Nonlinear Electron Magnetohydrodynamics Physics. I. Whistler Spheromaks, Mirrors, and Field Reversed Configurations

PHYSICS OF PLASMAS 15, 042307 ͑2008͒

Nonlinear electron magnetohydrodynamics physics. I. Whistler spheromaks, mirrors, and field reversed configurations ͒ R. L. Stenzel,a J. M. Urrutia, and K. D. Strohmaier Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA ͑Received 7 December 2007; accepted 3 March 2008; published online 22 April 2008͒ The nonlinear interactions of time-varying magnetic fields with plasmas is investigated in the regime of electron magnetohydrodynamics. Simple magnetic field geometries are excited in a large laboratory plasma with a loop antenna driven with large oscillatory currents. When the axial loop field opposes the ambient field, the net field can be reversed to create a field-reversed configuration ͑FRC͒. In the opposite polarity, a strong field enhancement is produced. The time-varying antenna field excites whistler modes with wave magnetic fields exceeding the ambient magnetic field. The resulting magnetic field topologies have been measured. As the magnetic topology is changed from FRC to strong enhancement, two propagating field configurations resembling spheromaks are excited, one with positive and the other with negative helicity. Such “whistler spheromaks” propagate with their null points along the weaker ambient magnetic field, with the current density localized around its O-line. In contrast, “whistler mirrors” which have topologies similar to linear Ͼ whistlers, except with Bwave B0, have no null regions and, therefore, broad current layers. This paper describes the basic field topologies of whistler spheromaks and mirrors, while companion papers discuss the associated nonlinear phenomena as well as the interaction between them. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2903065͔

I. INTRODUCTION magnetohydrodynamic ͑MHD͒ configurations such as spheromaks18–20 or mirrors. Whistler waves are possibly the oldest plasma waves Investigating the properties of nonlinear wave packets 1,2 known. Although their linear properties are well forms the basis for understanding strong whistler turbulence, 3,4 documented, the regime of large amplitude whistler is still which occurs in magnetic reconnection geometries,21,22 mag- a topic of current research in space and laboratory plasmas. netospheric and solar plasmas,23,24 cometary tails,25 and plas- Nonlinear phenomena occur when the wave perturbs those mas with weak background magnetic fields such as in the plasma parameters which affect its propagation, typically ionosphere of Venus.26,27 density and magnetic field for whistlers in cold plasmas. The work is presented in a series of several companion When collisions and kinetic phenomena are included, a papers. Part I describes the field topologies, Part II28 the change of the electron distribution by strong whistlers will nonlinear propagation and wave-wave collision phenomena, also create nonlinear phenomena.5 Density perturbations cre- Part III29 the electron energization, Part IV30 the secondary ate modulational instabilities like filamentation,6–9 envelope whistler instabilities created by modifying the electron dis- 31 solitons,10,11 and parametric instabilities.12,13 Less is known tribution, and Part V discusses triggered whistler emis- about nonlinear effects of whistlers with “large” wave mag- sions. The experimental setup and measurement techniques are mostly explained in Part I and referred to in the other netic fields, defined here as wave fields, Bwave, exceeding the papers. ambient static magnetic field, B0. Since the whistler disper- sion depends on the magnetic field, a large wave can be expected to propagate at a different speed and direction from a linear wave. It is not obvious how the wave would propa- A. Experimental setup gate if it creates a magnetic null point. Theoretical solutions for two- and three-dimensional vortices have been The experiments are performed in a large ͑1 m diam, published,14–17 but do not address wave propagation. The 2.5 m length͒ pulsed dc discharge plasma generated with a present work answers such questions from an experimental 1 m diam oxide-coated cathode shown schematically in ͑ Ӎ 12 −3 Ӎ viewpoint. It is demonstrated that whistler modes with Fig. 1. The parameter regime ne 10 cm , kTe 2 eV, տ ͒ Bwave B0 do exist, how they propagate and interact with B0 =5–10 G, 0.4 mTorr argon is that described by electron other waves, and how they energize the electrons and, finally, MHD ͑EMHD, magnetized electrons, unmagnetized ions32͒. that the modification of the electron distribution function Insulated magnetic loop antennas ͑2–4 turns, leads to whistler instabilities. The field topologies of 10–30 cm diam͒ are inserted into the plasma center and a wave packets are measured and compared with well-known charged ͑1200 V͒ capacitor ͑0.1–10 ␮F͒ is discharged into the loop using a fast, high current transistor. This results in a a͒ Ӎ URL: http://www.physics.ucla.edu/plasma-exp/. Electronic mail: weakly damped oscillatory current Imax 100–300 A, which / [email protected]. decays with period T=2␲͑LC͒1 2 Ӎ5–50 ␮s, an example of

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1 m cathode 160 A Langmuir Coil and anode Iloop and coax probe y Vloop 0 B z B0 1.5 m x 1kV 0.15 MW ≈ 12 −3 ne 10 cm ≈ Ploop 3D Magnetic kTe 3.0 eV probe B0 =5G 0 pn = 0.4 mTorr (Ar) 2.5 m 0.135 J Plasma

FIG. 1. Schematic of the experimental setup and relevant parameters. Uloop Vacuum which is shown in Fig. 2. In vacuum, the magnetic field in the center of the loop at the peak current can be as large as B B 0 =100 G 0 =5–10 G. 0 5 10 15 20 Figure 2 shows the measured antenna voltage and cur- t(µs) rent waveforms as well as the calculated instantaneous power and the energy deposited into the antenna. The dissipation in FIG. 2. Measured antenna current and voltage vs time at the antenna input vacuum occurs mainly in the resistance of the antenna and terminal. Calculated instantaneous power P͑t͒=I͑t͒V͑t͒ and energy ͐ transmission line since the free space radiation resistance is U= Pdt. In vacuum the energy loss by the circuit resistance is small compared to that by the radiation resistance in plasma. negligibly small. However, in the high-beta plasma, a large amount of energy is transferred into whistler modes which is evident from the increase in damping, measured by the time currents, B , from those created by antenna currents, ␶ / plasma constant =2L R. There is also a small increase in the ring- Bcoil, and to demonstrate important differences in the axial ͓␻ ͑ −1 −1 ␶−2͒1/2͔ ing frequency, = L C − , which is dominated by field direction relative to B0. a decrease in inductance L rather than the increase in damp- A Langmuir probe is also attached next to the magnetic ing. The radiation resistance due to whistlers is found to be probe so as to measure the plasma parameters in three- ͑␶−1 ␶−1 ͒Ӎ ⍀ ⍀ ⍀ Rrad=2L plasma− vacuum 7.1 −2.3 =4.8 . For a peak dimensional ͑3D͒ space and time. Typically, at a given probe 2 / current of 150 A, the wave power ImaxRrad 2=54 kW position, the probe voltage is fixed and the current is re- represents a significant fraction of the applied power corded versus time, then the voltage is incremented in small / ͑ ͒͑ ͒/ Ӎ Pin=ImaxVmax 2= 1200 V 150 A 2 90 kW. steps so as to obtain the time-resolved current-voltage traces. Further useful information is obtained in Fig. 2 from the time-resolved energy deposition. After the first half cycle of II. EXPERIMENTAL RESULTS the current waveform, only Ϸ5% of the peak energy is lost A. Single antenna current pulse in vacuum, while in plasma about Ϸ50% is deposited. In the second half cycle, when whistler spheromaks are excited, the We review briefly the excitation of EMHD fields by a fraction of energy deposition is even larger. These values single current pulse which have been reported in detail will later be compared with the internally measured magnetic earlier.33–36 Here, a current pulse in the form of a single field energy. half-sine wave is applied whose polarity is chosen so as to The local magnetic field is measured with a single mag- produce an antenna field opposing the ambient field. The netic probe containing three orthogonal small loops antenna field is larger than the ambient field, hence the net ͑5 mm diam͒ which can be moved along three coordinates, r, field on axis near the coil becomes reversed. ␪,z. The space-time dependence of the field B͑r,t͒ is ob- Figure 3 shows contour maps of the net axial magnetic ͑ ͒ ͑ ͒ tained from rapidly 1Hz repeated experiments which are field component Bz x=0,y,z at different times during the highly reproducible ͑␦n/nϽ3%͒. The probe signals are re- antenna current pulse. The time is indicated by the end of the corded with a four-channel digital oscilloscope with up to antenna current waveform inserted into the lower left corner 1 ns time resolution, averaged over 10 shots at each position of each frame. The total magnetic field consists of the axial ͑ ͒ to increase the amplitude resolution 10 bits . The data pro- dc magnetic field, B0, the field produced by the coil current, ͑ ϰ / ͒ cessing includes integration Vprobe −dB dt , coordinate and the field due to induced plasma currents. In the presence transformation from ͑r,␪,z͒ to ͑x,y,z͒, spatial differentiation of plasma, the penetration and decay of the time-varying ͒ ͑ ϫ /␮ٌ to obtain the current density, J= B 0, and integration to field-reversed configuration FRC is delayed. obtain magnetic flux, ⌽=͐B·ds, and magnetic energy, During the current rise ͓Fig. 3͑a͔͒, the induced magnetic ͐ 2 / ␮ U= B 2 0dV. At each probe position, we measure the field opposes the applied field, consistent with Lenz’s law. fields in vacuum and in plasma with different current polar- The net axial field is therefore enhanced at some distance ity. This allows us to distinguish the fields created by plasma from the antenna ͓͉z͉Ӎ15 cm in Fig. 3͑a͔͒, implying that the

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y 0 (cm) −10

15 G t = 4.2 µs −20 −40 −30 −20 −10 0 10 z (cm)

͑ ͒ FIG. 4. Vector field of the total magnetic field, By,z x=0,y,z after the end of the coil current pulse. The FRC field is maintained by electron currents flowing toroidally and preferentially in the magnetic null layer. The current is driven by a toroidal electric field associated with decreasing axial ͑poloi- dal͒ magnetic flux. The decay is due to dissipation which manifests itself by electron heating.

During the fast turn-off of the antenna current, electron currents are induced which, by Lenz’s law, try to maintain the FRC. They flow in the vicinity of the antenna across the magnetic field as electron Hall currents. Once the antenna current is zero, the FRC freely relaxes ͓Fig. 3͑c͔͒. The plasma is large compared to the FRC, con- ducting boundaries are far from the FRC, and the coil carries no current. The FRC expands axially and contracts radially ͓Fig. 3͑d͔͒. During the change from an oblate to a prolate ͑ ͒ elliptical shape, the center Bmax of the FRC does not move. Thus, the relaxation process cannot be described by a wave propagation phenomenon. As it relaxes, the prolate FRC also exhibits tilt. As seen with previous larger FRCs, tilting occurs early on in the FRC evolution and does not disrupt the structure. The tilt is not stabilized by wall currents. Two components of the FRC magnetic field are shown during its relaxation as a vector field in Fig. 4. The third component, the toroidal field Bx, is negligible in the center of ͑ Ӎ ͒ the FRC, small Bx 0.1Bz and of opposite sign near the expanding ends. The field has positive helicity density at the end which expands along B0, negative helicity density at the left end, such that the total helicity of the field is zero. The FRC is maintained by a solenoidal current ring centered near the magnetic null layer ͑yӍ Ϯ7cm,−10ՇzՇ10 cm͒ and dominated by electron Hall currents. An inductive toroidal FIG. 3. ͑Color online͒ Contour maps of the axial component of the total electric field develops in the plasma to try to maintain the ͑ ͒ ϫ magnetic field, Bz x=0,y,z at different times, t, after applying a half-cycle FRC. Radial E B drifts move field lines toward the dissi- sinusoidal current pulse to the magnetic loop antenna ͑20 cm diam͒. ͑a͒ pation region, converting magnetic energy into electron ther- During the current rise, a field enhancement is induced, while a field rever- mal energy. When the field reversal vanishes the magnetic sal grows inside the coil. ͑b͒ The perturbation, a “whistler mirror,” propagates axially in the whistler mode. The FRC is still present at the end of the perturbation can propagate as a low-frequency whistler wave antenna current pulse. ͓͑c͒ and ͑d͔͒ The FRC stretches axially during the free packet. relaxation but remains stationary, implying no propagation as in ͑b͒. B. Oscillatory antenna currents net field lines are compressed as in a single mirror coil. In Oscillatory antenna currents are observed to induce field time this field “perturbation” propagates in the whistler mode perturbations that always break up into oppositely propagat- axially away from the antenna ͓͉z͉Ӎ30 cm in Fig. 3͑b͔͒, ing wave packets, irrespective of polarity and amplitude. In hence the perturbation is appropriately called a “whistler EMHD plasmas, such waves propagate in the whistler mode. mirror.” It leaves behind a FRC near the antenna ͓Fig. 3͑c͔͒. However, the present waves are not linear waves and a dis-

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(a) (b) (c) (a) 2−D Toroidal Null Lines

v v whistler B whistler

FIG. 5. Schematic showing null lines and null points during the rise of the Electron Current antenna current in vacuum. ͑a͒ Toroidal null line is formed for weak antenna Separatrix Surfaces currents. ͑b͒ Null line collapses into a degenerate 3D null point on axis for ͑ ͒ (b) larger currents. c For still larger currents, a FRC with two 3D null points Degenerate on axis is formed. 3−D Null Point persion relation for nonlinear waves is generally not mean- vwhistler vwhistler ingful. Thus, it is more appropriate to describe such waves B by their space-time dependence. 3−D Radial Null Point C. Schematics of field topologies 2−D Toroidal Null Lines (c) We first describe the field topology near a loop antenna in a uniform background magnetic field. In order to produce ␪=0͒, the loop axisץ/ץ͑ a simple two-dimensional geometry has been aligned with the background magnetic field, B0 B ͑ ͒ = 0,0,B0 . Figure 5 schematically shows field lines at vari- ous phases of the formation of a FRC by coil currents. There exists a toroidal magnetic null line inside the coil for small ͓ ͑ ͔͒ currents Fig. 5 a . As the current grows, the null line moves v v radially inward until it collapses into a single three- whistler whistler Propagating Spheromaks dimensional null point on axis ͓Fig. 5͑b͔͒. Since this null point connects two separatrix surfaces, it is denoted as a FIG. 6. ͑a͒ Schematic of field topology when the antenna current reverses “degenerate” null point.37 Further current increase produces sign compared to the induced plasma currents that maintain the FRC. ͑b͒ two ordinary 3D null points at the intersection of the sepa- Intermediate stage of spheromak formation. The inner null line has formed a ͓ ͑ ͔͒ degenerate 3D null point on axis. ͑c͒ The degenerate null point is topologi- ratrix surface with the z-axis Fig. 5 c . In plasma, induced cally unstable and splits into two regular 3D radial nulls, thus forming two electron currents delay the field penetration, but do not fun- spheromaks on either side of the coil. damentally alter the near zone field topology. When the antenna current decreases, the process is re- versible in vacuum, but not in plasma. In the latter case, toroidal electron currents are induced and maintain the FRC D. Measured field topologies even in the absence of the coil current ͓see Fig. 3͑b͔͒. When the coil current reverses sign, the field lines near the coil The properties of a whistler mirror are summarized in produce two toroidal null lines inside the FRC, as shown Fig. 7 which shows snapshots of the field components in the ͑ ͒ ͑ ͒ schematically in Fig. 6͑a͒. This creates new separatrix sur- axial y–z plane. The vector field By ,Bz shown in Fig. 7 a , faces inside the FRC separatrix. The new surface separates confirms that when a growing reverse field is imposed by the field lines around the coil from those encircling the plasma antenna, two field enhancements are induced on either side currents. of the antenna ͓͉z͉Ӎ20 cm͔. The axial field component, The null lines move radially away from the antenna with shown in the contour plot of Fig. 7͑b͒ has been increased increasing antenna current. In this process, field lines sur- from B0 =5 G to Bmax=12.5 G forming a mirror ratio of rounding the coil reconnect with field lines inside the FRC RӍ2.5. The axial field is produced by a toroidal current or separatrix to form new field lines around the toroidal plasma electron EϫB drift. It twists frozen-in field lines in the di- current rings. When all the FRC flux has reconnected, the rection opposite to the toroidal current density, i.e., into the inner null line reaches the axis to form again a 3D degenerate plane in the upper half plane and out of the plane in the null point, as shown in Fig. 6͑b͒. lower half plane. A field line which for yϾ0 bends into the At this time, there exists two field-reversed regions ad- plane will develop a positive Bx component at the left mirror jacent to the coil. Further increase in antenna current and a negative component at the right mirror where it splits the degenerate null point into two regular 3D nulls as straightens out again. The signs are reversed for yϾ0. This shown in Fig. 6͑c͒. It will be shown that each field reversed quadrupolelike field pattern for the out-of-plane field compo- ͑ ͒ region develops magnetic helicity of the appropriate sign nent Bx is shown in Fig. 7 c . The field lines have a left- self-consistently, hence resembles the field topology of handed twist, or negative magnetic helicity, for the whistler spheromaks. mirror propagating against B0, and positive helicity for

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-tric field is balanced by magnetic forces, −␮ٌB. The poloi dal current loop contains radial Hall currents which are produced by toroidal electric fields. These exist at either side of the mirror where the poloidal flux increases or decreases due to the motion of the field. This inductive electric field is also responsible for the axial energy flow, Sz =E␪Br, while the radial electric field produces only a rotation in the Poyntings vector, S␪ =ErBz. The convective derivative of the toroidal flux also produces an inductive electric field. It has a radial component Er in the center that convects the toroidal magnetic energy, Sz =ErB␪, and contributes to the toroidal Hall ϫ current, Jz =−neEr Bz. Since the wave magnetic field is larger than the ambient field, the wave may be expected to travel in its own field. This will be explored and discussed in Part II.28 The corresponding properties of a whistler spheromak are shown in Fig. 8. The poloidal field in the y–z plane along the coil axis, shown in Fig. 8͑a͒, demonstrates that a field reversed topology detached from the antenna as it imposes a ͑ ͒ rising field along B0. Figure 8 b shows contours of the total axial field component Bz, clearly demonstrating the field reversal by plasma currents. Figure 8͑c͒ presents contours of ϯ ͑ ͒ the toroidal magnetic field component B␪ = Bx x=0 which ͑ ͒ peaks near the O-type nulls of the poloidal field By ,Bz , demonstrating field line linkage or magnetic helicity. Unlike in the FRC case, the orthogonal field components are of comparable strength. The toroidal field is produced by poloi- ͑ ͒ dal currents Jy ,Jz . The “out-of-plane” field is very common in EMHD and readily explained by the frozen-in concept of field lines.32 The toroidal electron drift, v␪ =−J␪ /ne, convects field lines out of the plane and generates the Bx component. The field lines upstream of the two spheromaks are unperturbed, and the downstream field lines are rotated, hence the B␪ components have opposite signs in each spheromak. The spheromak that propagates against the ambient magnetic field al- FIG. 7. ͑Color online͒ Properties of a whistler mirror. ͑a͒ Vector field of the ways has negative helicity while the one propagating along ͑ ͒ ͑ ͒ ͑ ͒ total poloidal magnetic field, By ,Bz ,inthey−z plane x=0 . b Contour B0 has positive helicity, as for whistler mirrors. Circularly ͑ ͒ plot of the total axial magnetic field component Bz x=0,y,z , indicating a polarized plane whistlers propagating along B also have Ӎ ͑ ͒ 0 mirror ratio R 2.5. c Toroidal field component Bx, showing that the whis- positive magnetic helicity as well as positive helicity of the tler mirror has the same helicity properties as linear whistlers ͑Ref. 41͒. polarization ellipse in time since the wave field rotates in the same sense as electrons orbit around B0. Finally, an example of the field line topology in 3D ͑ ͒ propagation along B0. The total helicity is zero integrated space is presented in Fig. 8 d , albeit from a different data set over the entire volume. There are no magnetic null points in ͑10 cm coil diameter͒. It resembles that of a spheromak or a whistler mirror. magnetic vortex, yet, in contrast to a stationary MHD The toroidal magnetic field B␪ is produced by a poloidal spheromak, it propagates in the whistler mode, hence is ap- ͑ ͒ current flow, Jy ,Jz which is partly field-aligned, partly a propriately called a “whistler spheromak.” This spheromak Hall current. It is linked with the toroidal current such that propagates along z or B0 with positive helicity. Field lines the net current lines are helical. The helicity of the current traced near the axial null points reverse direction in the cen- density lines has the same sign as that of the magnetic field ter of the spheromak. Note that a single spheromak leaves a lines. net twist to field lines near the separatrix. The second sphero- Hall physics explains most features of the wave packet. mak reverses this twist. A FRC has opposite twists at each A space charge imbalance is produced because a radially end but none in the center. Electron inertial effects are neg- /␻ outward magnetic force is exerted on the electrons but not on ligible since the spheromak size is large compared to c p. ϫ ͑ ͒ the ions, J␪ Bz. It creates a positive plasma potential inside Figure 9 a shows contours of the toroidal current den- ϫ͑ ͒/␮ٌ the whistler mirror irrespective of propagation direction. The sity Jx = By ,Bz 0 which peaks in the null region of the associated radial electric field produces the toroidal Hall cur- poloidal field or near B␪,max. The peak current density of Ӎ / 2 /͑ ͒ rent, J␪ =−neErBz. The associated parallel space charge elec- J␪ 6A cm yields an electron drift velocity vd =−J ne

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͑ ͒ ͑ ͒ FIG. 9. Color online Contour maps of a toroidal current density Jx and ͑ ͒ b helicity of the current density, JxBx. The sign of helicity is always negative for spheromaks propagating against the ambient ﬁeld and positive for propagation along B0.

ring current around the O-line. This differs from that of the whistler mirror, which has a broad current layer, unrestricted by null regions. The product J␪ ·B␪ makes the dominant contribution to the helicity, ͐B·ٌ͑ϫB͒dV, of the current density, J,orthe / ͑ ͒ electron fluid velocity, ve =−J ne. Figure 9 b shows that again the left-propagating spheromak has a negative sign for J␪ ·B␪ while the right-propagating one has positive helicity. Likewise the contribution from the poloidal current, Jz ·Bz,is ͑ ͒Ͼ negative for the left spheromak where Jz x=y=0 0, as de- Ͻ termined by Bx, while Bz 0 within the separatrix. For the right spheromak, Jz reverses sign but Bz does not, hence Ͼ Jz ·Bz 0. The remarkable stability of the toroidal current ring is worth pointing out. The ring does not radially expand as expected for a fluid conductor subject to a JϫB hoop force FIG. 8. ͑Color online͒ Vector field of the total poloidal magnetic field, and pressure gradients. The outward forces are balanced by ͑ ͒ ͑ ͒ ͑ ͒ By ,Bz in the y−z plane x=0 showing the topology of a detached whis- the inward force of the Hall electric field such that the sys- -tler spheromaks. Contour maps of ͑b͒ the axial net magnetic field compo- tem remains nearly force-free, JϫB−neE−ٌpӍ0. The cur ͑ ͒ ͑ ͒ nent Bz x=0,y,z , and c the toroidal field component Bx. Note that the ͑ ͒ rent ring shrinks during the propagation and relaxation, indi- linkage of the toroidal field B␪ and the poloidal field By ,Bz produces magnetic helicity which has opposite signs for the two spheromaks. ͑d͒ Field cating that the poloidal magnetic flux through the toroidal lines in 3D space for a whistler spheromak propagating along B0. current ring is not conserved. The decaying poloidal flux produces a toroidal electric field which drives the toroidal current, J␪ ʈE␪, causes dissipation, J␪ ·E␪ Ͼ0, and injects 7 tϾ0. Note that theץ/A·B͒͑ץ= Ӎ4ϫ10 cm/s which is smaller than the electron thermal magnetic helicity, −2E␪ ·B␪ Ӎ ϫ 7 / velocity ve,th 12 10 cm satkTe =4 eV. The drift could right and left spheromaks have the same sign of E␪, J␪, hence be even smaller if the current was carried by tail electrons. dissipation, yet opposite signs for B␪, hence helicity. The The spheromak current layer is concentrated in the form of a important question as to what determines the self-consistent

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͑ ͒ ͑ ͒ FIG. 10. Color online Contour maps of Bz x=0,y,z at different times t showing the creation and ejection of two whistler spheromaks. During the current rise, a whistler mode is emitted which enhances the net axial field. Near the antenna field-reversal occurs in which the antenna field forms an X-type toroidal null line that moves toward the axis ͑a͒. As the current decays, plasma currents maintain the FRC created by the antenna ͑b͒. After the current zero crossing, the plasma FRC is squeezed by the new antenna field ͑c͒, splitting it into two new structures ͑d͒. Once created, these field configurations travel away axially in the whistler mode ͑e͒. They develop helicity, hence have a spheromaklike topology. Then, the process begins anew ͑f͒.

electric field, current density, and dissipation will be investi- E. Space-time evolution of fields gated later. Analogous to the time evolution of the fields for a single For simplicity, the schematic diagrams assumed axial pulse ͑Fig. 3͒, we now describe the field evolution observed symmetry of the fields whereas the observed spheromaks fre- for an oscillatory current drive and compare it with the sche- quently are slightly tilted with respect to the axial back- matic pictures of Fig. 6. Figure 10 shows contour maps of ͑ ͒ ground magnetic field. The tilt has some important conse- Bz x=0,y,z at different times t of the antenna current. Dur- quences for the field topology which have been investigated ing the first current rise, a whistler mode is emitted which 38 theoretically for FRC fields ͑B␪ =0͒, and confirmed enhances the net axial field as in Fig. 3͑a͒. Figure 10͑a͒ experimentally.39 First, the tilt shifts the two 3D null points shows that near the antenna an X-type toroidal null line off the original axis through the center of the FRC. Second, moves toward the axis as field-reversal occurs. A FRC then the toroidal null line of a tilted FRC has two opposing toroi- develops as the potential field of the antenna turns the null ͓ ͑ ͔͒ dal magnetic fields. These cancel in two 3D spiral null line into two axial 3D null points Fig. 10 b . When the antenna current reverses sign, the FRC is pinched off under points40 located in the plane normal to the tilt axis and the coil ͓Fig. 10͑c͔͒ and two regions of field reversal are through the center of the FRC. The separatrix is not a closed formed ͓Fig. 10͑d͔͒. These fields then propagate axially away surface. A tilted spheromak with strong toroidal field has, to in the whistler mode ͓Figs. 10͑e͒ and 10͑f͔͒. They possess our knowledge, not yet been investigated. helicity, hence have a spheromaklike field topology, yet be- In ideal EMHD, the magnetic field is frozen into the have like large-amplitude whistler modes which do not inter- electron fluid and creates only rotational flows since the elec- act with the ions. Ӎ trons are incompressible when ne ni and the ions are stationary. To first order, E+vϫB=0, hence the electromag- III. CONCLUSIONS netic field is force-free, evidence for which is the fact that an FRC current ring does not expand. Consequently, a current In summary, we have shown experimental evidence that whistler modes with wave magnetic fields larger than the ring of magnetic moment m should also be torque-free in a ambient magnetic fields do exist. Field aligned structures, uniform magnetic field, mϫB =0. The observed tilt of a 0 called whistler mirrors, are large amplitude plasma responses current ring does indeed not grow. Instead, the axis of the that are carried by broad electron Hall currents. Of particular tilted current ring precesses since the transverse field com- interest are field topologies with magnetic null points where ponents are convected by the toroidal fluid drift. These pro- the linear whistler mode should not locally exist. These 34 cesses have been observed for EMHD FRCs, but appear to structures are called whistler spheromaks, due to the helicity hold also for propagating spheromaks which tilt back and carried by their fields as they travel. Observations show that forth during propagation. Such oscillatory tilts are simply the null points propagate with the wave structure in the di- projections of a 3D precession on a two-dimensional plane. rection of the weaker ambient magnetic field. Unlike whistler

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