J. Plasma Fusion Res. SERIES, Vol.5 (2002) 225-228

Eigenmode Analysis of Low Frequency Waves in a
Field-Reversed Gonfiguration (FRC)
IWASAWA Naotaka, OKADA Shigefumit and GOTO Seiichir
Satellite Venture Business Laboratory, Gunma University, Gunma 376-8515, Japan
IPlasma Physics Laboratory, Osaka (Jniversity, Osaka 565-0871, Japan(Received: 11 December 2001 / Accepted:29 May 2002)

Abstract
Global eigenmodes of low frequency waves in a FRC plasma are obtained using MHD model and asimple equilibrium model (1D). Dispersion relation and radial structure of the global wave fields for theazimuthal mode numbers m = 0,1 are shown. The results for m = 0 are compared with the results of a low
frequency wave heating experiment. The possibility of ion heating by the transit-time magnetic pumping
is discussed.

Keywords:
low frequency waves, eigenmode, MHD, transit-time magnetic pumping, field-reversed configuration

1. Introduction

expenmenB.

Low frequency waves have been used for plasma
heating. Recently a heating experiment of a FieldReversed Configuration (FRC) plasma has been
performed in Osaka University [1]. In this experiment,
low frequency (compared with the ion cyclotron
frequency in the external magnetic field) oscillating

2. Eigenmode Analysis

To investigate the low frequency waves
propagating through a FRC plasma, the single-fluid
MHD equations are used:magnetic field was applied to the FRC plasma. The
applied field was homogeneous in the azimuthal
direction. As a result, a fluctuation of the magnetic field
was observed to propagate in the direction parallel to theequilibrium magnetic field of the FRC plasma. Inaddition, increase of the plasma energy was observedand comparison of the total temperature and the ion
temperature suggests that the increase in the plasmaenergy was mostly due to the increase in the ion
temperature. This implies that the applied oscillating
magnetic field could excite low frequency waves andthe wave energy was absorbed by the ions. In this study
eigenmodes of low frequency waves in a FRC plasma isanalyzed for the azimuthal mode number m = O and l.

The results for m = 0 are compared with the! .v -(pv)=o(1)o(#.v'v)v=ixB-VpQ)

E+vxB=#(jxB)(3)

vxn=-48dt(4)

VxB=loi(5)V.B =0(6)(#.' 'v)no-r=0.

(7)@20O2 by The Japan Society of Plasma

Corresponding author's e-mail: e-mail: iwasawa@ kdlab.el. gunma-u.ac.jp
225

Science and Nuclear Fusion ResearchIwasawa N. et al., Eigenmode Analysis of Low Frequency Waves in a Field-Reversed Configuration (FRC)

Linearizing these equations and assuming that the FRC
plasma has no equilibrium flow, we have 6 independent

1.0.
PdP-u^ 0.5

equations for 6 perturbed quantities yl and El. Since we0.0oseek waves propagating in the 0 and z-directions with aio/@*l1.tar.'1 tx

global radial extension between the geometric axis andthe conducting wall, the perturbed quantities are assumed to have the following form in the cylindrical co-U
1B{8.

oordinate system:

0.0 0.2 0.4 0.6 0.8 1.0
r/r,f ,(r, 0, z, t)=/,Q) expfi (m0 + kz - alil] (8)This leads to an eigenvalue problem where the wave
number k is the eigenvalue and the global wave fields

Fig. 1 Radial profile of the One-dimensional FRC equilib-

rium for rJr*=9.4, K=0.52.

i-|), ircO), iuQ), Er,e). Eree), E,.(r; are theeigenfunctions for a given frequency rrl and a boundary
condition at r = 0 (geometric axis) and r - r- (perfectlyconducting wall). The problem is solved in the following way. Each eigenfunction is approximated in terms ofa finite series of basis functions p,(r) and a function
which satisfies the boundary condition. For example, ii1,0.3

3 0.10!

,,,{rfo cln Q .{r)(e)Q,t)="or[{, -l)nrlr.](10)atataatttaa
a- a- )- r a t at t
It a ,aaa tt-I tt- atl--ta
ta aaaaaoal
ttaatxltaa
at
aoar
atott
ltatataaatl! ttaalaaar"tot
aa
ttto aatt,ot' ' - lF- - ./- : 1>': -r-r- rr !

? 0.2

is expressed asi i ,,(r) =r,,tr1 y'r'0246810
f**kFig. 2 Dispersion relation for m = 0 low frequency
waves.

where F,,(r) is the function satisfying the boundary con-dition for 1i1,, and Clu') are the expansion coefficients.
the ion cyclotron frequency in the external magnetic
field and the wave numbe60 s normalized to 1/r-. The
eigenvalues /c are plotted fot ala,is = 0.015-0.3 inincrements of 0.015. The broken line corresponds to 80[kHz] which is the frequency of the applied field in theexperiment. The solid line shows the Alfv6n velocity v,a6
based on B, and pnu11 which is the mass density at thefield-null point. We see from Fig. 2 that the eigenmodes
are dense for alk ) y,co and sparse for alk 3 v;r,.
Figures 3 and 4 show the radial structure of theperturbed electric field, mass flow, and magnetic field of

In deriving the equations for the expansion coefficients,
we use the Galerkin method.
One-dimensional FRC equilibrium model known as
the rigid-rotor profile [2], which is homogeneous in 0and z-direction, is used in this study. The equilibrium
magnetic field and pressure are expressed aslr Plttri - 9l

(11)p oe) = p nutfecli lx pl'ti - t;](r2)

Bo(r) = B,tanhthe eigenmodes along the broken line in Fig. 2

where r, and B, are the separatrix radius and the magnetic field at the wall, and K is a parameter. pn,n is thepressure at the magnetic field-null point. Figure 1 shows(80[kHz]). The solid gray, solid black and broken black

3. Resultslines show the r, 0, and z components, respectively. Fora wave with the phase velocity much larger than vle theamplitude appears only outside the separatrix as shownin Fig. 3a. Namely this wave propagates in z-direction

3.1 Results for m = 0
The eigenvalue problem was solved for thedecreases, the amplitudes move to the inner region (Fig.

the equilibrium profile for r,/r, = O.4, K = 0.52.

only outside the separatrix. As the phase velocity3b), and finally the amplitude appears inside theseparatrix for (a/k)/vm = 1.4 (Fig.3c). In all theseazimuthal mode number m = O in the FRC equilibrium
shown in Fig. 1. Figure 2 shows the dispersion relation
for the low frequency waves propagating in z-direction.
The frequency is normalized to cl.r6 : eBnlm;, which ismodes (Figs. 3a-c), the magnetic field is largest in the0-direction for r > r,!/r as observed in the experiment.
226

lwasawa N. et al., Eigenmode Analysis of Low Frequency Waves in a Field-Reversed Configuration (FRC)a) r*k = 0.13, (a I k) / v16= 12a) r*k - 2.0, (a / k) / vn = 0.76'|E1l(vs1B.) O-l
I
p9v1/(pnu1v4e) 0BrlB,,_100o

B,tB..

0.0 0.2 0.4 0.6 0.8 1.0b) r*k = 0.33, (a I k) / ve6= 4.7-z

0.0 0.2
0.0
0.2 0.4 0.6 0.8b) r*k = 3.9, (a / k) | veo = 0.391

1.0E1(v664.) oEr(vno8r) o.o-1-1.050pevl/(poo11v4s) 0-50

pgvl/(pnutrvA6) 00'4

1.0

1-1

BrlBr, o.o

BrlBw o.o-o.4-1.0

o.4 0.6 0.8.o= I.4c)

0.0 0.2 0.4 0.6 0.8 1.0c) r*k = 5.4, (a I k) / Yeo = 0.28

1.0

0.5.<'

E1(va6,B.) o.o-0.5
1povl/(pnulryAo) 0pgv1/(poor1va6) 0.0-0.4-1AR1.0BtlB. o.o

BtlBn, o.o-0.5-1.00.0l-----,-"" p""*ilo.2 0.4 0.6 0.8

0.0

1.0

rlr*

l-"-rp""*t--- e-componentl

0.2 0.4 0.6
rlr*

0.8

1.0Io-componentll--- z-component|-||

Iz-componenl-I

Fig. 3 Radial structure of the eigenmodes for m = 0.

Fig. 4 Radial structure of the eigenmodes for rn = 0.

These modes may produce the waves observed in theexperiment. While for waves with much smaller phasevelocities, this tendency is not presented. For (a/k)lvo6=
1.76 (Fig. 4a), the amplitudes appear between inside andthe wave heating experiment for finite m mode has justbeen done recently for the frequency. For (alk)lvoo= 12
the wave propagates almost O.7 < rlr. < I where theplasma density is very small (Fig. 6a). For (alk)lvo, =outside the separatrix for the electric field and mass
flow, but not for the magnetic field. The magnetic field
propagates only inside the magnetic field-null point r =r,us a;rd is largest in the z-direction. For more smallerphase velocities, the amplitudes of the magnetic fieldappear both inside and outside the field-null point, and0.37 the mode shown in Fig. 6b appears. The number ofnodes in the mass flow increases as the phase velocity
decreases as observed in the modes with m = 0 for(alk)/vN < 0.39.

4. Discussions
In the experiment [1], the magnetic fluctuation

the number of nodes in the mass flow increases as thephase velocity decreased (Figs. 4b and 4c).

measurement was performed only around the separatrix.

The fluctuation in the azimuthal direction, BtB, was

3.2 Results lor m = 1

larger than the other components. The phase velocity ofthe observed waves is alk - 1-2 x 105 [m/s] andFigure 5 shows the dispersion relation for m = l.
Unlike the nt = 0 case, there is no eigenmode for 0.4vo6< alk < 9va6. Figure 6 shows the radial structure of theperturbed fields of the eigenmodes along the broken line
in Fig. 5 (95 tkHzl). This frequency is chosen because

increases with r.

For the eigenmodes shown in Figs. 3a-c, themagnetic field is largest in the azimuthal direction. The
mode which propagates near the separatrix has smaller
227

Iwasawa N. et al., Eigenmode Analysis of Low Frequency Waves in a Field-Reversed Configuration (FRC)

0.3

For a typical FRC plasma, Ti - 27" and B - L Thuslaat---t--aaal
aaaa
ataal
aaaaa
oaaaolaaatattaaaacttara a a a a a t aoa{Ita t lltl
ttaaataao aaaaaat
lt I aaI a a'aaoatartt
Itttaaaatatao
ttifraal i o iaa atrlte 0.2
3 0.1

?1rt.

46the thermal velocity of the ions is v,1,; - "'12/3vns 0.82v1s. The mode in Fig. 4a, whose phase velocity issimilar to vn.i, can have a strong interaction with theions. In addition, this mode has significant amplitude of81, inside the magnetic field-null point r = /,,a. Thus
there is a possibility that this mode heats the ions by thetransit-time magnetic pumpin7 l3l. Averaged change inkinetic energy of a plasma ion by this mechanism in a.

f-xkFig. 5 Dispersion relation
waves.

a)

Ior m= 1 lowfrequency

unit time is [4](* +) = -" :l: I r,,. r(f)(H),

r*k- 0.I4, (atk) vm= 12

where ,tt. is the magnetic moment of the ions in theequilibrium magnetic field,/(vo) is the velocity distribution function. The total change in the ion's kinetic energy is estimated quite roughly under the assumptions:(l) Ions inside the magnetic field-null point are heatedfor At - 50 its (the wave was applied for about 50 gs in
the experiment); (2) The distribution function is theMaxwellian; (3) The amplitude of the perturbed magnetic field 81. is set to be equal to the amplitude of 816 0.008 [T] observed in the experiment, because in the experiment we could not measure the magnetic field far
inside the separatrix. The total change is'I

E1/(vasB.) o-l

10
pgvr/(pnunva6) 0BrlB.

-10100o-1 00

0.0 0.2 0.4 0.6

0.8

1.0

0.2 0.4 0.6 0.8

1.0b) r*k - 4.9, (a I k) | vx = 0.37
0.5

E1/(vasB,) o.o-u.c'I

p6v1/(poonva6)

BtlB.
0.0

tr--.".p"*il]e-componentll--- z-component|-"=,.

I-z\i(rr1,,,(,)t, ( 4 ^ :' ) - zs u'\dt2lr/r,

where l" - 3.6 [m] is the plasma length in z-direction.
This value is comparable to the experimental results (85Ul), although the estimation is quite rough because theIFig. 6 Radial structure of the eigenmodes for m = 1.

magnitude of ,B1. is uncertain. However, this resultshows that the transit-time magnetic pumping could
have a significant effect on the ion heating.

phase velocity than that propagates away from theseparatrix (The mode in Fig. 3b has smaller phasevelocity than the mode in Fig. 3a, and the mode in Fig.
3c has the smallest phase velocity). This trend isconsistent with the experimental result. The Alfv6n
velocity voo - 105[m,/s] for the experimentsl parameters
B. = 0.04 [T] and fl.o" - fi = 4 x lOre [m-3]. Thus thephase velocity of the mode in Fig. 3c is alk - 1.4 x 10s[m/s]. This value is roughly the same as theReferences

t1l K. Yamanaka et al.,Phys. Plasmas 7,2755 (2000).
[2] W.T. Armstrong et al., Phys. Fluids 24, 2068
( l98l ).

t3l T.H. Stix, The theory of plasma waves (McGrawHill).

[4] Kenro Miyamoto, Plasma Physics for Nuclear Fusion (MIT Press).

experimental result.

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