J. Plasma Fusion Res. SERIES, Vol.2 (1999) 109-ll2

Vlasov Continuum Contribution to Anomalous
Plasma Transport
YAMAGISHI Tomejiro
Tokyo Metropolitan Institute of Technology, Hino, 191-0065, Japan(Received: 8 December 1998 / Accepted: 15 June 1999)

Abstract
The propagator for the Vlasov-Poisson system in inhomo-geneous plasma is applied for the spectral
decomposition of the perturbed distribution function, and the anomalous cross field flux which isdecomposed into three portions, the discrete mode determined by the dispersion relation, the ballisticmode, and the pure continuum contribution. The flux due to the discrete mode (instability) is ambipolar,
while the fluxes due to the continuum, in general, are not ambipolar.

Keywords:
propagator, spectral decomposition, anomalous transport

I is defined by
L=v'Y*ar'm- -l
dv

1. lntroduction

where the Louville operator

Particle transport process may consist of multiple
scattering with medium or fields, and free streaming [].
The former diffusion process may correspond to thediscrete mode determined from the dispersion relation[2], while the free streaming may correspond to thecontribution from the continuous eigenvalue of thetransport equation [3]. The importance of the spectral
decomposition of the transport process is that theanomalous transport induced by the discrete mode(instability) is ambipolar, while that induced by thecontinuum is not, i.e., the latter may be responsible to
the generation of the radial electric field. In this paper, aprevious attempt to apply the spectral decomposition toanomalous plasma transport flux [4] is reformulated
making use of the propagator for the Vlasov-Poisson
system in an inhomogeneous toroidal plasma.

Q)

The Lorenz force F is given byF =eE+lvxB(3)

The solution to Eq. (l) is formally given byf(r,v,t)=U,f(r,v,O)(4)

where the solution (semi-group) operator U is definedbYrr,\u,=exp[-J,otl6)

This operator can be expanded asu,=D*{ 1a,(" v *L*o*)) (6)2. Propagator for Vlasov-Poisson System
First we consider the Vlasov equation{*tf=,dt

which means the Taylor expansion operator with respect

to the spatial and velocity coordinates, (r, r) of the(l)@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author's e-mail: tyamagi@ cc.tmit.acjp
109

Yamagishi T., Vlasov Continuum Contribution to Anomalous Plasma Transport

initial distribution function j1r, ,,0). In the linearand tire dielectric function is given by

theory, the solution can be expressed by(r"\t/_. .\"t.. I d,r.r_ |I or mJ1r,v,r)=Jlr_ll".olI\"uJ€(k, o)= t +I )(,(k, a)a)(14)

where the susceptibility 7 is defined by

I dtve,a,(k.v)

The operator U is a shift operator to the initialdistribution. Denoting the spatial and velocity shifts
respectively by Ar and Av, expanding to the second
order, ensemble averaging, and making time derivative,

except the real axis and the pole ros given by thewe have a transport equation

dispersion relation

X,(k. a)=(15)The propagator is analytic in the complex c>plane€(k, ai=0# =(*,,*- r1or* . *o.r*) i ?, v, t), (8)(16)

i.e., it is discontinuous across the real axis, and haspoles at a = k-v and (o = 46. The pole at al = ft.y on thereal axis gives the ballistic mode. The discontinuity ofthe propagator is due to the velocity integration of thewhere D,,, D,, and D,, are diffusion coefficients, forexample D,, = I ( r)2 >/2t. The particle displacementsin space and velocity may be due to random scatterings
with waves, and/or free streaming. We here consider
how these processes may decompose the transport

ballistic mode. From the view point of the eigenvalue
problem of the Vlasov equation, the propagator isconvenient for the spectral decomposition of thequantities.

For the sake of simplicity, we assume theperturbed quantities i and fi in tne inverse Laplacetransformation. The pole ro - (o6 corresponds to thediscrete eigenvalue, while the pole or = t.y for all v

electrostatic approximation, and expand all perturbed
quantities in Fourier series of the formf (r, v, t1=\ 71r,v, /) exp (ik.r) .

J -*:^

corresponds to the continuous eigenvalue C.

(e)Operating the propagator (12) to the initial
distribution function j1r, v,0), we have the perturbed
transformed distribution function/(r, v, ar). The inverse

If we make Laplace transform to Eq. (1) with respect totime t:

Laplace transformation for f 1r, v, ar) by(k,v, ay= | dt exp 6aof &.v. t),
f-Jo

-|

f@+to

f (k.v. r)=:L
2t = J|

the solution can formally be written asi@, r, @) = (i@ - D-' i &, v, o).

-+ot

daexp (-iatt7j1k.

v. arr.

07)

yields the decomposed solution as shown in the next

In this case, the Poisson equation is written as

section.

r^ s4/te,
E=-Lf,a'r.
-'i*lrz
l'r(10)

If we take into account the higher order effect tothe Vlasov equation (1), "collision term", which isThe Louville operator can be expressed in the form [5]described in terms of the two point correlation functiong;, has to be included. The equation for the two point
correlation function can also be solved by applying the^Li(k,v)=ik' v+ 2 ,"U ],#,0*>",Ja',

propagator (12) t5l. In the inverse Laplacetransformation of gu, it is also found that the two point(1 1)

The Laplace transformed propagator can be written as

correlation function can be decomposed into the discretemode induced by the dispersion relation and thecontinuum contribution which is further decomposed
into the ballistic mode and pure continuum contribution.
In the time asymptotic limit, the ballistic mode
contribution to g; gives the collision term of theI

io-Li(at-k.v)
a;(k v) \5
f d'u . (12)ll ---r-l_t(/c,v) lL,eil\
J 'J i(o-k.v\|.

Balescu-Lenard equation [6]. That is the ballisticcontinuum contribution determines the equilibrium

where 0 is defined by

4tte, afat(k,v)=- --+ ik<r ,fr,k' ov.

(13)

distribution <"f> in the nonlinear asymptotic limit,
although it may damp in the linear theory [7]. Since thell0

Yamagishi T., Vlasov Continuum Contribution to Anomalous Plasma Transportcollision term gives the anomalous transport coefficientin velocity space through the Fokker-Planck equation,
the transport coefficient is also decomposed into thecan be expressed in the form:

i &, r, t) = foft , v) exp ( -iaro (t)r)+

discrete mode and continuum contributions.

+f .(k, v, t),

3. Cross Field Plasma Flux in NonuniformSystem(\O,\Q4)- a,(k, v)d've,"" e(k.v)Lt J|f a-a,(k.v\ltl,16,,--TI\ia;(k, v) 5

fa(k,v)=-L(.\(18)Ik'?^t r,(k,r\-J;(o, - arrx; I)/",tv; (19)k'\\r-

di=" jf"(k, v,/)='v is a nonlinear damping rate [7], @D = 2€,@*Q2rl2+ v2r1, a* = cTkyleBLn and are the curvature drift anddiamagnetic drift frequencies, respectively. The

where the species index has been omitted. The factor ll2in Eq. (25) is due to the half circle around the polea,(k, v), because e(k, a) and o(ft, ar) are discontinuous)(i(k,ot)=TI O',r2 |KJ)

-lI orr-''' 4kt
2nJc

I, - t'," ao,,'o.,a"(k.
?,iv),l+'-q'9e €(k. o)', e6)
\Jn|

I

t-\

susceptibility becomeso-a: \k'Qs)t.

t

f:'-',0,'i,o. r.o)- + tlw,{k.vr-ari) e s(k, ?,''(D,) \l"\/where a,(k, v) = k,v, + ap(v) - iv, the coefficient a, is

1.2

fi ao_'o-a,(k,D)?,] I {"',tX''"e e,(k, oi,f^(k, v)=5[' - ",''ru\'-.

gyrokinetic equation, we haveJ\,.2(distribution function, and introducing into theroQ3)

where the first term is the discrete mode (polecontribution) determined from the dispersion relation,
the second term is the ballistic mode given by the poleo = a,(k, v) and the third term is pure continuum
contribution given respectively:

We now develop the propagator to a non-uniform
Maxwellian plasma in a toroidal system. If we assume
the equilibrium distribution in the usual form/6(x, v) =N(x - vrlQ)fy(v) withfia(v) being the usual Maxwelliania - L(k, v)

f a(k, v) exp ( -ia;" ( k, v)t)

across C, and we can not close the circle in the residue

lt -J;------+" o- a'(k'v) l,f',(u)' Qo)

evaluation.
By the inverse Laplace transformation of Eq. (21),
we have the decomposed scalar potential:

where kp is the Debye wave number. The scalarpotential @ is expressed by

Q(k, t)= 0o(&) exp (-iruoot)s(k' a)o(k'tol=+n' k2 €(k, a)+ h(k, t)+ Q"(k, t),

Qr)

Q7)

wheres(k, ai

where s is defined bya3vi1*,v.01
ss(k,ot)-Le, Sli J O- A,W,v)

Qo&).
=_4np2 e'1k, raol'

12g)

Q2)
6"Uc, t)=-The transformed solution i &, v, a) is obtained byoperating the propagator (18) to an initial distributionf]r, u,0). Since the propagator (18) has the poles
determined from the dispersion relation (16) and o)= @,-'42
I a', -JJltl'9€(k,a,(k.v))k'i ",'J
exp (-iar" (k,v)t),Qe)s"(k' o,)Q,&,t)=-4 rc| o,exp ( -irot) ;\rk

embedded in the continuum C which is the set on thereal axis a: au(k, v) for all v, it is convenient for thespectral decomposition of the solution. By the inverse
Laplace transformation (17), the time dependent solution'(30)As compared with the previous study [4], the ballisticmode are separated from the continuum contributionsntYamagishi T., Vlasov Continuum Contribution to Anomalous Plasma Transport

5 ckrl l rQ,l,ta=-4k eElr- l;lfor/and @, because the ballistic contribution plays anessential role in the Balescu-Lenard collision term [6].
The ballistic mode (29) has been derived by changing
the order of integrations with respect to ar and v in theI)The continuum contribution I can be derived in thefunction s.
Let us now evaluate the cross field plasma flux:r=[ d'u (iU,,,tli1r,tl)

similar way [4]. Since the continuum contributions neednot satisfy the dispersion relation, I-s and Il for ions andelectron are, in general, different, i.e., the anomaloustransports due to the continuum are not ambipolar.

(31)

.

We assume the radial velocity fluctuation is given by
the E x B drift velocity i = -(c/ B)00/ Ay. If we substitutethe Fourier representations for/and 0 into Eq. (31), themixed mode terrns may vanish due to the phase mixing.
and the pairs of the same mode terms may survive, i.e.,
the flux is also decompose into three terms

4. Summary
The propagator for the Vlasov-Poisson system has
been derived for an inhomogeneous plasma in a toroidal

system, which has been applied for the spectral
decomposition of the perturbed distribution and scalarpotential. The distribution function has been
decomposed into the three portions; the discrete mode

corresponding to the spectral decomposition:

f=Ts+fo+1-".

Q2)

The discrete mode term fo is written as

determined from the dispersion relation, the ballisticmode, and the pure continuum contribution which
represents the interaction between the ballistic mode andplasma cloud (clump) trapped by the shielded potential[9]. The flux due to the discrete mode is basicallyambipolar. This does not hold for the flux due to therro=ic ,v,S krk2v2",,,4ffirw^k,ql\"l', (33)where cr" is a characterirti. f."qu"n.y, @. is the shielded
potential of the form of Eq. (21). The flux I-s grows intime in unstable plasmas [5]. It may tends to a steadycontinuum contribution, which may cause the generationof the radial electric field.

state when the growth rate 7 is balanced with certain
nonlinear damping rate DP.I-6 is the same for ions andelectrons, because lmb, = . Imx" from the dispersion
relation (16), i.e., the anomalous transport due to thediscrete mode (instability) is basically ambipolar. If thesaturation level of the normalized potential isleQ"lTl2 =Acknowledgment
This study is a joint research effort with theNational Institute for Fusion Science.

Q,D'2 I8l, the flux can be approximated by fs = -yl
k2N', i.e., the spatial diffusion coefficient D = ylkz isReferences[] T. Yamagishi and Sekiya, J. Nucl. Sci, & Technol.
s, 614 (1968).
t2l K.M. Case and P.F. Zweifel , Linear Transport

recovered.

For the flux due to the ballistic mode, if we takethe first term in Eq. (25), which is the direct beam mode

Theory (Addison-Wesely, New York, 1967).

[3] T. Yamagishi, J. Physics D 7, 510 (1974).
t4l T. Yamagishi, Proc. Int. Conf. Plasma Physics,

from source, combined with Eq. (29), we have-t,o'*tf","'= Ir e,B krJI' o', Jl' d3v' 6(a,,- r0,,,)s, ,, i,(k,r,o\r, i,Gk.r"o)=ttl' | €(k' a'(k' vDl- a'Ime(/c, rtt,(k,v)).

a'v' l'ola, - ro\)f un6(a" - o,,).

x(1996) Vol.l, p.338 (1996).

[5] P.H. Rutherford and Frieman, Phys. Fluids 6, ll39
( 1963).

t6l T. Yamagishi, Memoir of Tokyo Metro. Inst.
Technol. 12, lI7 (1998).
[7] J.K. Kim, Y. Kishimoto, W. Horton and T. Tajima,Q4)

For a single species case, expressing the shielded

Phys. Plasma 1,927 (1994).
T.H.
Dupree, Phys. Fluids, 15,334 (1972).
[8]T.
Yamagishi,
J. Phys. Soc. Jpn.68, 1526 (1999).
[9]

potential byand separate Ime(k, a1) from the v-integral, Eq. (34)
may approximately be written byn2