Determination of a macro- to micro-scale progression leading to a magnetized plasma disruption

Phys. Plasmas 

27

, 022109 (2020); 

https://doi.org/10.1063/1.5140348

27

, 022109

© 2020 Author(s).

Determination of a macro- to micro-scale

progression leading to a magnetized plasma

disruption

Cite as: Phys. Plasmas 

27

, 022109 (2020); 

https://doi.org/10.1063/1.5140348

Submitted: 26 November 2019 . Accepted: 15 January 2020 . Published Online: 12 February 2020

Byonghoon Seo

, Pakorn Wongwaitayakornkul

, Magnus A. Haw

, Ryan S. Marshall

, Hui Li

,

and Paul M. Bellan

Determination of a macro- to micro-scale

progression leading to a magnetized plasma

disruption

Cite as: Phys. Plasmas

, 022109 (2020);

doi: 10.1063/1.5140348

Submitted: 26 November 2019

Accepted: 15 January 2020

Published Online: 12 February 2020

Byonghoon

Seo,

1,a)

Pakorn

Wongwaitayakornkul,

Magnus A.

Haw,

1,b)

Ryan S.

Marshall,

Hui

Li,

and Paul M.

Bellan

1,c)

AFFILIATIONS

Applied Physics, Caltech, Pasadena, California 91125, USA

Theoretical Division, Los Alamos National Laboratory, Mail Stop B227, Los Alamos, New Mexico 87545, USA

Current address:

Department of Physics and Astronomy, UCLA, Los Angeles, CA 90024, USA.

Author to whom correspondence should be addressed:

bhseo@physics.ucla.edu

Current address:

NASA Ames Research Center, Building 230, Rm. 223, Moffett Field, CA 94035, USA.

Electronic mail:

pbellan@caltech.edu

ABSTRACT

We report the observations of a plasma jet evolving through a macro- to micro-scale progression sequence. This leads to a fast magnetic

reconnection that results in the jet breaking off from its originating electrode and forming a force-free state. A sausage-like pinching occurs

first and squeezes an initially fat, short magnetized jet so that it becomes thin. The thin jet then becomes kink unstable. The lengthening of

the jet by the kinking thins the jet even more since the kink is an incompressible instability. When the jet radius becomes comparable to the

ion-skin depth, Hall and electron inertial physics become important and establish the environment for a fast magnetic reconnection. This

fast reconnection occurs, disrupting the jet and establishing a force-free state. X-ray bursts and whistler waves, evidence of a magnetic recon-

nection, are observed when the plasma jet breaks off from the electrode. This experimentally observed sequence of successive thinning from

pinching followed by kinking is reproduced in a three-dimensional ideal Magnetohydrodynamic (MHD) numerical simulation. The results

of the experiment and the numerical simulation, together demonstrate a viable path from macro-scale MHD physics to micro-scale non-

MHD physics where fast reconnection occurs.

Published under license by AIP Publishing.

https://doi.org/10.1063/1.5140348

Magnetohydrodynamic (MHD) current-driven instabilities

1–3

have long been known to be fundamental to the behavior of magneti-

cally confined plasmas. Magnetic reconnection, another type of

instability, is also fundamental because it enables magnetic field

topology-changing events such as spheromak formation

4,5

and solar

eruptions.

6,7

These instabilities are also associated with tokamak stabil-

ity

8,9

and impulsive natural phenomena such as solar quasi-periodic

pulsations.

10,11

While finite resistivity enables reconnection in the

MHD framework, resistive MHD reconnection is too slow to explain

observations. Instead, in most cases of interest, reconnection is gov-

erned by much faster microscopic non-MHD processes involving the

Hall and electron inertia physics

missing from the more macroscopic

MHD description. These Hall and electron inertia effects only become

important at spatial scales smaller than the ion skin depth

which is microscopic and not resolved by MHD. However, because of

the large scale separation, it is unclear how MHD instabilities can cou-

ple to the Hall and electron inertia physics. One possibility is a cascade

of MHD instabilities to successively smaller scales until the ion skin

depth is reached.

13–15

Current-driven MHD instabilities are frequently observed in

both laboratory and space plasmas

3,5,16

and are known to be associated

with magnetic reconnection.

While it is unclear how the macro-

scopic current-driven instabilities can couple to microscopic magnetic

reconnection, previous theoretical and computational studies have

suggested the possibility of a cascade through a transition of succes-

sively smaller scale current-driven instabilities. For example, a compu-

tationalstudybyHaruki

et al.

on current-driven instabilities used a

3D particle-in-cell code to predict the possibility of a sausage-to-kink

cascade in the context of high energy particle production by a dense

plasma focus. Similarly, a recent analytic study by von der Linden and

Phys. Plasmas

, 022109 (2020); doi: 10.1063/1.5140348

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You

suggested a current-driven instability cascade but in an opposite

path, i.e., as a kink-to-sausage cascade. These studies suggest that if an

MHD cascade results in progressively finer scales, the ion inertia scale

and its associated fast non-MHD reconnection might be accessed.

We report here experimental observations showing a sausage-like

pinching-to-kink cascade that progresses to the ion inertia scale and

then triggers a mechanism that results in fast magnetic reconnection.

Here, “fast” reconnection is termed as a Hall-mediated reconnection

that could explain an observed magnetic reconnection rate which is

much faster than the classical resistive reconnection. The observations

show that the sausage-like pinching leads to a kink instability and that

because of the inherent incompressibility of the kink instability, the

kinking leads to further thinning. This thinning attains the ion inertia

scale and so results in fast magnetic reconnection. The observations of

a sausage-like pinching to kinking and thinning are reproduced in a

3D numerical MHD simulation.

The experimental configuration

5,13,19,20

creates an MHD-driven

plasma jet which spontaneously develops as a result of magnetic helic-

ity, plasma, and magnetic energy being injected by biased electrodes

intercepting poloidal magnetic flux. A cylindrical coordinate system {

} is used in this paper with the axis defined by the vacuum cham-

ber. The poloidal direction corresponds to {

}, and the toroidal

direction is denoted by

. This electrode setup is topologically identical

to that used in spheromak experiments;

21–25

what is new here is the

resolution of the distinct sequence wherein the plasma undergoes

sausage-like pinching, then kinks, and then detaches from the origi-

nating electrode resulting in a magnetic reconnection and, in a

sub-microsecond time scale, the formation of a force-free state

(spheromak). The sausage-like pinch to kink sequence followed by

the kinking leading to further thinning to attain the ion inertia scale

has been reproduced with high space and time resolution in a 3D

numerical MHD simulation.

Figure 1

shows the experimental setup. The electrodes consist of

a 0.2 m diameter copper inner disk and a coplanar, concentric 0.5 m

diameter outer copper annulus installed at one end of a 1.4 m diame-

ter, a 1.6 m long cylindrical vacuum chamber.

20,26

A bias coil located

immediately behind the disk electrode generates an initial dipole-like

poloidal magnetic field that links the inner and the outer electrodes

and creates the poloidal flux

gun

The bias coil radius is slightly

smaller than the radius of the inner electrode depicted by blue in the

top right inset in

Fig. 1

. The time scale of the bias coil current is several

milliseconds, so the linked flux is essentially constant on the experi-

mental time scale which is of the order of microseconds. A controlled

amount of hydrogen or nitrogen gas is puffed into the chamber from

eight gas nozzles on the disk electrode and eight gas nozzles on the

annulus electrode. High voltage from an electronically-switched capac-

itor bank breaks down this gas in a fraction of a microsecond to form

plasma. Eight plasma loops are initially formed where each loop fol-

lows the initial dipole poloidal magnetic field (see Ref.

for clear

images of the eight plasma loops). The inner parts of these loops

mutually attract and form a jet which propagates in the

direction

away from the electrodes and self-collimates via MHD forces.

19,27

The amount of injected helicity is controlled by the parameter

gun

,where

gun

is the injected current. Operation regimes

are classified based on

as follows:

regime I has low values of

resulting in the formation of a stable, straight plasma jet, regime II has

intermediate

involving the formation of a jet that then kinks,

and

regime III has high

which is the subject of this paper. The magnetic

FIG. 1.

Experimental setup. The inset at the top right shows a side view of the electrode, the bias coil, and the gas feeds and the inset at the bottom right shows t

he current

and the magnetic field geometry and the 2D plane used for

Figs. 4(b)

and

4(c)

. Red to orange stream lines in the inset indicate currents and blue to green stream lines indicate

the magnetic fields. MPA is the magnetic probe array.

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flux in regime III is relatively low at 0.7 mWb and so results in large

The regime III configuration is thus initially rich in toroidal magnetic

field energy (i.e., strong poloidal electric current) and so is susceptible

to the current-driven MHD instabilities that act to reduce the toroidal

magnetic field energy.

Diagnostics include a fast movie camera, a multi-cluster magnetic

probe array (MPA), a radio frequency (RF) magnetic probe, and a

plastic scintillator X-ray detector,

as shown in

Fig. 1

.TheMPAis

composed of 60 commercial chip inductors that measure at 20 loca-

tions, i.e., three inductors at each location provide three orthogonal

components of the magnetic field.

Current density is calculated from

the measured magnetic field using Ampe

`re’s law. The RF magnetic

field probe (quad probe in

Fig. 1

)iscomposedoffoursetsofthree

coils oriented to measure

;

with the sets having a tetrahedral

structure so that

can be determined. The coils are each a

single loop so as to provide the fast response time required for measur-

ing high frequency waves.

Most of the data reported here was obtained from hydrogen

plasma shots except for the visible-light images in

Fig. 2

which were

obtained from nitrogen plasma shots. The reason for using these two

gases is that as the plasma is radially compressed, its temperature

increases and it becomes fully ionized. Since a hydrogen ion is just a

proton and so has no bound electrons, a fully ionized hydrogen

plasma emits no line radiation, making it difficult to obtain a sharp

visible-light image, whereas a nitrogen plasma has a sharp image.

Figure 2

shows a sequence of visible-light images of an evolving

nitrogen plasma jet obtained using a fast movie camera. The electrode

plane is defined to be

0 (coordinate system shown in 5

sframein

Fig. 2

). At 1.5

s, eight initial plasma-filled arched flux tubes merge to

form a single axially moving current-carrying plasma-filled flux tube,

i.e., a jet propagating in the

direction (to the left in

Fig. 2

). At the

time of merging, the jet has a large radius

and a small length

so at

1.5

s the jet aspect ratio

is small. Initial small perturbations grow

exponentially during the time interval from 1.5 to 3.0

sandpinchthe

jet so as to reduce

while maintaining

constant; this constitutes a

sausage-like instability. This observed pinching is denoted sausage-like

because strictly speaking it is not a sausage instability since it does not

originate from a perturbation of an initial axisymmetric equilibrium as

in the textbook analysis of a sausage instability. However, since MHD

force-balance is lost as soon as a textbook sausage instability initiates,

the evolution of the observed sausage-like pinching is exactly the same

as that of a textbook theoretical sausage instability. Moreover, there is

by definition no initial steady state plasma in a coaxial gun configura-

tion powered by a capacitor bank because this configuration necessar-

ily involves a ramping up of a current and so cannot provide an initial

steady-state equilibrium. This situation is likely occurring in other

experiments having an analogous set-up;

21–25

i.e., none start from an

initial textbook MHD equilibrium. On the other hand, the results of

the MHD simulation, which will be described below, show that

although the initial condition is set to be the same as the experiment

so that there is no initial steady-state equilibrium, a short-lived equilib-

rium (i.e., force balance) briefly develops just before the onset of the

sausage instability.

As a result of the sausage-like pinching decreasing the jet radius

, the jet becomes a thin flux rope with increased aspect ratio

,as

seen at 3.5

s. The jet becomes kink-unstable at 4

s, and the helical

deformation caused by the kinking now substantially increases

Moreover, because the kink is an incompressible instability,

the jet

volume remains constant during the kink, so this increase in

necessi-

tatesafurtherreductionin

to maintain a constant volume

s, the jet length

is approximately three times longer than at 3.5

and

is reduced by approximately a factor of two. At 4.5

s, the jet

radius

decreases to be the order of the ion-skin depth; at this time,

the jet disrupts and detaches from the electrode, indicating that mag-

netic reconnection occurs. This disruption is manifested by several dis-

tinct simultaneous phenomena, namely, X-ray emission, whistler wave

emission, sudden change in the visible-light image indicating that the

plasma jet has detached from the electrode, and a change in magnetic

topology as indicated by magnetic probes.

The fast magnetic reconnection occurs when

v

v

’

where

v

is the electron drift velocity relative to ions and

v

the Alfv

en velocity which is comparable to the ion flow velocity; in

this limit, the Hall and electron inertia terms in the electron equation

of motion become important.

Using

and

FIG. 2.

Time series of a nitrogen plasma shot taken by the fast movie camera.

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gives the

v

v

ratio as

v

v

¼ð

Þð

ffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þð

Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

,where

is the

ionskindepth.Thenumerator

’

because

100 m

and

m. Thus, the regime

corresponds to

v

v

’

0and

the regime

’

corresponds to

v

v

’

;hence,shrinkingof

the flux rope radius to

satisfies the condition for fast magnetic

reconnection to occur. Stark-broadening spectroscopy of the hydrogen

plasma jet (see

Fig. 3

) shows that the electron density is 5

which implies a 3 mm ion skin depth. The observations show that

reconnection indeed occurs when the kink self-thinning reduces

be comparable to the ion skin depth. The resistive skin depth

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

is about 15 mm for

2eVand4.6mmfor

10 eV (

,ln

10, using 1

s), and so

the resistive skin depth is the same order of magnitude as the ion skin

depth. As discussed in Ref.

, finite resistivity will cause the reconnec-

tion growth rate to be altered from the collisionless growth rate. Since

the resistive skin depth is comparable to

, the magnitude of the

reconnection growth rate is thus presumed to differ to some degree

from the purely collisionless prediction. Because the thinned kinked

flux rope seen at 4.5

sin

Fig. 2

is so small (radius less than 3 mm),

the limited spatial resolution of the available diagnostics could not

resolve plasma parameters well enough to make a more definite state-

ment. However, the observation of X-rays and whistler waves dis-

cussed below strongly suggests that the whistler-related mechanism

underlying collisionless reconnection is operative, but possibly modi-

fied by a certain amount of collisionality.

Figure 4(a)

shows that current and voltage oscillations, whistler

waves,

and X-ray bursts

occur in hydrogen plasma shots at pre-

cisely the time the plasma is observed to detach from the electrode.

The coincidence of these transient phenomena with plasma detach-

ment supports the presumption that fast magnetic reconnection is

associated with the sequence of sausage-like pinching leading to kink-

ing that leads to

’

when

v

v

’

. The signals in

Fig. 4(a)

at 4

s occur when the jet detaches from the electrode. The 5.5

ssig-

nals are not understood at present but a possible explanation is that

the jet detaches from the outer electrode at 5.5

Figure 4(b)

shows

the spatial dependence of cos

ðj

jÞ

at 5

s; this indicates

that the current and the magnetic field are nearly parallel in the jet col-

umn, so the plasma achieves a nearly force-free state at 5

Figure

4(c)

shows the poloidal magnetic field (arrows) and the toroidal mag-

netic field (colors) at 5

s; the existence of closed poloidal magnetic

field lines indicates a spheromak-like formation (see

Fig. 5

for compar-

ison with the fields at 3.5

s, i.e., before the magnetic reconnection

occurs). In addition, ion heating was observed at the time

when magnetic reconnection happens (

s) and the jet detaches

40 mm, as shown in

Fig. 6

. This ion heating is also evidence of

magnetic reconnection since magnetic reconnection is expected to

convert magnetic energy into particle energy.

We have made a 3D ideal MHD numerical simulation of a

current-carrying flux rope to model the experimentally observed

MHD instability sequence. This simulation, done on the Los Alamos

Turquoise cluster, uses part of the Los Alamos COMPutational

Astrophysical Simulation Suite (LA-COMPASS).

31,32

The simulation

solves the 3D ideal MHD equations and traces the evolution of mass

density

, pressure

,velocity

v

,andmagneticfield

in a numerical

Cartesian box of size 2

, where 0

is the initial flux tube radius

and

14 cm. 40

grid points and a 0.01

time step were used

and the simulation was run from

0to

,where

s. The center of the Cartesian box is at

;

Initial conditions are based on the experimental jet parameters at 2

and are as follows: (i) a uniform skin current with

’

90 kA is intro-

duced at

5 cm; (ii) the plasma mass density in the flux tube is set

to be

u/m

,where

is the density in m

and

is the hydrogen mass in u and the temperature is 2 eV; (iii) a

-directed magnetic field

;

sim

224 G is uniformly applied in the

simulation domain; (iv) small perturbations are added to the density

to induce the instabilities. The total initial density is

Þ¼

main

sausage

kink

, where the main loop density and the respective per-

turbations for the kink and sausage modes are

main

exp

;

sausage

exp

cos

sausage

ÞÞ

,and

kink

exp

ðð

exp

. Here,

cos

kink

^

sin

kink

^

;

6cm,

36 cm,

2cm,and

7cm. Respective wavenumbers

kink

and

sausage

are chosen corresponding to the unstable modes

observed in the experiment. A small flow velocity, 6km/s at the top

and bottom and linearly decreasing on approaching the center (

0),

is initially imposed in the

6

directions to simulate the axial motion

of the jet. This flow velocity does not play a major role but provides an

imbalance between the periodic bulged structures to mimic the axial

motion of the pressure gradient in

6

direction.

Figure 7

shows the simulation results as a time sequence of the cur-

rent density iso-surfaces. A flux tube evolves with an initial skin current

configuration and an initial small aspect ratio

. The initial condition

wassettobethesameastheexperiment,sothefluxtubeisinitiallynot

in equilibrium and the inward radial force pinches the flux tube. Since

the magnetic and the thermal pressures increase by the radial pinching,

a radial force balance is established at 2.7

s and the flux tube becomes

stable (see

Fig. 8

). Then, the flux tube develops a sausage instability at

3.3

s. The sausaging increases

to form a highly collimated current

channel. A kink instability spontaneously starts at 3.6

s. The numerical

FIG. 3.

Position vs time plot of hydrogen density measured by the Stark broadening.

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simulation thus confirms that the flux tube can transition from being

sausage unstable to being kink unstable.

In addition, the simulation shows that regions where

v

v

develop when the jet kinks implying that non-ideal MHD

physics and resulting fast reconnection should occur at these locations

(see

Fig. 9

). Being MHD, the simulation does not replicate the fast

non-MHD physics, but simply indicates when and where the MHD

physics has accessed the ion skin depth scale.

In conclusion, we have experimentally observed a plasma jet tran-

sition from a sausage-like pinching to a kink instability which then leads

to magnetic reconnection. The sausage-like pinching occurs first and

pinches a fat, short magnetized jet to become a thin, long magnetized

jet. The thin, long jet then becomes kink unstable. The incompressible

kinked jet further lengthens and, to maintain a constant volume, the

kinked jet thins even more. Because of this thinning, the jet radius

becomes comparable to the ion-skin depth setting off Hall and electron

inertial physics that result in fast magnetic reconnection, jet disruption,

and the establishment of a nearly force-free state. The observation of X-

ray bursts and whistler waves, evidence of fast magnetic reconnection,

validate that non-MHD physics has been accessed. The experimentally

observed progression to ion-skin depth scales has been confirmed by a

three-dimensional ideal MHD numerical simulation. The experiment

and its numerical confirmation together establish a mechanism linking

macroscale ideal MHD (no reconnection, scale exceeding ion skin

FIG. 4.

(a) (From top) Time dependent current, voltage,

~

(waves) measured by the RF magnetic wave probe, and X-ray bursts obtained from hydrogen plasma shots. A

bandpass filter was applied to

~

in the frequency range of 1–5 MHz which is in the whistler wave regime. (b) Spatial dependence of cos

ðj

jÞ

at 5

s indicating

a near force-free state (c) poloidal magnetic fields (streamlines) and

(contour) at 5

s. The MPA was used to obtain (b) and (c).

FIG. 5.

Poloidal field (streamlines) and toroidal current density (color contour) at

s which is before magnetic reconnection occurs.

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FIG. 6.

(Left) Position vs time plot of ion temperature obtained from nitrogen ion Doppler broadening measurements at 434.8 nm; colorbar is in eV. (Right) Sam

ple Doppler

broadened spectrum.

FIG. 7.

Time sequence of the current density iso-surfaces obtained from simulation # 471. The two back planes at each time, respectively, show the cross secti

ons of the cur-

rent density at

0 and at

0. The surface plots contain three iso-surfaces at levels 0.19 (blue), 0.32 (green), and 0.41 (red) of the maximum current density at each time.

FIG. 8.

Time-dependent radial force obtained from the numerical simulation. Initially, the flux tube is not in equilibrium so the direction of the radial forc

eat2

s is inward. By

pinching the flux tube, a temporary radial force balance is established at 2

s. Sausage instability occurs at 3

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depth) to microscale physics (fast reconnection, scale less than ion skin

depth, X-rays, whistler waves).

This material is based upon the work supported by the NSF/DOE

Partnership in Plasma Science via the U.S. Department of Energy

Office of Science, Office of Fusion Energy Sciences Award No. DE-

FG02-04ER54755, by the USDOE ARPA-E Grant via Award No. DE-

AR0000565, by the NSF Division of Atmospheric and Geospace

Sciences via Award No. 1914599, and by the Air Force Office of

Scientific Research via Award No. FA9550-11-1-0184.

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FIG. 9.

Time sequence of

;

, and

v

v

in the

(

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v

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s because

becomes large at these locations as indicated by green arrows.

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