Helicity transformation under the collision and merging of two magnetic flux ropes Free

A magnetic flux rope is a bundle of twisted magnetic field lines generated by spiraling electric currents. It has been named for the magnetic field lines associated with its currents, which resemble threads in a rope and radially vary in pitch. Bundles of these structures are ubiquitous in astrophysical plasmas. The Solar Terrestrial Relations Observatory (STEREO),¹ the Transition Region and Coronal Explorer (TRACE),² and the Atmospheric Imaging Assembly (AIA)³ show multiple flux ropes extending from the solar surface into the solar corona. Similarly, the Cluster mission confirms the existence of magnetic flux ropes in the earth's magneto tail.^4–6 The behavior of these structures is of great interest, as the collision, merging, and tearing of multiple flux ropes can produce bursts of magnetic field line reconnection.^7,8 The unraveling of multiple flux ropes has been theorized to contribute to coronal heating.^9–11 The collision between multiple ropes can trigger a Coronal Mass Ejection.^12–14 

In astrophysics, flux ropes are modeled using the assumption of a perfectly conducting plasma. Under this condition, the motion of the fluid is bound by the conservation of the quantity

Here, A is the vector potential, B is the magnetic field (⁠$B=∇×A$⁠), and $HM$ is the magnetic helicity. Magnetic helicity has geometric significance. Equation (1) is a generalized expression for a topological linking number, describing the linkage of magnetic flux tubes. Equivalently, the expression characterizes magnetic twisting and writhe. This has been addressed by Moffatt,¹⁵ Pfister and Gekelman,¹⁶ Berger,¹⁷ Bellan,¹⁸ and Blackman¹⁹ among others.

When the ideal fluid picture breaks down, magnetic helicity is no longer conserved. However, Taylor conjectured that if the departure from perfect conductivity was small and rapid compared to the resistive diffusion time (⁠$τη=4πL2ηc2$⁠, where L is the size of the system and $η$ is the plasma resistivity), then magnetic helicity would be conserved on the time scales shorter than $τη$⁠.^20,21 This conjecture appears to be accurate.^{15,16,22–25} On time scales greater than $τη$⁠, the variation in magnetic helicity can be used to estimate plasma resistivity.^14,18,20

In this paper, we examine the magnetic helicity of two, experimentally generated, interacting flux ropes, using in situ magnetic field data. One must be careful using Eq. (1) to evaluate magnetic helicity because it involves the totality of all closed magnetic field lines. The field lines may be produced by currents in the plasma, but currents in the walls of the plasma chamber must also be considered. This limitation may be circumvented by using relative magnetic helicity, which will be discussed. We conclude that the dissipation of relative helicity in the flux ropes is balanced by the electric field generated during reconnection.

The experiment was performed on the Large Plasma Device (LAPD)²⁶ at UCLA. The LAPD is a cylindrical vacuum vessel much longer (L = 25 m) than it is wide (d = 1 m). The vacuum vessel is surrounded by 100 solenoidal electromagnets, which provide an axial magnetic field.²⁶ The repeatability of the LAPD's direct-current, plasma discharge, combined with its over 450 diagnostic ports, enables detailed measurements throughout the entire experimental volume. This experiment is the latest in a series of experiments on dynamical flux ropes done on the LAPD. Previous experiments have explored multiple coalescing flux ropes,²⁷ the first confirmation of a quasi-separatrix layer (QSL) between two flux ropes,⁷ and the investigation of chaotic behavior embedded in the system.²⁸ 

The operation of the experiment is as follows: at one end of the device (Fig. 1), an oxide-coated cathode generates a background plasma (I_D = 4 kA, V_D = 42 V, He, 50% ionized). The plasma and some ionizing electrons diffuse through a semi-transparent anode along the axis of the machine. This quiescent plasma is produced at a rate of 1 Hz. After 2 or 3 ms, the background plasma reaches a steady state, and a secondary cathode-anode pair is switched on. This cathode faces the BaO source located at the opposite end of the LAPD. Its anode is 13 m away. This cathode is masked with two circular cross-sections, each 7.5 cm in diameter and centers separated by 11 cm. This generates two flux ropes 11 m in length for 6 ms.

The flux ropes become kink-unstable once formed. This MHD instability^29–31 causes the displacement of each flux rope from its axis as the current in each flux rope (I_FR = 350 A) is above the threshold to trigger the instability (I_Kink = 150 A). As the kink instability saturates, the flux ropes begin to not only rotate about each other (due to $J×B$ forces) but also to move in an elliptical path (when viewed from an x-y cross-section). This motion is shot-to-shot repeatable, up to a phase, which can be resolved using a conditional trigger.

Several probe diagnostics were used to generate fully three-dimensional datasets of relevant quantities: a Mach probe measured the ion flow; magnetic pickup loops measured the magnetic field; an emissive probe measured the plasma potential;³² and a Langmuir probe measured the temperature and density of the plasma. Each probe was moved to forty-three thousand locations within the plasma; and at each location, repeatable time traces were recorded. The consistency of motion permits diagnostic probes, moved by a computer control drive, to reconstruct the magnetic field and the electric field point by point over the course of several weeks. The datasets and indirect quantities were calculated: the vector potential (A), current density (J), electric field (E), and plasma pressure (P_e).

The flux ropes are three times hotter (T_e = 12 eV) and five times denser (n_rope = 5 × 10¹²) than the background plasma with a current density as high as 5 A cm⁻². The perpendicular pressure forces point radially outward from the center of each flux rope and are balanced by the inward forces of J × B. The electrostatic field points inward toward the center of each flux rope as the ropes are electrostatically negative with respect to the plasma background. This electrostatic field generates E × B flows that spiral around the two flux ropes. In the parallel direction, an electrostatic field is responsible for electron streaming along the field to form the flux ropes. Similarly, axial pressure gradients lead to ions flows along the field. These quantities are used in the calculation of canonical helicity.

Magnetic helicity is only preserved within magnetic flux surfaces. When the volume integral of $A·B$ is not enclosed by a magnetic flux surface, the vector potential is not gauge invariant.¹⁹ For a straight flux rope enclosed in a cylindrical volume, the magnetic field penetrates the ends of the volume. Since the magnetic field was not measured past the walls of the experimental device, Eq. (1) cannot be used to evaluate H_M. A gauge invariant helicity is utilized instead

$A′$ and $B′$ are reference fields such that $B′=∇×A′$⁠. $KM$ is the relative magnetic helicity, a concept developed by Berger and Field,³³ Jensen and Chu,³⁴ Finn and Antonsen,³⁵ and others. For a cylindrical flux rope, a convenient reference field is $B′=Bz$⁠,³⁶ the background field which penetrates the ends of the volume and a constant in space and time.

When the flux ropes collide and reconnect, a small amount of magnetic field is annihilated. This produces an electric field, which can be identified in the data and used to locate times and regions of reconnection. This induced electric field is linked to changes in magnetic helicity, and these changes can be broken down into two integrals

In Eq. (3), the volume integral is a source (or dissipative) term of magnetic helicity, while the surface integral represents its transport. Each term in Eq. (3) was evaluated.

The volume integral in Eq. (3) may be temporally and spatially correlated with the squashing factor or q-value. A q-value is a topological description of magnetic field line divergence.^37,38 Using the fully three-dimensional magnetic dataset, hundreds of field lines are seeded and fixed at z = 0 m. Then, using spline coefficients calculated from the data, the field lines are followed axially until their x-y locations [denoted here as $Xx,y,t and Y(x,y,t$⁠)] are determined at z = 11 m. With the end positions of these field lines, q-values are calculated:

Equation (4) is the mathematical projection of field line topology onto a plane, having a unitless lower bound of 2.0. When q becomes large on multiple adjacent field lines, the collection of these field lines is called a quasi-separatrix layer or QSL.

The QSL helps identify reconnection regions. Previous flux rope experiments on the LAPD have demonstrated its effectiveness, showing that a QSL forms periodically, snaking in between two flux ropes that repeatedly collide and merge.²⁴ During the periodic formation of the QSL, two field lines diverge as they pass though the layer, suggesting a reconnection event. In this experiment, a QSL forms every 200 μs (or at a rate of 5.1 kHz) as long as the flux rope are driven kink unstable.

We focus on the changes to magnetic helicity inside the QSL. This can be done by integrating by $−dA/dt$ along the field lines. Mathematically, this is expressed as a dissipation

D, when viewed in a plane, is in general an m = 1 pattern (Fig. 2). This is the result of the flux rope's elliptical motion in the x-y plane. At the time of formation of the QSL, a tendril of dissipation reaches out and snakes between the two flux ropes. There is dissipation due to reconnection and is temporally and spatially correlated with the QSL. The remaining dissipation is from the resistive currents.

One of the benefits of presenting the data in this way is that it defines regions of interest that can be isolated from the rest of the plasma. Changes in helicity can be used to estimate plasma resistivity according to¹⁸ 

Instead of using the dissipation of helicity to estimate resistivity over the entire experimental volume, we isolate the volume of field lines located within the QSL. These are the field lines potentially undergoing the most change. Equation (6) yields a lower bound resistivity of 1.7 × 10⁻⁵ Ω m due only to the annihilated magnetic field. This is a resistivity approximately equal to the classical resistivity.

The amount of energy dissipation within the same region can be estimated by integrating $∫QSLE·JdV$⁠. The dissipation of energy inside the QSL peaks at approximately 100 Watts, a small fraction of the 100 kW used in the formation of the ropes. Over a reconnection event, this accounts for a change of 2 mJ of energy or 0.2 G of annihilated magnetic energy. While this change is small (less than 1% of the energy in the magnetic field), the energy released from reconnection is enough to increase the temperature of electrons between the flux ropes by 1 eV.

Equation (3) was evaluated using the volume of the QSL as the region of interest. The volume integral and the surface integral of Eq. (3) were calculated separately and are plotted in Fig. 3(a). The solid curve in Fig. 3(a) is the resistive dissipation of magnetic helicity inside the QSL. The dashed curve in Fig. 3(a) is the transport of magnetic helicity into or out of the QSL. The terms do not balance; something is missing. While magnetic helicity is moving into the QSL as the two ropes collide, it is not enough to supplant the magnetic helicity that is dissipated. A 2-D representation of this process is plotted in Fig. 3(b). This figure shows D at the time of peak dissipation. The contour of q = 100 is drawn to show spatial correlation with D. The arrows show helicity transport into the QSL, calculated using

Up to this point, helicity has been viewed in an MHD sense. But in the context of the experiment in question, MHD is an insufficient description: The field lines are not frozen to the flow, large pressure gradients exist, and the electric field has a dominating electrostatic component. These problems may be circumvented through the use of canonical helicity, as introduced by You.^39,40 Calculations of canonical helicity begin with canonical momentum, given by $Pq=mqvq+qA$⁠. Here, $mq$ and q are the mass and charge associated with a species, q, and $vq$ represents the fluid flow of that species. Generalized vorticity is introduced as the curl of canonical momentum (⁠$Ωq=∇×Pq$). By integrating $Ωq·Pq$ over the entire volume, canonical helicity is derived.

Canonical helicity is a generalized expression for a topological linking number that describes the self-linkage of canonical vorticity, $Ωq$⁠. The ion and electron species each have their own canonical vorticity. “Field lines” of $Ωe$ and $Ωi$ are shown in Fig. 4. Because the mass of the electron is negligible, $Ωe$ is equivalent to the magnetic field. On the other hand, $Ωi$ includes the addition of vorticity to the magnetic field. Therefore, the self-linkage of $Ωe$ and $Ωi$ differs from one another. Each will be considered separately.

To ensure that canonical helicity is gauge invariant, an arbitrary reference field is added to each physical quantity (⁠$B, A, E…$⁠) that penetrates the surface surrounding the experimental volume. This reference field $X′$ satisfies the criteria $X·∇S=X′·∇S$⁠, where $X$ is a stand-in for any experimental quantity and S is the surface of the experimental volume. Using this reference field, relative canonical helicity is defined in the same manner as relative magnetic helicity

As before, temporal changes in $Kq$ are broken down into a volume integral and a surface integral. This includes not only the induced electric fields but also temperature, density, plasma potential, and flows.³⁹ 

For convenience, the addition or subtraction of a reference field has been denoted by a subscript such that $X±=X±X′$⁠. The volume integral is the generalized source and sink of canonical helicity, where $Eq=−∇hq−∂Pq∂t$ and $hq=qϕ+1/2mqvq2+∫dPq/n$ for a single species.³⁹ The first term in the surface integral is the generalized helicity injection from an electrostatic potential. The second term in the surface integral is the AC injection of canonical helicity from the changing vector potential $∂A/∂t$⁠.

Considering first electron canonical helicity, the dissipative term (⁠$Ee·Ωe$⁠) is found to be spatially and temporally correlated with the QSL. This region of helicity decay snakes through the center of the two flux ropes as they begin to collide. By defining the region of interest to be within the QSL, the contribution to the resistivity from each term in $Ee$ is calculated. The contribution from the induced electric field is 1.7 × 10⁻⁵ Ω m. The contribution from the electrostatic term is 1.0 × 10^–5 Ω m, and the contribution from the pressure term is 1.8 × 10⁻⁵ Ω m. When added together, the volume-averaged resistivity within the QSL becomes approximately three to five times the classical value.

The arrows in Fig. 5(b) represent the influx of electron canonical helicity into the QSL. Visualizing the data in this manor is a useful technique to demonstrate that as K_e is being dissipated within the QSL [represented by the color bar in Fig. 5(b)], it is also being replaced by an influx of helicity across the boundary. Both the total dissipation and the total influx of K_e into the QSL are tracked during an experimental reconnection event, plotted in Fig. 5(a). Unlike Fig. 3(a), which used only the induced electric field, there is a balance between canonical helicity dissipation and canonical helicity influx. This results in $dKe/dt$ ∼ 0 inside the QSL. The inclusion of the electrostatic potential is crucial to the balancing of the two integrals in equation.

The temporal changes in K_i are shown in Fig. 6. Figure 6(a) plots the dissipation of K_i inside the QSL, and Fig. 6(b) plots the influx of K_i into the QSL. Both are similar in magnitude and shape, demonstrating that $dKi/dt ∼0.$ The figure includes not only the formation of a QSL as the two ropes collide (the first peak) but also the formation of a second QSL that occurs when the flux ropes break apart (the second peak).

The second peak is the consequence of the introduction of the ion flow into the calculations. The second peak corresponds to the times in which the flux ropes are in the process of breaking apart, when another QSL forms between the flux ropes. While there is no appreciable dissipation of K_e during this period, there is dissipation for K_i. This is caused by an increase in kinetic ion flow inside the QSL. Considering the energy density of kinetic ion flow (⁠$12minv2$⁠) within the QSL, the increase in kinetic energy as the flux ropes break apart approximately equals the energy dissipation, $∫QSLE·JdV$⁠, over the period of QSL formation—20 W over 10 ms or 2 × 10⁻⁴ J.

Two dynamical flux ropes were examined to determine the effects of magnetic reconnection on magnetic helicity. Volumetric measurements of this system were taken of the magnetic field, temperature, density, plasma potential, and ion flows. The importance of measuring each term, which included non-ideal MHD physics, was demonstrated from the fact that the electrostatic component of the electric field, the induced component of the electric field, and the pressure gradient term contribute equally to the resistivity within the region of reconnection.

By focusing on its temporal derivative, the dissipation of helicity and the influx of helicity were separately tracked within the QSL. If the changes in magnetic helicity are calculated using ideal single fluid MHD physics, helicity is not conserved. If a two-fluid approach that includes the electrostatic fields is used, conservation is demonstrated. Using this model, the dissipation of helicity is balanced by an influx of helicity into the QSL. Both electron canonical helicity and ion canonical helicity are individually conserved in a reconnection event.

This research was performed at the Basic Plasma Science Facility. The funding for this research comes from the Department of Energy: Office of Fusion Energy Science (DE-FC02-07ER54918), National Science Foundation (NSF-PHY 1036140), and the University of California (12-LR-237124).