J. Plasma Fusion Res SERIES, Vol.2 (1999) 50-53

Interesting Differences between Axisymmetric and
Non-Axisym metric Systems
NAKAJIMA Noriyoshi
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 4 January 1999 / Accepted:21 May 1999)

Abstract
Differences between axisymmetric and non-axisymmetric toroidal systems are reviewed from thevarious viewpoints of MHD equilibrium, particle orbits, neoclassical transport, and ideal MHD spectrum
and instability.

Keywords:
axisymmetric and non-axisymmetric toroidal system, MHD equilibrium, ideal MHD spectrum andinstability, neoclassical transportlarge size of magnetic islands and/or splitting of the

1. Introduction
Differences between axisymmetric and nonaxisymmetric toroidal systems come from whether

magnetic axis occur as pvalue increases, especially forlow magnetic shear systems. Although existence ofmagnetic islands by themselves brings about profoundsymmetry in the toroidal direction exists or not. In thatexistence of the ignorable coordinate reduc.rs three-axisymmetric toroidal systems are regarded asphysically degenerate states of non-axisymmetrictoroidal systems. Thus, phenomena inherent to non-accademic and practical problems, we could still findfruitful interesting differences between axisymmetricand non-axisymmetric toroidal equilibria under theassumption of clear nested flux surfaces. Under thisassumption, differences appears as the situation whether

axisymmetric toroidal systems exist together with many
common features to axisymmetric toroidal systems. In

equilibrium quantities depend on the toroidal angle ornot, when we use an appropriate magnetic coordinate

this paper, we will concentrate our attention on the
differences between both systems, or phenomena

system such as the Boozer coordinates. In other word,equilibrium quantities depend on not only poloidal but
also the toroidal Fourier modes in non-axsymmetrictoroidal systems. Hereafter, we consider L = 2 planar

dimensional problems into two-dimensional ones,

inherent to non-axisymmetric toroidal systems.

axis heliotron configurations with good flux surfaces
like LHD, where Z is the polarity of helical coils.

2. MHD Equilibrium and Particle Orbits
In axisymmetric toroidal systems, MHD equilibria
consist of clear nested flux surfaces and collisionlesssingle particle orbits have the invariant of motion. In
non-axisymmetric MHD equilibria, lack of the toroidal
symmetry leads to breaking up the flux surfaces into
magnetic islands, and to stochastic motion of particle
orbits. Thus, a kind of optimization of the nonaxisymmetric MHD equilibrium as was done in LHD
and W7-X is needed in order to avoid them as much as
possible. If such an optimization is not enough, then

3. Neoclassical Transport
The neoclassical transport is determined by particle
orbits under the Coulomb collision, which is expressedin terms of so called flux-thermodynamic force
relationship. Since the lowest order distribution function@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

50Nakajima N., Interesting Differences between Axisymmetric and Non-Axisymmetric systems

is the local Maxwellian, two kinds of theflow, but also toroidal flow, and the direction ofthermodynamic force appear for each particle species (ifno extemal field exists):

damping depends on the collisionality regime of eachparticle species. Thus, both poloidal and toroidal flows
have a component proportional to the radial electricfield, and the bootstrap current has a component directly
proportional to the radial electric field, when electrons
and ions exist in the different collisionality regimes.
Consequently, the determinable radial electric fielddP"(v) da(v)r,nJw W * drl, 'I dQ(tlt\
4"2(Vt\=l-,ea dv
A ,,,.\
t'at\Y)-

8.^, l rtt I|

where the standard notation is used, and @ and lrl are theelectrostatic potential and the label of the flux surface,

directly influences MHD equilibria through therespectively. The lowest order mean flows and heat
fluxes are determined by the flux-surface averaged

bootstrap current in the non-axisymmetric toroidal
systems. This influence becomes significant whenperpendicular and parallel force balances for eachparticle species. Both are incompressible ones on flux
surfaces. It are these lowest order quantities thatelectrons and ions exist in the llv and plateau
collisionality regimes, respectively. In this situation, theradial electric field determined by the ambipolar

determine the same order bootstrap current and the nextorder radial particle and heat fluxes. Differences ofboth
systems come from l) existence or non-existence of thecondition reduces the bootstrap current [1,2].

As is well known, the diamagnetic current andPfirsch-Schliiter current are independent of the radial

toroidal viscosity making a non-ambipolar radial flux,

electric field in both systems, because that the
diamagnetic current is created by the local gyration
motion around a magnetic field line, and that pfirsch_
Schliiter current flows so as to satisfy incompressibilityof the current. In contrast, bootstrap current comes fromthe global drift motion of the guiding center, which canand 2) the form of the parallel viscosity. In
axisymmetric toroidal systems with symmetry in thetoroidal direction, the toroidal viscosity disappears, sothat all the radial fluxes become intrinsically ambipolar.

It means that the ambipolar condition does notdetermine the radial electric field d@/dry. Toroidal

feel the global structure of the magnetic fieldsymmetry ensures the conservation of the lowest orderconfiguration, namely, symmetry and asymmetry.

total toroidal angular momentum, and hence only thepoloidal flow (flow in the direction without symmery)

4. ldeal MHD Spectrum and Instability
4.1 Shear Alfv6n continuum and spectral gaps
In one-dimensional systems such as straighttokamaks with circular cross section, equilibrium

is selectively damped by the parallel viscosity,
independent of the collisionality regime of each particle

species. The resultant poloidal flow has no component
due to the radial electric field, and only the toroidal flow

quantities describing shear Alfv6n branch are surface
quantities, which make shear Alfvdn continuum in thehas such a component. The fact that an undeterminedradial electric field appear only on the toroidal flow is
consistent to the conservation of the total toroidal

stable side of the ideal MHD spectrum. In axisymmetric(two-dimensional) toroidal systems, in whichangular momentum, because such a conserved quantity

equilibrium quantities have poloidal angle dependence,
this shear Alfv6n continuum has spectral gaps due topoloidal mode couplings between equilibrium quantities
and perturbations. In non-axisymmetric (threedimensional) toroidal systems, equilibrium quantities
have both poloidal and toroidal angle dependences, andshould be determined from an appropreate initialcondition, and hence an appropreate unknown quantity
must be included there. Moreover, the bootstrap current
does not include the component directly proportional to

the radial electric field, inspite of the form of thethermodynamic force A.r(V). Therefore, undetermined

hence the spectral gaps in the shear Alfvdn continuum

radial electric field does nor influence the MHD

are created by both poloidal and toroidal mode
couplings. The behaviour of the shear Alfv6n continuum

equilibria including the bootstrap current.
In non-axisymmetric toroidal systems, lack of thetoroidal symmetry leads to a non-ambipolar diffusion

is shown in Figs.l-(a) and (b), where an MHD
equilibrium in an L = 2/M = 8 planar axis heliotron
configuration is used (L and M are polarity and toroidal
field period of helical coils, respectively). In Fig.l-(a),due to the toroidal viscosity, which is inversely
proportional to the Coulomb collision frequency (l/vripple diffusion). Thus, ambipolar condition determines

the shear Alfv6n spectrum with both poloidal andtoroidal mode couplings is shown where Fourier modes
are used satisfying n * nr - nM, n1 = 2, n = 0, +1, +2.

the radial electric field. Due to lack of toroidal
symmetry, the parallel viscosity damps not only poloidal

5tNakajima N., Interesting Differences between Axisymmetric and Non-Axisymmetric Systems

(a)

(b)

Fig. 1 Shear Altu6n spectrum (a) with poloidal and toroidal mode couplings, and (b) without mode couplinglines) and with only poloidal mode coupling (stars).

(dsahed..., namely, n = "', -10, -6, -2,2, 6, 10, " '. From thesefigures, it is understood that the coexistence of poloidal

become unstable with different eigenvalues. Thus, aswell as one dimensional systems, interchange modes inand toroidal mode couplings dramatically changes the
spectrum gap structure in the shear Alfv6n continuum.
Toroidicity-induced shear Alfv6n Eigenmodes (TAE)axisymmetric equilibria will have countable infinite

number of eigenvalues. In axisymmetric (two-are considered to change compared with those ininterchange mode localizes at the different rational

dimensional) equilibria, each Fourier mode constructing
surface, and hence essential structure are same as one-axisymmetric toroidal systems.

dimensional one. In non-axisymmetric (threedimensional) equilibria, the situation is different,
because two modes with different combination of
Fourier modes, namely, (m, n) and (m', n') may have the

4.2 Gontinuous unstable spectrum of pressuredriven modes
The coexistence of poloidal and toroidal mode
couplings might lead to the possibility to make asame mode rational surface due to the toroidal mode
coupling, namely, = nlm = n'lm', whete n' satisfies n'|* n = iM for a particularly chosen integer i . The

continuous or quasi-point unstable spectrum.
Interchange modes have the properties to localize
around their mode rational surfaces. In one-dimensional
equilibria, both poloidal lzl and toroidal n mode numbers
are specified. For a particular selection (n, n), countablesuperposition of Fourier modes with the same rationalsurface could make a continuous variation of theinfinite number of eigenvalues cair exist, which comefrom countable infinite number of radial mode (node)continuous unstable spectrum, or quasi-point unstableeigenfunction and the eigenvalue, leading to aspectrum.

In the case of ballooning modes, the situation is

numbers. In axisymmetric (two-dimensional) equilibria'
only a toroidal mode number is specified, and hence anoverall structure of the interchange mode consists of thesuperposition of each Fourier mode localized around themode rational surface, namely, | = "'; nl(m - l)' n/m,
nl(m + l), ..., where r is the rotational transform. Notethat since the toroidal mode number is same, each
Fourier mode localizes at the different rational surface.

different from that of interchange modes. Ballooning
modes are driven by the locally unfavorable magnetic
curvature, and hence they have a tendency to belocalized around there. On the other hand, thesuperposition of Fourier modes with different toroidal
mode numbers, by changing their relative phase, leads
to the change of the localization in the toroidal direction.

Therefore, the change of the superposition of Fouriermodes with different toroidal mode number is easy tolead to stabilization, and hence ballooning modes might
not have a continuous unstable spectrum or a quasipoint unstable spectrum, execpt perhaps for the casewith extremety high toroidal mode numbers. Aboveconsideration is different from that in Ref. [3].

Since the interchange modes are driven by the averaged
unfavarable magnetic curvature, those amplitude arealmost constant along their mode rational magnetic fieldlines, and each Fourier mode is basically independent.
Therefore, the interchange mode constructed by thesuperposition of Fourier modes with different radialmode numbers and/or different relative phase can also52

Nakajima N., Interesting Differences between Axisymmetric and Non-Axisymmetric Systems

(a)

rt

(b) z

838

\
/
/

\
o.7

(cl z

r o't

o.2

\

1: \

t,

-__nI

--/?

-0.2t.0

1.0

Fig. 2 Ballooning modes inherent to non-axisymmetric systems. {a) radial distribution of Fourier modes of i.Vrlwith
their dominant toroidal mode numbers, and corresponding contours of the perturbaed pressure P = -V p .{ on thevertically {b) and horizontally {c) elongated poloidal cross section.

4.3 Pressure-driven (ballooning) instabilities
In the limit of a short perpendicular wave length(the local mode analysis), ballooning modes areanalyzed, by using the ballooning equation in thelevel surface and a topologically spheroidal levelsurface. In Mercier-unstable MHD equilibria, bothcylindrical and spheroidal level surfaces coexist, and in
Mercier-stable MHD equilibria, only spheroidal levelsurfaces exist [5]. Since the local mode analysis does

covering space (rg, 4, a), where ryand a are the labelsof a flux surface and a magnetic field line on it,

not lead to the global mode analysis due to arespectively, and 4 is a coordinate along the magnetic

dependence of the eigenvalue, it is conjectured that the
global modes corresponding to the eigenvalues withcylindrical level surfaces are tokamak-like ballooning

field line.
In axisymmetric toroidal systems, the eigenfunction( and the eigenvalue a2 have functional dependence: (modes or interchange modes, and that the global modes
corresponding to the eigenvalues with spheroidal level= 4@lV, 0*), a2 - @2(V,01), where 0s is the radialsurfaces are ballooning modes inherent to non-wave number stemming from the eikonal approximation.
Thus, level surfaces ofunstable eigenvalues af 1< 0; areaxisymmetric toroidal systems with localized in both thepoloidal and toroidal directions [5]. Recently, the above

topologically cylindrical with the axis in the a directionin (ty, 06 d) space. For a specified toroidal mode

conjecture has been confirmed by the global mode
analysis. One example of the ballooning modes inherent
to non-axisymmetric toroidal systems is shown inFig.2.
There are eight groups of Fourier modes for 6 .VVwith different toroidal mode numbers through the strong
toroidal mode coupling in Fig.2-(a). Each group has anumber n, semiclassical quantization condition
determines the eigenvalue of the global mode: o2 =a2(n), and the ballooning representation can determine
the global mode structure ( = (1i, n). Thus, the local
mode analysis leads to the global mode analysis.

In non-axisymmetric toroidal systems, the local

similar structure to that of ballooning modes inmagnetic shear having a stabilizing effects disappears
due to a large Shafranov shift even in the region withtokamaks by the poloidal mode coupling. Due to these
poloidal nad toroidal mode couplings, the perturbedpressure F = -Y P' i localizes in both the poloidal andtoroidal directions, leading to the localization on some
flux tubes as shown in Figs.2-(b) and (c).

the stellarator-like global magnetic shear, leading to thedestabilization of ballooning modes there [4]. Becauseof the toroidal angle dependence of the MHD equilibria
coming from helical coils, the magnetic curvature
consists of two parts due to both toroidicity and helicity

References[5]. The eigenfunction and the eigenvalue havefunctional dependence: ( = €(qlV, 0p a), a2 = a20{,

tll N. Nakajima and M. Okamoto, J. Phys. Soc. Jpn

06 a). The field line dependence a is due to the toroidal
symmetry breaking, and is mainly caused by the helicity
part of the local magnetic curvature [5]. Due to the field
line dependence of the eigenvalue, there are two typesof level surfaces of the unstable eigenvalue in (ry, 00, a)
space, namely, a tokamak-like topologically cylindrical

t2l K.Y. Watanabe, N. Nakajima, er.a/., Nucl. Fusion

61,833 (1992).
32, 1499 (1992\.

[3] R.L. Dewar and A.H. Glasser, Phys. Fluids 26,
3038 fl983).
t4l N. Nakajima, Phys. Plasmas 3,4545 (1996).
t5l N.Nakajima, Phys. Plasmas 3,4556 (1996).
53