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Three-dimensional magnetohydrodynamics simulations of counter-helicity spheromak merging in the Swarthmore Spheromak Experiment
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Image of FIG. 1.
FIG. 1.

(Color online) SSX in its prolate SSX-FRC configuration. (a) A z-r cross-section of the experiment showing the locations of the magnetic probes. A set of a set of calculated poloidal flux contours for BRCC  = 210 G is shown in gray. (b) A corresponding r-θ cross-section of the experiment. (c) A 3D rendering of the 96 magnetic field measurement locations.

Image of FIG. 2.
FIG. 2.

(Color) Time evolution of the measured n = 0 and n = 1 poloidal and toroidal magnetic energies in a typical counter-helicity discharge in SSX-FRC. In each plot, signals are shown from both the midplane and the off-midplane (east and west) probe locations. The discharge is demarcated into three phases: (I) injection; (II) merging; and (III) tilt.

Image of FIG. 3.
FIG. 3.

(Color online) 2D projections of the SSX-FRC magnetics measurements at t = 62.4 μs: (a) the raw vector magnetics data and (b) the n = 0 component showing the prevailing axisymmetric structure of the plasma at this time.

Image of FIG. 4.
FIG. 4.

(Color online) Sample initial conditions for the HYM-SSX simulations: (a) The 2D (z-r) spatial profile of the poloidal flux ψ p . (b) The 2D profile of the toroidal field Btor . (c) The q p ) profile moving outward from the magnetic axis of one of the spheromaks to its LCFS. (d) Radial line-tied flux profiles ψ p , LT (r) at the end caps of the flux conserver (z = ±Lc /2) for several values of the axial expansion factor ζLT . Note that this “line-tying parameter” is ζLT  = 4.7% for the sample initial conditions shown in (a)–(c).

Image of FIG. 5.
FIG. 5.

(Color online) The plasma configuration that forms during the HYM-SSX simulations. (a) Contours of the n = 0 poloidal flux ψ p at t = 70.2 μs after the merging process has stagnated. (b) Contours of the toroidal field at the same time showing that the these simulations produce a doublet CT. (c) Evolution of the merging fraction Rmerg , which saturates at . (d) Evolution of the total n = 0 toroidal magnetic energy showing that significant toroidal field persists into the tilt phase of the simulation.

Image of FIG. 6.
FIG. 6.

(Color online) Time evolution of the various toroidal modes during the simulation. Each curve represents the globally averaged energy density for a given toroidal mode number n. Note that the n = 1 tilt mode is the most virulent mode in the simulation and that its fitted linear growth rate is .

Image of FIG. 7.
FIG. 7.

(Color) Pressure and magnetic field line renderings of the simulated merging process. In each pressure plot, the displayed volume is bounded by the surface. The pmax value changes with time due to Ohmic heating in the plasma. The field lines in the corresponding magnetic plots are seeded near the same surface except where noted. (a) The initial conditions at t = t 0 = 25.0 μs. In each spheromak, one long field line traces out an irrational flux surface. (b) The doublet CT configuration at t = 62.4 μs. The outer field lines are 100 individual field lines that trace out an FRC-like shared flux surface. The two inner spheromak-like flux surfaces are each traced out by a single long field line that is seeded near the surface. (c) The tilt-dominated, non-axisymmetric configuration at t = 82.2 μs. The magnetic field line plot now contains only a single field line that traverses throughout the butterfly shaped configuration.

Image of FIG. 8.
FIG. 8.

(Color) Comparison between the radially averaged magnetic energy densities extracted from the simulation data and those measured in SSX-FRC (see Fig. 2). Strong agreement is found with the accumulation of n = 0 poloidal energy at the midplane due to reconnection and with the emergence of the n = 1 tilt mode late in time. On the other hand, the waveforms in (a) agree only qualitatively. Much better agreement is found in (b) where the off-midplane measurements of the simulation data are taken at zpr  = ±15.6 cm instead of zpr  = ±21.6 cm.

Image of FIG. 9.
FIG. 9.

(Color online) Comparison of magnetic vector plots from the experiment and the simulations. (a) Experimental n = 0 vector plots reprinted from Fig. 3(b). (b) Analogous vector plots extracted from the simulations with the off-midplane probes at zpr  = ±21.6 cm, which is the same location as in the experiment. The pitch and amplitude of these vectors indicate that this probe location cuts across flux surfaces that are farther outboard than the ones measured in the experiments. (c) A different set of vector plots from the simulations with the off-midplane probes at zpr  = ±15.6 cm instead. The structure of these plots is much more consistent with the experimental observations, indicating that effects from the coaxial magnetized guns, which are not included in the simulations, are likely important in this context.

Image of FIG. 10.
FIG. 10.

(Color online) Comparison of field lines in the reconnection region. The experimental field lines are reconstructed using measurements from a high density 3D magnetic probe array, which was implemented for merging experiments in a larger L/R = 2.4 flux conserver. (a) Reconnecting field lines early in time. (b) Tilted field lines late in the discharge.

Image of FIG. 11.
FIG. 11.

(Color) Reconnection region profiles of (a) current density and (b) flow velocity. Each plot on the far left contains the time evolution of the peak poloidal and toroidal components of the relevant quantity (i.e., J or v). Note that the peak current density is primarily poloidal, while the peak flow is instead primarily toroidal. The psuedocolor plots in the middle compare the magnitudes of the poloidal and toroidal components of these quantities at t = 55.2 μs. Additional plots on the far right examine the shear in the toroidal component of each quantity.

Image of FIG. 12.
FIG. 12.

(Color) Three-dimensional plots of the reconnection region at t = 55.2 μs. The plots in the bottom row are the r-θ projections of the corresponding isometric plots in the top row. Moving from left to right, each plot shows a series of reconnecting field lines that are progressively further away from the X-line in the outflow region. The release of the tension in the sheared field lines following reconnection is visible as they snap straight during this progression. The snapping motion “slingshots” the plasma toroidally in opposite directions on opposite sides of the X-line.

Image of FIG. 13.
FIG. 13.

(Color online) Comparison between a fully merging simulation and the SSX-like simulation. The SSX-like simulation in (a) has S = 700%, Re = 700%, and ζLT  = 4.7%, while the fully merging simulation in (b) has S = 1400%, Re = 700%, and ζLT  = 0.0%. With these parameter changes, the doublet CT structure is replaced by a fully merged FRC. (c) A comparison of the merging fractions Rmerg shows that Rmerg → 100% for the fully merging simulation (solid line), while it stagnates at for the SSX-like simulation (dashed line). (d) Concurrently, the toroidal field energy rolls off much earlier for the fully merging simulation (solid line) than for the SSX-like simulation (dashed line).

Image of FIG. 14.
FIG. 14.

(Color online) Contour plots of the merging fraction Rmerg and the stable lifetime τstab extracted from the 61-point simulation parameter scan. Each of the three sets of plots is a 2D cut of the S × Re × ζLT parameter space and the center of each cut corresponds to the SSX-like simulation with S = 700, Re = 700, and ζLT  = 4.7%. Complete merging is found at low resisitivty and line-tying. High resistivity (small S) inhibits merging and promotes stability. High viscosity (small Re) and high line-tying (large ζLT ) also inhibit merging, but they have a more limited effect on the plasma stability than high resistivity.

Image of FIG. 15.
FIG. 15.

(Color) Radial profile of total emissivity at . A spline fit of measured emissivity (black curve with error bars) is Abel inverted to generate a plot of the emissivity as a function of radius (blue curve with blue band for error bars). The emissivity profile as a function of impact parameter is reconstructed from the Abel inversion (red smooth curve). Note the enhanced emissivity at r = 0 and r = 12 cm.

Image of FIG. 16.
FIG. 16.

IDS flow measurement as a function of impact parameter. Plotted is the line of sight velocity for chords with impact parameters between r = 0 and r = 20 cm. The purely radial flow near r = 0 cm is consistent with r  = 0 km/s.

Image of FIG. 17.
FIG. 17.

Density measurements for counter-helicity merging in SSX-FRC. (a) A pair of density traces for two gas valve timings shows the range of accessible densities. (b) A scatter plot of Ti vs. ne for various discharges. During the axisymmetric phase of the discharge, the typical density is 2–5 × 1014 cm−3 and Ti  = 20–40 eV.


Generic image for table
Table I.

Parameters for the HYM-SSX simulations.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Three-dimensional magnetohydrodynamics simulations of counter-helicity spheromak merging in the Swarthmore Spheromak Experiment