J. Plasma Fusion Res. SERIES, Vol.2 (1999) 12-14

Parallel Plasma Transport at Long Mean-Free Path
HAZELTINE Richard D.
Institute for Fusion Studies, The University of Texas at Austin Austin, Texas 78712, USA(Received: 8 December / Accepted: 13 May 1999)

Abstract
Transport parallel to the magnetic field of a toroidal plasma confinement system is investigatedthrough kinetic theory, with emphasis on the long mean-free path limit. It is noted that a full transport
matrix does not exist in the collisionless limit. A collisionless transport law, involving a non-local
operator that accounts for toroidal topology, is derived for parallel heat conduction on ergodic magnetic
surfaces. In the rational surface case, perpendicular diffusion must be included in the kinetic equation toavoid singularity; this allows a calculation of the width and amplitude of resonant temperature
perturbations that will be excited by heat sources with sufficiently broad Fourier spectra.

Keywords:
plasma trasnport, long mean free pathgenerally driven by particle and energy sources; thedensity and temperature profiles mediate this driving
mechanism only when collisions dominate.
When the same problem is considered in toroidal
topology, with magnetic shear, the distinction between
rational and ergodic flux surfaces becomes crucial. On

1. lntroduction
Collisionless transport parallel to the magnetic fieldof a toroidal confinement system deserves study forseveral reasons. First, the transit frequency in many such
systems far exceeds the collision frequency, so that thelong mean-free path regime has practical importance,

an ergodic flux surface we derive an expression for theespecially outside the separatrix, where the assumptionof full parallel equilibration is not adequate to describe
relevant phenomena. Second, the topic bears on generalissues of fluid closure: under what circumstances can aheat flow closely analogous to that of the slab, with amodified kernel to account for toroidal topology. But
rational surfaces require separate treatment, because of
the limited ability of closed field lines to distribute heat.
Including radial diffusion to resolve the rational surface

nearly collisionless plasma be described by fluid (orfluid-like) equations? Finally the subject has intrinsic
interest, especially in comparison to more conventional(collisional) relaxation mechanisms.
We begin by reviewing the well-known [1] slabmodel description of collisionless parallel heat transport.
In this regard we note that the full transport matrix issingular in the collisionless limit. Indeed, the Chapman-resonance, we compute the temperature perturbation
corresponding to a localized heat pulse and compare its
amplitude to that of the ergodic case.

2. Slab Geometry
Perpendicular transport coefficients often change

Enskog model for fluid closure, even after it isgeneralized to allow for non-local effects, is nottheir functional form as the mean-free path (r,)
increases, but parallel transport changes more

appropriate outside the short mean-free path regime forwhich it was constructed. The point is that transport isfundamentally, losing its spatially local character when

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

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Hazeltine R.D., Parallel Plasma Transport at Long Mean-Free pathi. becomes comparable to the gradient scale length. Thus
the long mean-free path heat flux ft is no longer

entropy increase, even in the absence of collisions. Thus
entropy increases-information is lost-as heat spreads

proportional to.the local temperature gradient, but rather

out from a local energy source. In the absence ofresponds to the global temperature profile:physical boundaries, this spreading is irreversible.

,We denote the particle source in phase space by
Ss(x, v) and the heat source by S2(.r, v). Then theconventional rate of entropy production, denoted by(1)where n6 is the lowest order density, Z the temperatureand v, the thermal speed. This result, due to Hammett@=-[a,fic<r;,and Perkins [], suggests that transport coefficients
pertinent at short mean-free path become non-local
transport operators at long mean-free path.
Unfortunately, while reproducing (l) we find that nonlocality is only part of the story. Indeed the full transport
matrix becomes singular as i, -+ €.
We solve a one-dimensional kinetic equation,

where / is the perturbed distribution function, fia aMaxwellian distribution and C the collision operaror,
can be computed exactly for arbitrary collisionality. It
includes an entropy flow term, expressed as adivergence (or x-derivative in the one-dimensionalcase). We annhilate the flow term by performing a

representing collisions by a particle and energy
conserving Krook model. The transit and collisions
frequencies are assumed comparable-there is nocollisionality ordering-but the sources, necessary forthe presumed steady state, are taken to be weak. The

spatial average,(o)= and find that[_0,t,ot\.
@)=ls^4a+s\/'zrol

resulting solution is used to compute the2x 2 transport
matrix at arbitrary mean-free path. In the collisional
limit, this matrix is diagonal but otherwise agrees with\"'o(4)The kinetic equation shows that this quantity isconventional short mean-free path results. When thecollision frequency is allowed to vanish, the known
result of (1) is reproduced, but the corresponding formof the particle flux, I is surprising:r=-L vfino(vrcs, -#)(3)equivalent to the conventional entropy production rate

(@)=

(5)

This result is valid for arbitrary collisionality. In thecollsionless limit, the term involving f vanishes, as wehave remarked. But the heat-flow term remains finite
and can be seen, from (l), to make a positive-definite

Q)

This is Fick's law. with classical diffusion coefficient
D" = 0.5 v1z = gl2) v?tv (v is the collisional frequency)

and conventional driving force, A1 : Vlogp - eYaDlT.
What is surprising is that Fick's law represents the exact

contribution.

4. Toroidal Geometry
To treat toroidal geometry we introduce thesolution to the kinetic equation for any v > 0. In thecollisionless limit, D" -9 -, whence A1 -+ 0, as indeed
the kinetic analysis predicts: Vlogp -+ eYQIT as y -+0, corresponding to the Maxwell-Botzmann response.
Note that neither the flow, il nor the force 41, shows
singularity at v= 0.
Thus the transport matrix, even as a matrix of nonlocal operators, becomes singular in the collisionlesslimit. While it is always possible to relate the flows ofparticles and heat to corresponding sources, a nonsingular relation between flows and profiles is possible
only at short mean-free path.

poloidal and toroidal angles. 0 and (. use r for the radial
coordinate and denote that field line pitch or safety
factor by q. We also suppose a quasi-static state inwhich the net input of particles and energy to any flux
surface vanishes. Then the steady state response to
localized sources and sink on an ergodic surface can becomputed analogously to the slab case, with theresulting heat flux

hO- 6=-A/ .,"o n'zttro Tn

ft e'-e\A de'LT@',(+ Iql(o'- o))cotl ^ |r\213. Entropy Production
The heat flow described above is associated witht3(6)Hazeltine R.D., Parallel Plasma Transport at Long Mean-Free Path

where [4] is the largest integer contained in 4. This isthe version of (l) appropriate to toroidal topology; it hasexactiy; it is then used to compute the temperature
perturbation on a surface with a specified source, withsome interesting features, but does not address the mostthe result

important aspect of parallel heat transport in a torus:AT

resonance.

TOn rational flux surfaces r - r0, where e = molno
for integers ms and n6, the kinetic equation for steadystate parallel dynamics is singular for a general source

S.ono= a,ro"r{(r\)

where .l.ono is the Fourier component of the source, a)t is
the transit frequency and ar = (Drslnatq'(rs))l/3 is thewidth of the resonant layer. For comparable source-

configuration on the surface. The problem is that aclosed field line that makes contact with a localizedenergy source, say, may not connect to the balancing

strengths, this perturbation exceeds that on an ergodic

each closed line, rather than merely on the surface as asurface by about 2 orders of magnitude. Thus, if atokamak is penetrated by a narrow heating or coolingspike (as from an ECH pulse or pellet), (low-order)whole, is the well-known Newcomb condition.) To

rational surfaces are preferentially affected.

sink. (The requirement that sources and sinks balance on

avoid the resulting singularity we must include radial
diffusion, with diffusion coefficient D, as a perturbation
to parallel flow. The resulting 2-dimensional kinetic
equation can be reduced to an Airy equation and solved

References[]

t4G.W. Hammett and F.W. Perkins, Phys. Rev. Lett.
64,3Or9 (1990).