J. Plasma Fusion Res. SERIES, Yol.2 (1999) 501-504

On the Networking of Long-Living Filaments of Electric
Current in Magnetically Gonfined Toroidal Plasmas
KUKUSHKIN Aleksandr and RANTSEV-KARTINOV Valentin
INF RRC "Kurchatov Institute" Moscow. 123182. Russia(Received: 8 December 1998 / Accepted: 23 February 1999)

Abstract
The probable principles of the electric current networking in magnetically confined toroidal (MCT)
plasmas recently formulated are applied to suggest a novel qualitative view on MHD equilibria inmagnetically confined plasmas.

Keywords:electric current filament, networking of filaments, magnetohydrodynamic equilibrium

1. lntroduction

proceedings).

Recently the probable principles of the electric
current networking in magnetically confined toroidal(MCT) plasmas have been formulated [,2] which
inferentially extend recently formulated concept of theformation of the non-fluctuative (i.e. unexpectedly long-An application [,2] of the method of the multilevel
dynamical contrasting [3] to analyzing available datafrom tokamak plasmas revealed the signs of networking
in the visible light and soft X-ray images, and magneticprobing data. Also, implicit signs of networking haveliving) filaments of electric current, and their
networking, in Z-pinch plasmas [3] (and otherbeen found [1,2] in the Thomson scattering andmagnetoinertially-driven plasmas) to the case of MCT
plasmas (tokamaks, reversed field pinches etc.). Thisextension suggests a necessity to append conventional
picture of the non-filamentary plasma (which is nearly afluid described by the conventional MHD) with a"network" component which is formed by the strongest
long-living filaments of electric current and penetrate

similarity of tubular structuring of magnetoinertially-

The approach [1,2] opens new opportunities fordeveloping the model of the nonlocal (non-diffusive)
heat transport in plasmas by the electromagnetic carriersof the long mean free path - of the system's size andlarger. In particular, observations of fast nonlocal
responses to a localized perturbation and internaltransport barriers (ITBs) in tokamaks exhibit qualitative
correlation with the networking-based approach [1].
The present paper is focused at application ofprinciples [,2] to the problem of MHD equilibria in
MCT plasmas. The results give a novel qualitative view
on MHD equilibria and suggest a possibility to extend
J.B. Taylor's approach [5] to the case of non-force-freeequilibria in plasmas with a strong radial sectioning (or,confined and cosmic plasmas is suggested to sustain theequivalently, strong ITBs).

correlation refl ectometry.

the "fluid" component to produce a percolating network.

A qualitative approach which treats long-living
filaments as a classical plasma formation governed by

the long-range quantum bonds provided, at themicroscopical level, by nanotubes of elements ofoptimal valence (carbon) is outlined in [4]. The selfself-similarity of the electric current networking in a
broad range of length scales (see this conference@ I 999 by The Japan Society of Plasma

Science and Nuclear Fusion Research50r

Kukushkin A. et al., On the Networking of Long-Living Filaments of Electric Current in Magnetically Confined Toroidal Plasmas(i) few/several relatively thin layers which are

2. Probable Principles of Networking
2.1 Stair-Step Form of Networking offormed by the strongest stockings located atlaround loworder rational magnetic flux surfaces;(ii) the intermediate (thicker) layers with internal
networking which is weaker than that of strongest
stockings but still sufficient for the anomalous radialFilaments

The data analyzed in [1-3] suggest that thenetworking starts along with filamentation from the very
beginning of discharge. The first-generation filaments
are directed toroidally (to give a nearly force-freemagnetic configuration at low values of plasmatransport; and(iii) the regions of poorest networking, in closest
vicinity to strongest magnetic stockings, where theplasma to a largest extent resembles a fluid, because
here the "defect of the resonance" prevents not only thepressure) whereas the second-generation filaments tend

to weave the nested toroidal surfaces (called magneticstockings []), via toroidal and poloidal networking
which incorporates first-generation filaments, with anadmixture of radial networking of variable depth. In atokamak this leads, within each magnetic flux surface,
to a stair-step form of the rotational transform whichturns into canonical (i.e. smooth) one after spatialaveraging within respective magnetic flux surface.

self-closure of filaments but the good-quality
networking as well. The latter makes these regionshydrodynamically most unstable (e.g. sawteeth seem tobe ignited just in the region (4 - 1) -+ -0; similar MHD
activity, much less pronounced, exists around other, lessstrong stockings t6l).
The success of the reversed- or optimized-shear
regimes looks, in the networking-based approach, likebeing caused by avoiding the appearance of the (q - l)

2.2 Selective Survivability and Large-Scale

Structure of Magnetic Stockings-+ -0 region, even at the cost of non-achieving (fromThe individual stocking with simplest connection of
its filaments is most stable to various perturbations.

the side of larger values of4) the formation of strongest

be located at low-order resonance magnetic flux

stocking, at q - I surface. Note that achieving theformation of another strong stocking, at q - 2 surface,surfaces. Correspondingly, magnetic island looks like adoes make the discharge better [7].

Therefore the strongest stockings in a tokamak should

filamentary layer formed by the local pinching of

The above picture is in agreement withfilaments within respective magnetic stocking. Similarly
to stratification of Z-pinch plasmas in transverse
direction, with respect to Z-pinch axis (cf. Sec. 4 in ref.
[3]), in a tokamak there should be the regularly spaced

observations in the RTP tokamak [6], with the best everavailable resolution of 2", of electron ITBs at low-orderresonance magnetic flux surfaces. Therefore, it is not
surprising that the ITB position is not necessarily
coupled to qmin surface and may appear even outside thisbonds between neighboring magnetic islands. Thesebonds reside within respective magnetic surface and are
directed roughly perpendicular to magnetic islands. Thelatter is true of the large scale structuring which has
been found [,2] in processing available data in visible
light (from tokamak periphery) and soft X-ray images(from core plasma), and magnetic probing data (fromsurface [8].

3. Networking-Based Picture of MHD
Equilibria
3.1 Alternative View on Relaxed States
The analysis [l,2] enables us to treat, in unified
frames, the mechanisms of the relaxation to a force freeand non-force free magnetic configurations and to trace
the links between these equilibria.
First, we suggest an alternative explanation to thesuccess of J.B. Taylor's principle [5] of the relaxation toa minimum magnetic energy state (a relaxed state) undercondition of the conserved magnetic helicity. From theviewpoint [1,2], the principle of helicity conservation
may be treated as a consequence of the conservation of
networking of longJiving filaments of electric current.
The success of principle [5] may stem from the fact

plasma surface).

2.3 Radial Sectioning of MCT Plasmas
The MCT plasmas are suggested [,2] to becomposed of a finite-number set of nested toroidal
layers with strong (anomalous) radial transport of heatin their interior. The "network" components of theselayers are decoupled from each other to form theinternal transport barriers (the decoupling is selfconsistently supported by the shear of plasma rotation
velocity) whereas "fluid" component (partially orcompletely) washes out the jumps produced by thethat magnetic helicity K characterizes the knottedness ofmagnetic field lines. In ideal MHD, the integral of thesectioning. The set includes:502

Kukushkin A. et al., On the Networking of Long-Living Filaments of Electric Current in Magnetically Confined Toroidal Plasmas

scalar product (I , d) (*tt"r" A and d are vector
potential and magnetic field, respectively) over each
infinitesimal flux tube surrounding a closed field line is
an invariant (i.e. if one closed field line initially link
another N times then these two loops sustain thislinkage during any ideal MHD motion). In order toassure the conservation of total magnetic helicity K(tz)(i.e. integrated over entire plasma volume [, one need
the identity of magnetic flux tubes. Indeed, the identity

one more constraint:(i) in [10], a particular choice of the functional
dependence of the current densityj and plasma pressure
on safety factor q (in variational procedure the quantities

j and B, have been decoupled, because of allowed
reconnections, but the particular choice made themfinally interrelated by the conventional MHD equation);(ii) in I I,l2], the number of variables is reducedby using the Grad-Shafranov equilibrium equation.
Interestingly, in [10], an extended scheme, withaddition of helicity conservation, has also been
considered in the variational procedure.

leads to a force free configuration which is described bya single parameter rather than the infinity of parameters
as in the general case of ideal MHD. In [6], the identityof flux tubes results from allowed magnetic

The fruitfulness of the principle of magnetic
helicity conservation in describing a non-force-freereconnections (as they, on the one hand, conserve
magnetic helicity and, on the other hand, destroy all

equilibrium has been demonstrated in [3]. There thevariational principle minimizes !V. and IV,n, withother ideal MHD conservation laws). To makereconnections possible Taylor introduced finite
conductivity of plasma. The respective Ohmic
dissipation leads to the non-conservation of helicity,
however the decay rate of the helicity in turbulent
plasma has been estimated to be (much) less than that ofconserved helicity (rather than total electric current), andintroduces the Grad-Shafranov equation directly tovariational procedure. The results for cylindricalgeometry with the first-order toroidal corrections did
manage to give a peaked profile of electric current for

the analyzed there particular cases of the reversed field
pinch, screw pinch and tokamak experiments.

magnetic energy (note that the case of negligible plasma

thermal energy has been considered there).

The networking-based approach replaces theidentity of reconnectable magnetic flux tubes with theAn analysis, from the networking viewpoint, of theopportunities to extend the principle [5] suggests thatthe helicity conservation constraints could draw themost natural bridge between force-free and non-forcefree cases. Specifically, an extension of approaches[5,13] to the case of allowing for the sectioning of the
network component in MCT plasmas could be based onminimizing the magnetic, thermal (and stable plasmaflow) energies with the following constraints in theidentity of long-living (i.e. non-reconnectable) filaments.
In the case of MCT plasmas this implies the integrity of
network component of plasma in each radial section.

The identity of filaments is provided by the intrinsicmechanisms of networking, namely by thesymmetrization of sub-filaments within observed
individual filaments and by the similar mechanisms of
the symmetrization of networking processes in a broad
range of length scales (see [2,3] for more detail).

variational procedure:(a) Grad-Shafranov equation;(b) conservation of helicities integrated over majorradial sections of plasma, with a freedom for the plasma
macroscopic parameters to loose their smoothness at theITBs (i.e. at strongest stockings if they are considered as

3.2 On the Extension of Taylor's Principle
The networking-based view suggests the lines ofextending the Taylor's approach [5] to the case of afinite particle pressure and stable plasma flows, witha "singular" surface layers).

We are to comment on the following two issues.

allowing for ITBs.

A number of solutions to the problem of plasma

First, introduction of additional constraints

equilibria for a finite pressure gradient have been given
in literature for the case of smooth profiles of plasmapressure and electric current (i.e. plasmas without ITBs).
In tokamak case, a substantiation of the "profile
consistency" phenomenon [9] has been independently
given in [0-12]. All these schemes minimize magnetic
energy W. (of poloidal Be [0,] l] or total magnetic field[2] and thermal energy l4lrh [10,12] under condition ofconserved total (toroidal) electric current [10-12] and(conservation of partially-integrated helicities rather
than total one) doesn't lead to the "overweighting" ofthe problem because this is compensated by theappearance of new parameters introduced by thesectioning of profiles.
Second, plasma flow effects should be introduced,strictly speaking, everywhere: in total energy, plasma
equilibrium equation, probably even in the helicity tocombine the fluid plasma and magnetic field behavior503

Kukushkin A. et al., On the Networking of Long-Living Filaments of Electric Cunent in Magnetically Confined Toroidal Plasmas(cf. e.g. generalized helicity approach [4]). However,very preliminary analysis suggests that the role ofplasma flows is maximal in sustaining the shear (or,more precisely, the jumps) of rotation velocity (andconsequently, the existence of ITBs). Therefore, at thefirst step of analyzing the above scheme one couldand Particle Beams, 16,445 (1998); Preprint of theRRC "Kurchatov Institute", IAE 6045 /7, Moscow,October 1997.
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t6l N.J. Lopez Cardozo, et al., Plasma Phys. andsectioning as itself.

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t8l Y. Koide, er al. Proc. lTth Int. Conf. Plasma Phys.

4. Conclusions
The networking-based approach suggests a simple

and transparent interpretation of the success of theapproach [5] and opens new opportunities for extending
major idea of the concept [5], namely the dominant roleof the conservation of magnetic helicity, to much more
complicated cases which are under investigation now infusion systems with essential role of auxiliary heating,sheared plasma flows, and internal transport barriers.

Control. Nucl. Fusion Res., Yokohama, 1998,
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[10] B.B. Kadomtsev, Sov. J. Plasma Phys. 13, 443(1987).

tlll D. Biskamp, Comments Plasma Phys. Control.

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