Euro

ph

ysic

s N

ews

No

vem

ber

/Dec

emb

er 1

99

8

212 F U S I O N

The global pressure change in that volumewill be the sum of a positive contributionassociated with the increase in the elec-tron and ion density and a negative con-tribution associated with the decrease inion temperature. If the net effect is adecrease of the local plasma pressure(p = neTo + niTo + noTi < 0), the compres-sion will be amplified and an instabilitywill be generated. Since the driving termis proportional to the ion temperaturegradient, the instability takes the name of‘ion temperature gradient driven mode’.

A crude estimate of the turbulent ther-mal conductivity cturb (see Garbet, chapter3.5) can be given on the basis of quasilin-ear theory as cturb ≈ Lr

2/tc with Lr being theradial correlation length and tc being theautocorrelation time of turbulence, of theorder of the typical linear growth time ofthis class of instabilities tc ≈ a/vti, with vtithe ion thermal velocity and a the minorradius. Thus, the reduction in cturb can beobtained either by an increase in tc or by adecrease in Lr. As an example of self orga-nization mechanisms that lead to anincrease in tc we mention the formation oflarge scale coherent structures, observedin several numerical simulations. Thesestructures, which can be interpreted assolitary vortices, are coherent in the sensethat their life time tc is much longer thanthe eddy turnover time. Thus, they behaveas insulating structures and their occur-rence is marked by a drop in the turbulentheat flux.

Direct experimental measurements andtheoretical arguments show that plasmaturbulence can manifest itself both by longand short radial correlation lengths. Theturbulence radial correlation length Lr canbe determined by the competition betweenlinear coupling and nonlinear decorrela-tion. The linear coupling acts in such a wayas to produce radially elongated eddies.Indeed, the periodicity constraint in thepoloidal and toroidal directions impliesthat a generic perturbation must have theform exp{-iwt + imq - inf} with m and nbeing integers. Such a perturbation is radi-ally localized in a region of the order Ç(l, e, i)around the magnetic surface where thepitch of the magnetic field lines is equal tom/n (the so called rational surface). Sincein an axisymmetric equilibrium thetoroidal angle f is an ignorable coordinatewhereas the poloidal angle q is not, a lineareigenmode is characterized by a single nvalue and by the superposition of differentpoloidal harmonics with different m val-ues. The linear coupling between poloidalharmonics centred on neighbouring ratio-nal surfaces (ie the surfaces at which the

The heat and particle transport in fusionplasmas is generally due to turbulentprocesses associated with the presence ofsmall scale instabilities (typical wave-lengths of the order of the ion Larmorradius Çi) driven by the inhomogeneity ofthe density and temperature profiles in thedirection normal to a magnetic surface.The magnitude of turbulent transport isprobably the parameter affecting most theconfinement properties and hence theeconomical performance of a fusion reac-tor. Thus, it is important to some extent tobe able to control the turbulence level.This may occur in fusion plasmas, as wellas many other complex systems, by self-organization phenomena which affect theturbulence level and the associated trans-port. During the last 15 years several plas-ma regimes caracterized by enhanced con-finement have been experimentallyreached, although only a few of them wereobtained in steady state conditions. Thus,the achievement of these regimes insteady state conditions is an importantline of research in controlled thermonu-clear fusion.

The most important driving mecha-nisms for plasma turbulence were identi-fied at the beginning of fusion research.Experimental investigations began sys-tematically more than 20 years ago. As anexample of the complexity of magnetized-plasma turbulence with respect to theconventional, neutral fluid turbulence, weconsider the so called ion temperaturegradient driven mode. The most impor-tant destabilization mechanism for suchan instability is the so called ‘negativecompressibility’, which is briefly describedin the following. Let us consider a magnet-ic surface S, spanned by an equilibriummagnetic field B, and equilibrium density

and temperature spatial profiles n0 andTo(for the sake of simplicity we willassume equal electron and ion tempera-tures) with gradients in the direction nor-mal to S. Suppose that at time t=0 a per-turbation f in the electrostatic potential isproduced. Due to their small mass theelectrons will react immediately in orderto shorten out the electric field E = -—f.By moving along the magnetic field linesthey will produce an electron density per-turbation ne = no (ef/To). Since this createsa local inbalance in the electric charge, theion will also react in order to produce anion density perturbation ni = ne, in such away that local charge neutrality isrestored. If we neglect the existence of atemperature gradient, the result of theoriginal perturbation in the electric fieldwill simply be the increase in the plasmapressure (p = neTo + niTo). Following suchan increase, a restoring force associatedwith the plasma pressure will be producedand stable sound waves, propagating alongthe magnetic field, will be excited. Thedynamics described up to this point isvery similar to what happens in a neutralfluid. However, the situation changes dra-matically in a magnetized plasma if a gra-dient in the ion temperature is considered.Due to the presence of the electric fieldthe ions will move with a velocity vi suchthat the E + vi ¥ B ≈ 0, ie vi ≈ E ¥ B / B2.

The local value of the ion temperaturewill be perturbed by convection∂tTi = -vi—T0. Let us consider the situationat the time t = p/2w, with w the frequencyof the perturbation. Suppose that thephase velocity of the perturbation in thepoloidal direction is such that the volumein which the original compression wasproduced coincides with the position ofthe local decrease of the ion temperature.

Fusion Plasma as a

Non-Equilibrium System

Francesco RomanelliCentro Ricerche Frascati, Italy

Carlos HidalgoCentro de Investigaciones Energéticas Medioambientales y Tecnológicas, Madrid, Spain

Chapter 3

The Basic Science

3.1 Fusion Plasma as a Non-Equilibrium System Romanelli, Hidalgo

213

Euro

ph

ysic

s N

ews

No

vem

ber

/Dec

emb

er 1

99

8

F U S I O N

pitch of the magnetic field lines is equal tom/n and (m ± 1) / n) produces elongatededdies in the direction perpendicular to S.A rigorous analysis of the problem showsthat the typical radial width of the eigen-mode is of the order of (aÇi)1/2. Thus, if thelinear coupling dominates the radial corre-lation length can be estimated to be Lr ≈(aÇi)1/2. Nonlinear decorrelation mecha-nisms tend to break up this structure insuch a way that each poloidal harmonicbehaves independently from the others. Ifsuch a mechanism dominates we can esti-mate Lr ≈ Çi. Therefore, the rough estimateof the local thermal conductivity givenabove can be reduced to two different lim-its: cturb ≈ DB and cturb ≈ (Çi/a) DB, withDB = cT/eB the Bohm coefficient. The twodifferent limits are called Bohm and Gyro-Bohm, respectively.

Historically, the first example of mecha-nism leading to the reduction of Lr, andtherefore to a reduction of the turbulentflux, was proposed for the explanation ofthe transition to the High confinementmode (H-mode) which is obtained rou-tinely in tokamaks with a magnetic sepa-ratrix at the plasma edge and correspondsto the formation of a local transport barri-er. The improvement mechanism can beexplained as the formation of a shearedE ¥ B flow. If an electric field is producedin the direction perpendicular to the mag-netic surface, the plasma starts to rotatewith an E ¥ B velocity which lies on themagnetic surface and is directed in thedirection perpendicular to the equilibrium

magnetic field. If the radial electric field isspatially nonhomogeneous, plasma vol-umes on neighbouring magnetic surfacesrotate with different velocities. Such ashear in the rotation velocity tears apartturbulent eddies elongated in the radialdirection and reduces the transport level.Experimental support for this idea hasbeen obtained in many magnetic confine-ment devices (ie tokamaks, stellaratorsand reversed field pinches) either by theanalysis of spontaneous transitions or byusing polarization probe techniques forthe radial electric field generation. Duringthe transition, the experimental measure-ments show the formation of an inhomo-geneous radial electric field and the reduc-tion of the turbulence level. But, how cansuch a field can be maintained? The radialelectric field can be generated by the pres-sure gradient and poloidal and toroidalrotation. In the absence of substantialplasma rotation the radial electric fieldmust be proportional to the local ion pres-sure gradient. Thus, a simple bifurcationparadigm can be constructed. A steepen-ing of the pressure gradient produces aradial electric field which decreases theturbulence level and allows the maintain-ing of a steep pressure gradient. The close-ness to turbulence thresholds, the type ofinstability driving the turbulence and thepresence of sheared flows are consideredcritical elements in determining the tran-sition between different transport regimes(ie Bohm versus Gyro-Bohm).

In the last five years a second mecha-

Further readingL.I. Rudakov, R.Z. Sagdeev Dokl. Akad. Nauk. SSSR 138581 (1961)[ Sov. Phys. Dokl. 6 415 (1961) ]

M. Ottaviani, F. Romanelli, R. Benzi, M. Briscolini, P.Santangelo, S. Succi Phys. Fluids B2 67 1990

J.W. Connor, J.B.Taylor and H.R.Wilson Phys. Rev. Lett.

70 1903 (1993)

P.H. Diamond, T.S. Hahm Phys. Plasmas 2 3640 (1995)

nism for reducing the turbulent transporthas been experimentally found in toka-maks which involves the production ofnonmonotonic current density profiles(the current density is needed to producethe poloidal component of the equilibri-um magnetic field). If such a condition isachieved, a transport barrier is formedwith no turbulent transport. The under-standing of this phenomenology is still inprogress. A possible explanation might bethat in these plasma regimes the magneticconfiguration reduces the level of turbu-lence and/or the coupling between neigh-bouring poloidal harmonics. As a conse-quence, the power threshold for the tran-sition to enhanced confinement regimesvia sheared radial electric field effects isreduced. Thus, sheared radial electricfields could provide a universal mecha-nism to control plasma turbulence infusion plasmas.

With an eye on the future, the under-standing and development of methods forcontrolling plasma turbulence haveopened a new path in plasma physicsresearch and contributed to the design ofa nuclear fusion plant.

The Understanding ofOperational LimitsLaurent VillardEcole Polytechnique Fédérale de Lausanne, Switzerland

Given the high cost of experimentaldevices in magnetic fusion research it isimportant to have reliable and predictivetools for their operation. This, togetherwith the complexity of the geometry, hasprompted a major effort in the develop-ment of numerical techniques.

From single particles...The particular magnetic geometry stems

from the requirement thatthe particles should beconfined by the magneticfield. The particle motionis a fast race along the fieldlines added to a slower driftacross the magnetic surfaces.Therefore the topology istoroidal, with field lines helicallytwisting about the magnetic axis.

This ‘twisting’ can be createdby a plasma current (toka-

mak) or by currents in shapedcoils (stellarator). The field lines

should wind round nested mag-netic surfaces (figure 1).

3.2 The Understanding of Operational Limits Villard

Fig 1 Mesh used in the computation

of plasma equilibrium and stability,

showing a cross-section of an ITER

tokamak plasma with nested

magnetic surfaces