TRW Technical Report: Laboratory Detection of the Electrojet Instability R. L. Stenzel, C. F. Kennel, P. H. Lee, and D. Arnush TRW Systems Group, One Space Park, Redondo Beach, California 90278 The electrojet Hall current instability has been produced in a controlled steady-state experiment. The observed frequency and wavenumber spectra are consistent with linear fluid theory and auro- ral observations. 3-D probe correlation measurements show wave propagation in the theoretically predicted direction of maximum growth. PACS numbers: 52.35.Fp, 52.35.Qz, 52.72.+v, 52.75.-d When Δ e = ω ce /ν e >> 1 and Δ i = ω ci /ν i < 1, E × B fields can induce a relative drift v between elecrons (e) and ions (i) exceeding the ion-acoustic speed, c t . The resulting instability [1–3] has been detected in the au- roral and equatorial E-region ionospheres by rocket [? ]. Despite considerable theoretical work [1–3], to our knowl- edge no laboratory experiment has clearly identified the electrojet instability [4]. Our plasma is produced by two concentric cylindri- cal RF discharges in the center of a Helmholtz magnetic field coil (Fig. 1). The plasma parameters are shown in Fig. 2. At one axial boundary, a grounded copper cage shields the RF electrodes; at the other end, an anode is split into 18 concentric segments so that a staircase po- tential profile with 18 equal increments can be imposed on the anode. Coaxial probes, movable in either the az- imuthal, radial or axial directions measure the plasma parameters and wave properties. Floating potential fluc- tuations are displayed in the time and frequency domains and analyzed by an analog correlator consisting of a de- lay 0 <τ< 1.25 μsec, 10 MHz bandwidth delay line, a multiplier and an integrator. A radial electric field perpendicular to the axial mag- netic field leading to a closed azimuthal Hall current is imposed by the anode boundary conditions [? ]. With- out anode bias, the metallic walls float since the plasma potential V pl ’ 5kT e is above ground. With positive an- ode bias, V pl increases to keep the electron current equal to the ion saturation current. Since the electron conduc- tivity across B 0 is much smaller than that along B 0 one can establish different V pl on different field lines so that a radial anode potential gradient produces a radial plasma potential gradient. For Δ e >> 1 electrons drift in the E × B direction (see eqns. 3 and 4). The radial ion current implies plasma losses in the discharge center and a density increase in the gap and outer discharge [Fig. 2(a)]. Since the radially inward electron current is carried mostly by hot electrons with large Larmor radii and higher collision frequencies in a Ramsauer gas (Ar) there is a radial gradient in kT e [Fig. 2(a)]. By varying the rf power, ∇n and ∇kT e can be adjusted so that ∇nkT e ’ 0 (not shown here). In this case a Hall current instability is still observed. It arises at very low frequencies (f< 50 kHz) where collisional drift waves exist. Figures 2(a,b) compare the noise level with and with- out Hall current. The broad instability frequency spec- trum extends beyond the lower hybrid frequency, flh ’ 1 MHz up to the ion plasma frequency f pi ’ 1.5 MHz, consistent with auroral measurements. The autocorrela- tion function half-width, Δτ =0.6μs, is consistent with the bandwidth of the power spectrum. Fig. 3c indicates that the fluctuation level reaches δn/n ave ’ 25%. The spatial dependences of the cross-correlation func- tions C 12 of two probe signals yields azimuthal (Fig. 4a insert) and radial (Fig. 4b insert) half-widths Δξ ’ 0.95 cm and Δr ’ 1.25 cm, respectively. The axial corre- lation length is determined by displaying C 12 (τ,r,z) vs r at a fixed delay τ for different z (Fig. 4c). Long correla- tion lengths of Δz =8.5 cm are observed. By measuring C 12 (τ,ξ ) and C 12 (τ,r) vs ξ or r for different τ , we ob- tain time-of-flight diagrams ξ vs τ and r vs. τ (Figs. 4a and 4b). The peaks of the correlation functions yield the azimuthal and radial phase velocity components. Since separate measurements of the dispersion relation, ω vs k, with narrowband filters indicate that ω/k=const, such broad?band measurements of v ph are meaningful. Our spatial correlation measurements are summarized in Fig. 4d for comparison with theory [1–3]. A fluid theory of the electrojet instability yields the following dispersion relation ω vs k and linear growth rate γ L : (1) ω = (1 + α) -1 k · V De + α(1 + α) -1 k · V Di , (2) γ L = α[(1 + α)ν i (1 + Δ 2 i )] -1 ([k · V/(1 + α)] 2 (1 - Δ 2 i ) - k 2 c 2 T ), where V = V De - V Di , and (3) V De = -Δ e cE 0 /(1 + Δ 2 e )B 0 +Δ 2 e cE 0 × B 0 /(1 + Δ 2 e )B 2 0 , (4) V Di = -Δ i cE 0 /(1 + Δ 2 i )B 0 +Δ 2 i cE 0 × B 0 /(1 + Δ 2 i )B 2 0 , (5) α = (1 + Δ 2 i )/(Δ i Δ e )[1 + (k k Δ e /k ⊥ ) 2 ] and k k , k ⊥ are wavenumber components parallel and perpendicular to B 0 , respectively. The measurements and the predictions of Eq . (1) - (5) are compared in Table 1. Equation (2) predicts max- imum growth for waves with k ⊥ k V . As indicated in Fig. 4, the observed waves always propagate more or less along V. In addition, reversing B 0 is observed to re- verse only the direction of azimuthal propagation, while reversing E 0 reverses both v φ and v r . Theoretically, the wave with k ⊥ k V propagates at an angle between E 0 and E 0 × B 0 given by tan η = k φ /k r = -Δ i . For our
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TRW Technical Report: Laboratory Detection of the Electrojet Instability
R. L. Stenzel, C. F. Kennel, P. H. Lee, and D. ArnushTRW Systems Group, One Space Park, Redondo Beach, California 90278
The electrojet Hall current instability has been produced in a controlled steady-state experiment.The observed frequency and wavenumber spectra are consistent with linear fluid theory and auro-ral observations. 3-D probe correlation measurements show wave propagation in the theoreticallypredicted direction of maximum growth.
PACS numbers: 52.35.Fp, 52.35.Qz, 52.72.+v, 52.75.-d
When ∆e = ωce/νe >> 1 and ∆i = ωci/νi < 1, E×Bfields can induce a relative drift v between elecrons (e)and ions (i) exceeding the ion-acoustic speed, ct. Theresulting instability [1–3] has been detected in the au-roral and equatorial E-region ionospheres by rocket [? ].Despite considerable theoretical work [1–3], to our knowl-edge no laboratory experiment has clearly identified theelectrojet instability [4].
Our plasma is produced by two concentric cylindri-cal RF discharges in the center of a Helmholtz magneticfield coil (Fig. 1). The plasma parameters are shown inFig. 2. At one axial boundary, a grounded copper cageshields the RF electrodes; at the other end, an anode issplit into 18 concentric segments so that a staircase po-tential profile with 18 equal increments can be imposedon the anode. Coaxial probes, movable in either the az-imuthal, radial or axial directions measure the plasmaparameters and wave properties. Floating potential fluc-tuations are displayed in the time and frequency domainsand analyzed by an analog correlator consisting of a de-lay 0 < τ < 1.25 µsec, 10 MHz bandwidth delay line, amultiplier and an integrator.
A radial electric field perpendicular to the axial mag-netic field leading to a closed azimuthal Hall current isimposed by the anode boundary conditions [? ]. With-out anode bias, the metallic walls float since the plasmapotential Vpl ' 5kTe is above ground. With positive an-ode bias, Vpl increases to keep the electron current equalto the ion saturation current. Since the electron conduc-tivity across B0 is much smaller than that along B0 onecan establish different Vpl on different field lines so that aradial anode potential gradient produces a radial plasmapotential gradient.
For ∆e >> 1 electrons drift in the E×B direction (seeeqns. 3 and 4). The radial ion current implies plasmalosses in the discharge center and a density increase inthe gap and outer discharge [Fig. 2(a)]. Since the radiallyinward electron current is carried mostly by hot electronswith large Larmor radii and higher collision frequenciesin a Ramsauer gas (Ar) there is a radial gradient in kTe[Fig. 2(a)]. By varying the rf power, ∇n and ∇kTe canbe adjusted so that ∇nkTe ' 0 (not shown here). In thiscase a Hall current instability is still observed. It arisesat very low frequencies (f < 50 kHz) where collisionaldrift waves exist.
Figures 2(a,b) compare the noise level with and with-
out Hall current. The broad instability frequency spec-trum extends beyond the lower hybrid frequency, flh '1 MHz up to the ion plasma frequency fpi ' 1.5 MHz,consistent with auroral measurements. The autocorrela-tion function half-width, ∆τ = 0.6µs, is consistent withthe bandwidth of the power spectrum. Fig. 3c indicatesthat the fluctuation level reaches δn/nave ' 25%.
The spatial dependences of the cross-correlation func-tions C12 of two probe signals yields azimuthal (Fig. 4ainsert) and radial (Fig. 4b insert) half-widths ∆ξ '0.95 cm and ∆r ' 1.25 cm, respectively. The axial corre-lation length is determined by displaying C12(τ, r, z) vs rat a fixed delay τ for different z (Fig. 4c). Long correla-tion lengths of ∆z = 8.5 cm are observed. By measuringC12(τ, ξ) and C12(τ, r) vs ξ or r for different τ , we ob-tain time-of-flight diagrams ξ vs τ and r vs. τ (Figs. 4aand 4b). The peaks of the correlation functions yield theazimuthal and radial phase velocity components. Sinceseparate measurements of the dispersion relation, ω vs k,with narrowband filters indicate that ω/k=const, suchbroad?band measurements of vph are meaningful. Ourspatial correlation measurements are summarized in Fig.4d for comparison with theory [1–3].
A fluid theory of the electrojet instability yields thefollowing dispersion relation ω vs k and linear growthrate γL:
(1) ω = (1 + α)−1k ·VDe + α(1 + α)−1k ·VDi,(2) γL = α[(1 + α)νi(1 + ∆2
i )]−1([k ·V/(1 + α)]2(1 −
∆2i )− k2c2T ),where V = VDe − VDi, and(3) VDe = −∆ecE0/(1 + ∆2
e)B0 + ∆2ecE0 × B0/(1 +
∆2e)B
20 ,
(4) VDi = −∆icE0/(1 + ∆2i )B0 + ∆2
i cE0 × B0/(1 +∆2i )B
20 ,
(5) α = (1 + ∆2i )/(∆i∆e)[1 + (k‖∆e/k⊥)2]
and k‖, k⊥ are wavenumber components parallel andperpendicular to B0, respectively.
The measurements and the predictions of Eq . (1) -(5) are compared in Table 1. Equation (2) predicts max-imum growth for waves with k⊥ ‖ V . As indicated inFig. 4, the observed waves always propagate more orless along V. In addition, reversing B0 is observed to re-verse only the direction of azimuthal propagation, whilereversing E0 reverses both vφ and vr. Theoretically, thewave with k⊥ ‖ V propagates at an angle between E0
and E0 × B0 given by tan η = kφ/kr = −∆i. For our
2
FIG. 1. Schematic of the experimental setup. (a) Cross-sectional view. (b) Axial view. frf = 30 MHz, Prf = 100 W.
FIG. 2. Radial profiles of (a) the electron density, (b) the elec-tron temperature and the collision parameter ∆e = ωce/νenand (c) the plasma potential.
experiments shown in Fig. 4, basic collision data [5] pre-
FIG. 3. Spectrum of the floating probe signal (a) in absence,(b) in presence of the Hall current. (c) Langmuir probe tracein presence of the instability. Argon, 2 mTorr, B0 = 135 G,ωlh/2π ' 1 MHz, ωpi/2π ' 1.5 MHz.
dict ∆i =0.65 or η = −330, which is in good agreementwith the observed value for vr/vφ =0.82, ∆ξ/∆r =0.76,or η = −370.
Solving Eqs.(1) and (6) for k‖/k⊥ we find the expres-sion given in Table 1 which using the measured values forvr and vφ predicts k‖/k⊥ =1/15. The value directly mea-sured from the half-widths of the correlation functionsk‖/k⊥ = (∆ξ−2 + ∆r−2)1/2/∆z is about 1/12, assuming∆e '250. In view of the spatial variations of ∆e (see Fig.2b) this seems to be a reasonable agreement. Thus, themeasured azimuthal, radial, and axial wave numbers areconsistent with the dispersion relation.
Eqs. (1) and (2) combined yield a normalized growth
3
FIG. 4. Time-of-flight diagram of the peak spatial correla-tion function C12(τ, r, φ, z) in (a) azimuthal direction ξ = rφand (b) radial direction r. (c) C12 vs r at a fixed radial probedelay τr for different positions z which determines the axialcorrelation length (see right hand trace). (d) Vector diagramof drift velocities and measured phase velocity ω/k in theplane perpendicular to B0 (Te = 4.5 eV).
rateγL/ω = [αkc2T /viV (1 + ∆2
i )](Q2 − 1)
where Q = V (1−∆2i )
1/2/cT (1 + α)
For our experimental conditions a = 2.3, k⊥ =8.4 cm−1, cT = 3.5 × 105 cm/s, V = 1.66 × 106 cm/s,∆i = 0.65, Q = 1.033 and γL/ω = 21(Q2 − 1) = 1.4.
In view of the cumulative uncertainties in determiningQ, γL/ω may be subject to large errors. Nonetheless,it appears possible to account for a spatial growth fromthermal noise to large fluctuation amplitudes within the5 cm wide experimental region since the radial e-folding
FIG. 5. Table with measured and expected experimental pa-rameters.
length 1/ki = ∂ω/∂kr/[vrkrγL/ω] ' 1.8 mm.
The authors acknowledge valuable discussions with A.Y. Wong and B. D. Fried. Mr. M. Plummer contributedexpert technical support.
ACKNOWLEDGMENTS
Work supported in part by Rome Air DevelopmentCenter F3C602-73-C-0293, Air Force Office of ScientificResearch F44620-73-C-0007 and TRW Independent Re-search and Development funds.
[1] D.T.Farley, Phys. Rev. Lett. 10, 279 (1963).[2] O. Buneman, Phys. Res. Lett. 10, 285 (1963).[3] C. F. Kennel and D. Arnush, Proceedings of AGARD Con-
ference CTP-138-17a, Edinburgh, Scotland (1973).
[4] D. Jassby, Rev. Sci. Instrum. 42, 1355 (1971).[5] S. C. Brown, Basic Data in Plasma Physics, Vol. 78 (MIT
Press, 1959).
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