PHYSICS OF PLASMAS
VOLUME 7, NUMBER 11
NOVEMBER 2000
Electron magnetohydrodynamic turbulence in a high-beta plasma. II. Single point fluctuation measurements J. M. Urrutiaa) and R. L. Stenzel Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547
共Received 24 November 1999; accepted 3 August 2000兲 A magnetic void is created by high electron pressure in a large nonuniform laboratory plasma. A strong instability is observed in regions of high pressure and magnetic field gradients. It is associated with the electron diamagnetic drift through the essentially unmagnetized ions. Its spectrum is broad and peaks near the lower hybrid frequency. The coupled fluctuations in density, electron temperature, plasma potential, and magnetic field are measured with probes and cross-correlated. The temporal correlation extends only over 1–2 oscillations. The fluctuations propagate in the direction of the electron diamagnetic drift but at the lower ion acoustic speed. In the saturated regime of the instability, the fluctuation waveforms are highly nonlinear. Density cavities with ␦ n/n⯝⫺40% are formed with steepened density rise at the trailing edge. The associated high pressure gradient forms a diamagnetic current sheet. Positive density perturbations are smaller ( ␦ n/n⭐20%兲, broader, and produce regions of weak magnetic fields where the electrons become nearly unmagnetized. Amplitude distributions of nonlinear density, magnetic field, and current waveforms are evaluated. The three-dimensional magnetic field fluctuations are analyzed with hodograms. The direction of the average wave vector points essentially across the mean field in the direction of the diamagnetic drift. The magnetic fluctuations can be interpreted as highly oblique electron whistlers, the density fluctuations as sound waves, but both modes are coupled in a high-beta plasma. Fluctuations in the electric and magnetic fields lead to a time-averaged electron drift, i.e., anomalous transport, across the mean field. © 2000 American Institute of Physics. 关S1070-664X共00兲03211-0兴
I. INTRODUCTION
the experimental setup and measurement techniques in Sec. II, the experimental results, divided into various subsections, are presented in Sec. III. A conclusion in Sec. IV points out the relevance of the present findings to related observations.
Instabilities in high-beta plasmas are of broad interest in space and laboratory plasmas.1–4 The plasma beta is the ratio of the particle pressure to the external magnetic field pressure,  ⫽nkT/(B 20 /2 0 ), 5 and in the present case it is dominated by the electrons and exceeds unity. This causes the external magnetic field to be completely expelled from the interior of a dense, radially nonuniform discharge plasma column. Part I of three companion papers6,7 described the basic parameters of both plasma and field in the transition regime between electron diamagnetism and Boltzmann equilibrium. It also showed that near the plasma edge the electron diamagnetic drift through the unmagnetized ions creates a two-stream instability near the lower hybrid frequency. The present Part II focuses on the properties of the resultant turbulence of density and field fluctuations in the parameter regime between electron magnetohydrodynamics 共EMHD兲8 and unmagnetized plasmas. The density fluctuations have similarities to sound waves, while the magnetic fluctuations are similar to oblique low-frequency electron whistlers,9 but both are strongly coupled. Nonlinear wave phenomena such as the formation of density cavities, wave steepening, and current sheets are observed and discussed. The electron transport across the field is enhanced. Part II is organized as follows: After a brief summary of
II. EXPERIMENTAL ARRANGEMENT
The experimental setup, described in detail in Part I, briefly consists of a linear discharge plasma column (n ⭐1012 cm⫺3 , kT e ⭐5 eV, 50 cm diam, 200 cm length兲 in a weak axial magnetic field (B 0 ⫽5 G兲. The large normalized electron pressure,  e ⯝5, results in a complete expulsion of the magnetic field in the plasma interior as measured with a diamagnetic coil. The diamagnetic electron drift through the unmagnetized ions creates a strong cross-field instability near the lower hybrid frequency. Magnetic field fluctuations are detected with a three-component magnetic probe while basic plasma parameters 共density n, electron temperature kT e , plasma potential pl) and their fluctuations are obtained with Langmuir probes, all movable in a threedimensional 共3D兲 volume. The present part describes the analysis of fluctuations measured simultaneously in time for fixed probe positions. The pulsed experiment is repeated at a rate of 1 Hz and a large ensemble of data is acquired digitally for statistical analysis of auto and cross-correlations, amplitude distribution functions and their moments, and threecomponent magnetic hodograms.
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FIG. 1. Correlation between density and magnetic field fluctuations. 共a兲 Normalized density fluctuation vs time. 共b兲 Simultaneous measurement of normalized axial magnetic field fluctuations, showing a high anticorrelation.
III. EXPERIMENTAL RESULTS A. Cross-correlations between fluctuations in density, potential, magnetic field, and current density
Useful information about the physics of the turbulence can be obtained by correlating the fluctuations of two different physical parameters. For example, the correlation between electron pressure with electric and magnetic fields reveals the force balance, and correlations between components of fluctuating fields and current density indicate the field and current topology. If the measurements are performed at different spatial locations, the delay in the correlation maximum provides information about the propagation speed and direction and spatial coherence. Various correlation methods can be employed ranging from the simple inspection of waveforms to scatter plots and cross-correlation functions. We start with the relation between density and axial magnetic field fluctuations which, due to the electron diamagnetism, is expected to produce a pronounced anticorrelation, ␦ n⬀⫺ ␦ B z . Figure 1 shows simultaneous single-shot traces of fluctuations in the electron saturation current ␦ I e,sat and the axial magnetic field component ␦ B z . All fluctuating quantities have, by definition, zero means. Since relative fluctuations in the electron saturation current, ␦ I e,sat / 具 I e,sat典 ⯝ ␦ n/ 具 n 典 ⫹0.5␦ kT e / 具 kT e 典 , are mainly determined by density fluctuations and less by electron temperature fluctuations, which below are measured as ␦ kT e / 具 kT e 典 ⭐0.5␦ n/ 具 n 典 , one can consider to first order that ␦ n/ 具 n 典 ⯝ ␦ I e,sat / 具 I e,sat典 . The magnetic fluctuations have been normalized to the local mean field, ␦ B z / 具 B z 典 . The Langmuir probe and the magnetic loop are located on the same field line (⌬r⯝0) with minimum axial separation (⌬z⯝2 cm兲. The waveforms show qualitatively the expected anticorrelation but quantitatively, the magnetic fluctuations are too
J. M. Urrutia and R. L. Stenzel
small compared to the value expected from the pressure balance equation, B z2 /2 0 ⫹nkT e ⫽B 20 /2 0 , or its time derivative, ␦ B z / 具 B z 典 ⫹  /2␦ n/ 具 n 典 ⫽0. The normalized fluctuations should be of comparable magnitude for a local  ⯝2, but they actually differ by a factor of 2. This difference can again be explained by the presence of a fluctuating electric field whose ␦ E⫻B drift opposes the ⵜ ␦ p⫻B drift ( p⫽nkT e ), and thereby reduces the plasma diamagnetism. Alternatively, the electron pressure gradient is balanced not only by the magnetic forces ␦ J⫻B but also by the electric force ne ␦ E, which reduces the required current and produces smaller magnetic perturbations. This is the same finding as for the average magnetic field 具 B z 典 , which is also determined by electron diamagnetic and E⫻B drifts.6 The ratio of magnetic fluctuations to density fluctuations can be judged from a scatter diagram such as Fig. 2共a兲. Here the normalized fluctuations are plotted at equal time intervals (⌬t⯝2 s兲 over more than one fluctuation cycle. The data scatter around a straight-line fit whose slope is ⬃⫺0.25 for the large negative density fluctuations but close to ⫺1 for the smaller positive density peaks. Thus the electron confinement varies with amplitude and sign of ␦ n for the nonsinusoidal fluctuations. Fluctuating electric fields must be present to accelerate the unmagnetized ions that produce the observed density fluctuations. Experimental evidence for fluctuating electric fields is presented in Fig. 2共b兲. The current–voltage (I – V) characteristics of a Langmuir probe, examples of which are shown in the insert of Fig. 2共b兲, are swept fast compared to the time scale of the fluctuations (⌬t sweep⯝5 s⬍t fluct⯝50 s兲. Instantaneous values of density, temperature, and plasma potential are evaluated, and a large ensemble of plasma parameters is acquired This allows determination of the mean values ( 具 n 典 , 具 典 , 具 kT e 典 ) as well as the fluctuations around the mean. The normalized plasma potential e ␦ ( /kT e )⫽e( ⫺ 具 典 )/kT e is displayed vs ln(n/具n典) for purpose of comparison with the Boltzmann relation, ln(n/具n典)⫽e␦(/kTe). The latter describes the pressure balance in the absence of magnetic fields, ⵜp⫽neE, such as in the column center. For the large negative density fluctuations, the electrons are closer to Boltzmann equilibrium, while for the positive density peaks the electric force is too small to confine the electrons and a magnetic force is needed, producing somewhat larger magnetic fluctuations. The relation between density and electron temperature is presented in Fig. 2共c兲. The straight-line fit suggests an empirical relation kT e ⯝4.5 eV ⫹共1.5 eV/4⫻1011cm⫺3 )(n⫺6⫻1011 cm⫺3 ). The temporal coherence of fluctuating quantities is best described by correlation functions, e.g., the autocorrelation for a single variable and the cross-correlation for two random variables, such as density and magnetic field.10 The latter is ⬁ ␦ n(t) ␦ B z (t⫺ ) dt, where is a time defined as C( )⫽ 兰 ⫺⬁ delay between the second variable with respect to the first. If the two fluctuations are measured at different locations, the possible delay in the correlation maximum provides information about the propagation speed and direction and spatial coherence of the fluctuations. Figure 3 shows a crosscorrelation function between density and magnetic field fluctuations measured simultaneously at essentially the same lo-
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FIG. 3. Normalized cross-correlation function between density and axial magnetic field fluctuations. Anticorrelation yields negative peak at ⫽0; positive peak implies magnetic return flux is highly correlated and located downstream of the propagating perturbation.
FIG. 2. Scatter plots of fluctuations in different parameters. 共a兲 Normalized magnetic field vs normalized density. 共b兲 Normalized potential vs density. Insert gives sample Langmuir probe traces which provided the data. 共c兲 Correlation between electron temperature and density.
cation. The integration is performed over a time interval long compared to the typical fluctuation period (TⰇ ␦ t fluct), and the absolute cross correlation function C( ) is normalized by the square roots of the peak auto correlations of each variable, C n ( )⫽C( )/ 兵 兰 关 ␦ n(t) 兴 2 dt 兰 关 ␦ B z (t) 兴 2 dt 其 1/2. As expected, the correlation is negative and an extremum (C n ( ) ⫽⫺0.7) for ⫽0. However, a second positive correlation peak is observed at ⯝⫺25 s. It implies that when ␦ B z (t) is advanced by 25 s or the loop moved against the direction of wave propagation ( v x ⯝⫹1.4⫻105 cm/s兲 upstream by x⫽ v x ⯝3 cm, there exists a second, highly correlated magnetic field ␦ B z of opposite direction to the diamagnetic one at x⫽0. The two correlated field fluctuations ⫿ ␦ B z must
be created by a current ␦ J y that is not symmetrically distributed with respect to the density perturbation. The asymmetry in the diamagnetic current is related to the asymmetric density profile of the density perturbations. Typically, the rise from a density minimum is steeper than the fall from a density maximum 关see Fig. 1共a兲兴. Two phenomena can account for this nonlinear steepening of waves propagating at the sound speed: 共i兲 The electron temperature correlates with density, hence density maxima propagate faster than minima (c s ⬀ 冑kT e ); and 共ii兲 For large-amplitude sound waves, density maxima propagate faster than density minima leading to steepening of a rising wave front to an ion acoustic shock.11 Thus the pressure gradient and current density is larger on the trailing edge of a density minimum. A more detailed analysis of the current density will be presented further below. The short correlation time displayed in Fig. 3 implies that successive density/magnetic field fluctuations are poorly correlated. The saturated instability exhibits no coherent azimuthal eigenmode but a multitude of growing and decaying density/field perturbations, i.e., strong turbulence. The coherence changes in time as the instability evolves from growth into saturation. Time resolution can be obtained by forming a cross-correlation over a limited time span, ⌬t osc⬍⌬t ⬍⌬t growth . In order to improve the statisics, an average over a large ensemble of identical time intervals from repeated events is formed. Figure 4 shows a comparison of normalized cross-correlations during 共a兲 the growth and 共b兲 the saturation of the instability. The oscillatory pattern of C n ( ) during the growth phase shows higher coherence than the single oscillation during the saturation phase. The broadening of the main oscillation is due to the decrease in the mean magnetic field which determines the lower hybrid frequency, hence instability frequency, f ⬀ f LH⬀B z . In comparison with Fig. 3, the traces are time-shifted by ⌬ ⯝⫹25 s since the density was measured with a Langmuir probe mounted at a fixed distance ⌬x⯝3.5 cm from the magnetic probe. Since a posi-
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FIG. 4. Time-resolved cross-correlation functions between density and axial magnetic field fluctuations. 共a兲 In the early discharge, the oscillatory function implies long temporal coherence and a narrower spectrum. 共b兲 The coherence is limited to one oscillation in the later steady-state discharge, implying a broad turbulence spectrum. A separation between density and magnetic probes causes a shift of the correlation function in time from which propagation direction and speed is inferred.
tive delay in B z (t) is necessary to produce the previous correlation waveform, the magnetic loop is located upstream of the density probe, hence the fluctuations travel in ⫹x direction with a velocity component v x ⯝3.5 cm/25 s⯝1.4 ⫻105 cm/s. The coherent return flux that produces the positive correlation peak at ⯝0 is located on the downstream side of the density perturbation. Cross-correlations between magnetic field components such as 兰 ␦ B y (t) ␦ B z (t⫺ ) dt and 兰 ␦ B x (t) ␦ B z (t⫺ ) dt show nearly symmetric waveforms with a negative peak at ⯝0 and two positive sidelobes due to the coherent return flux. The symmetric autocorrelations of magnetic field components also have large side lobes adjacent to the maximum at ⫽0, but autocorrelations of the scalar density have negligible sidelobes. Cross-correlations between the orthogonal components of current density, e.g., 兰 ␦ J z (t) ␦ B y (t⫺ ) dt, where ␦ J z ⯝⫺( 0 v x ) ⫺1 ( ␦ B y )/ t, produce antisymmetric correlations with C n ( ⫽0)⫽0 since the field is antisymmetric to a current sheet. Finally, cross-correlations between fluctuations in density and current have been performed, 兰 ␦ n(t) ␦ J(t⫺ ) dt, which exhibit a peak shifted by ⌬ ⯝⫺10 s, consistent with the earlier conclusion that the peak current flows on the steeper density rise which trails the mimimum by ⌬x⫽ v x ⌬ ⯝1.4 cm.
J. M. Urrutia and R. L. Stenzel
␦ n(t)⫽n(t)⫺ 具 ␦ n(t) 典 , implying 具 ␦ n(t) 典 ⫽0. The density minima are observed to be larger than the density maxima and shorter in time, i.e., narrower in space. In order to quantify these properties, an amplitude distribution function has been evaluated12 as follows: The amplitude range of a fluctuation , e.g., ⫽ ␦ n/n, is divided into narrow bins 共typically 1% of max⫺ min), the time intervals ⌬t are counted during which the amplitude is observed to be at ⫾⌬ /2 and plotted as the count F( ) vs . The area under this unnormalized distribution function is 兰 F( )d⫽ 兰 T0 dt, where T is the total time span of the waveform. The distribution is normalized to 兰 F n ( )d ⫽1 and plotted against the normalized amplitude / , where ⫽ 关 兰 2 F n ( )d 兴 1/2 is the variance. This allows easy comparison with the normal or Gaussian distribution, (2 冑 ) ⫺1 exp关⫺(/2 ) 2 兴 and evaluations of the skewness sk⫽ 兰 3 F n ( )d / 3 and kurtosis kurt ⫽ 兰 4 F n ( )d / 4 ⫺3.10 Figure 5 shows waveforms and amplitude distributions F n ( ) for the normalized density fluctuation ⫽ ␦ n(t)/ 具 n(t) 典 and its derivative 关 ␦ n(t)/ 具 n(t) 典 兴 / t. The time derivative is a measure for the spatial density profile of convecting perturbations since / x⯝⫺( v x ) ⫺1 / t. The figure points out two nonlinear features: 共i兲 The formation of narrow, deep density cavities 关Fig. 5共a兲兴, which manifest themselves as a pronounced negative skewness in the amplitude distribution function 关Fig. 5共b兲兴; and 共ii兲 wave steepening on the trailing side of the cavity where the density rises producing large positive spikes in the derivative 关Fig. 5共c兲兴 and a positive skewness in the corresponding amplitude distribution 关Fig. 5共d兲兴. The density rise is typically three times steeper than the fall and corresponds to a width of ⌬x⯝1 cm ⭐c/ pe ⯝0.84 cm ⭐r ce ⯝1.9 cm. The wave front also steepened to ⌬x⯝1.2 mm ⭐r ce ⯝4 mm in earlier experiments with higher background magnetic fields (B 0 ⫽15 G兲.13 As the instability grows and saturates, the nonlinear wave steepening changes. Figure 6 shows a comparison of two normalized amplitude distributions, one taken during the growth of the instability 关Fig. 6共a兲兴, which shows little skewness, and a second one during the saturated steady-state 关Fig. 6共b兲兴 with pronounced skewness. The large kurtosis at early times is due to the relatively large number of small amplitudes in an exponentially growing oscillation. Figures 6共b兲 and 共d兲 summarize the continuous change of all moments of the amplitude distribution of ␦ n(t)/ 具 n(t) 典 . Since the instability is driven by a nonuniform pressure gradient, the turbulence is spatially inhomogeneous 共see Fig. 7 in Ref. 6兲. Figure 7 shows the radial dependence of the variance, skewness, and kurtosis of the normalized density fluctuations. The fluctuation amplitude and skewness decrease toward the column center where ⵜ(nkT e )→0. The increase in kurtosis toward the center reflects a relatively large number of small fluctuations. C. Current sheet formation
B. Amplitude distribution functions
The waveform ␦ n(t) is observed to be highly nonsinusoidal during the saturated state of the instability. The fluctuations are defined as deviations from the ensemble mean,
Wave steepening is not only visible in the density fluctuations but also in the magnetic field fluctuations, implying that the current density must be highly localized. Figure 8共a兲 shows simultaneous single-shot traces of normalized density
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FIG. 5. Waveforms and amplitude distribution functions for the density fluctuation and its time derivative. 共a兲 Single-shot trace of normalized density fluctuation, which exhibits pronounced amplitude asymmetry. 共b兲 Amplitude distribution function with abscissa ⫽ ␦ n/ 具 n 典 normalized to its standard deviation, , and ordinate normalized so that 兰 F( )d / ⫽1/冑2 for the purpose of comparison with a normal distribution of zero mean and ⫽1 共solid curve兲. Moments of the distribution show its deviation from a Gaussian. 共c兲 Time derivative of the waveform shown in 共a兲. For a convective perturbation, it is proportional to the spatial gradients of the density perturbation. 共d兲 Normalized amplitude distribution of 关 ( ␦ n/ 具 n 典 )/ t 兴 , which is skewed positively. This quantitative analysis shows that the instability preferentially forms deep negative density cavities with a steep density rise on the trailing side.
fluctuations and the components of the fluctuating magnetic field. The data are displayed in an expanded time window (0⭐t ⬘ ⭐300 s兲 during saturation. As demonstrated earlier 共Figs. 1 and 3兲, density and axial magnetic field are highly anticorrelated due to the electron diamagnetism. The transverse component ␦ B y is observed to be reasonably well correlated with ␦ n(t)/ 具 n 典 , but the small component ␦ B x in the approximate direction of wave propagation correlates poorly. At a density minimum, the magnetic vector lies in the y⫺z plane 关 ␦ B x Ⰶ( ␦ B 2y ⫹ ␦ B z2 ) 1/2兴. Steepening in the density fluctuation manifests itself by the rapid rise from ␦ n min to ␦ n max
compared to the slow fall from ␦ n max to ␦ n min . The magnetic fluctuations ( ␦ B y , ␦ B z ) rapidly change sign at the steepened density front. The dominant component ␦ B z has an essentially normal amplitude distribution 关Fig. 8共b兲兴, but its derivative is highly skewed to negative values due to wave steepening. The density and magnetic field fluctuations propagate in the direction of the electron diamagnetic drift, which, from conditionally averaged space–time measurements,7 is observed to be at an angle of ␣ ⯝⫺23° with respect to the
FIG. 6. Time evolution of the amplitude nonlinearity of density fluctuations. 共a兲 Amplitude distribution during the growth of the instability exhibits little skewness. 共b兲 During the saturated instability, the distribution is strongly skewed. 共c兲 Time variation of the standard deviation or rms density fluctuation showing transition from growth to saturation. 共d兲 Time dependence of skewness and kurtosis.
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FIG. 7. Spatial dependence of the moments of the amplitude distribution during steady-state discharge. 共a兲 Standard deviation vs radius showing a peak in regions of large pressure gradients. 共b兲 Skewness and 共c兲 kurtosis vs radius with larger magnitudes in regions of low magnetic field.
x-direction. To first order, the current density ␦ JÄⵜ ⫻ ␦ B/ 0 of a plane perturbation convecting at an observed constant speed v ⯝1.2⫻105 cm/s can be obtained from the time derivative, / r⯝⫺ v ⫺1 / t, which yields ␦ J z ⯝ ⫺( 0 v ) ⫺1 ( ␦ B⬜ )/ t, where ␦ B⬜ ⫽ ␦ B y cos ␣⫺␦Bx sin ␣ is normal to v, and ␦ J⬜ ⯝( 0 v ) ⫺1 ( ␦ B z )/ t. The time dependence of ␦ J⬜ (t) and ␦ J z (t) is shown in Figs. 9共a兲 and 共b兲, respectively. Both waveforms are well correlated and show large negative peaks of duration short compared to the typical oscillation period. Thus the total current density ␦ J ⯝( ␦ J 2y ⫹ ␦ J z2 ) 1/2 is localized in a narrow layer, a current sheet, at the steepened rise of the density perturbations. The average direction of the current density can be obtained from a scatter plot of ␦ J⬜ (t) vs ␦ J z (t), displayed at discrete time steps, such as shown in Fig. 10共a兲. A leastsquare fit to the data points yields an angle 具 典 ⯝56° with a regression coefficient R⫽0.92. The total fluctuating current density can now be calculated from ␦ J(t)⫽ ␦ J⬜ sin具典 ⫹␦Jz cos具典. Its waveform is close to that of either component 共Fig. 9兲, hence it is not displayed, but its negatively skewed amplitude distribution is shown in Fig. 10共b兲. The total current density maximizes in current sheets of typical full width at half maximum ⌬x⫽ v x ⌬t 1/2⯝1.2⫻105 cm/s ⫻17 s ⯝2 cm 共see Fig. 11兲. If this value was deconvolved from the spatial response of the magnetic probe 共1.4 cm diam兲, it would be closer to the earlier measured half width
J. M. Urrutia and R. L. Stenzel
FIG. 8. Waveforms and distribution of magnetic fluctuations. 共a兲 Simultaneous traces of fluctuating density and magnetic field components, all of which exhibit nonlinear waveforms. 共b兲 Amplitude distribution function of the axial magnetic field fluctuation and its derivative ( B z / t⬀ ␦ J y ). The former is nearly normal 共see superimposed Gaussian兲; the latter has a negative skewness indicating spikey currents.
of the steepened density front, ⌬x⯝1 cm 关Fig. 5共c兲兴. The peak current density J max⯝0.6 A/cm2 is about three times larger than the return current density, and produces a normalized electron drift v d /c s ⯝25 much larger than the average diamagnetic drift.
D. Turbulent transport
Since electron inertial effects are negligible on the time scale of the slow fluctuations, the electron pressure gradient must be balanced by electric and magnetic forces, ⵜ(nkT e ) ⯝⫺neE⫹J⫻B. From the knowledge of the density gradients, currents, and magnetic fields, one can obtain the fluctuating electric field whose dominant component is ␦ E x ⯝ 关 ␦ J y 具 B z 典 ⫺kT e ( ␦ n)/ x 兴 / 具 n 典 e. Transverse electric field fluctuations ␦ E x and axial magnetic field fluctuations ␦ B z produce a radial electron drift, ␦ v y ⯝ ␦ E x ␦ B z / 具 B z 典 2 , which is found to have a time-average negative value, 具 ␦ v y 典 ⯝⫺8 ⫻105 cm/s. Thus the fluctuations produce a radial electron outflow across the dc magnetic field. The drift speed slightly exceeds the Bohm speed 冑kT e /m i ⯝3⫻105 cm/s, implying a small net radial current 具 J y 典 ⫽ne 关 具 ␦ v y 典 ⫺c s 兴 ⯝32 mA/cm2 and an electron heat flux q⫽ 具 ␦ v y 典具 nkT e 典 ⯝0.2 W/cm2 .
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FIG. 9. Waveforms of current density components exhibiting correlated large negative current extrema.
E. 3D magnetic hodograms
As shown from the previous data analysis, single-point measurements of fluctuations, coupled with independent in-
FIG. 10. Properties of the total current density ␦ J⫽ ␦ J y sin具典⫹␦Jz cos具典. 共a兲 Scatter plot of measured components yields direction of total current density, 具 典 ⫽tan⫺1 ( ␦ J y / ␦ J z ). 共b兲 Amplitude distribution function of the total current density for a single trace, showing a large negative skewness compared to the normal distribution 共smooth curve兲.
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FIG. 11. Formation of current sheets inferred from spikes in waveform of ( ␦ J) 2 . Simultaneous waveform of density fluctuation shows that current sheets are localized on the steep rise from a density minimum.
formation of the wave propagation, can provide considerable insight into the physics of plasma turbulence. Single-point measurements of electromagnetic waves are often analyzed by constructing magnetic hodograms. The following evaluation will show that the magnetic turbulence can also be interpreted to consist of a broad spectrum of oblique lowfrequency whistlers. From simultaneous single-shot traces of magnetic fluctuations, ␦ B x (t), ␦ B y (t), and ␦ B z )(t), one of which is shown in Fig. 12共a兲, we construct the vector ␦ B(t) and plot it in 3D space at discrete time intervals (⌬t⫽5 s) between some time limits (t 1 ⬍t⬍t 2 ) 关Fig. 12共b兲兴. The tip of the vector traces out a curve, the hodogram, and the consecutive vectors define a surface. In the simplest case of a circularly polarized, plane, monochromatic whistler, the wave vector k is normal to the circular surface (ⵜ"B⫽0) and points into the direction defined by the right-hand rule. However, the present turbulence spectrum consists of a wide spectrum of waves and we start by analyzing the hodogram for one typical cycle. In Fig. 12共b兲, the surface normal is constructed from the cross product ␦ B(t)⫻ ␦ B(t⫹⌬t) and averaged over one cycle. The three projections of the hodogram surface 关Figs. 12共b兲–12共d兲兴 show that the surface is reasonably plane for one cycle. The normalized k vector makes an angle with respect to B0 and its projection into the x⫺y plane makes an angle with respect to the x-axis. When the hodogram is drawn for a large number of cycles 关Fig. 13共a兲兴, it reveals a multitude of different planes 关Fig. 13共b兲兴 indicating a broad spread in propagation directions. For each cycle the reasonably well defined k vector angles i , i have been evaluated and the ensemble is plotted in a scatter diagram i vs i 关Fig. 13共c兲兴. In spite of a broad scatter, the mean angle 具 典 ⯝90.6° shows that the fluctuations propagating nearly across B0 and the angle 具 典 ⯝⫺14° is consistent with the propagation direction of the
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FIG. 12. Construction of magnetic hodograms. 共a兲 Simultaneous waveforms of three magnetic field components for about one cycle of a fluctuation. 共b兲 Surface traced out by magnetic field vector ␦ B(t) and its outer surface normal kˆ indicating the wave vector direction for an electromagnetic wave. 共c兲, 共d兲 Different views of 3D hodogram showing a reasonably flat surface with an average normal kˆ⬜B0 almost aligned with the direction of the diamagnetic drift.
density perturbations in the x⫺y plane ( ⯝⫺23°). Since
具 典 ⬎90° the wave energy flows opposite to B0 and the EMHD fields are expected to have negative helicity, as will be confirmed in Part III. In order to check whether these propagation characteristics are consistent with the properties of oblique lowfrequency whistler waves, we examine their dispersion relation, given by (kc/ ) 2 ⯝ 2pe /( ce cos ).9 For the present plasma parameters ( pe / ce ⯝678), a phase velocity corresponding to the propagation speed of the density perturbations ( /k⯝1.2⫻105 cm/s兲, and a measured propagation angle ( 具 典 ⯝90.6°), oblique whistlers would have a normalized frequency / ce ⫽ 关 ( pe / ce )( /kc) 兴 2 (1/cos )⯝0.95 ⫻10⫺3 . Fourier transformation of the fluctuations6 showed a peak in the spectrum at / ce ⯝0.4⫻ LH / ce ⫽0.4 ⫻(m e /m i ) 1/2⯝1.5⫻10⫺3 , which is close to the value predicted by the theoretical dispersion relation. IV. CONCLUSIONS
A pressure-gradient driven instability has been observed in a high-electron- plasma. It produces large-amplitude
J. M. Urrutia and R. L. Stenzel
FIG. 13. Hodogram analysis for magnetic fluctuation over many cycles. 共a兲 Waveform of one fluctuating field component, ␦ B z (t). 共b兲 3D hodogram showing different surfaces for each fluctuation. The average kˆ vector over all cycles is indicated. 共c兲 Polar angles of the kˆ vector for each fluctuation cycle and its mean value 共open square兲.
density and magnetic field perturbations propagating at about the sound speed in the direction of the electron diamagnetic drift, v diaⰇc s . Detailed measurements of fluctuating waveforms of plasma parameters and fields have been performed with stationary probes. Correlations between density, potential, and magnetic field have shown that the electron pressure is balanced by both magnetic and electric forces. The electric field is of course required to produce the density perturbation of the unmagnetized ions. Due to its ambipolar nature, the ␦ E⫻B drift opposes the diamagnetic drift, hence the magnetic fluctuations are not as large as expected from the pressure balance equation. In the saturated regime, the instability exhibits interesting nonlinear effects. Density minima are typically twice as large as density maxima. The asymmetry is thought to be related to a magnetic field perturbation, which for ␦ n⬎0 nearly demagnetizes the electrons, while they are strongly magnetized for ␦ n⬍0. Nonlinear wave steepening is observed and related to the sound speed dependence on temperature and density, c s ⬀ T e (n), as well as known nonlinear sound wave phenomena 关v ( ␦ n) 兴 . In a high- plasma, wave
Phys. Plasmas, Vol. 7, No. 11, November 2000
steepening produces a localized high pressure gradient that forms a diamagnetic current sheet. Positive density perturbations are smaller ( ␦ n/n⭐20%兲, broader, and produce regions of weak magnetic fields where the electrons become nearly unmagnetized. Amplitude distributions of nonlinear density, magnetic field, and current waveforms are evaluated. The three-dimensional magnetic field fluctuations are analyzed with hodograms. The direction of the average wave vector essentially points across the mean field in the direction of the diamagnetic drift. The magnetic fluctuations can be interpreted as highly oblique electron whistlers, the density fluctuations as sound waves, but both modes are coupled in a high-beta plasma. Fluctuations in the electric and magnetic fields lead to a time-averaged electron drift, i.e., anomalous transport, across the mean field. ACKNOWLEDGMENTS
The authors gratefully acknowledge support for this work from National Science Foundation Grant No. PHY
Electron magnetohydrodynamic turbulence . . .
4465
9713240. The authors also thank the anonymous referee for her/his diligence while reviewing this series of papers. 1
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