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 vortex 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., spheromaks [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 spheromak-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 vorticity 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.