Apex Dips of Experimental Flux Ropes: Helix or Cusp?

The primary experiments of interest are the different iterations (Hansen & Bellan 2001; Stenson & Bellan 2012; Ha & Bellan 2016) of the Caltech single-loop experiment. All of these experiments exhibit the apex dip feature and have similar designs. The latest incarnation of the Caltech single-loop experiment (Ha & Bellan 2016) has a single pair of electrodes with internal solenoids and FGVs (Yee 1999). The system is mounted at the end of a 1.5 m long, 0.92 m diameter vacuum chamber with 10⁻⁷ Torr base pressure. The solenoids, located behind the electrodes, generate an arched background magnetic field ($\lt 0.1{\rm{T}}$). Fast valves then release cones of neutral particles over 5 ms (Yun 2008) through holes in the electrodes into the vacuum chamber. High voltage applied to the electrodes by a 59 μF capacitor ionizes the gas to form an arched plasma of density n ∼ 10²¹ m⁻³. The capacitor is typically charged to 2.5–5 kV, driving 30 kA of current for 10 μs. The schematic diagram of the experimental setup is shown in Figure 2.

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The FlareLab experiment at Ruhr University Bochum was designed based on the Caltech apparatus, and has similar gas supply, timescales, electrodes, and magnetic fields (Tenfelde et al. 2014). This experiment also observes a downward apex dip.

The Princeton Plasma Physics Lab (PPPL) apparatus is located in the MRX facility (Myers et al. 2016). It uses uniform background gas injection as well as fast-gas valve injection at a single footpoint. Plasma is also generated via high voltage breakdown from the electrodes. However, the timescale for this experiment is ∼1 ms, one hundred times longer than the Caltech loop, and is comparable to the gas diffusion time. This experiment does not observe apex dips.

The UCLA single-loop experiment (Tripathi & Gekelman 2013) is generated in a uniform pre-ionized plasma and utilizes LaB₆ electrodes with much lower currents (600 A). Additional density is added from laser ablation of targets at the footpoints to trigger eruptions. No apex dips are observed in this experiment either.

For comparison, Figure 3 shows the white light images for the solar flux rope in all four experiments.

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The Caltech experiment has detailed measurements of the neutral density profile emerging from a single footpoint (Yun 2008; Moser 2012). This measured profile is that of an exponential cone:

$$\begin{eqnarray}&&\rho (x,y,z)={\rho }_{0}\,{\left(\displaystyle \frac{{z}_{0}}{| z| +{z}_{0}}\right)}^{2}\exp \,\left[-\displaystyle \frac{K({x}^{2}+{y}^{2})}{{(| z| +{z}_{0})}^{2}}\right],\end{eqnarray} \tag{ 1 }$$

where $K=\tan \alpha \sqrt{\mathrm{log}2}=1.1$, $\alpha \approx 54^\circ$ is the half cone angle, z₀ = 0.01 m, and ${\rho }_{0}=2\times {10}^{-3}\,\mathrm{kg}\,{{\rm{m}}}^{-3}$ is the density at the footpoints.

The two gas nozzles (1 cm apertures) are equally spaced in the y direction (${{\boldsymbol{y}}}_{0}=\pm 4$ cm) and point in the z direction. This gas injection from both nozzles creates overlapping gas cones. If the mean free path is large, the neutral gas in the two cones will not interact and the final distribution is simply a linear superposition of the two cones (Figure 4(a)). However, if the neutral gas has a mean free path comparable to the system size (∼10 cm), the gases will interact and a density pileup will form between the two cones. The mean free path is defined as ${{\ell }}_{\mathrm{mfp}}={(\sigma n)}^{-1}$, where σ is the cross section and n is the number density. Calculations of ${{\ell }}_{\mathrm{mfp}}$ for the three main gases used in the experiment are shown in Table 1.

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Table 1.  Parameters for Density Pileup

	H2	He	N2
n (m−3)	1020–1021 (Moser 2012)	1020–1021 (Moser 2012)	1020–1021 (Moser 2012)
σ (10−19 m2)	2.62 (Ismail et al. 2015)	2.35 (Matteucci et al. 2006)	4.16 (Ismail et al. 2015)
${{\ell }}_{\mathrm{mfp}}$(m)	0.0027–0.027	0.003–0.03	0.0017–0.017

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Under standard experimental conditions, all three gases have a mean free path less than 3 mm. Since the overlap region is several centimeters wide, there should be significant interaction between the two cones.

Estimating the extent and magnitude of the pileup involves evaluating how the density from cone 1 penetrates into cone 2. To first order, this pileup should be confined to the scale of the mean free path, ${{\ell }}_{\mathrm{mfp}}$, and conserve mass. To satisfy these basic constraints, the pileup model is constructed such that the interpenetrating density is compressed to an exponential profile with local characteristic length ${{\ell }}_{\mathrm{mfp}}(z)$:

$$\begin{eqnarray}&&{\rho }_{\mathrm{pileup}}(x,y,z)=M(x,z)\displaystyle \frac{{e}^{-| y| /{{\ell }}_{\mathrm{mfp}}}}{{{\ell }}_{\mathrm{mfp}}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&M(x,z)={\int }_{0}^{\infty }\rho (x,y^{\prime} +{{\boldsymbol{y}}}_{0},z)\ {dy}^{\prime} .\end{eqnarray} \tag{ 3 }$$

This corresponds to integrating the density from cone 1 (e.g., left cone at $y=-{y}_{0}$), which penetrates the XZ-plane at each height, and redistributing it in an exponential profile with characteristic length ${{\ell }}_{\mathrm{mfp}}(x,0,z)$. The process is mirrored for cone 2.

The estimated density pileup from this exponential profile increases the apex density by a factor of 1.6, relative to the noninteracting case. Figure 4 highlights the difference between the case of no interaction, ${{\ell }}_{\mathrm{mfp}}\gt 0.1$ m, and the pileup region model, ${{\ell }}_{\mathrm{mfp}}\ll 0.1$ m. A pileup region creates peaked density contours as distinct from the flat profile of a direct superposition. For the gas cones and densities described here, this model predicts that the pileup effect is only significant for valve separation distances less than 12 cm.

Although this is an ad hoc model, 2D measurements of the FlareLab initial density profile (Tenfelde et al. 2012; Mackel 2014; Mackel et al. 2015) show a peaked density distribution, with contours very similar to those of Figure 4(b), indicating a comparable pileup region at the loop apex. This pileup model is used in Section 5 for the initial conditions of a 3D MHD simulation of the experiment.

The hoop force is an outward radial force present in all curved current channels. This force exists because the internal magnetic pressure of a current loop is greater than the exterior magnetic pressure. The equation of motion for an infinitesimal segment of a circular current (length ds, major radius R, minor radius a, and average mass density $\bar{\rho }$) is given by

$$\begin{eqnarray}&&{F}_{\mathrm{hoop}}\ {ds}=\ddot{R}\ \bar{\rho }\pi {a}^{2}\ {ds}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{F}_{\mathrm{hoop}}=\displaystyle \frac{{\mu }_{0}{I}^{2}}{4\pi R}\left[\mathrm{ln}\left(\displaystyle \frac{8R}{a}\right)-2+\displaystyle \frac{{l}_{i}}{2}\right],\end{eqnarray} \tag{ 5 }$$

where I is the current flowing through the plasma loop, and l_i is a constant of order unity related to the internal current distribution (Stenson & Bellan 2012; Ha & Bellan 2016). Approximating the term in square brackets in Equation (5) as constant and assuming a linearly rising current, the major radius expands quadratically with time: $R(t)\propto {t}^{2}/\sqrt{\bar{\rho }}$ (Stenson & Bellan 2012). However, sections of the loop with higher density will accelerate more slowly and lag behind the global expansion.

The simulation uses the initial density profile shown in Figure 4. In addition to the two gas cones, a high-density wall region is added below the footpoints to simulate the anchoring effects of the electrodes.

The initial background magnetic field (bias field) is generated by specifying a set of loop currents in a half-circle configuration below the electrode plane. This ensures that all field lines emerge and terminate at the footpoints. This topology is closer to that of the experimental field (i.e., where B_z does not change sign on a given electrode) than a simple dipole and is scaled to match the measured field strength at the loop apex. The resulting initial bias field is shown in Figure 9.

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Diffuse poloidal flux is continuously injected into the simulation domain corresponding to the electric current measured in the experiment. The diffuse current profile is constructed from the superposition of 110 thin circular current loops, where each loop has a simple analytic magnetic field expression (Simpson et al. 2001). To avoid singularities, the elliptic integrals are approximately evaluated using a truncated power series. This injected distribution, shown in Figure 10(a), is physically motivated by experimental current density measurements that indicate that the current profile begins as a flared diffuse structure and maintains this outer diffuse current during helicity injection. Figure 10(c) shows the current path of 10 loops in yz-plane with apexes equally spaced from $1.2{y}_{0}$ to $2.6{y}_{0}$. Figure 10(b) shows another view of the profile in xz-plane with 11 sets of 10 current loops distributed over ${\theta }_{{yz}}\in [-54^\circ ,54^\circ ]$, with respect to the yz-plane. The injected current profile is stationary throughout the simulation. Since we are principally interested in the formation phase, we do not attempt to model the helicity extraction or the decreasing current after ${t}_{\mathrm{off}}=20$ μs. The experimentally measured current undergoes a damped oscillation with a period of T = 40 μs, so we model the temporal dependence of the injection as

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}B(t)=\dot{{B}_{0}}\cos \left(\displaystyle \frac{2\pi t}{T}\right)H({t}_{\mathrm{off}}-t),\end{eqnarray} \tag{ 10 }$$

where $\dot{{B}_{0}}$ has only spatial dependency and H is a heaviside step function.

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Using an initial density with a pileup region, the simulation replicates the shape and expansion velocity of the loop. Figure 11(a) shows the image of the loop with the dip at the apex, and Figure 11(b) shows a synthetic image from the simulation where intensity is proportional to current density and number density squared ($I\propto \,J{\rho }^{2}$). Figure 12 shows that the apex position of the simulated loop also closely matches that of the experiment. The evolution comprises of three stages. First, after the initial brightening of neutral gas as shown in Figure 5, magnetic forces generate axial flow and pinch to form a collimated loop (Bellan 2003), resulting in the initial decrease in apex height. Subsequently, the apex is accelerated by the hoop force (Stenson & Bellan 2012), colliding with the neutral pileup region. Lastly, the apex is accelerated to its terminal velocity from the high magnetic curvature forces, illustrated in Figure 13, present in the cusp.

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Given the good match in shape and velocity, the simulation demonstrates that a pileup region is consistent with the observed loop evolution.

Most other experiments with FGVs do not have the appropriate densities or length scales necessary to create density pileup regions. However, as in our experiments, such an effect can greatly perturb the initial conditions and should be considered in the design of future plasma experiments. The high reproducibility and simple control of the feature suggests that future experiments could utilize such pileup regions to study the effect of density perturbations, instabilities, or localized collisions between plasma and neutral gas.

Since the plasma loop can be described by ideal MHD, its behavior can be scaled. This scaling allows for three free parameters a₁, a₂, and a₃ with the following invariant transformations: ${L}_{0}/{a}_{1}\to L^{\prime}$, $\ {\rho }_{0}/{a}_{2}\to \rho ^{\prime}$, $\ {B}_{0}/\sqrt{{a}_{3}}\to B^{\prime}$, $\ {P}_{0}/{a}_{3}\to P^{\prime}$, $\ \tfrac{1}{{a}_{1}}\sqrt{\tfrac{{a}_{3}}{{a}_{2}}}{t}_{0}\to t^{\prime}$, $\ \sqrt{\tfrac{{a}_{2}}{{a}_{3}}}{v}_{0}\to v^{\prime}$, and $\ \tfrac{{a}_{1}{a}_{2}}{{a}_{3}}g\to g^{\prime}$ (Ryutov et al. 2000). These transformations provide a one-to-one correspondence between systems, allowing simulated and experimental plasmas to be scaled to an equivalent system at the space plasma scale. Table 2 shows the characteristic parameters of the experiment, typical coronal loop parameters, and experimental parameters scaled to the solar environment using ${a}_{1}=2.5\cdot {10}^{-8}$, ${a}_{2}={10}^{8}$, and ${a}_{3}={10}^{4}$. With the notable exception of gravity, the experimental parameters scale well to the solar case. However, the effective gravity associated with the acceleration provides useful insight into gravitational effects.

The presence of a dense, cusp feature in the experimental flux rope is similar to common features of solar prominences. It is well established that solar prominences have inhomogeneous density along their axis and that the highest density is localized near the apex (Patsourakos & Vial 2002; Labrosse et al. 2010; Arregui & Soler 2015). Despite these measurements of density modulation, many models of coronal structures assume constant density (Tokman & Bellan 2002; Arnold et al. 2008; Wiegelmann & Sakurai 2012; Jiang et al. 2014; Inoue 2016). The experimental results indicate that density perturbations can result in large distortions of an erupting flux rope, even in the absence of significant pressure or gravitational forces. Consequently, a more realistic density profile should be considered when attempting to precisely model erupting flux ropes or coronal mass ejections.

Furthermore, this denser apex material is thought to sit in a shallow magnetic dip (Heinzel & Anzer 1999; Gunár et al. 2013), a similar but less extreme version of the experimental cusp. Many of the models simulating this apex density are purely hydrodynamic (Antiochos et al. 2000) and ignore magnetic effects from changes in minor radius. In future experiments, these theories could be tested by appropriate acceleration of the loop apex, imposing an effective gravity with appropriate scaling to solar gravity. The acceleration from loop expansion with current parameters (5·10⁷ m s⁻²) scales to an effective gravity of 10⁴ m s⁻² at the coronal scale, 40 times larger than solar gravity (270 m s⁻²).

The last mechanism of interest is the effect of the pileup region on the kink instability. The kink instability is a current-driven instability that drives exponential growth of long-wavelength helical perturbations. The instability threshold is reached when the magnetic field lines complete more than one twist around the major axis. The kink stability is usually defined with respect to the safety factor, q:

$$\begin{eqnarray}&&q=\displaystyle \frac{2\pi {{aB}}_{\phi }}{{{LB}}_{\theta }},\end{eqnarray} \tag{ 11 }$$

where q is the safety factor, L is the length of the major axis, a is the minor radius, B_ϕ is the toroidal field, and B_θ is the poloidal field. Full toroids and other line-tied flux rope experiments (Bergerson et al. 2006; Oz et al. 2011) become unstable for $q\lt 1$. B-field measurements of the loop from t = 10–14 μs give ${B}_{\phi }(a)=250\mbox{--}560\,{\rm{G}},{B}_{\theta }(a)=200\mbox{--}350$ G, a = 2–4 cm. From images we know that the length of the loop is between 40–56 cm in this time frame. These values imply an unstable safety factor of $q\approx 0.5$. Consequently, it is surprising that the loop does not exhibit more violent kinking behavior.

We propose that the high-density region at the loop apex suppresses the kink since the unstable kink mode has an anti-node at the apex, and the high-density region acts like a stationary node. This effectively halves the axial length available to kink and doubles the safety factor. Similar suppression of the longest wavelength kink modes by high-density regions has been seen before in astrophysical jet simulations (Nakamura et al. 2001) and experiments (Hsu & Bellan 2003).