Published, Physica Scripta T84, 112-116 (2000)

Vortices and Flux Ropes in Electron MHD Plasmas I.

R. L. Stenzel,∗ J. M. Urrutia, and M. C. Griskey Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547

Laboratory experiments are reviewed which demonstrate the existence and properties of three- dimensional vortices in Electron MHD (EMHD) plasmas. In this parameter regime the electrons form a magnetized fluid which is charge-neutralized by unmagnetized ions. The observed vortices are time-varying flows in the electron fluid which produce currents and magnetic fields, the latter superimposed on a uniform dc magnetic field B0. The topology of the time-varying flows and fields can be described by linked toroidal and poloidal vector fields with amplitude distributions ranging from spherical to cylindrical shape. Vortices can be excited with pulsed currents to electrodes, pulsed currents in magnetic loop antennas, and heat pulses. The vortices propagate in the whistler mode along the mean field B0. In the presence of dissipation, magnetic self-helicity and energy decay at the same rate. Reversal of B0 or propagation direction changes the sign of the helicity. Helicity injection produces directional emission of vortices. Reflection of a violates helicity conservation and field-line tying. Part I of two companion papers reviews the linear vortex properties while the companion Part II describes nonlinear EMHD phenomena and instabilities.

I. INTRODUCTION helicity generation by instabiliites are reviewed in Part II. Vortices are common phenomena in fluids, gases, and plasmas. They often arise from velocity shear such as II. EXPERIMENTAL ARRANGEMENT occur in flows past obstacles (for example, von Kar- man vortex streets [1], Kelvin-Helmholtz instabilities [2], drift waves [3], ion temperature gradient driven vor- The experiments are performed in a large laboratory tices [4], black auroras [5]) or differential rotations (cy- plasma device sketched schematically in 1. A 1 m diam × n < 12 −3 clones [6], rotating pure electron plasmas [7]). Two- 2.5 m long plasma column of density e 10 cm , kT < dimensional vortices form fluid rotations around one axis, electron temperature e 5 eV, Argon gas pressure p  × −4 three-dimensional vortices exhibit two linked rotations 3 10 Torr, is produced in a uniform axial mag- B < around orthogonal axes (e.g., Hills vortex [8, 9]). Vor- netic field 0 10 G with a pulsed dc discharge (50 V, t  t  tices in MHD plasmas (e.g., [10], Alfvenons 600 A, pulse 5ms, rep 1 s) using a large oxide-coated β > [11], flux ropes [12]) are more familiar than in electron cathode. High- ( 1) experiments are performed in the β MHD (EMHD) plasmas where their existence has been active discharge, low- studies in the quiescent, uniform, theoretically discussed [13, 14] but only recently stud- current-free afterglow plasma. The controlled excitation ied experimentally [15, 16]. The EMHD vortices are of EMHD vortices is accomplished with pulsed currents time-dependent three-dimensional (3-D) flows in an elec- to electrodes [18], pulsed magnetic fields to antennas tron fluid which is embedded in a stationary ion fluid (loops, toroids) [19], and heat pulses which produce lo- calized thermal currents [20]. The time-varying magnetic and a uniform dc magnetic field B0. The vortices oc- cur on time scales shorter than an ion cyclotron period fields associated with the plasma currents are measured or spatial scales smaller than an ion Larmor radius such with a triple magnetic probe, recording three orthogonal that the ions are effectively unmagnetized. The time- vector components versus time at a given position. By dependent electron flows produce currents, J = nev,and repeating the highly reproducible discharges and mov- perturbed magnetic fields, ∇×B(r,t)=µ0J, which prop- agate as whistler eigenmodes [17] along the dc magnetic field. EMHD vortices can be excited either from con- I(t) trolled sources or arise self-consistently in plasma insta- Magnetic 1 m 1.5 m Antenna Cathode bilities such as pressure-gradient driven instabilities in β β nkT / B2/ µ high- plasmas where = e ( 0 2 0) is the nor- 3D Vortex x malized electron pressure. After a brief description of Electrode z, B the laboratory experiment the basic properties of whistler y 0 vortices such as topology, helicity, propagation and reflec- I(t) tion are reviewed in Part I while vortex collisions, non- 3D ≤ 12 −3 Magnetic ne 10 cm linear phenomena, reconnection at 3-D null points and ≤ Probe kTe 2 eV ≈ B0 10 G ≈ 2.5 m pn 0.3 mTorr (Ar) ∗URL: http://www.physics.ucla.edu/plasma-exp/; Electronic address: [email protected] FIG. 1: Experimental setup and basic plasma parameters. 2 ing the probe to many positions in a three-dimensional volume, the vector field B(r,t) is obtained with high res- olution (∆r  1cm,∆t  10 ns). This allows us to cal- culate at any instant of time the current density J(r,t)= J(r) ∇×B/µ0 without making any assumptions about field symmetries or using ∇·B = 0. The plasma parameters are obtained from a small Langmuir probe which is also movable in three dimensions. B0

III. EXPERIMENTAL RESULTS

A. Properties of Linear Whistler Vortices

The penetration of the applied magnetic field into a FIG. 2: Measured current density lines, J(r,t = const), and plasma is theoretically described by Faraday’s law and surface of a current tube (I = const) for a pulsed current from Ohm’s law, which for an ideal uniform plasma domi- an electrode in a magnetoplasma. The flux-rope topology nated by the Hall effect yields ∂B/∂t = ∇×(v × B) arises from the superposition of an electron Hall current and when pressure gradients are absent. Here, v = −J/ne = the field-aligned current. In Electron MHD the current front −∇ × B/neµ0 is the electron fluid velocity and B = propagates at the speed of a whistler wave. B(r,t)+B0 the total magnetic field. Displacement cur- rents are negligible compared to conduction currents, 2 −7 Jdis/Jcond  (ω/ωp)  10 . Fourier analysis of the density lines[18]. equation yields the dispersion of low-frequency whistlers, Figure 3 shows an experimental verification of a short 2 ω  ωc(kc/ωp) . For small field perturbations, B(r,t)  3-D vortex in the perturbed magnetic field. It is ex- B0, propagating with wave velocity ±v = ∂z/∂t along cited by a current step applied to a toroidal magnetic the dc magnetic field, the linearized solution of the differ- antenna with Bθ ⊥ B0. Snapshots of (a) the toroidal B B ,B ential equation yields J/ne = ±vB(r,t)/B0.Theper- vector magnetic field θ =( x y) and (b) the linked turbed field has a positive (negative) self-helicity density poloidal or dipolar field (By,Bz) show a -like J·B(r,t) for propagation along (opposite to) the dc mag- field topology. The current density or electron fluid ve- netic field. The same holds for the magnetic self-helicity locity v = −J/ne = −∇ × B/neµ0 show a similar topol- density A(r,t) · B(r,t) and the kinetic helicity density ogy. The vortex can be viewed as a whistler wave packet v · ω ∝ J · (∇×J). The unique property of EMHD vor- consisting of a half cycle oscillation propagating predom- tices that their helicity densities depend on propagation inantly along the larger guide field B0. As the vortex direction is the result of the Hall effect, E = J × B/ne. propagates, it gradually becomes V -shaped since the off- The latter also shows that the electromagnetic fields are axial obliquely propagating fields are slower than central force-free and frozen into the electron fluid. When elec- field propagating along B0. Dispersion of whistlers can tron inertia is included, which becomes relevant for scale produce secondary vortices ahead and behind the main lengths shorter than the electron inertial length c/ωpe, vortex leading to nested vortices. the generalized Ω = ∇×v − (e/m)B is frozen Vortices can also be excited with a simple loop antenna into a collisionless electron fluid [21]. with dipole moment along B0 which excites the poloidal When a positive voltage pulse is applied to an electrode vortex field. The toroidal field develops self-consistently. the injected current in the plasma flows in the form of a It is worth pointing out that, in contrast to an isotropic spiral as shown in Fig. 2. The helical current flow can conductor, the toroidal current is not directly driven by be thought of as a superposition of the field aligned cur- the toroidal inductive electric field of the loop antenna, rent and an electron Hall current. The latter is produced but it is a Hall current due to a radial space charge elec- by a radial electric field which is due to the collection of tric field associated with the adiabatic compression of electrons at the electrode. Note that for B0 > 0 the Hall electrons. The incompressible electrons stream along the current JHall = B0 ×E/ne produces a right-handed helix dc field which produces current/field linkage similar to and for B0 < 0 a left-handed helix. The front of the spi- the case of electrode excitation. Space charge and induc- ralling current system propagates at the whistler speed tive electric fields have been determined separately [22]. along B0. Since the current must be closed (∇·J =0), the current density lines at the front return as outer helices to the negative return electrode. The length of the cur- B. Helicity Injection and Directionality of rent tube is determined by the applied pulse length and Propagation propagation speed. For short pulses a current vortex is formed which detaches from the electrodes and propa-  The helicity of a vector field A, defined as K = gates through the plasma. It exhibits knotted current A · (∇×A) dV , is of fundamental interest for 3-D vor- 3

10 ~~ (Bx,By) (a)

~~ −14 A •B= const (µG 2cm)

y −64 +14 0 +40 (cm) +64

5 cm

B0 B0 z = −10 cm −10 −10 0 10 x (cm) 2 G FIG. 4: Magnetic self-helicity density A˜ · B˜ of two oppositely 10 ~~ (By,Bz) propagating vortices excited by the loop. The helicity is posi- (b) tive (negative) for propagating along (against) B0.HereB˜ is measured and A˜ is calculated from ∇×A˜ = B˜ via transfor- mation in 3-D k-space and is defined in the Coulomb gauge.

y v B0 A r,t · B r,t 0 helicity density ( ) ( ), the other one opposite B (cm) to 0 with negative self-helicity density. Thus, the total self-helicity is zero, consistent with the fact that the loop does not inject net helicity. This also holds for the mutual magnetic helicity, A0 · B(r,t) dV = 0, where ∇×A0 = B0. The mutual helicity refers to the linkage of the vortex field B(r,t) with the mean field B0, and the total helic- x = 0 −10 ity is the sum of the self helicity and mutual helicity [25]. −20 −10 0 There is no mutual current helicity, B0 · J(r,t) dV =0. z (cm) Two vortices of opposite self-helicities are also excited by a torus antenna, which excites the toroidal magnetic FIG. 3: Three-dimensional magnetic vortex excited by a fields of the vortices. While the self-helicity vanishes, the toroidal antenna. (a) Vector plot of the toroidal field com- mutual helicity does not vanish since the applied toroidal ˜ ˜ ˜ ponent Btor =(Bx, By)inanx − y plane at the center of the field links with the uniform field so as to inject net he- z − B˜ vortex ( = 10 cm). (b) Poloidal field components pol = licity. Both vortices have the same Bθ, hence carry the ˜ ˜ (By, Bz)inay − z planealongtheaxisofthevortex(x =0). same mutual helicity, as expected from helicity conserva- ˜ ˜ Note the left-handed linkage of Btor and Bpol due to vortex tion. propagation against B0. A loop placed on the axis of a torus can be used to apply helical fields to a plasma, i.e., to inject helicity. The sign of the applied helicity depends on the relative tices. In ideal plasmas both magnetic energy and helicity direction of loop and torus currents. When positive he- are conserved quantities. Taylor [23] conjectured that in licity is injected a vortex with positive self-helicity is ex- weakly dissipative systems helicity is better conserved cited which travels along B0, for negative helicity the than energy. Measurements on propagating EMHD vor- propagation direction reverses. Thus, helicity conserva- tices have shown that in the presence of weak collisional tion implies directional radiation of EMHD fields. This damping both quantities decay at the same rate [18]. The new concept has been tested in a computer simulation generalized vorticity has also been obtained experimen- [26] and verified experimentally [27]. Figure 5 shows a tally and its helicity decays proportional to the magnetic snapshot of magnetic field components in two orthogonal energy, too [24]. However, Taylor’s conjecture applies to planes demonstrating that a loop-torus antenna radiates the problem of relaxation to a force-free state, whereas predominantly one vortex along B0 which has the same the present vortices are already in a force-free state. positive helicity as that of the antenna. The completely Helicity conservation between source and plasma is symmetric loop-torus antenna exhibits a high directivity generally satisfied. For example, when a current pulse is of 20dB (power ratio Pright/Pleft = 100). The antenna applied to a loop antenna as shown in Fig. 4 two vortices directionality also holds for receiving vortices. By mea- are excited, one propagating along B0 with positive self- suring the helicity of unknown vortices, their direction of 4

~ Incident Vortex Reflected Vortex (a) B⊥ = 60 mG (b) 10 ~ ~ ~ ~ 10 ~~ ~ B⊥, Bz (G) B⊥/3, B /3 (B ,B ) B z x y z 2 G

y y 0 0 −80 (cm) (cm) 0

B0 z=15 cm −10 −10 −1.5 −1 −0.5 0 −10 0 10 −10 0 10 −10 0 10 x x (cm) x (cm) x (cm) z = −10 cm (c) Loop−Torus Antenna 10 ~ FIG. 6: Helicity reversal of an EMHD vortex reflected by a B conducting boundary with surface normal along B0.Incident z 0 0 and reflected pulses are measured in a transverse plane at y B 0 ∆z 12 cm from the boundary. Vector fields (Bx,By)and 0 −100 contours Bz(r,t), show that the magnetic self-helicity density 0 (cm) of the incident vortex is negative (propagation against B0) 0 and of the reflected pulse it is positive (right-handed linkage). x=0 B −10 Note that upon reflection only θ reverses sign, implying that −20 −10 0 10 20 also the mutual, hence total magnetic helicity change sign z (cm) upon reflection. The same is found for the current helicity ~ · ,t · ,t Bz (mG) density J B(r ) [or cross helicity density v B(r )] and −100 0 100 fluid helicity density v · ω.

FIG. 5: Demonstration that helicity injection from a loop- torus antenna results in the directional radiation of a whistler x − y z  vortex. Snapshot of (a) the perturbed toroidal magnetic field and reflected vortex in an plane at 13 cm Bz component Btor(x, y) excited by the torus and (b) the axial in front of the plate. The poloidal field component field Bz(x, y) due to the linked loop, forming a vortex of pos- shown in contours does not change sign upon reflection itive helicity. (c) Axial field component in the central y − z while the toroidal field Bθ does reverse. Thus both the plane showing that the injection of positive helicity produces magnetic self-helicity and the mutual helicity, hence total only one vortex propagating along B0. The wave energy radi- magnetic helicity, are reversed. The boundary conditions ated opposite to B0 is two orders of magnitude smaller than imply that the normal magnetic field and the tangential along B0. electric field vanish at the conductor. Near the plate the magnetic field lines are predominantly radial due to induced toroidal currents in the conductor. The plate propagation can be determined. Transmission between draws no axial current from the plasma since Bθ  0at two antennas is non-reciprocal and unidirectional, where the surface. During the pulse reflection the axial mag- the direction can be selected by the antenna helicity or netic flux near the plate grows and decays which causes B the direction of 0. These interesting helicity properties the toroidal inductive electric field to reverse sign. This can lead to useful antenna applications, and explain the implies a sign reversal of the radial Hall current and axial asymmetric plasma production in helicon devices. current, i.e., the poloidal current system, which explains the reversal of the toroidal magnetic field component. The toroidal field component produces a twist in the C. Reflections of Vortices from a Conducting total field line B0 + B(r,t). Since the field line in the Boundary conducting boundary is stationary while the one in the plasma reverses its twist, the field lines cannot connect In this experiment we consider the conservation of he- across the boundary, i.e., field line tying does not occur. licity and energy as a vortex specularly reflects from a The concept of a continuous field line breaks down in the conducting boundary. It is expected that helicity is con- electron-depleted sheath at the boundary since the field served; however, helicity must change sign since reflection is not frozen into a conducting matter. Thus, global he- changes the direction of propagation along B0. The ob- licity need not be conserved. However, energy is highly servation shows that the latter is the case, i.e., helicity is conserved since the dissipation in the plate is negligible. not conserved. No fields are transmitted through the plate which is large A vortex is excited with a torus antenna and propa- compared to the vortex and thick compared to the resis- gates in the direction opposite to B0 against a conduct- tive skin depth. The plasma does have losses which cause ing plate whose surface normal is along B0. Figure 6 a decay of the vortex but there is no change due to the shows magnetic field components for both the incident reflection. 5

1.5 − + surements with and without plate. The incident vortex, ~~Φ Φ 300 dHs /dz = 2  d ⊥/dz propagating in −z direction, dissipates magnetic energy (a) 150 1 B L U /|dU /dz| B/˜ |dB/dz˜ | 0 on a scale length z = m m = 2 75 13 cm. No energy is lost at the conducting boundary 1 5 dHs /dz (σAl/σ  10 ). The reflected vortex, propagating in +z t 37.5 direction, decays at the same rate as the incident one. (G2 cm3) (µs) The magnetic self-helicity of the incident and reflected 18.8 vortex decays essentially at the same rate as the energy. 0.5 9.4 Of course, the sign the helicity reverses upon reflection.

4.7 The only exception is the decay of the mutual magnetic helicity which scales linearly with B˜θ, i.e., decays with 0 Lz  0 e-folding length 26 cm. Since the total magnetic energy, 1 |B + B˜ |2 2πr drdz = 1 B2 2πr drdz+ 1.5 2µ0 0 2µ0 0 ~ 4000  dH /dz = ∫A Bθ 2πrdr 1 ˜2 m 0 µ B 2πr drdz, changes at the same rate as that of (b) 2 0 3000 the vortex alone (Lz  13 cm), the total magnetic he- B0 licity decays at about half the rate of the total magnetic 1 2000 energy, a result coincidentally in agreement with Taylor’s t dHm/dz conjecture [23] for the relaxation in the presence of local- 1000 (µs) ized dissipation (e.g. in current sheets), which does not 2 3 (G cm ) apply to the present case. 0.5 0

−1000 The present result has interesting consequences on the reflection of ducted whistlers in the magnetosphere. −2000 0 Their helicity should reverse upon reflection from the −20 −10 0 10 z (cm) ionosphere, implying a lack of field line tying or frozen-in Plate Antenna field lines at the reflecting interface. FIG. 7: Contours of magnetic helicity per axial length, dH/dz, vs axial position and time. (a) Self-helicity dHself /dz = A˜ · B˜ 2πr dr 2 B˜θ dr B˜z 2πr dr,(b)Mu- dH /dz · ˜ πr dr B B˜ πr2 dr tual helicity mut = A0 B 2 0 θ . 1000 Both partial helicities, hence the total magnetic helicity, re- (a) verse sign upon reflection. dU dz

Figure 7 shows contours of the magnetic helicity per 100 axial length, dH/dz, in a time-position diagram for (a) the self-helicity and (b) the mutual helicity. The self- (G 2 cm 2) helicity is determined from the product of the two linked vortex fluxes, Hself = A˜ · B˜ dV  2ΦtorΦpol, where 10   1000 B˜θ drdz B˜z πr dr Φtor = and Φpol = 2 , selecting (b) B˜ only one sign of z so as to avoid counting the return dH

dz • flux. The mutual helicity is obtained from Hmut = A B A0 · B˜ dV = A0B˜θ dV , where A0 = rB0/2, hence  100 2 Hmut = πB0 r B˜θ drdz. The toroidal antenna located at z = 0 launches two vortices of opposite self-helicities and equal mutual helicities propagating in opposite di- (G 2 cm 3) rections to B0. The left-propagating vortex reflects from 10 the conducting plate at z −23 cm which reverses the −20 −10 0 sign of both the self helicity and the much larger mutual z (cm) helicity. FIG. 8: Axial decay of energy and helicities for incident and A comparison of the axial decay of magnetic energy reflected vortices. The former (solid dots) is obtained in the and helicity is presented in Fig. 8. The peak values absence of the plate, the latter (open circles) from the dif- z − v t along the propagation characteristics are plot- ference with and without plate. (a) Peak energy per axial 2 dU /dz ted vs z for (a) dUm/dz = (B˜ /2µ0)2πr dr and (b) length, m , showing same decay length for incident and dH/dz. The properties of the incident vortex are ob- reflected vortex and no energy losses at the plate. (b) Peak tained from measurements without the plate, those of magnetic self-helicity dHself /dz decays at the same rate as the reflected vortex from the difference between mea- energy but switches sign upon reflection. 6

IV. SUMMARY AND CONCLUSIONS Further properties will be described in the companion review,PartII. Basic laboratory experiments have shown that helicity is a fundamental property of fields and currents in EMHD plasmas. Our main findings are that transient EMHD Acknowledgments fields form 3-D vortex topologies which propagate in the whistler mode, have a unique sign of helicity depending The authors gratefully acknowledge support for this on propagation direction, and conserve helicity except for work by the National Science Foundation under grant a sign change upon reflection at a conducting boundary. PHY-9713240.

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