Computer Modelling of Magnetically ConfinedPlasmas

K. Lackner

Max-Planck Institut für PlasmaphysikBoltzmannstr. 285741 Garching, Germanykal@ipp.mpg.de

Summary Magnetic plasma confinement research poses multifaceted require-ments for computational modelling. The determination of plasma equilibria,the prediction of their global stability, the physics of plasma heating, the es-timation of the energy losses out of the plasma and the interaction of theenergetic plasma with the walls require all support by modelling, using dis-tinct approaches. In particular, the quantitative analysis of turbulent energytransport was for a long time exclusively based on semi-empirical approaches.In the last decade, however, ab-initio plasma models have become progres-sively more realistic. The contribution reports highlights and trends of thisdevelopment since the de-classification of fusion research, and describes thecomponents of a numerical tokamak1, expected to become, concomitantly withthe burning plasma experiment ITER, the main research tool of the fusionscience community.

1 Introduction

The needs of thermonuclear fusion research have traditionally been a strongdriver of computer modelling. A net energy gain from the reaction of the twohydrogen isotopes deuterium and tritium depends on producing a sufficientlydense, thermal plasma with ion temperatures Ti in the 10 keV range andconfining it over a sufficient time τE , so as to satisfy a Lawson-type crite-rion [1] for neTiτE (∼= 5×1021 m3keVs), with ne the electron density. Twovery distinct approaches have emerged for peaceful applications: impulsivecompression, driven by Lasers or fast particle beams [2, 3], with subsequentinertial confinement of the burning plasma on a nanosecond time scale, orquasi-stationary magnetic confinement of a plasma initially heated up within

1 The topomak is a toroidal chamber with magnetic coils as plasma confinementdevice.

E.H. Hirschel et al. (Eds.): 100 Vol. of ‘Notes on Num. Fluid Mech.’, NNFM 100, pp. 373–385.springerlink.com c© Springer-Verlag Berlin Heidelberg 2009

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some tens of seconds. The two lines have inspired also very different code devel-opments, with the inertial fusion effort profiting strongly from the conceptualproximity to military applications. We will concentrate in the following on themagnetic confinement approach and the progress of modelling in it during thelast five decades.

The most advanced toroidal confinement system, the tokamak [4] owns partof its early success to the fact that little computational sophistication is neededto design and operate a simple device, using the Joule heating intrinsicallyassociated with its toroidal current. Plasma confinement is due to the super-position of toroidal and poloidal fields producing axisymmetric flux surfacesin the form of nested tori. The complexity arises with the need to optimizeits configuration and to explore the limits of its performance. Figure 1 showsa schematic view of the plasma vessel and of the magnetic flux surfaces of amodern divertor tokamak (ASDEX Up-grade).

Fig. 1. Plasma vessel and flux surfaces of a modern tokamak.

The plasma pressure is balanced by the magnetic tension of field lines andthe gradient of the magnetic pressure. The magnetic field provides also avery effective thermal insulation perpendicular to flux surfaces and allows tomaintain a temperature difference between the core (the yellow region) andthe boundary (defined by the flux surface with a separatrix in the poloidalcross-section) of several keV over a distance of half a meter. Also the interna-tional fusion test facility ITER, to be built as a joint enterprise by the EU,Japan, Russian Federation, USA, China, South Korea and India in Cadarache(France), follows this design principle. It will be the first magnetic confine-ment device to rely on plasma heating by nuclear reactions, and is expectedto produce, by thermonuclear fusion, 10 times more power than externallysupplied to it for heating the plasma.

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Very advanced modelling tools were finally needed to quantitatively inter-pret and extrapolate the response of the plasma to powerful heating, and tounderstand and possibly control the nonlinear consequences of several insta-bilities. The advance of computational fusion science was of course ultimatelylimited by the pace of hardware and algorithm development, but reflects alsostrongly the changing needs of a continuously advancing experimental pro-gram. We classify in the following these developments into three phases, con-cluding with an outlook. Typically the needs of the tokamak program havesigned the pace of the developments and, except where specifically indicated,we refer to this configuration.

2 Early Modelling Efforts

After theoretical arguments had emerged at the beginning of the 1970s thatnon-circular plasma cross-sections should have significant performance advan-tages, "equilibrium" codes, relating the externally applied magnetic fields tothe plasma shape, were first needed. The conjecture was that such equilibriawould allow stable plasma operation over a larger parameter range, and there-fore also codes, testing these equilibria for linear stability, were needed. Finallyone had to solve some balance equations in the form of 1-d time-dependentdiffusion equations for particles, temperatures and current densities to relatethe observed profiles to the sources and sinks – in the latter including im-purity radiation. As it became soon evident that transport coefficients basedon laminar, collision-induced diffusion were generally optimistically low, ad-hoc coefficients with little theoretical justification were used to account forunexplained, turbulence-induced losses.

Joule dissipation in a plasma rapidly decreases with temperature and henceadditional heating methods are needed to approach the fusion-relevant 10 keVrange. Several such heating methods were developed with great success in the1980s, concomitant with modelling tools for the associated power depositionin the plasma. The increased heat fluxes highlighted the problem of plasma-wall interaction and the resulting wall damage and impurity influxes. For thispurpose the divertor concept (Fig. 1) had been incorporated into the newdevices, and a distinct new modelling discipline: divertor and scrape-off layercodes, emerged.

The increased heating power could also push tokamak discharges to lim-its of the sustainable ratio of plasma pressure to magnetic field pressure(β = 2pμ0/B2). Extensive work with magnetohydrodynamic stability codesidentified a relatively simple and universal expression for these ultimate lim-its, which was verified in impressive form by many dedicated experiments [5] .Below these limits, however, the actually achieved values of plasma tempera-ture and pressure were determined by turbulent transport, and no significantprogress was made in the theoretical understanding of the latter. An empir-ical and rather crude approach was adopted to extrapolate this transport –

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measured by the global energy confinement time τE (= plasma energy/appliedheating power) to future planned devices: a power law fit to the very extensivedata base produced – in later versions [6] – by up to a dozen, different sizetokamaks.

2.1 Emerging Fields of the 1980s

In the early 1980s powerful additional heating methods became available inthe form of neutral beam injection and different wave heating schemes. Theincrease in the power deposited in the plasma exasperated the problem ofimpurities, which are produced by plasma-wall contact, and – if penetratinginto the plasma core – produce radiative energy losses and dilute the po-tentially fusion-reacting species. Two experiments were constructed to test athigh heating powers the so-called "divertor" idea: to displace, by proper designof the magnetic field, the first plasma-wall contact into a separate chamber,relatively distant from the plasma. They demonstrated not only the feasibilityof this concept, but one of them discovered also that the divertor could alsolead to a dramatic reduction in the turbulent energy losses [7]. This initiateda complete conversion of the tokamak construction programs towards divertorequipped devices, but also motivated a large, dedicated modelling effort.

Divertor tokamaks, as shown in Fig. 1, maintain axisymmetry, but theplasma region in direct wall contact cannot be treated even approximately by1-d models. In fact the extend of the scrape-off layer (SOL) beyond the fluxsurface separating closed and open field lines (the "separatrix") is determinedby the competition of parallel and perpendicular transport. Whereas alongfield lines the plasma flow is essentially gasdynamical and can, in principle,become also supersonic, plasma convection perpendicular to field lines, bothwithin and perpendicular to flux surfaces is dominated by slow drifts anddiffusion. Likewise, heat conduction is extremely fast along field lines, butseveral orders of magnitude lower perpendicular to them. Neutral particlesare important – they are the source of the plasma – but are weakly coupledto the plasma, practically do not collide among each other and have to bedescribed by a Monte-Carlo approach. Impurities have to be included in avery comprehensive way, often by treating the individual ionisation stages asseparate, but interacting species, and accounting also for the radiative energylosses.

The geometry of the problem is complex, as can be seen from Fig. 1, andcoordinates following the magnetic field configuration have a singularity onthe separatrix. Modelling of this region is a multi-physics problem, and thecodes [8, 9, 10] which have emerged from this effort include also packagesdescribing the surface physics effects of the interaction of hydrogen and im-purity ions and atoms with the wall, and detailed packages for the atomicphysics. Like for transport codes in the plasma core, their basic limitation liesin the unsatisfactory description of the turbulent transport perpendicular tothe flux surfaces. In the SOL region this is further aggravated by the fact that

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fluctuations in density and temperature are not universally small comparedto the background values, and that intermittency and the irregular ejection of"blobs" of plasma often play a central role. At present these models thereforehave only a modest predictive value, but play an essential role in correlatingand interpreting the very extensive and diversified diagnostic measurementsin the plasma edge region.

Several methods of plasma heating – ITER will incorporate three of them,and is contemplating a fourth – have successfully been brought to matu-rity. Their physics is generally quite well understood and their performancecan be predicted by ab-initio models. For the case of low (tens of MHz) fre-quency electromagnetic wave heating, however, these models become compu-tationally quite demanding. The basic wavelength is not negligible comparedto the device dimensions, while at the same time essential physics (absorp-tion, mode conversion, reflection) happens over spatially very localized regionsand involves only a fraction of resonating particles, whose orbits have to betracked across the relevant interaction region. In the most widely used code[11] the electric wave field is decomposed into separating Fourier componentsin toroidal, and coupled ones in poloidal direction, with a finite element repre-sentation in the direction perpendicular to the flux surfaces. The requirementsbecome progressively more demanding with the size of the modelled device,and application of this code to ITER-like devices or its extension to higherfrequency, shorter wavelength heating schemes has become only possible nowby massive parallelisation of calculations even for an individual toroidal mode.(A realistic heating antenna spectrum excites several toroidal modes, but theirparallel treatment is trivial).

The above modelling developments were essentially driven by the needs ofthe tokamak. An alternative toroidal confinement geometry – the stellarator– has the principle advantage that it does not require a toroidal current flowin the plasma for the production of nested, closed flux surfaces, is intrinsicallystationary (the tokamak requires a transformer to induce these currents, orat least expensive additional systems to drive them by appropriate momen-tum input to electrons or ions) and not prone to sudden, instability drivendisruptions of the plasma current. It gains these advantages at the cost ofa substantially more complex coil geometry. Even more fundamental is thefact that by dropping a symmetry (the axial one) particle orbits lose one rig-orously conserved constant of motion (the generalised toroidal momentum)which in a tokamak ensures, in the absence of collisions and fluctuations, theconfinement of particles to closed surfaces.

It was a dramatic step forward that Boozer [12] formulated the theoreticalconstraint on 3-d magnetic field configurations to ensure a similar benign orbitbehavior as in tokamaks and that Nührenberg [13] succeeded to identify, bynumerical studies, actual stellarator plasma shapes satisfying these criteria.This was a major computational break-through, strictly linked to the arrivalof systems of the Cray-1 performance class. To arrive from this at the designof a stellarator experiment, substantially narrowing the performance gap to

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the then existing tokamaks, required to create the 3-d equivalents to mostof the above-mentioned codes. Three-dimensional plasma equilibria cannotbe simply found by the solution of a single partial differential equation, butrequire, by some procedure, the identification of 3-d stationary states of thetotal potential energy under suitable conservation constraints. Such codes [14]typically start from a given shape of the bounding plasma surface; identifyingsubsequently the external coil shapes producing these equilibria is, like in thetokamak case, not a well-posed problem, requiring regularisation procedures[15]. Due to the 3-d nature, the linearized analysis of MHD instabilities cannotmake use of a separation into non-interacting toroidal modes.

Collision driven ("neoclassical") transport for weakly collisional plasmas –a field that can be largely treated analytically in tokamaks – depends criticallyon the collision-free particle orbits, and can easily exceed, in stellarators, theturbulent transport found in tokamaks of comparable size. Its computation istherefore a very demanding discipline, with a large practical impact on exper-iment design. Codes developed for its analysis followed either a Lagrangianapproach to particle motion (Monte-Carlo codes) or an Eulerian descriptionof phase space. From the above it is evident that the design of a stellaratorrequires a computational effort literally of higher dimension than that of atokamak.

Figure 2 shows a conceptual picture of the plasma surface of the firststellarator experiment designed, incorporating all the above considerations(W7X) and of the complex shape of the magnetic field coils required to formit. This experiment, presently under construction in Greifswald, Germany,should drastically narrow the performance gap to present-day tokamaks, butis, in particular, also a milestone of computational physics.

Fig. 2. Plasma surface and coil system of an optimized stellarator (W7X).

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2.2 On the Way to a Numerical Tokamak

So far, the performance prediction of magnetic confinement experiments hasbeen nearly exclusively empirically. Even for the W7X stellarator, where nu-merical calculation played an unprecedented role in the layout, the large com-putational effort concerned mainly items which in a tokamak are either uncrit-ical, or can be calculated with relatively simple models. This empiricism basedapproach was adequate, as the performance target of all these devices (includ-ing ITER) are still essentially scientific, but becomes progressively more costlyand risky with each new and larger generation of experiments.

Since the mid-1990s, however, confidence has been rapidly growing, that ab-initio modelling of tokamak – and, subsequently also stellarator – performancewill be able to accurately explain ITER results and to fix the parameters ofits successors. Apart from the past and expected future growth of computerpower, and the algorithmic improvements of general-purpose routines (e.g.parallel matrix solvers), this is mainly due to the growing conviction that aplasma model has been identified containing all the necessary physics affectingturbulence in fusion devices, and that this plasma model can be implementedwith sufficient accuracy on upcoming computers to quantitatively explain theassociated energy and particle transport. This high expectance in computa-tional modelling has, moreover, also spread to other areas of fusion physics,so that the vision of a numerical tokamak, consisting of a complete ab-initiomodel of the plasma core as a design tool for future fusion power plants, isbecoming credible.

As reactor relevant plasmas are nearly collision-free and turbulence canalso include magnetic fluctuations, the most general model would require atime-dependent description of the distribution function in three geometricaland three velocity space coordinates, plus the full set of Maxwell’s equations,simplified only by the neglect of displacement currents. Some, usually well jus-tifiable simplifications render the problem of transport-inducing turbulence,however, more tractable. The most important one is the gyro-kinetic model,which reduces the phase space by averaging over the very fast gyro-motion ofparticles, while taking properly into account the possible spatial variation ofelectric and magnetic fields over the spatial scale of the circular gyro-orbit.

The second starts from the observation that the most important effectof magnetic field perturbations concerns the components perpendicular tothe equilibrium field, which allows to consider only one component of thevector potential. Furthermore, the extremely strong anisotropy of magnetizedplasmas implies a very long scale length for all perturbations along field lines.Using magnetic coordinates with one coordinate line aligned with the magneticfield allows to translate this into a much reduced resolution requirement inone coordinate direction. Finally, fluctuations are usually small compared tothe background quantities, suggesting a splitting of the distribution functioninto a smooth distribution Fo and a perturbation δf , reducing thereby e.g. inparticle-in-cell (PIC)-based methods the statistical noise.

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Kinetic equations can be treated either in an Eulerian or a Lagrangianframework, while the field equations, coupling the particles require a spatiallyfixed grid. Nevertheless the complexity of geometry and of particle orbitsappeared to favour a particle-following, Lagrangian approach, and the earlygyrokinetic turbulence simulations used PIC-Codes. Following the pioneeringefforts of Jenko and Dorland [16], Eulerian (also called continuum or Vlasov-)codes, based on a fixed grid in velocity space have established themselves asat least competitive, and at present comparable efforts are invested in thetwo lines of development. Also a hybrid (semi-Lagrangian) approach has beenemployed to combine the advantages of a fixed grid in velocity space with theuse of particle trajectories as characteristics.

Depending on parameters (ratios of electron to ion temperatures Te/Ti,gradients of Te, Ti and of electron density, collisionality, β, ratio of gyro-radiusto gradient length, structure of the magnetic field) different instabilities drivethe turbulence and necessitate, in particular, different sophistication in thetreatment of the electrons. A broad and relevant range of cases is covered byconditions where the smallest perpendicular space scale is of the order of theion gyro-radius ρI whereas the fastest time-scales are set by electron thermalmotion or by the (roughly comparable) Alfvén wave propagation along fieldlines.

Much of the underlying physics can be gathered from calculations coveringonly the spatial domain of a flux bundle less than 100 ion-gyroradii across inboth radial and poloidal directions. In this case artificial (usually periodic)boundary conditions have to be applied to the fluctuating quantities, bothwithin a flux surface, but also at the radial boundaries. The latter assumptionis mathematically consistent with a local approximation to the backgroundparameters neglecting their variation across the computational region. Suchmodels cannot give, however, full information about the scaling of turbulenttransport with the ratio of a/ρi across the regime covering present medium(a/ρi

∼= 200) to large (a/ρi∼= 400) size devices and onward to ITER (a/ρi

∼=800) and therefore a move to global simulations is everywhere ongoing.

Typically Lagrangian models can easily work on a global scale, but havemore difficulties refining the physics model and controlling statistical noise,whereas Eulerian models are pioneering more complete physics model, but areproceeding more slowly to a global coverage. All gyrokinetic turbulence sim-ulations are computationally extremely demanding undertakings as is man-ifested, for example by their inclusion into the Grand Challenges of the USDepartment of Energy initiative on Scientific Discovery Through AdvancedComputing (SCIDAC) [17].

Fig. 3 left and right, taken from a local simulation with a Eulerian Code[18], illustrate one of the most important and universal results of gyrokineticsimulations, for a situation dominated by ion temperature gradient (ITG)driven turbulence. The figures show two time-slices referring to the initialdevelopment of the instability and a later phase of saturated turbulence. Per-

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turbations are always aligned with the screw-shaped magnetic equilibriumfield and have a larger amplitude on the outer side of the torus.

Fig. 3. Nonlinear evolution of an instability driven by the temperature gradient ofthe ions. Left: linear phase of the mode, right: nonlinear evolution, affected by theformation of poloidal (zonal) flows.

Initially they tend to develop radially elongated streamers, which are veryeffective in mixing plasma across the magnetic surfaces (left). The emergingturbulence itself, however, drives sheared zonal flows in poloidal direction,which have a shorter radial wavelength, and tear apart these elongated struc-tures. Numerical simulations therefore, after some time (a few hundreds ofunits of the characteristic time L⊥/Cs, defined by the radial gradient lengthand the ion sound speed), settle down to a quasi-stationary state (right),with a significantly reduced transport compared to the initial overshoot. Suchturbulence simulations have succeeded capturing many features of the exper-imentally observed energy and particle transport.

The effects of finite extent of the ion-gyro radii can also be captured bya fluid model (called "gyro-fluid") [19, 20], which however, in the weaklycollisional case has to resort to ad-hoc assumptions to provide a fluid closurealong magnetic field lines. If the latter is properly chosen, a good agreementwith the gyrokinetic models can be reached even regarding the core regionsof plasmas. In particular, however, gyrofluid codes are the current state of artregarding the simulation of turbulence in the more collisional edge zone of theplasma, where geometry effects and boundary conditions are of dominatingimportance, and fluctuation amplitudes are comparable to the backgroundvalues.

At present, more than a dozen codes for the simulation of turbulence intoroidally confined plasma are in continuous development. This multitude ofmodels and algorithms is justified by the ambitiousness of the task – quanti-tative ab-initio modelling of turbulent transport – and the large savings thatwill result from a reliable model for the dimensioning of thermonuclear powerplants. It is necessary, however, also to establish the credibility of the outcome,

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as direct, unambiguous proof of the results by experimental measurements isstill limited in extent. Therefore, from the very beginning a significant efforthas been placed on the bench-marking of all these codes, starting with thedefinition of the "Cyclone" test case [21], which has since then been studiedwith each relevant code. For clearly defined conditions, such tests usually re-sult in a 10% agreement among the participating codes [22] for the predictedturbulent fluxes.

In the modelling of large scale, magnetohydrodynamic instabilities the in-terest originally was only in their prevention by a careful mapping of thelinearly stable plasma conditions. It was subsequently recognized that unsta-ble modes exist, which result only in a cyclic rearrangement of profiles andare compatible with an overall quasi-stationary discharge behavior. They can,however, influence the global energy confinement or the heating efficiency,or can be associated with pulsed heat loads to vessel structures and requiretherefore a nonlinear model for their full cycle. In some cases it has beendemonstrated or is expected that feedback-control can significantly extendthe operational range of tokamaks beyond the linearly stable regime. Finally,disruptive instabilities of the plasma current – expected to be rare, but design-base events in a fusion power plant – are potentially associated with large heatloads and electromagnetic forces onto the vessel structures that need reliableextrapolation from present devices.

These necessities and trends have given new impetus to the developmentof nonlinear, global MHD codes to describe such large scale, large amplitudeperturbations. Many aspects are tied to the question of fast reconnection ofmagnetic field lines, and a strong link exists therefore to the areas of spaceand astrophysical MHD modelling, including the common use of a test model[23]. Two large, parallel code developments are taking place in the US, bothinvolving several institutes, and both coordinated within a common project[24, 25] and two, less comprehensive models also in the EU [26, 27]. Contraryto turbulence simulations, the emphasis in this case is on significant changesin the magnetic topology, a global treatment of the whole plasma, and thegenerally longer characteristic time scale of the dynamics. In addition to thesevere problem of anisotropy (which is aggravated by the fact that the largetopological changes make the use of magnetic coordinates difficult), the largespread of the potential time-scales require the elimination or the implicit treat-ment of the fastest wave motions. The most common solution to this problemis the addition of a semi-implicit operator to the equation of motion [28, 29].

In some applications – notably the disruptive termination of tokamak dis-charges – one wants to model quite realistically the contact of the plasma withthe 3-d structure of the walls, including the passage of electric currents intothem. Other areas require an extension of the usual resistive MHD model bytwo fluid effects (to treat the smaller scale effects important for reconnection),additional current drive terms arising at low collisionality from the particu-lar nature of particle orbits in toroidal geometry (the so-called "bootstrap"current) and the electromagnetic coupling of the plasma to the suprathermal

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particles described below. The two US code developments are therefore truemulti-physics efforts, with different terms and packages – depending on thespecific application – added to the core of the resistive MHD equations.

Plasma heating often is associated with the presence of fast ions, injectedoriginally as neutrals from the outside, or accelerated by wave resonance pro-cesses, or – as final goal – produced by thermonuclear fusion reactions. Thesefast particles can influence the macroscopic stability of the plasma throughboth their gradients in geometrical and velocity space. In particular, fusion-α particles have at birth a velocity larger than the Alfvén velocity, and canresonantly interact and destabilize otherwise weakly damped plasma modesthrough the spatial gradients of their distribution function. Depending on thefurther nonlinear growth and interaction of these modes, they could lead toa significant loss of fast particles, reducing thereby the efficiency of the ther-monuclear plasma heating, and causing localized heat loads on the plasmavessel.

Fast-particle driven modes in different contexts have been studied on ex-isting experiments in some detail, but the combination of isotropy and super-Alfvénic velocity associated with fusion-α particles will be novel on ITER.As their effect will be of critical importance for the self-heating of a fusionreactor they are becoming a new focal point of computational modelling. Al-ready the linear stability analysis, requiring a spatial coverage of the wholeplasma and a detailed description of the particle distribution function is achallenging problem, which, in particular due to the resonances in frequencyleads to a difficult, non-standard eigenvalue problem. Early codes in fact useda simplified, perturbative approach, in which the eigenfunctions of stable,ideal MHD modes were used to compute only (complex) corrections to thefrequency arising from kinetic effects. When the energy content of fast parti-cles becomes a significant fraction of the plasma energy, however, new modescan arise, which have no correspondence in the ideal MHD spectrum, and arehence not captured by this approach. Latest-generation codes therefore treatfast particles and MHD effects on an equal footing, solving the linearizedequations either as an eigenvalue or initial value problem, or scanning theresponse of the system to an external excitation by an antenna with variablefrequency [30, 31, 32]. To determine the actual losses associated with theseeffects requires of course a nonlinear model, including the interaction of differ-ent modes, the modifications to the distribution function and ultimately alsothe nonlinear modifications to the magnetic perturbations. Also this work hasstarted, albeit so far with simplified models [32, 33].

3 Future Trends

The ultimate aim of fusion plasma theory is ambitious: to provide a model forthe plasma core of a fusion reactor that can be incorporated into design codesfor a power plant. This comprehensive model – sometimes termed "numerical

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tokamak" (or stellarator, respectively) – has to include modules covering allthe above areas in an integrated form taking into account their mutual in-teraction. The confidence is quite high that in most of these areas we haveadequate physics models at hand, and know – in principle – how to includethem into the codes. We still have to drop, however, some simplifying assump-tions in their implementation, include more realistic geometries and extendthe region in space and time covered by the simulations. Fortunately in mostareas, the computational effort required scales strongly with size, so that mod-els developed and tested for present-day experimental devices during the nextten years will find the computer power ready to run with ITER parametersonce the latter starts operating.

Prior to their combination in integrated models, we expect from the im-proved and extended codes the solution of a number of well-defined enigmas.The most important one regards the sudden improvement in energy confine-ment observed at high enough heating powers in divertor tokamaks [7], evi-dently associated with the suppression of turbulence in a narrow layer close tothe plasma boundary. This transition from L (low) to this H (high) confine-ment regime is a highly reproducible bifurcation phenomenon, with dramaticconsequences: it gives rise to a doubling of the plasma energy content, whichin a deuterium-tritium plasma would be associated with a 4-fold increase infusion power. It is expected that this phenomenon will spontaneously show upalso in turbulence simulations with a sufficiently realistic model, and achieve-ment of this would be viewed as the Holy Grail of fusion plasma theory. Asecond, similarly universal, and equally unexplained phenomenon is the ap-pearance of a density limit in tokamaks [34], which is not observed in ananalogous form in stellarators [35].

The integration of codes dealing with different aspects of tokamak operationis a growing necessity and task forces have been created explicitly for this task.A unification in efforts will, however, also arise form the fact that increasingcomputing power allows codes created for one purpose to extend their applica-bility to other areas. Simulations with codes originally developed for "micro"-turbulence cover increasingly larger regions of the total plasma, whereas non-linear codes for macroscopic MHD instabilities improve the physics modelsand the spatial resolution. A case in point for this kind of convergence arestudies of edge-localized magnetic perturbations ("ELMs"). They share somecharacteristics with global MHD instabilities, but are restricted to a small frac-tion of the plasma close to the boundary, and are presently treated, in parallelefforts by both turbulence [20, 36] and macroscopic MHD codes [25, 27].

The most significant trend in magnetic fusion research is, however, the gen-eral acceptance that first-principle based modelling will substitute the empir-ical and semi-empirical approaches used in the past [6] for the performanceextrapolation to next-generation devices. This is documented by the consid-eration of fusion related model developments as one of the grand challengesof computational physics, but, for example also by the decision of the EU andJapan to include a dedicated high performance computing centre into their

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joint "Broader Approach" effort towards fusion energy production, accompa-nying the construction of ITER.

Acknowledgements

The author is obliged to David Coster and Moritz Püschel for making availableunpublished illustrations.

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