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Different scenarios of transitions into improved confinement modes

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1997 Plasma Phys. Control. Fusion 39 B371

(http://iopscience.iop.org/0741-3335/39/12B/028)

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Plasma Phys. Control. Fusion39 (1997) B371–B382. Printed in the UK PII: S0741-3335(97)87179-5

Different scenarios of transitions into improvedconfinement modes

M TendlerAlfv en Laboratory, Royal Institute of Technology, Stockholm, Sweden

Received 13 June 1997

Abstract. This review addresses transitions into regimes with improved confinement. TheE × B paradigm is tested on the example of L–H transitions. It is demonstrated that L–Htransitions may emerge either due to an amplification of the diamagnetic drift term caused byenhanced pressure gradient or due to an increased poloidal rotation velocity at the separatrix.In general, it is asserted that the emergence and dynamics of transitions are very sensitive tofine details of prelude plasma profiles. The L–H transition occurs provided the radial electricfield changes dramatically within a few poloidal gyro-radii from the separatrix. The adoptionof a significant power flux to the separatrix as the trigger of the L–H transition appears fromthe theoretical point of view to be a simplistic ‘brute force’ solution to the issue of confinementcontrol. The propagation of the front of the barrier is very fast. The dynamics of interest consistsof threshold conditions for barrier formation, barrier propagation speed and profile steepeningrates, barrier limits and mechanisms for their relaxation and termination.

1. Introduction

There is accumulating evidence from tokamak experiments that regimes with improvedconfinement can achieve higher values of confinement, beta and bootstrap current than hadbeen thought plausible until recently.

Transport barrier dynamics is the key scientific concern at present. The interest isenhanced by the fact that a continuously operated tokamak will not be operated at asteady state, since for control purposes, it will necessarily require barriers to be createdor lowered from time-to-time in different portions of a plasma to facilitate the reactoroperation. Transport barriers will have to be controlled for operation purposes. This is bornout by stringent requirements for burn control. Furthermore, one must bear in mind thatthe location of transport barriers is an important issue because of the magnetohydrodynamicstability constraints. Steep gradients associated with transport barriers can be tolerated onlywithin the robust MHD stable regions of the plasma body.

At present, a large body of evidence obtained on many tokamaks shows that theformation of a transport barrier is closely related to the hypothesis that the increased shearin theE × B drift leads to a suppression of turbulence, thereby improving confinement.

This paradigm is of fundamental importance for fusion research. Indeed, it has amplifiedthe world-wide efforts put into studies of the origin of the self-consistent electric fields andalso encouraged methods by which to impose electric fields on plasma by various techniques.

There is little doubt that achieving an understanding and control of turbulent transportwill benefit the design of an economical fusion reactor. Significant progress in thisendeavour has been made by exploiting the spontaneous transition to a high confinement

0741-3335/97/SB0371+12$19.50c© 1997 IOP Publishing Ltd B371

B372 M Tendler

regime, H-mode. This occurs provided a certain threshold in power injected into edgeplasmas is exceeded. Transitions are envisaged to occur as a multistep process. First, at agiven temperature at the edge, a significant modification of the electric-field profile unfolds.Then the resulting shear of the electric field is responsible for the reduction of fluctuationlevels and transport. The bifurcation occurs during the transition as a discontinuous changein the equilibrium quantities such as electric field, temperature and density profiles. Thecornerstone of the concept is the link between the electric field and improved confinement.We first address the issue of the effect of the shear of the electric field on residual turbulenceand postpone the more difficult discussion concerning the origin of the electric field tofollowing sections.

2. Basic features of shear stabilization models

Important elements of the concept are the option of the linear stabilization of a possibleinstability in a tokamak or an alternative option of turbulence suppression due to enhancededdy-decorrelation induced by the shear of the electric field.

Rotation of a Navier–Stokes fluid resulting in turbulence saturation at a reduced level isa well known effect in fluid mechanics [1]. However, since a sheared flow also provides freeenergy which can drive the Kelvin–Helmholz instability the phenomenon is not frequentlyobserved in fluids. In contrast, in a magnetized plasma the impact of the sheared flow mayobtain two different forms.

The E × B shear can stabilize an instability within a given range of plasma volumedepending on plasma profiles. The complete stabilization of various modes occurs providedtheE × B shearing rate exceeds the linear growth rateγ of a given instability

γs > γMAX (1)

where

γs = RBθ

B

∂(Er/RBθ)

∂r(2)

andγMAX is the maximum linear growth rate. Equation (1) shows that the stabilization isa mode specific feature. The effect stems from the amplified damping of a given unstablemode due to an interaction between this mode with a nearby stable mode. The stabilizationresults from the enhanced Landau damping of an unstable mode due to this interaction.Furthermore, the form of the shearing rate given by (2) clearly depends not only on theelectric field profile, but also on the magnetic shear [2]. The dramatic impact of the magneticshear is a geometric effect occurring because of differences in the radial separation ofmagnetic surfaces. Hence, if the shearing rate significantly exceeds the growth rate of agiven instability one might expect stabilization followed by concominant improvements inconfinement.

However, the constraint given by (1) is rather stringent and can hardly be satisfied for allmodes, unstable within the entire plasma body. Therefore, in order to significantly reducethe turbulence level the shearing rate has to be at least of the order of nonlinear saturationof a turbulence level resulting from unstable modes. Quantitatively, in the absence of theE × B shear, the level of turbulence is determined by the balance between the growth rateof an instability and the damping provided by an anomalous diffusion. The timescale fornonlinear damping is of the order

τ−1 = ω ∼ γ = 4D/δ20 (3)

Transitions into improved confinement modes B373

whereγ is the growth rate,D is the coefficient of anomalous diffusion andδ0 is the radialcorrelation length of the turbulence. This value has to be compared with a scaleδ determinedby decorrelation of fluctuating plasma parameters (density, temperature, potential, etc)arising due to a differential poloidal motion on a scale of the poloidal variationk−1

θ .Introducing the shearing rate asγs ∼ kθ | dV0/dr|, whereV0 is theE ×B drift velocity,

one easily obtains the effective rateγc of the combined shearing and turbulent decorrelationrates as [3]

γc = (γ 2s1ωD)

1/3 (4)

where1ωD is the nonlinear decorrelation rate in absence of theE × B shear.Both linear and nonlinear mechanisms for turbulence suppression are beneficial for the

quality of confinement. The former is clearly more beneficial and dramatic, yet is subjectto more stringent constraints on the shearing rate. Recently the full nonlinear numericalanalysis of the effect of shearing rate on the ion-temperature gradient mode was carried out[4]. Unexpectedly, it has transpired that the ITG modes are completely suppressed whenγs ∼ γMAX .

This remarkable result indicates that in order to quantitatively assess the impact of theE × B shear only fully nonlinear models, addressing the effect in detail, suffice.

Models should include synergistic effects of prelude profiles, in their capacity to drivevarying modes unstable, theE × B and the related magnetic shear profiles. For example,negative or low shear current profiles allow stabilization of high-nMHD modes (ballooning),notoriously harmful to the confinement. Another example of the plausibly favourable effectemerges assuming prelude profiles with the safety factorq > 1 everywhere within theplasma body stabilizing sawteeth oscillations. In general we aim to show that the lackof these instabilities combined with the localized deposition of particle, momentum andheat sources or sinks allows pressure and/or rotation gradients to build up, thus amplifyinglocally the radial field gradient. Hence, the radial electric fieldEr can be changed locally byparticle, momentum and heat sources or sinks, thereby triggering multiple feedback loopsand leading to turbulence decorrelation and reduced transport. In summary, local transportbifurcations might occur resulting in the build-up of transport barriers due to the enhancedE×B shear decorrelation. Studies of different scenarios of transitions are presented below.

3. Origin of electric fields

The origin of electric fields is the important part of the concept. It is complementary to thephysics described in the previous section, yet more difficult to assess. In order to addressthe origin of the electric field the radial component of the momentum balance is invoked

Er = 1

eZini

∂pi

∂r+ BθVφ − BφVθ . (5)

Equation (5) is fundamental because it offers insights on how an electric field can be inducedwithin the plasma. Basically, there are three terms (‘knobs’), which can be employed. Theseare diamagnetic drifts, toroidal and poloidal rotations. Obviously, these three ‘knobs’ caneither reinforce or counteract each other. In the ideal situation, when all conditions arefavourable, all of the three terms can contribute to amplifying changes in the electric field.To start with, we consider the situation at the edge where the electric field is primarilydetermined by the diamagnetic drift and the poloidal rotation. Toroidal rotation is assumedto be effectively damped there by the anomalous gyro-viscosity.

B374 M Tendler

Then, employing the neoclassical value for the poloidal rotation velocity to be thefunction of the temperature gradient one easily obtains

Er = Ti

e

(d lnn

dr+ (1− k)d lnTi

dr

)(6)

where(1− k) is the reducing coefficient, well known from neoclassical theory [5]. Hence,the electric field is a linear combination of logarithmic derivatives of density and temperatureprofiles. This approach provides the well founded estimate for the evolution of theprofile of the electric field from first principles. Experimentally, it enables only standardlocal measurements of density and temperature profiles to be employed (without invokingadvanced diagnostics such as heavy-ion beam probe, charge exchange recombination, etc)in order to determine electric field profiles.

Equation (6) demonstrates that the electric field is amplified locally provided the densityand/or temperature profiles become steeper. The analysis of (6) together with particle, heat,momentum and turbulence balances offers a powerful means by which to follow in detailtransitions into regimes with improved confinement [6, 7].

However, although (6) yields the convenient approximation to determine the profile ofthe electric field, it remains rather crude in order to address transitions triggered by thepoloidal rotation because it does not include the transport of parallel flows in a tokamak,which might play the crucial role as will be shown later.

To this end, in general, it is well known that parallel dynamics is a very important factorfor many issues in tokamak physics. Therefore, it is likely that the parallel dynamics is alsorelevant and important for the issue of electric field profiles. The parallel component of theflux surface average momentum balance yields the equation describing the evolution of theE×B velocityV0 (see (2.25) in [8]). Obviously the approach employing (6) cannot provideinsights into a cause of the transition triggered by poloidal rotation because (6) implies therigid relation between the electric field and density and/or temperature gradients due to itsform. Therefore, it is mandatory to invoke the parallel component of the momentum balanceequation in order to address the causal relation between the electric field and the effect ofsteepening of density and/or temperature profiles within the context of different transitionscenarios.

4. Evolution of transport barriers

The formation of the transport barrier is to be modelled by the system of general diffusionequations taking the form

∂h

∂t− ∂

∂x

(D(α)

∂h

∂x− V (α)h

)= S (7)

where h is the density, energy, toroidal momentum and density of fluctuations,D is acorresponding diffusivity andV is the velocity of the inward pinch. Importantly, alldiffusivities and inward pinch velocities are decreasing functions of the shear of the electricfield α ∼ |dEr/dr|, shown in figure 1. Note that only the qualitative features of figure 1are crucial for an analysis of the transition, not the fine details of the curve.

The form of the curve shown in figure 1 emphasizes the importance of a prelude profile,existing prior to the transition. Indeed, if initially a profile of the electric field anywherein the discharge happens to be right at the pointα1 (this assumption has obviously tobe qualified by the form of profiles in existence in the L-mode), then an infinitely smalldeviation from the equilibrium profile may imply the start of a transition into the regimewith improved confinement.

Transitions into improved confinement modes B375

Figure 1. Tentative transport diffusivities versus the shear of the electric field.

Figure 2. The same as in figure 1 for a flux as a function of gradient.

Hence, employing the paradigm of shear stabilization one obtains the dependence of aflux versus a gradient shown in figure 2. The form of this dependence is reminiscent ofthe S-curves employed in the the theory of superconductivity in describing a transition ina superconductive state [9]. The physical meaning of figure 2 is simple. If a profile is onbranch I it corresponds to the L regime of confinement. In contrast, branch II describes theH-mode. Therefore, the direct transition occurs provided0 > 0th.

Among frequently asked questions pertaining to the issue of improved confinement, onemay wonder why H-modes were discovered first, long before the other types of transportbarriers such as VH-modes and all kinds of internal transport barriers were encounteredexperimentally. The answer probably lies in the sensitivity of the electric field profile tosubtle details of density and temperature profiles. Indeed, employing the formula given by

B376 M Tendler

(6) and ignoring, for simplicity, the temperature gradient one obtains

α ∼ dE

dr∼ d2 ln n

dr2∼ [−(n′/n)2+ n′′/n]. (8)

Focusing on the first term, this simple calculation indicates that for a given gradient of thedensity profile the constraintα > α1 is much less restrictive at the plasma edge, where theabsolute values of plasma density are much smaller than in the core. That is why transportbarriers are more resilient at the edge (the case for H and VH modes) than in the centre ofthe discharge. Another important point following from this argument is the utter sensitivityof the transition to a prelude profile due to the dependence on the second derivative, yieldedby the second term. In reality, there are some additional terms in (8). No significant changesare expected and the picture remains robust. This observation offers the option to trigger thetransition below the threshold value (0 < 0th). In other words, to employ thetunnelling inorder to materialize the transition. Indeed, the transition unfolds straightforwardly when theinflux exceeds a given value0 > 0th (see figure 3(a)). However, a more fine tune ‘knob’capable of triggering the transition emerges provided the density and/or temperature profilescan be dramatically altered locally, without, at the same time changing the absolute valuesof the density and temperature too much. This argument strongly favours localized sourcesand/or sinks of particles, energy and axial momentum as fine triggers of transitions intoregimes with improved confinement. Snapshots of the straightforward and the tunnellingtransitions are contrasted in figure 3. An example of the transition occurring below thresholdis shown in figure 3(b) where only a very small fraction of particles (around 2%) is addedvery locally (within the poloidal gyro-radius), at the edge of the plasma. Both transitionsunfold on approximately the same timescale. Yet, fine details of the tunnelling transitiondiffer from the transition triggered by the incoming flux. Indeed, the filamentation of regionswith improved confinement is observed shortly after the start of the transition. The bestargument corroborating this point is the fact obtained experimentally. Indeed, experimentsalong these lines were previously reported on TUMAN-3 [10]. The most recent experimenthas been carried out on TEXTOR [11], where a small pellet was injected into the bulkplasma during the radiative improved (RI) confinement mode. The quality of confinementhas been improved by 18% in response to the injection of a pellet.

5. Highlights of key H-mode work

Since there is no future without the past, it is important to review the short history of workcarried out, both experimentally and theoretically, aimed at understanding and controllingregimes with improved confinement. The record is also important because the dominantinterpretations have changed a number of times during this period, without actually statingthese changes clearly [12]. Furthermore, theoreticians often introduce special notions suchas ‘Stringer spin-up’, ‘Reynolds Stress Dynamo or Shear Reynolds Stress’ and so on. Thewide use of these terms is often confusing and complicates communications and the mutualunderstanding of the two communities, working on theory and experiments. Therefore, itappears timely to offer a brief explanation of the real meaning of the terminology.

The original discovery of the H-mode was achieved by Wagneret al on the ASDEXtokamak in 1982 [13]. Since then, the H-mode has proven to be one of the most robust andubiquitous modes of improved confinement in toroidal magnetic-fusion devices. Therefore,the physics of the L-mode to H-mode (L–H) has attracted a great deal of interest fromboth the experimental and theoretical communities. The next dramatic step forward for thedevelopment of experimental studies was the observation on the DIII-D tokamak that the

Transitions into improved confinement modes B377

(a)

(b)

Figure 3. Snapshots of transitions. (a) Evolution of the diffusion during the L–H transition0 = 0th. (b) Evolution of the diffusion coefficient during the pellet induced L–H transition.1, 2, 3 are snapshots taken at short times compared with confinement time in the L-mode withinthe barrier. Note the emergence of filaments in (b).

edgeimpurity ion poloidal velocity changes dramatically and abruptly during the transitionreported by Groebneret al [14]. The demonstration on the CCT tokamak that the H-modecan be triggered by applying internal biasing by means of an internal probe was the nextmajor step forward [15, 16]. Recently, regimes with internal transport barriers were obtainedon all major large tokamaks JET [17], TFTR [18], DIII-D [19] and JT-60 [20]. All thesesuccesses in the experimental work have benefitted from, and contributed to, the progressof the theoretical studies.

In theory, the idea that the electric field is the controlling parameter for the H-modewas brought up in [21]. The sign reversal of the poloidal rotation in the H-mode comparedto neoclassical predictions and the L-mode observed on DIII-D have led several theoriststo put forward and to advocate the ion loss mechanism as a steady-state scenario for thetransition [22, 23]. These models commanded the full attention of the community until veryrecently. Yet, six years later, in contrast to previous experiments, measurements of thepoloidal rotation velocity of the main ions were carried out, employing lines from mainspecies [24] and it was demonstrated on DIII-D that the sign of the poloidal rotation of themain ions and the impurity ions areindeed oppositeand, thesign of the poloidal rotationfor the main ions is in agreement with the neoclassical predictions. In other words, thesetheories were led to explain the wrong phenomenon.

B378 M Tendler

However, to their honour theoreticians were never satisfied with the artificial structureof the ion orbit loss theory for many reasons. First, this theory relied upon too manyphysical assumptions to occur simultaneously. It was obvious that such a coincidence ishighly improbable. Second, it did not offer a simple and harmonic picture expected froma plausible theory in physics, in general. Most importantly, in spite of the merits of theion loss theory, an attempt was made to compute fluid parameters within one poloidalgyro-radius from the separatrix, while only the kinetic approach is truly warranted. Inan important study [25], the kinetic approach has been employed for the same region inthe banana regime, yet neglecting anomalous transport. This work yielded only qualitativeresults for the value of the electric field at the separatrix. The electric field caused by the ionorbit loss has been obtained self-consistently by numerical methods for the banana regimein [26]. It was shown that the radial and poloidal electric fields emerge for a given densityprofile. Moreover, steeper density profiles imply stronger negative radial electric fields andvice versa. This value is also affected by the complicated flow pattern occurring in thescrape-off (SOL) plasmas outside the separatrix [28].

Hence, a different school of thought [27] emerged putting forward the bifurcation ofequilibrium plasma parameters as the result of an S-like dependence of heat fluxes on theneoclassical poloidal rotation velocity (therefore, also on the ion-temperature gradient inaccordance with the standard neoclassical theory) by analogy with the Landau–Ginzburgtheory of transitions into a superconductive state. However, the same experiment [24]has demonstrated that the measured value of the poloidal rotation in the H-mode differsdramatically from neoclassical predictions even after a few milliseconds following thetransition. Therefore, the need to address the issue of a torque capable of providing fora fast poloidal rotation velocity remained unanswered in [27]. Different models aiming atthe assessment of the fast poloidal rotation have been put forward adopting the idea thatthe turbulent radial transport of parallel flows might provide the poloidal momentum torqueby means of off-diagonal terms in the momentum balance [28–31]. The issue of paralleldynamics was addressed either from the neoclassical point of view [28, 29] or turbulencetheory [30, 31]. Again, it is important to bear in mind the relevance of the radial transportof parallel flows within the context of radial electric fields. Indeed, within the frameworkof neoclassical theory, parallel flows give rise to, due to the toroidal geometry to the wellknown Pfirsch–Schlueter flows and currents.

In general it is usually asserted that transport parallel to the magnetic field is in goodagreement with neoclassical theory. By contrast, cross-field transport of energy, momentumand particles are grossly anomalous. Some models put special emphasis on the subtlepoloidal dependence of anomalous cross-field transport in the momentum balance resultingin an additional flux-surface-averagehigher order products of harmonics of the radialtransport of angular momentum with parallel flow harmonics. This is the bottom lineof the notion ‘Stringer spin-up’ introduced in [29]. Off-diagonal terms (e.g. inertia) in themomentum balance stemming from flux surface average products of both anomalous radialand anomalousparallel flows were termed ‘Reynolds Stress Dynamo or Shear ReynoldsStress’ in [30, 31].

In hindsight, it turns out that the other important point ignored by most of the modelswas the impact of the ion diamagnetic drift (later found to be the major player for manyaspects of a transition, see the next section) on the electric field profile already followingfrom the dependence of neoclassical parallel flows on diamagnetic drifts. Since the electricfield at the edge within the transport barrier is well known to be negative, the diamagneticdrift is bound to reinforce this effect and therefore has the potential to contribute in the rightdirection, even in the absence of both poloidal and toroidal rotations. It should be borne

Transitions into improved confinement modes B379

in mind that diamagnetic drift was retained by the model [28] relying upon neoclassicalparallel dynamics because this drift is the only cause for the well known Pfirsch–Schluetercurrent.

In conclusion, although a complete detailed theory of confinement improvements doesexist, initial works based on a widely varying range of ideas and assumptions have mergedrecently into a more coherent and consistent path of development.

6. Different scenarios of L–H transitions

For some time there has been a lively debate between JET and DIII-D teams about therole of the poloidal rotation velocity in triggering the L–H transition. In more detail, theDIII-D team reported [14, 24] that the poloidal rotation changes dramatically during the L–Htransition while the measurements carried out on JET [32] did not record any significantchanges in poloidal rotation. The way to reconcile these two seemingly contradicting resultsis to consider the formation of transport barriers in detail. In general, the method of studyingthe evolution of transport barriers employs the system of balance equations (includingparticle, heat, toroidal and parallel momentum, turbulence) with diffusivities consistingof a small part independent of theE × B shear (ideally sub-neoclassical, collisional) andthe much larger part, which is strongly reduced as this shear is increased. Experimentally,there is a wide variety of range of opinions on the cause of the transition [33] because onsome machines such as JET the poloidal velocity at the edge does not show any changewithin the accuracy of the measurements. Therefore it is concluded that any changes ofEr are unlikely because the transport change would have to occur on a timescale shorterthan the measurement interval. In contrast, the measurements carried out on DIII-D havedemonstrated the significant change in the perpendicular velocity even 5 ms after thetransition.

Figure 4. Scenarios of two L–H transitions. Full curves are transitions triggered by poloidalrotation; broken curves are transitions triggered by density depletions,Vθ = 0.

Here, it is demonstrated that it is possible to reconcile these apparently contradictingdata by following two different scenarios, both resulting in the transition. Indeed, the radialelectric fieldEr can change veryfast due to the ballistic nature of the governing equations.

B380 M Tendler

Figure 5. Propagation of the barrier front. Random walk normalized to the barrier width againsttime in ms.

In detail, the transition is obtained by either changing the boundary and initial conditionsfor the poloidal rotation and consequently the electric field on the separatrix (full curves infigure 4) or adjusting the prelude density profile by self-consistent variation of a particlesource (by invoking a surplus or a depletion locally, in the vicinity of the separatrix) inorder to match the pointα1 necessary for the shear stabilization (broken curves in figure 4).Indeed focusing on the latter scenario first, poloidal rotation is intentionally kept to zero andfor a given source function in particle, energy or toroidal momentum balances the electricfield and the shearing rate resulting from it increases due to diamagnetic drift after reachingthe threshold for the nonlinear decorrelation of turbulent eddies, thereby locally decreasingdiffusivities. It is assumed that prelude profiles in the L-mode are close to the threshold forshear stabilization. Thereafter, the avalanche of propogation of the transport barrier unfoldson a very fast timescale of the order of less than tenths of a millisecond (see figure 5). Thepropogation within the barrier is very fast (much faster than the diffusion of the front shownby the linear curve) because of the ballistic nature of the solution. Correlations are carriedup and down the barrier. Here, the front of reduced diffusivity1x2 propogates inwards.The diffusion coefficient drops dramatically, thereby causing density and/or temperatureprofiles to steepen. The formation of the barrier is terminated because the shearing ratedecreases as the square of the local density. Once the barrier is formed, it affects thedensity profile within the entire plasma body on a much slower timescale. Density profilesbecome more peaked because of the pedestal whereas transport coefficients are suppressedwithin the barrier.

The other scenario (full curves) occurring due to the poloidal rotation may be envisagedif the prelude density profile is not so close to the critical shear levelα1. Then, of course,a small perturbation in density cannot trigger the transition. However, the transition canmaterialize provided the electric field is amplified due to the increased poloidal rotation overthe neoclassical level at the separatrix. For example, assuming that theE × B velocity atthe separatrix is of the order of the poloidal sound speedV0 ≈ 2Cs the transition unfolds.The profiles of the poloidal rotation velocity and the density are evolved and broaden quitesignificantly, thereby reducing transport coefficients within the barrier (see again the full

Transitions into improved confinement modes B381

curves in figure 4). The remarkable result is that at the end of the day the diamagnetic driftoverrides in importance the poloidal rotation (in reality, very quickly) and the evolutionof both scenarios with characteristically grossly different initial conditions merges into thesame form for profiles of the diffusion coefficient, convective velocity, density and theelectric field.

7. Conclusions

Discharges with transport barriers are a fascinating laboratory for turbulence studies since thedominant modes are suppressed and subdominant transport mechanisms can be investigatedall the way down to neoclassical levels and beyond. This calls for further improvements ofthe standard neoclassical theory with the emphasis on the so far academic effects of smallaspect-ratio portions of the discharge which are important for internal transport barriers,the squeezing of banana orbits by the varying electric fields [25] etc. To adopt power fluxto the separatrix as the trigger of the L–H transition appears from the theoretical point ofview a simplistic ‘brute force’ solution to the issue of the confinement control. The levelof understanding of the physical fundamentals has made sufficient progress to guide thepresent experiments and enable an extrapolation of trends to future devices. Although acomplete detailed theory of confinement improvements does not exist yet, initial work ona widely varying range of ideas and assumptions has recently merged in a more coherentand consistent path of development, mutually enriching and complementing each other. Afew outstanding problems remain to be addressed on the basis of first principles. Zero-dimensional models [34, 35] are inadequate to address the fundamentals of the dynamics ofthe transitions because of the plurality of scenarios and their sensitivity to the fine details ofprofiles. Specifically, a rigorous equation which is adequate to describe the propogation ofturbulence including the effect of theE × B shear has to be found. The progress justifiesfuture work in this area.

Acknowledgment

The author wishes to thank Professor R Hazeltine for comments.

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