocity of plasma ions with laserinduced fluorescence, the use of spheromaks for magnetic-confinement fusion and the study of Alfven waves in magnetized plasmas. Address: California Institute of Technology, Mail Code 128-95, Pasadena, CA 91125. Internet: email@example.com ly, astronomers developed instruments to observe these features (which they called prominences) when the sun was not eclipsed, and, most recently, spacecraft have returned stunning images of the ongoing solar fireworks.
Prominences consist of plasma (gas composed of free electrons and posilively charged ions) and thus are distantly related to the glowing matter in neon signs, lightning and auroras. Solar prominences usually evolve slowlyover days or weeks-but occasionally they burst apart in just a few minutes. Once every few years, such an eruption is strong enough to affect people rather profoundly: The huge cloud of magnetized plasma ejected by an erupting prominence can travel through interplanetary space, reach the neighborhood of the earth and destroy electronic circuits in orbiting satellites or cause large, disruptive currents to be induced in terrestrial power grids. (Such an event blacked out the entire province of Quebec on March 13th, 1989, for example.) For this reason, the study of the solar surface is of more than academic interest.
Why prominences sometimes erupt and why they commonly assume strange yet beautiful shapes have been mysteries for more than a century. About six years ago, it struck me that I might help to answer these questions by applying knowledge I had gained while working with plasmas in the laboratory. Along with colleagues at about a dozen other institutions, I was studying spheromaks, magnetic “bottles” that can hold high-temperature plasma by virtue of their geometry. Spheromaks offer a particularly attractive way of containing the very hot plasmas required for controlled thermonuclear fusion. Although no spheromak has yet held a plasma that is as hot or well confined as those produced in tokamaks (the most highly developed machines yet built for attempting controlled nuclear fusion), spheromaks offer the possibility of generating the necessary conditions for fusion using hardware that is much less expensive. (Research tokamaks typically cost a good fraction of a billion dollars, and future tokamak reactors will likely require several billion to build.) The potential economy arises because naturally occurring forces cause the spheromak plasma to organize itself spontaneously into the desired configuration.
Self-organization is also a critical feature of solar prominences. So even though I was not well versed in the particulars of solar plasmas, I was familiar with their underlying principles. After all, spheromaks and prominences both involve the interplay between magnetic fields and electric currents in a conductive fluid, effects described by a branch of physics called magnetohydrodynamics.
Indeed, the same fundamental equations govern laboratory plasmas and their astrophysical counterparts. But distinct jargon and research methods apply to each, which hampers communication between the two camps of physicists studying these different types of plasmas. I had first become aware of the possibility of using laboratory methods to simulate solar phenomena in 1991, on hearing a highly speculative talk given by Teruo Tamano (then at General Atomics in San Diego, now at the University of Tskuba in Japan) at a workshop in the beautiful village of Varenna on the shores of Lake Como in Italy. Yet because the experimental apparatus he envisioned seemed awkwardly large and very pricey, no one had followed through on his proposal. I paid only modest attention to Tamano’s words at the time, but they must have been on my mind in 1994, when I realized that the techniques I had developed for creating rather compact spheromaks might provide a practical method for simulating solar prominences. Graduate student Freddy Hansen joined my group in 1995 and started working with me on turning this idea into reality.
The plasmas we now create in the laboratory are more than a billion times smaller and evolve more than a billion times faster than real prominences. Yet they exhibit many of the features that solar astronomers routinely record-including expanding arches, twisting fibrils and S-shaped bends. To understand how we reproduce the surface of the sun in miniature requires some knowledge of magnetohydrodynamics. Fortunately, sufficient comprehension can be gained without having to master Maxwell’s equations or the theorems of vector calculus. All that is needed are a few helpful analogies and a little imagination.
Although the plasma found in space is extremely tenuous, it is an excellent electrical conductor. In fact, the electrical conductivity of plasma compares with that of metals and, contrary to intuition, does not depend on density. How is it that a thin, wispy plasma can conduct electricity as well as a thick, concentrated one? The answer is easy to understand. As density increases, the frequency of drag-inducing collisions between particles rises, which alone would lower conductivity. But the number of current-carrying charges also mounts with increasing density, which boosts the ability of the material to conduct electricity. Acting together, the two effects cancel, leaving the conductivity of a plasma approximately constant for a wide range of densities.
The plasmas used in fusion research and those that form prominences high above the sun are both many orders of magnitude less dense than the air at sea level. Yet these plasmas conduct electricity extremely well, which means that the principles of magnetohydrodynamics apply to them.
One of the fundamentals of electromagnetism, Faraday’s law of induction, states that when a conductor moves across magnetic-field lines, an electric field is established in the material. (This phenomenon provides the basis for electric generators.) But the application of Faraday’s law to a plasma seems to contradict another electromagnetic principle, namely that charged particles inside a perfect conductor always move in just such a way as to cancel out any internal electric field. The only way to satisfy both Faraday’s law and the property of being a perfect conductor (that is, that no internal electric field can arise) is for the magnetic-field lines to “freeze” to the plasma-like spaghetti noodles caught in cold honey. This way the frozen magnetic-field lines never cut across the conductor and so do not generate an electric field. Thus one can expect that a plasma and any magnetic-field lines that permeate it should move together.
What makes them move at all? The answer lies in another fundamental tenet of electromagnetism–that parallel currents attract one another. Plasma physicists have dubbed this mutual attraction the “pinch effect,” because a distributed electric current can be thought of as the flow of charges through a bundle of parallel wires that would tend to pinch together, shrinking the cross section of the overall conduit. It follows then that antiparallel currents repel. Physicists call this repulsion the “hoop force,” because the segments on opposite sides of a loop of current push away front each other, tending to increase the diameter of a current-carrying “hoop.”
To understand plasmas further, one also needs to remember that the charged particles they contain are pushed in a direction that is at right angles to the magnetic field and to their own velocity vector. Thus electrons and ions tend to orbit around magnetic-field lines–something called cyclotron motion because this sort of particle trajectory was used in early atom smashers called cyclotrons. Although the magnetic field influences the movement of particles perpendicular to the lines of force, the parallel component of their motion is unaffected. Thus, in addition to making cyclotron orbits, charged particles also move freely along field lines. When the cyclotron circling is combined with the free drift along the field lines, the net result is helical motion.
If the ambient magnetic field is very intense, the radius of the cyclotron orbits becomes microscopic, and the helical motion effectively becomes just a streaming along the field lines. The magnetic fields in solar prominences, spheromaks and our experimental simulations are sufficiently strong that their lines of force channel charged particles (and hence electric currents) in just this fashion.
How, one might reasonably ask, do solar magnetic fields arise in the first place? Specialists remain somewhat sketchy about the answer, but they believe that the fundamental source is the solar dynamo–the churning convection within the sun that acts as a kind of electric generator. The dynamo action normally takes place well below the surface. But from time to time some of this electromagnetic commotion bubbles up, giving rise to prominences.
The character and behavior of solar prominences hinge on a subtle, yet important distinction regarding the location of the currents that generate the magnetic fields. In physicists’ parlance, the magnetic field in the region exterior to the currents that produce it (for example, outside the wires of an electromagnet) is called a “vacuum” magnetic field, because such a field can exist in a vacuum. It is fundamentally different from the magnetic field internal to a region of current flow (say, inside the wires of the electromagnet).
Like a rope slung between two supports, a vacuum magnetic field has a shape that is the lowest energy state. Deviations from this configuration are possible but require currents in what was originally the vacuum region. Magnetic-field lines deformed in this way have a greater energy and, in general, have a tendency to return to their vacuum geometry. So in a sense, deformed magnetic-field lines are like stretched rubber bands: One can imagine an elastic tension acting along the lines, forcing them to snap back to their vacuum shape. But this is not the whole story, because if a current flows along a stretched, curved field line, the hoop force generated by this current opposes the tension. The balance between hoop force and magnetic tension allows some interesting, higher-energy equilibria to exist in magnetized plasmas.
To understand how these stable, non-vacuum states come about, suppose a plasma is immersed in a strong vacuum magnetic field. If the plasma has only small electric currents flowing within it, the magnetic fields they produce are negligible compared to the background vacuum field, which acts as a rigid guide for the flow of current.
current flowing along vacuum magnetic field If, however, the currents become large enough to create a local magnetic field that rivals the vacuum field, the situation becomes more complicated.
To visualize what happens, imagine a small electric current flowing along one of the lines of force of a vacuum field. As the current grows in size, it begins to generate an appreciable local field of its own, which encircles the original current path. The combination of local and vacuum fields thus creates a helical total field. Because electric charges stream along this spiraling total field, the current also becomes helical. This situation, where a helical current follows a helical magnetic field, is called a “force-free” configuration. Although the magnetic field is not in the lowest energy state possible (the vacuum state), it is in a local minimum energy state and so represents a stable equilibrium. If for some reason the topology of the magnetic field does not start out in this equilibrium condition, instabilities generally develop that quickly move the system into such a state.
Helical geometry thus turns out to be important in the magnetohydrodynamics of plasmas. But because of the complexities involved in modeling all three spatial dimensions, until recently most studies of solar prominences adopted two-dimensional simplifications (that is, they assumed that physical properties did not vary along the third dimension). The people doing this research now recognize that the workings of prominences depend very much on the three-dimensional effects of helicity and that two-dimensional models miss the essence of this physics.
This realization is perhaps somewhat late in coming: In the 1950s, Lodewijk Woltjer, then at the University of Chicago, now at the Observatoire de HauteProvence in France, had noted the importance of such helical topology to astrophysical plasmas. But his discussion of the topic was quite abstract, and it was not applied to any specific phenomena. So Woltjer’s point of view received only modest attention until the 1970s, when Bryan Taylor of the Culham Laboratory in England invoked and extended it to explain a decade-old mystery concerning a device constructed for fusion research called the “ZETA reversed field pinch.” (ZETA is a curious acronym standing for Zero-Energy Thermonuclear Assembly.) Taylor argued that it was energetically favorable for the plasma inside the ZETA machine to arrange itself into a force-free helical state and that any kind of turbulent, unstable motion would, in fact, bring about this self-organization.
The beauty of the Woltjer-Taylor theory is that the end state of the plasma depends only on a few simple conditions prevailing at the outset. It is not necessary to follow the complicated dynamics that take place along the way to predict the final “relaxed” state. This simplification helped physicists understand magnetized plasmas that evolve into helical toroids-like a spring bent fully around so that the two ends connect. Spheromaks, which are based on this simple configuration, have been under study since the 1980s.
In 1984 Jean Heyvaerts, then at the Observatoire de Meudon near Paris and now at the Observatoire Astronomnique de Strasbourg, and Eric Priest at the University of St. Andrews in Scotland postulated that solar prominences are also a manifestation of this tendency for current-carrying, magnetically dominated plasmas to settle into a helical, force-free state. According to this concept, prominences that are highly twisted must have substantial currents coursing through them.
But only so much current can flow through the body of a prominence before force-free equilibrium is lost. This limit arises because the current generates hoop forces, which tend to make the field bulge out from its original vacuum configuration. There thus becomes a contest between the hoop force arising from local currents and the effective tension of the field lines generated by distant currents. The rubber-bandlike field lines can resist only so much bulging–just as overblown balloons can resist only so much gas pressure–before things break up.
The strong correlation observed between the geometry and the stability of solar prominences fits this picture well. Untwisted or slightly twisted prominences can be stable for weeks on end, whereas extremely twisted ones are highly volatile and can burst apart in minutes. When such a prominence erupts, it sheds its excess helicity and reforms itself into a less twisted state.
Meanwhile, Back at the Lab
Five years ago, I set out with Hansen, my graduate student, to reproduce such intriguing behavior in the laboratory. To mimic a solar prominence, an experimental plasma has to satisfy several conditions: It must support large magnetic fields and electric currents, so that the magnetic forces overwhelm other effects, such as pressure and gravity. The plasma should conduct electricity well, so that the magnetic field remains frozen to the plasma, as it does on the sun. To make our simulations reasonably realistic, we set up a magnetic field that is initially archshaped, and we arrange for driving a large electric current along it. Finally, we allow plenty of room for the plasma to expand, so as to approximate the ample empty space above the solar surface.
We generate a strong vacuum magnetic field–one that is several thousand times more intense than the ambient magnetic field of the earth–using a U-shaped electromagnet with 12 centimeters’ spacing between the two poles, which are electrically insulated from each other so that we can apply a high voltage across them. Placing the poles flush with the wall of a vacuum chamber produces a smoothly arched field on the inside. The chamber is many times the size of the electromagnet and so provides our miniature prominences plenty of room to expand. To keep it from overheating, we power our electromagnet only for a brief period (about 10 milliseconds at a time). But because the rest of the experiment takes place within an even smaller interval (a few microseconds), the vacuum magnetic field from the electromagnet can be considered as essentially constant in time.
The short duration of our experiments is required for another reason: It makes the magnetic field act as if it were completely frozen into the plasma. Experiments performed at a more leisurely rate would reveal that the supposedly frozen-in field lines can, in reality@ slowly diffuse across the plasma. So we must set the pace of our experiments judiciously. If the timing is too slow, the plasma does not act as a perfect conductor; if the timing is too fast, the plasma does not have a chance to relax into a force-free state.
Where does the plasma come from in the first place? After the electromagnet sets up the initial arched field, an electrically operated valve squirts a cloud of hydrogen gas between the two poles of the magnet. Once this gas is in place, we connect a large capacitor across the poles of the magnet using a high-power switch called an ignitron. The high voltage (about 6,000 volts) accelerates a few stray electrons to substantial velocities. Some of the fastmoving electrons collide with atoms of hydrogen gas, knocking off more electrons and leaving behind positively charged hydrogen ions. The newly freed electrons then accelerate and collide with yet more hydrogen atoms, sparking an ionization chain reaction that transforms the entire cloud of gas into a conductive plasma within a fraction of a microsecond.
The formation of conductive plasma creates what is in essence a short-circuit for the discharging capacitor. So the current flow swiftly mounts, rising to as much as 60 kiloamperes within a tiny fraction of a second. Early on, when the current is still small, the magnetic field it generates remains modest. At this stage, the net magnetic field differs little from the original vacuum magnetic field produced by the electromagnet. Thus the current travels directly along the vacuum field, and highspeed imagery reveals only a simple arch glowing in the darkened chamber.
As the amperage rises over the next few microseconds, several distinct changes take place. First, the pinch effect causes the current to concentrate itself in a narrow channel. Second, the encircling magnetic field of this current becomes appreciable with respect to the vacuum field, and so the net magnetic field becomes helical. The electric current, which must flow along the total field, then follows a helical course. Next, the hoop force causes the segments on opposite sides of the spiral channel to repel each other, creating a general bulging. Finally, the current channel breaks up into smaller braided filaments.
This evolution is extremely reproducible in our experiments and happens in less than 10 microseconds. To capture these rapid changes, we must use cameras with extremely high shutter speeds. Although a movie camera able to film at several million frames per second would in principle be needed to photograph these events unfolding, we can make do with a single high-speed still camera by creating a series of identical plasmas and then taking snapshots at progressively delayed times. Using two cameras arranged side by side and triggered simultaneously, we have also made stereoscopic movies of the plasma in action. Such stereo images help us to discern whether the corkscrew pattern of electric current turns to the right or left and also whether the more complicated braiding of the current filaments exhibits the same sense of overall twist.
Golden Arches Everywhere
An interesting problem arose during our first set of experiments: We discovered that the current could travel in more than one path between the two poles of the electromagnet. It could either flow through the arched main field or skirt along fringing magnetic fields that connected with the conductive wall of the chamber. We generally found that early in the experimental run the current traveled directly from one pole to the other, but later it preferred to detour along the fringing fields, thereby creating two side arches. These secondary features pinched to smaller cross sections, twisted up, bulged out and affected each other in curious ways. Although this situation was interesting, the geometric complexity made it hard for us to follow what was going on.
With our more recent experiments, we avoided these complications by extending the electrodes on the faces of the magnet poles. This arrangement stops any voltage drop from developing along the fringing field lines and so prevents current from being driven along them. Now almost all of the current flows directly between the two poles of the electromagnet, resulting in a single well-defined structure. Each miniature prominence in our simulated universe begins its life simply, starting as an untwisted arch, evolving through a series of increasingly contorted shapes and, in the end, erupting.
Our simulations help to explain quite neatly some hitherto enigmatic aspects of prominences. For example, until now, no one could account for the relatively uniform cross section seen in many solar prominences. Were this glowing plasma simply following lines of force emanating from the surface of the sun, it would be expected to widen substantially in the center. Our experiments suggest that the pinch effect squeezes what would otherwise be a fat middle into a slender conduit of current. And even our struggles to avoid fringing magnetic fields in our vacuum chamber proved useful, because they made us realize that similar fringing fields can explain some of the wilder shapes seen on the sun.
If nothing else, these experiments clearly demonstrate that the physical mechanisms controlling solar prominences are consistent with well-known principles of electromagnetism. Although I expected as much, it was nevertheless thrilling to witness for the first time a tiny solar prominence erupting an arm’s length away–something made possible by the great range of scale to which the equations of magnetohydrodynamics apply.
Sadly, the equations of magnetohydrodynamics cannot provide all the answers. Because this formulation treats plasmas as perfect conductors, it predicts that there will never be any internal electric fields. Solar physicists, who observe powerful x rays and rapidly accelerated particles spawned by erupting prominences, know full well that large electric fields must somehow arise. So to understand the physics of solar prominences more fully, investigators must be willing to delve more deeply into the subject than the magnetohydrodynamic description allows. It is my hope that laboratory simulations will ultimately be able to shed light on such complexities–and that the period of illumination will not be as fleeting as the microsecond-long flashes my apparatus provides!