Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)606–609
000–000
J. Plasma Fusion

Measurement of Supersonic Rotation Accompaniedwith a Plasma HoleOKAMOTO Atsushi*, YOSHIMURA Shinji1 and TANAKA Masayoshi Y.1
Nagoya University, Nagoya, 464-8602, Japan
1National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 10 December 2003 / Accepted: 12 February 2004)

Abstract
A supersonic ion flow has been measured with a directional Langmuir probe (DLP) in a plasma hole, which is amonopole vortex with density cavity and is characterized by supersonic ion flow. The measured flow velocity is compared to the E × B drift velocity. It is revealed that the higher order components of Fourier mode with respect to theprobe angle increases in the DLP signal as the flow velocity exceeds the sound velocity. The previously proposed DLP
method is extended to apply supersonic flow measurements.
Keywords:
supersonic flow, flow measurement, directional Langmuir probe, E × B drift rotation, ECR plasma

urements.
In this paper, the effect of supersonic flow on the DLP
current is presented, and our previously proposed method isextended to the supersonic region. The experimental setup isdescribed in the next section, the results are shown in section
3, followed by discussions in section 4.

1. Introduction
Plasma flow has been one of interesting topics in laboratory plasmas[1], confined plasmas as well as astrophysicalplasmas. Vortical flow, especially, attracts much attention inconnection with the formation of nonlinear structure and thefundamental structure in highly developed turbulences.
Recently, a monopole vortex with a co-axial density cavity(referred to as “plasma hole”) has been observed in an electron cyclotron resonance (ECR) plasma[2]. When the plasmahole occurs, a sharply-peaked electrostatic potential is spontaneously formed, generating a supersonic E × B rotation of
the plasma. Also a steep density gradient is formed near thesupersonic rotating region, and this structure seems to be ashock front. Thus an investigation of the interplay between
the supersonic rotation and generation of shock-like structure
is an important subject to understand the physics of plasma
hole.
From the experimental point of view, determination offlow structure depends on the method of measuring flow
velocity in plasmas. One of the simplest methods for lowtemperature plasmas uses a directional Langmuir probe(DLP) or a Mach probe, for which many theoretical models
have been proposed[3,4,5]. Commonly, these theoreticalmodels are valid for subsonic flow cases because they adopt
the Bohm criterion for the evaluation of ion saturating current, and may not be extendable to a supersonic flow. On theother hand, our model[6] is based on the property of symmetry of ion current in a flowing plasma, and hence there is apossibility of applying this method to supersonic flow meas-

2. Experimental setup
The experiments have been performed in the high-density plasma experiment (HYPER-I) device at National Institutefor Fusion Science[7]. The HYPER-I device consists of acylindrical chamber (30 cm in diameter, 200 cm in axiallength) and ten magnetic coils, which produce magnetic
fields of 1–2 kG on the chamber axis. Plasmas are produced
and sustained by electron cyclotron resonance (ECR) heating.
A microwave of frequency 2.45 GHz is generated by a klystron amplifier (80 kW CW maximum) and is launched at anopen end of the chamber, where the high-field side condition(ωce > ω, ωce: electron cyclotron frequency, ω : wave frequency) is satisfied. The magnetic field configuration is a socalled magnetic beach with the ECR point at 90 cm from themicrowave injection window. An electron cyclotron wave isexcited in the plasma and fully absorbed before reaching theECR point[8]. The plasma volume is 30 cm in diameter and200 cm in axial length. The typical electron density and temperature are about 1011 cm-3 and 10 eV for the operation pressure 6 × 10-4 Torr (Helium), respectively. The microwave
input power in the present experiment is ≤ 8kW.

Corresponding author’s e-mail: aokamoto@flanker.q.t.u-tokyo.ac.jp* Present address: The University of Tokyo, Tokyo, 113-8656, Japan
1606©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchOkamoto A. et al., Measurement of Supersonic Rotation Accompanied with a Plasma HoleFig. 2 Ion saturation current of DLP as a function of probe
angle in a supersonic flow.

Fig. 1 Cut view of directional Langmuir probe and coordinate system. The x-axis is in the direction to the magnetic field, while y-axis azimuthal direction. The angle
θ is the angle of normal of probe surface, and θv theangle of ion flow velocity with respect to the x-axis,respectively.

in the DLP current at the angle 90° and 270°, i.e., the directions perpendicular to the magnetic field. The ion Larmor
radius is comparable to the probe radius, (ρi/rp) ~ 1, for thehelium plasma, thus the ion is weakly magnetized. The twodips are attributable to the effect of magnetic field. On theother hand, the ion saturation current in supersonic region (r= 30 mm) has no dips at the directions perpendicular to themagnetic field, which is shown in Fig. 2. There are one peak
_ 30° and one dip at θ ~_ 290°. These two angles seem toat θ ~have no symmetry and the profile is rather distorted from theaverage.
Although the ion saturation current I (θ ) includes theeffect of the magnetic field, it can be completely eliminatedwith the use of eq. (1)[2]. The Fourier component of ∆I/〈I〉(∆I = I(θ + π) − I(θ ) and 〈I〉 = [I(θ + π) + I(θ )]/2) for boththe subsonic and supersonic cases are shown in Figs. 3 and 4,respectively. In these figures, the averaged components (m =0) are not shown. In the subsonic flow case, the amplitude offundamental mode 2a1 (see eq. (3)) and the flow velocity areconnected with the relation |v/Cs| = 2a1/α. The amplitude of
the fundamental mode is 0.56 (Fig. 3), thus the ion flow
velocity is determined to be 0.24Cs. The higher modes in Fig.

In these experimental conditions, a plasma hole is generated spontaneously[2]. The flow velocity field of ions accompanied with the plasma hole consists of azimuthal rotation,
axial flow and radial inward flow. The azimuthal rotation isdriven by the E × B drift induced by the sharply-peaked electrostatic potential, where the radial electric field reaches themaximum value of 2 kV/m. The corresponding E × B drift_2×velocity is comparable to the ion sound velocity, Cs ~106 cm/s. We have measured the ion flow velocity at two
radial positions: (a) r = 30 mm, supersonic region, and (b) r =70 mm, subsonic region.
The ion flow velocity was measured with a directional
Langmuir probe (DLP), which collects a directed ion current
through a small opening (1 mm diam) made on the side wallof the ceramic insulator (3 mm diam). The coordinate system
is shown in Fig. 1. The ion flow velocity at a certain
angle θ with respect to the reference axis, v (θ ), is obtainedby measuring two ion saturation currents, Is(θ ) and Is(θ + π),and by using the following relation:[6]v(θ ) 1 Is(θ + π) − Is(θ )=,Csα [Is(θ + π) + Is(θ )]/2

(1)

where θ is the angle between the normal of electrode of DLP
and the reference axis, Cs the ion sound speed. The factor α
is of the order of unity and is determined by comparing thesame E × B drift velocity obtained from the potential measurement. In this experiment, an E × B drift velocity of a subsonic flow region is used to determine the value of α, giving
α = 2.3.

3. Experimental Results
The ion saturation current as a function of the angle ofprobe normal, I(θ ) , was measured at two radial positions. In
the subsonic flow region (r = 70 mm), we observed two dips

Fig. 3 Fourier amplitude of [I(θ) − I(θ + π)]/〈I 〉 in a subsonicflow.

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Okamoto A. et al., Measurement of Supersonic Rotation Accompanied with a Plasma Hole

which is mainly expressed by the contribution from m = 3
mode.
The phase of the higher mode (m = 3) has a characteristic feature in the supersonic flow case. It is nearly 180° different from that of fundamental mode. This result sharply
contrasts with that of the subsonic flow case, where the phasedifference between these modes is not clear and the amplitude of m = 3 mode is negligibly small. The appearance of m
= 3, 5,··· modes may be related to the existence of supersonicflow, and therefore be utilized to evaluate the flow velocity
in the supersonic region, which is discussed in the next section.

4. Discussions
Fig. 4 Fourier amplitude of [I(θ) − I(θ + π)]/〈I 〉 in a supersonicflow.

Here we consider the effect of supersonic flow on a DLPcurrent. For a plasma with a finite ion flow velocity v and amagnetic field B, the ion saturation current of DLP is modified from I0 , and is written asI(θ ) = I0[1 + Fv(v · n)] [1 + FB((B · n)2)],(2)

where I0 is the ion saturation current without the flow and themagnetic field, Fv and FB are the correction term due to theeffects of ion flow and magnetic field, respectively. In subsonic cases, the correction term Fv is expressed as Fv (v · n) =−a1 cos(θ − θ v), while the term FB is expressed as FB((B ·n)2) = b cos 2θ , where a and b are positive and θ -independent factors of the order of a1 ~ v/Cs, b ~ (ρi/rp)2, respectively.
We have assumed so far that the ion flow is too small to be
disturbed by the existence of probe.
However, the probe immersed in a supersonic flow may
change the flow field around the probe, because compressibility becomes important. In this situation, the factor Fv may
have the higher order correction terms, which are in generalθ-dependent. Considering the symmetry of the flow past acylinder (probe), the effective correction term Fv may be
written asFig. 5 Polar plot of Fourier components m = 1, 3 in a supersonic flow.

3 are negligible compared to the fundamental mode, showing
the validity of DLP method. In the supersonic flow measurement, however, the amplitude of higher odd modes, m = 3,5,···, are not negligible (see Fig. 4).
The polar plot of the Fourier modes m = 1, 3 of thesupersonic case are plotted in Fig.5, where the magnetic fieldlies in the x-axis (180°), and the y-axis (90°) corresponds tothe azimuthal direction. The ion flow has both the rotational
component (v y<0) and axial component along B (v x<0).
Considering the radial electric field is outward, we can conclude that the rotational flow is due to E × B drift. Actually,
the rotational component of fundamental mode in the subsonic case is 0.39, which corresponds to |vy /Cs| = 0.17, while the_E × B drift velocity at the same position is 2.8 × 105 cm/s (~0.18Cs). In the supersonic flow measurement (Fig. 5), however, the rotational component of the fundamental mode is
1.27 corresponding to |vy /Cs| = 0.55. On the other hand, the E_× B drift velocity at the same position is 1.4 × 106 cm/s (~0.9C s). The disagreement in the supersonic flow case isattributable to the distortion of flow field around the probe,

F(v · n) = − [a1 + a2 cos(θ − θ v)
+ a3 cos2(θ − θv) + · · ·] cos(θ − θ v).

(3)

Here we include the correction term a2 cos(θ − θ v) + a3 cos2(θ − θ v) for simplicity. With the help of the even property ofb cos 2θ , we can eliminate the effect of magnetic field bymaking of the quantity ∆I/〈I〉∆ I/〈I 〉 = C1cos(θ − θ v) + C3 cos3(θ − θ v)
+ C5 cos5(θ − θ v),(4)where ∆I = I (θ + π) − I (θ ) and 〈I 〉 = [I (θ + π) + I (θ )]/2,
and the Fourier amplitudes are C1 = 2a1(1 + 34 a2 + 58 a22) +a3(1 + a2), C3 = 12 a1a2(1 + 54 a2) + a3(1 + 34 a2) and C5 = 18
a1a22 + 14 a2a3, respectively. The expansion is taken to theorder of 5θ . The eq. (4) is reduced to eq. (1) when |a1/a2| << 1and |a3/a1| << 1 . Thus the factors a2 and a3 represent the disturbance around the probe caused by the compressibility of3608

Okamoto A. et al., Measurement of Supersonic Rotation Accompanied with a Plasma Holethe flowing plasma. Solving these equations with the experimental result (C1 = 1.47, C3 = −0.40), we have a1 = 1.1,where the value C5, which is too small to be distinguishedfrom the experimental error, is determined by a1 and a2 in thefirst iteration. Then the flow velocity is given by v/Cs = 2a1/α= 1.0. The rotational component of flow velocity is |vy /Cs| =_ 0.9, showing
0.86, while the E × B drift velocity is vE × B /Cs ~a good agreement.
A DLP measurement has been carried out for both asupersonic and a subsonic flow, and the results are comparedto the E × B drift velocities obtained by the potential measurements. The higher modes (m = 3, 5, ···) in Fourier amplitude of the quantity ∆I /〈I 〉 appears as the flow velocity
approaches to the ion sound velocity because of compressibility in the supersonic flow. A simple model including thehigher modes due to flow field distortion is formulated toevaluate the supersonic flow velocity, extending applicabilityof the previously proposed DLP method. There is a good
agreement between the predicted value and experimental one.

References[1] A. Okamoto et al., Phys. Plasmas 10, 2211 (2003).
[2] K. Nagaoka et al., Phys. Rev. Lett. 89, 075001 (2002).
[3] M. Hudis and L.M. Lidsky, J. Appl. Phys. 41, 5011
(1970).
[4] P.C. Stangeby, Phys. Fluids 27, 2699 (1984).
[5] I.H. Hutchinson, Phys. Fluids 30, 3777 (1987).
[6] K. Nagaoka et al., J. Phys. Soc. Jpn. 70, 131 (2001).
[7] M.Y. Tanaka et al., Rev. Sci. Instrum. 69, 980 (1998).
[8] M. Tanaka et al., J. Phys. Soc. Jpn. 60, 1600 (1991).

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