Three-Dimensional Electron Magnetohydrodynamic Reconnection

PHYSICS OF PLASMAS VOLUME 10, NUMBER 7 JULY 2003

Three-dimensional electron magnetohydrodynamic reconnection. II. Tilt and precession of a field-reversed configuration M. C. Griskey, R. L. Stenzel, J. M. Urrutia, and K. D. Strohmaier Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547 ͑Received 28 August 2002; accepted 7 April 2003͒ Further observations are presented on a reconnection experiment involving a three-dimensional magnetic field reversed configuration ͑FRC͒ in the parameter regime of electron magnetohydrodynamics ͑EMHD͒. The stability of the FRC that relaxes in a large ambient plasma free of boundary effects is investigated. No destructive instabilities are observed. However, the EMHD FRC performs a precession around the axis given by the ambient magnetic field after a tilt develops. The precession velocity corresponds to the electron drift velocity of the toroidal current. The phenomenon is explained by the convection of frozen-in field lines in a rotating electron fluid. It is a new phenomenon in EMHD plasmas. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1578999͔

I. INTRODUCTION II. EXPERIMENTAL RESULTS A. Poloidal field evolution A field-reversed configuration ͑FRC͒ describes a magnetized plasma whose field topology can be described by a The initial FRC is set up with a Helmholtz coil, whose poloidal ͑dipolar͒ field, driven by toroidal plasma currents, field, BH , is directed opposite to a weaker uniform field, B0 . embedded in an opposing uniform field.1 Unlike in sphero- As described in Part I,20 the current to the Helmholtz coil is maks, there are no toroidal fields present. Plasma devices turned on sufficiently slow to allow BH to penetrate into the with such fields are of interest for the confinement of fusion plasma and establish a steady-state FRC. At tЈϭ0, the coil ␤ϭ 2 ␮ ЈӍ ␮ plasmas due to a high plasma beta ͓ nkT/(B /2 0)͔ and current begins to turn off and is zero by t 3 s. From then freedom of toroidal coils.2,3 FRCs are also potentially useful on, the time-varying magnetic field structure is solely pro- for plasma propulsion.4 However, magnetohydrodynamic duced by plasma currents coupled to whistler waves. The 5 Ӎ ͒ ͑MHD͒ theory predicts a destructive tilting instability. The cold ions (kTi 0.1 eV do not move significantly during the tilting is well known in spheromaks,6 which need a conduct- relaxation phase, which occurs on a time scale (р15 ␮s͒ ing flux conserver for stabilization.7 Surprisingly, the tilting long compared to the whistler transit times inside the FRC 8–10 ␶ ϭ Ӎ ␮ Ӎ Ӎ ␮ ͒ is not seen in many other FRC experiments. One expla- ( w L/vw 2 s for L 30 cm, vw 15 cm/ s , but short ␶ϭ␮ ␴ 2Ӎ nation for the FRC stability is the non-MHD properties of compared to the classical diffusion time ( 0 ЌL 500 11,12 Ϫ1 Ϫ1 ions with large Larmor radii. However, this hypothesis is ␮s for ␴ЌӍ50 ⍀ cm ). The initial size of the EMHD still under investigation.13 The lifetime of FRCs is also ex- FRC is determined by the size of the Helmholtz coil ͑30 cm tended by applying rotating magnetic fields14–16 to prevent diameter͒, and is embedded in a larger plasma column ͑1m tilting. In the limit of completely unmagnetized ions but diameter͒ in an even larger vacuum chamber ͑1.5 m diam- magnetized electrons, known as electron MHD ͑EMHD͒,17 eter͒. This renders insignificant any boundary effects. Fur- FRCs have also been studied.18,19 To date, little is known ther, no induced currents are formed in the Helmholtz coil about the stability of a freely relaxing EMHD FRC. Al- because it is in an open-loop state after tЈӍ3 ␮s. The elec- though the focus of this series of papers20–22 is the basic tromagnetic field evolves self-consistently subject to Max- physics of reconnection in the EMHD regime, the present well’s equations and Ohm’s law. The latter is predominantly paper addresses the stability of an EMHD FRC in an un- governed by EMHD physics except in the toroidal magnetic bounded plasma. It describes a new phenomenon, i.e., the null layer. Time and space resolved measurements of the precessing motion of a slightly tilted FRC.23 When the po- magnetic field indicate how the field relaxes. loidal field becomes inclined relative to the uniform field, the First, the evolution of the poloidal field in the midplane ͑ ͒ 20 FRC precesses around an axis defined by B0 . The precession of the FRC is investigated. As noted in Fig. 4 a of Part I, occurs in the direction and at the velocity of the toroidally the FRC is nearly azimuthally symmetric, and it remains so drifting electrons, and is the result of convection of frozen-in throughout the free relaxation of the FRC. Figure 1͑a͒ shows field lines in a drifting electron fluid. contours of the axial magnetic field Bz(x,y,0) late in the The present paper is organized as follows. Since the ex- relaxation process, tЈϭ10 ␮s. The center position and axial 20 perimental setup was described in Part I, we start with the symmetry of the distribution of Bz(x,y,0) have not signifi- experimental results, divided into various subsections, in cantly changed from those of the initial FRC. In hindsight, Sec. II. The conclusion, Sec. III, points out the new findings this is not surprising since the initial current distribution that and implications to other related research. drives the FRC is axisymmetric. It depends on the existence

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B. FRC tilt The development of a small tilt of the FRC axis is observed when the poloidal field is displayed in the y–z plane ϭ 2 on axis (x 0). Figure 2 shows unit vectors (By ,Bz)/(By ϩ 2 1/2 ⌽ Bz) and contours of constant poloidal flux (y,z) ϭconst at different times during the relaxation of the FRC. As described in Part I,20 the flux is calculated by first iden- tifying the axis of symmetry of the poloidal field structure and then integrating the flux perpendicular to that axis, ⌽ ϭ͐r Ј ˆ ␲ Ј Ј ˆ (r,z) 0B(r ,z)•a 2 r dr , where a is identical to a vector parallel to the inner part of the separatrix. The magnetic dipole moment of the FRC, defined as mϭ(1/2)͐r ϫ ϭٌϫ ␮ J dV, where J B/ 0 and r is the radius vector from the origin, cannot be used to determine this vector because the current density distribution is not exactly symmetric about the separatrix. Flux contours, as noted in Part I,20 closely follow traced field lines, but offer the advantage of quantifying a frozen-in field line whose motion can be observed. Once established, MHD FRCs can be affected by a well- documented tilting instability.5,25 According to Ref. 26, sta- 2у␥ ⍀ bility can be expected if (krci) MHD / ci , where k ϳ ␥ ⍀ ϳ 1/rs and MHD / ci 2rci /ls . In this expression, rs is the ⍀ separatrix radius, ls the separatrix length, ci the average ␥ internal ion gyrofrequency, and MHD the growth rate of the FIG. 1. Stability of the poloidal field distribution during the free relaxation of the FRC. ͑a͒ Contours of B (x,y,0) at a fixed time tЈϭ10 ␮s. ͑b͒ tilting stability. The expression can be rewritten in terms of z ␧ϭ ϭ ϭ͐ ˆϩ ˆ two dimensionless parameters, the ellipticity, ls /2rs , and Hodograph of center position rc ͗xc ,y c͘ (xx yy)Bz(x,y)dxdy/ ͐ Bz(x,y)dxdy showing negligible translation of Bz,max from the origin. s, the ratio of the size of the system to thermal orbit size, i.e., ␴ϭ ␴ ␴ ϭ 25 Hodograph of mean square deviation from center, ( x , y) ͕͗(x the ion gyroradius. Some workers define the size of the Ϫ 2͘1/2 ͗ Ϫ 2͘1/2͖ xc) , (y y c) showing that the field profile remains circular and system as the major radius of the torus defined by the FRC, its width varies by only Ϯ10%. Thus the EMHD FRC is not subject to a 26 destructive instability. which is determined by the null layer radius. More recently, it is defined instead as the minor radius, which is the distance ϳ between the null layer and the separatrix, a rs/4. Taking this definition, the stability condition can be rewritten as of Hall currents, which are maintained by EϫB drifts. In the s/␧р1/4. present case, the dominant magnetic field is along the sys- In the absence of a formal theory for EMHD FRCs, we tem’s axis, forcing the currents to be axisymmetric. Quanti- simply use the same concepts as Ref. 26, except that we tatively, we determine the center position of the field distri- employ the electron Larmor radius, rce . For our experimen- ϭ ϭ͐ ˆ bution from its first-order moment, rc ͗xc ,y c͘ (xx tal parameters, the length of the FRC is defined as the dis- ϩ ˆ ͐ ͑ ͒ ͓ yy)Bz(x,y)dxdy/ Bz(x,y)dxdy, and plot it in Fig. 1 b tance between the two cusp-type magnetic null points Figs. ͑ ͒ ͑ ͔͒ Ϸ as a hodograph throughout the decay time. Hodographs were 2 a and 2 b . Hence, ls 53 cm. The radius of the FRC is devised by W. R. Hamilton to better express Newtonian given by the radial distance from the origin to the separatrix mechanics,24 and are graphical representations of the path ͑defined as the ⌽ϭ0 contour͒. Although the ⌽ϭ0 contour traced by the tip of a temporally evolving vector. The at tЈϭ0 and zϭ0 lies outside the measured y–z plane, the hodograph presented in Fig. 1͑b͒, therefore, allows inspec- calculated position in steady state ͑i.e., vacuum-like condi- ͒ Ӎ ␧ϭ tion of the entire temporal evolution of the distribution center tions is rs 25 cm. Thus, the initial ellipticity is 53 in one simple graph. The peak of the poloidal field exhibits a cm/(2ϫ25 cm͒ ϭ 1.06. Finally, the size of the system is the negligible translation compared to its half width. The latter difference between the radial distance between the magnetic ␴ ϭ can be defined by the second moment of Bz , x ͗(x null layer (r0 roughly equal to the Helmholtz coil radius, 15 Ϫ 2 1/2 ϭ ͐ Ϫ 2 ͐ 1/2 ͒ ⌬ ϭ Ϫ ϭ Јϭ Ӎ xc ) ͘ ͓ ( x xc ) Bz (x,y) dxdy/ Bz (x,y) dxdy͔ cm and rs , r rs r0 10 cm. At t 0, kTe 3eV,so ϭ ϭ ϭ⌬ Ӎ for the x direction and similarly for the y direction. Figure rce 1.2 cm for B0 5 G yielding se r/rce 8.3 and ͑ ͒ ␴ϭ ␴ ␴ ␧Ӎ 1 b shows a hodograph of both components, ( x , y). se / 7.9, which would predict a highly unstable FRC. Rotation of ␴ would imply noncircular contours. Throughout However, the MHD model does not apply here, because the the decay, the vector ␴ does not rotate, and its length varies magnetic force and pressure gradients are balanced by the ϫ Ϫ Ϫٌ Ӎ by less than Ϯ10%. This implies that the profile remains electric force in EMHD, J B neE pe 0. Therefore, a circular, and there is little broadening. Thus, observation dipole in an opposing background field is not in unstable shows that the tilting exhibited by the EMHD FRC is not equilibrium and will not destructively tilt as is observed un- destructive. der MHD conditions.26

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͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 2ϩ 2 1/2 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ⌽ϭ ͑ ͒ FIG. 2. a , c , e , g Unit vectors, (By ,Bz)/(By Bz ) , and b , d , f , h contours of constant poloidal ﬂux, const magnetic ﬁeld lines , in the central y–z plane at different times during the free relaxation of the FRC. The initially horizontal FRC ͑a͒,͑b͒ performs a small tilt ͑c͒,͑d͒ which recovers at later times ͑e͒,͑f͒.

Nevertheless, a small tilt in the entire field structure is tilting instability observed under MHD conditions. There, the observed to develop. The cause for the initial tilt is still under instability initially develops at the center of the FRC, for ␧ investigation, but is not random as its observation involves Ͼ1, and spreads throughout the entire structure, destroying many repeated experiments. Magnetic fluctuations within the FRC. In contrast, the EMHD FRC observed in this ex- one event are negligible (␦B/BӍ0.5%, see Part IV22͒. The periment tilts is as if it were a semi-rigid structure. This is flux contours prior to turn off, in Fig. 2͑b͒, show an initial tilt likely one of the reasons why the tilt is not destructive in toward increasing z near the coils, but not on axis. However, EMHD. the FRC tilts in the opposite direction, as shown in Fig. 2͑c͒, Flux contours are calculated with respect to the inclined during the relaxation. Thus, coil alignment with the ambient axis. In time, the tilt decreases while the entire FRC is ob- background field, B0 , is not the cause of the FRC tilt. This is served to precess until the reverse field is again nearly Ϫ Ӎ ␮ ͓ ͑ ͒ ͑ ͔͒ further confirmed by experiments performed with a tilted aligned along B0 at tЈ 9 s Figs. 2 e and 2 f . During dipole field, which did not trigger any destructive tilting in- the relaxation, the ellipticity of the EMHD FRC increases, stability. The largest tilt occurs at tЈӍ5 ␮s as shown in Figs. and the magnetic flux decreases. At tЈӍ10 ␮s, the FRC is ͑ ͒ ͑ ͒ ␧ 2 c and 2 d . The average field direction in the center of the still stable, further proof that the stability parameter, se / ␪ Ӎ Ӎ ϭ FRC is found to be inclined at an angle tilt 20° with re- 5/2.03 2.46, derived for electrons in analogy to that of Ϫ spect to B0 . This effect is markedly different from the ions does not apply for our experiment. Ultimately, the sepa-

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FIG. 4. Schematic and measured current density J(x,y,0) at tЈϭ7.3 ␮s. ͑a͒ Schematic picture of a tilted current layer whose magnetic dipole moment precesses around the z axis. ͑b͒ Intersection of the tilted current cylinder on ϭ FIG. 3. Linear spaced contours of Jx in the mid y–z plane at four consecu- the mid x–y plane (z 0) produces an annular distribution for the toroidal tive times during the FRC relaxation showing the axial expansion, the radial component J␪ and two opposing currents Jz where the current cylinder contraction, and discrete structure in the current layer. intersects the x–y plane along the line orthogonal to m. ͑c͒ Contours of the 2ϩ 2 1/2 current density (Jx Jy) show the annular proﬁle of the toroidal current. ͑ ͒ d Contours of the axial component Jz,m exhibit two opposing currents ͑ ͒ ratrix collapses at tЈӍ13 ␮s ͓Figs. 2͑g͒ and 2͑h͔͒ into a similar to those sketched in b . single null point at the origin.

27 Dreicer field (E␪ӍE ϭm␯ v /eӍ0.2 V/cm͒ but no run- C. Rotating current density D ei th away electron distributions are found in the current sheet. As discussed in Part I,20 the FRC contains an internal However, strong current-driven ion sound turbulence is ob- ␪ ␦ Ϸ current layer that generally flows in the direction. The served in the neutral sheet ͓ n/n 5%, note that v f Ӎ 1/2 Ӎ cross section of the current layer is shown as contours of 30(kTe /mi) , Te 10Ti], which can produce anomalous ␯ Ϸ␻ ␦ 2Ϸ 8 Ϫ1 28 Jx(y,z) at four time steps in Fig. 3. The axis of revolution collisions ͓ * pe( n/n) 10 s ] and explain the ob- Ӎ ⍀Ϫ1 Ϫ1Ӎ ␴ of the current layer is slightly inclined at an angle which served low ratio J␪ /E␪ 5 cm 0.1 Spitzer for kTe varies in time. Also, the current density is asymmetrically Ӎ3 eV. The electron temperature increases during the FRC ϭ → ͒ distributed above and below the axis. The tilted vectors relaxation (kTe 2 4eV. Further details about heating and flux contours in Fig. 2 result from this uneven distribu- and turbulence are presented in Parts III21 and IV.22 tion. The positive and negative current densities have been The tilt of the FRC causes the cylindrical current to in- separately integrated and the net currents are identical to tersect any of the measured planes, as schematically shown ͉͐ ͉ ͑ ͒ within 1% during the lifetime of the FRC, yϾ0Jx dydz in Fig. 4 a . There, the direction of the FRC is assumed to be Ӎ͉͐ ͉ yϽ0Jx dydz . The current structure expands in the axial determined by the magnetic dipole moment of the FRC. direction while contracting radially. Peak levels of Jx form at Hence, a measured x–y plane intersects the inclined layer at zӍ0 and 18 cm which roughly coincides with the position of an oblique angle, and the observed current density has not the magnetic islands that form later during field line annihi- only a toroidal but also an axial component, (J␪ ,Jz). The lation observed in Fig. 9 in Part I.20 As the axial expansion former produces an annulus, the latter forms two opposing occurs, the peak current positions remain fixed, moving currents where the current cylinder intersects the x–y plane roughly in synchronization with the tilting of the zero con- ͓Fig. 4͑b͔͒. tour. Note that if the magnetic force density (JϫB) were This is indeed observed when the current density ϭ 2ϩ 2 1/2 unbalanced, the rings of current at different axial positions J(x,y,0) is displayed as contours of J␪ (Jx Jy) in Fig. ͑ ͒ ͑ ͒ would take on the order of a microsecond to merge. Because 4 c , and contours of Jz,m defined in the following in part this is not observed, we must conclude that the force free ͑d͒ at tЈϭ7.3 ␮satanx–y plane located in the center of the condition of EMHD is satisfied to a large extent. FRC (zϭ0). The toroidal current layer intersects the plane It is interesting to note that Fig. 3 shows that the current as a nearly circular ring, similar to the current sketched in ͑ ͒ layer shrinks radially but the current density does not vary Fig. 4 a . However, the distribution of Jz is found not to be significantly during the relaxation of the FRC. Additional symmetric about a plane defined by the boundary between experiments have shown the current density does not change positive and negative Jz . The distribution could be repre- appreciably with different initial values of BH , indicating a sented by the superposition of several line currents as in Fig. current-limiting process controls the decay. The self- 4͑b͒, but each slightly offset in ␪ and r and decreasing in consistent electric field in the neutral layer is close to the magnitude as r increases ͑not shown͒. Adding the individual

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FIG. 5. Contours of Jz,m in the mid x–y plane at four consecutive times during the FRC relaxation. The current system and its magnetic dipole mo-

ment (mx ,my) rotate as the FRC precesses.

͑ ͒ ϭ 1͉͐ FIG. 6. a The amplitude of the magnetic dipole moment m 2 r ϫ ͉ ͑ ͒ ␪ magnetic dipole moments of each line current system leads J dV of a freely relaxing FRC. b Precession angle of m, prec ϭtanϪ1(m /m ). ͑c͒ Tilt angle of m, ␪ ϭtanϪ1͓(m2ϩm2)1/2/m ͔. to an overall dipole moment that is not exactly aligned with y x tilt x y z the moment of the strongest and most central line currents. According to the model of a rigid toroidal current distribu- D. Oscillation, rotation, and precession of magnetic tion whose axis is tilted with respect to the z axis, the Jz in dipole moment the x–y plane is only seen where J␪Þ0. This concept is In order to quantify the direction of the FRC, we calcu- applied to the measured data by using J␪ normalized to its late the magnetic dipole moment, m. Because the measured peak value at each ␪, J␪͉ , as a mask to filter out the max data consists of an x–y plane and a y–z plane, not an entire portions of J not contained in the cylindrical structure. The z volume, the various components of m are calculated sepa- masked J is defined as J (x,y)ϭJ (x,y)J␪(x,y)/J␪ at z z,m z ,max rately. Assuming axial symmetry and using the dominant to- each time. Although the peak values of J do not signifi- z,m ˆϩ ˆ roidal current, Jx , the y–z plane data yield myy mzz cantly differ from the peak values of Jz , the values of Jz near Ӎ 1͐ ˆϩ ˆ ϫ ˆ ␲ the edge of the measurement volume are reduced. This af- 2 (yy zz) xJx 2 ydydz. From the x–y plane data we ˆϩ ˆӍ 1⌬ ͐ ˆϩ ˆ ϫ ˆ fects the value of the calculated moments of the distribution have mxx myy 2 z (xx yy) zJz,mdxdy, where the ͑ ⌬ of Jz,m in the x–y plane. The magnetic dipole moment per length scale in z is calculated from the y–z plane data z ͒ ˆϩ ˆ ⌬ Ӎ 1͐ ˆϩ ˆ ϭ͐J (yϭy ,z) dz/J (yϭy ,zϭ0) with y the radial unit length , defined as (mxx myy)/ z 2 (xx yy) x max x max max ϫ ˆ ͑ ͒ position of the peak Jx . zJz,mdxdy, has its direction indicated in Fig. 4 d .Itis found to be nearly aligned with the boundary between the Figure 6 shows the magnitude and angles that define the ϭ ␪ ␪ ˆϩ ␪ ␪ ˆ extrema of Jz,m , and defines the projection of the axis of the direction of m m sin tilt cos precx m sin tilt sin precy ϩ ␪ ˆ ͑ ͒ tilted cylindrical current structure in the x–y plane. m cos tiltz. The magnitude, m, is shown in Fig. 6 a , the ␪ ͓ ͑ ͔͒ As time progresses, the magnetic dipole moment re- precession angle, prec Fig. 6 b , is the angle between the mains nearly aligned with the boundary between the rotating projection of m in the x–y plane and the x axis while the tilt ␪ ͓ ͑ ͔͒ axial current peaks. This is clearly shown in Fig. 5 where angle, tilt Fig. 6 c , is the angle between m and the z axis. ␪ Ӎ ␮ contours of Jz,m together with the magnetic dipole moment The maximum of tilt occurs at tЈ 5 s, as seen in Figs. per unit length are shown at four consecutive times. At the 2͑c͒ and 2͑d͒. The decay of m is predominantly due to the earliest chosen time, tЈϭ5 ␮s, the projected m vector is radial contraction of the toroidal current structure and par- ␪ nearly perpendicular to the line joining the peak Jz,m con- tially due to its weakening in time. prec varies uniformly for ӍϪ 6 Ϫ1 ␪ ץ ␪ץ ϭ ␮ ␮ tours. By tЈ 6 s, direction of m and the peak values of about 7 s at a rate of prec / t 10 s and tilt never Jz,m have rotated clockwise by a bit less than 60°. For the exceeds 20° from the z axis. ␶ ͑ ͒ ͑ ͒ chosen interval, the period of rotation, osc , is approximately Figure 7 a shows a two-dimensional 2-D hodograph ␮ 6 s. However, it varies as the FRC relaxes. The magnitude of the components (my ,mz). The vector oscillates during the տ ␮ ͒ ␶ Ӎ ␮ of Jz,m and the radial distance from the origin to the peak free relaxation (tЈ 3 s with period osc 6 s while de- value of Jz,m is also seen to decrease over this time period. caying in magnitude. No oscillations are observed when the ʈ ͒ The detailed evolution of the current density is discussed in Helmholtz coil field is reversed (BH B0 , not shown . Figure Part I.20 7͑b͒ shows a second 2-D hodograph for the axial current

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FIG. 8. Electron fluid velocity v f and velocity of the rotating magnetic field vm in the toroidal current layer. Both are comparable and do not change significantly during the decay of the FRC. Thus, the orthogonal field com-

ponents (Bx ,By) are frozen into the toroidal electron ﬂow causing the slightly tilted FRC to precess.

rise of the coil current. The sense of rotation is in the opposite direction since the toroidal electron ﬂuid drift is reversed.

F. Relation to rf driven FRCs The precession has been explained as a manifestation of frozen-in field lines in a rotating EMHD plasma. The rotation is due to the toroidal current and the field is a component ϭ 1͐ ϫ FIG. 7. Hodograph of the magnetic dipole moment m 2 r J dV of a Ϫ ͑ ͒ normal to the z axis or B0 . An alternative view, leading to freely relaxing FRC. a 2-D hodograph (my ,mz) exhibits an oscillation ␶ Ӎ ␮ ͑ ͒ the same conclusion, is to consider fields and flows with with period osc 6 s. b Projections (mx ,my) form a rotating 2-D hodograph. The rotation is in the direction of the toroidal electron flow. ͑c͒ respect to the axis ␰ of the poloidal field tilted at an angle ␪ Ϫ 3-D hodograph shows that m precesses around the z axis or B0 . with respect to the z axis. With respect to ␰, the FRC is now ␪ located in a uniform field with two components, B0 sin along ␰ and B cos␪ perpendicular to ␰. The perpendicular system, (m ,m ). This hodograph performs a rotation in the 0 x y component should be frozen into the electron fluid which same sense as the toroidal electron flow ͑shown in the fol- rotates around the ␰ axis. In order to rotate B cos␪, the ␰ lowing͒. Combining the three components allows the con- 0 axis must precess around the z axis at the speed of the drift- struction of the three-dimensional ͑3-D͒ vector m ing electrons, as is observed. ϭ(m ,m ,m ) displayed as a hodograph in Fig. 7͑c͒. The x y z In the present experiment, the toroidal electron flow ro- magnetic dipole moment m precesses around the z axis or tates the perpendicular magnetic field. The reverse process is ϪB . The oscillation and rotation of the 2-D hodographs are 0 used in rotamaks14 and rotating magnetic field ͑RMF͒ just projections of a precessing vector. Next, we address the FRCs.16 There a magnetic field is applied perpendicular to an direction and speed of the observed precession of m. ␻ FRC and rotated at a frequency in the EMHD regime, ci Ͻ␻Ͻ␻ . The electron fluid is rotated by the frozen-in E. Convection of field lines by fluid flow ce magnetic field resulting in a quasi-steady-state toroidal cur- Ϯ ϭ ϩ The opposing axial currents Jz produce magnetic field rent. The net external magnetic field, Btot B0,z Brf,r ,is components (Bx ,By) which form a line dipole. From the tilted with respect to the FRC axis and precesses around it. ץ ␪ץ rotation of m, the angular velocity, prec / t, and the real Vice versa, the FRC precesses in the frame of Btot . The velocity of the field lines are determined inside the current difference between RMF FRCs and the present relaxing ץ ␪ץ ϭ cylinder, vm rcurrent prec / t. The velocity of the electron EMHD FRC lies in the energy flow: In RMF FRCs, the ϭϪ Ӎ fluid in the current cylinder is given by v f J␪ /nee 8 external rf magnetic field supplies the energy to overcome ϫ 6 Ӎ 2 Ӎ ϫ 11 Ϫ3 ͑ 10 cm/s for J␪ 1 A/cm and ne 6 10 cm . Figure dissipative losses manifested by a lag of flow with respect to 8 shows a comparison of the field line velocity and the fluid field͒ so as to maintain a steady-state FRC. In the relaxing flow versus time during the free relaxation of the FRC. The EMHD FRC, the free energy source is the poloidal magnetic fluid velocity slightly exceeds the field line velocity but in field. It drives the toroidal electron flow, which decays by general the field lines rotate with the electron flow, i.e., are dissipation ͑manifested by a lag of field with respect to flow͒. frozen in. The slippage may be caused by the magnetic null In the absence of tilt/precession, the O-type magnetic ␪Ӎ0. The current in thisץ/ץ ,region where EMHD does not hold. However, much of the null layer has axial symmetry 20 toroidal current is a Hall current and the tilt produces field layer is an Ohmic current, J␪ϭ␴ʈE␪ . When the FRC is in- components normal to the original null layer. clined with respect to B0 , the toroidal null line degenerates It should also be noted that the magnetic field compo- into two 3-D null points. Most of the toroidal current is now nents (Bx ,By) rotate during the growth of the FRC, i.e., the a Hall current. In ideal EMHD, the dissipation should be

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