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Transcript
Pulsed currents carried by whistlers. VIII. Current disruptions and instabilities caused by plasma erosion R. L. Stenzel and J. M. Urrutia Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547 ~Received 21 August 1996; accepted 15 October 1996! In a large magnetized laboratory plasma (n.1012 cm23 , kT e >1 eV, B 0 >10 G, 1 m 3 2.5 m!, the transient processes of switch-on currents to electrodes are investigated experimentally. The current rise time lies between the ion and electron cyclotron periods ~electron magnetohydrodynamics!. The initial current scales linearly with applied voltage and is not limited by the electron saturation current of the positive electrode, but by the ion saturation current of the return electrode. The collection of electrons in the flux tube of the positive electrode gives rise to a space charge electric field, which expels the unmagnetized ions, erodes the density, and disrupts the current. Repeated current oscillations arise from a feedback between current, density, and potential oscillations. The dependence of the transient and unstable electrode currents on externally variable parameters is investigated in the present paper. A companion paper @Urrutia and Stenzel, Phys. Plasmas 4, 36 ~1997!# presents in situ measurements of plasma currents, plasma parameters, and microinstabilities. These results are relevant to the physics of pulsed Langmuir probes, current collection from tethered electrodes in space, and plasma erosion switches. © 1997 American Institute of Physics. @S1070-664X~97!02801-2# I. INTRODUCTION Time-dependent plasma currents between electrodes in magnetized plasmas arise in many applications in laboratory and space plasmas. For example, the current-voltage characteristics of diagnostic Langmuir probes is often swept rapidly in time so as to obtain time-resolved plasma parameters. A well-known feature is a current ‘‘overshoot’’ which occurs near the plasma potential for positive voltage ramps.1 It has been explained by the inertia of ions in a sheath which changes from ion-rich to electron-rich.2 Alternatively, the rapid variation of the plasma potential relative to a constant probe potential in rf discharges can drastically change the current-voltage characteristics of the probe.3 In space plasmas, the rapid motion of satellite-borne electrodes leads to time-dependent currents in the stationary magnetoplasma.4 If the time variation lies between the electron and ion cyclotron periods, the physics of the transient electromagnetic processes is described by electron magnetohydrodynamics ~EMHD!.5 In this regime, detailed laboratory experiments have shown that the transient current is transported in the whistler mode, forms force-free configurations, and behaves linearly up to field energy densities exceeding the particle energy density.6–12 Nonlinearities arise when the fields change the plasma parameters. Electron heating is the fastest process in collisional plasmas, occurring on the time scale of electron-ion collisions. It locally increases the Spitzer conductivity and causes current filamentation.11,13 The next nonlinearity arises from the motion of the ions which occurs on a time scale of an ion transit time across the current channel. The associated nonlinear effects and instabilities are the topic of the present work. It describes the transition from the transient current front to steady-state currents in weakly magnetized plasmas where the ions are mobile but unmagnetized ( n in . v ). For currents small compared to the electron saturation current, the behavior is linear, i.e., current and voltage waveforms are identical step 26 Phys. Plasmas 4 (1), January 1997 functions. The current density exhibits helicity, J•B Þ 0, even in steady-state. For electrode potentials exceeding the plasma potential, the current waveform exhibits a transient overshoot and a relaxation oscillation. The peak current is not limited to the initial electron saturation current. However, such large currents last only for the duration of an ion transit time across the current channel. The collapse of the transient current produces an overshoot effect similar to that described earlier1 but of fundamentally different properties calling for a new explanation. For example, the duration of the overshoot is not an ion transit time across the sheath but across the entire electrode. The height of the overshoot is so large that it cannot result from excess ions in a sheath.2 The initial current is driven by an electric field that is not confined to the Debye sheath but penetrates as a whistler wave field over many collisionless skin depths into the plasma. For such time-dependent fields, the current-voltage characteristic is governed by an ac Ohm’s law rather than dc Langmuir probe theory. The large current collapses due to a major density depletion in the flux tube of the electrode. The density ‘‘erosion’’ produces essentially a plasma opening switch.14 After the collapse of the transient current, the smaller ‘‘steady-state’’ current can exhibit relaxation oscillations with a frequency approximately given by the ion sound speed divided by the electrode size. This relaxation oscillation is not a ringing response to the transient current overshoot, but a genuine instability which also arises when the voltage is turned on slowly, i.e., without initial current overshoot. Steady-state oscillations generated by positively biased probes in weakly magnetized discharge plasmas have been observed earlier but without explaining the instability mechanism.15 In strongly magnetized Q-machine plasmas,16 positively biased ‘‘buttons’’ can excite current-driven ion cyclotron instabilities which have been studied extensively.17–22 At larger bias voltages, the button modifies 1070-664X/97/4(1)/26/10/$10.00 © 1997 American Institute of Physics FIG. 1. Experimental setup, diagnostics and basic plasma parameters. the plasma production and losses and generates ‘‘potential relaxation instabilities,’’ which modulate the button current.23 However, in spite of 60 years of research on probes and their use, there is surprisingly little information available about the type of instability discussed in the present work. Related current disruptions by plasma erosion and double layer formation have been observed in magnetic reconnection experiments.24 They have also been studied in low density plasmas without electromagnetic effects.25,26 Recently, their importance to electromagnetic tethers in space has been pointed out.27,28 The present work describes many new features of the EMHD current transient and instability not reported earlier. The variety of measurements and parameter scalings contributes to the understanding of the nonlinear, transient current phenomena. The paper is organized as follows: After describing in Sec. II the experimental setup and measurement techniques, the observed results are presented in the various subsections of Sec. III. The conclusion, Sec. IV, points out the relevance of the present findings to related observations and applications. While the first of the two companion papers focusses on external current-voltage measurements, the second one29 describes internal measurements of plasma parameters, fields, currents, and microinstabilities. (Dr'1 cm, Dt'50 ns!. From Ampère’s law, the conduction current density, J5¹3B/ m 0 @ e 0 ] E/ ] t, is calculated without making any assumptions about field symmetries or relying on ¹–B50. The measurement accuracy of B can be expressed as u ¹–Bu / u ¹3Bu .5%. The plasma parameters are obtained from a small Langmuir probe ( p r 2 .2.6 mm2 ), which is also movable in three dimensions. Typical Langmuir probe errors are ¹n/n.10%. Miniature coaxial probes ~0.1 mm diam 3 1 mm! are used to detect and correlate fluctuations excited by current-driven instabilities. II. EXPERIMENTAL ARRANGEMENT III. EXPERIMENTAL RESULTS The experiments are performed in a large laboratory plasma device schematically shown in Fig. 1. A 1 m diam 3 2.5 m long plasma column of density n e .1012 cm23 , electron temperature kT e .2 eV, and argon gas pressure p n .0.4 mTorr, is produced in a uniform axial magnetic field (B 0 .30 G! with a pulsed dc discharge (V dis .55 V, I dis .1200 A, t pulse .5 ms, t re p .1 s! using a large oxidecoated cathode.30 In the quiescent, uniform, current-free afterglow, switched plasma currents (t rise <1 m s, I<100 A! are applied with disk electrodes (<2 cm diam! biased positively (V 0 <250 V! with respect to the end chamber wall. The time-varying magnetic fields associated with the plasma currents are measured with a triple magnetic probe, recording (B x , B y , B z ) versus time at a given position. By repeating the highly reproducible discharges and moving the probe to . 15 000 positions in a three-dimensional ~3-D! volume, the vector field B(r,t) is obtained with high resolution A. Rapidly swept Langmuir probe traces Phys. Plasmas, Vol. 4, No. 1, January 1997 FIG. 2. Overshoot features of rapidly swept Langmuir probe characteristics. Slow voltage ramp, V(t) ~dashed line!, produces a small overshoot in the current, I(t), near the plasma potential in a cold plasma ~lower trace!. In a warm plasma ~upper trace!, no overshoot arises since it takes longer to sweep out the retardation region. ~b! Faster voltage ramp produces a larger current overshoot in the cold plasma of parameters as in ~a!. Timedependent peak current does not yield the electron saturation current. We start with the familiar observation that a rapidly swept voltage ramp applied to a Langmuir probe creates a current overshoot near the plasma potential.1 Figure 2 shows probe voltages and currents vs time. The former varies linearly in time and is shown for two different sweep rates @ dV/dt.0.5 V/ m s in ~a! and dV/dt.2.5 V/ m s in ~b!#. The currents correspond to plasmas at two different temperatures but same densities. At the higher temperature (kT e .1.4 eV!, the retardation region is swept out within Dt.10 m s, which is sufficiently slow to produce no current overshoot. At the lower temperature (kT e .0.2 eV!, the corresponding time is much shorter (Dt.2 m s! and an overshoot is visible. The current overshoot grows as the sweep time is decreased @ Dt.1 m s in ~b!#, implying that the peak current is not the electron saturation current. The duration of the overshoot (Dt.1 m s! roughly corresponds to an ion transit time at the R. L. Stenzel and J. M. Urrutia 27 FIG. 4. Time-resolved current-voltage characteristics for a 4 mm diam disk electrode to which a step-function voltage is applied at t50. Insert shows typical large-amplitude current waveform, I(t). During the collapse of the current overshoot, the I(t)2V el (t) characteristics exhibits a negative differential resistance. At late times, a ‘‘normal’’ Langmuir probe characteristics is observed but the saturation current is lower than the theoretical value, I e,sat .0.52 A. FIG. 3. Current, I(t), to a disk electrode produced by step-function voltages of different amplitude, V 0 . At voltages below the plasma potential (V 0 ,F pl .8 V!, current and voltage have the same waveforms. With increasing V 0 , a transient current overshoot and subsequent relaxation oscillations develop. Insert shows peak overshoot current, I max vs V 0 . It is not limited to the theoretical electron saturation current, I e,sat 52 p r 2el ne(kT e /2p m e ) 1/2.24 A. The subsequent time-average current, ^ I(t) & , is smaller than I e,sat . sound speed (c s .105 cm/s! across the size of the probe (Dr.1 mm! rather than across the Debye sheath (s.5l D .0.05 mm!. Thus, the overshoot phenomenon cannot be due to an adjustment of the sheath but is possibly caused by a density perturbation on a scale of the probe dimension. The saturation current observed after the collapse of the initial current yields the locally perturbed plasma properties rather than those in the absence of the probe. A swept probe voltage makes it difficult to separate time-dependent and voltage-dependent phenomena. An alternate approach to ramping the probe voltage is to apply a step-function waveform @ V(t<0)50, V(t.0)5V 0 5 const#, record the current vs time, I(t), and to repeat the experiment at different voltages V 0 . Figure 3 shows singleshot traces of I(t) for a relatively large disk electrode (r el 51 cm.r ce .1.3 mm at 30 G, 1.4 eV! with surface normal along B0 . Electrons are collected from both sides of the disk. At voltages smaller than the plasma potential (F pl .8 V!, the current has the same step-function waveform as the voltage. But as the voltage exceeds the plasma potential, an initial current overshoot develops. The peak current, I max , exhibits no saturation at the Langmuir limit, I e,sat 52 p r 2el ne AkT e /2p m e . 24 A for the initial plasma parameters, and increases almost linearly with applied voltage 28 Phys. Plasmas, Vol. 4, No. 1, January 1997 as shown in the insert of Fig. 3. The current rise is limited by the circuit inductance to dI/dt.V/L.V 0 /2.8 m H . 0.36V 0 A/Vm s. The duration of the current overshoot decreases with voltage but is always much larger than the voltage rise time (t rise .100 ns! or the ion plasma period ( f 21 pi .30 ns!, which excludes capacitive and sheath effects as the cause for the overshoot. After the current has collapsed to a value smaller than I e,sat , it exhibits strong fluctuations which consist of repeated current spikes similar to the first one but at a lower amplitude. The period of the repeated current spikes (t re p .6 m s! corresponds roughly to an ion acoustic transit time across the current channel of diameter determined by that of the electrode. Before going further into the physical processes of the current disruption, it may be of interest to examine the current-voltage (I-V) characteristics from which one usually derives the plasma parameters. Figure 4 shows I-V curves for a smaller probe (r el 52 mm! and finer voltage increments (DV el .2 V! than in Fig. 3. At the time of the current maximum @ t.0.4 m s, trace ~a!#, the I-V characteristic exhibits no saturation. This feature indicates that the electric field is not initially limited to the Debye sheath but extends into the plasma and drives currents obeying an Ohm’s law. The slope of the I-V curves is also influenced by the inductance of the wire leading to the probe. This will be examined further below. During the current collapse, the I-V characteristic has a region of negative differential resistance (dI/dV,0). This can lead to possible instabilities with external circuit elements. Apparently ‘‘normal’’ Langmuir characteristic is only observed after the collapse of the transient current and in the absence of subsequent current spikes. However, the inferred density is too low compared to the initial unperturbed value. Averaging the time-dependent current does not yield the correct electron saturation current either. Thus, large planar probes are of limited use for plasma diagnostics. The objecR. L. Stenzel and J. M. Urrutia FIG. 5. Characteristic differences of the transient current overshoot and the subsequent current oscillations. ~a! Step-function voltage applied to a onesided 2 cm diam disk electrode in an active discharge plasma of indicated parameters. ~b! Single trace of electrode current showing both initial transient and subsequent relaxation oscillations. ~c! Ensemble average over N520 repeated experiments shows that the transient overshoot is highly reproducible while the current oscillations phase-mix due to variations in amplitude and timing. ~d! Slowly rising voltage waveform with same peak value (V 0 563 V! produces current waveform ~e! without transient but unstable relaxation oscillations. Oscillation period increases during growth due to widening of the density-depleted current channel. tive of the present work is to understand the mechanism of the perturbations, to find conditions which minimize the errors for diagnostic applications, and to optimize the disruptions for switching applications. In the following subsections, the dependence of the transient current on various externally variable parameters will be discussed, such as voltage waveform, electron density and temperature, magnetic field, ion species, neutral density, electrode size, geometry, orientation with respect to B0 , properties of the external circuit, and role of the return electrode. While this information is highly useful, conclusive evidence requires in situ measurements of plasma currents and parameters, which are presented in the companion paper.29 B. Current overshoot, disruptions, and relaxation oscillations First, we point out some important differences in the properties of the first current overshoot and the subsequent current spikes. The former is a transient phenomenon, the latter is a current-driven instability. Characteristic features are summarized in Fig. 5. A rapidly rising step function voltage @Fig. 5~a!# applied during the active discharge to a 2 cm Phys. Plasmas, Vol. 4, No. 1, January 1997 diam, one-sided disk electrode produces both an initial current transient and subsequent relaxation oscillations as shown in the single current trace of Fig. 5~b!. An ensemble average over N520 repeated traces, ^ I(t) & N520 @Fig. 5~c!#, shows that the initial current transient is highly reproducible while the unstable current oscillations phase-mix due to variations in amplitude and timing. The current oscillations do not depend on the existence of the initial current overshoot. In Fig. 5~d!, the voltage rises slowly to the same dc value as in Fig. 5~a!. In this case, the current @Fig. 5~e!# has no transient overshoot, but the same current oscillations as in Fig. 5~b! are observed to grow. The oscillation frequency decreases as the amplitude increases. It is shown below that the frequency depends on sound speed and radius of the density-depleted current channel. Thus, the frequency pulling suggests that the depleted channel grows radially to an asymptotic size given by the electrode radius. No frequency pulling occurs when the current transient rapidly erodes the density profile. Under certain conditions, the current waveform exhibits only a transient but no oscillations. This occurs ~i! at lower voltages ~see Fig. 3, V 0 520 V!, indicating a threshold for the current instability, ~ii! for different electrode sizes and geometries as discussed below, and ~iii! in H1 -Ar1 plasmas and in the late afterglow where kT e .kT i , suggesting that damping of ion sound waves quenches the instability. The latter point is demonstrated in Fig. 6. At an early afterglow time @Fig. 6~a!, top trace, kT e .1 eV#, the current, I(t), exhibits both a transient overshoot and relaxation oscillations while at a later time, i.e., lower electron temperature @Fig. 6~a!, lower trace, kT e .0.3 eV#, the oscillations are absent. The loss of density @25% for t n 52n/( ] n/ ] t).2.5 ms# cannot explain the disappearance of the oscillations, since density variations by an order of magnitude at higher temperatures have no influence on the instability. In a pure argon plasma at kT e >1 eV, the current relaxation oscillations are present @Fig. 6~b!, top trace#, but they are absent in a mixture of argon and hydrogen @Fig. 6~b!, bottom trace#. The addition of the light ions does not lower the electron temperature, but can cause ion Landau damping of sound waves in argon.31 C. Dependence on plasma parameters The above described current waveforms have been observed over a wide range of magnetic fields (4,B 0 ,80 G! and plasma densities (1010,n e ,1012 cm23 ) in both afterglow and active discharge plasmas. They are present in various gases ~H, He, Ar, Kr! over a wide range of pressures (1024 , p,1023 Torr!. The influence of ion mass on the current waveform has been investigated in krypton and argon at the same neutral pressure. The time scales of the current transient and the instability period are found to increase with the square root of the ion mass ( v Ar / v Kr.1.4). The frequency is nearly independent of neutral density, plasma density and magnetic field, which is characteristic of an ion acoustic oscillation. A new physical effect arises at both high neutral densities and high electrode voltages which changes the current waveform dramatically. Ionization, accompanied by excitaR. L. Stenzel and J. M. Urrutia 29 FIG. 7. Modification of transient currents by ionization. ~a! Current and light emission at low neutral pressures in the absence of ionization. ~b! At higher neutral pressures light emission indicates excitation/ionization. Electrons are energized at an anode double layer. Current rises when ion production exceeds ion expulsion by electric fields. Neutral pressure drop near electrode causes quenching of ionization and current collapse. FIG. 6. Suppression of the relaxation oscillations due to damping of ion acoustic waves. ~a! Current, I(t), at different electron temperatures, controlled by the afterglow time. No oscillations are observed at low values of T e /T i ~bottom trace!. ~b! Current, I(t), in an argon plasma ~top trace! and in a hydrogen-argon plasma ~bottom trace! where current oscillations are suppressed by ion Landau damping. tion, is clearly identified from light emission measurements with a photodiode. Figure 7 presents a comparison of electrode current and light emission at two different neutral gas pressures. At the lower pressure @Fig. 7~a!#, no light emission is observed and the ensemble-averaged current exhibits the typical transient and oscillations. At the higher pressure @Fig. 7~b!#, the observed light emission indicates ionization. The current collapse of the initial transient is reversed and large currents (I max .40 A! are collected. Probe measurements near the electrode show that a potential double layer is formed near the positive electrode, which accelerates electrons to energies exceeding the ionization energy. The current rises as long as the production of ions by ionization exceeds their loss by acceleration in the electric field. At a constant voltage, the current and light decay at t.30 m s indicates a loss of ionization. The loss arises when the outflow of fast ions exceeds the inflow of slow neutrals such that the gas pressure drops and the ionization is quenched. Enhanced current collection by ionization is used for contactors of electrodynamic tethers.32,33 Anodic double layers with ionization are well described in the literature.34–36 Figure 8 shows that the current relaxation oscillations are also affected by ionization phenomena. While keeping electron density and temperature approximately constant (1.231012,n e ,1.331012 cm23 , 1.4,kT e ,1.6 eV!, an increase in neutral density is found to delay the onset of the 30 Phys. Plasmas, Vol. 4, No. 1, January 1997 secondary current pulses. From in situ density measurements reported in the companion paper,29 it is shown that the current oscillations arise when large currents are flowing in a density-depleted flux tube. Ionization compensates for the density loss and delays the onset of the instability. Since plasma production is proportional to the neutral density,37 one finds a linear increase of the delay time, Dt, between the first and second current pulse and the gas pressure ~Fig. 8, insert!. FIG. 8. Dependence of current waveform, I(t), on neutral gas pressure, p n . With increasing pressure, the onset of relaxation oscillations is delayed since ionization refills the depleted current channel. Insert shows linear increase of delay time, Dt, between the first and second current pulse vs neutral pressure. The delay results because plasma production increases linearly with pressure. R. L. Stenzel and J. M. Urrutia FIG. 9. Dependence of current waveform, I(t), on electrode size and orientation. ~a! Currents to three planar electrodes of different cross sections with surface normals along B0 . Voltage and initial plasma parameters are constant. For electrodes smaller than an electron Larmor radius ~top trace!, the narrow overshoot (Dt.0.15 m s! is followed by a dc current close to the theoretical electron saturation current. The overshoot widens for larger electrodes, and is followed by oscillations whose period increases with electrode radius. ~b! Voltage and current waveforms for a one-sided, 2 cm diam disk electrode with surface normal along and across B0 . In the latter case, the oscillations vanish while the transient current remains. D. Dependence on electrode size and geometry The electrode size and its orientation with respect to B0 have a significant effect on the current waveform. Figure 9~a! shows the current waveforms for three electrodes of different sizes. Their surface normals are along B0 , the same step-function voltage is applied (V 0 51100 V, t rise <0.1 m s!, and the plasma parameters are identical (n e .1012 cm23 , kT e .1.3 eV, B 0 530 G!. The smallest electrode (r.0.65 mm, 2 p r 2 .2.6 mm2 , top trace! whose size is less than an electron Larmor radius (r ce .1.3 mm! exhibits only a short initial current transient (I max .0.35 A, Dt.0.15 m s! followed by a dc current (I dc .80 mA! close to the theoretical electron saturation current (I e,sat .83 mA!. However, for electrodes large compared to r ce ~middle and bottom traces!, the overshoot is followed by multiple current peaks whose widths and repetition times increase with electrode radius. For the large electrode, the half width of the current overshoot (Dt.5.6 m s for r52.5 cm! corresponds approximately to an ion transit time across the electrode radius at a speed v i 5r/Dt.4.53105 cm/s or energy 1 2 2m i v i .4.1 eV. The initial current rise time is determined by the applied voltage and the inductance of the 1.5 m long wire connecting the pulse generator to the probe (L'2.8 m H!, which limit the initial current rise to dI/dt<V/L537 A/m s as observed. Figure 9~b! shows the effect of electrode orientation with respect to B0 . When the disk is oriented with its Phys. Plasmas, Vol. 4, No. 1, January 1997 surface normal n̂'B0 , the current oscillations vanish but the transient overshoot is almost as large and lasts as long as when n̂i B0 . In the latter case, the current waveform is not affected if n̂ is anti-parallel to B0 , i.e., whether the one-sided collector faces toward the cathode or away from it. The ratio of peak currents, I',max /I i ,max .0.84, varies only by 65% when the magnetic field is varied by 600% ~10 G,B 0 ,60 G, not shown!. The current ratio decreases with increasing electrode dimensions. The large cross-field current can neither be a Pedersen current @ I P5 p r 2 s' E' .60 mA at E' .1 V/cm, s' . s i ( n / v ce ) 2 . 0.02 V 21 cm21 ], nor a parallel current to the small surface area projected along B0 (2rd.0.03p r 2 ). Direct measurements of the current density near the electrode, shown in the companion paper,29 demonstrate that the cross-field current is an electron Hall current @ s H. s' ( v ce / n )@ s' # produced by a space charge electric field E'n̂, B0 in front of the electrode outside the sheath (Ei n̂ at the electrode!. The duration of the transient current is approximately given by an ion transit time across the electrode along E. Since the Hall electric field does not energize electrons, no ionization phenomena are observed for identical conditions as in Fig. 7~b!. Further detailed investigations of the current dependence on electrode size and orientation with respect to B0 have been performed with a probe of continuously variable cross section. It consists of a rectangular tantalum plate ~3.2 cm 32.2 cm30.4 mm!, which can be retracted between two insulating mica sheets so as to vary the double-sided collecting area in the range 0,A,14 cm2 . With the surface normal along B0 and V 0 5 const, the measured current has been normalized to the collector area and plotted as current density vs electrode area @Figs. 10~a! and 10~b!#. The ion current density is found to be independent of surface area as expected from probe theories.2 Basic properties of the ion current are shown in Fig. 10~c!. Its temporal behavior exhibits no nonlinearities or instabilities comparable to those observed at similar positive voltages ~see Fig. 3!. There is a capacitive current transient, I5CdV/dt, which is associated with the sheath and circuit capacitance (C.250 pF, C sheath . e 0 A/5l D . 178 pF, C line .72 pF. The ion current follows classical I-V characteristics.2 In earlier experiments in the same device, a test electron beam has been used to measure accurately the electron plasma frequency.38 It was shown then that the ion saturation current yields, to within 62%, the same density when an empirical Bohm factor of 0.43 is used, i.e., I i,sat 50.43Ane(kT e /m i ) 1/2. Thus, under the present conditions, switched ion currents are reliable for time-resolved plasma diagnostics. Returning to the electron currents, one observes that, for V 0 @kT e /e, the peak overshoot current density, J e,max @Fig. 10~a!#, and the ensembleaveraged current density obtained at a late time (30,t,50 m s!, ^ J e & @Fig. 10~b!#, decrease with area indicating that the collection of large electron currents perturbs the plasma. The most probable perturbation is a density loss, i.e., plasma erosion. This explanation is supported by direct measurements discussed in the companion paper.29 For small areas, the average electron current density appears to converge to the theoretical electron saturation current density, J e,sat 5ne(kT e /2p m e ) 1/25 0.93(m e /m i ) 1/2J i,sat , indicated R. L. Stenzel and J. M. Urrutia 31 FIG. 10. Comparison of plasma perturbations by electron and ion currents for an electrode of variable area, A. Current densities, J5I/A, vs electrode area for the ~a! peak transient current, ^ J e,max & , ~b! the theoretical, J e,sat , and measured, ^ J e,sat & , electron saturation currents, and the ion saturation current, J i,sat . The observed loss of electron current density with increasing area indicates a perturbation in the plasma density. ~c! Time variation of the ion current, I i (t) ~insert!, shows no instabilities and the transients are caused by sheath and cable capacitances, I5CdV/dt. The I-V characteristics follow theory indicating that ion current collection does not perturb the plasma and is suited for time-resolved plasma diagnostics. The same appears to hold for electron collection with electrodes smaller than the electron Larmor radius. in Fig. 10~b! by a dashed line. This convergence and the fact that there are no fluctuations in the electron saturation current on probes smaller than the electron Larmor radius ~Fig. 9~a!, top trace! indicate that the I-V characteristics of small electrodes appears to be useful for diagnostic purposes. For large currents, the resistive voltage drop along the copper wire between the pulse generator and the electrode has to be taken into account (R wire .0.5 V). As the area increases, the electrode voltage, V el 5V 0 2I max R wire , decreases and, since I max } V el , the current density also decreases. After correcting for the resistive voltage drop, the current density, J e,max (V 0 /V el ), displayed in Fig. 10~a!, still decreases strongly with area. The voltage correction is negligible (V 0 /V el ,1.03) for the smaller average electron current ( ^ I e & ,5 A!. Inductive voltage drops are absent at the current maximum. The nonlinear scaling again suggests that the loss of current density is caused by a modification of the plasma parameters, e.g., a loss in density, which occurs on the time scale of the overshoot. Figures 11~a! and 11~b! present a comparison of the current collection along and across the magnetic field (B 0 550 32 Phys. Plasmas, Vol. 4, No. 1, January 1997 G! with the variable-area probe. The transient current is displayed vs time and electrode area in linear contour plots. When the surface normal, n̂, is along B0 @Fig. 11~a!#, both the peak current and the disruption time increase linearly with surface area. The initial current (t,0.5 m s! is inductance-limited, hence independent of electrode size provided A Þ 0. When the surface normal is across B0 @Fig. 11~b!#, the peak current is reduced and occurs earlier than in the case n̂i B0 . Furthermore, when the adjustable electrode length l (0,l,3.2 cm, across B0 ) exceeds its constant width w (w52.2 cm, along B0 ), the peak current and the disruption time do not further increase with electrode size. In this case, the electrode becomes ‘‘magnetically’’ insulated. Again, this property cannot be explained by resistive crossfield currents or edge currents but indicates a limitation in the electric field, E, parallel to l, which controls the electron E3B drift into the electrode. Considering the global circuit @see insert in Fig. 11~b!#, the current waveform is determined by a constant external inductance (L.2.8 m H! and a timedependent plasma resistance, R pl (t). The latter acts like an opening switch, i.e., its rapid increase produces the current disruption. From the circuit equation, the plasma resistance, R pl (t)5(V 0 2LdI/dt)/I, is readily evaluated and plotted vs time and electrode area for both n̂i B0 @Fig. 11~c!# and n̂'B0 @Fig. 11~d!#. The logarithmically spaced contours ~6 dB/contour! indicate that in the former case the resistance varies exponentially both in time, with scale t 5R/(dR/dt).1 m s (A.4 cm2 ), and electrode area, with scale a 5R/(dR/dA).21.2 cm2 (t.2 m s!. In both orientations, the initial resistance is comparably low (R pl .2 V at A.4 cm2 ), but it does not further decrease when l.w for n̂'B0 . The plasma resistance exceeds the resistance of the copper wire of the external circuit. A more detailed analysis of the circuit effects is given below. E. Dependence on external circuit parameters The inductive and resistive voltage drops in the external circuit can cause a significant difference between the voltages at the generator and the electrode. By connecting a voltage sensor directly to the 2 cm diam disk electrode, the voltage, V el (t), has been recorded simultaneously with the current, I(t), and both are shown in Fig. 12~a!. From the schematic circuit diagram @Fig. 12~a!, insert#, one infers that V 0 5IR1LdI/dt1V el where V 0 590 V is the open-loop voltage or charging voltage of a large capacitor connected to a transistor switch. The effective resistance and inductance of the circuit are found from a linear least-square fit of (V 0 2V el )/I vs (dI/dt)/I, which yields R.1.6 V, L.2.9 m H during the current rise. The main contribution comes from the 1.5 m long tantalum wire between the generator and the electrode. However, the internal resistance of the transistor switch becomes important at larger currents. The effect of these circuit elements is to limit the short-circuit current (I max 5V 0 /R.56 A! and its rise time (L/R.1.8 m s!, as well as to produce large inductive open-loop voltages at the electrode during current disruptions (V max .V 0 ). For larger electrodes, hence larger currents, the inductive voltage, LdI/dt, during the current disruption can far exceed the applied dc voltage.24 Alternatively, rather high open-loop voltR. L. Stenzel and J. M. Urrutia FIG. 11. Contours of transient current, I, and plasma resistance, R pl 5V el /I, vs variable-electrode area and time for two different electrode orientations with respect to B0 ~see inserted sketches!. ~a! Current, I(A,t), for surface normal n̂i B0 showing a linear increase in peak current and disruption time with electrode area, A52lw @left scale as in ~c!#, or length, l ~right scale!. ~b! Current with n̂'B0 shows similar properties as in ~a! but the waveform becomes independent of electrode size when l>2 cm .w. ~c! Plasma resistance @see equivalent circuit in ~b!# for surface normal n̂i B0 increases by over an order of magnitude at the time of current disruption. The disruption timing depends on electrode size. The plasma resistance has the characteristics of an opening switch. ~d! Plasma resistance for n̂'B0 shows similar behavior as in ~c! except when the resistance becomes essentially independent of electrode size. ages are required in order to generate fast-rising, highcurrent pulses, e.g., V 0 >4500 V for I max 5150 A, t rise .0.1 m s, L.3 m H.12 The ratio V(t)/I(t), plotted in Fig. 12~b!, describes the impedance behavior of the electrode-plasma-wall circuit element, consisting of two nonlinear sheaths and the timevarying, nonuniform, anisotropic plasma. Capacitive sheath currents are negligible compared to electron conduction currents. Inductively driven currents are also not significant since the voltage and current waveforms at early times (t,1 m s! preclude a relationship of the form V el 5LdI/dt. Interpreting the ratio as a plasma resistance, R pl 5V el (t)/I(t), one finds, as in Fig. 11, that the resistance rises from a low initial value of R pl (0).2 V gradually to R pl (1 m s).4 V at the current maximum, and then exponentially to a maximum value of R pl,max (1.3 m s).20 V at the current disruption. Note that the plasma is not inductive since during the disruption dI/dt,0 while V el (t).0. It is shown in the companion paper29 that the plasma density near the electrode during this time interval is highly depleted and Phys. Plasmas, Vol. 4, No. 1, January 1997 current-limiting space-charge layers are created. The loss of conductivity by plasma erosion is the principle of a plasma opening switch.14 In those devices, inductive energy stored in a transmission line can be rapidly transferred to a resistive load in parallel to the plasma switch. In our experiments, the resistance decreases during the slow recovery period of the current (2,t,3 m s) but does not return to its original value. Since the plasma impedance is predominantly resistive, the product V el (t)I(t)5 P pl (t) represents the instantaneous power dissipated in the plasma. Figure 12~b! shows that the peak dissipation ( P pl,max .1250 W! occurs in the early phase of the current disruption (t.1.2 m s!. At this time, the energy dissipated, * P(t)dt.500 m J, in a volume p r 2 l.10 cm2 at a density n e .531011 cm23 would increase the particle energy by 1 m J/cm3 /(831028 As/cm3 ) .12.5 eV. This value represents an upper limit of observable heating, since much of the energization occurs in the sheath rather than in the plasma volume. Hence some of the heat is convected by electrons into the electrode. Some energy is required to accelerate ions out of the current channel, while R. L. Stenzel and J. M. Urrutia 33 FIG. 12. Effect of the external circuit properties on electrode current and voltage waveforms for a 2 cm diam, one-sided disk electrode with n̂i B0 . ~a! Current, I(t), open-loop voltage, V 0 5 const, and electrode voltage, V el (t), which differs from V 0 by inductive and resistive voltage drops along the line between source and electrode ~see the inserted circuit diagram!. ~b! Time-dependent power delivered to the plasma, P pl 5I(t)V el (t), and plasma resistance, R pl 5V el (t)/I(t), vs time. Resistance and power maximize near the current disruption. part of the energy goes into the electromagnetic fields of the whistler mode radiated away from the electrode, thus contributing to the radiation resistance. The inductance of the transmission line between voltage source and electrodes has a profound effect on the I-V characteristics for time-dependent currents, created either by switched voltages as in the present case or by timedependent plasma parameters when V 0 is held constant. For example, Fig. 13 shows I-V characteristics of the 2 cm diam disk electrode at various times during the rise of the current, I(t), shown in the insert. This early phase was not discussed in Fig. 4, which was focussed on the current disruption and transition to steady state. In Fig. 13~a! the voltage is measured at the electrode (V el ), while in Fig. 13~b! it is the open-loop voltage at the generator (V5V 0 ). In the latter case, the I-V 0 curves are nearly straight lines whose slopes increase in time. These characteristics are dominated by the external circuit inductance, I/V 0 .t/L, which limits the current rise (V 0 @V el ). In the former case, the I-V el characteristics are also nearly straight lines, but with much higher slopes which gradually decrease in time. These characteristics depend only on the plasma properties. Initially, the plasma conductance is high (I/V el .1 V 21 ), but in time it decreases due to density erosion in the current channel as earlier shown at a single voltage @Fig. 12~b!#. It has been demonstrated that the current depends on the applied voltage, the external circuit, the plasma parameters, and the electrode. However, to close the circuit, a return electrode is required which is usually the grounded chamber 34 Phys. Plasmas, Vol. 4, No. 1, January 1997 FIG. 13. Current-voltage characteristics of a 2 cm diam, one-sided disk electrode during the rise of the current ~see the insert!, demonstrating the effect of the inductance of the transmission line to the electrode. When the voltage is measured at the electrode ~a!, the slope of the curves reflects a decreasing plasma conductance caused by plasma erosion. When the current is plotted vs open-loop generator voltage ~b!, the curves are dominated by the external circuit inductance, I/V 0 .t/L, i.e., form nearly straight lines whose slope increases in time. wall. Alternatively, floating electrodes of different sizes have also been used. As in any passive double-probe system, the negative electrode collects an ion current equal to the electron current collected at the positive electrode, and whichever is limited determines the maximum current. Due to the large surface area of the present plasma column @ 2 p r pl (r pl 1l pl ).105 cm2 , r pl .50 cm, l pl .250 cm#, the ion saturation current to the chamber wall (I i,sat .500 A! exceeds the peak electron currents of the small positive electrodes. However, for smaller return electrodes or smaller plasma devices, the electron overshoot current may be limited by the return ion saturation current, and no overshoots are observable when (I i,sat ) return ,(I e,sat ) collector or A return /A collector , Am i /m e .271. In this case, most of the applied potential drops off at the ion-rich sheath and the positive electrode remains biased below the plasma potential. Instead of a passive ion collector, the return electrode can also be an active electron emitter, i.e., a cathode. Such emitter-collector systems have been used for electrodynamic tether experiments in space.4 Laboratory models have shown the same current transients and instabilities as described above.39 IV. CONCLUSION Switched electron currents collected by biased electrodes in a magnetized plasma present a challenging problem in nonlinear plasma physics. To our knowledge, the theoretical analysis has only been performed in the linear regime for a two dimensional problem by solving the equations via nuR. L. Stenzel and J. M. Urrutia merical techniques.40 Classical Langmuir probe theories are inadequate to explain the phenomena observed. These include a transient current overshoot and a continuous, largeamplitude current instability. The present investigations have been concentrated on external current and voltage measurements as a function of several experimental variables. While these results provide valuable information and can lead to plausible explanations, a conclusive physical picture arises only in conjunction with internal measurements of fields, currents and plasma parameters, which are presented in the companion paper.29 Nevertheless, one can firmly conclude from the present observations that the scaling of the transient current overshoot with voltage and electrode size involves the penetration of electric fields far beyond the Debye sheath, that a density perturbation occurs on the scale length of the electrode, and that the current-driven instability involves an ion acoustic mode with wavelengths set by the electrode size. Several useful conclusions for applications can also be drawn: Diagnostic probes should be of size less than an electron Larmor radius or employ ion saturation currents. Transient electron currents well in excess of the electron saturation current can be drawn provided a sufficient return current can be collected. The duration of the current transient is determined by the electrode dimension across B0 and the sound speed, and the current rise time is limited by electrode voltage and circuit inductance. The current disruption caused by the perturbation to the plasma can be used to transfer the magnetic energy stored in the external inductance to a load in parallel with the plasma with a voltage transformation ratio 11(LdI/dt)/V 0 . For disk electrodes collecting crossfield currents, there is no ‘‘magnetic insulation’’ due to the Hall effect. ACKNOWLEDGMENTS The authors gratefully acknowledge discussions with Dr. Christopher L. Rousculp. The authors also gratefully acknowledge support for this work by the National Science Foundation under Grant No. PHY93-03821. D. G. Bills, R. B. Holt, and B. T. McClure, J. Appl. Phys. 33, 29 ~1962!. F. F. Chen, in Plasma Diagnostic Techniques, edited by R. H. Huddlestone and S. L. Leonard ~Academic, New York, 1965!, p. 113. 3 J. E. Allen, Plasma Sources Sci. 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