Nonlinear electron magnetohydrodynamic physics. VII. Magnetic loopantenna in a field-free plasma

R. L. Stenzel, J. M. Urrutia, and K. D. StrohmaierDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA

�Received 2 September 2008; accepted 24 December 2008; published online 6 February 2009�

Nonlinear whistler phenomena near a magnetic loop antenna in a field-free plasma have beeninvestigated experimentally. The loop field oscillates at a frequency far below the electron plasmafrequency, hence all linear electromagnetic modes are cut off. However, the peak antenna field is solarge that the electrons become magnetized allowing whistler modes to exist in the near zone of theantenna. The shielding magnetic field propagates at a speed which increases with magnetic fieldstrength and decays slower than the rf period, resulting in a remnant field when the antenna fieldvanishes. A field-reversed configuration �FRC� is produced when the antenna field reversesdirection. The FRC expands into the magnetized plasma and produces self-consistent magnetichelicity consistent with that of whistler modes. Thus, the new field penetrates in the whistler modein a background field left over from the previous half-cycle. The electrons become unmagnetized atlarge distances, and the field convection goes over into field diffusion. Observations of lightemission indicate electron energization. Current-driven instabilities produce magnetic oscillationswith frequencies much higher than the applied one. Small amplitude, high-frequency whistlerspropagate in the large amplitude fields of the low-frequency whistler, but become absorbed at largerdistances where the cyclotron frequency decreases below the wave frequency. The nonlinearphenomena observed may be relevant to inductively coupled plasma sources, laser-plasmainteractions, plasma opening switches, and active experiments in space. © 2009 American Instituteof Physics. �DOI: 10.1063/1.3073674�

I. INTRODUCTION

Magnetic loops are used for many purposes in plasmaphysics, such as antennas for wave excitation,1–4 heating andionization for processing plasmas,5,6 creation of pinchplasmas,7 and induction of currents in magnetic fusion de-vices. In the present work, we are interested in nonlinearinteractions of a loop antenna with an unmagnetized plasma.It is an extension of previous studies of the excitation oflarge amplitude whistler modes in magnetized plasmas de-scribed in several companion papers.8–11 Strong nonlineari-ties were observed in those experiments when the inducedfield exceeded the ambient field. The present case of an un-magnetized background plasma may be considered the ex-treme case where the fields of the antenna and inducedplasma currents magnetize the electrons such that wavepropagation becomes possible.

Linear cold plasma theory predicts that a wave magneticfield oscillating far below the electron plasma frequency inan unmagnetized collisionless plasma penetrates on the scalelength of the skin depth c /�p, where �p is the electronplasma frequency.12,13 In highly collisional plasmas, the scalelength increases to the collisional skin depth,�2� /��1/2�c /�p�, where ��� is the collision frequency. Inhot, collisionless plasmas, the anomalous skin effect furtherincreases the field penetration.12,14 Fast field penetration hasbeen observed in plasma opening switches and ascribed toelectron magnetohydrodynamic �EMHD� modes in nonuni-form plasmas.15–17 Magnetic bubble experiments have been

proposed in space to study a variety of field-plasmainteractions.18

In the present case, the large magnetic field oscillates,and penetrates into a completely unmagnetized plasma in itsfirst half-cycle.19 The field remains partly frozen-in when thesecond half-cycle imposes an opposite magnetic field. Afield-reverse configuration �FRC� is then generated, expand-ing along the background field.20 The FRC develops helicitydensity consistent with that of whistler modes. A new FRC isinjected at every half-cycle into a decaying outer FRC. Itspenetration depth decreases in subsequent cycles as the am-plitude of the coil magnetic field decays. These topologychanges occur in an amplitude-determined volume aroundthe antenna where the electrons are magnetized. Each FRChas a toroidal magnetic null line with toroidal electric fieldsand currents. Electrons are energized and current-driven in-stabilities of oblique high-frequency whistler modes are ob-served. At large distances, the electrons lose magnetization,and the waves are absorbed collisionally or by cyclotronresonance.

In Sec. II we start the paper with a very brief review ofthe experiment as it has been described in detail elsewhere.8

The experimental observations in Sec. III describe externalmeasurements, such as the radiation resistance and radiatedpower, followed by internal magnetic probe measurements ofthe low and high frequency whistler waves. Based on theobservations, physical processes for the whistler emissionsare discussed. The results are summarized and related to rel-evant applications in Sec. IV.

PHYSICS OF PLASMAS 16, 022103 �2009�

1070-664X/2009/16�2�/022103/10/$25.00 © 2009 American Institute of Physics16, 022103-1

II. EXPERIMENTAL ARRANGEMENT

The experiments are performed in a large �1.5 m diam,2.5 m length� laboratory plasma �density ne�2�1011 cm−3,electron temperature kTe�2 eV, argon pressure pn

=0.4 mTorr� produced by a pulsed dc discharge with setupshown in Fig. 1, Part I.8 In contrast to previous experiments,no external magnetic field is applied. Without magnetic con-finement, electrons expand spherically from the cathode andproduce a broad plasma of lower density than in a magne-tized plasma column.

An insulated magnetic loop antenna �15 cm diam, 4turns� is inserted into the plasma center coaxial with thechamber axis, z. A charged capacitor �1000 V, 0.1 �F� isdischarged into the loop using a fast, high current transistorproducing a weakly damped oscillatory coil field ofBcoil,center�50 G at a frequency f �200 kHz.

The time-varying magnetic field is measured with asingle, movable magnetic probe containing three orthogonalsmall loops �6 mm diam�. The space-time dependent fieldsare assembled from repeated experiments. Measurements areperformed in plasma and vacuum so as to separate the fieldcontributions from antenna and plasma currents.

III. EXPERIMENTAL RESULTS

A. Power absorption

Figure 1 shows a comparison of the externally measuredcoil current in vacuum and in both an unmagnetized andmagnetized plasma. From the known capacitance �C=0.1 �F�, the measured radian frequency �= ��LC�−2

− �R /2L�2�1/2, and constant decay time ��=2L /R� one findsthe circuit inductance L and series resistance R. In vacuum,we obtain L�5.98 �H, R�0.58 �, and L�5.7 �H, and R�1.3 � in the unmagnetized plasma. Thus, the plasma en-hances the circuit losses by �R�0.72 � and decreases the

inductance by �5% due to the partial shielding of the an-tenna field by the plasma. Note that the measured inductanceincludes that of the transmission line to the antenna, which isnot affected by the plasma. The initial power deposited in theplasma, Imax

2 �R /2�8 kW, must be dissipated since theplasma supports no electromagnetic waves at the applied fre-quency. Although described in detail elsewhere,8–11 the caseof a magnetized plasma �B0,z=5 G� is also shown. It indi-cates an even stronger initial damping or higher plasma re-sistance ��R�1.3 ��, which is partly due to the formationof whistler spheromaks on every other half-cycle and partlydue to a higher plasma density. The initial damping of thelarge current oscillations is noticeably stronger than that ofthe small currents at later times. In the unmagnetized plasma,the first cycle does not fit to the exponential decay of thesubsequent oscillations possibly because magnetic energy isdeposited into an initially unmagnetized plasma. Hence, itpropagates over a shorter distance, less energy is deposited,and this manifests itself as lower damping.

B. Electron energization

Large amplitude whistler waves can energize the elec-trons, in particular in the presence of magnetic null lines.21

The electron energization in the present experiment is quali-tatively inferred from light emission measurements as con-firmed elsewhere by Langmuir probe measurements.10 Lightemission arises from inelastic collisions of electrons withmore than 10 eV energy with neutrals and ions. The experi-ments are performed in a dark 2 eV “afterglow” plasma.Light is detected with a photomultiplier tube with fast timeresponse ��t�50 ns� and sufficient spatial resolution trans-verse to the line of sight ��z�2 cm�,10 which is radial alongthe x-axis through the middle of the coil �y=z=0�.

Figure 2 shows waveforms of the coil current and rela-tive light emission. Contrary to observations in a magnetizedplasma,10 emission peaks occur at every half-cycle. The rea-son is that identical field topologies are produced every half-cycle, which is not the case when there is an ambient mag-netic field. In such case, one half-cycle produces whistlerspheromaks with copious light emission, while the next half-cycle excites whistler mirrors without null lines producingnegligible light. Th light source moves with the whistlerspheromak away from the antenna, but in the present caselight is only produced close to the antenna. No light is de-

0 10

|Icoil|

(A)

200

220 30 40

t (µs)

Plasma, B0 = 5 G

1020

100

Plasma, B0 = 0

Vacuum(a)

(b)

(c)

τD ≈ 21 µs

τD ≈ 9.4 µs

τD ≈ 12.5 µs

τD ≈ 9 µs

τD ≈ 6.5 µs

FIG. 1. Absolute value of coil current waveforms �a� in vacuum, �b� in anunmagnetized plasma, and �c� in a magnetized plasma.

0 5

Icoil

Light(arb.units)

0

010 15 20

t (µs)

z= 50 cm

z= 0

100 A

FIG. 2. Waveforms of the coil current Icoil�t� and the light emission due toenergetic �10 eV� electrons at two axial locations.

022103-2 Stenzel, Urrutia, and Strohmaier Phys. Plasmas 16, 022103 �2009�

tected at a distance �z=50 cm from the antenna where fieldsand currents are negligibly small.

By comparing the timing of the light emission peaksrelative to the coil current some conclusions about the heat-ing mechanism can be drawn. If the fast electrons were pro-duced by the toroidal electric field the light emission peaksshould coincide with the zero crossings of the coil current.However, the light emission peaks at each current extremum,where the inductive electric field vanishes, and the lightminimizes when the electric field maximizes. The scaling ofelectron energy with current or magnetic field suggests per-pendicular electron heating by adiabatic compression,mv�

2 /2=�B, where � is the adiabatic invariant of the elec-trons. During the early current oscillations, the magneticfield changes by an order of magnitude and so should theelectron temperature �2→20 eV�, which is sufficient to pro-duce the light emission. During each current decrease, theelectron energy is lost which creates the rapid drop in lightemission. Collisions scatter electrons from perpendicular toparallel velocities. This leads to real heating and is observ-able as a slowly varying light background on which the lightoscillations are superimposed. The light background reflectsa gradual temperature rise during the first few oscillationsfollowed by a slow decay as the coil current decreases.

C. Field penetration

We start with observations of the penetration of the time-varying antenna field into the plasma because it explains theformation of more complicated field topologies shown later.Figure 3 shows a comparison of the magnetic field compo-nent Bz in the central y−z plane in �a� vacuum and �b�plasma at identical times in the current waveform. Invacuum, the field penetrates at the speed of light and has asimple axial dependence, Bz=B0�1+ �z /rcoil�2�−3/2, where B0

is the field in the center of the coil.In plasma, the field distribution depends on time and

amplitude, an example of which is shown in Fig. 3�b�. At atime when the coil current has changed sign �see insert�,there is both a remnant magnetic field from the previoushalf-cycle and a new field imposed by the antenna. Thisleads to a FRC which does not exist in the initial turn-onwhere the field penetrates into an unmagnetized plasma. The

Earth’s magnetic field �B�0.3 G� is shielded by the plasmawhose pressure, nkTe, exceeds the magnetic pressure,B2 /2�0, by two orders of magnitude.

Next, we show that the penetration of the applied fielddepends on its amplitude. Figure 4 shows contour plots ofBplasma�0,0 ,z , t� at two different time spans or, equivalently,amplitudes �b, c� as indicated on the coil current waveform�a�. The crest of the field strength propagates with an axialpenetration velocity, which increases with amplitude andcannot be explained by linear field diffusion. One explana-tion is that due to the magnetization of the electrons theinduced fields propagate as an eigenmode, the whistler modein the present parameter regime. Its velocity increases withfield strength. The remnant field of the previous half-cycleprovides the background field for the whistler mode, albeit itis weaker than the expanding field.

A second nonlinearity observed is displayed in Fig. 5.The induced peak plasma field on axis, Bz,plasma at x=y=0,z=−5 cm is displayed versus the peak applied coil field invacuum in the coil center, Bcoil�0,0 ,0�. The shielding fieldfirst increases linearly with increasing applied field, thensaturates and eventually decreases. The weaker shielding athigher fields can be explained by a decrease in plasma con-ductivity as the electrons become magnetized. At small am-plitudes, the ratio J /E is given by the field-free value �=ne2 /m�ei, but the shielding current becomes a cross-fieldcurrent as the electrons become magnetized. The Hall con-ductivity, �H=ne /B decreases as the magnetic field in-creases.

As explained below, the magnetized electrons self-

−20 −10 10 20−20

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x = 0, t = 3 µs

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2010

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3

−0.31

FIG. 3. �Color online� Contours of the axial field component Bz�0,y ,z� in�a� vacuum and �b� plasma. In plasma, the remnant field from the previoushalf-cycle of the current waveform �see insert� opposes the penetration ofthe growing antenna field.

32 40

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20

0

z

(cm)

t (µs)30

(c)

(a)

t (µs)0 30 402010

100 A

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Icoil

34 36 38

v = 13 cm/µs

2 10t (µs)

0 4 6 8

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20

0

z

(cm)

(b)Bz,plasma (G)

0

0.1250.25

0.3750.5

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−0.25−0.375−0.5−1

Bz,plasma (G)

v = 36 cm/µs x = y = 0

x = y = 0

0

1

10

01.25

2.53.75

5

7.5

−1.2

5

−2.5

−3.75−5−7.5

−10

FIG. 4. �Color online� �a� Coil current waveform and comparison of thefield penetration speed at �b� large and �c� small field strengths. Contour plotof Bz,plasma�0,0 ,z , t� shows slower propagation, hence shallower penetration,with decreasing amplitude. The contour spacing corresponds to an e-foldingamplitude change.

022103-3 Nonlinear electron magnetohydrodynamics physics. VII… Phys. Plasmas 16, 022103 �2009�

consistently twist magnetic field lines producing a field com-ponent B or Bx in the y−z plane. Figure 5 shows its scalingwith applied field. It requires a small threshold of a fewGauss before the applied field magnetizes the electrons. Sub-sequently, Bx increases without saturation as the electronmagnetization increases.

Magnetization of electrons usually refers to conditionswhen their Larmor radius becomes smaller than either thecollisional mean free path or the gradient scale length of thefield, whichever is smaller. For the present parameters�n�2�1011 cm−3, kTe�2 eV�, the mean free path due tothe dominant Coulomb collisions is ve /�ei�21 cm, whichequals an electron Larmor radius at B�0.23 G. The axialgradient scale length of the antenna field is Bz / �dBz /dz�

1.5rcoil�11.3 cm. For B0.43 G, the Larmor radius be-comes smaller than the gradient scale length, hence the elec-trons become magnetized. If the field penetrated the plasmaas in vacuum, the electrons would be magnetized up to z�36 cm along the coil’s axis when antenna field is maxi-mum �B=�0NIcoil,max /2rcoil�50 G�. In plasma, this distancevaries with time and topology, but the electrons are magne-tized over much of the measurement volume.

Whistler modes exist when ���c, which for an appliedfrequency of f =200 kHz requires B0.07 G and is mostlysatisfied. Since the observed scale lengths of B are largerthan c /�p�1.2 cm, electron inertial effects are unimportant.Group and phase velocities of low frequency whistlers in-crease with magnetic field strength �vgroup�2�c /�p�����c�1/2�36 cm /�s at 10 G�, implying faster propagationat larger field strengths. Only at very small amplitudesshould the field penetrate by diffusion with scale lengthgiven by the collisional skin depth �2 /��0��1/2�3 cmrather than c /�p since �ei�.

D. Field topologies

The magnetic field undergoes fundamental changes intopology during the oscillation cycle. These are explained inFig. 6 with the aid of both schematic field lines �a-c� andactually measured fields �d-f� at three characteristic timeswhen the coil current reverses sign.

Due to the inertia of the magnetic field, a previouslyimposed antenna field remains partially embedded when the

0 20

Bx

(G)

10

040Bcoil (G)

Bz

(G)

15

5

0

3

2

1Bx

Bz

FIG. 5. Scaling of the peak induced shielding field Bz,plasma and the out-of-plane field Bx,plasma vs applied coil field Bcoil,vacuum. The shielding field de-creases at large applied fields due to the magnetization of electrons. Mag-netic helicity density increases as the electrons become magnetized.

(a) t = 19.07 µs

(d) (e)

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−10

10

20

0

y

(cm)

z (cm)0

(f)

Btotal (G), (By, Bz)total

(c) t = 20.05 µs(b) t = 19.27 µs

0.37

1

0.13

0.05

2.71

7.36

FIG. 6. �Color online� Schematic field lines �a–c� and measured fields �d–e� in the y−z plane �x=0� for a magnetic loop antenna in an unmagnetized plasmaat different times. �a, d� Field lines and magnitude contours just before a zero crossing of the antenna current �see insert� showing a remnant dipole fieldembedded in the plasma. �c, d� Schematic and measured fields just after coil current reversal. Field lines near the antenna produce an X-type toroidal null lineinside the loop while plasma currents produce an O-type null line outside the coil. �c, f� With increasing coil current the 2D toroidal null line has convergedon axis, formed a degenerate 3D null point and split into two regular 3D null points on axis forming a FRC. In time, the outer dipolar field moves out of themeasurement plane �dashed rectangle� and decays.

022103-4 Stenzel, Urrutia, and Strohmaier Phys. Plasmas 16, 022103 �2009�

coil current vanishes �Figs. 6�a� and 6�d��. As the currentchanges sign, two opposing dipolar fields are present. Thisleads initially to the formation of an X-type toroidal null lineinside the coil, r�rcoil=7.5 cm �Figs. 6�b� and 6�e��. Withfurther increasing current, the null line converges to thez-axis, forms a degenerate three-dimensional �3D� null point,which breaks up into two regular 3D null points on axis oneither side of the antenna. The result is a FRC �Figs. 6�c� and6�f��. In time, the FRC grows and the outer dipolar fieldmoves radially outward.

When the coil current has passed its extremum and de-creases, plasma currents are induced near the antenna in thesame direction as the coil current, i.e., opposite to the previ-ously induced currents well outside the coil. The two nestedand opposing toroidal currents produce a plasma field whichalso exhibits a FRC topology, as shown below. It persistsuntil the coil current reverses sign.

In ideal EMHD, the magnetic field lines are frozen intothe electron fluid which is a visual description of Faraday’slaw with convective electric fields, �B /�t=−��E=�� �v�B�. The lines are convected in the direction of theflow when the flow is across magnetic fields and varies or-thogonal to v�B. In the present case, the toroidal electrondrift v=−J /ne �Fig. 7�e�� bends the poloidal field lines�By ,Bz� �Fig. 7�a�� in the -direction. A field line that is bentout of or into the y−z plane creates a pair of opposing out-of-plane field components Bx that represent cuts through to-roidal field lines �Fig. 7�b��. The convection of frozen-infield lines can also be expressed more mathematically: Thev�B field is radial and has an axial dependence, hence itscurl has a -component and creates a B field.

The toroidal and poloidal fields are linked. The linkagein the left hemisphere �z�0� between the toroidal and poloi-dal field lines is right-handed; in the right hemisphere thelinkage is left-handed. The net linkage or magnetic helicityin the volume is zero. Quantitatively, the helicity density,A ·B �Fig. 7�d�� is obtained from the vector potential A �Fig.7�c�� whose toroidal component is found from the poloidalflux, -A ·dl=2 rA=�Bz2 rdr. There is also a poloidal vec-tor potential which is parallel to the poloidal magnetic fieldfor z�0 and antiparallel for z0. The sign of the helicitydensity agrees with that of whistler modes:22 positive forpropagation along the background magnetic field, negativefor propagation against the field �Fig. 7�d��. The same holdsfor the helicity of the current density, J ·B �Fig. 7�f��. At thechosen instance of time �Icoil=0, see insert in Fig. 7�a��, themagnetic field is the remnant field from the first half-cycle. Itis strong enough to magnetize the electrons so as to allowwave propagation in the whistler mode.

It is worth mentioning that the polarity of the toroidalmagnetic field remains the same at alternating zero crossingsof the coil current while that of the poloidal magnetic fieldreverses sign. This implies that a time-average dc field isgenerated.23,24 The helicity density alternates sign but re-mains consistent with whistler mode propagation along theremnant field �Bz. The only exception is the turn-on of thecoil current where the field diffuses into an unmagnetizedplasma. The formation of magnetic helicity is a clear indica-tion that the field penetration at large amplitudes involves

wave convection. If diffusion dominates, the fields are notfrozen into the plasma and the field lines would not becometwisted.

E. Electric fields and currents

Electric fields arise in EMHD plasmas from both mag-netic induction and space charge imbalances. The latter aremostly responsible for electron Hall currents, the formermostly energize the electrons, particularly along closed mag-netic null lines. Assuming toroidal symmetry, the inductiveelectric field can be obtained from the measured poloidal fluxchange, E=−��� /�t� /2 r=−����Bz /�t�2 r�dr�� /2 r. It isinstructive to determine its space-time dependence and tocompare it with the current density J, which is shown inFig. 8.

A strong toroidal electric field �Fig. 8�a�� and toroidalplasma current density �Fig. 8�b�� are induced when the an-tenna current has its peak derivative, i.e., during sign rever-

(d) AxBx (G2−cm)(c) Ax (G−cm)

20

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(e) Jx (A/cm2) (f) JxBx (G−A/cm2)

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.5−1

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(b) Bx (G)(a) B (G), (By, Bz)

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(cm)

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0.5

0.0312

0.0156

0

−0.5

−0.2

5

−0.125

1

−1

t = 2.5 µs

x = 0

FIG. 7. �Color online� Helicity properties of magnetic fields and currentsnear an antenna in a plasma with B0=0. �a� Total magnetic field streamlinesand magnitude contours. Insert shows coil current waveform. �b� Out-of-plane toroidal magnetic field component Bx. �c� Vector potential Ax, calcu-lated from the poloidal flux �Bz2 rdr. �d� Magnetic helicity density AxBx,showing positive helicity for propagation along B in the left hemisphere andnegative helicity for z0. �e� Toroidal current density Jx, calculated from�� �By ,Bz�. �f� Helicity density of the current density JxBx showing thesame sign properties of whistler mode helicity as the magnetic field.

022103-5 Nonlinear electron magnetohydrodynamics physics. VII… Phys. Plasmas 16, 022103 �2009�

sals. Current densities up to 4 A /cm2 can be induced. Thetoroidal current ring produces a strong dipolar or poloidalmagnetic field, �By,plasma,Bz,plasma� �Fig. 8�c��. Just after coilcurrent reversal �t=2.81 �s�, the antenna imposes an oppo-site dipolar field, which produces a FRC topology when su-perimposed with the opposing plasma dipolar field �Fig.8�d��.

The antenna produces no flux change at the peak of thecoil current, but there remains a broadly distributed toroidalelectric field and current density in the plasma associatedwith the decay of the poloidal plasma field. When the coilcurrent decreases �t=4.25 �s�, the inductive electric field re-verses sign, starting near the coil as shown in Fig. 8�e�. Areverse toroidal current is produced around the coil �Fig.8�f��. The new inner current grows and the old outer currentdecreases, creating a magnetic shielding field with FRC to-pology �Fig. 8�g��. The two opposing currents do not exertforces or torques since the magnetic force in ideal EMHD isbalanced by the electric force, J�B=neE. Because of thelarge coil current, the total magnetic field is dominated bythe coil field, exhibiting a FRC topology with null points farfrom the coil �Fig. 8�h��.

Frequently, the ratio of E /J is identified with a resistiv-ity, but this is not appropriate for nearly collisionless elec-trons drifting across magnetic field lines. The observed ratiois of order �1 V /cm� / �1 A /cm2�=1 � cm which is closer tothe Hall “resistivity” ��=B /ne�3 � cm for B=10 G, n=2�1011 cm−3, than the field-free resistivity �� =m� /ne2

�0.03 � cm for ��2�106 s−1. The toroidal current nearthe coil wire is not a pure Hall current since there are twoorthogonal electric field contributions: a radial space chargeelectric field normal to the coil and an inductive electric fieldparallel to the coil. The latter produces an E�B drift normalto the coil which cannot flow since it would be divergent.The remaining toroidal current flows parallel to the inductiveelectric field which leads to dissipation, E ·J=��J20.

Besides the toroidal current there is a poloidal current�Jy ,Jz� associated with the formation of the toroidal field B

and maximizing during coil current reversals. The poloidalcross-field currents are also consistent with Hall currents dueto the toroidal inductive electric field. Assuming toroidalsymmetry, the poloidal currents are approximately given byJy ���Bx /�z� /�0 and Jz�−��Bx /�y� /�0 and streamlines of�Jy ,Jz� coincide with contours of Bx=const �not shown�. Thepoloidal electron drift, E /Bz, is radially outward along they-axis and axially inward along the z-axis, producing a stag-nation point at the origin.

If the toroidal flow convects field lines, so does the po-loidal flow. On the y-axis, the flow convects field lines ver-tically outward such that the O-type null point also movesradially outward at the time when the coil current approacheszero. After coil current reversal, the FRC develops radialfield components near its two 3D null points, which are thenconvected inward axially leading to a highly oblate FRC.

The induced plasma field peaks at the origin when thecoil current falls between a zero crossing and an extremum,e.g., at 2.81 �s. The plasma field has a weak axial amplitudedecay while the total field has a steep axial gradient. Al-though the toroidal plasma current broadens axially, it neversplits into two current rings, which occurs when there is anaxial dc bias magnetic field.8–11

F. High-frequency oscillations

The waveforms of the magnetic field components in thenonlinear regime contain high frequency oscillations unre-lated to the low frequency of the coil current or its harmon-ics. As explained elsewhere,11 these small-amplitude oscilla-tions are extracted by smoothing the measured signal andsubtracting the smoothed waveform from the original one

(g) (h)

29.43

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Btotal (G), (By, Bz)total

0.54 1.

473.

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t = 2.81 µs

t = 4.25 µs

(b)

Jx (A/cm2)

0

0.06

2

−0.0

62

−0.1

25

0.125 0.25

0.51

−0.25−0.5

−1

(f)(e)

10.8324

−2

−4

FIG. 8. �Color online� Electric fields, current density, plasma and total magnetic fields at two characteristic times. When the coil current reverses sign atoroidal electric field �a� and toroidal current density �b� are induced by the antenna. The latter produces a dipolar plasma field �By,plasma,Bz,plasma� �c�. The totalmagnetic field �d� has a FRC topology since the growing antenna field is reverse to the frozen-in plasma field. When the coil current decays from �I�max theelectric field �e� reverses sign, creating two opposing toroidal plasma currents �f�, a FRC topology in the plasma field �g� while the total field �h� remainsdipolar since it is dominated by Icoil.

022103-6 Stenzel, Urrutia, and Strohmaier Phys. Plasmas 16, 022103 �2009�

which effectively high-pass filters the waveform while mini-mizing phase shifts or aliasing effects associated with Fou-rier transforms.

We start by showing evidence of the oscillations andtheir nonlinear character. Figure 9�a� shows the magnetic os-cillations at the center of the coil �x=y=z=0� together withthe coil current for timing reference. The switch-on of thecoil current excites a transient response which is a linearwave phenomenon. The turn-on transient is the only high-frequency component produced by the coil current, as veri-fied by analyzing �B in vacuum. Therefore, all subsequentoscillations are created by nonlinear plasma processes. Theoscillations peak near the zero crossings of the coil current.At high antenna currents, a small amplitude, high frequencyoscillation follows the lower frequency large amplitude os-cillation. A fast Fourier transform of the oscillations reveals aspectrum from 2 to 4 MHz �not shown�, where the large am-plitudes create the low frequency end and the initial ��5 �s�small oscillations produce the high frequency components.The amplitude of the main oscillations scale nonlinearly withantenna current as shown in Fig. 9�b�. No oscillations arefound below a certain threshold �Icoil�10 A�, while both theamplitude and spectral width increase progressively with in-

creasing current. The nonlinearity is even more pronouncedfor the high frequency oscillations as they disappear afterthree half periods of the coil current waveform. Figure 9�c�displays a contour plot of the wave current density, �Jx, ver-sus position y and time t. This time-of-flight diagram alsoshows that the main oscillations start near the coil, propagateradially outward and inward as discussed in more detail be-low. The high frequency oscillations propagate predomi-nantly radially inward. There is an asymmetry between posi-tive and negative half-cycles of the coil current waveformwhich is discussed further below.

In order to determine the origin and propagation charac-teristics of the oscillations, two-dimensional space-time mea-surements have been performed. The wave propagation iscomplicated by the fact that the background magnetic fieldalso varies in space and time, in particular, when the coilcurrent reverses sign and the FRC topology is formed. Forthis time span, Fig. 10 shows a series of snapshots of thebackground field �Figs. 10�a�–10�f��, the oscillating magneticfield component �Bz �Figs. 10�g�–10�l��, and the wave mag-netic field vectors ��By ,�Bz� �Figs. 10�m�–10�r��. All mea-surements are taken in the central y−z plane �x=0�. Thetiming is indicated by inserts of the coil current waveform inthe upper panels. Note that the wave amplitude is three or-ders of magnitude weaker than the background field, henceare linear wave phenomena.

Prior to reversal of the coil current �Figs. 10�a�, 10�g�,and 10�m��, the background field is dipolar, magnetic oscil-lations originate near the coil and form multiple loops withperiodicity in the y-direction. In time �Figs. 10�b�, 10�h�, and10�n��, the oscillations grow and propagate radially outwardin the �y-direction, which coincides with the elongation ofthe induced electron current sheet �see Fig. 7�e��. Just aftercoil current reversal, a new wave activity starts inside thecoil �Figs. 10�c�, 10�i�, and 10�o��. It is caused by an inducedelectron current opposing the growing coil current. It leads toelectron acceleration at a radial location inside the coil wherean X-type magnetic null line forms �Figs. 10�d�, 10�j�, and10�p��. The current in the O-type toroidal magnetic null linewell outside the coil weakens and produces little wave ac-tivities. On a time scale short compared to the applied oscil-lating period, the X-type toroidal null line collapses on axisand forms two 3D null points which expand axially to formthe FRC �Figs. 10�e�, 10�k�, and 10�q��. The main magneticoscillations are excited during this phase of the FRC cre-ation. As the FRC elongates axially, the induced currentspreads axially, the current density decreases, and the waveactivity decays �Figs. 10�f�, 10�l�, and 10�r��. While the col-lapse of the X-type null line appears to be a transient phe-nomenon, neither the subsequent high frequency oscillationsnor the previous radially travelling oscillations can be ex-plained by transient effects.

Distinctly different wave propagations are observedprior to and after coil current reversal. Prior to FRC forma-tion, the propagation is purely radial across the dipole field.Figure 11 shows a sequence of snapshots of the magneticfluctuations �Bz. Peaks of alternating sign, representing theaxial component of field line loops, propagate outward in the�y-direction with a typical speed of 25 cm /�s. Weak oscil-

Icoil

0

x = y = z = 00

0

100 A

50 mG

δBz

5 10 15 20t (µs)

0 20

20

δBz

(mG)

040 60

Icoil (A)

40

0 2

0

y

(cm)

−204 6

t (µs)

20

8

δJx (× 50 mA/cm2) x = z = 0

0

20

2−2

2−4

2−8

2−16

−20−2−2−2−4−2−8−2−16

(a)

(b)

(c)

FIG. 9. �Color online� High frequency magnetic oscillations generated by alarge amplitude low-frequency magnetic field of a loop antenna in an un-magnetized plasma. �a� Waveforms of the coil current and magnetic oscil-lations. �b� Nonlinear scaling of the oscillation amplitude with coil current.�c� Time-of-flight diagram for the wave magnetic current density �Jx�y , t�.Large amplitude oscillations propagate from coil outward and inward. Highfrequency oscillations are mainly observed inside the coil region ��y��8 cm� in the first three half-cycles.

022103-7 Nonlinear electron magnetohydrodynamics physics. VII… Phys. Plasmas 16, 022103 �2009�

lations start near the antenna and grow as they propagateradially outward. The wave propagation is across a back-ground magnetic field, hence must involve a highly obliquewhistler mode. The wave also travels in a plasma with aradial electron drift. The poloidal electron drift, associatedwith the toroidal magnetic field �Fig. 7�b�� is radially out-wards and can convect the transverse wave field �Bz. Theelectron drift and wave propagation are radially outward atevery coil current reversal.

It is also instructive to display the current density of thewave magnetic field, �Jx= ��� ��By ,�Bz��x /�0. Figure 12shows contours of �Jx�y ,z� for �a� the high-frequency oscil-lations observed at early times and �b� the large amplitudeoscillations propagating vertically outward prior to a coilcurrent reversal. In the latter case, the wave current formstoroidal current loops of alternating sign which expand radi-ally outward. Since the �Jx contours are symmetric with re-

spect to the z-axis and have opposite signs, they are toroidalcurrent rings. These oscillations have predominantly radialwavenumbers, k�ky.

The source current for the higher frequency oscillationsstarts with multiple current rings which merge into nestedcylindrical current shells, shown in Fig. 12�a�. In time, thesenested coaxial current layers propagate axially and radiallyoutward from the vicinity of the antenna. The oscillatingcurrents or fields propagate only over a limited distancewithin the magnetic “bubble,” defined as the volume of mag-netized electrons. The high frequency modes ��4 MHz�propagate the shortest distance, the larger low-frequencymodes propagate further, suggesting that the propagation dis-tance is determined by cyclotron resonance �fc /B=2.8 MHz /G�.

While a single current loop forms a dipolar magneticfield topology, two opposing and displaced loops form a

−20

−10

10

20

0

y

(cm)

x = 0

(δBy, δBz)

20 mG

−20 −10 10 20z (cm)

0

δBz (mG)

0

0.78

0.391.56

6.25

3.12

−0.

78−

1.56

−3.

12

−6.

25

12.525

−12.5−25

t = 6.92 µs t = 7.14 µs t = 7.4 µs t = 7.6 µs t = 7.8 µsB (G), (By,Bz ) t = 7.49 µs(d)(b)(a) (c) (e) (f)

1.47

0.54

3.98

10.83

(p)(n)(m) (o) (q) (r)

(j)(h)(g) (i) (k) (l)

FIG. 10. �Color online� Snapshots of the background magnetic field lines and strength �a–f�, wave magnetic field �Bz �g–l�, and wave vector field �m–r�before, during, and after coil current reversal.

−20 −10 10 20−20

−10

10

20

0

y

(cm)

z (cm)0

δBz (mG)

0

0.78

0.39 1.56

6.25

3.12

−0.39

−0.7

8

−1.5

6

−3.12

−6.25

t = 4.3 µs t = 4.4 µs t = 4.5 µs t = 4.6 µs t = 4.7 µs t = 4.8 µs

x = 0

vy ≈ 25 cm/µs

FIG. 11. �Color online� Sequence of snapshots of the magnetic fluctuations �Bz just prior to coil current reversal at t�4.9 �s. The oscillations originate nearthe coil, grow, and propagate radially outward with speed �vy��25 cm /�s.

022103-8 Stenzel, Urrutia, and Strohmaier Phys. Plasmas 16, 022103 �2009�

quadrupole field. The growth of current rings in the sourceregion creates a dynamic multipole magnetic field ��By ,�Bz�similar to that shown in Figs. 10�o� and 10�r�. Wave phasefronts and wavelengths can only be identified at some dis-tance from the source region. In Fig. 12�a�, the separation ofcurrent shells of equal sign would indicate wavelengths of��10 cm. The wavelengths increase with decreasing back-ground magnetic field. It is worth noting that the half-widthof the wave current density layers observed in Fig. 12�a� areclose to the inertial limit, �y��z�2 cm�1.5c /�p. Belowthis scale, the EMHD approximation begins to break down.

While the radially outward propagating oscillations havethe same properties for each half-cycle, the high frequencymodes differ on alternating half-cycles as already noted inFig. 9�c�. A hodogram of the wave magnetic field ��Bx ,�By�is found to rotate in the direction of the toroidal electron driftwhich reverses on alternating half-cycles �not shown�. Thesewaves may therefore also have a toroidal wavenumber caus-ing them to spiral radially inward, which can produce theasymmetries observed in Fig. 9�c�. A symmetric inwardpropagation implies that the current ring contracts, an asym-metry implies no toroidal current closure, hence a toroidaleigenmode.

IV. SUMMARY AND CONCLUSIONS

The present work has investigated the penetration of astrong magnetic field into an unmagnetized plasma. The ap-plied magnetic field oscillates at a frequency for which alllinear electromagnetic modes are cut off. However, at largefield strengths, the electrons become magnetized inside amagnetic bubble such that the time-dependent field canpropagate in the whistler mode. The penetration depth,speed, and topology become nonlinear. The superposition ofa dipolar field from the loop antenna and an opposing field,frozen into the plasma from a previous half-cycle, produce aFRC topology. The topology determines the current profilesin the plasma. A narrow current layer with half-width closeto the inertial scale is observed. The FRC develops locallymagnetic helicity density, but is globally helicity-free. Thesign of the helicity density is that of whistler modes, i.e., themagnetic field is frozen into the drifting electron fluid.

The electron drift creates instabilities. High-frequencymagnetic oscillations are observed and their origin, propaga-tion, and field topologies have been investigated. Since thereis no uniform background field, the high-frequency wavespropagate in the time-varying, nonuniform magnetic field ofthe applied low frequency field. The waves originate fromtoroidal current layers around the coil. They propagatewithin the bubble and are damped at the magnetic beach��c=��. Alternating current rings are shed off in radial di-rection across a strong dipolar background field. When thelatter changes to a FRC topology, shells of waves propagateboth axially and radially away from the FRC. In the sourceregion, the wave topology changes from dipolar to quadru-polar when current rings of opposite polarity are created atseparate locations.

Since the magnetic oscillations are associated with cur-rent layers, the electron drift could be the source of the in-stabilities. However, the mechanism does not appear to be aLandau wave-particle resonance since the radial wave vectoris orthogonal to the toroidal drift.25 The excited current ringshave no toroidal wavenumbers. In contrast, in the presenceof a background magnetic field,26 Landau resonance is pos-sible since wave propagation in the toroidal direction is ob-served. In the present case, electron tearing or velocity shearmay create the observed shedding of current rings.

Although of intrinsic interest, the waves are too small toplay a significant role in the energy transport. External mea-surements show that a large fraction of the applied power�8 kW /15 kW� is dissipated in the plasma. Evidence forelectron heating is obtained from visible light emission, pro-duced by electrons with energy 10 eV colliding inelasti-cally with neutrals and ions. The light pulsates indicating thatfast electrons gain and lose energy rapidly, suggesting adia-batic compression to account for most of the energization.The bulk electron temperature rises slowly, indicating that inthe absence of electron confinement the heating is smallsince the energy is distributed over a large reservoir ofplasma.

The properties of magnetic bubbles involve many inter-esting nonlinear phenomena which should be of general in-terest to rf penetration in unmagnetized plasmas, whistlerinstabilities, and nonlinear antenna phenomena.

ACKNOWLEDGMENTS

The authors gratefully acknowledge support for thiswork from NSF/DOE Partnership Grant No. ER54905.

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−20 −10 10 20−20

−10

10

20

0

y

(cm)

z (cm)0

x = 0

(a)

δJx (mA/cm2)

(b)

−20 −10 10 200z (cm)

500.4

−1.6

0

t = 0.65 µs t = 7.14 µs

0.8

−0.4−0.8

−50

0

0.4

0.81.6

3.16.312.5

−0.4

−0.8

−3.1 −12.5

FIG. 12. �Color online� Contours of the wave current density �Jx�y ,z� fortwo different types of waves, �a� high-frequency modes propagating radiallyand axially in strong coil fields, observed at early times, and �b� oscillationspropagating in the �y-direction prior to coil current reversals.

022103-9 Nonlinear electron magnetohydrodynamics physics. VII… Phys. Plasmas 16, 022103 �2009�

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