Nonlinear Electron Magnetohydrodynamics Physics. III. Electron Energization K

PHYSICS OF PLASMAS 15, 042309 ͑2008͒

Nonlinear electron magnetohydrodynamics physics. III. Electron energization K. D. Strohmaier, J. M. Urrutia, and R. L. Stenzel 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 Wave-particle interactions of low-frequency whistler modes with wave magnetic fields exceeding the ambient field are investigated experimentally. These highly nonlinear modes are excited with magnetic loop antennas in a large magnetized afterglow plasma. While the nonlinear wave properties are described elsewhere, the present paper focuses on the modification of the electron distribution function by the whistler waves. When the electron current flows in regions of magnetic nulls, such as in spheromak and field-reversed configurations ͑FRCs͒, strong electron energization is observed. When the whistler modes are created by electron Hall currents, such as in whistler mirrors, no significant energization occurs. The electron temperature can be raised locally by an order of magnitude. Non-Maxwellian distributions with energetic tail electrons are observed. Electron energization to տ10 eV produces visible light emission whose time and space dependence is mapped. The light source travels with the subthermal speed of whistler spheromaks. When counterpropagating spheromaks collide, the resultant FRC produces strong local heating and light which dissipates its free magnetic energy. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2903070͔

I. INTRODUCTION and perform nonadiabatic motions adjacent to the null line. Comparatively little electron heating is observed in large- The resonant interaction of electrons with whistler amplitude whistlers without null lines. waves has been studied for a long time.1,2 Two wave-particle resonances can occur: Whistlers propagating oblique to the II. EXPERIMENTAL SETUP background magnetic field B0 have parallel electric fields which can give rise to Landau resonance when the electron The experimental device and the method for measuring magnetic fields have been presented in Part I.8 Here we only velocity matches the parallel phase velocity; i.e., v=␻/kʈ. These resonant electrons can gain energy via Landau damp- describe the diagnostic techniques for measuring the proper- ing or give energy to the wave if the distribution function has ties discussed in the present manuscript; i.e., the electron a positive slope at the phase velocity. energization by strong whistler modes and related param- A second resonant interaction arises from cyclotron reso- eters. nance even though the wave frequency is below the cyclo- ␻ A. Fast Langmuir probe diagnostics tron frequency c. It occurs when electrons move along B0 opposite to the wave vector k, so as to experience a Doppler- Langmuir probes are a well-established diagnostic tool ␻ ␻ 10–12 upshifted frequency; i.e., −k·v= c. These resonant elec- in plasma physics. Although the theory has been formu- trons gain perpendicular energy from the wave or transfer lated for steady-state conditions,13 probes are often used in energy to the wave if the electron distribution has excess time-varying plasmas or with rapidly varying probe voltages. perpendicular energy. Two limitations arise under these conditions: First, the exter- In large amplitude waves, stochastic rather than resonant nal probe circuit has to be designed to provide the fast time 3–7 wave-particle interactions are dominant. In the present resolution. Second, an inherent problem arises when the time case, where the wave magnetic field exceeds the ambient variations are faster than the establishment of the dc sheath field, one has to start from first principles and analyze local which is of order of the ion plasma period. The current- heating processes within the nonuniform field. In general, voltage characteristics on such fast time scales differs electron heating can be expected in regions of large electric from the dc current-voltage ͑I-V͒ curves, resulting in an in- fields along magnetic fields or magnetic null lines. Conse- correct evaluation of the plasma parameters. Well-known ex- quently, these will be regions of large current densities. Hall amples are the broadening of the I-V characteristics in rf electric fields ͑EЌB͒ or currents will not energize electrons. plasmas14 and current overshoots on rapidly swept or pulsed Not every magnetic null point topology produces current probes.15,16 sheets, reconnection and dissipation. For the present field The approach taken in rf plasmas is to let the probe topologies of whistler spheromaks and mirrors,8,9 the domi- potential oscillate with the plasma potential while varying nant heating occurs in a toroidal O-type null line with in- the dc potential so as to extract only the dc plasma proper- duced toroidal electric field. Electrons are directly acceler- ties. In the present case, however, we are interested in the ated to high energies, scattered by collisions and fluctuations, instantaneous plasma potential and other plasma parameters

and have taken the following approach: A single planar (a) Langmuir probe of area 1.5 mm2 is mounted on the same shaft as the magnetic probe so as to measure simultaneously Iris Light−emitting diode electromagnetic fields and plasma parameters in three- (movable, dimensional ͑3-D͒ space and time from repeated experi- Photomultiplier pulsed) ments. The radial probe shaft is perpendicular to the toroidal inductive electric field so that no voltages are induced on the transmission line to the probe tip. The probe is biased with a (b) (c) dc voltage. However, the voltage is applied via a transistor Light switch just prior to the excitation of whistler modes. This minimizes perturbations to the afterglow plasma density (Arb. when drawing electron saturation current.17 The probe cur- Units) rent is measured versus time on a properly terminated trans- 0 0 50 100 150−4 −2 0 2 4 mission line between the probe and a digital oscilloscope t (ns) x (cm) with as little as 1 ns time resolution. Simultaneously, the probe voltage is also recorded since it is slightly loaded by FIG. 1. ͑a͒ Schematic, ͑b͒ temporal, and ͑c͒ spatial resolution of the optical different probe currents. At a given probe position, the volt- diagnostics to measure light emission generated by electron acceleration in nonlinear whistlers. age is incremented from ion to electron saturation currents in typically 300 steps ͑⌬V=0.5 V͒. At each voltage, the experi- ment is repeated ten times and the data are averaged. Subse- ͑ ͒ quently, the current-voltage characteristics are constructed at able point light source light-emitting diode . The temporal ͑ ␾ resolution is obtained by pulsing the light source with a fast each time sample, and the plasma parameters potential pl, electron temperature kT , and electron density n ͒ are evalu- rising current. The observed rise and fall times of the optical e e ⌬ Ӎ ated. Incrementing the probe voltage adds effectively a system are t 50 ns. The optical setup is mounted on a fourth dimension to the data set, making 3-D measurements movable platform so as to scan radially or axially across the of plasma parameters extremely time consuming and two- heated plasma. dimensional radial and axial scans more practical. The photomultiplier output is proportional to the light Although the probe voltage is held constant, rapid varia- intensity, which has been verified with calibrated absorption tions of the plasma potential produce the same effects as filters. The measured light intensity depends on various fac- rapidly varying probe voltages in a dc plasma. In the latter tors such as the electron energy, velocity distribution func- case, a positive voltage ramp creates a current overshoot near tion, density, dimensions along the line of sight for transpar- the plasma potential where the sheath changes from ion-rich ent plasmas, and emission spectrum relative to the spectral to electron-rich. Due to ion inertia, there is temporarily an response of the detector. Thus, quantitative measurements of ion excess above the plasma potential that allows larger elec- the electron properties are difficult to obtain except in special tron currents to be collected. The same occurs for a probe circumstances; for example, in a Maxwellian afterglow with fixed bias when the local plasma potential rapidly de- plasma. There, the light intensity Ilight has been plotted vs creases. Likewise, when the probe voltage sweep is reversed electron temperature measured with a Langmuir probe, ͑ ͒ or the plasma potential rises rapidly, the collected current is which yields a nonlinear relation Ilight kTe . However, such a reduced near the plasma potential compared to steady-state calibration may not be applicable to wave heating, which can conditions. Experimentally, pronounced overshoot phenom- produce runaway electrons. Thus, the main usefulness of the Յ Ϸ ␲/⍀ optical diagnostics is to verify the existence of energetic ena are observed on time scales of t 100 ns 2 p. A fundamental limitation of Langmuir probes is their electrons and to determine the light source in space and time. inability to resolve anisotropic distribution functions that can occur in collisionless plasmas in the presence of strong fields III. EXPERIMENTAL RESULTS and currents. Although directional velocity analyzers do ␾ exist,18 their use in complex magnetic and electric fields is A. Probe measurements of kTe, pl, and ne

limited. Non-Maxwellian distributions can be inferred from a We first show basic Langmuir probe log10 I-V character- non-constant slope of the log10 I-V characteristic, but one istics and their modifications by strong whistler modes. Prior cannot obtain the angular dependence of, e.g., energetic elec- to the excitation of waves ͓t=−5 ␮s in Fig. 2͑a͒, current is tron tails. applied to the loop antenna at t=0͔, the afterglow electron temperature is low ͑2eV͒, uniform and almost constant on the whistler time scale. At tӍ1.2 ␮s, the whistler mirror B. Light emission diagnostics arrives, which increases the electron temperature to 5 eV and Visible light emission from energetic electrons raises the plasma potential to +35 V on axis. It is followed at ͑ / 2 Ͼ ͒ 19 1 2mev 10 eV colliding with neutrals and ions is mea- tӍ3.2 ␮s by a whistler spheromak, which heats the elec- sured with a photomultiplier setup schematically shown in trons dramatically to 20 eV and lowers the plasma potential Fig. 1͑a͒. Simple collimating optics provides for spatial res- to −25 V. Even stronger heating is observed closer to the olution transverse to the line of sight. The width at half- antenna where the whistler spheromak fields are more in- intensity is measured at ⌬xӍ2cm͓Fig. 1͑b͔͒ using a mov- tense. 042309-3 Nonlinear electron magnetohydrodynamics physics. III… Phys. Plasmas 15, 042309 ͑2008͒

10 (a) Spheromak, Iprobe 20 eV, µ 1 t = 3.2 s Mirror, (mA) 6 eV, t = 1.2 µs 0.1 Afterglow, 2 eV, µ x=y=0 t=−5 s ∆z=23cm

10 (b) 1 y=10cm Bulk, 5 eV Iprobe y=−2cm 0.1 y=14cm Spheromak µ (mA) t = 2.5 s x=0 Tail, 32 eV ∆z=7cm 0.01 −40 0 40 80 Vprobe (V)

͑ ͒ FIG. 2. Color online Langmuir probe traces log10 I vs V at different positions and times during the passage of intense whistler modes. ͑a͒ At an axial distance ⌬zӍ23 cm from the antenna three traces are shown on axis: one in the unperturbed afterglow plasma prior to the start of the antenna current ͑t=−5 ␮s͒, a second in the presence of a whistler mirror, and a third in the presence of a whistler spheromak. The plasma potential is positive in a mirror and negative in a spheromak. ͑b͒ At a closer distance from the antenna ͑⌬zӍ7cm͒, three traces at different radial positions are shown during a whistler spheromak. Non-Maxwellian distributions with energetic tails ͑ ͒ ͑ Ӎ ͒ 32 eV and hot bulk Maxwellians kTe 30 eV at y=10 cm are produced. FIG. 3. ͑Color online͒ Basic plasma parameters, derived from Langmuir probe traces, vs radial position y and time t at x=0, z=23 cm, antenna at z=0. ͑a͒ Antenna current waveform for timing reference. ͑b͒ Bulk electron ͑ ͒ temperature. Strong heating is produced only by whistler spheromaks that Figure 2 b shows three I-V traces at different radial Ϸ / Ͼ ͑ ͒ ⌬␾ are excited when Icoil 0 with dIcoil dt 0. c Plasma potential change pl positions cutting through the toroidal current layer of a Ϸ / Ͻ increases in whistler mirrors excited when Icoil 0 with dIcoil dt 0, and Ӎ Ϸ / Ͼ spheromak at z 7 cm. Within the current layer decreases in whistler spheromaks excited when Icoil 0 with dIcoil dt 0. ͑ ͒ ͑ ͒/ Ͻ ͑yӍ10 cm͒, one finds almost monoenergetic electrons with d Electron density ne shows an inherent density gradient d ne dy 0, but Ӎ ͑ ͒ no significant change by either whistler mirrors or spheromaks. kTe 30 eV. Adjacent to the current layer y=14 cm and near the center ͑y=−2 cm͒, the bulk temperature drops to 5 eV but energetic electron tails still remain. Note that the Larmor radius of 30 eV electron in a field of 10 G is only electrons out of mirror fields and into spheromak fields. Al- 1.8 cm; hence, the energetic electrons are still highly magne- ternatively, the pinched magnetic field lines of a mirror tized and do not simply spiral out of the toroidal current would like to straighten out and carry the frozen-in electrons layer. axially outward, leaving an excess positive charge or positive Next we show in Fig. 3 the plasma parameters, evaluated plasma potential inside the mirror. Conversely, the bulged from the Langmuir probe traces, versus radial position y and field lines in a spheromak push the electrons inward and time t at x=0, z=23 cm. For reference, the antenna current is create a negative charge/potential. In detail, the force balance shown in Fig. 3͑a͒. The electron temperature ͓Fig. 3͑b͔͒ has is probably more complicated since the potential minimum is been evaluated in the electron retardation region, i.e., just slightly delayed with respect to the heating maximum; i.e., below the knee of the I-V characteristics; hence, refers to the toroidal current layer. There is also an inductive poloidal bulk electrons rather than the tail. The largest electron heat- electric field due to the time-varying toroidal magnetic field. Ϸ ing occurs in whistler spheromaks excited when Icoil 0 with For a whistler mirror, the radially outward pointing space / Ͼ ͑␾ ͒/ dIcoil dt 0. Since there is little heat confinement, the elec- charge electric field, i.e., Ey =−d pl dy, has the correct di- tron temperature is raised only in the presence of the sphero- rection to produce the toroidal Hall current. The axial space mak and decays rapidly after it has propagated way. The first charge electric force balances the outward magnetic force .spheromak is stronger than the second one; hence. it heats −␮ٌB more and extends over a larger radius. The alternating sign of the space charge is not a Figure 3͑b͒ shows that the change in plasma potential nonlinear feature as it is also present in the linear regime. As relative to the undisturbed plasma is positive in whistler mir- the antenna current decays, whistler modes without null rors and highly negative in whistler spheromaks. The sign of points are excited and they must also exhibit positive poten- ϫ ʈ the space charge is produced by Eind B drifts, which carry tials when Bz,wave B0 and negative potentials when 042309-4 Strohmaier, Urrutia, and Stenzel Phys. Plasmas 15, 042309 ͑2008͒

The density enhancement propagates radially outward with decreasing speed and amplitude. The former is due to the probe moving oblique to the wavefront that originates at the antenna. The latter is due to the geometric expansion of a density perturbation starting at the antenna. It has earlier been shown that pulsed currents in an insulated antenna produce a density depletion around the antenna wire.20 It is caused by a magnetic JϫB force on the electrons, which in electron magnetohydrodynamics ͑EMHD͒ produces a radial space charge electric field and accelerates ions together with electrons away from the antenna. The antenna wire is located at ͉y͉ =7.5 cm, z=0. The main perturbation arrives at t Ӎ7 ␮s after the start of the antenna current and must have traveled axially at a supersonic speed of v=⌬z/⌬t =106 cm/s; i.e., the ions initially travel ballistically with about 20 eV of energy. In time, this perturbation steepens into an ion acoustic shock with asymptotic velocity v =⌬y/⌬tӍ2.9ϫ105 cm/s, slightly larger than the ion sound ͑ / ͒1/2 Ӎ ϫ 5 / speed cs = kTe mi 2.2 10 cm s in argon at kTe Ӎ2 eV. Figure 4͑b͒ shows a contour plot of the absolute density vs radius and time. Both outward and inward propagating density perturbations are visible. Geometric effects enhance the inward propagating perturbation and decrease the outward perturbation. No comparable perturbations are seen at larger axial distances ͑⌬z=23 cm͒ from the antenna; hence, they are not produced by propagating whistler spheromaks. FIG. 4. ͑Color online͒ Density perturbations during and after the antenna ͑ ͒ ͑ ͒ ͑␦ ϰ␦ ͒ current oscillations top trace . a Density perturbation ni Ie,sat vs time at different radial positions y and at x=0, z=−7 cm. A density perturbation B. Plasma parameters in collisions that steepens into an ion acoustic shock is created at the antenna. ͑b͒ Density contours vs radius and time showing that the perturbation originates near the of whistler spheromaks ͑ Ϯ ͒ coil y= 7.5 cm and propagates radially outward and inward. As shown in Fig. 8 of the companion paper Part II,9 the collision of two counterpropagating whistler spheromaks leads to a stationary field-reversed configuration ͑FRC͒, ʈ Bz,wave −B0 so as to be consistent with Hall electric fields, whose magnetic field energy dissipates within approximately currents, and wave magnetic fields. For example, when half an rf period. The magnetic energy is converted into elec- Bz,wave opposes B0, the space charge electric field must point tron thermal energy, which leads to strong heating as con- radially inward, so as to produce a toroidal Hall current J␪ firmed by probe measurements. Any directional acceleration Յ ʈ 0, which produces Bz,wave −B0. of electrons can be excluded since it would produce currents Finally, the electron density profile is displayed in Fig. and measurable magnetic fields. Figure 5͑a͒ shows a contour ͑ ͒ ͑ ͒/ Ͼ 3 d . There exists a vertical density gradient d ne dy 0 plot of the bulk electron temperature versus radial position prior to and after the excitation of whistlers. Most of the and time approximately in the midplane between the anten- density contours at ⌬z=23 cm from the antenna is not af- nas, where the FRC is formed and decays. The bulk electron fected by the intense whistler modes. The main reason is the temperature rises to more than 15 eV over a larger radius and short duration of the electric field during which the heavy longer time than observed for a single propagating sphero- ions do not respond. For example, in a radial electric field of mak ͓see Fig. 3͑b͔͒. Two temperature peaks are observed on 1V/cm with duration 1 ␮s the ions gain a velocity axis at different times. The first one is due to the arrival of ⌬ ⌬ /͑ ͒Ӎ ϫ 4 / v=eE t mi 2 10 cm s, which is smaller than the ion the two spheromaks at the point of measurement. Given that sound speed and displaces them by a negligible distance both arrive almost simultaneously, both spheromaks merge ⌬r=⌬vϫ⌬tӍ0.2 mm. The observed small perturbations in leading to the canceling of their opposing poloidal fields. The the density contours are due to the aforementioned problem second occurs when the resulting toroidal current ring de- of the probe response to rapidly changing plasma potentials. creases in radius and heats a smaller volume to a higher Closer to the antenna, where electric and magnetic fields temperature. It is shown in Part IV that this leads to high are stronger, density perturbations are unambiguously frequency whistler instabilities.21 The second peak is defi- present. The most obvious feature is a propagating density nitely not due to the whistler mirror since it arrives later and perturbation observable well after the end of the whistler drives much less heat than the FRC. The second FRC pulses. Figure 4͑a͒ shows traces of the density perturbation ͑8ϽtϽ10 ␮s͒ is weaker than the first one since the antenna ␦ ϰ␦ ni Ie,sat versus radius at z=7 cm from the antenna on a current decays. time scale long compared to the antenna current ͑top trace͒. Figure 5͑b͒ shows the plasma potential change relative 042309-5 Nonlinear electron magnetohydrodynamics physics. III… Phys. Plasmas 15, 042309 ͑2008͒

I 0 coil (a) 0.2 Light 0.1

0 0 51015 t(µs) 0.4 (b) ∆z=25cm 0.1 Light (Arb. Units) ∝e−(t/11.5 µs) 0.01 0.004 0102030 t(µs)

FIG. 6. Time dependence of coil current and light emission at ⌬z=25 cm from the antenna. ͑a͒ Coil current ͑top trace͒ and light emission ͑bottom trace, showing strong light emission during spheromak excitation when Ӎ / Ͼ ͑ ͒ Icoil 0 with dcoil dt 0. b Light intensity on a semi-log scale showing a slow exponential decay of light after the passage of several whistler spheromaks.

FIG. 5. ͑Color online͒ Plasma parameters vs radius and time during the sign of the plasma potential is the same as in the nonlinear ʈ collision of nonlinear whistler modes. Probe located at x=0, z=−23 cm; regime; i.e., positive when Bwave B0 and negative when 15 cm diam loop antennas located at z=0,−50 cm; antenna current shown B ʈ −B relative to the ambient potential. Thus, the net ͑ ͒ ͑ ͒ wave 0 in Fig. 3 a . a Electron temperature kTe, showing strong heating during ͑ ͒ ⌬␾ ͑ ͒ field reversal by the wave field does not change the sign of decay of FRC. b Plasma potential change pl 0,y,−23 cm,t , which is negative inside the FRC. ͑c͒ Plasma potential change on an extended time the space charge. scale where the late oscillations are linear modes without null points or Density perturbations by colliding nonlinear whistler heating, yet having the same sign of space charge as the first nonlinear modes are again negligible in the midplane on short time modes. scales ͓see Fig. 3͑d͔͒, but are noticeable on longer times and near the antenna as shown in Fig. 4͑b͒. The lack of initial density perturbations confirms that the nonlinear properties ⌬␾ ͑ ͒ to the unperturbed plasma, i.e., pl 0,y,−23 cm,t , for the of the first few spheromaks are not due to modulational above conditions. Positive potentials are observed in collid- instabilities22–24 but to wave magnetic field nonlinearities. ing whistler mirrors, negative potentials in colliding spheromaks. As in propagating spheromaks, the temperature C. Light emission from whistler spheromaks maxima and potential minima in the stationary FRC do not and mirrors exactly coincide. The second potential minimum coincides with the collapse of the toroidal null line on axis forming a In the present parameter regime, light emission arises minimum B. On axis, the inward pointing magnetic force from inelastic collisions of energetic ͑Ͼ10 eV͒ electrons ͑ ␮ٌ ٌ␾ ͑ Ϸ ϫ 12 −3͒ − B is balanced by an outward force e pl such that no with neutrals density n0 9 10 cm and ions density Ͻ 12 −3͒ 25 axial field-aligned currents are driven. Prior to the collision, ni 10 cm . There is little light emission in the after- the magnetic field is pinched in the center and reverses ␮ٌB glow plasma at 150 ␮s after the discharge voltage is turned ␾ٌ and pl. Radial gradients in pressure and potential produce off. The excitation of linear whistler modes produces no cross-field drifts or currents consistent with the reversed light. The excitation of nonlinear whistler modes with ʈ Ͼ magnetic fields. Thus, space charge fields develop self- Bz,wave B0 and Bz,wave B0 produces some light near the consistently with magnetic topologies. Potential wells or antenna, but copious light emission is seen within the plasma ʈ hills are not the source for closed, field-aligned currents. The volume for the opposite polarity; i.e., Bz,wave −B0. latter are driven by rotational electric fields that also deter- When the antenna axis is rotated by an angle ␪ with ␪ mine the dissipation of magnetic energy. The former are nec- respect to B0, the light emission decreases. At =90°, no essary to satisfy the force equilibrium or generalized Ohm’s light is observed in the plasma volume ͑⌬z=25 cm͒ for ei- law. ther polarity, although weak light emission occurs directly at Figure 5͑c͒ extends the time scale of Fig. 5͑b͒ to a time the antenna at every half-cycle of the rf current. when current no longer flows on the antenna and, therefore, Figure 6 shows the temporal behavior of light emission heating and null points are no longer created. The current at a fixed axial distance ͑⌬z=25 cm͒ from the antenna with a waveform is the same as the one displayed on Fig. 4͑a͒. The radial line of sight through the center of the plasma where 042309-6 Strohmaier, Urrutia, and Stenzel Phys. Plasmas 15, 042309 ͑2008͒

200 Light (a) Icoil x = 7.5 cm (Arb. ∆z=0 0 units) (A) 0 25 cm −200 0 50 cm x=0 0 75 cm Light 0 0 048 0 20 40 µ t(s) t(µs)

FIG. 7. Light emission vs axial distance from the antenna. The light peak (b) propagates approximately at the whistler speed ͑30 cm/␮s͒ and is much 26.4 slower than the thermal speed of 10 eV electrons ͑80 cm/␮s͒. Thus, the Light source of the light lies in the whistler spheromak rather than hot electrons (Arb. 26.7 26.2 streaming away from the antenna. 26.9 Units) 26 27 t = 24.8 µs 0 whistler modes propagate. Light pulses are excited from −10 010 whistler spheromaks, which are launched during coil current x (cm) ͓ ͑ ͔͒ / Ͼ reversal top trace in Fig. 6 b with dIcoil dt 0 and propa- Ӎ /␮ FIG. 8. ͑Color online͒ Spheromak light emission vs radius at different times gate at vz 20 cm s. Growth and decay of the light pulse ͑ ͒ ͑ ͒ during spheromak formation. a Coil current Icoil t and light emission at are comparable and much faster than the light decay of the coil radius and axis x=y=0. ͑b͒ Light intensity vs radius at different times ͑␶ /͓ / ͔Ӎ ␮ afterglow plasma =Ilight dIlight dt 180 s, not shown showing that light originates at the antenna and subsequently contracts to- here͒. On a logarithmic scale ͓Fig. 6͑b͔͒, the light decay ward the axis like the toroidal current layer. shows a second time scale of ␶Ӎ12 ␮s after the passage of several spheromaks. This slower light decay reflects the decay of heat deposited by the spheromaks in the background the antenna region is the largest. It also shows a peak when ͑ Ӎ / Ͼ ͒ plasma. The duration of the light pulses is comparable to the whistler mirrors are excited Icoil 0 with dIcoil dt 0 . transit time of the spheromak through the line of sight. Thus, Since the light from mirrors disappears at zϾ25 cm from the electrons are energized only in the spheromak and “lose” antenna, it is not the result of wave currents but by electron their energy rapidly thereafter. This could occur due to sev- acceleration in the toroidal magnetic null line at z=0 just eral reasons: The energetic electrons could stay within the outside the coil ͓see Fig. 6͑c͒ in Part I.8͔. With increasing toroidal current ring and travel axially with it, an unlikely axial distance the main light peak is delayed and decays scenario since the electrons would have to be trapped in the similar to the whistler spheromak intensity. The axial propa- Ӎ /␮ spheromak, which travels slowly compared to the electron gation speed vz 30 cm s is much smaller than the thermal thermal velocity. Another possibility could be that the elec- speed of the 10 eV electrons producing light ͑180 cm/␮s͒. trons are energized unidirectionally so as to carry the toroidal Thus, the source of the light is the propagating whistler current. As they travel out of the current layer, they could spheromak, not the thermal expansion of an initial heat pulse transfer their energy back to the fields. This is also unlikely at the antenna. Heat transport causes the light pulse to spread since the peak current density ͓see Fig. 8͑c͒ in Part I8͔ is too out with distance. small for a bulk drift at Ͼ10 eV. It would also imply con- The radial temperature profile can be inferred from axial servation of magnetic energy which is not the case. The most light measurements that provide radial spatial resolution but likely scenario is that the energized electrons simply convect are axially averaged. Figure 8͑a͒ shows antenna current and out of the current layer and are replaced by cold electrons. light emission from the center of the plasma ͑r=0͒ and from ͑ ͒ ͑ Ӎ ͒ ͑ ͒ This lowers the light intensity for two reasons: i The den- the coil wire rcoil 7.5 cm versus time. Figure 8 b shows sity of hot electrons decreases as they expand along poloidal the light emission versus radius at different times within the field lines into a larger volume ͑except toward the center of light pulse. Initially, the light peaks near the coil, where the the spheromak where the temperature often peaks͒ and ͑ii͒ toroidal current rings of the two spheromaks originate. The the electrons transfer energy by collisions with the mass of current layer peaks in the toroidal magnetic null region, colder surrounding electrons as well as by inelastic collisions whose radius decreases as the total current decays. Thus, the with neutrals and ions; e.g., by excitation of light. In this light peaks move toward the axis and the hollow light/ situation, the electrons provide for a true dissipation channel temperature profile disappears. Note that the light pulse is for the wave magnetic energy which is observed to decay. much shorter than the long rf period chosen here. The rela- The wave damping is much stronger than by elastic electron- tion of spheromak duration and rf period has been briefly neutral collisions which dominate in whistler mirrors or lin- discussed in Part II,9 but is expanded below. ear whistler waves. In whistler spheromaks or FRCs, the di- Simultaneous probe and light measurements confirm the rect acceleration of electrons effectively produces an coincidence between light, heating and the presence of the “inertial” resistivity that is not present in other topologies. spheromak. For the current waveform shown in Fig. 8͑a͒, the Figure 7 shows the light intensity versus time at different time dependence of the light emission, viewed radially at axial distances from the antenna ͑z=0͒. Light emission from ⌬z=25 cm from the antenna, is shown in Fig. 9͑a͒ on an 042309-7 Nonlinear electron magnetohydrodynamics physics. III… Phys. Plasmas 15, 042309 ͑2008͒

2 (a) Light Half−width 1 (µs)

Light Intensity (arb. units) 0 0 50 100 150 Icoil (A) 60 (b) Rf Period (µs) Light Intensity 40 (arb. units) Light Half−width µ × 20 ( s 10)

0 01025 15 0 f × Icoil (A/µs)

FIG. 10. ͑Color online͒ Variation of intensity and half-width of light pulses with coil current waveform. ͑a͒ Variation of coil current at a fixed frequency ͑ ͒ / ␻ ϰ 0.2 MHz . Note that dIcoil dt= Icoil Icoil determines the applied inductive electric field, hence spheromak current, heating, and light intensity. A current threshold is required for spheromak formation. ͑b͒ Variation of fre- ͑ Ͻ Ͻ ͒ / ϰ␻ quency 18 f 300 kHz also changes dIcoil dt Icoil. The spheromak amplitude rises rapidly with applied field, while the width does not and can be much shorter than the rf period.

FIG. 9. ͑Color online͒ Temporal coincidence of light and plasma parameters in whistler spheromaks. Antenna current as in Fig. 8͑a͒. ͑a͒ Light intensity ert a small inward magnetic force to balance the electron vs time at ⌬z=25 cm from the antenna along a radial line of sight. ͑b͒ pressure gradient and to create a small excess electron ͑ ͒ ͑ ͒ Radial bulk electron temperature profile kTe x=0,y,z=25 cm,t . c Toroi- ϫ ͑ ͒ charge. The Jtor B0 force points radially inward irrespective dal current density Jx x=0,y,z=25 cm,t , indicating the location of the ͑ ͒ ␾ ͑ ͒ of propagation direction and wave intensity. The axial wave spheromak. d Plasma potential profile pl x=0,y,z=25 cm,t . propagation in the presence of a radial electron drift may cause the lag between the potential minimum relative to the spheromak center. expanded time scale. Figure 9͑b͒ shows the radial profile of the electron temperature from Langmuir probe measure- ͑ ͒ ͑ ͒ ments, i.e., kTe x=0,y,z=25 cm,t ; Fig. 9 c displays the ͑ ͒ D. Parameter dependence of light emission toroidal current density Jx x=0,y,z=25 cm,t obtained from ͒ ͑ ͒ ϫ /␮ٌ ͑ magnetic probes J= B 0 ; and Fig. 9 d presents con- By varying the parameters of the plasma and of the ⌬␾ ͑ tours of the plasma potential change, i.e., pl x=0,y,z applied field, we have investigated what determines the =25 cm,t͒, from unperturbed plasma. No significant density spheromak strength and size. The spheromak current is perturbations are observed. The data show that heat and light driven by the applied electric field near the antenna; i.e., Ӎ /͑ ␲ ͒ϰ / peak in the center of the whistler spheromak. The tempera- E␪ Vcoil 2 rcoil dIcoil dt. For a sinusoidal current, ͑ / ͒ ␻͑ ͒ ␻ ture does not peak in the toroidal current ring, but on axis dIcoil dt max= Icoil max, such that for =const, one expects ϰ ͑ ͒ where the peak poloidal current flows. The nearly symmetric Iplasma Icoil. Figure 10 a shows the intensity and half-width light and temperature profiles indicate that the electrons are of the light pulse observed radially at 25 cm from the an- only energized within the spheromak but lose their energy tenna versus peak coil current. The current has to exceed a after leaving the wave fields. The plasma potential change threshold before field reversal and spheromak formation are inside the spheromak is negative relative to the ambient possible. Thereafter, the light intensity increases rapidly with potential, albeit with a small delay. Thus, the electron ener- Icoil. The width of the light pulses also increases with Icoil.As gization cannot arise from space charge electric fields. Fur- shown elsewhere ͑Fig. 8, Part II9͒, the radius of the current thermore, the electrons are not in Boltzmann equilibrium ring increases with increasing plasma current while its axial ␾ ٌ ͒ ͑ 9ٌ͒͑ pl kTe , but must be forced into the spheromak by propagation speed decreases Fig. 3, Part II , both of which magnetic force. Although an ideal spheromak has a force- contribute to increasing the spheromak light intensity and free field ͑JϫB=0͒, the actual whistler spheromak must ex- duration. 042309-8 Strohmaier, Urrutia, and Stenzel Phys. Plasmas 15, 042309 ͑2008͒

B Spheromak

∆r=0 Light Rcoil ∆r 1.3 Rcoil

(Arb. Mirror Units) ∆r>2Rcoil

0 048 t(µs)

FIG. 12. ͑Color online͒ Light emission vs time for collisions of radially displaced spheromak ͑radial view at z=25 cm between antennas͒. The longest light emission occurs for head-on collisions where the two spheromaks merge into an EMHD FRC. The interaction is weakened when the spheromaks are radially displaced ͑see inset͒. When the spheromaks no longer ͑ ⌬ տ ͒ overlap radial separation r 2Rcoil =15 cm , the light emission is a linear superposition of two counterpropagating spheromaks.

versus B0 and time at two different axial distances from the antenna. For purpose of comparison, the current waveform is FIG. 11. ͑Color online͒ Contours of light intensity as a function of back- also shown ͓Fig. 11͑a͔͒.At⌬z=25 cm ͓Fig. 11͑b͔͒, the light ground magnetic field strength B0 and time at two different axial distances emission arises from propagating whistler spheromaks. ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ⌬ from the antenna; i.e., Ilight x=y=0,z,t . a Coil current Icoil t . b At z For very low background magnetic fields ͑B Ͻ3G͒, the to- =25 cm from the antenna, light emission from spheromaks has an optimum 0 Ӎ tal field is essentially that of the antenna, which does not at B0 10 G and vanishes for much larger or smaller fields where magnetic field reversals do not occur. The contour crest curves since the propagation have a spheromak topology. For large background fields → ͑ ͒ ͑ ͒ speed decreases as B0 0. c The light from the antenna z=0 peaks when ͑B Ͼ20 G͒ and a given coil/plasma current, the whistler Ӎ 0 both whistler mirrors and spheromaks are emitted. For B0 0 light emission peaks at current maxima and minima when hot electrons are confined by the mode cannot reverse the external field to form a spheromak. ͑ Ӎ ͒ antenna magnetic field. Thus, there is an optimum field B0 10 G for the most efficient heating and light emission. The graph also shows / that with decreasing B0 or increasing Bwave B0, the propaga- Figure 10͑b͒ shows light intensity and width versus tion speed decreases and the half-width increases, as earlier ͑ ͒ ͑ ͒ 9 → applied frequency and coil current. With increasing fre- shown in Figs. 10 a and 3 , Part II. However, as B0 0, quency the current decreases; hence, the dependence on the delay is better described by field diffusion than convec- ␻ ϰ / Icoil dIcoil dt is shown. Again, the light intensity increases tion by a wave. / with dIcoil dt, but the spheromak length changes little and Figure 11͑c͒ shows the light emission at the antenna, can be much shorter than the rf period. Thus, the whistler z=0. In addition to the strong spheromak light, one can also spheromak has very different properties than linear whistler see light emission when whistler mirrors are emitted. Since modes or nonlinear whistler mirrors whose wavelengths vary the light is only seen near the antenna, the energetic electrons with frequency. are not produced in the wave field but in the magnetic null As shown in Figs. 6͑a͒ and 6͑b͒ of Part II,9 two sphero- line just outside the antenna ͓see topology in Fig. 6͑c͒, Part maks start to form when a null line inside the antenna arises I8͔. Another interesting observation is that copious light is → from the reversal of the coil current. With increasing coil produced as B0 0 on every half-cycle of the applied cur- current, the null line moves to the axis, the FRC flux recon- rent. Electrons are energized in the toroidal null line inside or nects and becomes the toroidal flux of two oppositely ejected outside the coil and confined by the coil magnetic field. spheromaks. This time scale depends on antenna radius and The production of fast electrons, measured by the light radial field penetration speed, which is comparable to, or intensity, has also been investigated as a function of plasma slower than, the axial spheromak propagation speed and neutral densities. As expected, the light intensity in- /␮ ⌬ Ͼ / 10 cm s, resulting in a time scale t rcoil vwhistler creases with neutral gas pressure but surprisingly little with Ӎ /͑ /␮ ͒Ն ␮ 7.5 cm 10 cm s 0.75 s, which is comparable to plasma density, which is varied with discharge current Idis. the observed half-width of the light pulses. Comparing The half-widths of the light pulses increase with ion density ͑ pulses from antennas of different diameters 2rcoil and gas pressure. The latter also changes the plasma density ͒ =10,15,20,30 cm show that smaller antennas produce when Idis=const. shorter light pulses or spheromaks, while larger antennas ex- We have also measured the light emission from colliding cite longer spheromaks. spheromaks and mirrors. Figure 12 shows light intensity vs Since the propagation speed of the whistler mode de- time in the midplane between two identical loop antennas pends on frequency, magnetic field, and density, we have separated axially by 50 cm. The loops can be displaced ra- ⌬ next investigated the dependence on background magnetic dially by a distance r across B0. The line of sight is through field B0. Figure 11 shows contour maps of the light intensity the center of the spheromaks along the direction of radial 042309-9 Nonlinear electron magnetohydrodynamics physics. III… Phys. Plasmas 15, 042309 ͑2008͒

m <0 verted into electron heat, which is a fundamental process of 100 z Light (Arb. magnetic reconnection or annihilation. Non-Maxwellian Icoil mz <0 Units) electron distributions are produced. The scaling of light 0 0 emission with plasma parameters, antenna size and orienta- mz >0 (A) tion has been explored. Acceleration processes near the an- −100 tenna have been explored, such as expulsion of ions by space m >0 z charge electric fields and the acceleration of electrons in null 0102030 40 lines of the antenna near zone field. These results are of t (µs) intrinsic interest in nonlinear wave physics and of practical FIG. 13. ͑Color online͒ Light emission from a single coil current pulse. In importance for efficiently exciting whistler modes from mag- ͑ Ͻ ͒ one polarity magnetic moment mz 0 , it excites a decaying FRC, which netic antennas. produces strong and long-lasting ͑4 ␮s half-width͒ light emission. In the opposite polarity, it excites a propagating whistler mirror which produces negligible light or electron heating. ACKNOWLEDGMENTS The authors gratefully acknowledge support for this work from National Science Foundation Grant No. PHY displacement. For a head-on collision ͑⌬r=0͒, the light in- 0076065 and Air Force Contract No. FA8718-05-C-0072. tensity is the largest and lasts the longest ͑⌬tӍ3 ␮s͒, similar to the electron temperature measured with probes ͓Fig. 5͑a͔͒. 1 R. A. Helliwell, Whistlers and Related Ionospheric Phenomena ͑Stanford During this time, the magnetic energy of the FRC is con- University Press, Stanford, CA, 1965͒. verted into electron thermal energy. For glancing collisions 2G. Schmidt, Physics of High Temperature Plasmas ͑Academic, New York, of spheromaks ͑⌬rϾ0͒, the interaction weakens, the light 1966͒. 3S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett. 42,1189͑1979͒. pulses become shorter, and the interaction vanishes for 4A. N. Kaufman, in Intrinsic Stochasticity in Plasmas, edited by G. Laval ͑⌬ Ͼ ͒ r 2rcoil . The light emission becomes the same as that of and D. Gresillon ͑Les Editions de Physique, Orsay, France, 1979͒, a propagating spheromak, which is governed by its transit pp. 131–157. 5 time ⌬t=⌬z/v . C. R. Menyuk, A. T. Drobot, and K. Papadopoulos, Phys. Rev. Lett. 58, z 2071 ͑1987͒. Spheromak collisions are the strongest and most interest- 6S. J. Tanaka, Plasma Phys. Controlled Fusion 29, 1067 ͑1987͒. ing wave-wave interactions. Mirror collisions produce negli- 7W. J. Wykes, S. C. Chapman, and G. Rowlands, Phys. Plasmas 8,2953 gible heating and mirror-spheromak collisions are dominated ͑2001͒. 8 by spheromak heating. R. L. Stenzel, J. M. Urrutia, and K. D. Strohmaier, Phys. Plasmas 15, 042307 ͑2008͒. Finally, it should be recalled that EMHD FRCs can also 9J. M. Urrutia, R. L. Stenzel, and K. D. Strohmaier, Phys. Plasmas 15, be produced by pulsed rather than oscillatory coil currents 042308 ͑2008͒. ͑see Fig. 3, Part I8͒. In this case the decaying FRC also 10F. F. Chen, in Plasma Diagnostic Techniques, edited by R. H. Huddlestone ͑ ͒ produces long lasting light emission as shown in Fig. 13, and S. L. Leonard Academic, New York, 1965 , pp. 113–200. 11N. Hershkowitz, in Plasma Diagnostics-Surface Analysis and Interactions provided the coil field is opposite to the ambient field. 2, edited by O. Auciello and D. L. Flamm ͑Academic, Boston, MA, 1989͒, pp. 113–183. IV. CONCLUSIONS 12V. Godyak, R. B. Piejak, and B. M. Alexandrovich, J. Appl. Phys. 73, 3657 ͑1993͒. This paper continues the experimental investigation of 13H. M. Mott-Smith and I. Langmuir, Phys. Rev. 28, 727 ͑1926͒. 14 nonlinear whistler modes. Its focus is the energization of D. Blackwell and F. F. Chen, Plasma Sources Sci. Technol. 10,226 ͑2001͒. electrons in whistlers whose wave magnetic field exceeds the 15D. G. Bills, R. B. Holt, and B. T. McClure, J. Appl. Phys. 33,29͑1962͒. ambient field. The plasma parameters are directly measured 16R. L. Stenzel and J. Urrutia, Phys. Plasmas 4,26͑1997͒. with Langmuir probes and the presence of energetic elec- 17J. M. Urrutia and R. L. Stenzel, Phys. Plasmas 4,36͑1997͒. 18 trons is indirectly inferred from light emission measure- R. L. Stenzel, W. Gekelman, N. Wild, J. M. Urrutia, and D. Whelan, Rev. Sci. Instrum. 54, 1302 ͑1983͒. ments. A newly discovered heating process differs from fa- 19A. Yanguas-Gil, J. Cotrino, and L. L. Alves, J. Phys. D 38, 1588 ͑2005͒. miliar damping processes such as Landau and cyclotron 20J. M. Urrutia, M. C. Griskey, R. L. Stenzel, and K. D. Strohmaier, Phys. damping. The observed electron acceleration depends on the Plasmas 10, 2801 ͑2003͒. 21 topology of the magnetic field, which in turn depends on J. M. Urrutia, R. L. Stenzel, and K. D. Strohmaier, “Nonlinear electron magnetohydrodynamics physics. IV. Whistler instabilities,” Phys. Plasmas field amplitude and direction. For a sinusoidal excitation, the ͑to be published͒. heating process greatly varies from one half-cycle to the 22H. Washimi, J. Phys. Soc. Jpn. 34,1373͑1973͒. next. The strongest electron heating is found when the wave 23V. I. Karpmann, R. N. Kaufman, and A. G. Shagalov, Phys. Fluids B 4, ͑ ͒ produces magnetic null lines such as in spheromak and FRC 3087 1992 . 24I. Kourakis and P. K. Shukla, Phys. Plasmas 12, 012902 ͑2005͒. topologies. Electrons are accelerated along the null lines by 25H. R. Griem, Principles of Plasma Spectroscopy, Cambridge Monographs an inductive electric field. The wave magnetic energy is con- on Plasma Physics ͑Cambridge University Press, Cambridge, UK, 1997͒.