Three-dimensional MHD simulations of counter-helicity spheromak mergingin the Swarthmore Spheromak Experiment

C. E. Myers,1, a) E. V. Belova,1 M. R. Brown,2 T. Gray,2 C. D. Cothran,2, b) and M. J. Schaffer31)Princeton Plasma Physics Laboratory, Princeton, New Jersey 085432)Department of Physics and Astronomy, Swarthmore College, Swarthmore, Pennsylvania 190813)General Atomics, San Diego, California 92186

(Dated: 7 September 2011)

Recent counter-helicity spheromak merging experiments in the Swarthmore Spheromak Experiment (SSX)have produced a novel compact torus (CT) with unusual features. These include a persistent antisymmet-ric toroidal magnetic field profile and a slow, nonlinear emergence of the n = 1 tilt mode. Experimentalmeasurements are inconclusive as to whether this unique CT is a fully-merged field-reversed configuration(FRC) with strong toroidal field or a partially-merged “doublet CT” configuration with both spheromak- andFRC-like characteristics. In this paper, the SSX merging process is studied in detail using three-dimensionalresistive MHD simulations from the HYbrid Magnetohydrodynamics (HYM) code. These simulations showthat merging plasmas in the SSX parameter regime only partially reconnect, leaving behind a doublet CTrather than an FRC. Through direct comparisons, we show that the magnetic structure in the simulations ishighly consistent with the SSX experimental observations. We also find that the n = 1 tilt mode begins asa fast growing linear mode that evolves into a slower-growing nonlinear mode before being detected experi-mentally. A simulation parameter scan over resistivity, viscosity, and line-tying shows that these parameterscan strongly affect the behavior of both the merging process and the tilt mode. In fact, merging in certainparameter regimes is found to produce a toroidal-field-free FRC rather than a doublet CT.

I. INTRODUCTION

Spheromaks1,2 and field-reversed configurations3–6

(FRCs) are compact, simply-connected magnetic fusionplasmas. Fusion reactor designs that are based on thesecompact torus (CT) plasmas are much simpler to de-velop and build than standard tokamak-based reactordesigns. FRC-based reactors in particular are attractivebecause of their natural divertor geometry and large βlimit. The large β limit reduces the external magneticfield requirements for the reactor and accommodates theuse of cleaner aneutronic fusion fuels.

In order to realize the potential of FRC-based fusionreactors, however, the performance of experimental FRCplasmas must be significantly improved. Two key ar-eas for improvement are (1) formation and (2) stabil-ity. With regard to formation, much of the experimen-tal success to date has been achieved with the field-reversed θ-pinch (FRTP) formation process.3 FRTP-formed FRCs are prolate (elongated) and highly kineticwith S∗ . 10. Here, the kinetic parameter S∗ is de-fined as S∗ ≡ Rs/λi ' Rs/ρi,ext, where Rs is the FRC’sseparatrix radius, λi ≡ c/ωpi is the ion skin depth, andρi,ext is the ion gyroradius in the external field. Unfor-tunately, FRTP formation requires high voltage pulsedpower supplies that do not scale favorably to reactor size.Additionally, a reactor-grade FRC will likely operate ina more MHD-like regime with S∗ & 40. Thus, in orderto progress toward a feasible FRC-based fusion reactor, a

a)Electronic mail: cmyers@pppl.govb)Present address: Global Strategies Group (North America)

scalable formation technique that can produce MHD-likeFRCs must be developed.

With regard to stability, both spheromaks and FRCsare theoretically unstable to an ideal MHD n = 1 tiltmode.7 Experimentally, FRCs produced by FRTP for-mation have been observed to be relatively stable to thistilt mode, likely due to finite Larmor radius (FLR) ef-fects and nonlinear stabilization;8,9 however, some evi-dence of tilting has been observed even in highly-kineticFRCs.10 Spheromaks, which are inherently more MHD-like, are typically stabilized against the tilt mode using acombination of equilibrium field coils and a close-fittingflux conserver.7,11,12 It has been shown that a cylindricalflux conserver is stabilizing if it is oblate with a length-to-radius ratio L/R ≤ 1.67.13–15 In order to improve theperformance of fusion-relevant CT plasmas, we must con-tinue to develop our understanding of virulent instabili-ties such as the tilt mode.

The SSX-FRC device is designed to study advancedformation and stabilization techniques that will hope-fully lead to higher-performance FRC plasmas. Forma-tion in SSX-FRC is accomplished using counter-helicityspheromak merging.16 Originally developed by Yamadaand Ono in the TS-3 device,17 counter-helicity mergingis initiated by generating two side-by-side spheromaks ofopposite helicity. The parallel currents in the two sphero-maks draw them together and drive them to reconnect.Ideally, this process annihilates the oppositely-directedtoroidal field in the two spheromaks and leaves behinda toroidal-field-free FRC plasma. Also, because poloidalflux is amplified during relaxation,11 a large amount ofpoloidal flux is entrained in the resulting FRC.

Stabilization in SSX-FRC is provided by a quasi-cylindrical copper flux conserver. Though several differ-

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ent flux conservers with various shapes and elongationshave been implemented in SSX, no central conductor hasever been used in the device. Thus, all SSX plasmas aretrue simply-connected compact toroids. In this paper,we focus on counter-helicity merging experiments thatwere conducted in a prolate (L/R = 3) cylindrical fluxconserver. This flux conserver, which is (intentionally)tilt-unstable, is designed to study the development of thetilt instability in various CT plasmas.

The primary goal of these prolate SSX-FRC experi-ments was to produce and study fully-merged FRC plas-mas. During the experimental campaign, however, proto-typical FRC plasmas with only poloidal field were neverobserved. Instead, a novel CT with unusual featuresis formed.16 Experimentally, this CT is characterizedby a strong antisymmetric toroidal field profile and aslow, nonlinear emergence of the n = 1 tilt instability.These results are surprising given the successful forma-tion of toroidal-field-free FRCs via counter-helicity merg-ing in the TS-3 device.18 Consequently, understandingwhy spheromak merging is inherently different in SSXis of great interest to recent experimental campaigns indevices such as TS-3/4,19 the Magnetic ReconnectionExperiment (MRX),20 and the C-2 device at Tri-AlphaEnergy.21 In C-2, two FRTP-formed FRCs are superson-ically merged to form a single flux-amplified FRC.

Several recent numerical simulation results may ex-plain the SSX-FRC observations. On one hand, hybridsimulations by Omelchenko et al. with the FLAME codehave shown that an antisymmetric toroidal field profilecan develop in large-S∗ FRCs.22,23 It is found that thepresence of this toroidal field has a stabilizing influenceon the tilt mode. Alternatively, resistive MHD simula-tions by Belova et al. with the HYbrid Magnetohydrody-namics (HYM) code24 have shown that large ion viscosityin the plasma can slow both the merging process and thegrowth of the tilt mode.25 The stalled merging processleaves behind a partially-merged configuration with twosets of spheromak-like flux surfaces embedded in a singleset of FRC-like flux surfaces. In this “doublet CT” con-figuration, the measured toroidal field profile is simply aremnant of the initial toroidal field that is carried in bythe two merging spheromaks.

Despite these simulation efforts, it remains unclearwhich of these scenarios accounts for the merging behav-ior that is observed in SSX-FRC. In this paper, we re-solve this debate by showing that nearly all of the salientfeatures of the SSX-FRC merging process are consistentwith the formation of a partially-merged, tilt-unstabledoublet CT. This conclusion is drawn from direct com-parisons between the SSX-FRC experimental data andnew three-dimensional resistive MHD simulations fromthe HYM code. These simulations are designed to be asrealistic as possible, and their evolution is highly consis-tent with the experimental data. This work also includesdetailed analyses of the simulation results and an explo-ration of the role of key plasma parameters including re-sistivity, viscosity, and line-tying. Thus, not only do we

establish what happens in the SSX-FRC merging process,we also identify the most important physical mechanismsthat drive this behavior.

The paper is organized as follows: In Section II we mo-tivate the present work by describing the relevant SSX-FRC experimental results. Then, in Section III we detailthe computational framework that has been implementedin the HYM code to accurately simulate the SSX-FRCmerging process. Section IV contains the simulation re-sults for SSX-like plasmas, including direct comparisonsto the experimental data. Detailed analysis of MHD phe-nomena in the reconnection region is reported in SectionV, and Section VI presents a multi-dimensional scan ofthree important plasma parameters (resistivity, viscos-ity, and line-tying). This parameter scan, which helps toplace the simulation results in a broader context, revealsthat the merging behavior is quite responsive to changesin the plasma parameters.

II. MOTIVATION: SSX-FRC EXPERIMENTS

In this section, we describe the SSX-FRC device andthe experimental results that motivate the simulationwork presented in this paper. The magnetic diagnos-tics used in SSX-FRC are emphasized because data fromthese diagnostics will later be directly compared to datafrom the HYM simulations. Results from additional SSX-FRC diagnostics are discussed in the appendix.

A. The SSX-FRC device

The SSX-FRC device in its prolate configuration is de-scribed elsewhere16 so we only briefly review it here. InSSX-FRC, opposing coaxial magnetized plasma guns pro-duce spheromaks of either helicity at either end of thedevice (see Fig. 1). Each plasma gun is surrounded by a“stuffing flux” coil that seeds the poloidal field for eachspheromak during formation. The guns are typically op-erated at ∼ 1 mWb of magnetic flux, but the poloidal fluxin each of the spheromaks is amplified to∼ 3–4 mWb dur-ing relaxation. In the counter-helicity merging configu-ration, the gun flux diverges from one gun and convergesback to the other, producing a mirror-like backgroundfield and spheromaks with parallel toroidal currents.

Spheromak merging is unique in SSX because fully ion-ized spheromaks are formed remotely and injected intoa high vacuum, low magnetic field merging zone withno solid central column. This merging zone is boundedby a quasi-cylindrical copper flux conserver that is sus-pended inside the vacuum vessel. Several different fluxconservers have been used in SSX with radii ranging fromR = 8.5–25.0 cm. The prolate flux conserver that wasused in nearly all of the experiments discussed here hasR = 20.3 cm and L = 61.0 cm (a length-to-radius ratioof L/R = 3, which is deliberately tilt unstable).

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(a)

(c)

StuffingFlux Coil

CapacitorCircuit

CoaxialMagnetizedPlasma Gun

ReconnectionControl Coil

FluxConserver

MagneticProbes

Vacuum Vessel

(b)

z

WestMid

East(c)

z

WestMid

East

z

FIG. 1. (Color online) The Swarthmore Spheromak Exper-iment (SSX) in its prolate SSX-FRC configuration. (a) Az-r cross-section of the experiment with the magnetic probeshighlighted in blue/green/red and a set of calculated poloidalflux contours for BRCC = 210 G shown in gray. (b) A corre-sponding r-θ cross-section of the experiment. (c) A 3D ren-dering of the 96 magnetic field measurement locations.

The primary diagnostic that is used to characterizethe spheromak merging process in SSX-FRC is an ar-ray of twelve linear magnetic probes (see Fig. 1). Eachprobe measures all three components of the magneticfield (Bz, Br, Bθ) at eight radial locations along its lengthfor a total of 12 × 8 = 96 measurement locations insidethe plasma. The radial spacing between the measure-ment locations is 2.5 cm. The twelve linear probes aredistributed across three axial locations within the fluxconserver: one at the midplane (z = 0 cm) and theothers at the “west” and “east” locations (z = ±21.6cm). At each axial location, four probes are inserted atequally spaced toroidal locations to permit toroidal modedecomposition (n = 0, 1). The measurements from themagnetic probes are digitized at 0.8 µs intervals and canthus be used to examine both the temporal and spatialevolution of the plasma.

B. Experimental results

The counter-helicity merging process in SSX-FRC ismediated by two “Reconnection Control Coils” (RCCs)located near the midplane of the device. Calculationsfrom the finite element code EQLFE26 show that thecharacter of the expected post-merging equilibrium de-

pends on the strength of the applied RCC field, BRCC .In particular, modest-to-high RCC values (BRCC & 140G) should produce a partially-merged “doublet CT” withtwo distinct regions of private flux surrounded by a regionof shared flux (see Fig. 1a); low RCC values (BRCC . 70G) on the other hand are expected to produce a fully-merged FRC. As shown in Cothran et al.,16, the amountof reconnected (shared) flux at the midplane is reduced(as expected) by high values of BRCC . At low values ofBRCC , however, the expected fully-merged FRC equilib-rium is not achieved.

To illustrate this behavior, we present the magneticsdata from a typical low-BRCC discharge (BRCC = 70G). First we examine the time evolution of the distri-bution of magnetic energy in the plasma. The toroidalarrangement of the magnetic probe array (see Fig. 1c)permits the Fourier decomposition of the measured mag-netic field components (Bz, Br, Bθ) into their constituentn = 0, 1 toroidal mode amplitudes. These mode ampli-tudes are converted to poloidal and toroidal mode ener-gies, wnm,pol and w

nm,tor, before being radially-averaged

to reduce the data to just four waveforms (n = 0, 1 ×pol, tor) at each of the three axial probe positions. Thisanalysis procedure, which is more thoroughly explainedin Cothran et al. 16 and in Section IV B of this paper,produces a total of twelve radially-averaged mode energydensity waveforms, 〈wnm,pol〉 and 〈wnm,tor〉.

The mode energy waveforms from a typical SSX-FRCcounter-helicity discharge are shown in Fig. 2. The“injection” phase of the discharge (Phase I) begins att = 0 µs with the firing of the capacitor banks. At thistime, the neutral gas in the plasma guns breaks downand the spheromaks begin to form. Strong J×B forceswithin the guns drive the spheromaks out into the fluxconserver. Phase I lasts until t ' 25 µs when the off-midplane probes begin to measure the n = 0 poloidaland toroidal magnetic energy that is carried toward themidplane by the two injected spheromaks.

The axisymmetric “merging” phase of the discharge(Phase II), which lasts from t ' 25–80 µs, begins withthe detection of large n = 0 energy (both poloidal andtoroidal) at the off-midplane probe locations. This mea-sured energy indicates that the two newly-formed sphero-maks have entered the flux conserver. When the sphero-maks collide shortly thereafter, n = 0 poloidal magneticenergy begins to accumulate at the midplane. This be-havior indicates that pairs of flux surfaces from the twospheromaks are reconnecting to form shared flux surfacesthat straddle the midplane. In prototypical counter-helicity merging, this process continues until all of theprivate flux surfaces have reconnected and the axisym-metric toroidal field is completely annihilated. In theseSSX-FRC discharges, however, significant n = 0 toroidalfield is detected by the off-midplane probes even after thegrowth of the midplane poloidal energy (and hence thereconnection process) has stagnated.

Phase III of the SSX-FRC discharge, the “tilt” phase,begins at t ' 80 µs and is dominated by a sharp decline in

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FIG. 2. (Color online) Time evolution of the measured n = 0and n = 1 poloidal and toroidal magnetic energies in a typicalcounter-helicity discharge in SSX-FRC. In each plot, signalsare shown from both the midplane and the off-midplane (eastand west) probe locations. The discharge is demarcated intothree phases: (I) injection; (II) merging; and (III) tilt.

both the poloidal and toroidal n = 0 mode energies. Thisdecline is driven by the emergence of the n = 1 tilt mode,which is evident between t ' 80–100 µs with the rise inthe measured n = 1 energy densities (see, in particular,the midplane waveform in Fig. 2c). The damped n = 1energy measured prior to this time is attributed to in-herent non-axisymmetries introduced during the sphero-mak formation process. Note that additional probes wereadded to the off-midplane measurement locations in or-der to resolve the n = 2, 3 modes, but that the measuredamplitudes of these higher-n modes were found to be in-significant compared to those of the n = 0, 1 modes. Itis interesting to note that the n = 1 tilt mode takesmore than 15 radial Alfvén times (5 axial Alfvén times)to emerge macroscopically. Here, the radial Alfvén timetA0 ' 3 µs is defined as tA0 ≡ Rc/vA0, where Rc = 20.3cm is the radius of the flux conserver and vA0 ' 70 km/sis the typical Alfvén velocity in SSX-FRC. When the tiltmode is finally detected, it has a characteristic growthtime of γ−11 ∼ 6–8 tA0, which is much slower than theexpected linear growth time of γ−11 ' tA0.

In addition to the mode energy waveforms shownabove, it is also useful to examine single-time snapshots ofthe magnetic structure of the plasma. One such snapshot(at t = 62.4 µs) is shown in Fig. 3. Vector plots of theraw magnetics data are shown in (a), while plots of the fil-tered n = 0 components are shown in (b). The n = 0 vec-tor plots reveal the salient features of the merged plasma.In particular, the residual antisymmetric toroidal field isevident from the off-midplane r-θ plots, while the gen-eral poloidal field structure is visible in the two z-r plots.

FIG. 3. (Color online) 2D projections of the SSX-FRC mag-netics measurements at t = 62.4 µs: (a) the raw vector mag-netics data and (b) the n = 0 component showing the pre-vailing axisymmetric structure of the plasma at this time.

Unfortunately, the sparse distribution of magnetic mea-surements in the axial direction does not reveal whetherthe merged plasma has private flux regions on either sideof the midplane. The interesting and unexplained fea-tures of the SSX-FRC merging process described heremotivate the simulation work presented in this paper.

III. THE HYM-SSX COMPUTATIONAL FRAMEWORK

The HYbrid Magnetohydrodynamics (HYM) code is aFortran-90-based plasma simulation code that employsMPI (Message Passing Interface) and domain decompo-sition in up to three dimensions to perform massivelyparallel simulations of fusion-relevant plasmas. HYM isa fourth order finite difference code with an explicit sec-ond order centered-in-time stepping scheme. Simulationscan be performed in any orthogonal coordinate system,and the code is written with a modular structure that fa-cilitates the simulation of a wide range of configurations.Accordingly, the code has been adapted to run in manydifferent experimental geometries including that of theSSX-FRC L/R = 3 flux conserver introduced here.

Fundamentally, HYM is a hybrid code that can imple-ment various combinations of the MHD, two-fluid, and

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kinetic plasma models. For the simulations in this pa-per, however, only the resistive MHD model is used. Thevalidity of the resistive MHD model in the SSX param-eter regime has been verified with an additional set oftwo fluid simulations (see Section IV C). The simulationspresented here were run on the Franklin computer at theNational Energy Research Scientific Computing Center(NERSC) using up to 1000 parallelized cores. For pro-duction runs, meshes with 513×129×32 (Nz×Nr×Nθ)nodes are used. High resolution runs such as these con-sume approximately 3000 processor-hours each.

In its MHD-only operational mode, the HYM codesolves the nonlinear resistive MHD equations with finitescalar resistivity and viscosity:

∂ρ

∂t+∇ · (ρv) = 0 (1)

∂(ρv)

∂t+∇ · (ρvv)= J×B−∇p+ µ∇2v (2)

∂p

∂t+ v · ∇p+ γp (∇ · v) = (γ − 1)×[αOH

(ηJ2

)+ αV H

(µ(∇× v)2 + µ(∇ · v)2

)] (3)E + v ×B = ηJ (4)

B = ∇×A (5)∇×∇×A = µ0J (6)

∂A

∂t= −E (7)

In these dimensionless equations, ρ is the mass density,v is the fluid velocity, J is the current density, B is themagnetic field, p is the plasma pressure, E is the electricfield, and A is the magnetic vector potential. The scalarresistivity η is parameterized in the code as Rc/S, whereS is the Lundquist number and Rc is the typical MHDscale length of the plasma in units of the ion skin depthλi ≡ c/ωpi. For these HYM-SSX simulations, Rc = 20.3cm ' 28.2λi is the radius of the SSX-FRC flux con-server. Similarly, the scalar viscosity µ is parameterizedas Rc/Re, where Re is the fluid Reynolds number. Bothη and µ are prescribed constants in time and space.

The pressure evolution equation used in the resistiveMHD version of HYM (Eq. 3) is formulated so as toconserve energy throughout the simulation. In particu-lar, the terms on the right hand side of Eq. 3 representOhmic heating (OH) and viscous heating (VH), respec-tively. Additional terms that account for loss mecha-nisms such as heat conduction and radiative transportare neglected. In order to partially compensate for theseneglected loss terms, the constants αOH and αV H are in-troduced into Eq. 3. Setting either of these constants toa value less than unity inhibits the corresponding heat-ing term, which, in turn, crudely accounts for some ofthe neglected loss mechanisms. For the simulations pre-sented in this paper, αOH = 0.5 and 0.0 ≤ αV H ≤ 1.0.It was found here that the precise value of αV H is unim-portant in these simulations because the OH term tendsto dominate even for αV H = 1.0.

A. HYM-SSX boundary conditions and initial conditions

An important part of accurately simulating the SSX-FRC merging process is to set up realistic boundary con-ditions and initial conditions for the simulations. Ac-cordingly, the boundary of the HYM-SSX simulation do-main is defined by a cylinder with the dimensions of theL/R = 3 flux conserver in SSX-FRC. This cylindricalboundary is perfectly conducting, which is appropriategiven the long skin time of the SSX-FRC copper flux con-server (> 1 ms) as compared to the plasma discharge time(∼ 100 µs). Contributions from several external magneticfield coils (e.g., the stuffing flux coils and the reconnectioncontrol coils) are frozen into the flux conserver boundaryfor the duration of the simulation.

Due to the simulation boundary conditions, the injec-tion of the spheromaks by the coaxial magnetized gunsthrough the ends of the flux conserver is not simulated.Instead, the initial conditions for the simulations are setup to approximate the state of the spheromaks shortly af-ter they exit the plasma guns and relax inside the ends ofthe flux conserver. To do this, we use a quasi-equilibriumapproach to calculating the initial conditions. We firstsolve the Grad-Shafranov (GS) equation for a sphero-mak equilibrium in half of the SSX-FRC flux conserver.The resulting solution is then reflected across the mid-plane so that a second spheromak is placed in the otherhalf of the domain. This reflection preserves the directionof the toroidal current in the second spheromak but re-verses its toroidal field so that initial conditions with twocounter-helicity spheromaks are produced. Note that thetwo side-by-side spheromaks are not in true equilibriumbecause their parallel toroidal currents will pull them to-gether when the simulation begins.

The HYM-SSX initial conditions are composed of ax-isymmetric 2D (z-r) spatial profiles of several quanti-ties including the poloidal flux ψp, toroidal field Btor,pressure p, mass density ρ, and current density J. Sam-ple poloidal flux and toroidal field profiles are displayedin Figs. 4a and 4b, respectively. This arrangement ofpoloidal flux and toroidal field gives rise to a monotonicq(ψp) profile within each spheromak (see Fig. 4c). Inthis setup, the last closed flux surface (LCFS) is limitedby the end cap of the flux conserver.

A potentially important experimental effect is that ofaxial line-tying from field lines that stream back into theplasma guns. Line-tying is incorporated here by firstsolving for the quasi-equilibrium GS solution in a fluxconserver that is slightly longer than the SSX-FRC fluxconserver. When this modified GS solution is placed backinside the original flux conserver, the last few flux sur-faces of each spheromak are cut off and tied to the per-fectly conducting wall. The relative amount of line-tyingthat is added using this technique is quantified by the ax-ial expansion factor ζLT ≡ 2∆z/Lc, where Lc/2 is half ofthe length of the original flux conserver and ∆z is widthof the axial expansion at each end of the domain. Radialprofiles of the resulting line-tied flux ψp,LT are displayed

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FIG. 4. (Color online) Sample initial conditions for theHYM-SSX simulations: (a) the 2D (z-r) spatial profile of thepoloidal flux ψp; (b) the 2D profile of the toroidal field Btor;(c) the q profile as a function of ψp moving from the magneticaxis of one of the spheromaks out to its last closed flux surface(LCFS); and (d) radial line-tied flux profiles at the end capof the flux conserver for several values of the axial expansionfactor ζLT . For the sample initial conditions shown here, this“line-tying parameter” is ζLT = 4.7%.

in Fig. 4d for several values of ζLT .The final component of the HYM-SSX initial condi-

tions is a 3D perturbation to the initial vector fluid ve-locity profile. Note that all of the initial conditions de-scribed thus far have been 2D, so it is this initial velocityperturbation that introduces 3D behavior into the simu-lations. For the simulations described in this paper, thisinitial velocity perturbation is turbulent (i.e., random-ized at each location). It is also scaled spatially with themagnitude of the poloidal flux. Note that this type ofperturbation does not drive the two spheromaks towardthe midplane; instead they are simply left to acceleratetoward each other as a result of the attractive force be-tween their parallel currents.

B. HYM-SSX simulation parameters

A primary goal of this work is to accurately simulatethe merging process in SSX-FRC. In order to do this, theavailable set of simulation parameters must be chosen tobe as close as possible to the known plasma conditions inSSX-FRC. The key parameter values that were used forthese simulations are displayed in Table I. Many of theseparameters such as B0 are well established by measure-ments from the experiments. Additional experimentalmeasurements that help to constrain parameters such as

TABLE I. Parameters for the HYM-SSX simulations.

Magnetic Field (Edge) B0 = 1.0 kGDensity (Peak) n0 = 10

15 cm−3

Nominal Temperature Ttot = Ti + Te ' 25 eVIon Skin Depth λi = c/ωpi ' 0.72 cmMHD Scale Length (Radial) Rc = 20.3 cm ' 28.2 λiAlfvén Velocity vA0 = 69.0 km/sAlfvén Time (Radial) tA0 = Rc/vA0 ' 2.94 µsAdiabatic Index γ = 5/3Resistivity η = 0.040 (S ' 700)Viscosity µ = 0.040 (Re' 700)Applied RCC Field BRCC = 70 GLine-Tying Parameter ζLT = 4.7%Init. Velocity Perturbation v0,max = 0.05 vA0

n0 and Ttot = Ti + Te are detailed in the appendix.Other simulation parameters are not so easily con-

strained. These include the resistivity η, the viscosityµ, and the line-tying parameter ζLT . From a classical re-sistivity estimate, S ' 500–1000 for SSX-FRC, whichagrees with the experimentally-observed plasma decayrate. Thus, the simulation value of S = 700 is in theproper regime. Obtaining an accurate estimate of theviscosity is more challenging. There are several estimatesthat could be employed. Using Ti ∼ 12 eV and the otherrelevant parameters from Table I, the Braginskii unmag-netized ion viscosity gives an estimate of Re ' 120, whilethe Braginskii weakly magnetized ion viscosity gives Re' 600. Finally, the Braginskii gyro viscosity gives Re' 200. This wide range of estimates gives little indica-tion as to the appropriate value of Re. The simulationvalue of Re = 700 was chosen based on various detailedcomparisons between the simulations and the SSX-FRCmeasurements. The same is true for the line-tying pa-rameter ζLT , which at the value of ζLT = 4.7% has amodest effect on the simulations (see Section VI).

IV. SIMULATIONS OF SSX-LIKE PLASMAS

A. Simulation results

In this section, we present results from HYM simula-tions of merging plasmas in the SSX parameter regime.Because the initial conditions described in Section III Aroughly correspond to the end of the spheromak injectionprocess, the simulations begin at the start of Phase II ofthe discharge (t ' 25 µs). Over the next∼ 45 µs, flux sur-faces from the two counter-helicity spheromaks proceedto reconnect. Before the merging process can complete,however, the reconnection process stagnates. The result-ing configuration is a doublet CT with regions of bothpublic and private flux (see Fig. 5a). The two privateflux regions retain most of the antisymmetric toroidalfield that is initially carried in by the two merging sphero-maks (see Fig. 5b).

To quantify the degree to which the two spheromaks

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FIG. 5. (Color online) The plasma configuration that formsduring the HYM-SSX simulations. (a) Contours of the n = 0poloidal flux ψp at t = 70.2 µs after the merging process hasstagnated. (b) Contours of the toroidal field at the same timeshowing that the these simulations produce a doublet CT.(c) Evolution of the merging fraction Rmerg, which saturatesat Rmerg ' 50%. (d) Evolution of the total n = 0 toroidalmagnetic energy W 0m,tor showing that significant toroidal fieldpersists into the tilt phase of the simulation.

merge, we introduce a simulation metric that we call themerging fraction Rmerg:

Rmerg(t) ≡ 1−[z+(t)− z−(t)z+0 − z−0

]2, (8)

where z+ (z−) is the axial location of the poloidal fluxpeak in the right (left) half of the flux conserver. Atthe beginning of the simulation, Rmerg = 0% becausez+ = z+0 and z− = z−0. If, on the other hand, thespheromaks fully merge at some later time, then z+ = z−and Rmerg = 100%. This metric is very useful for com-paring the relative behavior of simulations with differentplasma parameters. For SSX-like plasmas, Rmerg climbssteadily until t ' 70 µs when it saturates at Rmerg ' 50%(see Fig. 5c). During this time, the total n = 0 toroidalmagnetic energy, W 0m,tor, slowly declines due to both re-connection and resistive decay (see Fig. 5d). Becausethe merging process remains incomplete, however, signif-icant toroidal magnetic energy persists until the n = 1tilt mode emerges around t ' 80 µs.

The emergence of the tilt mode marks a transition tonon-axisymmetric behavior in the simulations. As de-scribed in Section III A, non-axisymmetry is seeded bya turbulent initial velocity perturbation. Fig. 6 showsthat this turbulent perturbation quickly organizes intocoherent toroidal modes. The most virulent of these

FIG. 6. (Color online) Time evolution of the various Fouriermode energy densities during the simulation. Each curve rep-resents the globally-averaged energy density 〈wnm〉 for a giventoroidal mode number n. Note that the n = 1 tilt mode isthe most virulent mode in the simulation and that its fittedlinear growth rate is γ−11 ' 2.9 µs ' tA0.

modes, the n = 1 tilt mode, begins to grow almost im-mediately. The fitted linear growth time for this modeis γ−11 ' 2.9 µs (γ1 ' t

−1A0), which is consistent with the

ideal MHD estimate25 of γ1 = 0.7–1.3 t−1A0. The tilt mode

transitions to the nonlinear regime around t = 60 µs,which is around the time that it becomes experimentallydetectable (∼ 10−2 kJ/m3). Figure 6 also shows the de-velopment of the n = 2, 3 modes. The growth of thesehigher-n modes, which is initially driven by numericalcoupling to the n = 1 tilt mode, is eventually driven byphysical nonlinear coupling between the various modes.

The different characteristics of the simulations de-scribed above are summarized in the pressure and mag-netic field line plots of Fig. 7. These plots, which areproduced with the VisIt software package, show (a)the axisymmetric initial conditions; (b) the partially-merged doublet CT structure; and (c) the tilt-dominatedstructure late in time. In the initial condition plots(t = 25.0 µs), two field lines are used to trace out anirrational flux surface that is near the q = 1/3 rationalsurface within in each spheromak. In the doublet CTplots (t = 62.4 µs), we see that the public regions ofthe doublet CT contain a non-negligible amount of con-fined plasma pressure. Despite this, the highest pressurecontours (red) remain within the private flux regions ateither end of the plasma. Note that the peak pressurepmax increases from the initial conditions due to Ohmicheating in the plasma. The corresponding t = 62.4 µsfield line plot shows the magnetic topology of the dou-blet CT. The outer set of field lines is composed of 100primarily-poloidal field lines that trace out an FRC-likeflux surface. In the figure, these shared field lines, whicheach close in a single turn, surround two additional flux

8

FIG. 7. Pressure and magnetic field line renderings of the simulated merging process. In each pressure plot, the displayedvolume is bounded by the p ' 0.5 pmax surface. The pmax value changes with time due to Ohmic heating in the plasma. Thefield lines in the corresponding magnetic plots are seeded near the same p ' 0.5 pmax surface except where noted. (a) Theinitial conditions at t = t0 = 25.0 µs. In each spheromak, one long field line traces out an irrational flux surface. (b) Theparitally-merged doublet CT configuration at t = 62.4 µs. The outer field lines are 100 individual field lines that trace out anFRC-like shared flux surface. The two inner spheromak-like flux surfaces are each traced out by a single long field line that isseeded near the p ' 0.9 pmax surface. (c) The tilt-dominated, non-axisymmetric configuration at t = 82.2 µs. The magneticfield line plot now contains only a single field line that traverses throughout the butterfly-shaped configuration.

surfaces that are still spheromak-like. It is this combi-nation of FRC- and spheromak-like characteristics thatdefines the doublet CT configuration.

Though the field lines within the doublet CT are ini-tially closed, they eventually become stochastic duringthe non-axisymmetric phase of the simulation. This be-havior is manifest in the field line plot at t = 82.2 µs,where a single field line fills a stochastic volume thattraverses throughout the configuration. Note that thetwo private flux regions have tilted in opposite directions,forming a butterfly-like topology with n = 1 character.Similar non-axisymmetric features are visible in the cor-responding pressure plot.

B. Comparison to experimental measurements

The HYM-SSX simulation results described in the pre-vious section will now be compared directly to experi-mental data from SSX-FRC. To do this, magnetic mea-surements are extracted from the simulation data at lo-cations that correspond to the magnetic probe locationsin SSX-FRC. We then use the same analysis techniquesthat are applied to the experimental data in order tofacilitate direct comparisons between the two. The firstcomparison concerns the mode energy density analysis in-

troduced in Section II B. We begin by providing a moredetailed description of this analysis procedure.

The first step in the magnetic energy density analysisprocedure is the toroidal mode decomposition of the vec-tor magnetic field components (Bz, Br, Bθ) that are ac-quired from each measurement location (zi, rj , θk). ForBz, the mode decomposition gives

Bnz (t | zi, rj) =1

Nθ

∣∣∣∣∣Nθ∑k=1

Bz(t | zi, rj , θk) e−inθk∣∣∣∣∣, (9)

where i = 1 . . . 3, j = 1 . . . Nr, and k = 1 . . . Nθ. Notethat in the experiments Nr = 8 and Nθ = 4, while in thesimulations Nr = 128 and Nθ = 32. Also, we restrict thetoroidal mode number to n ≥ 0, so an extra factor of 2must be included in Eq. 9 when n > 0 to account for thecontribution from the corresponding n < 0 mode. Af-ter computing the Fourier amplitudes (Bnz , B

nr ,B

nθ ), the

localized poloidal and toroidal magnetic energy densitiescan be calculated at each (zi, rj) location:

wnm,pol(t | zi, rj) =1

2µ0

[(Bnz)2

+(Bnr)2]

, (10)

wnm,tor(t | zi, rj) =1

2µ0

(Bnθ)2. (11)

9

FIG. 8. (Color online) Comparison between the radially-averaged magnetic energy densities 〈wnm〉 extracted from the simulationdata and those measured in SSX-FRC (see Fig. 2). Strong agreement is found with the accumulation of n = 0 poloidal energyat the midplane due to reconnection and with the emergence of the n = 1 tilt mode late in time. On the other hand, the w0m,torwaveforms in (a) agree only qualitatively. Much better agreement is found in (b) where the off-midplane measurements of thesimulation data are taken at zpr = ±15.6 cm instead of zpr = ±21.6 cm.

The final step is to radially average these localized mag-netic energy densities to reduce the data to one waveformper axial measurement location zi and toroidal modenumber n:

〈wnm〉(t | zi) =

2

NrRc

Nr∑j=1

rj−1/2[wnm]j−1/2, (12)

where

rj−1/2 ≡1

2

[rj + rj−1

], (13)[

wnm]j−1/2 ≡

1

2

[wnm(t | zi, rj) + wnm(t | zi, rj−1)

]. (14)

Note that in the experiments, [wnm]1/2 is approximated aswnm(t | zi, r1) because there are no probe measurementson axis. In the simulations on the other hand, the j = 0mesh nodes are used to properly compute [wnm]1/2. As inthe SSX experimental analysis, the vacuum RCC mag-netic energy density is subtracted from w0m,pol.

The various 〈wnm,pol〉 and 〈wnm,tor〉 waveforms thatare obtained from the above analysis of the simulationdata are shown in bold in Fig. 8a. The correspond-ing experimentally-measured mode energies from Fig. 2are reprinted in relief for comparison. It is clear thatthese two sets of waveforms have many features in com-mon. This is especially true in subplot (i) of Fig. 8a,which shows the evolution of the n = 0 poloidal magneticenergy density. Both of the green midplane waveformsin this subplot indicate similar accumulation of poloidal

magnetic energy as a result of reconnection between thetwo spheromaks. This behavior is followed by the stag-nation of the merging process and the eventual roll-off ofthe n = 0 waveforms late in time.

In subplot (ii) of Fig. 8a, we find that there is quali-tative but not quantitative agreement between the n = 0toroidal field waveforms. The measured toroidal fieldenergy in the experiment is much stronger than thetoroidal field energy found in the simulations at the samezpr = ±21.6 cm off-midplane probe locations. We hy-pothesize that this discrepancy is due to the fact thatthe formation and evolution of the plasma in the gunregions is not simulated. Thus, it is possible that the ex-perimental plasma extends some distance back into thegun regions that are absent in the simulations. The sim-ulated doublet CT configuration would then be axiallyshorter than the experimental doublet CT. We test thishypothesis by taking additional off-midplane probe mea-surements from the simulations, this time at zpr = ±15.6cm (Fig. 8b). This second set of measurements showsmuch better agreement between the n = 0 toroidal fieldwaveforms, which circumstantially supports the coaxial-gun-effect hypothesis described here.

We can continue to investigate this hypothesis by look-ing, as we did in Fig. 3, at magnetic vector plots of themerged configuration. Figure 9 contains three sets ofthese magnetic vector plots, each showing the filteredn = 0 magnetic structure from t = 62.4 µs. Theyare (a) from the experiments; (b) from the simulationswith zpr = ±21.6 cm; and (c) from the simulations with

10

FIG. 9. (Color online) Comparison of magnetic vector plotsfrom the experiment and the simulations. (a) Experimentaln = 0 vector plots reprinted from Fig. 3b. (b) Analogous vec-tor plots extracted from the simulations with the off-midplaneprobes at zpr = ±21.6 cm, which is the same location as in theexperiment. The pitch and amplitude of these vectors indi-cate that this probe location cuts across flux surfaces that arefarther outboard than the ones measured in the experiments.(c) A different set of vector plots from the simulations with theoff-midplane probes at zpr = ±15.6 cm instead. The structureof these plots is much more consistent with the experimentalobservations, indicating that effects from the coaxial magne-tized guns, which are not included in the simulations, arelikely important in this context.

zpr = ±15.6 cm. It is clear that the pitch and ampli-tude of the vectors in (b) do not closely resemble thosefrom (a). In fact, the vectors in (b) are consistent withflux surfaces that are farther outboard (i.e., stronger Brand weaker Bθ) than those in the experimental measure-ments. At the shifted off-midplane probe locations shownin (c), however, the vectors are in much better agreementwith the experimental observations. This provides addi-tional evidence that the simulated doublet CT is axiallyshorter than the experimental configuration.

It should be noted that the peak toroidal field am-plitude in the simulations is smaller than the experimen-tal measurements regardless of the measurement locationzpr. While both the temporal and spatial toroidal fieldprofiles are qualitatively consistent (see Figs. 8b(ii) and9a/c, respectively), the amplitude discrepancy remainsunexplained. In fact, additional confined toroidal field inthe simulations would upset the MHD quasi-equilibrium,so it is not clear how such large quantities of toroidal fieldare confined by the experimental plasma.

The final remaining element of the data comparisonsin this section is to examine the behavior of the n = 1 tiltmode. Returning to Fig. 8, it is clear that the nonlineargrowth of the tilt mode in the simulations coincides withthe experimentally-measured increase in the n = 1 energydensity around t ' 80 µs. In fact, the behavior of then = 1 mode is nearly identical at the two off-midplaneprobe locations. The experimentally-measured n = 1energy prior to t ' 80 µs is attributed to inherent non-axisymmetries that arise during the spheromak injection

process. The conclusion, then, is that the tilt mode inthe simulations, which grows linearly at small amplitudewith γ−11 ' tA0, is entirely consistent with the slowerexperimentally-observed mode that grows with γ−11 ∼ 6–8 tA0: it simply transitions into the slower-growing non-linear regime before it is detected experimentally.

C. Validity of the MHD model in the SSX regime

The strong agreement between the HYM-SSX simu-lations and the SSX-FRC experiments that is demon-strated in the previous section clearly indicates that theresistive MHD model used for these simulations providesa good description of the global dynamics of the SSX-FRC plasmas. In order to further verify that the re-sistive MHD model is valid in the SSX-FRC regime, anadditional set of simulations has been performed to studythe effects of two-fluid physics on the observed mergingdynamics. Overall, it was found that the global dynam-ics of the SSX-FRC merging process do not change sub-stantially in the two fluid simulations. In particular, thereconnection timescales and the global magnetic field andplasma profiles are similar. Also, the magnitudes of themacroscopic flows are not strongly affected. This is con-sistent with the observation that the Hall MHD descrip-tion of magnetic reconnection is appropriate when thecondition λi & δSP is satisfied,27 where δSP = L/

√S

is Sweet-Parker width of the reconnection current sheetand λi is the ion skin depth. For the SSX-FRC plasmas,

11

λi . δSP such that the reconnection process is predomi-nantly classical.

Some notable changes that do occur in the two fluidsimulations include a ∼ 25% increase in the peak recon-nection rate and changes in the magnetic field and ve-locity profiles in the reconnection region. In particular,the radial component of the reconnection outflow velocityis enhanced in the two-fluid simulations.28 These changesare primarily due to the inclusion of Hall effects that pro-duce the well-known out-of-plane quadrupole field. Thisquadrupole field alters the local geometry of the recon-nection layer from a quasi-2D MHD layer to a 3D layerwith locally-bent field lines. These findings are supportedby experimental results that show that Hall effects mod-ify the radial outflow profiles and radially shift the X-point during counter-helicity spheromak merging.29

V. RECONNECTION PHENOMENA

Though the HYM-SSX simulations capture the globaldynamics of spheromak merging in SSX-FRC, it is notclear that their applicability should extend to the re-connection region. Of concern are the aforementionedHall and two fluid effects whose impact on the recon-nection dynamics is well known. Despite this, compar-isons between the simulation data and experimental mea-surements again show respectable agreement (see Fig.10). Here, field line reconstructions from a high den-sity 5× 8× 5 (Nz ×Nr ×Nθ) magnetic probe array16,30at the midplane of SSX are compared to field lines fromthe same region in the simulations. Note that the experi-ments with the 3D probe array were conducted in a largerflux conserver with L/R = 2.4 (R = 0.25 m, L = 0.60 m)instead of L/R = 3.

The field lines in Fig. 10 are compared at two times.The first is at t = 31.2 µs (Fig. 10a), which is shortlyafter the beginning of the merging process. In the twoplots, the current sheet length (`cs ' 3 cm) and aspectratio (`cs/δcs ' 5) are very similar. At this early time,the reconnection is relatively two-dimensional (z-r) be-cause the reconnecting field lines from the edges of thetwo spheromaks are still mostly poloidal. The second setof field line plots (Fig. 10b) comes from the tilt phase ofthe discharge at t = 82.2 µs. Here, the remnants of thereconnection layer have tilted out of the local field linetracing region, leaving behind nearly-poloidal field linesfrom the outer regions of the doublet CT. The simulationplot is again similar to the experiment, though it shouldbe noted that the non-axisymmetry of the tilted configu-ration means that this plot shows one of several field linetopologies that exist in the simulations at this time.

We now proceed to analyze the dynamics of the simu-lated reconnection layer in more detail. Local current andvelocity profiles from the simulations are shown in Fig.11. Note that when the reconnection rate peaks aroundt ' 55.2 µs, the reconnection current density is primarilypoloidal (radial at the midplane) while the outflow ve-

FIG. 10. (Color online) Comparison of field lines in the recon-nection region. The experimental field lines are reconstructedusing measurements from a high density 3D magnetic probearray, which was implemented for merging experiments in alarger L/R = 2.4 flux conserver. (a) Reconnecting field linesearly in time. (b) Tilted field lines late in the discharge.

locity is instead primarily toroidal. If only poloidal fieldwere reconnecting, the current sheet would be two dimen-sional with toroidal current density and radial outflows.As the counter-helicity spheromaks reconnect, however,increasing amounts of toroidal field enter the reconnec-tion region, creating a current sheet that is obliquely-inclined in the r-θ plane. Thus, the reconnection currentdensity, which remains normal to the current sheet, willhave both poloidal and toroidal components.

This obliquely-inclined current sheet, in turn, drivesstrongly-sheared toroidal outflows on either side of theX-line. In particular, the distribution of reconnectingpoloidal and toroidal field creates field lines that enter thereconnection region with a sheared profile in the r-θ plane(see Fig. 12d). As the progression in Fig. 12 shows, thetension in these sheared field lines is released as they re-connect and move away from the X-line. Their ends snapstraight, thereby spinning up the plasma toroidally inopposite directions on opposite sides of the X-line. Thisso-called “slingshot effect” was first identified experimen-tally by Yamada et al. 17 and is the primary driver of thetoroidal flows found in these simulations. These toroidalflows do not, however, account for bursty radial outflowsthat have been observed in the SSX-FRC experiments.31

In fact, the reconnection process is relatively quiescentthroughout these simulations and does not produce anysuch impulsive events.

12

FIG. 11. Reconnection region profiles of (a) current density and (b) flow velocity. Each plot on the far left contains the timeevolution of the peak poloidal and toroidal components of the relevant quantity (i.e., J or v). Note that the peak currentdensity is primarily poloidal, while the peak flow is instead primarily toroidal. The psuedocolor plots in the middle comparethe magnitudes of the poloidal and toroidal components of these quantities at t = 55.2 µs. Additional plots on the far rightexamine the shear in the toroidal component of each quantity.

FIG. 12. Three-dimensional plots of the reconnection region at t = 55.2 µs. The plots in the bottom row are the r-θ projectionsof the corresponding isometric plots in the top row. Moving from left to right, each plot shows a series of reconnecting field linesthat are progressively further away from the X-line in the outflow region. The release of the tension in the sheared field linesfollowing reconnection is visible as they snap straight during this progression. The snapping motion “slingshots” the plasmatoroidally in opposite directions on opposite sides of the X-line.

13

VI. SCAN OF VARIOUS PLASMA PARAMETERS

In order to further investigate the physics of thecounter-helicity merging process studied in this paper, wehave carried out a simulation parameter scan over threekey parameters: resistivity, viscosity, and line-tying. Asshown in Fig. 13, moving one or more of these param-eters out of the SSX-FRC regime can produce entirelydifferent merging behaviors. The fully-merged poloidalflux contours shown in Fig. 13b are from a simula-tion with lower resistivity (S ' 1400) and no line-tying(ζLT = 0.0%). When compared to the SSX-like simula-tion (Fig. 13a), we see that the merging fraction Rmergapproaches 100% instead of stagnating at Rmerg ' 50%(Fig. 13c). Also, the completed merging process anni-hilates all of the toroidal field energy W 0m,tor before thetilt mode emerges in the simulation (Fig. 13d), leavingbehind an axisymmetric, toroidal-field-free FRC.

The data shown in Fig. 13 comes from just 2 of the61 data points in the aforementioned multi-dimensionalparameter scan. The scan covers three 2D cuts of theS × Re × ζLT parameter space with each cut centeredabout the SSX-like simulation (S = 700, Re = 700, andζLT = 4.7%). Figure 14 summarizes the results of thescan with contour plots of the merging fraction Rmergand an additional simulation metric tstab. The “stablelifetime” tstab systematically characterizes the relativestability of a given simulated plasma, regardless of thenonlinear details of its tilt mode. We define the stablelifetime tstab as

tstab ≡ t110% − t0, (15)

where t110% is the time when the amount of magneticenergy in the n = 1 mode becomes more than 10% of theenergy in the n = 0 mode (W 1m/W

0m ≥ 10%). Thus, tstab

represents the length of time over which the configurationis predominantly axisymmetric.

In the parameter scan data shown in Fig. 14, wefind several interesting trends. Simulations with com-plete merging (Rmerg → 100%) are found only in theextrema of the scan, particularly in the low-resistivity,low-line-tying limit. High resistivity (small S), high vis-cosity (small Re), and high line-tying (large ζLT ) are eachfound to inhibit the merging process. With regard tostability, resistivity is found to have the most significanteffect. At high resistivity, the configuration decays soquickly that the growth of the tilt instability becomesnonlinear at small amplitude. This leads to simulationswith very long tstab for S . 500. High viscosity and highline-tying are also found to promote stability, though to alesser extent than high resistivity. These results indicatethat a wide variety of merging behaviors are accessiblein the S × Re× ζLT parameter space of counter-helicitymerging. Thus, we conclude that the specific location ofan experimental device in the broader parameter spaceof counter-helicity merging is very important. This im-plies that experimental details such as the plasma forma-tion technique and the geometry of the flux conserver can

FIG. 13. Comparison between a fully-merging simulation andthe SSX-like simulation. The SSX-like simulation in (a) hasS = 700, Re = 700, and ζLT = 4.7%, while the fully-mergingsimulation in (b) has S = 1400, Re = 700, and ζLT = 0.0%.With these parameter changes, the doublet CT structure isreplaced by a fully-merged FRC. (c) A comparison of themerging fractions Rmerg shows that Rmerg → 100% for thefully-merging simulation (solid line), while it stagnates atRmerg ' 50% for the SSX-like simulation (dashed line). (d)Concurrently, the toroidal field energy W 0m,tor rolls off muchearlier for the fully-merging simulation (solid line) than forthe SSX-like simulation (dashed line).

have far reaching impacts on the behavior of the mergingprocess in a given device.

VII. SUMMARY

In this paper, we present 3D resistive MHD simu-lations that explore unexpected results from counter-helicity spheromak merging experiments in the SSX-FRCdevice. The simulations, which are highly consistentwith experimental measurements, show that a partially-merged “doublet CT” configuration forms instead of theanticipated toroidal-field-free FRC. This doublet CT hasboth FRC- and spheromak-like characteristics that con-fine an antisymmetric toroidal field profile. We also findthat the experimentally-observed slowly-emerging n = 1tilt mode is consistent with the nonlinear phase of the tiltmode in the simulations. The mode is found to grow lin-early at small amplitude with γ1 ' t−1A0 before becomingexperimentally detectable. Investigation of reconnectionphenomena in the simulations shows that large shearedtoroidal outflows are driven by the “slingshot effect” thatoccurs due to the counter-helicity reconnection topology.Finally, a simulation parameter scan over resistivity, vis-

14

FIG. 14. Contour plots of the merging fraction Rmerg and the stable lifetime tstab extracted from the 61-point simulationparameter scan. Each of the three sets of plots is a 2D cut of the S × Re × ζLT parameter space, and the center of each cutcorresponds to the SSX-like simulation with S = 700, Re = 700, and ζLT = 4.7%. Complete merging is found at low resisitivtyand 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.

cosity, and line-tying shows that a wide range of mergingbehaviors are possible, including full merging at lower re-sistivity and line-tying. These results indicate that theparameter space for counter-helicity spheromak mergingis quite large and that diverse merging behaviors are tobe expected in different devices.

VIII. ACKNOWLEDGEMENTS

The authors gratefully acknowledge the technical con-tributions of D. Cohen, V. Chaplin, S. Palmer, and J.Haldeman and thank V. S. Lukin for useful discussions.This work is supported by Contract Number DE-AC02-09CH11466 with the U.S. Department of Energy and bythe Center for Magnetic Self-Organization (CMSO), aNational Science Foundation (NSF) Frontier Center es-tablished in coordination with the Department of Energy.

Appendix: Additional SSX-FRC experimental results

In order to better constrain the HYM simulations pre-sented in this paper, we have performed line-averagedexperimental measurements of plasma density, ion tem-perature, and emissivity. Our conclusions, shown earlier

in Table I, are that the SSX-FRC plasmas are in theMHD regime with peak density ne = 10

15 cm−3 andAlfvén speed of 70 km/s. The minimum ion skin depthis c/ωpi = 0.7 cm. Temperatures are in the range of20–50 eV, indicating relatively low resistivity.

Line-averaged ion flow and temperature Ti at the mid-plane is monitored with a 1.33 m ion Doppler spectrom-eter (IDS). We achieve high spectral resolution (0.0075nm per pixel) with an Echelle grating operating at 25thorder.32 We achieve high temporal resolution using a 32channel photomultipler tube array. The SSX IDS instru-ment measures with ≤ 1 µs time resolution the width andDoppler shift of the CIII impurity 229.687 nm line to de-termine the temperature and line-averaged flow velocityduring spheromak merging events. Velocity resolution(obtained by fitting the lineshape) depends on both sig-nal strength and the thermal Doppler width, but it canbe as good as a few km/s (a few percent of the Alfvénspeed); the width (FWHM) of the instrument functioncorresponds to 5 km/s (3.4 eV for C ions). In additionto the IDS measurements, energetic ions have been di-rectly measured with electrostatic probes in SSX.33,34

Using the IDS, we can measure line shapes along anumber of chords across the device at the midplane. InFig. 15, we plot the total emissivity of the CIII line asa function of impact parameter for 10 different chords.

15

FIG. 15. Radial profile of total emissivity at t ' 55 µs. Aspline fit of measured emissivity (black curve with error bars)is Abel inverted to generate a plot of the emissivity as a func-tion of radius (blue curve with blue band for error bars). Theemissivity profile as a function of impact parameter is recon-structed from the Abel inversion (red smooth curve). Notethe enhanced emissivity at r = 0 and r = 12 cm.

The intensity of the light is integrated across the entireline and 10 shots are averaged at each chord. Data isalso averaged into 5 µs time bins to smooth shot-to-shotvariability (black curve with data and error bars). A cu-bic spline is then fit to the data. Two matched splinesare used, one that vanishes at the wall and another withvanishing derivative at r = 0. The spline fit is Abel-inverted to generate a plot of the emissivity as a functionof radius (blue curve with blue band for error bars). Fi-nally, the Abel inverted data can be used to re-constructthe emissivity profile as a function of impact parameter(red smooth curve). If the inversion is correct, then thereconstruction should overlie the original data. For thedata presented in Fig. 15, the fit is excellent.

Note that at the time depicted (t = 55 µs), the CIIIemission is enhanced at r = 0 and r = 12 cm. Magneticdata show that at the midplane, the magnetic axis is atr = 14 cm16 where we expect the particle confinementto be best and the plasma the hottest. For comparison,earlier in time during the reconnection process, emissionis a factor of two higher and comes primarily from thecore of the plasma (around r = 12 cm). From magneticmeasurements, we know that at t = 55 µs, reconnec-tion dynamics have settled down but the plasma is stillaxisymmetric. Abel inversion requires that the data beclose to axisymmetric. This measurement indicates thatemissivity (related to n2e and Te) varies only by a factor oftwo across the plasma radius late in time and that CIIIis a good tracer of bulk properties (not just edge effects).We find that emissivity is highest immediately after thespheromak merging process (around 40–50 µs) and lowduring early formation and late decay.

In Fig. 16, we show a plot of the line-of-sight velocityprofile as a function of impact parameter. Data from 10

FIG. 16. IDS flow measurement as a function of impact pa-rameter. Plotted is the line of sight velocity for chords withimpact parameters between r = 0 and r = 20 cm. The purelyradial flow near r = 0 cm is consistent with vr = 0 km/s.

shots are averaged at each chord in 5 µs bins. For thechord labeled r = 0 cm, the component of the line-of-sight velocity is vr and our measurement is consistentwith vr = 0 km/s there (as it should be). Note thatat the time depicted (t = 60 µs) the data show thatin general, the line-of-sight velocity is small (≤ 5 km/s)across the radius of the plasma and that there is little orno shear. There is some indication of toroidal flow at r =19 cm (line-of-sight velocity is mostly vθ there). Early intime, we have observed high velocity outflows (±40 km/s)associated with reconnection.31 Note also that an Abelinversion of this velocity data would be difficult since theline of sight velocity consists of a different combinationof vr and vθ at each chord and vz is not measured.

Density is measured with a quadrature HeNe interfer-ometer. A modified Mach-Zehnder interferometer usesa linearly polarized scene beam and a circularly polar-ized reference beam to generate signals proportional tothe sine and cosine of the phase shift introduced by theplasma.35 We are able to vary plasma density by adjust-ing gas valve timing while limiting recycling off the wallswith baking and glow discharge conditioning. In Fig.17a we plot a pair of density traces for different gas valvetimings. Plasma densities range from 1–10×1014 cm−3.In Fig. 17b we show a scatter plot of ion temperatures(from gaussian fits of IDS data) measured at a variety ofplasma densities. Temperature and density are measuredsimultaneously (within a 10 µs window) immediately af-ter the doublet CT is formed. Because optical access islimited at the midplane, densities were measured on thewest end of the machine while ion Doppler measurementswere performed at the midplane. Note that while thereis a large scatter in the data, we obtain our highest iontemperatures at the lowest plasma densities. During theaxisymmetric doublet CT phase discussed in this paper,the typical density is 2–5×1014 cm−3 and Ti = 20–40

16

(a) (b)

FIG. 17. Density measurements for counter-helicity merging in SSX-FRC. (a) A pair of density traces for two gas valve timingsshows the range of accessible densities. (b) A scatter plot of Ti vs. ne for various discharges. During the axisymmetric phaseof the discharge, the typical density is 2–5× 1014 cm−3 and Ti = 20–40 eV.

eV. Line averaged electron temperature is inferred froma model fit to data from a soft x-ray array as well as avacuum ultraviolet spectrometer. We found Te increasesfrom 10 eV to 20 eV after reconnection.36

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