Proceedings of ITC/ISHW2007

Viscosity Estimation Utilizing Flow Velocity Field Measurements
in a Rotating Magnetized Plasma
Shinji YOSHIMURA1) and Masayoshi Y. TANAKA2)
1)

National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan
2)

Kyushu University, 6-1 Kasuga-koen, Kasuga 816-8580, Japan

The importance of viscosity in determining plasma flow structures has been widely recognized. In laboratory
plasmas, however, viscosity measurements have been seldom performed so far. In this paper we present and discuss
an estimation method of effective plasma kinematic viscosity utilizing flow velocity field measurements. Imposing
steady and axisymmetric conditions, we derive the expression for radial flow velocity from the azimuthalcomponent of the ion fluid equation. The expression contains kinematic viscosity, vorticity of azimuthal rotationand its derivative, collision frequency, azimuthal flow velocity and ion cyclotron frequency. Therefore all quantities
except the viscosity are given provided that the flow field can be measured. We applied this method to a rotating
magnetized argon plasma produced by the Hyper-I device. The flow velocity field measurements were carried out
using a directional Langmuir probe installed in a tilting motor drive unit. The inward ion flow in radial direction,
which is not driven in collisionless inviscid plasmas, was clearly observed. As a result, we found the anomalous
viscosity, the value of which is two orders of magnitude larger than the classical one.
Keywords: viscosity, flow, vorticity, fluid equation, directional Langmuir probe, Hyper-I device

1. Introduction
In various fields of plasma research such as fusion
plasmas, space and astrophysical plasmas and laboratory
plasmas, the importance of plasma flow has been widely
recognized and many studies on plasma flow related
phenomena have been performed extensively. In somecases, viscosity plays a crucial role in determining plasma
flow structures. In torus plasmas, for example, poloidal
plasma flow driven by the radial electric field produces
toroidal plasma flow through the effect of viscosity.
Anomalous viscosity has been found experimentally bothin helical devices and tokamaks [1]. In addition, strong
viscosity, the value of which is up to 108 times larger than
predicted by classical collision theory, has been observedin magnetized pure-electron plasmas [2]. In spite of its
importance, little attention has been paid to the viscosity oflaboratory plasmas so far.
In a laboratory plasma, however, spontaneous
formation of a stationary vortex with a density hole aroundthe central axis, which is referred to as plasma hole [3-5],
has been observed, where the viscosity has an essential role.
The flow velocity field associated with the plasma holeexhibits a monopole sink vortex, and the vorticity
distribution is identified as Burgers vortex [6], which isintrinsic in viscous fluids. Scale length of the Burgers
vortex is determined by the kinematic viscosity and theauthor’s e-mail: yoshimura.shinji@LHD.nifs.ac.jp

inward convergent flow. The viscosity estimated from thesize of the vortex is found to be up to four orders ofmagnitude larger than the classical value. Therefore we canconclude that the anomalous viscosity does occur inlaboratory plasmas and its measurement is necessary forthe deeper understanding of flow structure formation.
In this paper, we present and discuss an experimental
estimation method using flow velocity field measurements
and apply it to a rotating magnetized argon plasma as anexample.

2. Radial flow of ion fluid with finite viscosity
The equation of motion of ion fluid with finiteviscosity in an external constant magnetic fieldB 0 = (0, 0, B0 ) is written as∂υ1e+ ( υ ⋅∇ ) υ = [ −∇φ + υ × B 0 ] −∇p∂tMMn1+ ν ∇ 2 υ + ν∇ ( ∇ ⋅ υ ) − ν c υ(1)3, where υ = (υr ,υθ ,υ z ) is the ion flow velocity, n theion density, M the ion mass, e the elementary charge,
φ the plasma potential, p the pressure, ν thekinematic viscosity, and ν c the collision frequency. Here
we neglect the second viscosity. If the fluid may be
regarded as incompressible, the term which contains ∇ ⋅ υ
is vanished, and the equation becomes considerablyProceedings of ITC/ISHW2007
simpler. In cylindrical polar coordinates r , θ , z , theazimuthal component of the fluid equation is given as
follows [7].
∂υθυυe ⎡ 1 ∂φ⎤+ ( υ ⋅ ∇ )υθ + r θ =−− υ r B0 ⎥⎢∂tr
M ⎣ r ∂θ⎦1 1 ∂p2 ∂υ υ ⎤⎡+ ν ⎢ Δυθ + 2 r − θ2 ⎥ −ν cυθ . (2)∂θ r ⎦Mn r ∂θr⎣Imposing the steady ( d / dt → 0 ) and axisymmetric( d / dθ → 0, d / dz → 0 ) condition into eq. (2), we have−⎡ ∂υθ υθ ⎤+ ⎥ = −υr Ωi +r ⎦⎣ ∂rυr ⎢⎡ ∂ 2υθν⎢+1 ∂υθ υθ ⎤− ⎥ −ν cυθr ∂r r 2 ⎦(3)⎣ ∂r, where Ωi is the ion cyclotron frequency. Note that theterms in the square bracket on the left and on the right areequivalent
tothevorticity
ω = (∇ × υ) z =(1/ r )(∂ / ∂r )(rυθ ) and its derivative, respectively. Finally,
we have considerably simple expression for radial flow
velocity asν (∂ω / ∂r ) −ν cυθ
υr =.
(4)
ω + Ωi2

It is obvious from eq. (4) that the radial flow is not driven
in collisionless inviscid plasmas, which have axial
symmetry. In other words, the existence of radial flow
indicates the nonnegligible effect of finite viscosity
and/or collisions. Eq. (4) contains kinematic viscosity,
vorticity of azimuthal rotation and its derivative, collision
frequency, azimuthal flow velocity and ion cyclotron
frequency. Since all quantities except the viscosity are
given when the flow field can be measured, we canestimate the magnitude of viscosity from eq. (4).

collects a directed ion flux through a small opening madeon the side of insulator body. The Mach number of theflow velocity component at a certain angle θ is given by
the difference between upstream ( I (θ ) ) and downstream
ion currents ( I (θ + π ) ) as follows:υ (θ )I (θ + π ) − I (θ )=K(5)Cs
I (θ + π ) + I (θ ), where K is the calibration constant of the order of unity.
We set K = 1.9 in the present experiments, which wasjustified by comparing the flow velocity measured with theDLP to the E × B drift velocity determined from thepotential measurement with an emissive probe.
The DLP was installed in a tilting motor drive unit,
which can vary the insertion angle up to ±23 degrees.
The schematic of the DLP measurement system is shownin fig. 2. Scanning simply in a straight line passing throughthe cross-sectional center of the device twice withcollecting azimuthally-opposed DLP currents, we canobtain the continuous υθ (r ) profile. On the other hand,several steps are needed to measure υr (r ) profile. At first,
the insertion angle of the DLP is varied by tilting
mechanism of the motor drive unit. Then the DLP positionis set to the tangent point of the circumference of a circleof which radius is r0 and the center is identical to thedevice center. By measuring radially-opposed DLP current,
we have one data point of υr (r = r0 ) . Repeating thisoperation with changing insertion angles, discrete data setof υr (r ) can be obtained.

3. Experimental setup
The experiments were performed in the high density
plasma experiment (Hyper-I) device [8] at National
Institute for Fusion Science. The Hyper-I is a cylindricalplasma device (30 cm in diameter and 200 cm in length)with ten magnetic coils (Fig. 1). Plasmas are produced byelectron cyclotron resonance (ECR) heating with amicrowave of frequency 2.45 GHz. The magnetic field
configuration is a so-called magnetic beach structure (1.2kG at z=10 cm (measured from the microwave injection
window), 875 G (ECR point) at z=130 cm). In the present
experiment, the microwave input power was 7 kW and theoperation pressure of an argon gas was 1.3 × 10−4 Torr. A
Langmuir probe was used to measure the electron
temperature and the electron density, typical values ofwhich are 13 eV and 1× 1011 cm −3 , respectively.
The ion flow velocity measurements were carried out
with a directional Langmuir probe (DLP) [9]. The DLP

Fig. 1. Schematic of the Hyper-I device.

Fig. 2. Schematic of the DLP measurement system.

Proceedings of ITC/ISHW2007

4. Results and Discussions

8Density (a.u.)

(a)

0.4
0.3

Vθ / Cs

0.2
0.1
0.0
-0.1
-0.2

ExB
DLP

-0.3
-0.4
-0.5
-6-4-202

468x (cm)
Fig. 4. Comparison between the azimuthal velocity
measured with the DLP (open circle) and the E×Bdrift velocity determined from the potential profile(filled circle).
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10

602468r (cm)4Fig. 5. Profile of the z-component of vorticity.

20-6-4-202

468x (cm)
70Plasma Potential (V)0.5ω / Cs (cm-1)

To demonstrate the validity of the viscosity
estimation method proposed in Sec. 2, a plasma which
exhibits a good axial symmetry is needed. Density andspace potential profiles of the argon plasma we studied in
the present experiments are shown in fig. 3(a) and (b),respectively. It is confirmed that both profiles, which arehollow in the centers, show good symmetry. The electron
temperature is 13 eV, which is approximately constantover the experimental region.
The hollow potential profile shows that an inward
electric field exists in this plasma, which drives
counterclockwise E × B rotation. Figure 4 shows theazimuthal flow velocity profiles, where the open circles
connected with solid lines indicate the velocities obtainedby the DLP and the filled circles denote the E × Bvelocities calculated from the potential profile. Bothvelocities, which are normalized to the ion sound speed
Cs, show quite a good agreement, where the calibration
factor K in the eq. (5) is set to 1.9. The azimuthal flow
profile shows an axial symmetry. Rigid like rotation isfound around the central axis, and the maximum Mach
number is υθ / Cs ∼ 0.4 .

(b)

60
50
4030
20
100-6-4-202468x (cm)Fig. 3. (a) Density (ion saturation current) profile
measured by Langmuir probe. (b) Plasma
potential profile measured by emissive probe.

Once the azimuthal flow velocity profile is obtained,
the vorticity profile can be easily calculated when theaxisymmetric condition is satisfied. The vorticity profileis shown in fig. 5, where open circles, which denote theexperimental data, are depicted with fitted curve (solidline). When we evaluate the radial flow velocity, thederivative of this fitted curve is substituted to eq. (4).
The ion cyclotron frequency is determined by themagnetic field strength (900 G) and the ion mass (40 u),
so that we have Ωi = 2.16 × 105 s −1 . Adopting the charge
exchange with neutrals as the dominant collisional
process, the collision frequency is given ν c ∼ 3 × 103 s −1 ,where the cross section of the charge exchange collision
for low energy argon is given by Sheldon [10]. Now wehave all quantities except the kinematic viscosity on theright of eq. (4).
Figure 6 shows the comparison between the radialflow velocity measured with the DLP system (opencircle) and the eq. (4) (solid curve, where anomalouskinematic viscosity ν = 2 × 105 cm 2 /s is assumed). In
this figure, the inward flow, which implies the finite

Proceedings of ITC/ISHW2007

References

0.15

[1]
[2]

DLP data
Theoretical curve

0.10

Vr / Cs

0.05
0.00
-0.05
-0.10

ν = 2x105 cm2/s

-0.15
-0.20

024

68

r (cm)
Fig. 6. Comparison between the radial flow velocity
measured by the DLP (open circle) and theoreticalcurve given by eq. (4) with anomalous kinematicviscosity.

viscosity, is evident.
If the plasma may be regarded as inviscid, the radialflow velocity can be estimated as

υr / Cs ∼ (ν c / ω +Ωi ) × (υθ / Cs ) ∼ 0.002.
This radial flow speed is considerably small and cannot
explain the present experimental result. In contrast, thesolid curve in fig. 6 well agrees with the DLP data andshows similar tendency. Note that the kinematic viscosity
predicted by classical theory is ν cl ∼ 3 × 103 cm 2 /s . The
value of the viscosity we used in fitting curve is twoorders of magnitude larger than the classical one, andtherefore we can conclude that the anomalous viscosity
was found in a laboratory magnetized plasma.
To make the presented estimation method more
powerful, an accurate absolute velocity measurement
system is required. Since there is an ambiguity indetermining absolute velocity using the DLP or Machprobes, we are now preparing laser induced fluorescence(LIF) Doppler measurements for velocity field
determination [11]. Detailed study of viscosity effects onflow structure formation utilizing LIF measurements will
form our future work.

Acknowledgments
This work was partially supported by Japan Society
for the Promotion of Science, Grant-in-Aid for Young
Scientists (B) (No. 19740348) and by the National Institutefor Fusion Science, a budgetary Grant-in-Aid NIFS07
ULPP103.

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