Chapter 2: Plasma confinement and transport
ITER Physics Expert Groups on Confinement andTransport and Confinement Modelling and Database∗
ITER Physics Basis Editors†
ITER EDA, Naka Joint Work Site, Mukouyama,Naka-machi, Naka-gun, Ibaraki-ken, Japan
Abstract. Physics knowledge in plasma confinement and transport relevant to design of a reactor-scale
tokamak is reviewed and methodologies for projecting confinement properties to ITER are provided.
Theoretical approaches to describing a turbulent plasma transport in a tokamak are outlined andphenomenology of major energy confinement regimes observed in tokamaks, including those with edge
and internal transport barriers, is described. The chapter is focused on the energy confinement in thehigh confinement regime (H-mode) with the edge localized MHD modes, the basic operational regimeof ITER. Three approaches are being pursued: (i) derivation of empirical global scaling laws; (ii)
non-dimensionally similar studies; and (iii) one dimensional transport modelling codes, with the first
approach recommended as the most robust at the present time. Special attention is paid to analysis
of uncertainties in confinement predictions. Empirical scaling relations for projecting the L-mode to
H-mode power threshold based on regression analysis of an extensive database are discussed. Particle
and toroidal momentum confinement and their relation to energy confinement are reviewed.

Contents∗ ITER Physics Expert Group on Confinement and Transport: M. Wakatani (Kyoto U., chair), V.S. Mukhovatov (ITER
JCT, co-chair), K.H. Burrell (General Atomics), J.W. Connor (UKAEA, Culham), J.G. Cordey (JET), Yu.V. Esipchuk(Kurchatov Inst.), X. Garbet (CEA, Cadarache), S.V. Lebedev (Ioffe Inst.), M. Mori (JAERI), K. Toi (NIFS), S.M. Wolfe(MIT). ITER Physics Expert Group on Confinement Modelling and Database: J.G. Cordey (JET, chair), D. Boucher(ITER JCT, co-chair), A.N. Chudnovskii (Kurchatov Inst.),J.W. Connor (UKAEA, Culham), J.C. DeBoo (General Atomics), W.A. Houlberg (ORNL), S.M. Kaye (PPPL), Y. Miura(JAERI), Y. Ogawa (U. of Tokyo), M.V. Osipenko (Kurchatov Inst.), F. Ryter (IPP Garching), T. Takizuka (JAERI).
Additional contributing authors: G. Bateman (Lehigh U.), G.
Bracco (ENEA, Frascati), D.J. Campbell (NET Garching),
T.N. Carlstrom (General Atomics), Yu.N. Dnestrovski (Kurchatov Inst.), T. Fujita (JAERI), A. Fukuyama (Okayama U.),
P. Gohil (General Atomics), M.J. Greenwald (MIT), T. Hoang(CEA, Cadarache), G.M.D. Hogeweij (FOM Inst.), A.E. Hubbard (MIT), O.J.W.F. Kardaun (IPP, Garching), J.E. Kinsey (Lehigh U.), J.A. Konings (General Atomics), M. Marinucci (ENEA, Frascati), Y.R. Martin (CRPP/EPFL, Lausanne), D.R. Mikkelsen (PPPL), J. Ongena (EERM-KMS),
T.H. Osborne (General Atomics), F.W. Perkins (ITER JCT),
A.R. Polevoi (Kurchatov Inst.), M.N. Rosenbluth (ITERJCT), D.P. Schissel (General Atomics), J.A. Snipes (MIT),
P.M. Stubberfield (JET), E.J. Synakowski (PPPL), A. Taroni(JET), K. Thomsen (JET), M.F. Turner (UKAE, Culham),
N.A. Uckan (ORNL), M. Valovic (UKAE, Culham), G. Vlad(ENEA, Frascati), R.E. Waltz (General Atomics), J. Weiland(Chalmers U.), H. Zohm (Stuttgart U. ).
† ITER Physics Basis Editors: F.W. Perkins (ITER JCT),
D.E. Post (ITER JCT), N.A. Uckan (ORNL), M. Azumi(JAERI), D.J. Campbell (NET), N. Ivanov (RRC-Kurchatov),
N.R. Sauthoff (PPPL), M. Wakatani (Kyoto Univ.). Additional contributing editors: W.M. Nevins (LLNL), M. Shimada(JAERI), J. Van Dam (Univ. Texas).

Nuclear Fusion, Vol. 39, No. 12

1. Introduction

2176

2. Mechanisms of transport in atokamak2178

3. Experimental description ofconfinement and transport2184
3.1. Introduction . . . . . . . . . . . . . . . 2184
3.2. General results for ohmic and L-mode 2185
3.3. Regimes with edge transport barrier(H-mode) and recommended regime
for ITER . . . . . . . . . . . . . . . . 2186
3.4. Regimes with internal transport barriers2187
3.4.1. Barriers
associatedwithreversed or weak magnetic shear2188
3.4.2. Other improved core confinement regimes without edge
radiation . . . . . . . . . . . . 2189
3.5. Enhanced confinement with edge radiation . . . . . . . . . . . . . . . . . . . 2190
4. L–H and H–L transition physics
2192
4.1. Physical processes of transition . . . . 2192
4.2. Edge pedestal . . . . . . . . . . . . . . 2193
4.3. Power threshold scaling . . . . . . . . 2195
5. Impact of global instabilities ontransport
2198
5.1. Sawteeth . . . . . . . . . . . . . . . . . 2198
5.2. Edge localized modes . . . . . . . . . . 2200

c 1999, IAEA, Vienna

2175

ITER Physics Basis
6. Global energy confinement scalings 2201
6.1. Introduction . . . . . . . . . . . . . . . 2201
6.2. H-mode global confinement database . 2202
6.3. Power law scaling expressions . . . . . 2203
6.3.1. ELMy H-mode thermal confinement scalings . . . . . . . . 2204
6.3.2. ELM free H-mode thermal
confinement scaling . . . . . . . 2205
6.3.3. L-mode thermal confinement
scaling . . . . . . . . . . . . . . 2205
6.4. Point and interval estimation for theconfinement time in ITER . . . . . . . 2206
7. Scaling studies with similar dimensionless physics parameters2209
7.1. Basics . . . . . . . . . . . . . . . . . . 2209
7.2. Discharges with identical dimensionless parameters . . . . . . . . . . . . . 2211
7.3. Results of experiments with similardimensionless parameters . . . . . . . 2211
7.4. Limitations on transport scaling studies with similar dimensionless parameters . . . . . . . . . . . . . . . . . . . 2213
7.5. Summary . . . . . . . . . . . . . . . . 2215
8. One dimensional transport models
2215
8.1. Introduction . . . . . . . . . . . . . . . 2215
8.2. Theoretical and semi-empirical transport models . . . . . . . . . . . . . . . 2216
8.3. Plasma profile database . . . . . . . . 2219
8.4. Results of one dimensional modelling
tests . . . . . . . . . . . . . . . . . . . 2219
8.5. Predictions of transport models for
ITER . . . . . . . . . . . . . . . . . . 2222
9. Particle transport: hydrogen,helium, impurities

2225

10.Momentum confinement

2228

11.Summary

2231

A ITER energy confinement
projection
2232
A1. Point and interval estimation basedon log-linear and log-non-linear scaling expressions . . . . . . . . . . . . . 2232
A2. Framework for interval estimation . . 2235

1.

Introduction

The most developed concept for realizing continuous energy (electricity) production from fusion
2176

reactions is the tokamak, based on an axisymmetric toroidal plasma. In recent years there has been
considerable development of databases and accumulation of knowledge of the behaviour of tokamak plasmas around the world, and these make it possible todesign an experimental fusion reactor for energy production. However, a degree of uncertainty still exists
in predicting the confinement properties and plasma
performance in such a device.
A precise theory of the classical collisional transport losses has been developed. Since this does notcompletely explain the transport processes acrossmagnetic surfaces (as discussed in Section 2), additional processes driven by plasma turbulence are
required to be taken into account. Significant theoretical efforts are being devoted to understanding thecross-field transport in tokamaks due to the turbulence, and a few models are broadly consistent withpresent experiments. On the other hand, since tokamaks with a range of sizes, operating parametersand heating powers have been constructed, empirical scaling laws derived from these are useful for
predicting plasma performance in any new device.
Furthermore, the empirical scalings serve as a benchmark for theoretical models. One expects that predictions with such scaling laws will be improved if
one imposes dimensional constraints on the form of
these laws in the scaling studies. It is also recognizedthat transport codes solving radial transport equations numerically are also useful for obtaining quantitative predictions. As a result, three approaches
are being pursued at the moment for providing predictions for confinement, these are: (a) derivation of
empirical global scaling laws; (b) non-dimensionallysimilar studies; and (c) one dimensional transportmodelling codes.
The main strengths of the empirical energy confinement scaling method (described in Section 6) are
its simplicity and the fact that all of the physicalprocesses are contained within the data. Its main
weakness is that the modelling of the energy confinement time τE by a simple log-linear form, oreven by more sophisticated forms, can only, at best,
be a very approximate description of the physicalprocesses taking place, since no knowledge of theheating, temperature or density profiles, or atomic
physics for that matter, is built into the analysis. The
log-linear form itself is equivalent to assuming thata single turbulence mechanism with one scale size is
responsible for the transport. This seems unlikely tobe the case for the ELMy H-mode, the operationalmode foreseen for ITER, where the core region may
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
be dominated by short wavelength turbulence of the
gyroBohm type, and the behaviour in the edge region
is possibly determined by MHD events such as theedge localized modes (ELMs). These two processes
will scale differently with the main scaling parameter ρ∗ , the normalized Larmor radius ( ≡ ρi /a).
There is also the possibility that there are hidden
parameters that are not being taken into accountin the analysis. One parameter that may affect theconfinement in some situations, but which is notpresently being taken into account, is the edge neutral pressure; another possible important parameter is the Mach number of the toroidal rotation M( ≡ Vφ /cs ). Both these parameters are currently
being added to the database, but their effect on the
ITER predictions is, as yet, unclear. The degradationof confinement associated with proximity to operational limits, such as the Greenwald density limit, βlimit and H-mode power threshold, is still an area ofactive studies and is not yet well quantified by theavailable database.
One can assess the statistical uncertainty in the
ITER energy confinement time τITER using the techniques that are described in Section 6.4. The 95%
interval estimate for the prediction of τITER from the
ELMy H-mode log-linear scaling Eq. (26) is τITER =4.4–6.8 s. With this narrow uncertainty interval,
ITER would definitely ignite, although operation at
higher current or higher density might be required
to preserve ignition at the lower end of this interval. A larger interval, τITER = 3.5–8.0 s, has been
derived by taking various non-log-linear forms andother considerations into account. At the lower endof this range the minimum Q ( ≡ thermonuclear
power/input power) would be above 6.
The dimensionless physics parameter scaling technique (described in Section 7) overcomes one of
the shortcomings of the energy confinement scaling
approach, namely the influence of the plasma profiles on the confinement. In fact, in the simple caseof a single dominant turbulence mechanism, one canobtain directly, by a simple projection, from thoseof current experiments, the temperature and density
profiles in ITER, provided the source profiles of heatand particles are the same. A further key advantageof this approach is that the MHD stability properties(i.e. the ELM and sawtooth behaviour, discussed in
Section 5) would be the same in ITER as those of
the similarity discharge in the present device.
The technique does not entirely overcome one of
the major possible shortcomings of the log-linear
energy confinement scaling approach, that is the exisNuclear Fusion, Vol. 39, No. 12 (1999)

tence of different types of turbulence dominating theedge and core regions. In this situation, a more complex projection of the profiles of a current device to
ITER would be required.
Another concern with this technique is whether
all of the key dimensionless parameters have been
identified. Several theoretical papers have proposedthat the turbulence in a tokamak can be quenched byshear in the toroidal flow. This implies that the Mach
number M is an important dimensionless parameter.
This is certainly not being kept constant in the ρ∗
scaling experiments completed so far, which use neutral beam injection heating. Clearly, experiments inwhich the Mach number is varied around the values anticipated for ITER, while other dimensionlessparameters (i.e. ρ∗ , β and ν ∗ ) are kept fixed, arerequired. (Toroidal momentum confinement is discussed in Section 10.)
The range of uncertainty of the ITER prediction
using this technique is quite large for extrapolationfrom a single machine (see Table 8). This uncertainty
comes about from the narrow range in ρ∗ available ona given machine (typically a factor of 2) comparedto the distance in ρ∗ (a factor of 5.5 in JET) by
which one has to extrapolate to ITER. To reduce
this uncertainty, a database with joint ρ∗ scans fromdifferent devices will be required.
Work on the full one dimensional modelling
approach is progressing well and there are currently
several one dimensional models, of both the purely
theoretical and the semi-empirical type, which arereasonably successful in reproducing the temperature
profiles in the core (Section 8.4). The edge region
0.9 < r/a < 1 is still a problem, since, at the present
time, there is no tested model for this region. Coupled with the strong sensitivity of some of the ‘stiffer’
one dimensional models to the edge conditions, thismakes the prediction of the performance of ITER
using this technique rather difficult, as is shown in
Section 8.5. An additional uncertainty is the differing manners in which the models treat stabilizationof the turbulence by E × B velocity shear.
The three techniques do at least have overlapping
confidence intervals, and efforts need to be directedin the next few years at reducing the confidence interval of each technique.
There is a further uncertainty associated with theuse of the ELMy H-mode as the operating regime,in that there is a power threshold that has to be
exceeded before the ELMy H-mode is achieved. Thisis discussed in detail in Section 4; the presently projected power in ITER would appear to be sufficient2177

ITER Physics Basisto exceed the threshold, provided the transition to
the H-mode is achieved at a low density.
Besides the ELMy H-mode, several otherimproved confinement modes have already been realized, as discussed in Section 3. One of the mostpromising ones is the negative, or reversed, shear
configuration, which involves an internal transportbarrier. It can be used to realize a continuous operation of the tokamak, since there is a significant fraction of bootstrap current with a self-consistent profile. The RI-mode also has attractive features such asa high confinement at very high density, approaching or exceeding the Greenwald density limit, withthe major portion of plasma energy radiated fromplasma periphery.
Furthermore, in order to achieve continuous operation, particle control is essential, in addition to heatexhaust. Thus, for DT burning plasmas, helium ash
exhaust is an important requirement; the helium
density as well as the impurity ion density must be
kept sufficiently low in order to minimize the dilution of fuel ions. To solve these problems, the particle
transport properties in fusion plasmas must be characterized; this is discussed in Section 9. In particular,
the sawtooth seems crucial for controlling the helium
ash and impurities in the central region and ELMs
at the plasma edge; these are discussed in Section 5.
Finally, it should be noted that there are several
interesting, but unresolved, challenges to our understanding of transport, such as profile resilience, theisotope effect, non-local effects, fast time-scale phenomena, etc. A better understanding of these processes could contribute to improving the predictionof the fusion performance of reactor grade plasmas.

2.

Mechanisms of transport
in a tokamak

There is a precisely formulated theoretical modelfor the transport of plasma particles and energy
across the magnetic surfaces due to Coulomb collisions in a stable, quiescent toroidal confinement system. This process, which takes account of the particle orbits in an axisymmetric inhomogenous toroidal
magnetic field [1] and leads to higher levels of collisional transport than in a cylindrical plasma (whichis called classical transport), is known as neoclassical transport [2–5], and will always be present in
a tokamak. The classical diffusion coefficient can be
estimated by Dc ≈ νei ρ2e for plasma particles, andthe ion thermal diffusivity by χi ≈ νii ρ2i , which2178

is greater than both Dc and the electron thermal
diffusivity χe ≈ νee ρ2e by a factor of (mi /me )1/2 .
Here, ν is a 90◦ collision frequency and ρ is a Larmor radius, ρ = Vth /Ω, which is proportional to theratio of a thermal speed Vth to a cyclotron frequencyΩ = eB/mc, where B = |B| is the magnetic fieldstrength and m is the particle mass. The neoclassical transport levels exceed the classical ones bygeometrical factors: q 2 ε−3/2 in the low collision frequency ‘banana’ regime (ν ∗ < 1.0) and q 2 in the collisional fluid limit, as a result of the toroidal geometry. Here, ε = r/R is the inverse aspect ratio, with r
and R being the minor and major radii of the magnetic surface, and q ≈ rBT /RBp is the safety factor,
where BT and Bp are the toroidal and poloidal magnetic field components, respectively. The collisionality parameter is ν ∗ ≡ νeff /ωb , where νeff ≡ νei /ε isthe effective collision frequency for particle detrapping, and ωb ≈ ε1/2 Vth /(Rq) is the trapped particle
averaged bounce frequency.
In spite of its larger value, neoclassical transport cannot normally explain the observed perpendicular transport in a tokamak. For the usual low
confinement regime (L-regime) in auxiliary heatedtokamaks, the neoclassical transport theory predicts
confinement times longer than those experimentallymeasured by up to two orders of magnitude for theelectrons and one order of magnitude for the ions.
Plasma turbulence is probably a dominating effectin causing plasma cross-field transport. However, forsome improved confinement regimes and in localized
regions of the plasma, transport can be reduced to
the neoclassical level [6–8].
The objective of the transport calculations is thedetermination of the magnitudes and radial profilesof the plasma parameters (such as density, temperature, etc.). To do so, it is useful to formulate thetransport problem in a macroscopic way [9]. On
transport time-scales, a fluid representation is generally used. Axisymmetry is assumed and plasma
observables are averaged on a flux surface over thepoloidal angle. In this way, a set of one dimensional
fluid like equations can be derived to represent theevolution of the density, momentum and energy foreach plasma species:
1 ∂
∂na
=−
(rΓa ) + Sa
∂t
r ∂r
1 ∂
∂Via
=−
(rΦia ) + Mia
ma na
∂t
r ∂r
1 ∂
∂Ta
3
5
2 na ∂t = − r ∂r [r(Qa + 2 Γa Ta )] + Pa

(1)
(2)
(3)

Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
and the poloidal magnetic field, Bp , diffusion equation:∂η ∂(rBp )∂∂Bp=(ηJb ).
(4)−∂t∂r µ0 r ∂r∂rHere, the subindex a indicates the particle species,
the subindex i indicates the components of the velocity vectors, Γa , Φa and Qa are the particle, momentum and energy fluxes, and Sa , Ma and Pa are thesource, sink and transfer terms for particles, momentum, and energy. The radius r is a generalized minor
radius that labels each magnetic surface.
In the magnetic field diffusion equation, Eq. (4), ηis the parallel resistivity and Jb is the bootstrap current predicted by neoclassical theory. To understandthe origin of this neoclassical effect, we observe thatthe circulating electrons carry the parallel current
in response to the toroidal electric field and othercurrent drive inputs. This parallel current response
is determined by the balance between the collisional
friction of circulating electrons with ions and the viscous drag on their poloidal flow produced by collisions with the trapped electrons. The predictedparallel resistivity, η, agrees well with experimental
results. Another element of the parallel transport,
the bootstrap current [10],
√
ε dp
Jb ≈
Bp dr
has also been shown to be in good agreement withtheoretical predictions [11, 12]. Therefore, the theoretical picture of neoclassical parallel transport isconsistent with experiment.
The fluxes contain all the dynamic informationon the transport processes. While neoclassical theory gives an explicit expression for them, there is
not a general form for turbulent transport. In practice, these fluxes have been parametrized by analogy
with the structure given by neoclassical theory. For
instance, the simplest assumption of diffusive fluxes
would give
∂na
(5)
∂r
∂Via
(6)
Φia = −χia
∂r
∂Ta
.
(7)
Qa = −χa∂r
In practice, each of the fluxes depends on all thethermodynamic forces (profile gradients) and a fullmatrix of fluxes must be considered, including nondiffusive and non-diagonal terms. For advanced tokamak scenarios, it is important to include the radialΓa = −Da

Nuclear Fusion, Vol. 39, No. 12 (1999)

electric field [13] in the transport model, since thegradient of the radial electric field is predicted toreduce transport coefficients [14], and the coupling of
its evolution to the set of equations given above. The
calculation of the plasma profiles using Eqns (1)–(4)requires the knowledge of these diffusivities and also
the appropriate boundary conditions, particularly
as the predictions of some ‘stiff’ transport models
involving marginally stable profiles can be sensitive
to these [15]. The boundary conditions will be discussed at the end of this Section and in Section 4.2.
The determination of the diffusivities has long been
the goal of transport studies. There are many existing reviews on this topic [16–23] and, here, we give
only a very simplified overview. In understanding thetransport coefficients, there are three possible waysof making progress: (1) theoretical; (2) numericalsimulation; and (3) experimental. We discuss theseapproaches in the remainder of this section.
We have seen that the transport theory basedon particle collisions can incorporate the geometryof the tokamak magnetic system, but neoclassicaltheory still assumes that the plasma is in equilibrium and is axisymmetric. Real tokamak plasmas
always show the presence of a broad spectrum of fluctuations, e.g. in plasma density, temperature, andelectromagnetic fields [17, 24]; thus, real tokamak
plasmas are turbulent and intrinsically three dimensional. The turbulent fluctuations give rise to transport across the equilibrium magnetic surfaces and itis necessary to incorporate their effect in a comprehensive transport theory. From the theoretical pointof view, most of the instabilities that we think areresponsible for the observed plasma turbulence havea very small component of the wavenumber vector
parallel to the magnetic field, compared to the perpendicular component. That is, most of the turbulent
eddies are quasi-perpendicular to the toroidal magnetic field. Therefore, we can expect that turbulence
dominates perpendicular transport, but the influenceof the plasma turbulence on the parallel transport israther small, as experiment confirms.
Hydrodynamic turbulence is an unresolved complex physical problem [25]. Plasma turbulence is evenmore complex. Since we can describe plasmas asa combination of several fluids (an electron fluid
plus ion fluids), we can imagine plasma turbulenceas the result of these multiple turbulent fluids coupled through electromagnetic, friction, and energy
exchange effects. It is not surprising, therefore, that
there is not yet a theory or even a comprehensive
approach to this problem.
2179

ITER Physics Basis
Plasma turbulence differs in many ways from fluid
turbulence, starting from the structure of sources andsinks. Plasma turbulence is driven by the free energy
sources of the many plasma micro-instabilities, essentially the gradients of the density and temperature[26–29]. Thus, in the core of the plasma there aremicro-instabilities driven by the ion temperature gradient (ITG); these are the electrostatic ITG drift
modes and trapped ion modes. These modes areoften characterized by a value ηi = d(ln Ti )/d(ln n),
and referred to as ‘ηi modes’. Others are driven
by the electron temperature gradient: electrostatic
trapped electron and shorter wavelength electromagnetic ηe drift modes, and micro-tearing modes.
There are also fluid like instabilities driven by pressure gradients: the current diffusive ballooning [30]and neoclassical tearing modes [31–33]. At a collisional plasma edge, there is a range of fluid instabilities driven by gradients in pressure, resistivity
and current. In Table 1, we give a summary viewof plasma instabilities that are possible underlying
mechanisms for plasma turbulence in tokamaks. In
the edge region, it is possible that atomic physics
processes can play a part, so that in addition to thoseinstabilities shown in Table 1, another candidate at
the plasma edge is the ionization instability [34–36],
even if the validity of the linear theories has beenquestioned [37]. The plasma drives couple directly,
through these instabilities, to a broad range of turbulence scale lengths. Therefore, in general, one cannot
separate between the driving and inertial ranges, as
is done in hydrodynamic turbulence. The dissipative
terms do not have the simple form of a collisional
viscosity. The complicated magnetic geometry of atokamak adds even more difficulties to this problem.
In spite of this, research on tokamak plasma turbulence has proceeded at several levels: (1) renormalization of simple sets of equations modelling the turbulence; (2) scale invariance and dimensional analysis
techniques; and (3) numerical solution of turbulencemodels.
The renormalization approach starts from perturbation theory. When summing the perturbation
series, the characteristic time-scale associated with
the non-linear processes, τc , must be determined ina self-consistent manner [38, 39]. If it is approximated by the linear time-scale (inverse of the linear
growth rate, γ, of the appropriate plasma instability), conservation laws are violated. The renormalization approach introduces a recursive determinationof the non-linear time-scale (in a way similar to thedetermination of the electron mass in quantum elec2180

trodynamics). The most common closure scheme isthe direct interaction approximation. The adequacyof this closure scheme has been tested with numerical calculations within the accuracy achievable withpresent numerical resolution. The determination of
the basic turbulence scale length, ∆T (mixing length
like length-scale) [40], and the decorrelation time, τc ,allows the construction of an effective turbulent diffusivity, DT ≈ ∆2T τc . In practice, to apply renormalization techniques, the problem must be reduced toa one, or at most two, equation model. Otherwise itis practically impossible to solve the problem analytically, or even numerically. Analytical approaches
are also limited because the approximations needed
to solve the equations restrict the applicability ofthe solutions to a small range of plasma parameters,
while experiments cover, in general, a broad parameter range. However, these techniques are essential
for making progress in understanding. In some simple models [41–43], it has been shown that ∆T is
proportional to the inverse perpendicular wavenumber, k⊥ , and the following form for the turbulence2, where Λ isdiffusivity can be adopted DT ≈ Λγ/k⊥a weak function (logarithmic or just a constant) of
the plasma parameters [44, 45]. This formulation of
the turbulent diffusivities has been adopted in an ad
hoc fashion in some of the transport models.
The scale invariance approach [46, 47] is an alternative approach for the determination of the transport coefficients. It is based on the idea of intermediate asymptotics [48], and, in some simple models,
it has been shown to be equivalent to the renormalization group technique [49]. In this approach, one
identifies all the independent transformations underwhich the dynamical equations are invariant. It has
been applied to several types of turbulence dynamics
and it can be a powerful tool in identifying the main
dependences of the diffusivities. The minimal formof the scale invariance approach is the dimensional
analysis that will be discussed in the context of theempirical determination of the transport coefficients.
Another analytical approach invokes marginal stability criteria, e.g. [50–52]. Yet another approach is thestudy of the saturation of turbulence spectra by nonlinear plasma dynamics [53–55].
The numerical solution of the primitive equations
allows us to cover a broader parameter range than by
the analytical approaches. It is a promising method
when computer capabilities allow calculations with
sufficient resolution. The numerical calculations canbe carried out at different levels as follows.
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
Table 1. Main instabilities that contribute to the anomalous transport losses in tokamaks [29] (ηj = density
scale length/temperature scale length for species j).

Group

Instability

Source offree energy

Ion
Instabilities

ηi modes∇Ti

Slab modes
Toroidal modes
Trapped ion modes

Electron Drift Waves∇ne

Slab modes
Toroidal modes

Dissipative trappedelectron modes∇Teεω < νe ≤ ε3/2 Vthe /qR
εn q < k⊥ ρs ≤ νe Ln /εcs

Collisionless trappedelectron modes∇Teνe < εω ≤ ε3/2 Vthe /qR
ε n q < k⊥ ρ s ≤ 1

ηe modes∇Te

EM drift waves∇ne

Resistive
ballooning modes∇P

Current diffusive
ballooning modes∇P

Electroninstabilities

Fluid likeinstabilities

Subspecies

Slab modes
Toroidal modes

Propertiesω ≤ ω∗i
ηi > ηic
LTi /R < (LTi /R)critω ≈ ω∗eωpe /c < k⊥ < ρ−1ekk Vthe , ωbe < ω < ω∗eω ≈ ω∗e , k⊥ ρs ≤ 1

Fast modes
Slow modesω ≈ ω∗ekk Vthe < ω
kk Vthe < ωHere ω, kk and k⊥ are the frequency and parallel and perpendicular wavenumbers of the instability, respectively;ω∗j is the diamagnetic frequency of species j, ωpe the electron plasma frequency, ρs the ion Larmor radius at theelectron temperature, cs the sound speed, LT the temperature scale length of species j, εn the ratio of density
scale length Ln to major radius R and ηj = Ln /LTj .

(1) Particle calculations. In this case, the motionof individual particles in the plasma is followed in time. This approach requires considerable computational resources. A great deal ofprogress has been made by using the gyrokinetic approach. In this approach, the particle
motion is averaged over the fastest time-scale,Ω−1 , the inverse cyclotron frequency.
(2) Fluid calculations. Moments of the distributionfunction for each plasma species are followedin time as fluid quantities. A closure scheme isneeded to limit the number of moments. Whenthe closure scheme incorporates Landau damping, the approach is called gyrofluid [56]. These
are absent from reactive fluid models [57].
Considerable progress has been made in recent
years through gyrokinetic [58, 59] and gyrofluid [60]calculations, and it has been shown that these calculations can be used for the determination of thescaling of ∆T and τc with plasma parameters [15].
At present, there is not good agreement between
these two numerical models; active work is underNuclear Fusion, Vol. 39, No. 12 (1999)

way to understand and resolve the reasons for thedifferences.
The turbulent transport picture seems to fit someof the qualitative features of perpendicular transport
in a tokamak. To provide a sound basis for this mechanism, research effort has been concentrated on identifying the connection between the observed plasma
fluctuations and transport. In doing so, it is convenient to distinguish between two plasma regions:the core and the edge. At the plasma edge, the fluctuations are, in general, large (of the order of theequilibrium quantities). They are dominantly electrostatic, unless the plasma pressure approaches theideal magnetohydrodynamic (MHD) stability limit,
and, because of the low plasma temperature, detailed
measurements can be made with probes. From thesemeasurements, it has been inferred that the edge particle loss can be explained by fluctuation induced
transport [24].
Core fluctuation levels are low (a few per centof the equilibrium values of density, etc.) [17, 24];
their measurement by non-intrusive techniques iscomplicated. Several of these techniques are used:
2181

ITER Physics Basisfor instance, beam emission spectroscopy [61], reflectometry [62–64], and correlation measurements ofelectron cyclotron emission (ECE) signals [65, 66].
The first two measure density fluctuation levels andthe third electron temperature fluctuations; therefore, we have no knowledge of the turbulence driven
fluxes. The heavy ion beam probe [67] can determine
both density and potential fluctuations, and, therefore, the particle flux. However, its use has been limited. The fluctuation measurements indicate that the
spectrum is peaked at long wavelengths and radial
correlations lengths are of the order of a centimetreor less. In some experiments, fluctuation levels seemto correlate well with global, and even local, transport [68], but without determination of the fluxes,
no unambiguous correlation with transport can beestablished. The turbulence characteristics are consistent with calculations of ITG driven turbulence,but comparisons have only been carried out for alimited number of discharges. Plasma core turbulence may have a significant magnetic component.
Although there are measurements of the magnetic
field fluctuations at a fixed wavelength [69], there is
not yet a direct measurement of this component fora broad range of wavenumbers, and its role in transport is controversial [70]. Recently, other fluctuation
measurement techniques have become available thatenable us to measure ion temperature and parallel
velocity fluctuations [68]. Hopefully, these measurements will help in clarifying the plasma turbulence
drive and will shed some light on turbulence induced
energy transport.
The turbulence driven transport theory has also
been effective in the interpretation of improved confinement regimes. For instance, there is much theoretical and experimental evidence that E × B velocity shear (i.e. radially varying profile of the E × B
plasma flow) can reduce turbulent transport, as will
be discussed in Sections 3.3 and 4.1. The theories
seem consistent with the observation of a causal correlation between the buildup of the sheared radial
electric field and the suppression of turbulence [71].
Some of the limitations of the transport based on
turbulence are discussed at the end of this section
and in Section 8.
Since fully convincing transport predictions cannot yet be made from first principles, it is necessary
to use empirical approaches to predictive transport.
Even in this case, the transport coefficients cannot
be arbitrarily chosen. There are dimensional constraints that follow from invariance principles [72,73],
discussed more fully in Section 7. These constraints
2182

reduce the number of experiments to be performed
in order to determine these coefficients. To establish
these constraints, we need to identify the relevant
invariance principle and dimensionless parameters.
This can only be done within some given theoreticalframework. The broader the framework, the less constrained are the transport coefficients. On strict theoretical grounds, even assuming transport is due toquasi-neutral plasma physics processes, the numberof dimensionless parameters for a confined plasma islarge, up to 19 have been identified [74]. They include
plasma physics parameters such as β, the ratio of theplasma kinetic pressure to the magnetic pressure, thecollisionality, ν ∗ ≡ νeff /ωb , and the normalized ion
Larmor radius, ρ∗ ≡ ρi /a. There are also parametersdescribing the magnetic field geometry, such as thesafety factor q, aspect ratio A ≡ R/a, the ellipticity
κ, and triangularity δ of the plasma cross-section,and parameters representing the plasma composition: Te /Ti , me /mi , Zeff , etc. For a local diffusion
coefficient, we have to include the parameters related
to plasma profiles, such as the ratios of scale lengths,
LT /R, Ln /R, etc. From dimensional arguments, andassuming a power law dependence on ρ∗ , the diffusivities can be expressed in the following form:
D = cs ρs (ρ∗ )α F (ν ∗ , β, q, A, κ, δ, LT /R,
Ln /R, . . . , me /mi , Te /Ti ) (8)p
Te /mi is the sound speed, ρs =where cs ≡√2mi Te /eB is the ion Larmor radius evaluated at
the electron temperature, and F is a function of
the dimensionless parameters that is to be determined. The main change in plasma parameters ingoing from present experiments to the fusion reactor is in the value of ρ∗ . Therefore, determining
the transport scalings with ρ∗ is critical. Two limits that are often discussed in transport studies aredefined by Eq. (8). When α = 1, the scaling law iscalled gyroBohm. This is the expected scaling frommost local turbulence theories for which the turbulent scale length ∆T is proportional to ρi . For α = 0,
the scaling is called Bohm, when the turbulence scale
length involves the macroscopic size of the plasma.
Of course the confinement time can be determinedby boundary conditions on the transport equations,as well as the thermal diffusivities themselves; this
could introduce a different dependence on ρ∗ [75] asdiscussed in Section 4.2.
A series of experiments have been performed toresolve the scaling of D with different dimensionlessparameters, in particular with ρ∗ . They will be disNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
cussed in Section 7. In a given device, tokamak discharges that have the same dimensionless parametersapart from ρ∗ must be produced. The level of control
required for these experiments makes them very difficult. In present devices, only a limited range of variation in ρ∗ can be achieved (less than a factor of two).
A larger range of variation in ρ∗ can be obtained byusing different devices, but in this case the experiments are even more difficult. At present, and for the
low confinement L-mode, some experimental results
for the energy confinement time, τE , are consistent
with, or close to, Bohm type scaling [74, 76, 77],
while others are between Bohm and gyroBohm [78].
In the high confinement mode (H-mode), the diffusivities seem to be consistent with gyroBohm scaling. However, the problem is more complicated thanstated because the electron diffusivity may be different from the ion diffusivity. The separation between
both transport channels requires more diagnostics
and makes the analysis of the experimental results
more difficult. Only a few experiments separating
electron and ion channels have been performed; they
will be discussed later in this section. Scalings of Dwith β and ν ∗ have also been carried out.
The basis of this empirical approach to the determination of the transport coefficients needs to beestablished. The non-dimensional parameter scalings are based on plasma physics. However, plasma
experiments are affected by edge conditions, plasma
wall interactions, etc. If these boundary conditions
modify confinement in an essential manner, thenon-dimensionally similar discharges would not beenough to determine the scaling of the transportcoefficients. To test the basis of this approach, discharges with the same dimensionless parametersmust be set up in devices of different sizes to test if
they are equivalent. Experiments comparing similar
discharges from DIII-D and JET have given positive
results [79].
The basic picture and scalings for plasma transport presented here are challenged by some experimental observations. These observations underline
the fact that we are far from having resolved all
the scientific issues in plasma transport. Examplesof these open questions are as follows.
1. Bohm scaling and local fluctuations
Since the basic scale length of the microinstabilities is ρi , a simple random walk argument
leads to transport coefficients that are characterized by a gyroBohm scaling. However, this seems
to disagree with the experimental observation in LNuclear Fusion, Vol. 39, No. 12 (1999)

mode plasma confinement. To explain a departure of
the core turbulence from a gyroBohm scaling, twokinds of models have been proposed so far. First,
many instabilities are sensitive to the toroidal geometry. This geometrical effect introduces a coupling
between poloidal components. In a homogeneous system, this results in eigenmodes that extend over thesize of the device, similar to phonons in a crystal. In the linear regime, this leads to the conceptof ‘global’ eigenmodes that exhibit non-gyroBohm
radial lengths and growth rates [80–83]. However,
it is expected that these correlations over large distances are destroyed in a fully developed turbulence,
which should, therefore, exhibit a gyroBohm scaling. This may not be true close to the turbulence
threshold when the turbulence level is low. Indeed,a simplified numerical simulation shows long lived
global modes in this case. More precisely, the correlation lengths scale as ρi , but the correlation time
does not follow the gyroBohm prediction [84]. The
second kind of model relies on the concept of selforganized criticality [85], for which the paradigm
is the sandpile. Such a system exhibits large scale
events (avalanches), which dominate the transportscaling in spite of their scarcity [86,87]. This leads toa non-gyroBohm behaviour. A velocity shear results
in a recovery of a gyroBohm scaling through a
decorrelation process of the large scale events [88].
The correlation lengths scale as a gyroradius in thismodel. Moreover, the gradients are below the critical value for moderate fluxes. Most of these models predict the scaling of the correlation lengths andtimes. Therefore, the experimental determination of
the scaling of correlation functions should clarify thisissue.
2. Isotope effect on plasma confinement
In Eq. (8), if we only consider the scaling of D
α/2with the ion atomic mass, we obtain D ∝ Ai F (Ai )(α = 0 for Bohm models, α = 1 for gyroBohm).
Unless there is a strong dependence of F on the ion
mass, this result does not agree with experiment. One
observes in almost all tokamaks that plasma properties depend on the plasma gas [89], and, contrary
to what is expected from the simple scaling, confinement has a positive dependence on Ai . Although
there are some preliminary indications supportinga positive dependence of confinement on Ai fromgyrokinetic calculations and theories based on nonlinear ion Landau damping [53], there is not yet asatisfactory theoretical explanation of this effect. For
ELMy H-mode, ELMs could cause an isotope effect2183

ITER Physics Basisto appear through boundary conditions. Phenomenological observations will be described in Section 3.
3. Fast time-scales and non-local effects
When a transient local perturbation in plasma
density n or temperature T is applied to a tokamak,
one might expect its effect to propagate across themagnetic surfaces on a time-scale that is related tothe diffusion coefficients characterizing steady stateparticle and energy balance. In fact, observations ofheat pulses generated by, for instance, sawtooth collapses or localized auxiliary heating such as ECRH,
show that these signals propagate as if the diffusion
coefficient exceeded that determined from steadystudies by a factor typically in the range of 1–5 [90].
This could be understood if the diffusion coefficientsare themselves functions of n and T and their radial
gradients, i.e. the transport fluxes are non-linear
functions of the driving gradients. In fact, many theories of turbulent transport have this feature. L-mode
transient transport experiments on ASDEX-Upgrade[91] indicate that for small, ECH generated electron
temperature transients, the response is adequately
described by such diffusivities.
However, a number of dramatic observations
involving transient behaviour pose serious challenges
to our understanding of the processes underlying
transport in tokamaks. Three examples are:(i) the fast propagation of the cold pulse [92] associated with the injection of pellets into a tokamak;(ii) the almost instantaneous effect of the lowto-high (L–H) confinement transition at theplasma edge on the transport coefficients in theplasma core as observed in JET [93, 94]; and(iii) the heating of the plasma centre within a
few milliseconds of injecting impurities at theplasma edge in TEXT [95] and TFTR [96].
These transient effects can propagate radially withvelocities ∼102 m· s−1 , two orders of magnitude
faster than results from a normal diffusive process.
Evidence that the magnetic configuration remains as
nested toroidal surfaces is needed to support thesetransient tests of thermal diffusivities.
An entirely satisfactory mechanism to explainthese surprising results is not yet available. However,
several conjectures have been proposed. The ones
based on long correlation lengths of either fluctuations or the transport events given above for Bohm
2184

scaling can be applied here. The rational surface
at q = 1 appears to play a role in the fast propagation of the cold pulse [97]. This might suggestthat some MHD activity is triggered by the cooling there and, possibly through toroidal coupling,affects the radial transport. Another model involves
the non-linear growth and interaction of neoclassical magnetic islands on different rational surfaces.
This produces regions of stochastic magnetic field
when the islands overlap spatially. The resulting
rapid plasma transport allows fast communication
across the plasma radius [98]. A further proposal isstimulated by the empirical observation that tokamak plasmas have a tendency to take up certain preferred radial profiles, termed profile consistency orresilience. This model assumes that turbulent processes associated with MHD instabilities cause theprofiles to relax to these special ones on a fast timescale. Theoretical arguments have been advanced forthese preferred profiles and the idea has been incorporated in the so-called ‘canonical temperature profile’ transport model, which can also describe theinward heat pinches sometimes observed in tokamaks[99,100]. Similar fast propagation may be anticipatedfrom stiff transport models such as those that drive
profiles to marginal stability [15].

3.

Experimental description ofconfinement and transport

3.1.

Introduction

Early tokamaks were heated ohmically, but it
became clear that ohmic heating to ignition was notefficient, so the confinement properties of tokamaks
with auxiliary heating by energetic neutral beams orradio frequency waves have been thoroughly investigated. Nevertheless, there will be phases in which
ITER will be without additional power and theconfinement properties of ohmically heated devices
can shed light on our understanding of confinementin general. Thus, in Section 3.2, we first characterize confinement in ohmically heated tokamaks.
The application of auxiliary power, while producing
higher energy content in the tokamak, is associatedwith a degradation in energy confinement time relative to that in the ohmic phase; this so-called L-mode
is also discussed in Section 3.2. As a consequence, anignited tokamak operating in L-mode would have to
be unacceptably large. Fortunately, it was discoveredthat when sufficient auxiliary power was applied, atransition to an improved confinement mode, the HNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transportmode, occurred. This mode, which involves the formation of an edge transport barrier, forms the basis
for the design of ITER, and its confinement properties are described in Sections 3.3 and 6.2. A number of other improved confinement modes have since
been discovered. In Section 3.4, improved confinement modes associated with internal transport barriers, including those involving negative, or low, magnetic shear in the centre of the plasma, are described,
and in Section 3.5, improved confinement regimes
involving edge radiation are discussed.
3.2.

General results for ohmic and L-mode

An ohmic plasma is one that is resistively heatedwith a power given by Ip Vres , where Ip is the plasma
current and Vres is the resistive portion of the loop
voltage. The electrons are heated directly, while theions are heated by the equipartition energy flow fromthe electrons. There are two principal ohmic regimes.
At low density, the confinement time is found toincrease linearly with density (χe ∝ 1/ne ) up tosome critical value; this is the linear ohmic confinement (LOC) regime. Beyond this critical density,
the confinement time remains constant with density; this is called the saturated ohmic confinement(SOC) regime. The coupling between the electrons
and the ions becomes stronger in the LOC as density
increases and electron temperature decreases, causing Ti to increase [22, and references therein].
In the LOC regime, the energy confinement timeis also found to increase with device size and withsafety factor. The confinement time in the LOC isconveniently represented by the neo-Alcator scaling[101]τE (s) = 0.07ne (1020 m−3 )qκ0.5 R(m)2 a(m)where q is the safety factor at the plasma surface,κ is the elongation of the plasma cross-section, R is
the major radius and a is the minor radius. Froma series of ohmic experiments on the Japanese tokamaks, JFT-2, JFT-2M, JT-60 and DIVA, the parametric dependence of the critical density for transition from LOC to SOC with gas fuelling was foundto be [103]ncrit (1020 m−3 ) = 0.65A0.5
i BT (T)/qR(m)where BT is the toroidal magnetic field and Ai is ion
mass number in AMU. The energy confinement time
was found to be higher in deuterium plasmas relative
to hydrogen ones, i.e. τE ∝ Aα
i with α = 0.5 in small
Nuclear Fusion, Vol. 39, No. 12 (1999)

tokamaks [22] and with α = 0.2 in large machines
such as JET [103] and TFTR [104].
Although in some devices (e.g. Alcator A [105],
ISX-A [106] and T-11 [107]), the level of energy
confinement at high density can be explained by
the neoclassical ion thermal conduction losses, inother experiments with gas fuelling, an anomalous
high ion conduction was identified as a reason forthe confinement saturation. Density profile peaking resulting from pellet injection on Alcator C wasassociated with confinement time increases in otherwise SOC discharges up to values predicted by the
LOC scaling for intermediate densities [108]. Thisre-establishment of LOC scaling at high density was
also seen on other tokamaks with pellet injection andresulting density profile peaking. The strong influence of edge conditions on core confinement in ohmic
plasmas was shown in ASDEX experiments in which
a reduction in the gas feed at a certain time in the
discharge led to a slow peaking of the density andcurrent profiles, and re-establishing the linear dependence of confinement time on density [109].
While confinement times given by the LOC scaling could be extrapolated to values acceptable forreactor scenarios, it was recognized early that theohmic regime was inefficient for achieving the necessary temperatures. As temperature increased, theresistive heating decreased, and, therefore, auxiliary
heating would help to increase the plasma temperature to that level required for significant fusion power
production. Auxiliary heating was performed by avariety of techniques, including neutral beams andRF (ion and electron cyclotron, lower hybrid, andion Bernstein waves). While the temperature andstored energy increased with this auxiliary heating,the incremental increase in stored energy was lessthan that expected from the ohmic scalings, resulting
in a degradation of global confinement. This mode ofoperation, with degraded energy confinement time, iscalled the L-, or low confinement, mode. Characteristic features of L-mode plasmas are the low temperatures and temperature gradients near the plasma
periphery.
The L-mode is typified by not only a degradation in confinement from the ohmic value, but acontinuous degradation of confinement with increasing auxiliary heating power. While the confinement
was found to be a strong increasing function of q inohmic discharges, for auxiliary heated discharges theconfinement is not found to be a function of q, butrather scales linearly with the plasma current. Any
direct dependence of confinement time on plasma
2185

ITER Physics Basiscurrent in ohmic discharges may be masked by the
explicit dependence of the ohmic heating power on
the plasma current. The global confinement of auxiliary heated plasmas, unlike that of ohmic discharges,
shows no explicit dependence on the toroidal magnetic field [101]. Under direct electron heating, such
as ECRH on T-10 [110] or LH on Tore-Supra [111],
when the electron transport losses dominate, the confinement increases linearly with density. This suggests that (χe ∝ 1/ne ) may be intrinsic to theelectron transport dominated regime (or to regimes
with no fast ion component). With ion heating andion transport losses predominant (e.g. with NBI,
ICRH), the global confinement depends only weakly
on plasma density, although more of a dependence
on plasma density is found for the thermal energy
confinement time [112]. Increased plasma elongationand plasma size result in higher confinement.
The effect of the plasma isotope on confinement is ambiguous. While most experiments show
some improvement of confinement on Ai , the magnitude of the effect varies from device to device andfrom isotope to isotope (see [113, 114] and [22, andreferences therein]). Comparing discharges with H
and D for either the working gas or beam species,
no dependence of confinement on ion mass wasobserved in L-mode discharges in DIII-D or ToreSupra. A weak dependence (A0.3
i ) was observed in
ASDEX, PDX, TFTR and JET. A stronger dependence (Aai , α ≥ 0.45) was observed in DIII, DIII-D
at ECRF heating, and in JT-60U. In more recent
DD/DT L-mode experiments on TFTR, the isotope
effect was found to be strong for both the thermal
plasma and beam component, with Wth ∝ A0.5–1.0
i
and Wbeam ∼ A0.7
i [115]. The isotope effect was foundto be stronger at higher beam power. In TFTR
experiments with ICRF heating, the isotope scaling[116]
in L-mode discharges was found to be A0.35–0.5i(see Section 6.3).
It is important to note that the L-mode confinement parametric dependences are generally consistent with the Bohm and high-β scaling constraints[112].
Perturbative experiments (i.e. current ramps)
have shown that the plasma current global variable is
not necessarily an appropriate parameter with which
to describe the energy confinement. In these experiments [117, 118], the energy confinement time wasfound to change on time-scales much slower thanthat of the current ramp and close to that of thecurrent profile relaxation. This led to the recognition that the current profile, not the global plasma
2186

current, may be the controlling factor in determining the plasma energy transport, with higher confinement being associated with the more peaked current
profile for this transient phase.
As with ohmic plasmas, the anomalous transport
loss is governed by microturbulence whose source,
however, is still not known. Measurements of density
fluctuations during parametric scans show changes
in fluctuation levels and radial correlation lengths
consistent with the resulting change in energy confinement [119].
3.3.

Regimes with edge transportbarrier (H-mode) andrecommended regime for ITER

The high confinement mode (H-mode) associatedwith a spontaneous formation of an edge transportbarrier was first discovered in ASDEX [120] and, as
is discussed in Section 4, has now been seen on awide variety of magnetic confinement devices undera wide range of conditions.
As is discussed in Section 6, the general forms
for the global energy confinement scaling in L-modeand H-mode are similar, although the former scaling is Bohm like, while the latter is more gyroBohm.
The H-mode exhibits global energy confinement values about a factor of two better than L-mode. Part ofthis is due to the formation of the edge transport barrier, as is discussed further in Sections 4.1 and 4.2.
Another part of this improvement is due to a reduction in local transport throughout the plasma after
the L–H transition. Experiments comparing L-modeand H-mode local transport rates under as similar conditions as possible have shown reductions in
the electron thermal diffusivity, ion thermal diffusivity and angular momentum diffusivity, with the
reduction in the electron thermal diffusivity being
especially prominent. An example of these results isshown in Fig. 1 [121].
There are several reasons why the H-mode has
been chosen over the other improved confinement
modes as the primary operating mode for ITER.
(1) As is discussed further in Section 4, the Hmode is robust, having been seen under awide variety of conditions in a large number of devices. Many of the other improved
confinement regimes (e.g. TFTR ‘supershot’[122], TEXTOR RI-mode [123, 124], ISX Zmode [125], ASDEX counter-injection mode[126]) have been seen in only one or a few
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
12

10

6

a)
χe
χϕ

H-Mode L-Mode

χi
χiNEO

8
χ (m2/s)b)
H-Mode L-Mode

4

6

2

4

2

0
0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

ρ

ρ

Figure 1. Comparison of electron thermal, ion thermal and angular momentum
diffusivity in the limiter L-mode and divertor H-mode discharges in DIII-D. The discharges were prepared to have very similar plasma shapes and identical line averaged
densities, plasma currents, toroidal fields and input powers at the time of comparison [121].

devices under a limited range of operating conditions. In addition, steady state operation in
many of these modes remains to be demonstrated. The H-mode with edge localized modes(ELMy H-mode) has been run for as long as
20 s on JET [127], with the duration limited
only by power supply considerations.
(2) The H-mode requires little in the way of special wall conditioning, unlike, for example, thesupershot regime, although it does benefit fromreduced particle recycling.
(3) The H-mode exhibits flat density profiles in
the plasma core, which are consistent with
reduced peaking of impurities and helium ash.
The DIII-D results [128], for example, show flat
helium density profiles in H-mode plasmas withne (r)/nHe (r) constant.
(4) The H-mode exhibits good confinement evenin high density cases where the electron andion temperatures are equilibrated; this is consistent with the alpha particle heating andhigh density operation that will be needed for
ITER. Several of the other improved confinement modes (high βp mode [129], supershot
and enhanced confinement modes associatedwith weak or negative central shear) have only
been seen so far in cases where Ti  Te .
Nuclear Fusion, Vol. 39, No. 12 (1999)(5) The H-mode requires no special current profile control for long pulse operation, unlike
the operating modes with negative central
magnetic shear. Although the negative central
shear operating regime may ultimately improvethe characteristics of a tokamak reactor, it is
not sufficiently investigated to be consideredas the main operational regime for ITER.
The physics of the L–H-mode transition, including scalings for the heating power required for thetransition, are discussed in Section 4, and the effectsof large scale MHD phenomena (ELMs and ‘sawteeth’) on the H-mode confinement are described in
Section 5.
3.4.

Regimes with internaltransport barriers

Recently, core or internal transport barriers(ITBs) have been discovered that lead to significant enhancements in confinement and plasma performance. Transport barriers associated with weakor negative shear have been observed on all of thelarge tokamaks (the enhanced reverse shear (ERS)
mode on TFTR [6], negative central shear (NCS)
mode on DIII-D [130], optimized shear scenario [131]and PEP mode [132] on JET, and reverse shear mode
on JT-60U [133]). Internal transport barriers are also
observed in plasmas with monotonic q profiles.
2187

ITER Physics Basis
3.4.1. Barriers associated with reversed
or weak magnetic shear
For most of the regimes with reversed magnetic
shear, shear reversal is obtained through a combination of a rapid current ramp and auxiliary heating
early in the plasma discharge. The enhanced performance of these plasmas is, at present, transient,however, and experiments are now being designed
to extend this regime of operation to steady state.
Recently, ELMy reversed shear H-mode discharges
were sustained in DIII-D [134] and JT-60U [135] for
1.5 s, although with a relatively low figure of merit
FF us = βN H/q 2 = 0.5, where H is the enhancement
factor above the L-mode confinement scaling [136]and βN = β/(I/aBT ). The highest transient valueof FF us achieved so far in negative central shear
discharges is about 1.2 [6], which should be compared with FF us = 0.64 expected in ELMy H-mode
in ITER.
In TFTR ERS plasmas [6, 137–139], dramatic
reductions in the ion thermal conductivity and particle diffusivity are found shortly after the onset ofnearly balanced neutral beam injection. Both quantities fall to or below present day standard neoclassicalpredictions. Evidence for exceptional particle confinement is obtained through particle balance analysis [6,137,138], as well as tritium and helium gas puffing and lithium pellet injection [140]. The transport
reductions are accompanied by local reductions in
the fluctuation levels [139]. Interestingly, the electron
thermal transport appears to be relatively unaffectedby the reduction in fluctuations. This observation,
coupled with the observed improvement in particle
confinement, may have practical implications withrespect to the issue of helium ash accumulation insuch regimes. Pressure and density profiles in ERS
plasmas are characterized by strong central peaking,with the transport barrier located at or near the location of the minimum in the q profile. Steady state
operation is limited by plasma disruption, triggered
either by qmin falling below 2, or by the high pressure
gradient region slowly expanding through the radiusof weak magnetic shear.
Significant reductions in transport are also
observed on DIII-D in NCS plasmas with ITBs
[130, 141, 142]. As in TFTR, the thermal transport
reduction is observed primarily in the ion channel.
Large increases in the ion temperature, electron temperature and electron density are observed inside theradial location of the ITB, with large gradients in
the ion temperature and electron density developing2188

in the region of the ITB. The ion thermal diffusivities are below standard neoclassical levels in the corefor NCS plasmas with L-mode plasma edge properties. For NCS plasmas with H-mode plasma edges in
DIII-D, the ion thermal diffusivities are below standard neoclassical levels for nearly the entire plasma
cross-section [143], and fluctuation levels, measured
by far infrared scattering, are dramatically reduced
as well. L-mode edge NCS plasmas are characterized by a dramatic peaking of the density profile,
much like TFTR ERS plasmas. Large reductions in
the particle diffusivity are observed inside the ITB.
The cores of these discharges are in the second stable regime for ballooning modes due to the weak or
negative magnetic shear and high central q. However, L-mode edge NCS discharges normally disrupt
when normalized β, βN = β/(I/aB), exceeds 2.0–2.5
as a result of becoming unstable to global resistive
interchange modes. In general, for H-mode edge NCS
plasmas, the broad density profile results in a relatively broad pressure profile that is better alignedwith the weak shear and exhibits greater stability.
Larger βN values are obtained in plasmas with Hmode edges than with L-mode edges, which is consistent with ideal MHD calculations [144]. Contrary
to the observations on TFTR, these plasmas are stable to the passing of qmin through 2.
On JT-60U, transport barriers are also observedwith [133] and without shear reversal. With shear
reversal, the region of small transport is again
located near qmin . It is inferred that the electron
thermal transport can also be reduced in these plasmas [145], in contrast to the conclusions drawn fromTFTR and DIII-D core transport barrier formation
with reversed shear. From power and particle balance
analysis, it also appears that the improvement in confinement is constrained to regions of steep gradients.
Radial expansion of the transport barrier after formation is observed. It has been demonstrated on JT60U that the barrier can be sustained if RF power issubstituted for neutral beam power, indicating thatthe physics of transport barrier sustainment does notdepend on particle fuelling or density gradients per
se [146]. Like TFTR ERS plasmas, but unlike DIII-D
NCS plasmas, these plasmas exhibit high disruptivity near qmin = 2.
In the JET tokamak, transport barriers are also
observed with reversed or weak magnetic shear [131,147–149] through use of lower hybrid current drive,
ICRH heating, or pellet fuelling. The transport barriers are characterized by strong gradients in theion and electron temperature, electron density, andNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
toroidal rotation. Power balance analysis indicatesthat thermal conduction in both the electron andion channels is reduced in these plasmas. The particle diffusivity is reduced, and the confinement ofelectrons deposited in the core is sometimes similar to that observed in DIII-D NCS, TFTR ERS,
and JT-60U ERS plasmas. A characteristic of theplasma evolution is the large radial expansion of thetransport barrier once the barrier is formed. Using
the radial location of the minimum in the q profile
obtained from equilibrium calculations, it is inferredthat the barrier moves far outside qmin , in contrast
to the observations made on other machines. It isinferred that the optimum q profile for the formation of the barrier has slightly negative or flat shearwith q > 1 everywhere.
Candidate theory based models for transport barrier formation have been proposed that rely on theshear reversal itself [150, 151], E × B shear suppression [6,130,137–141,152], strong Shafranov shift gradient [151, 153], or a combination of two or all threeof these mechanisms.
There is experimental evidence that E×B shear isnecessary and Shafranov shift effects and shear reversal are not sufficient to sustain enhanced confinement [154]. Transport and fluctuation levels remain
low until the characteristic shearing rate of turbulence [155] is reduced through the applied torquebelow a threshold value. Recent observations of local
core poloidal velocity excursions prior to the onset
of ERS confinement point to a possible similarity inthe bifurcation physics of core and edge barrier formation [156].
In DIII-D high performance NCS plasmas, there
is both a temporal and spatial correlation between
the reduction in transport and the reduction in electrostatic fluctuations when the E × B flow shear
exceeds the local microinstability growth rates [121].
This large flow shear results from the strong peaking
of toroidal rotation inside the region with weak or
negative magnetic shear, in contrast to TFTR ERS
plasmas, where the origin of E ×B flow shear is largepressure gradients formed in part by strong central
fuelling.
Although Shafranov shift gradient stabilization
extrapolates well to a reactor scale device, the ρ∗
scaling of the diamagnetically driven E×B shear stabilization does not extrapolate favourably unless anexternal source of rotation (for instance the neutralbeam injection or ion Bernstein wave (IBW) radio
frequency heating) is applied. However, further theoretical and experimental confirmation of these ideas
Nuclear Fusion, Vol. 39, No. 12 (1999)

for transport barrier creation, dynamics and control
is required before fully quantitative extrapolations
from present devices can be made for ITER.
3.4.2. Other improved core confinement
regimes without edge radiation
To this class belong the supershot regime in
TFTR, the VH-mode in DIII-D, the high-li (internal
inductance) regime in TFTR and Tore-Supra, thehigh βp mode of JT-60U, improved L-modes in JFT2M, the LHEP (lower hybrid enhanced performance)mode of Tore-Supra, and the PEP (pellet enhanced
performance) mode obtained in several machines.
The supershot regime in TFTR [122] is characterized by extremely peaked density and pressure
profiles, along with high Ti  Te (Ti (0) = 45 keV,
Te (0) = 14 keV) and confinement enhancements of
up to 3 over L-mode. The supershot regime shows astrong isotope scaling of confinement; in DD versus[115].
DT plasmas, τE was found to scale as A0.85
i
The single most important controlling parameter for
supershot generation is minimizing the particle influxfrom the wall.
VH-mode operation on DIII-D represents anenhanced confinement regime that is not linked withreversed or weak magnetic shear [157]. Magnetic
braking experiments, i.e. experiments where a reduction of the toroidal rotation by an externally appliedresonant magnetic field is brought about, suggestthat toroidal flow shear is important for enhanced
core confinement in these cases [158].
The high-li and LHEP regimes on Tore-Supra[159] are both characterized by an enhancement factor for the electron energy content of up to 1.7 over
the Tore-Supra L-mode scaling. LHEP discharges up
to 2 min in duration have been obtained in ToreSupra with the lower hybrid current drive [160]. The
formation of transport barriers and a reduction ofelectron thermal diffusivity coefficients to near neoclassical values have been observed in the plasmacore, where the magnetic shear is weak or negative.
DT plasmas with increased plasma inductancehave been produced in TFTR [161] by expanding
the plasma minor radius during the current rise in
the startup phase. These plasmas show an enhanced
stability due to an increase in the maximum possible βN , proportional to li , and stay in the L-mode,
unlike the high-li discharges obtained by a current
ramp down. The li values up to 2, corresponding toβN values of up to 3, and pressure peaking factors
from 3 to 6.2 have been observed.
2189

ITER Physics Basis
In JT-60U, improved L-mode confinement has
been observed in deuterium high-βp discharges at low
density with centrally peaked beam deposition [162].
Boronization is essential to get a low density target
plasma and sawtooth activity, and locked modes aresuppressed by careful control of the internal inductance and the toroidal rotation. It is obtained atvalues for the cylindrical equivalent safety factor q ∗between 4 and 11, and is characterized bythreshold can be obtained by (i) directing the ion∇B drift [171] away from the X-point, (ii) by H0
injection, or a combination of both. Power degradation is weaker than that of the usual L-mode
scaling and confinement approaches that of the Hmode with increasing heating powers, as shown in
Fig. 2, independent of how the high threshold was
obtained. An isotopic dependence of confinement has
been observed proportional to A0.5i .

(i) confinement improvement over the L-mode bya factor of about 3, increasing linearly with εβpup to about 0.5;

0.15(ii) bootstrap current fractions of about 60%;

0.10(iv) high fusion neutron rates.
It is further characterized by a highly peaked Ti
and ne profile, a broad Te profile, and a relatively
broad current density profile, with li < 1.2, where
the weak shear develops in response to the generation of bootstrap current [163]. The existence ofan ITB is deduced from the shape of the ion temperature and toroidal rotation profile [164]. This
regime is terminated by β collapse due to ideal
low-n kink ballooning modes. The β collapse canbe avoided through the beneficial effects of an Hmode pedestal and pressure profile broadening [165],and through increased plasma triangularity [166]. As
contrasted with reverse shear operation on JT-60U,
nearly steady state operation of these plasmas has
been demonstrated [8].
In JFT-2M, the improved L-mode is transiently
obtained just after an H–L transition [167]. The
other improved L-mode is transiently obtained withcounter neutral beam injection associated with apeaking of the electron density.
A transient increase in confinement has beenfound in L-mode discharges in several machines usingpellet injection in ohmic and additionally heated discharges (PEP mode) [168–170]. A strong peakingof the electron density and temperature profiles isobserved, together with reduced core (r/a < 0.5)transport and a reduction in χeff . Enhancement factors versus L-mode scaling of up to 1.6 have been
observed, deteriorating with increasing heating powers.
Improved L-mode without additional impurity
radiation occurs in ASDEX-Upgrade when the Hmode power threshold is high. The high H-mode
2190

τE (s)(iii) high central temperatures (Ti (0) > 40 keV,
Te (0) > 10 keV, Ti (0)/Te (0) ∼ 4–5); andITERH92–P (ELMy, Wtot)

ITER89-P
0.05

0
0

2

Low Pthres

H–modeHigh Pthres

L–mode4

6
Ptot (MW)

8

10

12

Figure 2. Confinement time versus total heating powerin ASDEX-Upgrade: pure deuterium plasmas, both ion∇B drift directions, Ip = 1 MA, BT = ±2.5 T.

3.5.

Enhanced confinementwith edge radiation

Included in this class are the RI-mode of
TEXTOR-94 [172–174], the improved L-mode
regime of ASDEX-Upgrade [175] and the IL-mode[176] and RI-mode like regimes of DIII-D [177–179].
All these modes are characterized by strong radiationin the edge caused by edge impurity seeding.
The RI-mode on TEXTOR-94 has been obtained
in deuterium discharges heated by neutral beam deuterium co-injection (NBI-co) alone or in combination
with ICRH and/or counter injection. The RI-mode
can be obtained with Ne, Ar, Si and Si + Ne seeding,and has been observed (i) with PNBI -co /Ptot ≥ 25%;(ii) at sufficiently large densities such that the Greenwald number n̄e /nGR ≥ 70% (where nGR = Ip /(πa2 )with units 1020 m−3 , MA, m); and (iii) with impurity seeding such that the radiated power fraction,
γ = Prad /Ptot , exceeds about 50%.
The RI-mode combines simultaneously many
attractive features that are promising for application
to a future fusion reactor:(i) high confinement as good as ELM free H-mode;
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport(ii) high plasma β, up to βN = 2.1 with simultaneous values for product βN × H89P up to 4.5;

100

80

(iv) confinement increasing linearly with density
given by τRI = (n̄e /nGR )τITERH93 -P , both for
Ne and Ar seeding, as illustrated in Fig. 3 (hereτITERH93 -P is the energy confinement time predicted by ITERH93-P scaling for ELM free Hmode [180]);60(v) long and quasi-stationary pulses up to 160
times the energy confinement time, which isvery close to the ratio of the burn time to theconfinement time foreseen for ITER, or about
the skin resistive time without impurity accumulation (versus time) in the centre of the discharge;(vi) no difficulty with operation at low qa (checkedpresently down to qa = 2.7) leading to values
for the figure of merit for ignition margin up to
H89P /qa = 0.8;(vii) promising heat removal capabilities by edge
radiation with a radiated power fraction up to
0.95;(viii) no ELMs, no power threshold observed so far;and(ix) concentration of the seeded impurity sufficiently low such that the neutron yield is notdecreasing with respect to discharges withoutneon seeding.
Injecting pellets into RI-mode shots can lead to a further quasi-stationary increase in confinement [181].
At the highest currents explored up to now (Ip =520 kA), where the highest densities can be reached,central Zeff values around 2 are observed.
The extrapolation of this regime to a future fusion
reactor has to be assessed experimentally in orderto gain more knowledge on the influence of larger
machine size and different plasma parameters on
the transport of the energy and particles (D, T
and impurities). First encouraging results with Ne
and Ar seeding in deuterium discharges have beenobtained recently on TFTR [182], Tore-Supra and
DIII-D [183, 184].
ASDEX-Upgrade has obtained improved confinement in L-mode discharges by Ne impurity seeding,studied in D+ or H+ plasmas heated with D0 and
Nuclear Fusion, Vol. 39, No. 12 (1999)

τ E, EXP (ms)(iii) high density, presently observed from 0.75 up
to 1.2 times the Greenwald density;

Si
Ne
Ar
Si+Ne

40
fH93 ≥ 0.85
1.5 < Ptot < 4.5 MW
0.45 < γ < 0.95
0.25 < PNBI-CO /Ptot

20

3.5 < ne < 8.0 x 10 19 m -3
0
0

40

20

τ RI (ms) =

ne
nGR

60

80

100

-0.67
τ ITER-93H ∝ I p0.06 n e1.17 P tot

Figure 3. Comparison between the experimentallyobserved confinement times and the predictions of the
RI-mode scaling law τRI = (n̄e /nGR )τITERH93 -P ∝−0.67
I 0.06 n1.17
Ptot
for Ne, Ar, Si and Si+Ne seeded RIe
mode discharges in TEXTOR-94.

H0 NBI respectively, at power levels between 1 and
10 MW and q95 between 3 and 4. The confinement
increases with radiated power and, at high radiation
powers, reaches 80% of that of the H-mode [175].
The IL-mode of DIII-D has been obtained with Ne
seeding in both single null and double null configurations after an H–L transition. Confinement of the
IL-mode is between 1.4 and 1.8 times L-mode. Bothelectron and ion thermal transport is reduced, witha doubling of the central electron temperature. The
IL-mode can make a transition to the H-mode, anda confinement has been observed in this subsequent
H-mode phase by a factor of 3 higher than that in
L-mode.
Recently, enhanced confinement with impurity
seeding has been observed in DIII-D in both ELMing H- and L-mode deuterium plasmas in diverted
and limited configurations. Both Ne and Ar havebeen used as radiating impurity and all plasmas were
heated with NB co-injection. A variety of confinement modes have been observed ranging from Lmode to H-mode and even VH-mode. The results
obtained so far in the parameter range 1.2 < Ip <1.4 MA, 1.5 < Bt < 2.05 T, 3 < qa < 4.5, 4.5 <Padd < 13.4 MW, n̄e /nGR ≤ 1, can be summarized
as follows.
(i) In inner limiter plasmas the elongation was κ =1.4–1.6; maximum values obtained for the Greenwald number and the radiated power fraction aren̄e /nGR ≈ 80% and γ = Prad /Ptot ≈ 80%. At the2191

ITER Physics Basishighest densities, values of the energy confinement
time between those characteristic for ELMing and
ELM free H-mode were observed with moderately
peaked density profiles.
(ii) In a divertor configuration, the H-mode plasmas with n̄e /nGR ≈ 70–80%, radiation fractions
Prad /Ptot ≈ 95% and a confinement characterized
by fL89 ≥ 1.6, equivalent to fH93 ≥ 1 (i.e. better
than ELM free H-mode confinement) was observed.
A promising scenario leading to quasi-stationary conditions is the so-called ‘puff and pump’ scenario
with impurity seeding [179]. Under those conditions,
phases have been obtained with a duration of about
3.5 s or 30 × τE , βN values up to 2.3 and βN × fL89
product up to 4.5.
(iii) In addition, the VH-mode was observed inupper single null plasmas at high target densities andradiation levels. A value of βN × fL89 = 6 has beenobtained for up to 1.6 s, and for nearly the entire
period βN stayed at the stability limit (≈4 × li ).
In transient phases, confinement improved to values fL89 = 4 equivalent to fH93 = 2, at densities up to n̄e /nGR ≈ 60% and radiation fractions
Prad /Ptot ≈ 50%. In these high performance discharges, it was difficult to increase the radiation further by impurity seeding, possibly due to the very
high gradient of the density profile at the edge, leading to radiating mantles with a limited radial extent.
Like the TEXTOR RI-mode, these regimes needfurther studies before extrapolation to ITER can be
made.

4.

L–H and H–L transition physics

The H-mode of confinement is one of the mostrobust and reactor compatible of the improved tokamak confinement regimes, combining good energy
confinement [185] with high beta [186] and, in thepresence of ELMs, with acceptable particle transport
rates for the control of density, impurity and helium
exhaust [128,187]. In addition to its practical importance, the attempt to explain the turbulence reduction and confinement improvement that occur in Hmode have led to fundamental insights into plasma
physics.
The H-mode, first discovered in ASDEX [120], hasbeen obtained in all divertor tokamaks that haveoperated since 1982, in limiter discharges in several tokamaks [188–192], in a current free stellarator [193–195], in a heliotron/torsatron [196,197], andin a linear tandem mirror machine [198]. H-mode
2192

has also been produced with a wide variety of techniques: heating with neutral beam injection; electron
cyclotron heating [193–195, 199, 200]; ion cyclotron
heating [188, 201]; lower hybrid heating [202]; andOhmic heating [203–206]. Furthermore, H-mode has
also been produced by biasing the plasma using an
external electrode [207, 208] or by biasing the limiter [198].
There is a set of common features that are seenin all devices that obtain H-mode. The first to be
identified was the formation of a transport barrier at
the plasma edge [209] where the density and temperature gradients steepen after the transition. The formation of this barrier is associated with a drop in theDα radiation all around the plasma, indicating a significant decrease in the particle outflux. In addition,
later work showed that the density fluctuation amplitude decreases in the region where the transportbarrier forms [210–212]. A reduction in the amplitude of magnetic fluctuations has also been observed[213, 214]. Finally, at the same time as the formation of the transport barrier and the reduction influctuations, a steep gradient region develops in theradial electric field Er at the plasma edge [215–219].
These features have been seen in all tokamak discharges where diagnostics capabilities allow [220,221]
and have also been seen in the stellarator and mirror
results [195, 198].
4.1.

Physical processes of transition

Because the H-mode confinement improvement
appears in many configurations and has been produced by many means, an understanding of the confinement improvement requires some universal mechanism. The leading hypothesis to date involves the
reduction of turbulent transport by sheared E × B
flow [14, 222, 223]. The fundamental idea is thatthe sheared flow tears apart the turbulent eddies in
the plasma, reducing their radial extent and, hence,
reducing the transport that they cause. Both a nonzero first radial derivative [14, 222, 223] or second
radial derivative [224, 225] of the E × B flow canreduce transport. Because the shear in the E × B
flow can have the same effect on a wide variety of turbulence, this mechanism has the universality needed
to explain the transport decrease in a wide range ofconditions. This same sort of shear decorrelation by
E × B flows has also been seen in the edge of ohmically heated limiter discharges in TEXT [226]. It has
further been seen on the open field lines beyond theseparatrix in diverted plasmas [227] when a sheared
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
electric field is created with divertor bias.
Decorrelation of turbulence by sheared flows is
a mechanism that also functions in ordinary fluids [228, 229]. However, because the sheared flowis also a source of free energy, which can drive
Kelvin–Helmholtz instabilities, situations in which
net reduction of turbulence occurs are infrequent.
In magnetized plasmas, the stabilization of Kelvin–
Helmholtz instabilities by shear in the magnetic field
allows the flow shear decorrelation to produce a net
turbulent transport reduction [14]. Accordingly, this
explanation of the H-mode confinement improvement
has led to a fundamental contribution to the understanding of the physics of fluids.
A full understanding of the L–H transition
requires an understanding of the physics that controls the radial profile and magnitude of Er that isgoverned by the radial force balance equation for ions
Er + Vp BT − VT Bp = −(Zi ni e)−1 dpi /dr.

(9)

There has been a large theoretical effort devoted
to this issue, which has led to the formulation ofequations to describe Er near the plasma boundary
in toroidal geometry with neoclassical and anomalous effects included [230–233]. The experimental
assessment of theory is very difficult due to the factthat many of the quantities of interest cannot bemeasured with the existing diagnostic capability. The
ideas that are being actively pursued can be divided
into four categories as follows.
(1) Ion orbit loss [222, 223, 234, 235]. Ions are preferentially lost from the plasma edge because
ions in the loss cone intersect material surfaces. Thus, the plasma is charged up negatively. Alternatively, one can see this as causing a torque generating poloidal flow in competition with the damping due to neoclassical parallel viscosity and charge exchange processes [236, 237].
(2) Stringer spin up [238–240]. A large poloidally
asymmetric sink or source of particles overcomes the natural damping of poloidal rotation
and allows a large poloidal rotation to develop.
The relationship between poloidal rotation and
Er has not been specified in this model.
(3) Pressure gradient drive [241–243]. A toroidal
equilibrium naturally develops a negative
radial electric field to balance the ion pressure
gradient. Thus, the large and negative Er is,effectively, a consequence of good confinement.
Nuclear Fusion, Vol. 39, No. 12 (1999)(4) Anomalous viscosity [221, 222, 230, 244, 245] or
turbulent Reynolds stress [199, 233, 246–251].
Transport of momentum can modify the average flow profile of the plasma. The relation
between plasma velocity and Er has been specified for these models.
Although we have a model of turbulence stabilization and confinement improvement due to E × B
shear stabilization, we still need to validate a modelthat can predict the Er value under given conditions.
Considerable experimental effort is being devoted tothis area, but quantitative tests of the various models
require, in many cases, development of novel diagnostics. In addition, since there is no fundamental
understanding of energy and particle transport in
the plasma edge, even if we had such a model for
Er , we could not predict the edge plasma conditions
in ITER with sufficient certainty to utilize it. Furthermore, some models [236, 237, 248] are dependenton, or modified by, the presence of neutrals, which
can be affected by details of the magnetic configuration. Accordingly, as is discussed in Section 2.4.3,
estimation of the power threshold for ITER is being
done empirically. Nevertheless, the L–H transition isan edge phenomena, and the transition conditions
should be expressible in terms of edge parameters
rather than global ones like heating power. Empirical relations of this type are emerging (as discussed
in Section 4.3 and in Chapter 4, Sections 3.7 and 3.8)
and will help to provide experimental tests of theory.
4.2.

Edge pedestal

A defining feature of the H-mode is the existenceof a transport barrier near the plasma boundary.
Although the H-mode edge transport barrier can bequite narrow (2% of the minor radius in DIII-D),
the characteristics of this layer are significant in theoverall plasma performance and in divertor effects.
Stiff ITG-mode turbulent transport models [75,243] predict that the core temperature is strongly
linked to the edge temperature, suggesting that
ITER may require relatively high edge temperaturefor ignition. This result is in qualitative agreementwith data from DIII-D [252] and C-MOD [253]. On)0.58 /BT0.93 , where
DIII-D, HH93 ∝ (TePED )0.55 (nPEDe
HH93 is the energy confinement enhancement factor
relative to the ITER93H-P scaling [180], and PED
refers to values at the top of the H-mode pedestal(see Fig. 4 and [252]). ASDEX-Upgrade also showsa direct relation between stored energy and pedestalpressure gradient (Fig. 5) [254].
2193

ITER Physics Basis

1.6

highest values of the electron temperature at the topof the H-mode transport barrier occur just prior tothe crash of a type I ELM. The maximum pressure
gradient is usually consistent with ideal ballooning
mode stability at the edge [255]. However, data from
DIII-D shows that the edge gradient can exceed thenominal infinite-n ballooning limit [252] by factorsof 2–3. Type III ELMs often, but not always, havelower pedestal pressures and lower confinement (e.g.
Fig. 4).

ITER shape, q95 = 3.2, Ip = 1.5MA

HITER93–H

1.2

0.8

Type I ELMs

0.4

Type III ELMs
L-mode
0
0

2

4

6

pePED averaged over ELMs

Figure 4. H-mode energy confinement enhancement
factor relative to ITER93 H-mode scaling increases withincreasing H-mode pedestal pressure (kPa) averaged over
ELMs in DIII-D [252].
0.8Type I ELMs
Type III ELMs

1.0MA

1.2MA

0.6
Wth (MJ)

0.8MA
0.4
0.6MA
0.2

0
0

50

100

150

200

∇pe (kPa/m)

Figure 5. Relation between thermal stored energy andedge electron pressure gradient for discharges in ASDEXUpgrade near the ideal ballooning limit at the edge [254].

The study of the H-mode pedestal parameters
can be separated into the analysis of the scalingof the width of the steep pressure gradient region,
which is expected to be set by turbulence suppression physics [220], and the magnitude of the edge
gradient. The height and width of the H-mode transport barrier depends on the type and level of ELM
activity (Chapter 3, Section 2.6, and Chapter 4, Section 3.8; see also Section 5.2 of this Chapter). The
2194

There is considerable variation in the experimental results for the scaling of the pedestal width.
Experiments on JT-60U have reported that electron and ion temperature pedestal widths δ scale asδ ∝ ρpi , the poloidal ion Larmor radius, in the ELM
free phase [256], and that it is 2–3 times greaterin the ELMy phase. Results from JET, in which
the pedestal pressure is measured and it is assumed
that the steep edge gradient is limited by the ideal
MHD high-n ballooning limit [257], imply a scaling δ ∝ (ρpi L)1/2 , where L is a macroscopic length.
Direct measurements of the width in JET [258] yield
different results. In ELM free H-mode, the electron temperature barrier width δT e is nearly constant at 3–4 cm; between ELMs it is 5–6 cm. A scaling δT e ∝ (TePED )−0.16 independent of plasma current I, is found, i.e. not correlated with ρpi . The
electron density barrier width δn ∼ δT e /2. During
ELM free H-mode, the ion temperature barrier widthscales as δT i ∝ TiPED with δn = δT i . Recent JET
experiments on the isotope scaling indicate that thepedestal height increases strongly with the isotope
mass [259]. In ASDEX-Upgrade, the barrier width δis fairly constant (δ ∼ 2–2.5 cm) and is independentof I, again implying it is not related to ρpi [254].
Experiments on DIII-D with parameters chosen tobe similar to ITER indicate that δ can be fit with twoforms, δ/R ∝ (ρpi /R)2/3 (stronger than the otherexperiments) or δ/R ∝ (βpPED )1/2 . These results canbe used to estimate the edge temperature in ITER
by assuming type I ELMs will occur at the sameα (the normalized pressure gradient parameter for
ideal MHD ballooning modes) as in DIII-D or otherexperiments at the ITER value of q. Scaling from
DIII-D discharges with a strong scaling with ρpi ,δ/R ∝ (ρpi /R)2/3 gives T PED ∝ (LBT /n3G ) = 1 keV
for ITER (this scaling also implies δ ∝ (L/nG BT )1/2 ,and pPED ∝ (BT3 /nG L)1/2 ). The form of the scaling more consistent with the other experimentsof δ/R ∝ (βpPED )1/2 gives a significantly higher
pedestal temperature T PED ∝ (LBT /nG ) = 5 keV
for ITER (here δ ∝ L, and pPED ∝ BT2 ). Similar valNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transportues (∼4 keV) are obtained from extrapolations fromother experiments (Chapter 4, Section 3.8).
4.3.

Power threshold scaling

The H-mode is reached above a certain threshold power, Pthr , which depends on plasma conditions
and machine size, and it is essential to predict whatvalue is needed for ITER. The threshold dependence
on plasma configuration and parameters, studied insingle devices during the past years [171, 260–268]
can be summarized as follows. The threshold poweris about 2 times lower for the single null (SN) configuration with the ion ∇B drift towards the X-point,
than for the opposite direction or double null (DN)
configuration; the threshold is about 2 times lower
in deuterium than in hydrogen; reduction of neutral density and impurities by appropriate wall conditioning and good divertor retention is favourable
for achieving low threshold powers. The studies insingle devices also show a rather clear linear dependence of Pthr on n̄e and BT . However, the size dependence, an essential element for extrapolation, cannot
be deduced from experiments made on single devices.
Therefore, since 1992, the ITER H-mode threshold
database has been constructed [269] and presently
includes 10 divertor tokamaks [270–273]: Alcator
C-Mod, ASDEX, ASDEX-Upgrade, COMPASS-D,
DIII-D, JET, JFT-2M, JT-60U, PBX-M and TCV.
The present version of the database contains about
150 variables, which describe the magnetic configuration, the core and the edge regions of the plasma.
The latter is believed to play a major role in transition physics as shown in Section 4.1. Care wastaken to include in the database data representing
the lower threshold from each device and, in particular, from discharges with good wall conditions.
Apart from allowing studies of the size scaling, thedatabase provides a unique framework for systematically comparing threshold data from several tokamaks. The database confirms the earlier results fromsingle devices, in particular the BT dependence, butalso shows differences for the density dependence.
The threshold power generally shows a minimum ata density n̄e,min in the range 0.1–0.25 × 1020 m−3(sometimes even exhibiting an apparent density
threshold [265]), except for Alcator C-Mod, for whichn̄e,min ≈ 0.8 × 1020 m−3 . The value of n̄e,min isaround 20–30% of the Greenwald density limit, but
no clear relation could be established so far. Aboven̄e,min , the power threshold increases linearly withdensity up to a density around 80% of the Greenwald
Nuclear Fusion, Vol. 39, No. 12 (1999)

density limit. Above this value, the power thresholdincreases with a strong non-linear dependence [274].
Scaling expressions for Pthr are obtained fromthe database by performing a linear regression of
the net heating power Pl = Pheat − dW/dt versus n̄e ,BT and machine size and geometry using R, a, S
and/or κ (major radius, minor radius, plasma surface area, elongation). The time slices are taken justbefore the L–H transition, for conditions providing
low power threshold: deuterium plasmas, ion ∇Bdrift toward X-point. Previous results were presented
in [269–273]. More recent analyses from the last version of the database (ITERTH DB2.3, September1997) are given in Table 2. Results show that the root
mean squared error (RMSE) is significantly reduced
when the size regressors R and a (expression 2) or S
(expression 3) are used, instead of R only.
Using κ as an additional regressor does not modify
the results significantly. Moreover, the κ dependence
is not well assessed because it is only provided by
the fact that ASDEX has a circular cross-section,
whereas all the other tokamaks are elongated with
1.3 ≤ κ ≤ 1.7.
Further analyses show that two groups of tokamaks can be distinguished: ASDEX-Upgrade, Alcator C-Mod, DIII-D, JFT-2M, JT-60U, JET, PBX-M
on one hand, ASDEX, COMPASS-D and TCV on
the other hand. Making a regression with data fromthe first group yields expressions 4 and 5, which havea low RMSE and are represented in Figs 6 and 7. The
seven tokamaks of this group agree with the fit within
one standard deviation. The three tokamaks of thesecond group are on a line parallel to that of expression 4 but higher by 60% (Fig. 6). For COMPASS-D
and TCV, the reason for this effect may be attributed
to the small size of the devices in which the influence of neutrals is expected to become important.
For ASDEX, it may be attributed in addition to thecircular cross-section. When expression 5 is considered (Fig. 7), only COMPASS-D and TCV are above
the line defined by the other seven tokamaks, whereas
ASDEX agrees with this fit.
At present, there is no strong reason allowing adecisive conclusion to be drawn from Table 2 orFigs 6 and 7. The values given here represent theextrapolation uncertainties of the H-mode threshold power in ITER based on the present database.
Expressions 4 and 5 of Table 2 are conservative, take
into account plasma geometry as complete as can beobtained from the database, and have a good RMSE.
For these reasons they might be recommended forextrapolation to future devices.
2195

ITER Physics Basis
Table 2. Summary of the threshold analyses with global parameters

Eq.

Excludedtokamaks

Obs.

Num.
factor

ne,20

BT

R

(1)
(2)

none

512

0.70

0.94

0.80

2.12

none

512

1.79

0.78

0.76

1.14

(3)

none

512

0.057

0.64

0.83

(4)

ASDEX, TCVCOMPASS-D

432

0.041

0.69

0.91

(5)

ASDEX, TCVCOMPASS-D

432

1.38

0.77

0.92

1.30

a

S

RMSE
%

Low
95%
MW

ITER
pred.
MW

Up
95%
MW

Constr.
eq.

30.5

67

124

230

3.04

28.3

53

95

169

2.36

0.89

28.8

49

88

157

2.15

0.96

25.2

70

116

192

2.39

25.1

79

132

224

2.54

0.78

0.76

The columns from left to right indicate: the expression numbering, the tokamaks not included in the regression, thenumber of time slices included in the analysis, the numerical factor, the exponents of density, magnetic field, major
and minor radius, plasma surface area, RMSE of the regression, the lower values of the 95% confidence interval (usual
definition) of the ITER prediction, the threshold power predicted for ITER, the upper 95% confidence interval, theresults of the sum determined by constraint with dimensionless parameters (see later in the text). The units used are
m−3 , T, m, m2 , MW.

ASDEX
ASDEX Upgrade
Alcator C-ModCOMPASS-D
DIII-DJET
JFT-2M
JT-60U
PBX-M
TCV

1

1

0.1

0.1
0.1

1.0

10

0.041 n e,200.69 B T0.91 S 0.96 (MW)Figure 6. Comparison of experimental power thresholdswith the scaling expression 4 (solid line). The dashed line
is expression 4 multiplied by 0.66, and the dotted line,
which fits ASDEX, COMPASS-D and TCV data, is 60%
above expression 4.

The H-mode being determined by conditions at
the plasma edge, the power flux across the edge,core, is a global parameter
PL = Pheat − dW/dt − Pradbetter suited to describe the power threshold. The
core, can be subradiation inside the separatrix, Prad
tracted only for some of the analyses because of
the limited data available. This excludes devices,increases the RMSE of the regression [272,273] and is
not taken into account here. Further data and work
are necessary to obtained a reliable result.
To analyse the database one may also be guided
by the observation that, for fixed values of the con2196

ASDEX
ASDEX Upgrade
Alcator C-ModCOMPASS-D
DIII-DJET
JFT-2M
JT-60U
PBX-M
TCV

10

P L (MW)P L (MW)

10

0.1

1.0
1.38 n e,200.77 B T0.92 R 1.23 a 0.76 (MW)10

Figure 7. Comparison of experimental power thresholdswith the scaling expression 5, same meaning of lines as
in Fig. 6.

trollable plasma parameters, the H-mode transition
occurs at the plasma edge and that a minimum
power flux across the separatrix is required. Starting from this assumption, using the usual dimensionless plasma variables, ν ∗ , ρ∗ , β and assuming that a power law expression for the threshY Z
old power, Pthr = n̄Xe BT R , satisfies the high-β(Kadomtsev) constraint, one obtains the following
relation between the exponents: 8X + 5Y − 4Z = 3[270, 271]. This approach implies that the L–H transition is only governed by plasma physics parameters, which is not necessarily the case at the plasmaedge where atomic physics might also play a role.
The exponents of the variables in the scaling expresNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
sions (2)–(5) in Table 2 yield values for 8X +5Y −4Z
between 2.15 and 2.54, as indicated in Table 2, therefore approaching, but not quite meeting, the condition derived with dimensionless variables.
Under the assumption of a linear BT dependence(Y = 1) as observed in all the tokamaks, Pthr =R2 is dimensionally correct and in roughCBT n̄0.75
eagreement with the experimental observation within
the uncertainties. The uncertainties can be formulated by the quantity (n̄e R2 )α , where α is determinedfrom the analysis of the database. This leads to thefollowing expression (10), given in [271], where thereader can find a complete derivation:
R2 (n̄e R2 )αPthr = C(R/a, κ, q, α)BT n̄0.75
e(10)

where C is a non-dimensional coefficient. Using thedatabase version ITERTH DB2.1 (September 1996),
one finds: −0.25 ≤ a ≤ 0.25 and C = (0.45 ± 0.1) ×
0.6α . The range in α is obtained by analysing thedata scattering of the database while varying thedensity exponent X in the Kadomtsev constraint,
8X + 5Y − 4Z = 3, and maintaining Y = 1,
as explained in [271]. Expression (10) leads to arange for the threshold power prediction for ITER
between ∼50 MW and ∼200 MW. The uncertainty
is dominated by the R dependence, R1.5 –R2.5 . Since
this work was performed, efforts have been made to
reduce the scattering of the data, and are expected tosomewhat reduce the uncertainties of future results.
The power threshold in ITER predicted by the
database at present has a considerable uncertainty(from ∼50 MW to ∼200 MW) and the upper value
exceeds the currently planned heating power. The
high required power and the large range of the uncertainty are due mainly to the exponent on the size
dependence, which is close to 2 in all the cases. It
must be stressed here that, extrapolating from JET
and JT-60U to ITER, the size dependence is the
major contribution compared to those from n̄e andBT . For the extrapolation to ITER, one assumes
that the conditions for a low threshold, as required
in present tokamaks and listed above, will be fulfilled. This will, most probably, be the case: a SN
configuration with favourable ion ∇B drift is foreseen in ITER, high divertor retention and low recycling wall conditions will provide low neutral density. The expressions given in Table 2 are obtainedby a free fit through the standard dataset. According to present knowledge, effects causing a particularly high threshold in ITER are not expected. On
the contrary, one may expect the threshold powerin ITER to be kept low for the two following reaNuclear Fusion, Vol. 39, No. 12 (1999)

sons. First, if one assumes that the above conditions
that provided the lowest threshold values in present
tokamaks can be achieved in ITER, one may reduce
the numerical factor of expressions 4 and 5 suchthat the corresponding line goes through the lower
point boundary, as suggested by the dashed line inFigs 6 and 7. This decreases the threshold prediction
for ITER by about 30%. Second, as was mentioned
above, the power threshold is lower in deuterium
than in hydrogen. Very recent experiments in JET
with tritium indicate a ∼25% decrease of the power
threshold in DT (50%–50%) plasmas [275]. These
results yield a power threshold that is inversely proportional to the effective plasma mass. This mightalso lower the power threshold in ITER by about
25% in DT operation. It may even be valuable tofirst reach the H-mode in pure tritium to take advantage of the lower threshold and then add the necessary deuterium. In several devices a power hysteresis
has been observed for the H-mode threshold: about
1.5–2 times more power is required to achieve the
H-mode than to sustain it. However, the hysteresis
is not observed in JET [276] and it disappears when
the density is increased towards the density limit, asshown in ASDEX-Upgrade [274]. The latter effect is
attributed to confinement degradation observed athigh density [277]. Therefore, one cannot rely on thehysteresis for ITER prediction in the present statusof understanding, and further investigations are necessary on this topic.
A more precise assessment of the ITER threshold power demands further understanding and quantification of the influence of plasma geometry, edge
parameters and neutrals on the threshold, as well as a
reduction of the data scatter for each tokamak. Suchstudies are being actively pursued in several tokamaks and in the framework of the database activity.
The understanding of the effect of neutrals is still
controversial, in particular for the region around the
X-point. For the rest of the main plasma it is clearthat smaller devices and low density cases are sensitive to viscous damping of rotation by neutrals,
whereas in larger devices at high density, neutrals
only affect the very edge of the plasma, and possibly do not reach the region where the L–H transition happens. Significant progress has been made on
threshold studies with edge data and their comparison with theory (see the Sections above). It appears
in several devices that the edge electron temperature
at the L–H transition consistently depends linearlyon BT , seems to increase with Ipγ , with 0.5 ≤ γ ≤ 1,
and depends only weakly on n̄e . Note that these2197

ITER Physics Basisdependences are in agreement with the global scaling
presented above, even for Ip and n̄e , as demonstrated
by the following explanation. The required edge temperature depends on heating power and edge transport. Therefore, the higher edge temperature necessary at higher Ip is provided, at least partly, by
the increase of confinement with Ip (L-mode confinement). Thus, Pthr depends weakly on Ip at given BT
and n̄e . Similarly, the weak dependence of the edge
temperature on density is consistent with the linear
dependence of Pthr on n̄e : at higher n̄e , the edge
density is also higher and more power is required tosustain the same edge temperature for given Ip andBT .
However, the large scatter of the edge data
presently available in the threshold database prevents one from making a meaningful prediction for
ITER; this still requires additional work. It is also to
be underlined that a prediction of the required edge
values to achieve the L–H transition in ITER will beof practical significance only if one is also able to predict the associated heating power. For this purpose,
reliable transport modelling will be necessary.

5.

Impact of global instabilities
on transport

A number of large scale MHD phenomena,
described more fully in Chapter 3, Section 2, canhave an impact on global confinement. Two of these
are the periodic sawtooth instability, which can havea significant effect on the profiles of temperature,density and impurities in the central core region, andthe ELMs, which periodically affect the plasma edge
region. These are discussed more fully in Sections 5.1
and 5.2, respectively. A third candidate is the transport induced by low-m, n magnetic islands, wherem and n are poloidal and toroidal mode numbers;
such an island is located at the resonant surface rs
where m = nq(rs ), q being the safety factor. Because
the temperature is rapidly equilibrated along thereconnected magnetic field lines within islands, theyeffectively short-circuit the normal transport acrossnested toroidal magnetic surfaces, decreasing theeffective size of the plasma. An expression for thedeterioration in plasma energy δW arising from thepresence of an island of width w located at a rationalsurface rs in a plasma of minor radius a is [278, 279]



3 
20
rs w
rs2
rs2
δW
=
1− 2 1− 1− 2
W
3
a
a
a a
2198

where W is a stored energy without the magnetic
island. These magnetic islands can arise from tearing
modes, possibly of a neoclassical origin (Chapter 3,
Sections 2.2 and 2.3). They will, therefore, tend tooccur near operational limits, e.g. low q, or higher βN(the normalized β); in particular, they could lead toan onset of confinement degradation when βN > 2.
The transport effects of low-m, n modes have been
observed in TFTR supershots [279].
5.1.

Sawteeth

When the central value of the safety factor q
falls below unity, relaxation oscillations are normallyobserved in the core of a tokamak plasma. These
appear on a number of plasma parameters but are
particularly evident in the central electron temperature Te (0) [280]. The oscillations in Te (0) exhibit atime trace with a distinctive sawtooth shape consisting of a slow rise or ‘ramp’ phase, during which theplasma inside q = 1 heats up, followed by a rapid
collapse or ‘crash’ when the plasma energy is redistributed from the core to the region outside q = 1, i.e.
over a region within the so-called mixing radius rm .
This then propagates as a heat pulse to the plasma
periphery. The position where the perturbation in Techanges sign is known as the inversion radius ri . Thispattern repeats with a period τsaw . This mechanism
has the effect of degrading the global energy confinement time τE as rm /a becomes significant, typicallyfor q95 ≤ 3. Experiments on DIII-D (discussed in[281]) show that the increase of H-mode confinementwith current saturates for Ip /BT ≈ 1 MA· T−1 , corresponding to q95 ∼ 3, although JET shows no appreciable degradation down to q95 ∼ 2.0 [282]. Varying
q95 in DIII-D by means of scan over elongation κshows that it is not q95 that determines the confinement degradation due to sawteeth. Rather, experiments on DIII-D and JET indicate that this degradation increases with rm /a. JT-60U has demonstrated a deterioration in confinement as the sawtooth period τsaw decreases [283, 284]. Chang andCallen [278] have proposed a model for the sawtoothdegradation factor fsaw depending on Am , the relative area of sawtooth mixing, and x = τsaw /τE . For aconstant thermal diffusivity and central heating, thistakes the form
fsaw = (1 − Am )g(x) + 1 − g(x)with
g(x) = (1 − e−x )/x.
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
Sawteeth have an effect on the plasma density andalso the distribution of impurities. There is evidence
that some plasma density is removed from the sawtoothing region, leading to an outwardly propagating
density pulse. However, the effect is less than in thecase of temperature; for example, the density profilein the core becomes somewhat less peaked, rather
than flat, in the TEXT tokamak [285]. The impact
on impurities is more significant. The sawtooth canboth effectively purge accumulated core impurities
and allow impurities diffusing inwards to rapidly penetrate the region inside q = 1. Reference [286] provides a source of references on the experimental evidence for this. In particular ASDEX-Upgrade, operating in the CDH (completely detached H-mode),demonstrates a density peaking, improved confinement and impurity accumulation when sawteeth areabsent [287].
Because the thermonuclear power depends nonlinearly on plasma pressure p (Pf us ∝ p2 ), the redistribution of plasma energy due to sawteeth would
cause a periodic overall power loss in an ignited tokamak. If the scale-length for the central pressure is rp
so that p = p0 (1 − r2 /rp2 ), the fractional loss of power
at a sawtooth collapse would be ∼ 21 (ri /rp )6 . This
would typically imply a power loss of tens of MW.
Thermal pulses from the sawtooth collapse could
lead to undesirable transient heat loads on divertor
plates when the mixing radius is large. A large sawtooth crash could excite other MHD phenomenon,
e.g. ELMs or neoclassical tearing modes.
The sawtooth phenomenon is believed to be associated with an instability having an m = n = 1
structure, which arises when a q = 1 surface ispresent. The q profile also oscillates with a sawtooth
behaviour, falling during the ramp as the central current density increases with increasing Te , and rising
sharply at the crash, typically by a few per cent. Measurements of the central q vary; typically q0 ∼ 0.7
is observed [288], but there are cases with q0 ∼ 1reported.
Kadomtsev proposed a resistive MHD model toexplain early observations of sawtooth behaviour.
During the ramp phase, as Te (0) heats up and q0
falls, a magnetic island begins to grow at the q = 1
surface as the result of an m = n = 1 instability. The
resulting magnetic reconnection occurs on a crashtime-scale τc ∼ (τR τA )1/2 where τR is the resistive
diffusion time and τA the Alfvén time. As a consequence, there is a redistribution of poloidal magnetic
flux that terminates when q > 1 everywhere. Associated with this there is a similar redistribution of
Nuclear Fusion, Vol. 39, No. 12 (1999)

energy leading to a flattening of the√ n and Te profiles out to the mixing radius rm ∼ 2r(q = 1) [289].
However, later experiments exposed a number of
weaknesses in this model:(i) it is unclear why the instability does not growthroughout the period when q < 1;(ii) the event that triggers the onset of the crashand sets the period τsaw of the sawteeth is amystery;(iii) the time-scales for the crash in larger, hotter
tokamaks are much shorter than predicted bythis resistive model; and(iv) the small changes in q0 arising from resistive
diffusion are incompatible with q0 ∼ 0.7, suggesting that the very centre does not undergomagnetic reconnection; on the other hand, Te
is flattened throughout the core region.
All these topics are areas of active research and possible explanations have been proposed.
Nevertheless, the Kadomtsev description offers abasis for incorporating the effects of sawteeth intransport codes. The basic features of such a modelfor the effect of the sawtooth are as follows. Whenq0 falls below unity (or some other critical value, say0.7), a repetitive flattening of Te and Ti (and possibly n) over a specified region r < rm is performed
each sawtooth period τsaw . Both rm and τsaw canbe regarded as parameters√ to be explored, thoughprescriptions like rm ∼ 2r(q = 1) and determining τsaw from kinetic stabilization criteria for them = n = 1 mode have been invoked [290]. Morecomplex prescriptions have been proposed to explain
why q0 remains well below unity [291]. A model forthis redistribution of impurities, based on the sameideas as in [291], has been given in [286].
The interaction of sawteeth and energetic particles, arising as fusion products and from RF heating, is another topical research area. While there
is conflicting evidence from JET on whether suchparticles undergo the redistribution experienced by
the thermal particles [292], a substantial redistribution of alpha particles has been observed on TFTR,
although losses are small [293]. Transport codes for
simulating burning plasmas can prescribe an instantaneous loss of alpha heating power at the sawtoothcrash to represent a possible loss of alpha particles,
which then recovers over an alpha particle slowing
down time. On the other hand, there is experimental
evidence from JET [294, 295] and TFTR [296] and2199

ITER Physics Basistheoretical arguments [290,297] that energetic particles can stabilize sawteeth for long periods. In such acase, the tokamak experiences large amplitude ‘monster’ sawteeth [294]; these effects can also be incorporated in modelling codes [290]. The collapse of suchmonster sawteeth could have serious implications fordivertor target heat loads.
Advanced tokamaks could possibly achieve
improved performance by stabilizing sawteeth by: (i)current profile control to maintain q > 1; (ii) local
current profile control near q = 1; and (iii) energetic
particle stabilization.
Edge localized modes

The ELM is a relaxation oscillation triggered by
an MHD instability, which leads to a fast (ms)loss of particles and energy from the plasma edge.
ELM physics has been summarized in recent reviews[298, 299] and is discussed more fully in Chapter 3,
Section 2.6. From the ITER viewpoint, ELMs are
beneficial because they lead to impurity and heliumash expulsion from the plasma edge, allowing cleaner
plasmas. However, as is discussed in Chapter 4, Sections 3.8, 4.2.4 and 6.4, the cost of these benefits isthe need to handle the heat pulses to the divertor
plates produced by the ELMs. Although ELMs do
reduce the global energy confinement by 10–20%, aswill be seen presently, they have a much larger effecton the particle confinement than on the energy confinement. Accordingly, the use of ELMs to control
impurities and helium ash does not impose a large
energy confinement penalty.
There are two major types of ELMs of interest
to ITER. Type I ELMs exhibit a repetition frequency that increases with increasing input power,
while type III ELMs have a repetition frequency that
decreases with increasing input power. In general,
type III ELMs occur when the edge electron temperature is fairly low, while type I ELMs occur at higher
edge electron temperatures [298]. The exact MHD
mode associated with ELMs is, as yet, unknown. Theedge density and temperature parameter range forthe various types of ELMs is illustrated in Fig. 8 fora case from DIII-D [252]. The type I ELMs appear
to be driven primarily by the edge pressure gradient [255], while the type III ELMs depend bothon the edge density and the edge electron temperature [252, 300] suggesting a role for resistivity oredge current [264, 298]. As is seen in Fig. 8, there
appear to be two branches for the type III ELMs,
one at low density and one at low temperature. Clear
2200

1.5

ITER shapeq95 = 3.2, Ip = 1.5= Type I ELMs
= Type I ELMs(< 3ms before ELM)
= Type III ELMs
= L-mode

1.0
TePED (keV)

5.2.

MHD precursors have been seen for the type I andtype III ELMs, although the precursors for the type I
are much more difficult to detect [301]. In small
machines (e.g. ASDEX), type III ELMs can be seen
even at input power levels adequate to drive theplasma to the MHD beta limit. In larger machines(ASDEX-Upgrade, DIII-D, JET), the power needed
to approach the beta limit is sufficient to heat theedge to the point where type I ELMs occur. Basedon this observation, ELMs in ITER will probably betype I ELMs, although it may be possible and moredesirable to operate with smaller type III ELMs,
which occur if the edge density is sufficiently high.

0.5

0
0

2

4

6

8

nePED (1019 m–3)

Figure 8. The edge electron density and temperaturein DIII-D discharges with different types of ELMs [252].

An examination of the global power balance intype I ELMing H-modes in DIII-D [302] showed that
PELM , the power lost through ELMs, is less than
20% of the total input power PT . In DIII-D, thisfraction was found to decrease with increasing PT
so that the energy loss per ELM, δE, decreased withPT , while the ELM frequency νELM was proportional
to PT . Accordingly, PELM was roughly independentof PT over the power range studied [302]. However,in later work, PELM was found to be proportional toPT [252]. A similar behaviour is found in ASDEXUpgrade [303], where δE is independent of PT andνELM increases with PT , leading to PELM /PT ∼ 25–
40%. The difference between the two behaviours
may be the longer duration of the ELMs in theearlier DIII-D study. There, the plasma appears totransiently return to L-mode after each ELM, withthe duration of the L-mode decreasing with increasNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
ing PT . Above a certain heating power, which will
depend on the H-mode threshold power, the durationof the ELM is roughly constant. Thus, the statementthat the energy loss per ELM is roughly constant,would be valid only above a certain heating power.
Since type I ELMs can induce a return to L-mode[304], the ITER design must assure that there is sufficient power flow through the plasma edge to allow
a return to H-mode after an ELM at the parameterscharacteristic of the burn phase.
The energy loss due to ELMs causes a reductionof the global energy confinement time τE . Studies on
DIII-D showed that, in type I ELMing discharges,
τE is reduced by 10–15% [304,305], although JT-60U
shows larger reductions in the presence of higher frequency ‘giant ELMs’ [283,306]. A more general investigation of the influence of ELMs on τE was done byestablishing a scaling of τE in ELMy H-mode [305].
A comparison with a similar scaling for the ELM
free H-mode shows that, for present experiments, thereduction in τE is indeed around 10–20%. For larger
devices, especially for the ITER parameters, the two
scalings agree within the statistical errors. Such a
result is expected if ELMs can be considered a surface effect. However, the scaling given in [305] mixes
data with type I and type III ELMs with their different power dependences of the repetition frequency.
Accordingly, we need better data without this confusion to make a firmer conclusion for ITER. In any
case, it appears that the confinement reduction willprobably not be larger than 20%.
A different way to estimate the confinement degradation due to ELMs is to quantify the reduction due
to ELMs by a factor η, defined byscaling of η for future machines critically depends
on the scaling of rELM , which also enters into thescaling of δE and, therefore, PELM . A model for theloss of energy by ELMs based on transport due tothe stochastic magnetic field caused by the ELM
precursors has successfully described results from
COMPASS-D [309, 310] and also ASDEX-Upgrade.
The effect of the ELMs on the particle confinement time is also of major interest. A quantitative analysis of the effect of ELMs on particle confinement time has not yet been given. Experimental results on a number of machines clearly showthat the density is constant in ELMy H-mode, while
it increases monotonically in ELM free H-mode. In
both modes of operation, the total energy content
reaches a steady value. These results indicate that
ELMs reduce global particle confinement time much
more than they reduce τE . As was shown in DIII-D,
the particle density control in ELMy H-mode also
makes removal of helium ash from the plasma possible [128, 187].
The difference in the effect of ELMs on particle
and energy loss can be due to the different source
profiles of energy and particles: as was pointed out
in [311], an edge localized loss phenomenon is most
severe for a quantity whose source profile is also edge
localized, as is the case for particles. For a central
source profile, which is mostly the case for plasma
heating, the source is not directly affected and theeffect on global confinement is less severe. This willbe also true for ITER, where the alpha heating profile will be comparable to present day neutral beam
heating profiles.

η = τEELMy /τEELM −free .

6.

Global energy confinement scalings

The precise value of η has to be found by an analysisof the profile effects due to the ELM. Assuming a
diffusive process and spatially separated source andsink profiles, which is a reasonable assumption for
centrally peaked heating profiles, one can calculate
[307]

6.1.

Introduction

η = 1 − [1 − (rELM /a)2 ]PELM /PT .
Accordingly, due to the localization of the ELMs, theconfinement degradation is appreciably lower than
the fraction of power transported across the separatrix by ELMs. Typical values from ASDEX-Upgrade
are PELM /PT = 0.3 and rELM /a = 0.8 [308], resulting in η = 0.9. This agrees reasonably well with theη = 0.85 result from DIII-D mentioned above. The
Nuclear Fusion, Vol. 39, No. 12 (1999)

Due to the complexity of the processes determining heat and particle transport in thermonuclear
plasmas, it is not yet possible to provide a firstprinciples derivation of the dependence of energy
confinement properties on plasma parameters. The
description of the global energy confinement time byempirical scalings that are based on relevant datasets
within specific operating regimes such as L-modeor H-mode has, therefore, become the key tool in
extrapolating plasma performance to ITER. As wellas predicting the performance of a next step device,
such as ITER, in terms of its basic design parameters, such scalings can also be used as a normalization for plasma energy in one dimensional simulation
2201

ITER Physics Basis
codes, which use various transport models that predict the plasma behaviour in ITER, or as an approximate constraint on the form of theoretical models.
When expressed in terms of dimensionless plasma
parameters, scaling expressions can also serve as a
guide to modellers by emphasizing different types oftheory based transport loss mechanisms, for example
electrostatic versus electromagnetic loss mechanisms,
which differ in their dependence on beta.
The present database activity originated in aninternational L-mode database, initiated by S. Kaye,in the early eighties, that led to the ITER89-P scaling [136] during the ITER conceptual design activity (CDA) phase. The initial multi-tokamak H-mode
confinement database was assembled by an H-mode
database working group in 1989, with contributions
from six machines (ASDEX, DIII-D, JET, JFT-2M,
PBX-M and PDX). Basic documentation and analysis of this dataset can be found in [312].
An extended version of this database [180,313,314]
formed the basis of such scaling expressions as
ITERH92-P(y) [305] and ITERH93-P scaling [180],
describing the global thermal energy confinement
time in ELMy and ELM free discharges, respectively.
This database included extended plasma parameter
ranges for most of the constituent devices, as well as
improved estimates of the thermal energy confinement time. It also addressed several detailed issues,
such as the combination of different energy measurements (from diamagnetic measurements and MHD
equilibrium fits) and the correct B (or q) and κ [314]exponents.
Most recently a new working version of thedatabase has been established (‘DB3’), which
includes data from additional tokamaks. Details of
the new dataset are discussed in Section 6.2 andin [315]. This has necessitated the development of
a new set of selection criteria, for example to include
RF heated as well as NBI heated plasmas. The subset of ELMy data in the new database has been
improved considerably in terms of the uncertainty
in projecting to ITER, as is discussed in Section 6.3
and 6.4.
In Section 6.3, by using physical relationshipsthat hold ‘on radial average’, confinement scaling
expressions given in engineering variables (which are
directly under the control of the experimentalist or
machine designer) are transformed into expressions
in dimensionless physical variables, which have aclose connection to physical theories, e.g. ρ∗ the normalized ion Larmor radius, β the normalized plasmapressure, and ν ∗ the normalized plasma collisional2202

ity [72, 73]. On a logarithmic scale, this corresponds
to a linear transformation of the response and regression variables. Within the class of simple power law,or log-linear, models, one then has the same scaling
expressed in two different sets of variables [313, 316].
In Section 6.4 the uncertainties associated withthese ITER projections are discussed and functional
forms other than the simple power law form aretouched upon (a more detailed discussion is givenin the Appendix).
Issues concerning the impact on confinement of
ITER’s proximity to operational limits, such as theGreenwald density limit and β limit, are areas for
future study as new data near these limits is added
to the database. Although difficult to achieve, good
H-mode confinement has been obtained above theGreenwald density, for example n/nGW < 1.5 withpellet injection in DIII-D [317], indicating that there
is no fundamental obstacle to operation in this density regime (see Chapter 3, Section 3; Chapter 4, Sections 3.3, 3.4 and 6.2; and Chapter 9, Section 3.4, fordetails).
6.2.

H-mode global confinement database

The assembly of the latest version of the ITER Hmode confinement database, ITERH.DB3, was completed in September 1997. This version contains
data from 12 different tokamaks: ASDEX∗ , ASDEXUpgrade, COMPASS-D, JET∗ , TCV and TEXTOR
from Europe; JFT-2M∗ and JT-60U from Japan;
Alcator C-Mod, DIII-D∗ , PBX-M∗ and PDX∗ fromthe USA (∗ indicates contributors to the old database
ITERH.DB2). All the ITERH.DB2 data are in the
new database and a detailed description of these data
can be found in [313]. The main characteristics ofthe new H-mode data specific to ITERH.DB3 aredetailed in Table 3.
It should be noted that the number of discharges
contributed by some of the new machines is verysmall. Whereas the data from each of the six contributing tokamaks in ITERH.DB2 included both
ELM free and ELMy NBI heated H-mode data, it
is apparent from Table 3 that the ELM free subset of ITERH.DB3 does not contain data from all
the machines. A second new feature is that no single
heating method is employed to obtain H-mode on all
devices. As a result, it has been necessary to redefine the selection criteria used to delimit the standard analysis dataset of ITERH.DB2 [313] to avoid
excluding some machines from the standard analysis
dataset of ITERH.DB3. The new selection criteria,
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
Table 3. Main characteristics of the new H-mode data specific to the new ITER H-mode confinement database,
ITERH.DB3
Alcator C-ModICRF heated ELM free and ELMy data

ASDEX-Upgrade

ICRF and NBI heated type I ELMy data

COMPASS-D

Ohmically heated ELM free and ELMy data
ECRF-heated ELMy data

DIII-DECRF and NBI heated ELM free and type I ELMy dataJET

ICRF and NBI heated ELM free and type I ELMy data

JT-60U

NBI heated ELMy data

TCV

Ohmically heated ELM free and ELMy dataTEXTORICRF and NBI heated RI-mode data (for comparison with H-mode)

which are comparable to the previous criteria butallowing all heating schemes, are listed in Table 4.
Applying these criteria to ITERH.DB3 results in astandard ELM free H-mode dataset of 1131 observations with contributions from nine tokamaks (Alcator C-Mod, ASDEX, COMPASS-D, DIII-D, JET,
JFT-2M, PBX-M, PDX and TCV), and a standard
ELMy H-mode dataset of 1398 observations from11 tokamaks (Alcator C-Mod, ASDEX, ASDEXUpgrade, COMPASS-D, DIII-D, JET, JFT-2M, JT60U, PBX-M, PDX and TCV).
6.3.

Power law scaling expressions

The power law scaling expressions for thermalenergy confinement time, τth , can be expressed eitherin ‘engineering’ variables as
f it= CI αI B αB P αP nαn M αM RαR εαε κακ
τth

xτ

x

x

(12)

The ‘engineering’ variables are R = major radius(geometric centre), I = plasma current, B = toroidal
magnetic field (at major radius R), P = loss power(corrected for charge exchange and orbit losses),n = line average density, κ = elongation, ε =inverse aspect ratio, and M = average ion mass.
The ‘physics’ variables are the Bohm time, τB , normalized toroidal Larmor radius, ρ∗ , normalized collisionality (with Zeff = 1 assumed), ν ∗ , normalized
plasma pressure, β, cylindrical safety factor, q, andare defined byτB ==(minor radius)2
Bohm diffusion coefficienta2 B∝ ε2 R2 BT −1
T

Nuclear Fusion, Vol. 39, No. 12 (1999)

β=

(14)

plasma pressure
magnetic pressure

2µ0 ne(Te + Ti )∝ nT B −2(15)
B2
connection lengthν∗ =trapped particle mean − free path
1/2  3/2
MiR
qR ∝ nRT −2 qε−3/2
= νii
eTi
a
(16)
=

qcyl =RBf (κ, δ) ∝ BRI −1 ε2 κ
ε2 I

(17)

(11)

or in ‘physics’ variables as
f it= C1 τB B ρ∗ ρ∗ ν∗ ν∗ β xβ M xM q xq εxε κxκ .
τthion gyroradiusminor radius
1/2

2eTi
Mi
∝ (M T )1/2 /εRB
=
Mi
eBa

ρ∗ =

(13)

with Ti in eV and f (κ, δ) a function of the plasma
shape parameters. These obviously vary across theplasma profile, but for a global analysis temperature,
T can be replaced by [(P τth )/(6π 2 enε2 κR3 )], using
the power balance relation. (See Section 7 for a moredetailed discussion of dimensionless parameters.)
On a logarithmic basis the power law expressions
are linear and the above relations for the ‘physics’
variables define a linear transformation between the‘engineering’ and ‘physics’ variables and between
their exponents in the scaling expressions. In the following subsections, ordinary least square regression
techniques have been used to determine the exponents in the scaling expressions. This requires thatthe errors on the independent (regressor) variables
are negligible compared to that on the confinement
time. As this is not satisfied for the ‘physics’ variables, the regressions must be done using the ‘engineering’ variables. In addition, various ‘physics con2203

ITER Physics Basis
Table 4. Selection criteria for the standard analysis dataset of ITERH.DB3
1. H-mode data only, with no restriction on heating scheme
2. All essential data available
3. Pellet discharges are excluded
4. Limits on dW/dt
5. Limits on total radiation
6. Limits on q95
7. Limits on fast ion energy content
8. Limits on β9. Hot ion H-mode data are excluded
10. 1987 JET data are excluded

strained’ scalings can be tested against the data.
A constrained scaling has one or more constraints
imposed on the exponents in the scaling. For example, for the high-β [73], or Kadomtsev [72], constrained scaling, the constraint is xτB = 1 in physics
variables (or 4αR − 8αn − αI − 3αP − 5αB = 5 inengineering variables). The gyroBohm constrained
scaling [47] has two constraints imposed, that of
the high-β constraint and xρ∗ = −1 (or 6αR −22αn − 9αI − 12αP − 15αB = 0). The Bohm constrained scaling also has two constraints imposed,that of the high-β constraint and xρ∗ = 0 (orαR − 7αn − 4αI − 7αP − 5αB = 0). See also [316] forfurther details. In the above definitions, the quantities τB , ρ∗ and β are based on toroidal quantities,
but definitions based on poloidal quantities can also
be formulated.

tokamak in turn by 10%, the application of equal
tokamak weightings, and the use of various formsof open/closed divertor corrections to the ASDEX
and/or PDX data, do not change the prediction toa large extent. In only a few cases do the ITER
predictions differ by as much as 20%. Moreover,in contrast to ITERH.DB2, the ITERH.DB3 ELMy
H-mode standard dataset satisfies the high-β constraint. The addition of Alcator C-Mod data seems
to be responsible for this last result. The resulting
high-β constrained ELMy H-mode scaling expression
for ITERH.DB3 is, in ‘engineering’ variables,
ELMy
τE,th
= 0.0365I 0.97B 0.08 P −0.63 n0.41

× M 0.20 R1.93 ε0.23 κ0.67(18)(in s, MA, T, MW, 1019 m−3 , AMU, m), which translates to the ‘physics’ form

6.3.1. ELMy H-mode thermal confinement scalings
The new ELMy H-mode standard dataset as
defined in Section 6.2 for ITERH.DB3 is significantly
better conditioned than that of ITERH.DB2. Not
only is the database mean of each of the engineering parameters closer to the ITER parameters, but
the ranges in R, n, I, P and B are larger. Only threecorrelation coefficients are larger than 0.7 (between
I and P ; I and R/a; I and κ). Principal component analysis shows that the extrapolation to ITER
is greater than four standard deviations in only one
direction and that this is along the largest principal
component. This implies that the uncertainty in the
ITER prediction using log-linear scalings is reduced.
The new ELMy H-mode standard dataset provides the basis for a robust confinement prediction
for ITER. Even large perturbations to the dataset,such as removing each tokamak in turn, systematically increase or decrease the confinement of each
2204

ELMy
∝ τB ρ−0.83
β −0.50 ν∗−0.10
τE,th∗× M 0.97 q −2.52 ε−0.55 κ2.72 . (19)The RMSE for this (Eq. (18)) fit, which is shownin Fig. 9, is 15.8%. For ease of future reference, wecall this scaling IPB98(y). Its confinement time prediction for ITER is 6.0 s. As described in Section 6.4,
other empirical log-linear scalings have been derived
that are based on subsets of the standard dataset andwhich use another definition of kappa to account for
the relatively high confinement in the bean shaped
PBX-M tokamak. One such scaling, IPB98(y,2), is
mentioned in particular. It is expressed in engineering variables as
ELMy= 0.0562I 0.93B 0.15 P −0.69 n0.41
τE,th× M 0.19 R1.97 ε0.58 κ0.78
a(20)

Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport(in s, MA, T, MW, 1019 m−3 , AMU, m), and in‘physics’ variables as

10
ASDEX
DIII-D

ELMy∝ τB ρ−0.70
β −0.90 ν∗−0.01 M 0.96 q −3.0 ε0.73 κ2.3
τE,th
∗
a .

JET
JFT-2M

(21)

PBX-M
1

6.3.2. ELM free H-mode thermal
confinement scaling
The ITERH.DB3 ELM free H-mode standard
dataset of 1131 observations, as defined in Section 6.2, satisfies both the high-β and the gyroBohm
constraints, as was the case for ITERH.DB2. The
Nuclear Fusion, Vol. 39, No. 12 (1999)

PDX
ASDEX UpgradeτE,th (s)The RMSE of this scaling (Eq. (20)) with respect
to the ITERH.DB3 standard dataset is 15.6%, andits prediction for ITER is 4.9 s. It should be mentioned that the Kadomtsev constraint is not satisfied
when Alcator C-MOD is removed from the dataset
on which IPB(y,2) was based. Everything else being
kept the same, this leads to, instead of Eq. (20), the
IPB98(y,3) scaling in Table 5, which gives a very
similar point prediction for ITER as IPB98(y,2). At
present, available physical empirical evidence is felt
not to be conclusive enough to justify making a preferential recommendation between the just mentioned
log-linear scalings.
The scalings (18) and (20) are not very different from the ITERH.EPS97(y) scaling [315], which
was based on an earlier version of the DB3 databaseand on using TAUC93 rather than TAUC92. (Asin previous regression analyses, a correction factor
TAUC92 [305] or TAUC93 [180] has been used tonormalize the data from closed divertor configurations in ASDEX and PDX to data from the more
ITER like configurations found in the other devices.
TAUC92 and TAUC93 differ only in the method fornormalizing the PDX data, see [180,305].) Eqns (18)and (19) were developed using TAUC92. However,when the TAUC93 normalization is used instead, thedata also satisfy the gyroBohm constraint, and theconfinement time prediction for ITER increases byless than 5%.
Although the new dataset is clearly better conditioned than the previous dataset, some of the existing problems remain and new complications havebeen added. For example, a limitation is that it isstill not possible to establish distinct scalings forthe various ELM types with the current database.
It should also be noted that the different heating
schemes may introduce new systematic differences
between the machines through heating profile effects
that are not dealt with in this global database.

ALCATOR C-ModCOMPASS-D
JT-60U
0.1

TCV
ITER

0.01

RMSE = 15.8%
0.01

0.1

τ E,IPB98(y) (s)110Figure 9. Comparison of H-mode thermal energy confinement time with the scaling expression in Eq. (18) for
ELMy data in the ITER H-mode database version DB3.

resulting high-β constrained ELM free H-mode scaling expression for ITERH.DB3 in ‘engineering’ variables is
ELM free= 0.0314I 0.94B 0.27 P −0.68 n0.34
τE,th× M 0.43 R1.98 ε0.10 κ0.68(22)(s, MA, T, MW, 1019 m−3 , AMU, m), which translates to the ‘physics’ form
ELM free∝ τB ρ−0.89
β −0.92 ν∗−0.13
τE,th∗× M 1.78 q −2.77 ε−1.17 κ2.90 . (23)The RMSE for this fit is 15.6% and the distribution of the fit is shown in Fig. 10. Equation (22) issimilar to the scaling developed from ITERH.DB2,
referred to as ITERH93-P [180], and the projections
to ITER are almost the same.
6.3.3. L-mode thermal confinement scaling
The present L-mode database [112] consists of
2938 observations from 14 tokamaks (Alcator CMod, ASDEX, DIII, DIII-D, FTU, JET, JFT-2M,
JT-60U, PBX-M, PDX, TEXTOR, TFTR, ToreSupra and T-10), of which 1881 are L-mode points.
The remainder relate to ohmically heated andenhanced L-mode operation. The L-mode database2205

ITER Physics Basisexactly constrained ‘physics’ expression

250
RMSE = 15.6%

LτE,th∝ τB ρ0.15
β −1.41 ν∗0.19 M 0.67 q −3.74 ε−0.09 κ3.22 .
∗
(25)

150

100

50

0

0

0.5

1.0

τth/τfit

1.5

2.0

Figure 10. Distribution of H-mode thermal energy confinement time about the scaling expression in Eq. (22)for ELM free data in the ITER H-mode database version
DB3.

contains sufficient fast ion information, in the case ofneutral beam injection, to calculate both the thermal
and the global (i.e. including fast ions) confinement
times. This subsection is concerned with the results
for the thermal confinement time.
In the L-mode database, 1312 observations contained enough information to determine τth . Of
these, 861 were from limiter discharges and 451 werefrom divertor discharges. The latter came predominantly from more modest sized devices with greater
shaping capability, for example DIII-D and JFT-2M.
The limiter subset contained 627 observations from
JET, JT-60, TFTR and Tore-Supra, while the divertor subset contained no TFTR or JET L-mode. Discharges with helium gas were excluded to avoid difficulties with the species dependence (i.e. both A andA/Z).
A standard power law regression gives the following fit to the thermal confinement data for the combined limiter and divertor data subsets(24)(s, MA, T, MW, 1019 m−3 , AMU, m). The Kadomtsev constraint is satisfied within statistical uncertainty. Multiplication of (24) by R−0.05 leads to the
2206

1.0
Alcator C-ModASDEX
DIII-DJET
JFT-2M
JT-60
PBX-M
PDX
T-10
TEXTOR
TFTR
Tore-Supra

0.1

0.01
R2 = 0.97
RMSE = 15.8%

0.001
0.001

0.01

τITERL 96 (s)

0.1

1.0

Figure 11. Comparison of L-mode thermal energy confinement time with the scaling expression in Eq. (24) in
the ITER L-mode database version DB1.

6.4.

L
= 0.023I 0.96 B 0.03 P −0.73 n0.40
τE,th× M 0.20 R1.83 ε−0.06 κ0.64

The RMSE is 15.8% (see Fig. 11) and the thermal energy confinement time extrapolation for ITER
is 2.2 s. Virtually no overall difference between thedivertor and limiter data with respect to the fit to
the scaling expression has been found.
A comparison of the H-mode thermal confinement
times from the ITERH.DB2 database with the Lmode scaling expression, shows that the ELM free
data have an average enhancement factor (over Lmode) of 1.72, while ELMy data have an averageenhancement factor of 1.48. However, as is apparent from Fig. 12, the enhancement factor varies withthe size of the machine, tending to be larger in larger
devices. The transformation between the engineeringand plasma physical parameters naturally inflates
the difference in the exponents between the L-modeand H-mode scalings [315, 320].

τE,th (s)Frequency

200

Point and interval estimation forthe confinement time in ITER

Statistical forecasting is sometimes viewed as agame against nature, considered as an intelligent
opponent [318]. Given the possible consequences
associated with losing such a game, it goes without saying that a reliable empirical prediction of the
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport

1.2
ITER

1.0
0.8

In (τ E,th/τ ITERL-97P)

0.6
0.4
0.2
0.0
ASDEX
ASDEX Upgrade
Alcator C–Mod
COMPASS–D
DIII–D
JET
JFT–2M
JT–60U
PBX–M
PDX
TCV

-0.2
-0.4
-0.6
-0.8
-1.0
-2.5

-2.0

-1.5

-1.0
In (

-0.5

0

0.5

1.0

1.5

)

Figure 12. Confinement enhancement factors of the
ELMy discharges in the working dataset of ITERH.DB3,
as a function of minor radius, a(m). The different symbols
denote the various tokamaks.

confinement time of a device with the importanceof ITER requires a thorough search for alternative
possibilities, based on several data analyses. In thisvein, in the present section we investigate, in somewhat more detail than in Section 6.3:(a) point prediction from log-linear scalings basedon various subsets of the working dataset;(b) regression fits to the data by log-non-linear
scalings; and(c) construction of an interval estimate for the confinement time.
The main results are presented here, while further
details of a more technical nature are given in the
Appendix.
As can be seen in the Appendix, the full DB3
working dataset contains several additional devices
and is less homogeneous with respect to the additional heating than the DB2 standard dataset. Two
influences on the point prediction for ITER, the variation due to several different subsets of the standard dataset and the impact on aspect ratio scaling as a result of using an alternative definition ofκ that de-emphasizes the extreme (i.e. bean) shaping of PBX-M, are summarized in Table 5. The tablecontains exponents and predictions for ITER for several log-linear scalings. Scalings IPB(y) and IPB(y,1)
are based on the full working dataset, IPB(y,2) is
Nuclear Fusion, Vol. 39, No. 12 (1999)

based on the dataset DB2.8 (i.e. DB3 restricted to
NBI discharges only, but including Alcator C-Mod),
IPB(y,3) on DB2.5 (i.e. DB3 restricted to NBI only,and excluding Alcator C-Mod), and IPB(y,4) on
DB2.8(IS) (i.e. DB2.8, restricted to its five ITER
similar devices). The point predictions in Table 5vary between 4.9 and 6 s with an average (roundedwithin ±2%) of 5.5 s. Since the Kadomtsev constraint was satisfied within statistical uncertainty,
the dimensionally restricted form of the scalings ispresented in Table 5 for IPB(y), IPB(y,1-2) andIPB(y,4). In the case of IPB(y,3) the Kadomtsev constraint is not satisfied, and the free regression fit isshown.
Comparison of the first two cases in the tableshows, for the full DB3 working dataset, the impact
on aspect ratio scaling associated with redefining
elongation. PBX-M has systematically high residuals
with respect to usual log-linear scaling expressions,such as Eq. (18), which corresponds to the first casein Table 5. To account for this, the elongation has
been defined in the remaining four cases in the tableas κa = area/πa2 , which leads to a similar aspectratio dependence as when PBX-M is dropped fromthe dataset. The ratio κa /κ is 1.3 ± 0.1 for PBX-M,
and 0.9 ± 0.1 for all other tokamaks in the standard
dataset, the difference being directly related to theindented shape of the PBX-M plasmas. The first twolines in the table indicate that redefining kappa inthis way does not notably change the ITER prediction. On the other hand, restricting the full working dataset to its three subsets defined above gives arange in ITER confinement prediction between some
5 and 6 s, and only a small variation in the aspectratio dependence. On average, the IPB98(y)(1–4)
scalings in Table 5 lead to a 10% more conservative
prediction for ITER and to a stronger inverse aspectratio dependence than the IPB98(y) scaling.
Another topic is the variation of the predictions
with respect to weighting the observations (between
equal weight per observation and per tokamak). Thisis addressed in Appendix A and summarized inFig. 27. In summary, this aspect increases the rangeof the predictions, even at a weighting exponent 13 ,which is somewhat closer to equal observation thanto equal tokamak weighting.
Even with these facets taken into account, the loglinear scalings do not reflect the full range of statistically ‘admissible’ point predictions that stem from
log-non-linear scalings. While this topic is further
addressed in the Appendix, we give a brief discussion here.
2207

ITER Physics Basis
Table 5. Exponents of the several empirical log-linear scalings based on ITERH.DB3

ScalingC (10−2 )

I

B

n

P

R

κa

a/R

M

N

RMSE
(%)

ITER
τE (s)(1)

IPB98(y)

3.65

0.97

0.08

0.41

−0.63

1.93

0.67

0.23

0.20

1398

15.8

6.0

IPB98(y,1)

5.03

0.91

0.15

0.44

−0.65

2.05

0.72

0.57

0.13

1398

15.3

5.9

IPB98(y,2)

5.62

0.93

0.15

0.41

−0.69

1.97

0.78

0.58

0.19

1310

14.5

4.9

IPB98(y,3)

5.64

0.88

0.07

0.40

−0.69

2.15

0.78

0.64

0.20

1273

14.2

5.0

IPB98(y,4)

5.87

0.85

0.29

0.39

−0.70

2.08

0.76

0.69

0.17

714

14.1

5.1

(1)

The quantity κa has been defined as area/πa2 in the scalings IPB98y(1,2,3 and 4) and as κ = b/a in the scaling
IPB98y. For IPB98y(4), the exponent κ is underlined to indicate that it imposed (as the average of IPB98y(1,2and 3)) on the regression fit. For the ITER predictions, a cross-sectional area of 39.1 m2 was used; see [319].
Wherever it is compatible with the data, the Kadomtsev constraint has been applied. For IPB(y,3) the free fit ispresented.

Since a log-linear model corresponds to a similar
scaling of the core plasma and of the edge plasma,
the possibility that the global ELMy H-mode confinement is actually better described by a log-nonlinear model than a log-linear model should not bedismissed lightly. To estimate its impact on the prediction for ITER, we consider a variety of alternative
empirical models. One (‘DK-96’) is an interaction
model [321], which contains a significant cross term
between qcyl on the one hand, and (na2 )1/2 (q95 /qcyl )on the other. Secondly, an offset linear scaling [440]based on the (DB2) ELM free dataset is taken andmultiplied with an empirical factor cELMy , foundby looking at the residuals from the ELMy (DB2.5)
dataset with respect to this scaling. The resulting
scaling is labelled as OK-96. Finally, an ELMy offset
non-linear scaling [320] based on DB2 plus JT-60U(‘TT-96’) is employed. These scalings and their predictions are described more fully in the Appendix,
Section A1. The ensuing point estimates for the confinement time in ITER vary between some 4.4 and7.3 s. If one wants to be pessimistic, it is possible to obtain considerably lower predictions (some3 s or below) by optimizing non-linear models withrespect to the RMSE of the fit and, simultaneously,
towards a low value of ITER confinement. However, such a procedure is not considered to be statistically admissible and is not pursued any furtherhere.
In addition to investigating the variation of thepoint estimates, it is interesting to look at statisticalconfidence intervals associated with each of the models and data subsets. This is easily done for log-linear
models where, according to [312, 315], the classical2208

statistical interval can be written asp Xλ̂2IT ER,j 1/2
ln(τ̂E ) ± c.
1+
λ̂2pc,jj=1(26)

As derived in [312], this equation is based on the
representation in which the data from the p explanatory variables are geometrically described by a datacloud of N observations in p dimensional space.
This cloud is approximated by a family of concentric ellipses that match all first and second ordermoments. The projections of the data on the principal axes of the ellipses are called the principal components. In Eq. (26), λpc,j denotes the standard deviation of the jth principal component and λIT ER,jis the distance of the centre of the database to the
ITER reference operating point in the direction along
the jth principal axis.
A delicate point is the proportionality factor
c in√
this expression, which is traditionally 2σ̂/ N for a(two sided) 95% interval, with σ̂ the RMSE of the fit.
In a simple approach, the total number of observations, N , is replaced by Neff , representing the number of independent data points. As a first approximation, it has been assumed [320] that Neff = N/4,
where the factor 4 roughly accounts for the correlations between data points, e.g. stemming from thefact that several have been taken during the samedischarge. A more thorough assessment of the proportionality constant has been developed in [321] andis summarized in the Appendix. This approach yields
a 95% log-linear uncertainty interval for the confinement time of (+25%, −20%), to be interpreted as
the range of values into which 95% of a large number(say 1000) of discharges performed at the ITER stanNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
dard operating point would fall. This interval, which
includes the variation in point prediction from thedifferent weightings in Fig. 27, presupposes, however,that a log-linear model is ‘essentially’ correct andall major influences have been taken into account.
Since we know this is only approximately true, wehave to consider a larger range of possibilities. This
has led to several ‘definitions’ of a 95% interval estimate [320, 321], each definition describing a particular aspect of the uncertainty. From the discussion
in the Appendix, we recall the variation due to the(point) predictions from ‘admissible’ non-linear models, which is about twice as large as the interval
above, see Table 14. In addition, ‘jackknife type’interval estimates can be considered, based on thevariation of point estimates from log-linear models
fitted to subsets obtained by deleting data from one
tokamak in turn from the database. The statisticaljustification of this somewhat automatic approach israther subtle. In fact, there are two versions of thistype of estimate, which give intervals close to theabove mentioned log-linear and log-non-linear intervals, respectively, see the Appendix and [321,322] forfurther details.
The summary interval estimate from the
Appendix is graphically represented in Fig. 13. In
this figure, the large interval is interpreted as a
95% log-non-linear interval, and the smaller interval
as both a 23 log-non-linear and a 95% log-linear
interval. It should be noted that all log-linear and
log-non-linear scalings discussed in the Appendix
give point predictions for the nominal ITER parameters [323], which are within or above the smaller
interval (see Table 14).
With respect to other factors that may influence the confinement time, but which are poorly
accounted for in the dataset, we mention that,for practical reasons, the power lost by radiation
inside the separatrix of the existing devices has been
neglected when deriving the scalings. However, for
ITER, such radiation is subtracted from the loss
power when calculating the projected energy confinement time. This approach has been motivatedby the fact that ITER, in contrast to the present
day tokamaks, will have a substantial amount ofbremsstrahlung and cyclotron emission from theplasma centre. The ensuing somewhat (10–15%) conservative effect on the ITER prediction is qualitatively counterbalanced by the difference in heating profile between the high-Z wall material small
devices (ASDEX, JFT-2M, PDX, PBX-M) and lowZ wall material larger devices (ASDEX-Upgrade,
Nuclear Fusion, Vol. 39, No. 12 (1999)

DIII-D, JET) in the DB2.5 (NBI only) dataset. Some
aspects of the role of wall conditioning (by analyses
similar to those in [324]), and of difference between
closed and open divertor machines, have been dealt
with in the correction factor TAUC92. However,analysis of ASDEX L-mode confinement [321] suggests that the latter influence is more complicatedthan is assumed in the present simplified approach.
An element not present in these simple log-linear
scalings is a possible reduction of the confinement
time for plasmas near the H-mode existence region.
Further issues are that the possible effects of plasma
rotation on local transport, and, hence, global confinement, have not been addressed, nor are thosefrom the profile shapes of the plasma current, magnetic configuration and electron density. On the one
hand, such effects can lead to modifications of thepresent scalings, while on the other they could possibly provide additional flexibility to optimize theenergy confinement in ITER over the accessible operating range.

7.

Scaling studies with similardimensionless physics parameters

7.1.

Basics

Quasi-neutral plasma turbulence is believed togovern transport processes in the core of tokamak plasmas. This led Kadomtsev to observe [72]that transport in the plasma core should be fundamentally governed by three physical dimensionless plasma parameters, denoted by ρ∗ , ν ∗ and β, aswell as other geometrical and engineering parameters{pi } defined below, which, at least in principle, areunder the control of the experimental physicist and,in present machines, can be made close to ITER values. The non-dimensionally similar approach to confinement scaling is to create, in present machines,
discharges that are as similar to ITER as possible,
with fundamental dimensionless parameters being
the measure of similarity. These have become known
as ITER demonstration discharges. For those parameters that cannot be matched, experimental scans arecarried out to determine both the actual value of theenergy confinement time as well as its scaling withdimensionless parameters. An analogy has often beendrawn between this approach and wind-tunnel testsof aircraft designs.
Mathematically, the non-dimensionally similar
approach implies that the non-dimensional energy
2209

ITER Physics Basis

8

8.0s

7

τITER (s)

6.8s

6

5

4.4s
4
3.5s
Figure 13. Interval estimation of the energy confinement time in ITER at the reference operating point. The
inner box indicates a ‘95% log-linear’ interval, whereas
the region enclosed by the wiskers is an estimate of a
‘95% log-non-linear’ interval. In this case, the inner interval corresponds roughly to a ‘ 23 log-non-linear’ interval.
The centre of the intervals approximately coincides withthe average of the IPB98(y) and IPB98(y,2) scalings.

confinement time Ωi τE can be expressed asΩi τE = F (ρ∗ , ν ∗ , β{pi }).

(27)

The list of other parameters {pi } is long (see Eq. (8)).
Table 1 in [74] presents a list of 18 dimensionlessparameters. Representative examples are q, κ, R/a,
Ti /Te , Zeff , single null divertor, H-mode edge, as well
as density profile shape and auxiliary power deposition profiles. Contemporary theories of flow shear
stabilization suggest that the toroidal Mach number Vφ (T /M )−1/2 is a key dimensionless parameter,
which has previously been neglected. The definitions
we adopt are as defined in Eqns (13)–(17), and thedegree to which these parameters can be made identical to ITER will be discussed below.
The concept of discharges with similar dimensionless parameters arises from dimensional analysis of
the equations governing microinstability plasma turbulence [73], which proceeds in a five dimensional2210

phase space – only the fast gyration motion of particles around the field lines can be averaged over.
The fundamental equation is a Boltzmann equation
for a five dimensional distribution function coupledwith the constraint that the divergence of the current
density moment must vanish. Definitions given by
Eqns (13)–(17) are motivated by the fact that whena characteristic spatial length of the ion gyroradius
and a characteristic velocity – the diamagnetic drift
velocity – are introduced, as well as a typical fluctuating amplitude ñ/n ∼ ρ∗ [47, 326–328], then thedimensionless, non-linear Boltzmann equation contains coefficients of order unity, indicating that a selfconsistent scaling has been obtained. A secondary
expansion in collisionality, ν ∗ , is often made, because
ITER will lie in the region where ν ∗  1. A key stepin this procedure is to assume that the turbulence
is local and that its saturation level is governed by
local quantities, such as T , and their gradients.
An examination of the requirements for matching
dimensionless parameters shows that ITER can bematched in the ν ∗ and β parameters, but not in theparameter ρ∗ . ITER will have ρ∗ values 5–8 times lessthan present tokamaks. In order to match ITER ν ∗and β values in present tokamaks, the discharge density and temperatures should be scaled from ITER
values according to [329]n ∝ B 4/3 R−1/3 ,T ∝ B 2/3 R1/3 .

(28)

Table 6 lists representative discharge conditions
with the same β and ν ∗ values as ITER. They arecompatible with standard operating regimes in thevarious devices, indicating that present tokamaks canbe in the same physics regime as ITER.
For the remaining parameter ρ∗ , the standard
assumption is that the function F can be taken to
be a power law in ρ∗
F = (ρ∗ )−(2+α) F̄ (ν ∗ , β, {pi })(29)

where the exponent α is expected to lie in the range
0 ≤ a ≤ 1, the upper and lower limits corresponding to Bohm or gyroBohm scaling, respectively. The
power law form is equivalent to the assumption that
there is no characteristic value of ρ∗ , and, hence, nocharacteristic length, that governs microinstability
turbulence scaling apart from the particle gyroradiusand plasma size a. The value of the exponent a thendetermines a characteristic turbulence scale size ` ≈1−α. Even though one can identify other charραi aacteristic microscopic lengths in the core plasma, forconstant β and ν ∗ these lengths scale as ρi and, thus,
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
would not change the relation between α and `. Thisis fortunate because it is planned to operate ITER
near its β limit so that a power law assumption for βscaling is not generally valid. Instead, physics arguments suggest that the confinement should be independent of β at low β, where the turbulent transport
is a consequence of electrostatic micro-instabilities
and should degrade dramatically as the MHD β limit
is approached. On the other hand, because ν ∗  1,magnetically trapped particles bounce many times
before detrapping occurs, so one would expect transport to be governed by collisionless physics and to
be only weakly dependent on ν ∗ .
From Table 6 we note that the ρ∗ value for ITER
is 4.6 times smaller than the values achieved on JET,
and that the range of ρ∗ available is roughly a factorof 3. (Recognize that many of the devices in Table 6
can be operated at lower fields than indicated and,hence, at greater ρ∗ .) The additional range in ρ∗ represented by Alcator C-Mod and COMPASS-D would
serve to reduce the uncertainty of ITER projections.
Smaller values of ρ∗ could be attained by 4 T operation on JET (reducing ρ∗ /ρ∗ITER to ρ∗ /ρ∗ITER = 3.3)
if there were sufficient auxiliary heating power toattain ITER β and ν ∗ values.
This approach to the analysis of transport andconfinement based on similar dimensionless physics
parameters addresses the fact that, in principle,
the gyroradius scaling of plasma turbulence could
depend on β, collisionality, or any other nondimensional parameter. For example, different scalings might arise in the collisionless (ν ∗  1) andcollisional (ν ∗  1) regimes. Expressed in termsof Eq. (29), this implies that the exponent α could
be a function of β, ν ∗ or density profile index αn(= [(a2 − r2 )/2rne ] × dne /dr). Indeed, experiments
find that the exponent α differs for L-mode and Hmode discharges, indicating a dependence on othernon-dimensional parameters. For this reason, it isimportant to determine the ρ∗ scaling exponent at
ITER relevant ν ∗ and β values and with ITER likeflat density profiles.
While most of the scaling studies with similardimensionless parameters have focused on the ρ∗
scaling of nominal ITER discharges, it is also of interest to determine how ITER confinement will dependon other parameters under experimental control, e.g.
q, β and ν ∗ . These studies are needed to project
the response of ITER to variations in operating conditions. Scans in these parameters can be carriedout at fixed ρ∗ , which, while much larger than the
ITER value, is nonetheless sufficiently small that one
Nuclear Fusion, Vol. 39, No. 12 (1999)

can argue that there is a common physics governingtransport. The case would be strengthened by observation of common β and ν ∗ scalings at several ρ∗
values.
7.2.

Discharges with identicaldimensionless parameters

An evident question for transport scaling withsimilar dimensionless parameters is: do experiments
support Eq. (27)? The answer lies in preparing discharges with identical dimensionless parameters, but
distinctly different physical values. The value of Ωi τEshould remain invariant. Such comparisons have been
made for circular ohmic tokamaks [74] and for auxiliary heated, ITER like tokamaks [315, 330], albeit at
β values below the planned ITER value. Good agreement was obtained. Table 7 presents the results for
an ITER like JET/DIII-D comparison [330].
Further comparisons are planned for Alcator CMod, DIII-D and JET. In addition, an extension ofthis approach to the H-mode power threshold is also
under consideration. For discharges with identicaldimensionless parameters, the relation Pthresh a3/4 =const. is predicted. One can also note that unconstrained, power law regression analyses of global confinement scaling fulfil a constraint on the exponents
derived from Eq. (27). Based on present evidence,
one can conclude that experiments are indeed inaccord with Eq. (27) [315].
7.3.

Results of experiments withsimilar dimensionless parameters

Scans in ρ∗ with similar dimensionless parameters
β and ν ∗ have been carried out for ELMy H-mode
ITER demonstration discharges on ASDEX-Upgrade[91], Alcator C-Mod [331], DIII-D [76,330,332], JET[276], and JT-60U [322]. Results for L-mode scaling are available for DIII-D, ASDEX-Upgrade andJT-60U. A principal conclusion is that the confinement scaling exponent, α, depends on the confinement mode.
A successful ELMy H-mode ρ∗ scan requires careful matching of β and ν ∗ profiles, an operating regime
that attains Te ≈ Ti , and similar, preferably flat,
density profile shapes. Figure 14, taken from [76],illustrates the excellent agreement obtained on DIIID ρ∗ scans. Similar success was achieved on JET and
JT-60U ρ∗ scans, as well as JET and DIII-D β andν ∗ scans. Table 8, adopted from [315], presents thevalues of α obtained in the DIII-D and JET exper2211

ITER Physics Basis
Table 6. Representative discharge parameters scaled from ITER at constant β, Zeff and ν ∗Parameter

ITER

JET

JT-60UASDEX-Upgrade
DIII-D

Alcator
C-Mod

COMPASS-D

B (T)
R (m)
n20
T (0) (keV)ρ∗ /ρ∗ITER

5.7
8.1
1.0
20.0
1.0

2.5
2.9
0.47
8.2
4.6

1.8
3.2
0.3
6.8
4.7

2.1
1.67
0.45
6.1
7.3

5.0
0.67
1.9
8.0(1)
8.7

2.2
0.56
0.68
4.3(1)
17.4(1)

Alcator C-Mod and COMPASS-D need additional auxiliary heating to reach this value.

Table 7. JET and DIII-D non-dimensionally identical discharges
B (T)

Ip (MA)n19

W (MJ)

Ptot (MW)τth (s)

Bτth

0.56
0.97

2.10
1.07

1.14
1.0

7.6
2.4

0.60
0.84

6.1
4.25

0.10
0.20

0.21
0.21

2212

0.02

a)

0.95T
1.9T

10.0

b)

β (%)

0.01

ρ*i

1.0

0.002
1.0 c)

Te/Ti

0.1
2

0.1

d)

1

0.01
3 e)

0
6

2

4

f)

αTi

ν*i

iments, together with a simple estimate of the 2σ
uncertainty.
The errors in δτ are assumed to arise entirely fromerrors in the determination of the parameter α inthe ρ∗ scaling experiments. For a standard error of±15% (2σ) in the stored energy, the 2σ error in theparameter α will be δα = ±0.4 for DIII-D, and δα =±0.3 for JET. The reason the errors are so large isdue to the fact that the range in ρ∗ is very small inthe experiments (DIII-D; ρ∗1 T /ρ∗2 T = 1.6 and JET;
ρ∗1 T /ρ∗2.6 T = 1.9).
To reduce the errors in the prediction of confinement, it will be necessary to complete a joint ρ∗ scanon at least two machines of different sizes to increase
the range of ρ∗ . Table 6 indicates that Alcator CMod and COMPASS-D ρ∗ scans would be valuableadditions to the database.
JT-60U [232] has also carried out an ELMy Hmode ρ∗ scan at βN = 0.8 and qeff = 4.4. Profilesof β and ν ∗ were well matched. High triangularity
was used to reduce the effect of ELMs on confinement. The results show α = 0.8 and, like DIII-D,
an appreciable difference between L-mode and Hmode in the ρ∗ scaling of the ion heat transport.
ASDEX-Upgrade [91] finds gyroBohm scaling in Hmode when local profiles are matched, but Bohm
scaling in L-mode. ASDEX-Upgrade was unable toreach a density low enough to match ITER ν ∗ values.
Initial H-mode scaling studies on Alcator C-Mod find
α = 1.1 ± 0.7, i.e. gyroBohm scaling with appreciable
uncertainties.

αnDIII-D
JET

a (m)

1

2

0

0
0 0.2 0.4 0.6 0.8 1.0
r/a

0 0.2 0.4 0.6 0.8 1.0
r/a

Figure 14. Radial profiles of (a) relative ion Larmor
radius, (b) thermal beta, (c) ion collisionality, (d) ratioof electron to ion temperature, (e) non-dimensional density scale length, and (f) non-dimensional ion temperature scale length at 1.9 T (solid lines) and 0.95 T (dashedlines) for H-mode discharges in DIII-D [76].

Figure 15 portrays the results of JET and DIII-D
ρ∗ scans compared to the ITER93H-P scaling relaNuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
Table 8. Experimental determination of the ρ∗ scaling exponent α for ELMy H-mode discharges
TokamakβNρ∗ /ρ∗ITERαδατth,ITER (s)δτ (σ)

DIII-D
JET

2.1
2.2

7.7
5.5

1.1
0.7

±0.4
±0.3

28
6.4

±18
±3

tion, which has α = 0.7. It is clear that, for discharges
carefully constructed to be as ITER like as possible
with present devices, this value of α = 0.7 describes
the experimental situation well.

non-dimensional parameter that differs from L-mode
to H-mode.
It is illuminating to analyse the parameters of
ITER demonstration discharges locally. In a ρ∗ scan,
one can compute, as a function of minor radius, theratio of thermal conductivity (or heat flux) between
member discharges of a ρ∗ scan. With the aid oftransport analysis codes this can be done for the totalheat flux or for the electron and ion channels separately. Analysis of data from JT-60U [333] and DIIID [76] indicates that the major change between Hmode and L-mode discharges is the scaling of the ion
thermal flux, which improves from worse than Bohm
in L-mode to close to gyroBohm in H-mode. Interestingly, the electron channel exhibits gyroBohm scaling for both L-mode and H-mode. The overall Bohm
scaling in L-mode results from the combined electron
and ion channel scaling. Under the assumption of
local transport, the difference between L-mode andH-mode must result from a change in some otherlocal, non-dimensional parameter, such as the density gradient index αn .

Figure 15. Comparison of BτE measured in ρ∗ scans on
JET and DIII-D with the ITER93H-P scaling relation.

7.4.

JET and DIII-D have also carried out an investigation of β scaling and ν ∗ scaling. The results
indicate that confinement is independent of β for
βN < 2.0. Moreover, the ν ∗ dependence is nonexistent for L-modes and weak, Ωi τE ∝ (ν ∗ )−0.3 , for
H-modes. The DIII-D team argued that the weakν ∗ scaling could be attributed to ion neoclassicalphysics, which would be negligible in ITER. The βscaling experiments found little dependence of confinement on β, which disagrees with global scaling
relations. Collinearities in the global database are apossible explanation [334]. Alcator C-Mod reported
a 1/ν ∗ dependence on collisionality [335].
Turning to L-mode results, global L-mode ρ∗
scaling experiments, carried out by TFTR, DIII-D,
ASDEX-Upgrade and JT-60U, have concluded thatthe τE scaling is Bohm like, which again indicatesthat the scaling exponent, α, must depend on some
Nuclear Fusion, Vol. 39, No. 12 (1999)

Limitations on transport scaling studies
with similar dimensionless parameters

Conceptually, the non-dimensional approach todetermining confinement scaling is simple and direct:
create a discharge as close to ITER conditions as possible and then experimentally determine the scaling
in the single remaining parameter, ρ∗ . Limitations
arise because of bias and correlations that are generated in attempting to create ITER like discharges.
A discussion of some of these limitations follows.
Arguably the most important potential source ofbias is toroidal rotation and rotational shear arisingfrom the directed NBI heating commonly used on
tokamaks. It is argued that this shear will suppress
micro-instabilities and improve confinement. Comparison of discharges with radio frequency heating,which has effectively no source of angular momentum
input, versus those with directed NBI on the samedevice should elucidate, and perhaps resolve, the flow
shear issue. The JET facilities are well suited to thisinvestigation.
2213

ITER Physics Basis
Successful scaling experiments with similardimensionless parameters call for appropriately
scaled plasma densities and identical plasma density
profile shapes; an area where experimental control
is limited at best. Plasma fuelling via gas puff, NBI
and inside/outside launch pellets can act to produce
variations in the density profile, which could affecttransport scaling properties. Because only peripheral
pellet or gas puff fuelling is anticipated for ITER,
very flat density profiles are likely, in accord withobservations on JET. Planned upgrades to present
experimental capabilities, including both highly
baffled divertors with domes and efficient insidepellet fuelling, may serve to increase the degree ofexperimental control on the profile shape. Active
control of the density profile shape would elucidatea presently poorly characterized aspect of transport
and its scaling.
A continuing anomaly in tokamak transport arises
from the observation, on the one hand, of gyroBohm
like core confinement scaling and, on the other, theobserved isotope effect wherein heavier hydrogen isotopes exhibit somewhat superior confinement. Simple gyroBohm scaling is well known to have a weak
inverse isotope effect. Some additional mechanism
and its associated non-dimensional parameter must,
therefore, be at work. Suggestions include ELM
induced confinement degradation, impurity modes
and impurity concentrations, flow shear effects, nonlinear ion Landau damping processes, and fast ion
concentrations. In fact, the recent JET experimentson the isotope scaling have shown that the positive
mass scaling comes from the pedestal [259] and not
the plasma core. Although there has been a strong
observational program focused on identifying that an
isotope effect exists [116], the problem has not been
approached from the view of validating candidate
mechanisms. More fully developed theoretical models of the various proposed mechanisms are needed sothat a set of predicted consequences is available forexperimental tests. The goal is to achieve an intellectual framework that can support both gyroBohm
scaling and a (albeit weak) positive isotope effect; asis currently employed for ITER confinement projections. One should keep in mind that isotope effects
may involve more than one mechanism.
Scaling experiments with similar dimensionlessparameters focus on the thermal plasma and neglect
any influence from energetic ions arising from NBI orradiofrequency heating. In present devices, the ratioof total fast ion energy to thermal energy is of order∼0.2, while in ITER, the fast ion energy content will2214

be negligible because the ratio of ion slowing downtime to the energy confinement time will be much lessthan unity. At present there is no systematic understanding of whether or how the presence of fast ions
could affect transport.
Indeed, one can generalize this potential source ofbias to the question: what is the appropriate definition of collisionality? The conventional choice of ν ∗governs whether trapped particles can complete a fullbanana orbit. The quantity τE νie , which measures
the importance of electron–ion energy exchange andthe fast particle concentration, could provide anotherdefinition. This parameter assumes importance insupershot plasmas, where the value τE νie ∼ 1 permits Ti  Te , a condition known theoretically andexperimentally to reduce transport. Such plasmas are
not accessible for ITER, which will have τE νie  1.
In ITER demonstration discharges, operational steps
are taken to assure Te ≈ Ti .
A current and key issue for tokamak transport is:how ‘stiff’ is the transport arising from ITG modes,
where the ion heat diffusivity depends on the departure of the ITG from a critical gradient. In a ‘stiff’system, where the ion heat flux increases rapidly
when the critical gradient is exceeded, small relative
deviations from the critical gradient will be observed.
This causes difficulties for non-dimensional scaling
analyses and, in its extreme form, makes the coretemperature directly proportional to the boundary
temperature, which may not have the same physics
scaling. Due to measurement inaccuracies and theoretical uncertainties in the critical gradient, it is difficult to apply dimensionless scaling to stiff systems.
Some ITG theories result in a stiff system [75]. Transient L-mode transport experiments on ASDEXUpgrade, however, suggest the system is not stiff [91].
ELMs constitute a rather uncontrolled boundary condition for heat transport. To minimize ELM
effects on transport scaling, it is desirable that onlya small fraction of the power outflow across the separatrix occurs as a result of ELMs. A key scalingdifficulty is that present ITER demonstration discharges have a power loss through the separatrix
well above the H-mode power threshold, resulting instrong type I ELMs. The difference in the scalingof the threshold and transport loss powers indicatethat ITER will operate close to the H-mode power
threshold, which can affect ELM activity.
Summarizing, dimensionless transport studies canbenefit from experiments on present facilities, which
can remove or reduce the limitations describedabove.
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
7.5.

Summary

Dimensionless scaling studies are performed under
the assumption that the problem of scaling transport
from present experiments to ITER can be reduced
to the determination of a single exponent, α, definedin Eq. (29). Experiments to determine the scaling
exponent on several devices conclude that α is nota constant, but can depend on the transport regime
and, thus, on the other non-dimensional variables ofthe discharge. Therefore, it is important the transport scaling for ITER be determined by discharges
with ITER like non-dimensional parameters. For Hmodes, the overall scaling in dimensionless similarity
experiments is close to gyroBohm, with α = 0.9±0.3,
in accord with the concept that plasma transport
is dominated by low-β, electrostatic, collisionlessinstabilities. L-mode discharges, on the other hand,exhibit Bohm scaling, which is further found to be
a combination of gyroBohm electron transport andworse than Bohm ion transport. To a good measure,these observations are common to several tokamaks,
engendering confidence in the results. Nevertheless,a local control parameter that governs the transition
to gyroBohm scaling remains unidentified.
Reducing the uncertainty in α expressed inTable 8 will require experiments with a greater−2/3 −1/6
R. Therefore, it would seem
range in ρ∗ ∝ Ip
that the largest and smallest tokamaks – JET and
COMPASS-D – can generate the greatest difference,and this emphasizes the importance of 4 T operation on JET and the beginning of ρ∗ scans on
COMPASS-D. JT-60U could also lower the ρ∗ ratioto ρ∗ /ρ∗ITER = 3.3 at ITER like β and ν ∗ , if operation at n20 = 0.6 and B = 3.0 T were possible.
Because confinement scaling depends on the confinement mode, it becomes important to determine
what dimensionless parameter(s) is the key to Hmode scaling. Two candidates are: (i) the presenceof an edge transport barrier; or (ii) the magnitudeof the density gradient relative to the temperature
gradient. The latter parameter meshes with the concept of local turbulence, while an edge criterion that
affects core transport scaling is decidedly non-local,
but can be realized in a stiff transport model via
sensitivity to the edge boundary condition. Observations indicate that the Bohm like or worse ion
scaling in L-mode correlates with regions of strong
density gradient [336]. If it is a relatively steep density gradient that is responsible for L-mode confinement with Bohm scaling, then this could imply thatdensity gradients introduce long wavelength, trapped
Nuclear Fusion, Vol. 39, No. 12 (1999)

ion modes, which would not exhibit scale separation between the equilibrium and the turbulence andwould, therefore, yield Bohm scaling for transport.
Since ITER will likely have flat core density profiles because of its size, even with an L-mode edge,
the influence of the density gradient on confinement
scaling must be clarified, particularly in view of thefact that there are several counter examples of steep
density gradients with good confinement, e.g. TFTR
supershots [337] and the ERS mode [338].
From a non-dimensional scaling perspective a key
question is: does ITER need an H-mode transportbarrier at the edge to attain gyroBohm core transport scaling?

8.

One dimensional transport models

8.1.

Introduction

Calculations of the expected fusion power fromITER can be carried out at various levels. The
simplest is to use a global energy balance, taking
the confinement time τE from extrapolating empirical scaling relations, as discussed in Section 6, or
employing dimensionally similar scaling studies, as
in Section 7. These calculations take some accountof profile effects by assuming particular forms (e.g.
T = T0 (1 − r2 /a2 )αT , treating αT as a parameter tobe chosen or explored), and specify levels of impurities in terms of a global Zeff and the fraction of∗. The next simplest step
He ash using a lifetime τHe
would be to use profiles that crudely represent known
physics. Thus, one could take n and T flat within theq = 1 surface, or the mixing radius, to represent sawtooth effects (see Section 5.1), invoke edge pedestals
to represent the H-mode transport barrier (see Section 4.2), and then use a simple, say linear, radial
interpolation between; this would produce a trapezoidal shape. A more realistic step would be to use
a one dimensional transport model, which contains
models for heating due to fusion power and additional heating, losses due to radiation, and sources ofparticles and impurities, including He ash, with simple empirical forms for particle transport (see Section 9). A semi-empirical approach to such modellingof the energy transport can be employed in which one
chooses forms for the radial profiles of χi and χe thatare known to reproduce experimental profiles well,
but one fits their overall magnitudes to ensure someparticular global scaling expression for τE is reproduced; again, models for sawteeth and the H-mode
barrier can be added. The ultimate aim of this one
2215

ITER Physics Basis
dimensional modelling, however, is to have a complete transport model that predicts both the temperature profile and confinement time. (It is worth
noting that profile modelling is not only relevant toenergy confinement, but can potentially provide realistic profiles with which to test the MHD stability of
ITER.)
Transport models can themselves be subdivided.
Both energy and particle transport models canbe semi-empirical, in which, while physical ideas
like dimensional analysis and critical gradients are
employed, the choice of the structure and the magnitude of diffusivities is guided largely by experimental
comparisons. The extrapolation to ITER of results
using such models is subject to reservations similar
to those encountered in extrapolating global scalinglaws for τE . On the other hand, it is also possible toderive physics based models that permit extrapolation to ITER with more confidence, since they automatically respond to differing conditions; provided
that no new phenomena unanticipated by the model
appear. Not only does this apply to extrapolation insize for a particular confinement mode (say ELMy Hmode) but, if the physics model is adequate, the samemodel could simultaneously describe other modes,
such as reverse or optimized shear, which might eventually be invoked for ITER. These physics basedmodels can be derived entirely from first principles
or contain just a few fitting parameters that arise
from estimates in theories that are not quite complete (e.g. turbulence saturation levels). A physics
based model that is also capable of providing a good
fit to data is an aim of transport modelling activity.
There is a range of transport models that havebeen proposed and partly tested against various
tokamaks. If these are to be used for predicting theperformance of ITER, a considerable extrapolationfrom existing devices, it is important to understand
how well they represent as wide a range as possible ofexisting tokamaks. This has led to the developmentof the ITER profile database [339], which contains
fully analysed profile data, specified in a standardized manner, from many tokamaks and covering avariety of confinement modes. By defining transportmodels in a standard form, using the same variables
as defined in the profile database, and using transport codes that are also written in a standardized
form and benchmarked against each other, it is possible to carry out reliable and verifiable testing oftransport models. All the resulting modelled profiles
are available to each modeller and various ‘figures2216

of merit’ have been defined to help quantify how
successfully each model performs. To avoid the needfor a H-mode transport barrier model, still an active
research topic, the testing employs an experimental
boundary condition for temperatures at r = 0.9a.
Results of predictive modelling of specified ITER reference cases, which prescribe the edge temperatureand the mean density as parameters to be explored,
are also available. Predictive codes can also be used
to investigate models for the sawtooth cycle (Section 2.5.1) and its impact on confinement and profiles.
8.2.

Theoretical and semi-empiricaltransport models

Within the framework of neoclassical collisionaltransport theory, it has proved possible to obtain
precise expressions for transport fluxes [4, 340, 341]
(Section 2). However, most tokamaks experience alevel of anomalous transport that exceeds neoclassical values, although the ion neoclassical thermal
diffusivity may sometimes play a significant role,
particularly in enhanced confinement regimes withtransport barriers (modifications to the basic theory to account for such features as the steep gradients in these cases are being developed). Thisanomalous transport is believed to be caused by thefine scale turbulent fluctuations that are observedin tokamaks. These, in turn, are believed to be theresult of the non-linear saturation of various microinstabilities: electron drift waves, ITG modes, pressure driven ballooning modes, etc. Many estimates(e.g. mixing length ones) of the turbulent transport coefficients associated with these fluctuations
have been made over the years [17, 28, 29] (Section 2). Such models, which are inevitably gyroBohm
in nature, usually contain a single overall constantavailable for fitting. Particular models can often capture some of the basic scalings of global confinement.
For example, a number of features of ohmic confinement (Section 3.2) result from electron drift wave
and ITG models. In the LOC regime, the increaseof confinement (and decrease of electron temperature) with increasing density is consistent with areduction in transport with density due to trappedelectron modes [22]. The improved ohmic confinement (IOC) associated with a steeper density gradient has been modelled using ITG models; the ultimate saturation with density has been attributed
to ion neoclassical transport [108]. However, these
simple models are sometimes challenged by other
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
experimental results, such as the current scaling,
and often fail to predict the correct shape of profiles [342, 343].
The Weiland–Nordman reactive drift wave model[344] is a more complete version of this approach,calculating the whole transport matrix (including
impurity fluxes) according to the quasi-linear theory,using a particular mixing length rule; the validityof this approach has been supported by two dimensional fluid mode–mode coupling simulations [345].
Features of this model are the important role of a
critical ITG for the excitation of ITG turbulence, andpinch terms due to toroidal geometry. As a result,
this model is able to overcome some of the weaknesses above [346–348].
The multi-mode model has evolved from earlier
versions [349, 350] and now brings together a version of the Weiland–Nordman model, a resistive ballooning model due to Guzdar and Drake, a kinetic
ballooning mode model, and neoclassical transport[351]. The multi-mode model currently treats the κdependence of the transport coefficients in an empirical manner, guided by global scaling laws. Full profile and global features are reproduced with themulti-mode model, partly as a result of the role
played by the resistive ballooning mode contribution,
which becomes important near the more resistive
plasma edge [350–353]. It is interesting to note that
this model, which is inherently gyroBohm, models
well the Bohm like L-mode discharges. In the modelling, this is attributed primarily to variations inneutral penetration and edge temperature profiles,
which give rise to non-gyroBohm behaviour near theplasma edge.
A particularly sophisticated extension of thisphysics based approach is the GLF23 model [354,355], which aims to capture the anomalous transport due to the whole of so-called drift ballooning
physics; the only free parameters in this model arechosen by fitting to more detailed theory, not experiment. The model captures many features of tokamak behaviour, including the formation of ITBs.
An important step in the development of physics
based models is the use of extensive numerical simulations of turbulence to determine the parametricdependence of transport coefficients. The IFS/PPPL
model [15,75] is a leading example of this, combining
gyro-fluid simulations of ITG turbulence in a representative thin annular region of the tokamak poloidal
cross-section with more complete gyro-kinetic calculation of the critical ITG for instability. The model
was first successfully tested on TFTR L-mode disNuclear Fusion, Vol. 39, No. 12 (1999)

charges. Recent advances in the stabilizing effect ofradial electric field shear (see Section 4.1) have led tomodifications to the IFS/PPPL model. The GLF23
and IFS/PPPL models, as opposed to the multimode model, have the property of being ‘stiff’, i.e.
the thermal diffusivity (due to the ITG turbulence)
becomes very large once a critical ITG is exceeded,
and this inhibits departure of the ion temperature
profile from this marginally stable profile. However,this feature of the GLF23 and IFS/PPPL models is
an active research topic. These models are based on
gyro-fluid simulations in a local annular region of theplasma cross-section; related simulations using more
fundamental, but more computationally challenging,gyro-kinetic simulations predict considerably lower
transport. Possible reasons for this discrepancy are:the treatments of poloidal flow damping; particle
noise; non-linear wave particle resonances; and linear theory characteristics. Careful cross-checking ofthese codes is underway addressing these possibilities[356–359]. Results from predictive modelling efforts
have indicated that features of the core plasma are
well modelled using the ITG mode as a basis fortransport there [15, 75, 348–356].
A somewhat different physics based model is thecurrent diffusive ballooning mode (CDBM) model[360]. This is based on a one point renormalizationof pressure driven ‘resistive MHD’ turbulence, butwith the important difference that a self-consistentturbulent electron viscosity due to electron inertia
replaces collisional resistivity in Ohm’s law and sustains the turbulent transport. In this theory, the turbulence has a sub-critical nature, which is supported
by direct numerical simulations [361], and the transport is not particularly dependent on the linear instability criterion. The model incorporates effects of alarge Shafranov shift in the equilibrium and reflectsfavourable aspects of ideal MHD ballooning stability: reduced transport for low (or negative) and highmagnetic shear and high pressure gradients; transport reductions due to sheared radial electric fields
can also be included [362]. The theory involves one
undetermined numerical coefficient, which is chosen
once and for all to optimize the fit to a dataset. The
model has captured satisfactorily the essential features of the ohmic, L-mode, the ITB for the high βp
mode of JT-60U [363], and current profile control by
LHCD [364].
One of the challenges for theory based models
is to recover the isotope effect observed in experiments (experimentally, confinement often appears to
improve with increasing ion mass (Sections 3 and 6)).
2217

ITER Physics BasisSeveral drift wave based models are gyroBohm innature and would be expected to predict the opposite effect (Section 2); the models can only predict
the observations through some indirect effect (forexample, through boundary conditions or a correlation between the density profiles and the isotope
employed, which might result as a consequence of different particle fuelling characteristics). However, thestabilizing tendency of the velocity shear on turbulent transport increases with isotopic mass, and thismight provide a possible explanation. The CDBM
model does have an explicit isotopic dependence,
which is in the same direction as that observed.
Another approach to modelling is the semiempirical one, based on a view that we cannot yet
satisfactorily calculate turbulent transport fluxes,
or even fully identify the underlying cause. Thus,
expressions for these are proposed that embody theoretical concepts like dimensional analysis and criticalgradients, but whose particular form is partly influenced by experimental evidence. Thus the gyroBohm
Rebut–Lallia–Watkins (RLW) model [365] is based
conceptually on the excitation of microscopic magnetic islands when a critical electron temperature
gradient is exceeded, but is tuned to describe a wide
range of experimental results from JET. This has
been modified by Boucher (RLWB) to contain Bohm
like ion transport in the light of evidence for the
Bohm like scaling in L-mode. The Culham model isinfluenced by ideas from collisionless skin depth [366]and pressure driven turbulence, and is constructed
to represent a number of L-mode discharges from anearly version of the ITER profile database. Taroni
and co-workers [367] have devised the so-called mixed
model, a combination of simple Bohm and gyroBohm
terms suggested by drift wave driven transport, with
the Bohm term containing a non-local element that
responds to edge conditions. In this way, it is able
to describe a diversity of steady and transient experiments on JET. This model has been modified tothe mixed shear model by Romanelli and co-workers[368, 369] to embody theoretical predictions that the
Bohm contribution will be reduced for low magnetic shear. The T-11 model contains a combination of ion neoclassical transport and a gyroBohm
electron term suggested by experimental studies on
the T-11 tokamak, but which is also close to thetransport expected from fluctuations on the collisionless skin depth scale [107]. Recent extensions toinclude anomalous ion transport guided by dimensional analysis (uniquely, in that it allows the Debye
length to enter) and experiment have led to the2218

semiempirical transport (SET) model [370]. Finally,
we mention the canonical profiles transport model(CPTM), which contains a combination of empirical background anomalous transport and additionalterms that tend to force the profiles back to so-called‘canonical’ ones suggested by MHD energy minimization arguments [371].
All the models above, which are representative ofmodels available in the fusion literature, are being
actively tested against the ITER profile database atpresent, at least as far as energy transport is concerned (particle transport is discussed briefly below
and in Section 9). It is to be anticipated they willdevelop further in the light of further research and it
is desirable that future (and indeed other existing)models be similarly tested.
Complete modelling of a discharge requires transport equations for temperature and particles, valid
across the whole plasma profile. While some models (e.g. the multi-mode model) prescribe the particle transport, a number of the above models donot; in this latter case experimental density profiles
are taken. However, some general theoretical arguments for the density profile can be advanced [372].
Thus, in the presence of low frequency ionic turbulence, the electrons, particularly the trapped ones,
diffuse conserving their adiabatic invariants: µ, themagnetic moment; and J, the longitudinal invariant.
The implication of this is that, if particle sources
are weak, the electron density takes up a ‘canonical’ profile that is peaked on axis: the degree of
peaking depends on the relative responses of trappedand passing electrons to the turbulence. Calculations [373] show that, for an ITER like magnetic
geometry, ne (0) ∼ 1.5ne (ped), if only trapped particles respond; alternatively, using a ratio for thisrelative response that fits DIII-D data, one finds
ne (0) ∼ 2ne (ped) (here ne (ped) is the density at the
top of the H-mode edge density transport barrier).
Since not all models attempt to model the edge
region and H-mode barrier (say, r > 0.9a) at the
moment, or the central sawtooth region (say r <0.2a), testing is restricted to 0.2a < r < 0.9a. Thus,
it has been agreed to prescribe the experimental temperature at r = 0.9a as a boundary condition forthe models. However, some models do represent theedge region (e.g. multi-mode [350, 351]), and others
contain an explicit H-mode barrier model (e.g. [367]involves a narrow neoclassical layer); the inclusionof this physics is necessary if one is to give a firstprinciples transport model for the H-mode. Sometransport models contain relatively simple sawtooth
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transportmodels (e.g. periodic Kadomtsev mixing within thesawtooth region, as in the Culham model [366]), butothers have proposed prescriptions for the sawtooth
period based on MHD stability criteria and rules for
its periodic effect on the profiles based on reconnection and relaxation ideas [290,374,375] (Section 5.1).
It is interesting that including the stabilizing effect ofalpha particles allows a long period for the sawtooth(∼100 s), leading to ‘monster’ sawteeth [290].
8.3.

Plasma profile database

The ITER profile database is being developed toprovide a facility for testing and developing transport models against reliable, well documented data
in an open and verifiable manner [339, 376]. One
objective of the database is, therefore, to provide
all the profile and global data required for comparing transport predictions with experimental observations in a readily accessible form. By September
1997, 141 discharges from nine tokamaks were available from the database. These discharges are not all
up to the same standard: limited diagnostic capability prevented some tokamaks from providing all thenecessary information; the safety factor (q) and effective charge (Zeff ) are notoriously difficult to provide.
As a consequence, the descriptions of the discharges
are still evolving with time as more information ismade available.
The choice of discharges provided in the profile
database results from a balance between the need tocover as wide a range in plasma parameters as possible – tokamak sizes and range of parameters withineach tokamak – and the need to include discharges
that emphasize a specific transport phenomena, suchas cold pulse experiments, reversed central shear configurations, supershots or parameter scans. A briefsummary of the discharges available in the databaseis given in Table 9.
8.4.

Results of one dimensionalmodelling tests

The existence of the ITER profile database (Section 8.3) provides an opportunity to carry out extensive and verifiable testing of transport models suchas those described in Section 8.2. Table 10 displays
models and modellers who have placed simulations
on the ITER profile database. A large variety oftransport codes using various procedures – ranging
from fully predictive (sources/sinks computed by thecode) to fully interpretive (sources/sinks taken fromthe database) – have been used to test the models
Nuclear Fusion, Vol. 39, No. 12 (1999)

against experiments. Because of the different implicit
assumptions made by these codes, the model testing outputs were found to depend on the particulartransport code being used. To overcome this difficulty, a standard procedure has been defined andused as a reference. This standard procedure prescribes a reference set of transport equations andstandard inputs (sources and sinks) to these equations. The models themselves have been standardized
by specifying exactly how each quantity entering in
the model expressions is to be made consistent withthe standard equations and inputs. The procedure
has been implemented in three different transport
codes to ensure that the tests of the models wereindeed totally transport code independent.
A standard subset of discharges that are relevant
to ITER operation (i.e. L-mode and ELMy H-mode)and have all necessary variables properly available
in the database, has been defined. It consists of 55
discharges from four tokamaks (JET, DIII-D, TFTR,
JT-60U). To check that the standard transport equations have been properly implemented, two typesof benchmarking have been carried out. In the firstplace, the results have been compared with a set ofanalytic solutions to the transport equations, using
an artificial model χ = ne (0)/ne , by Mikkelsen; themaximum error is less than 0.5% from both SMC,
the standard code of Boucher, and the HYPED code.
Secondly, the results from separate codes (e.g. theMLT code of Waltz, the SMC code and the HYPEDcode) using another artificial model χ = 1 m2 · s−1
and a special benchmark dataset have been compared. Finally, the predictions of the codes for themodels discussed in Section 8.2 have been calculated
for the standard dataset and placed on the ITER
profile database server.
Analysis software is also available on this server togenerate the various figures of merit for testing models shown in Table 11. We have chosen to concentrateon(i) figure of merit 1 in Table 11, which represents
the ability of the models to simulate the experimental energy contents W (we present comparisons for the total energies above the edge
pedestal, i.e. the ‘incremental’ energies W inc ,since the pedestal energy is an input throughthe edge boundary condition and is particularly important for H-modes); and(ii) figure of merit 6a in Table 11, which represents the distance between the modelled andsimulated electron or ion temperatures profiles
2219

ITER Physics Basis
Table 9. Database discharges
R/a(m)

Heating
(MW)

Field
(T)

Current(MA)

Alcator
C-MOD

0.67/0.22

ICRH
0–2.5

5.2

DIII-D1.69/0.63

NBI
0–15

JET

3.0/1.1

JT-60U

Phase(1)Comments

0.8–1.0

L,
H,
HSELM

High magnetic field

1.0–2.0

0.5–2.0

L,
HSELM

Scans: temperature,density, elongation, ρ∗

NBI
0–18

1.0–3.2

1.0–3.0

L,
HSELM,
Hot ion H

Scans:
ρ∗ , β, ν ∗3.2/0.89

NBI
5–23

2.4–4.2

1.0–3.5

L,
Hot ion H

TFTR

2.5/0.87

NBI
4–36

2.1–5.5

0.8–2.3

L,
ERS,
H,
Supershot

Impurity injection:
Xenon, krypton
scans:
current, ρ∗ , β, ν ∗ ,powerComparison between
DD and DT discharges
Cold pulses experiments.

RTP

0.72/0.16

ECH
0–0.352.2

0.077

Ohmic,
L

Hollow Te profile

T-10

1.5/0.32

ECH
0–1.7

2.8

0.2–0.4

Ohmic,LTEXTOR

1.75/0.46

2.25

0.4

TORESUPRA

2.3/0.7

NBI
2.8
ICRH
2.8

2.2

0.4

L,
RI-modeL

Tokamak(1)

L, L-mode; H, ELM free H-mode; HSELM, ELMy H-mode (small ELMs); ERS, enhanced reversed shear.

(this is taken over the ‘transport region’ 0.2 <ρ < 0.9, where ρ is the normalized toroidal flux
excluding the sawtooth region and the edge
region, where additional transport mechanisms
might need to be incorporated in some of themodels).
The results are displayed in Figs 16–19. Figures 16
and 17 show the mean and the mean square deviations of the predictions for W inc from each transportmodel, expressed as inc
1 X Wsim−1
hRw i =
inc
N
Wexp
s

and
∆Rw =

2
 inc
1 X Wsim
−
1
inc
N
Wexp

where N = number of discharges modelled, displaying them separately for (i) H-modes with giant ELMs
2220

Transition RI-modeto L-mode
Enhanced performancemode(HGELM); (ii) H-modes with small ELMs (HSELM);
and (iii) L-modes from the standard dataset. The
standard dataset comprises 3 HGELM, 14 HSELM
and 38 L-mode discharges. Standard code results forall of these discharges are not yet available for allmodels, as shown in Table 12. Each entry in the
figures for a given model corresponds to an average over the results from the standard codes foreach discharge and then averaged over the discharges
modelled. (However, in the case of the Weiland andT-11/SET models, no standard code results wereavailable so the modeller’s own results were used.
For the multi-mode model an average of standard
code results and those from the authors’ own modelling with the BALDUR code were employed.) Figures 18 and 19 show equivalent results for STD(Te )and STD(Ti ), respectively.
On the basis of these tests it would appear thatthe multi-mode and the IFS/PPPL E × B models
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
Table 10. Models and modellers
Model

Modeller

Physics

WeilandJ. Weiland (EU), D. Mikkelsen (US), R. Waltz (US)ITG

Multi-mode

J. Kinsey (US), G. Bateman (US), D. Mikkelsen (US)Drift waves, RBM,
kinetic ballooning,
neoclassicalWaltz GLF23

R. Waltz (US), J. Kinsey (US)ITG

IFS/PPPL, no
IFS/PPPL, ×

ITG

RLWB, RLW

M. Turner (EU), S. Attenberger (US), B. Dorland (US),D. Mikkelsen (US), R. Waltz (US), Y. Ogawa (JA),D. Boucher (JCT)A. Fukuyama (JA), S. Attenberger (US), D. Mikkelsen,
R. Waltz (US), D. Boucher (JCT), J. Kinsey (US),Y. Ogawa (JA)D. Mikkelsen (US), D. Boucher (JCT)Semi-empirical

CulhamM. Turner (EU), S. Attenberger (US), D. Boucher (JCT)Semi-empirical

Mixed

A. Taroni (EU)Semi-empirical

Mixed-shear

Semi-empiricalT-11/SET

G. Vlad/M. Marinucci (EU), D. Boucher (JCT),Y. Ogawa (JA)A. Polevoi (RF)CPTM

Yu. Dnestrovskij (RF)Semi-empirical

E × B;E B

CDBM

Current diffusive
ballooning modes

Semi-empirical

Table 11. Figures of meritP

1: Ratio of incremental total stored energy:
inc
inc
3
Wsim
/Wexp
, where W inc =(ne T̃e + ni T̃i )dV2f
and T (ρ) = T (ρ) − T (0.9)
inc
inc
inc
inc
2: (Wsim/Wexp)e and (Wsim/Wexp)i (separating e and i)

3: [{(ni,ρ=0.3 Ti,ρ=0.3 W )s }/{(ni,ρ=0.3 Ti,ρ=0.3 W )x }]P[(T − T ) /N σ ], where σ is the experimental error and N the number of observations
P n T dV5: β /β , where β =pP(T − T ) )/(pP T )],6a: STD = [(pP T )] for electrons and ions separately.
P
6b: OFF = [( (T − T ))/(
4: χ2 =
∗2
s

s

x

∗2
x

2

2

∗2

2 2
i i

s

s

x

x

2

2
x

2
x

perform the best from amongst the physics basedmodels; both predict incremental stored energy toan accuracy of within 24% overall. (Simulations withthe multi-mode model using the BALDUR code giveslightly better predictions with an accuracy of within22%.) Of these, the multi-mode model gives a better
prediction for the electron temperature profiles withan average STD value of 13%, as against 25% for the
IFS/PPPL E × B model, but both perform equallywell in the prediction of ion temperature profiles withan STD value of 18% overall. However, several of

Nuclear Fusion, Vol. 39, No. 12 (1999)

the semi-empirical models are competitive with thesetwo physics based models, particularly the mixed
shear and Culham models with accuracies of 26%
for the predictions of the incremental stored energy.
It could be that uncertainties in experimental inputs
could generate discrepancies of these magnitudes. We
note that some credit might be given to a model on
the grounds that it is physics based, particularly if it
is able to correctly predict results in other regimes,
such as reversed shear, without modifications to themodel. Such a model can improve its performance

2221

ITER Physics Basis
Table 12. Number of discharges from standard dataset used in testing of each model

E B
E B

H-mode withsmall ELMs

L-Mode

All

1
3
3
3
2
3
3
3
3
3
0

5
14
14
14
14
14
14
14
14
14
13

3
25
30
38
34
38
38
38
38
38
14

9
42
47
55
50
55
55
55
55
55
27

by artificially adjusting its coefficients in the thermal
diffusivities. For instance, the GLF23 model achieved
a reduction in the mean square deviation on a 46 discharge subset of the database from 26.9% to 19.6%
using such a recalibration. The CDBM model hasa single overall constant multiplier, which is to bechosen to provide the best fit to the data; a renormalization of this model could clearly improve its
performance. In fact, a consequence of this modelling
exercise using the ITER profile database is that someof the semi-empirical models have evolved, improving
their performance in the process.
The testing procedure described above has not yet
proved decisive in choosing preferred models. It was
the simplest and most direct exercise that could beattempted to complement the global database activity in support of ITER. The true value of the investment in the profile database is that it will facilitate
physics based investigations: specific scaling studies on ρ∗ , ν ∗ , β, flow shear, Te /Ti , etc., on the one
hand, and comparisons with perturbative and transient experiments on the other. For example, there isa proposal on DIII-D to compare models with pulsed
localized ECH experiments that will test the stiffnessof models such as the IFS/PPPL one. The discharges
supplied by TFTR provide a number of cases involving perturbative data for such tests.
8.5.

Predictions of transport models
for ITER

The use of local transport models can effectively
complement the two methods detailed earlier in thischapter: global confinement scaling expressions andnon-dimensionally similar discharges, by providing
additional information such as temperature profiles,
2222<Rw>= ( (Wsim/Wexp – 1))/N

WeilandMulti-mode
GLF23
IFS/PPPL no ×
IFS/PPPL ×
CDBM
RLW
RLWB
CulhamMixed-shear
T-11/SET

H-mode withgiant ELMs

0.6
HGELM
HSELM
L-Mode

0.4
0.2
0
-0.2
-0.4
-0.6

Weiland MultimodeGLF23 IFS/PPPL IFS/PPPL CDBM
no ExB
ExB

RLW

RLWB

Culham Mixed- T11/SET
shear

Model

Figure 16. Ratio of simulated to experimental incremental stored energy using figure of merit 1.
Rw= SQRT( (Wsim/Wexp – 1)2/N)

Model

0.7

HGELM
HSELM
L-Mode

0.6
0.5
0.4
0.3
0.2
0.1
0 Weiland MultimodeGLF23 IFS/PPPL IFS/PPPL CDBM
no ExBExB

RLW

RLWB Culham Mixed- T11/SET
shear

Model

Figure 17. Mean square deviation of the ratio of simulated to experimental incremental stored energy using
figure of merit 1.

the ratio between ions and electrons transport ultimately determining the ratio Ti /Te , and the determination of the total fusion power consistently withheat and particle sources and sinks.
The transport models presented in this document, however, are still evolving or are not necessarily suitable for extrapolation to ITER. The
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport

HGELM
HSELM
L-ModeMean STD (Te)

0.4
0.3
0.2
0.1
0

WeilandMulti- GLF23 IFS/PPPL IFS/PPPL CDBM
mode
no ExBExB

RLW

RLWB Culham Mixedshear

T11/SET

Model

Figure 18. Distance between modelled and experimental electron temperature profiles using figure of merit 6a.
0.5
HGELM
HSELM
L-ModeMean STD (Ti)

0.4
0.3
0.2
0.1
0

Weiland MultimodeGLF23 IFS/PPPL IFS/PPPL CDBM
no ExB
ExB

RLW

RLWB

Culham Mixedshear

T11/SET

Model

Figure 19. Distance between modelled and experimental ion temperature profiles using figure of merit 6a.

RLWB model, for instance, was essentially developed to fit L-mode discharges using a purely Bohm
like ion transport, which would not be suitable for
an ELMy H-mode regime that exhibits gyroBohm
scaling. Other empirical models, which by definition
include adjustable parameters, have not yet been
fully calibrated against the experimental data – as isvisible on Fig. 16 – without this necessary step, thepredictions of fusion power in ITER are premature.
Extrapolations are more meaningful, however,with transport models such as the multi-mode model,
which has been carefully fitted against experiments
and models without adjustable parameters such as
IFS/PPPL and GLF23. But in this latter case, as
well as fitting against experiments, the models need
to be fitted against the most accurate numerical turbulence simulations if they are to be used as valid
predictors of the transport level predicted by theory.
The independent work from the Cyclone group [356]
indicates that this might not be the case.
Nevertheless, as a guide toward the further development of these transport models, empirical or theory based, it is instructive to study their predictionof ITER fusion power performance.
Nuclear Fusion, Vol. 39, No. 12 (1999)

Since most of the models being tested addressenergy transport and not particle transport or thephysics of the transport barrier in H-mode, a set of
ITER target density and current profiles and boundary conditions has been prepared to allow the comparison of ITER predictions using the various models in an objective fashion and with the same input
parameters. To simplify the testing procedure andfacilitate the comparisons between models, the range∗of the scanning parameters was reduced to τHe/τE =10, Paux = 100 MW, hne i = 1.5nGW where nGW is
the Greenwald density (nGW = 8.5 × 1019 m−3 at
21 MA), and values of T (0.9) were chosen to cover awide range: from 1 keV to 5 keV. Figure 20 summarizes the range of the fusion power predicted by a representative set of models both empirical and theory
based. Not surprisingly, at this early stage of model
testing and calibration, the range is large: about offactor 6 between extremes. The multi-mode model
prediction is very close to the reference fusion power
found independently using global scaling expressions
for energy confinement time. The models based on
gyro-fluid numerical treatment of electrostatic turbulence occupy the lower range and predict a similar amount of fusion power. The multi-mode model,
although also based on similar drift wave electrostatic turbulence and giving a better fit to experimental data, predicts about three times the fusion
power. This higher level of fusion power, quite in linewith other independent projections, is also found ingeneral if one uses the more complete gyro-kinetic
numerical simulation of drift wave electrostatic turbulence [356].
3
Fusion power (GW)

0.5

2.5

Theory based
Theory based (gyro-fluid)
Empirical

2
1.5
1
0.5
0
IFS/PPPL IFS/PPPL GLF 23
no ExBExB

CulhamMixed- Multi-Mode CDBM
Shear

RLW

Model

Figure 20. Fusion power predicted by various models
with an edge temperature T (0.9) = 3 keV.

An important aspect of the fusion power predictions is their sensitivity to the assumed edge temperature. Figure 21 illustrates how the predicted fusion
power – normalized for clarity independently for each2223

ITER Physics Basismodel to its value at T (0.9) = 3 keV – varies withthe edge temperature as it is varied from 1 to 5 keV.
Combining the fusion power predictions from
Fig. 20 and the edge dependence of Fig. 21, one candeduce what edge temperature would be required foreach model in order to achieve a given fusion power:
1.0 or 1.5 GW. Figure 22 indicates that, despite thewide disparity between models, an edge temperature
up to 4 keV would ensure at least 1.0 GW (Q = 10)
from all models, with the exception of the original
version of the IFS/PPPL model that did not takeE × B stabilization into account. An edge temperature up to 5 keV would ensure 1.5 GW (Q = 15).
CulhamCDBM
IFS/PPPL no ExB
IFS/PPPL ExB
Mixed-Shear
RLW
GLF23
Multi-Mode

3
2.5
2

5

1.5

Mixed-Shear

1
0.5
0
0

1

2

3
4
T(ρ=0.9) keV

5

6

Figure 21. Relative sensitivity of the predicted fusion
power to the edge temperature. The fusion power is normalized, independently for each model, to its value atT (0.9) = 3 keV.
6
1.0 GW
1.5 GW

T(ρ=0.9) keV

54

Normalized ion temperature

Normalized fusion power

3.5

on fusion performance of the temperature profiles,
one can define the quantity [Ti (ρ) − Ti (0.9)]/hTi (ρ) −Ti (0.9)i, which represents the normalized ion temperature profile inside ρ of 0.9. Figure 23 plots thisquantity for the various models.
Despite self-consistent calculation of sources andsinks, the normalized profiles are remarkably close,with the exceptions of the CDBM and mixed shear
models, whose central peaking can be attributed to
the weak shear inside the q = 1 surface assumed in
the target safety factor profiles. Nevertheless, if one
computes the fusion power that would be produced
from these shapes – after imposing T (0.9) = 3 keV
and forcing all profiles to correspond to the same
12 keV volume average temperature – it is found that
it would differ by less than 10% between extremes.

4

GLF23

3

CulhamMulti-Mode
CDBM
DIII-D, JET

2

RLW
IFS/PPPL
ExB

1

3
2
1
0

RLW

CDBM

Multimode

Mixed
shear

CulhamGLF23

IFS/PPPL IFS/PPPL
no ExBExB

Figure 22. Edge temperature in keV at ρ = 0.9 required
to achieved a given fusion power, 1.0 or 1.5 GW, for eachtransport model. RLW and CDBM would achieve thefusion power for any value of the edge temperature.

The fusion power predictions for local transportmodels come from a combination of dominant factors: the edge temperature, the magnitude of thetransport coefficient – χe and χi – in the core, and,
finally, the shape of these coefficients. To study the
influence of the shapes and, therefore, the impact
2224

0
0

0.2

0.4
ρ

0.6

0.8

Figure 23.
Normalized ion temperature profiles,[Ti (ρ)−Ti (0.9)]/hTi (ρ)−Ti (0.9)i, compared between various local transport models and for DIII-D and JET ITER
demonstration discharges.

In conclusion, the incomplete nature of the calibration of the models, either against experiments for
empirical models or against detailed numerical turbulence simulations for theory based models, makes
the prediction of fusion power in ITER from the
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transportmodels still premature. Nevertheless, a number ofconclusion can be drawn: the multi-mode model,
which gave the best fit to experimental data, predicts a fusion power for ITER close to that independently predicted by global scaling expressions. Anedge temperature, at ρ = 0.9, above 4 keV is consistent with more than 1.0 GW of fusion power fromall models considered, the IFS/PPPL model without
E × B correction being about 20% lower.
‘Stiff’ models, such as IFS/PPPL or GLF23, arefound to be very sensitive to the edge temperature,with the fusion power increasing by about a factorof 4 when the temperature is raised from 2 to 4 keV.
Clearly, a reduction of both sources of uncertainties –edge temperature scaling and treatment of drift ballooning driven turbulence – are required before suchmodel predictions can be relied upon for estimationof the fusion power in the ELMy H-mode in ITER.

9.

While most attention has been focused on energy
transport, issues of particle transport also impact thedesign of the ITER device. In particular, the designand operation of the fuelling and pumping systems
will depend on the anticipated transport propertiesof the fuel species, helium ash and impurities.
A key factor determining performance projections
for ITER is the fraction of helium ash accumulation
during long pulse or steady state operation. The primary effect is dilution of the fuel and reduction of
the fusion power. The impact of helium ash can bequantified in terms of the simple ratio of the effective helium particle confinement time to the global∗∗/τE , where τHe
takes
energy confinement time, τHe
into account finite recycling and helium pumping;that is(30)

where τHe is the particle confinement time for helium
nuclei, and Reff is the effective recycling coefficient(<1) to the core plasma. Steady state ignited solu∗/τE <tions exist only for values of the ratio η = τHeηcrit , where ηcrit is a number of order 10. A fullderivation has been given by Reiter [377, 378]; here
we will merely sketch the main ideas of that derivation.
The key point is that the fusion power production
rate and the helium production rate are exactly proportional to each other. Therefore, we can write the
Nuclear Fusion, Vol. 39, No. 12 (1999)

3 ne (1 + fi + fz + fHe )T= 14 n2i hσνiEα − n2e Rrad
2
τE
(31)
where fi = ni /ne is the fraction of hydrogenic
species, fHe = nHe /ne is the helium fraction, fZ
is the fraction of other impurities, Eα is the alphaparticle energy, hσνi is the DT fusion reaction rate,and Rrad is the radiation cooling rate from all
species. Here we have simplified the problem by taking nD = nT = ni /2, and by neglecting spatial profiles of n and T . Considering only a single, low-Z
impurity, the charge neutrality condition becomes
1 = fi + ZfZ + 2fHe . In the same terms, the helium
particle balance becomesfHe = ne τE ( 14 hσνi)fi2 η.

Particle transport:hydrogen, helium, impurities∗= τHe /(1 − Reff )τHepower balance and helium particle balance each interms of the usual fusion parameter ne τE so that theignition condition can be written as(32)

Using the charge neutrality condition to eliminate fi
and equating the two expressions for ne τE , we obtaina cubic equation for fHe involving η, T and fZ . On
substituting back the physically relevant solutions,
we obtain ignition loci in the ne τE –T plane, with∗
/τE as a parameter [377]. Thesethe ratio η = τHeloci exist and form closed contours only for valuesof η less than some number of order 10, the precise
value depending on assumptions about the impurity
concentration and profile effects.
From the above discussion, the important quan∗/τE , which depends on both the coretity is τHeparticle transport properties and the pumping efficiency. Experimental studies with injected helium
have demonstrated that satisfactory values of thisratio can be obtained in present tokamaks in L-mode[379–382], supershots [382], and in ELMy H-mode[128]. Expectations of efficient helium ash removal
in supershots were confirmed with recent measurements of helium ash production and pumping in
TFTR DT plasmas [383]. In the ELMy attacheddivertor experiments, helium was introduced into an
H-mode discharge by gas puffing, and the concentration in the core measured by charge exchange recombination spectroscopy. Pumping was accomplishedby means of an in vessel cryopump conditioned with∗
/τE between 8 and 10 wereargon frost. Values of τHe
inferred from these measurements. The shape of thehelium profile remained essentially unchanged during the active pumping phase, indicating that theeffective exhaust of helium from the core is limited
2225

ITER Physics Basisby the pumping speed and not by core He transport. Modelling using the MIST code [384] indicatesa best fit to the data using a time dependent effective recycling coefficient for helium with a starting
value RHe = 0.95 and a spatially varying diffusivity
in the range 0.5 ≤ D ≤ 1.75 m2 · s−1 . It is reported
that over a range of ELMing discharges in DIII-D∗/τE is in the range of 10–20.
the inferred value of τHe∗/τE between 6 and 8 are also reported
Values of τHewith enhanced wall pumping in high-βp ELMy Hmode on JT-60U [385]. These results are encouraging for ITER. By contrast, experiments in ELM free
H-mode discharges in TEXTOR [386] indicate ratios∗/τE ≈ 70, which would not be acceptable. Thisof τHeresult emphasizes the importance of the confinement
regime and coupling to the pumping system in determining the performance. Helium pumping has been
demonstrated in completely detached H-mode experiments in ASDEX-Upgrade, in which feedback controlled neon injection was used to create a steadydetached plasma with type III ELMs [387]. However, in ASDEX-Upgrade, the divertor and pumping
geometry (outboard turbomolecular pumping) differs from the ITER design (pumping from the private
flux region) and the core impurity concentration was
unacceptably high.
Transport of intrinsic or injected impurities ofhigher charge is also of critical importance to ITER.
Although a strongly radiating boundary would be adesirable means of distributing plasma power losses,
such a boundary leads to the potential danger ofexcessive core radiation and fuel dilution if impurities
tend to accumulate in the core. Perhaps because of
the availability of spectroscopic techniques, studies of
impurity transport have been carried out for a longtime, and, in some respects, it is better characterized
than majority transport. The workhorse technique islaser blow-off [388], which uses a high power laser
pulse incident on a target material, usually in theform of a thin film, to create a population of energetic
neutrals directed toward the plasma. Spectroscopic
techniques are used to follow the impurities into andback out of the discharge. Codes such as MIST [384]may be employed to model the atomic physics andtransport phenomena. A scaling for impurity confinement time in ohmic plasmas [389], which fits datafrom a number of early experiments, isτimp (s) = 0.75aR3/4 (Zeff /q)(mbg /Zbg )(33)

where R (m) is the major radius, a (m) is the minor
radius, q is the safety factor and mbg and Zbg are themass number and charge of the background (major2226

ity) species. The impurity transport appears to be
independent of the impurity species, and, somewhat
surprisingly, of the target density. Studies of impurity
confinement in JET and Tore-Supra [390] found thatEq. (33) yields an impurity confinement time that ismuch too large for those devices, and proposed thealternate scaling for ohmic and L-mode plasmas
τimp = 7.4Vp0.70 Ip0.31 (Pin /ne )−0.57 (s)(34)

where Vp (m3 ) is the plasma volume, Ip (MA) is theplasma current, Pin (MW) is the ohmic plus auxiliary heating power, and ne (m−3 ) is the volume averaged electron density. In H-mode and other improved
energy confinement regimes, impurity confinement
is also improved. These improvements are not easy
to capture in global scaling relationships. Considerations of gross particle confinement time provide
the coarsest representation of particle transport phenomena. For more detailed performance projections,which rely on profiles of reacting species and impurities, it is necessary to consider the local transport
properties (although sawteeth and ELMs can produce more macroscopic effects, as discussed in Section 5). The simplest useful characterization of local
transport in tokamak plasmas identifies the local particle flux as a combination of a diffusive term and a‘pinch’ term. A simple example, using a fixed radial
form for the relationship between the pinch and diffusive terms, isΓ = −D∇n + nV = −D(∇n + Cv n2r/a2 )(35)

where D and V are each supposed to be functionsof local plasma parameters. Here, the velocity V isa convenient way of representing those componentsof the particle flux that are not driven directly by
the particle gradient; this notational simplification
conceals substantial physics, some aspects of which
are better understood than others. The most common example of such an off-diagonal transport fluxis the so-called neoclassical pinch [4], which is an(inward) particle flux driven by the parallel electricfield. Other terms, more difficult to interpret, mayarise in the form of particle fluxes driven by temperature gradients.
Experimentally, studies of local particle transport
rely on two sorts of measurement: quasi-steady flux
balance analysis and perturbative (transient) techniques. From quasi-steady profile analysis, we can,at best, determine only the (perhaps spatially varying) ratio Cv of transport coefficients in Eq. (35).
Determination of both coefficients requires analysis
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transportof the time response of the system, using perturbative techniques. These may rely on intrinsic perturbations, such as sawteeth or ELMs, or on active
methods such as oscillating gas puffs, pellet injection,or, in the case of impurity transport, laser blow-off.
Numerous examples [391–399] of such experiments,
and associated analyses, exist in the literature.
Despite the large number of experiments, a clear
description of majority species particle transport intokamaks has not emerged; indeed, the level of consistency with respect to particle transport appears
to be even less than that which typifies the energy
transport problem. Nevertheless, some general observations can be made. Many experiments indicate an
inward pinch in the outer region of the core thatis coupled with a high diffusivity in order to reproduce the dynamics of the plasma density evolution
for r/a ≥ 0.7. However, in this region, the error bars
are large because of uncertainties in the sources. The
presence of ELMs further complicates edge particle
transport analysis. Deeper in the core, the evidence
for a particle pinch is mixed. In most larger devices,which may be more relevant to ITER, analysis ofdensity transients indicates that the particle pinch isextremely small and not inconsistent with the neoclassical value. For example, measurement of the DT
neutron generation profile in TFTR tritium injection
in supershots [400] has provided a first and uniquedetermination of the local transport properties ofhydrogenic species. The results indicate a negligible
anomalous pinch in the core.
Typically, but not always, D exhibits an inverse
dependence on density. A strong temperature dependence of the electron particle fluxes was found in atemperature scan at constant density in TFTR Lmode plasmas [401]. Both D and V typically increasein magnitude toward the edge of the plasma. As thedensity limit is approached, both D and V exhibit
dramatic increases, particularly in the outer regionof the plasma. Pellet injection experiments [92] indicate that fuelling beyond this edge layer increases
fuelling efficiency dramatically and can lead to density limits that are higher than achieved with gas
injection or high recycling. However, further pellet experiments in conjunction with active divertor
pumping are needed to clarify the roles of the fuelsource, recycling and edge transport properties for
ITER projections. The dependence of effective particle confinement on the source distribution (e.g. pellets or gas) is important for plasma density and fuelisotopic control [402].
Experiments indicate a strong correlation between
Nuclear Fusion, Vol. 39, No. 12 (1999)

local thermal and particle transport properties in thecore plasma, with the particle diffusivity more closelyaligned with the ion thermal diffusivity. Ratios of
χ/D vary, but in DIII-D, χeff /D is reported to be
typically of order unity in both L-mode and H-mode,as shown in Fig. 24 [187]. In TFTR L-modes andsupershots, He diffusivities were comparable to, andcorrelated with, the ion thermal diffusivity [401].
The T (tritium) diffusivity was also comparable tothe ion thermal conductivity in supershots [401]. In
Alcator C-Mod χe /D ∼ 1–2 in L-mode [403]. In
JET L-mode plasmas, the local diffusivity of impurities is strongly decreased in the core where thedimensionless shear parameter, s = d(ln q)/d(ln r),
drops below 0.5 [404]. Enhanced confinement mode
operation often results in enhanced fuel and impurity confinement. Injection into ELMy H-mode discharges in Alcator C-Mod resulted in impurity particle confinement times of a few hundred ms, compared to typically 20 ms in L-mode. In ELM free
H-mode discharges, impurity confinement times areeven longer than in the ELMy case. While the impurity transport in L-mode is consistent with purely
diffusive behaviour, with Dimp ∼ 0.5 m2 · s−1 , the Hmode transport is characterized by greatly reduced
Dimp , especially in the outer portion of the plasma,
and a strong inward convection; the edge particle
transport coefficients are comparable to neoclassicalvalues [405]. In PEP mode discharges in the samedevice, impurity accumulation consistent with neoclassical transport was observed, with inferred diffusivities Dimp ≈ 0.25 m2 · s−1 (about a factor of 2
below De ), and pinch velocity at the half radius up
to 30 m· s−1 , or about an order of magnitude larger
than the electron velocity Ve . Neoclassical impurity
transport was inferred following pellet injection on
Alcator C [406]. In reverse shear plasmas [6], particle transport can also be reduced to neoclassicallevels in the core.
For the purpose of modelling ITER discharges, it
is recommended that a reference particle diffusivity∼ 1similar to the ion thermal diffusivity Dan /χanibe used, with Dan independent of charge or mass for
fuel, He and low-Z impurities (e.g. Be). The implias low as 0.3 should be concations of Dan /χanisidered for its impact on the design of the fuellingand pumping systems. However, because the relevant factor in He accumulation is D/χeff , a smallvalue of D/χe may lead to deleterious accumulationof He in enhanced confinement regimes, where theion conductivity and particle diffusivity are selectively reduced. In the edge region, 0.7 ≤ r/a ≤ 1.0,2227

ITER Physics Basis

2.5

L-mode

1.5
1.0

0

0
5

(m2/s)(m2/s)

3

ELMy H-mode

3
2

0.2

0.4

ρ

0.6

0.8

VH-mode

2

χeff
DHe

1

χeff
DHe

1
0
0

χeff
DHe

0.8
0.4

χeff
DHe

0.5

4

ELM-free H-mode

1.2(m2/s)(m2/s)

2.0

1.0

0
0

0.2

0.4

ρ

0.6

0.8

1.0

Figure 24. Comparison between DHe and χeff in DIII-D: (a) L-mode, (b) ELM free
H-mode, (c) ELMy H-mode, and (d) VH-mode.

a particle pinch and ELM activity strongly affectthe coupling between the scrape-off and core plasma.
The neoclassical contributions to the particle fluxes
and ion thermal conductivity must also be included
because these have been shown to be importantin the core under improved confinement conditions.
Although models for ELMs are just now being developed and not well tested, experimental evidence supports a periodic partial expulsion of the particles
from the plasma edge to the scrape-off and divertor.

tion are rather high, i.e. Vφ ∼ 100–400 km· s−1 ,which correspond to a Mach number Vφ /cs up to
about 0.3, where cs is the ion sound velocity. The
observed rotation velocities are, however, very small
compared to predictions based on the neoclassicaltransport theory [407, 408].
Measurements of the momentum confinement
time τφ are based on the global angular momentum
conservation equation
1/2 
Z
2mb
Rtan
1
(36)
dV ni mi Vφ = Pb
τφ
Eb
R

10.

where Rtan is the tangency for the injected neutral beam, Pb is the beam power, mb and Eb are
the mass and energy of beam particles, and ni miis the plasma mass density. Rotation velocities areusually measured by the Doppler shift of emission
lines of hydrogen-like impurity ions resulting fromcharge exchange reactions of fully stripped ions with
injected fast hydrogenic particles.
Many tokamaks have reported similar magnitudes
for the momentum confinement time τφ and theenergy confinement time τE [409–416]. Figure 25
shows a comparison of τφ with τE measured in JET
during tangential neutral beam injection [416]. One
can see that τφ and τE are approximately equal for
steady state L-mode and ELMy H-mode discharges,
in agreement with earlier results obtained on ASDEX[413] and Doublet III [411]. Experiments on Doublet III have shown that τφ and τE scale in the same

Momentum confinement

Injection of toroidally oriented neutral beams into
tokamak plasmas adds toroidal angular momentum
to the plasma and results in toroidal plasma rotation. The study of the confinement of toroidal angular momentum and plasma rotation is of interestfor several reasons. First, the beam induced toroidal
plasma rotation can suppress the growth of the error
field instability in a tokamak and improve the stabilizing effect of the resistive wall for low-n kink likemodes. Second, the toroidal plasma rotation can contribute to a flow velocity shear suppression of microinstabilities. Third, angular momentum confinement
investigations provide further insight into the general
problem of confinement in tokamak plasmas.
The toroidal rotation velocities attained in experiments with tangential high power neutral beam injec2228

Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport

Nuclear Fusion, Vol. 39, No. 12 (1999)

1.5

+
+

0.5

0
0

+

+

1.0
τφ (s)

way with plasma current and neutral beam power.
These results, together with the identical shapes ofthe beam energy and the toroidal angular momentum
deposition profiles, suggest that heat and momentum
transport at steady state conditions are governed byrelated mechanisms [22].
There are, however, experimental conditions whenτφ deviates from τE . In the transient ELM free phaseof hot ion H-mode discharges in JET, the angularmomentum confinement time reaches a value of only
about 0.6 times the energy confinement time (seeFig. 25). The authors explain this effect by J × B
transfer due to particles injected into trapped orbits
and producing the radial current of fast ions between
the point of birth and the first orbit average [416].
In high-βp H-mode discharges in PBX-M, the ratio
τφ /τE was observed to increase significantly (from∼1 to ∼3), at the same heating power and unchangedτE , when the direction of the neutral beam injection
was changed from tangential to nearly perpendicular[414]. The authors explain the observed behaviour ofτφ by a strong negative dependence of τφ on the net−0.8. Another reason for thetorque Tnet , i.e. τφ ∝ Tnet
increase in τφ is a possible change in a beam induced
radial electric field Er , which can affect the toroidal
rotation velocity according to Eq. (9).
The MHD activity can strongly affect toroidal
rotation. Experiments on DIII-D have shown thatsome discharges that never exhibit sawtooth oscillations have essentially zero rotation velocity, in spiteof the input neutral beam torque [411]. In these discharges, the connection between τφ and τE is clearlybroken, since τφ drops to zero while τE decreases
by a factor of 2 from its value with sawteeth. Similar phenomena have been observed in ASDEX [417]and JET [418, 419]. They have been explained interms of the growth of large, stationary magnetic
islands (i.e. ‘locked’ modes) that are capable of transferring angular momentum through an electromagnetic interaction between the MHD mode and either
the vacuum vessel or a fixed stray field. A model
describing the toroidal force balance for MHD mode
locking, consistent with observed toroidal momentum loss in ASDEX, is suggested in [420]. The model
includes both the electromagnetic forces due to interaction with a resistive wall and error fields as wellas the viscous coupling between the island structure
and the bulk plasma. Other similar works appear
in [421,422]. Suppression of the growth of the locked
modes by the beam induced plasma rotation is discussed in more detail in Chapter 3, Section 2.5.
In most experiments, Er is not measured directly+++++++++++++ ++++++++++++++++++ +
+++
++
+

+
+

+

0.5

1.0

1.5

τE (s)

Figure 25. Toroidal angular momentum confinement
time of thermal particles during neutral beam injection
versus simultaneously measured energy confinement time
for steady state L-mode and ELMy H-mode discharges(crosses), and for transient ELM free phase of hot ion
H-mode discharges (squares) in JET [415].

because of lack of appropriate techniques. It canbe found, however, from Eq. (9), if other terms in
the equation are known. Recently, the toroidal andpoloidal rotation velocities and the densities andtemperatures of the main ions (He2+ ) and impurity ions (C6+ and B5+ ) have been measured in Hmode helium plasmas in DIII-D [423]. It was shownthat the values of Er deduced from the main ion
and impurity ion measurements coincided within theerror bars, confirming the validity of Eq. (9). It was
shown also that the toroidal rotation velocities ofthe main ions, Vφi , and impurity ions, VφI , werequite similar in the plasma core, but significantly
different in the edge region. The measured difference(Vφi − VφI ) was found to agree with the neoclassical prediction, which relates the velocities to radial
gradients of densities and temperatures [408].
The radial electric field plays an important role
in toroidal rotation characteristics observed in ICRF
heated plasmas in the absence of direct input of
toroidal momentum. Toroidal rotation velocities of
Ar16+ impurity ions greater than 120 km· s−1 in the
co-current direction have been observed in ICRF
heated H-mode discharges in Alcator C-Mod [424].
The magnitude of the rotation velocity increases withthe stored energy increase. The rotation velocity ofthe main (deuterium) ions is estimated to be about
1.5 times higher than that of the Ar16+ impurity ionsand three times higher than the main ion diamag2229

ITER Physics Basis2230

χφ = 1.3(ρ2s cs /Ls )(1 + ηi )/τ , where Ls is a magnetic shear length, τ = Te /Ti and ρs = cs /Ωi [432].
This gyroBohm theory leads to χi = χφ , which isobserved in some experiments. Another model basedon small scale turbulence leads to very low values ofτφ = τE me /mi βe [433] that contradict experiments.
10

Diffusivity (m2/s)

netic drift velocity, ∇pi /eni Bp , with the dominantterm proportional to the radial electric field (Er up
to 300 V· cm−1 was deduced from Ar16+ ion rotation
velocity neglecting its diamagnetic and poloidal rotation terms). The mechanism for the observed rotation is obscure. Similar results have been obtained
during ICRF heated H-mode in JET [425]. The
toroidal rotation velocities up to 60 km· s−1 in the cocurrent direction have been observed in these experiments with a good correlation of the local angular momentum density and the ion pressure. The
authors conclude that the ion pressure gradient may
be the major driving mechanism for toroidal rotation.
Let us now consider results of measurements ofradial profiles of the toroidal momentum diffusivity
χφ (r). Detailed measurements of χφ (r) and the electron and ion thermal diffusivities χe (r) and χi (r)have been performed on DIII-D in the hot ion Lmode and H-mode [121, 426]. The results presented
in Fig. 1 show that the most dramatic improvement
at the L-mode to H-mode transition is in χe (r) andχφ (r), which improve by about a factor of 3 throughout the plasma. Outside of ρ = 0.3, χe and χφ arebasically equal, within the error bars, in both L-modeand H-mode. χi is significantly less than χe and χφin the centre of both the H-mode and L-mode plasmas, being close to the predictions of the neoclassical theory inside of ρ = 0.3 for the H-mode. On
the other hand, similar measurements in the hot ion
L-mode on TFTR have shown that χφ is close toχi and DHe (the helium diffusion coefficient) and isgreater than χe (Fig. 26) [380]. Similar results havebeen reported for TFTR L-mode with Ti ≈ Te [427].
The values of χφ ≈ χi /Z have been deduced in JET
experiments [428].
The above results are related to effective valuesof χφ . Transient toroidal momentum transport has
been examined in JT-60U [429] by using a momentum source modulation technique. Assuming thatthe toroidal momentum flux consists of diffusive andconvective terms, it has been found that there is a
non-diffusive inward flux of the toroidal momentum,
similar to the pinch term in particle transport (Section 9), comparable in absolute value with the diffusive flux.
Experimental values of χφ discussed above are
significantly higher than neoclassical toroidal viscosity, which is χφ = 65 νii ρ2i in the Pfirsch–Schlüterregime [407], or χφ = 0.1νii ρ2i q 2 in the banana
regime [430,431]. Turbulence based theoretical models for χφ include that due to ITG turbulence:

1.0

χφ
He

χi
χe
0.1
0

0.2

0.4
Minor radius (m)

0.6

0.8

Figure 26. Similarity of the toroidal momentum diffusivity χφ , helium diffusion coefficient, DHe , and ion andelectron thermal diffusivities, χi and χe , in a hot ion Lmode discharge in TFTR [379].

The toroidal plasma rotation frequency in a reactor at a moderate injection power of ∼50 MW andat Eb ∼ 1 MeV is expected to be significantly lower
than in present day experiments, i.e. fφ,N BI ∼ 1 kHz
for the ITER basic parameters, as follows from
Eq. (36) assuming τφ = τE . In practical units thisrelation can be written as

 

Pb
τφ
2000 m3fφ,N BI ≈ (1200 Hz)
50 MW
6s
Volume1/2 

 20 −3 
1 MeV
mb
10 m
×
ni
Eb
mi

2
1/2 

8m
Rtan
mD
.
(37)
×
mb
6.5 m
R
Results of one dimensional modelling of toroidal
plasma rotation in ITER for various plasma densities and beam energies at χφ = χi are given in [434].
The natural diamagnetic frequency [435], which isconsidered as a measure of the toroidal rotation frequency in the absence of direct toroidal momentum
input, is given by



hTi i
21 MA
ρip
νti ≈ (100 Hz).
fφ,dia ≈2πRr
10 keVI(38)
It should be noted, however, that estimates basedon Eqns (37) and (38) can be inaccurate in reactor
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transportconditions, where Pb  Pa and alpha particles canstrongly contribute to the radial electric field.
A more precise prediction of the toroidal plasma
rotation needs the creation of a toroidal momentum confinement database and the development of atheoretical model for momentum transport validated
against the database.

11.

Summary

The basic theory of the transport of heat and particles in a tokamak by both classical and non-classicalturbulent processes is becoming mature, althoughunresolved characteristics of turbulent transport still
remain. Qualitative models describing most of theobserved phenomena in tokamaks, such as the transport in the plasma core, the L–H transition, sawteeth, and ELMs are available. However, at thepresent time, we do not have a reliable model describing the transport across the entire plasma profilethat could be used in ITER extrapolations. Good
progress has been made in testing theoretical transport models in the plasma core region (Section 8)
and some of the models give a reasonable fit to thedata. This is encouraging and suggests that with further theoretical development, especially for the edge
region (e.g. the pedestal temperature), it may be possible to produce a model that gives a good enoughfit to the data to be reliably used in ITER predictions.
At the present time, the main approach used in
predicting the performance of ITER in its main
regime of operation, the steady ELMy H-mode, is theglobal confinement time scaling approach describedin Section 6. In the last two years the global confinement steady state ELMy H-mode database has
been considerably expanded with the inclusion ofdata from five new machines and new data fromexisting devices. This has reduced the uncertainty in
the ITER prediction. Based on the log-linear models
applied to various subsets of the data in Section 6.4,
the interval estimate is 4.4–6.8 s. This comes close
to a classical statistical interval estimate based on
the fit for the standard working dataset from Section 6.3, allowing for a multiplication factor (roughly
accounting for some of the modelling imperfections).
Allowing for non-linear models, and some additional
considerations as presented in Section 6.4, the 95%
interval estimate is 3.5–8 s. In the latter case, thesmaller interval 4.4–6.8 s corresponds, roughly, to a
66% interval estimate.
ITER is projected to ignite throughout the narrow
Nuclear Fusion, Vol. 39, No. 12 (1999)

confidence interval although steps might have to betaken such as operation above the Greenwald density,
nGW , using pellet injection, or operation at higher
plasma current, to preserve ignition at the very lower
end of this confinement interval 4.4–4.8 s. With thewider interval, the lowest confinement range 3.5–4.4 swould require driven operation. The performance indriven mode depends on accessibility of densities
above nGW , the amount of available heating power
and/or higher plasma current. Nevertheless, undernominal conditions (n/nGW < 1, Paux = 100 MW,
I = 21 MA) the minimum Q in this range would beabove 6.
To reduce the confidence interval, an understanding of the reasons for the systematic differences
between the confinement in different devices must
be obtained. The identity experiments on different
machines used to verify the dimensionless scaling
approach will be particularly useful in this respect.
The non-dimensional similarity approach of Section 7 is closely related to the global confinement
scaling approach, in that the data themselves areused to determine the performance of ITER. The
95% interval estimate of the dimensionless parameter scaling approach is still larger than that of
the global scaling approach. To reduce it, a multimachine database of these types of pulses will need tobe constructed, ensuring similarity of toroidal rotation, and the range of operation of the largest device,
JET, be extended to 4 T with 40 MW of heating.
The comforting aspect of these three approaches
is that the confidence intervals are overlapping andthere is a program to narrow these intervals.
Our knowledge of particle transport is rather
weaker than that of energy transport; however, therecommendation of Section 9 that the best procedure
is to take D ∼ χi means that there will be adequatetransport of the helium ash for helium poisoning andthe consequent reduction in the fusion power not to
be a problem.
The MHD events, such as sawteeth, described in
Section 5 will undoubtedly reduce the fusion performance of ITER, but are not thought to have a dominant effect on the confinement projections and may
have a beneficial effect on removing the helium ashfrom the plasma central region. On the other hand,the ELMs that are beneficial in controlling density
and impurities also control the height of the edge
pedestal, and do, clearly, have an effect on confinement, the type III ELMs leading to lower confinement than those of type I. Although at the present
time we do not have good model for this edge region,
2231

ITER Physics Basisboth model development and testing are proceeding
very rapidly.
Turning to the H-mode threshold, we also findthat although there is a reproducible phenomenology of the L → H transition, we do not have a tested
quantitative theory. The scaling studies of Section 4give a rather large range for the required power for
an H-mode in ITER. Further effort to understandthe reasons for the large scatter in the data and aquantification of the influence of the plasma geometry, edge parameters and neutrals on the H-mode
threshold are needed.
Thus, in summary, a qualitative understandingof the energy confinement process taking place in atokamak is now available, and fairly firm confidence
intervals have been given for the ITER predictions;the challenge in the next few years is to narrow theseintervals. A further positive feature is that new operational regimes with improved confinement, such as
those with reversed shear described in Section 3, continue to be developed, and the use of these regimes
will further improve the existing ignition margins of
ITER.

Appendix A
A

ITER energy confinement
projection

A1.

Point and interval estimation
based on log-linear and
log-non-linear scaling expressions

As noted in Section 6.2, the working dataset
of DB3, consisting of data from 11 tokamaks,
leads to an ELMy log-linear scaling (Eq. (18))
that is similar to ITERH-92P(y), the ELMy loglinear scaling derived from DB2 [305]. Restricting attention to the statistically significant differences only, this new ELMy scaling is proportional to n0.11 M −0.2 κ0.13 times ITERH-92P(y), or
n0.24 M −0.21 B −0.24 ε0.34 times ITERH-93P. The predictions for the ITER engineering design activity(EDA) standard operating point are approximately
the same (assuming 15% confinement degradation
due to ELMs in ITER).
It is important to realize that such log-linear scalings only provide a first order approximation to
the true regression surface by a linear regression
plane on a logarithmic scale. In fact, several typesof empirical log-non-linear scalings have been developed, from which one can get some insight into the2232

accuracy of this approximation. In one approach[180,305,324], interaction models are considered that
express, on a logarithmic scale, the confinement time
as a sum of not only (first order) linear but also (second order) cross-product terms with respect to thebasic plasma parameters, while in another approach[136, 436–438], offset linear type scalings have been
derived, which describe the thermal plasma energy
as the sum of two power law scalings. Offset linear
scalings have been studied for many years [436] andposses a practical physical and empirical motivation.
However, actual fitting of such scalings to the data
is a fairly high dimensional non-linear problem.
In Section A1, we discuss in somewhat more detailthan in Section 6.4, the variation of point predictions from several log-linear scalings (based on various subsets of the working dataset), as well as fits
to the data by log-non-linear scalings. In Section A2,
we provide statistical background and address some
practical aspects of interval estimation for the confinement time in ITER.
The ELMy H-mode confinement scalings basedon DB3 in this appendix are based on the working datasets characterized in Section 6.4. An equivalent alternative description is as follows. There are,in fact, several intermediate steps in the transition between DB2 and DB3. In roughly increasingorder of heterogeneity with respect to an overall
log-linear scaling: DB2.2 (DB2+ASDEX-Upgrade),
DB2.5 (DB2.2+JT-60U), DB2.8 (DB2.5+Alcator CMod), DB3 (DB2.8+TCV+COMPASS-D). In each
step, the tokamak has been added that is most inagreement with the log-linear scaling based on thedataset under consideration. Finally, we consider the
restricted subset DB3r(IS) consisting of the large
to medium size tokamaks that are similar in shape
to ITER (JET, ASDEX-Upgrade, DIII-D, JFT-2M,
Alcator C-Mod).
There are several issues, related to the coherence of the dataset, which, for simplicity, are notaddressed in the present discussion:(i) the question of how closed the Alcator C-Mod
divertor is with respect to the other machines
and the difference between the two divertor
types at JET;(ii) the inclusion of a part of the ELM free JFT2M dataset that exhibits small ELMs in theDα signal [439];(iii) the influence of different heating methods on
the observed confinement times;
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport(iv) the effect of systematic inconsistencies between
various (diamagnetic, equilibrium and kinetic)
measurements of the plasma stored energy.
With respect to point (iii), it can be stated,
qualitatively, that inclusion of the ohmic and
ECH discharges from the smaller machines (TCV,COMPASS-D, albeit not for Alcator C-Mod) tends
to lead to higher predictions for ITER, whereas inclusion of the JET ICRH discharges tends to produce
somewhat lower predictions for ITER than thosefrom the dataset with only NBI heating.
It should also be acknowledged that the present
ELMy dataset constitutes a mixture of type I andtype III and possibly other types of ELMs. To someextent, the type of ELMs is controlled by the engineering plasma parameters, including the proximity
to operational limits. (This is one of the reasons thatthe difference between ELM free and ELMy scalings
is more complicated than a simple multiplication factor, and compatible with the fact that the regression
surface of the ELMy confinement time seems to bebetter described by a simple power law, i.e. to beless curved on a logarithmic scale, than that of the
ELM free confinement time [320,438].) Insofar as thetype of ELMs is controlled by the engineering plasma
parameters, the effect of the change in ELM type on
confinement is, in part, reflected by the scalings.
Table 14 describes how well a number of scalings
predict the observed confinement times in two of theworking datasets and what their prediction is for
ITER (at the standard operating point). The firsttwo scalings are log-linear scalings based on DB2.
The next scaling, IPB98(y), is based on DB3 anduses κ = b/a. The following four scalings are based oneach of the four working datasets, respectively, anduse κa = area/πa2 . The last four scalings in the table
are log-non-linear scalings. The most notable difference between the log-linear scalings using κ = b/a
and the other ones is the aspect ratio dependence.
As is shown in Table 14, the standard log-linear
fits to the above restricted versions of the standard
dataset give ITER predictions of 4.8 s (DB2.8, i.e.
with Alcator C-MOD) and 4.9 s (DB2.5, i.e. without
Alcator C-Mod), respectively. According to the subset DB3r(IS) of the ‘ITER similar’ tokamaks, the loglinear point prediction is 5.6 s. Compared with the
6 s standard estimate of DB2 [180, 305, 320], about
1
3 of the slightly more than 1 s reduction in confinement time in the first two cases is due to a change
in the ITER operating point (i.e. ne from 13.0 to
9.7 × 1019 m−3 , P from 192 to 180 MW) and about
Nuclear Fusion, Vol. 39, No. 12 (1999)

2
3 is due to the data from the additional machines. If

the ohmic and ECRH H-mode data from COMPASSD and TCV as well as the JET ICRH discharges areadded to DB2.8, then the predictions for the new
operating point increase to 5.6 s (IPB98(y,1) withκa = 1.53) and 6 s (IPB98(y) with κ = 1.73), respectively.
The number of data points contributed by each of
the tokamaks varies considerably over the database.
This raises a question about how one should weightthe data points in the regression. Figure 27 displays
the sensitivity of the ITER predictions with respect
to weighting the observations from tokamak j by−a−afor 0 < a < 1, where Ntok,jstands for the
Ntok,jnumber of observations (i.e. time slices) contributed
by tokamak j. (For a = 0 all observations and for
a = 1 all tokamaks are weighted equally.) One cansee that the dissimilarity between the ITER predictions tends to become larger when a increases. The
correct value of a would be 0 provided all systematic differences between the tokamaks affecting theconfinement have been accounted for. Since we know
this is only approximately true, there exists no hardstatistical rule for the choice of a. Practical experience with analysing the present dataset data may
suggest that a moderate value of a ( 13 or 12 ) could
lead to somewhat improved estimates with respect
to the choice a = 0 used in the standard analysis.
As a preamble to a more detailed discussion ofseveral log-non-linear models, a study performed in[437, 438] is illustrated: two models of the offset linear type were fitted to the ELM free DB2 dataset,
while using, extensively, the principle of (non-linear)
least squares minimization. Since the full offset linear model contains eight geometrical parameters andonly six devices were available in DB2, the aspectratio exponents in both the linear and the offset termin [438] were not fitted to the data, but left as freeparameters. A sensitivity study of the change in theother regression exponents with respect to these two
unknown aspect ratio exponents over a certain range,
led to the compact class of two term power law scalings of the form Wth = W0 + τinc P for ELM free Hmode plasma energy content, with the exponents inboth terms represented graphically in Fig. 28. These
scalings are based on DB2 and, since the ELM freedataset has not been extended very much, essentiallyalso on DB3. For ELMy confinement, an offset nonlinear scaling, based on the DB2 dataset extended
with confinement data from JT-60U, has been proposed in [439].
We consider in this section three non-linear scal2233

ITER Physics Basis7τE,th (s)

6

5

4

3
0.0

0.2

0.4

DB 3.0DB 3.0r(Is)0.6

DB 2.8
DB 2.00.8

1.0
DB 2.5Figure 27. Sensitivity of the ITER confinement time
prediction with respect to weighting the data from toka−a−amak j proportional to Ntok,j, 0 < a < 1 (where Ntok,jstands for the number of time slices included from tokamak j) for log-linear regressions based on each of the four
subsets of ITERH.DB3. For a = 0 all time slices and for
a = 1 all tokamaks are weighed equally in the regression
fits.

ings (on a logarithmic scale). First, as one of themodels that tend to give a rather low confinement prediction for ITER, Dorland and Kotschenreuther have investigated the influence of the interaction between the engineering safety factor qcyl ∝a2 Bκ/IR and the ratio of the square root of adimensionless, normalized pressure gradient, α ≡−2µ0 (q 2 R/B 2 )dp/dr, to the normalized Larmor
radius, ρ∗ = ρi /a. In engineering variables, this leads
to a termp
2 h(ε, κ)
na2 q95(39)
log g2 ≡ aint log qcyl log
qcyl
added to a log-linear model, where h is a shape factor. For the ELM free dataset, the interaction coefficient aint is significant, 0.3 ± 0.04, and this has theinterpretation that for large machines, and at higher
density, the exponent of qcyl becomes larger. This
leads to a lower ITER prediction, about 4 s for 0.85
times ELM free confinement, based on DB2, because
the value of qcyl for ITER is below that of the averageof the database. (Alternatively, it can be interpreted
as a decrease with qcyl of the beneficial effect of the2234(na2 )1/2 factor on the confinement time.) A breakdown of this interaction term into its constituentshas been analysed in [321]. From this, it appears thatln(q95 /qcyl ), related to the plasma shape, exhibits anempirical influence on confinement, in addition to theusual engineering variables. It should be noted thatfor the ELM free dataset, additional, statisticallysignificant, interactions are present [313, 440]. Thisis also apparent from the large difference between
the corresponding exponents of the offset and linear
term of the offset linear scaling (see Fig. 28). Including these additional interactions increases the prediction for ITER again, to some 5.0 s or above. For
the DB2.5 ELMy dataset, the interaction describedabove is less pronounced, aint = 0.18 ± 0.04, andleads to an ITER prediction of 5.4 s. With Alcator
C-Mod included, an additional interaction between
current density and power per surface area is present,
which suggests a more favourable power dependence
at higher current density (which is higher for Alcator
C-Mod than for ITER). Including these two interactions leads to an ITER prediction of 5.0 s.
5
4

Value of exponents

8

3
2
1
0
–1
–2
–3

aI

bI

aB bB an bn aR bR aκ bκaεbε aM bM

Figure 28. Estimated exponents (±2 standard deviations shown as shaded parts) in both terms of an ELM
free offset linear scaling (OL-95), together with their sensitivity with respect to a variation of the two aspect ratio
exponents. The exponents aε and bε (which correspondto the inverse aspect ratio exponents of the linear andthe offset part, respectively) have been varied independently over a range, as simultaneously indicated by theirthree sensitivity lines. This induces a change in the leastsquares estimates of the other exponents, as shown by
the other sensitivity lines.

We consider next a prediction for ITER basedon an ELM free offset linear scaling while making corrections for the presence of ELMs. In theabsence of an ELM characterization, we use for
ELMy confinement the OL-95 (ELM free) scaling[441] times a multiplier cELMy , which has been
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
found from DB2.5 empirically to be proportional−0.5 −0.1
I(n19 /j)0.3 κ−0.2 ε0.4 , where j denotes theto qcyltoroidal plasma current density. The first two factors indicate a difference in current and magnetic
field dependence, the third factor a tendency of
ELMy and ELM free confinement to merge somewhat near the Hugill–Greenwald limit, and the lastfactor a favourable influence of a small aspect ratio
on ELMy versus ELM free confinement. The multiplier cELMy is an indication to consider any fixed
multiplier, e.g. cELMy = 0.85, as merely a zerothorder approximation to the true ELMy scaling. Furthermore, the ELMy data show a different dependence on the net absorbed power (approximated by
PL0 = PL − PCX − POL in [180]) for some of thetokamaks, the effective exponent being 0.3 higher for
ASDEX, 0.3 lower for PDX and DIII-D, and 0.15
lower for JET (with mixed divertor data) than thatpredicted by OL-95 ELM free. The reasons for suchdifferences (divertor type, neutrals, ELM type) are a
subject for further investigation. The ELM free OL95 scaling leads to a prediction of 6.9 s for ITER
if one assumes that ELMs lead to 15% confinement
degradation in ITER, and 7.3 s if one uses the more
complicated multiplier cELMy , described above.
Finally, we consider the offset non-linear scaling
derived in [439], based on the standard ELMy DB2
dataset minus PBX-M plus JT-60U. This scaling
accommodates well the Alcator C-Mod data. Whenderiving this scaling, it has been assumed (in accordance with the discussion above) that PBX-M canbe considered as an outlying machine with especiallygood confinement. The formula is
W = 0.082IBaRκ(BR1.25)−0.1+ 0.043(In19PL )0.6 aR1.3 (BR1.25 )−0.15

A2.
(40)

which gives a relatively low prediction for ITER.
The scaling has been derived without correcting the
ASDEX and PDX data to account for their ‘closed’
divertor. This leads to some 10% prediction bias
in Table 14. (It is noted that the correction factor
TAUC92, motivated in [305], had a somewhat conservative influence on the ITER predictions of the
ITERH-92P(y) and ITERPBH-98P(y) scalings.)
The goodness of fit of the various scalings basedon the several alternative datasets, and their confinement predictions for ITER, are given in Table 14.
The goodness of fit (expressed in ‘bias’ and ‘standard
deviation’) indicates how well each scaling expression agrees with the observations from two different
datasets, DB2.5 and DB3.0. With respect to the preNuclear Fusion, Vol. 39, No. 12 (1999)

dictions for ITER, in this table no allowance is madefor a confinement reduction close to an operational
limit (notably density and H-mode threshold).
For reference, Table 13 gives the number ofobservations from each of the tokamaks in thedatasets DB2, DB2.8 and DB3. (DB2.5 is just DB2.8
minus Alcator C-Mod.) In addition, Table 13 shows,
approximately, the fraction of ‘essentially different’timeslices per tokamak. These fractions are, in Section A2, employed to estimate the proportionality
factor c in the expression for the log-linear interval
estimate (Eq. (26)).
The three non-linear scalings discussed here,
which give quite similar confinement time predictions
for the data in the database, complement the existing log-linear models. While being based on different
considerations and a varied empirical basis, they give
an impression of the way in which the estimate of theenergy confinement time in ITER is liable to deviate
from the (standard) log-linear ones because of a different functional form of the scaling.
It is noted that two of the three log-non-linear
scalings lead to lower confinement time predictions
for ITER than the log-linear ones. Because the projection from the present devices corresponds primarily to a change in ρ∗ , an important difference between
a log-linear and a log-non-linear model is that thefirst implies a simple power law in ρ∗ with a fixed
exponent, while the second contains an exponent
that depends on the other variables in the model.
This could have the character of a critical value ofρ∗ at which there is a change in the ρ∗ dependenceof confinement, as discussed in Section 7.1.
Framework for interval estimation

Based on [320], in this appendix some of the basic
concepts of interval estimation are described andapplied to the ITERH.DB3 dataset. When making
predictions of the energy confinement time that will
be achieved by ITER, one has to be aware of the factthat there exist different scalings for the energy confinement time that are all compatible with large andrelatively high quality datasets, reasonably in accordance with the available (additional) experimental
evidence, and not in direct contradiction with basic
principles of plasma theory. This ambiguity is evidently related to the fact that the basic transportmechanism(s) in reactor relevant plasmas is/are notsufficiently well understood, while at the same time anumber of competing energy loss processes (their relative strength depending on the plasma parameters)
2235

ITER Physics Basis
Table 13. Distribution of the number of data points in the standard working datasets of DB2,
DB2.5 and DB3 over various tokamaks
Tokamak

DB2

DB2.5

DB3

Neff /NASDEX
ASDEX-Upgrade
Alcator C-ModCOMPASS-D
DIII-DJET
JFT-2M
JT-60U
PBX-M
PDX
TCV
ALL

298
—
—
—
168
88
59
—
59
97
—
769

431
102
—
—
270
246
59
9
59
97
—
1273

431
102
37
17
270
306
59
9
59
97
11
1398

0.48
0.45
0.54
0.82
0.71
0.73
0.32
0.89
0.78
0.89
0.64
867/1398

Note: DB2.5 consists of additionally heated discharges by NBI only; DB2.8 equals DB2.5 plus thedischarges from Alcator C-Mod. The last column denotes approximately the fraction of ‘essentiallydifferent’ time slices in DB3 (i.e. of which the mutual distance, in all variables, is at least threeestimated standard errors of the experimental accuracy).

are playing a role in different parts of the plasma.
This makes the global confinement time a multifactorial quantity from which the separate influences
are difficult to disentangle. Nevertheless, predictionsof energy confinement time in ITER have to be
made, in the face of uncertainty, based (as always)
on incomplete information, and using the available
evidence. During the ITER CDA phase, it was sufficient to concentrate mainly on an accepted scaling ofa simple and quite robust type (so-called log-linear
scaling or simple power law) and its point prediction
for ITER to provide an initial orientation [136]. However, since any point prediction will be different fromthe value that will actually be achieved, the ITER
EDA requires some type of interval estimation. We
focus in this section on a 95% interval estimate for
ITER. As described in [320], the intuitive meaningof such an interval has been well characterized by T.
Takizuka during the ITER CDA phase. However, inorder to obtain an operationally useful interval estimate, it is necessary to be precise about the scientificinterpretation, and, related to that, about the way
such an interval is constructed. We restrict attention
to an interval estimate of the average confinement
time of a large number (say 1000) of ITER discharges
all performed at the same operating point. Two definitions are given, each of which covers a (complementary) part of the complicated real situation andhas to be incorporated in some form into an estimateof the prediction margin of ITER.

2236(i) According to classical frequentist statisticaltheory, a 95% confidence interval for τE is
a random interval that covers the ‘true’ (i.e.
average) confinement time with 95% probability under the hypothesis that (a) a specific
type of model, e.g. a simple power law scaling, is correct; (b) all essential regression variables are included; and (c) the residual data
variation can be, for practical purposes, modelled as independent realizations of haphazard(‘random’) events. For log-linear models, such atype of interval is routinely calculated by basic
statistical software packages (such as SAS, SPLUS and JMP) [442, 443]. As explained in[321], it is based on error propagation from thecentre of gravity of the data to the ITER operating point. A geometrical interpretation yielding formula (26), based on a simple summationof projections in principal axes, was derived
in [312], and a linearized projection formulaaround a standard operating point in [320].
We know, however, that the stated hypothesis is
not well satisfied in our situation, and that the classical intervals are too narrow to be physically realistic. To cope with this problem, √one can increase thetraditional scale factor c = 2σ̂ N in the standard
error propagation formula to provide some – albeit
a general and, hence, imperfect – safety margin. In
addition, one can make some assessment about thequality of the approximate character of the fitted
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
Table 14. Goodness-of-fit of various empirical confinement time scalings for the DB2.5 and DB3.0 standard ELMy
datasets and their prediction for ITER
Prediction of DB2.5(N = 1273)
Bias (%) Std dev. (%)

Scaling

Type

Based on

ITERH-P92y
ITERH-P93
IPB98(y)IPB98(y,1)
IPB98(y,2)
IPB98(y,3)
IPB98(y,4)
DK96(y)DK96(y)OK96(y)TT96(y)

ll
ll
ll
ll
ll
ll
ll
ia
ia
ol
onl

DB2 (ELMy)
DB2 (ELM free)(1)DB3 (ELMy)(2)DB3 (ELMy)
DB2.8 (ELMy)
DB2.5 (ELMy)
DB3r(IS) (ELMy)(3)DB2.5 (ELMy)
DB2.8 (ELMy)(4)DB2 (ELM free)(5)DB2 (ELMy)(6)
(7)

2.9
0.7
−0.8
0.4
0.6
−0.6
5.8
1.1
−0.8
—
10.4
12.8

15.1
18.5
15.2
14.7
14.5
14.2
16.7
14.4
14.6
16.8
20.1
19.4

Prediction of DB3.0(N = 1398)
Bias (%) Std dev. (%)
2.0
0.8
−0.6
0.6
1.2
−1.0
6.3
0.3
−0.4
2.0
10.6
12.5

17.5
19.3
15.8
15.3
15.6
16.1
17.3
16.5
15.3
18.7
22.8
19.3

ITER
(ref.)
τE (s)
5.7
6.1
6.0
5.9
4.9
5.0
5.1
5.4
5.0
7.3
4.4
—

(1)

Multiplied by 0.85 to account for ELMs.
Eq. (18); (negligible) bias due to rounding and to satisfying the high beta constraint.
(3)
Zero bias and 13.5% standard deviation for the dataset DB3r(IS).
(4)
Including a second interaction between current density and input power per surface area, which is not in a favourable
direction for ITER.
(5)
Adjusted to DB2.8 (ELMy), see text; ITER prediction is 6.9 s for 0.85 times OK-95 ELM free confinement; thevalues are 9.3 and 8.0 s, respectively, for the ITER IDR parameters (192 MW, 1.3 × 1020 m−3 ).
(6)Plus JT-60U (ELMy); scaling based on analysis that does not include a correction for ASDEX and PDX withrespect to the closeness of their divertor in comparison to the other tokamaks.
(7)
The same scaling fitted to DB2.5 and 2.8, respectively, without PBX-M.
Bias means the predicted minus the observed confinement time (on a natural logarithmic scale), averaged over thedataset. Extension (y) indicates that a scaling pertains to ELMy energy confinement. DB3 is the working dataset
from Section 6.3; DB2.8 excludes ohmic H-mode and additional heating other than NBI (except for Alcator C-Mod).
DB2.5 equals DB2.8 minus Alcator C-Mod.
Abbreviations: ll, log-linear; ia, interaction; ol, offset linear; onl, offset non-linear.
(2)

log-linear models. The last element involves (besides
graphical residual analysis) issues related to the functional form of the regression surface and to the systematic influence of factors not accounted for in the
regression model (usually called ‘hidden variables’).
The issue of the functional form leads to the following interval definition (even if idealized and departingfrom classical statistics):(ii) A 95% interval is formed by 95% (i.e. almostall) ‘admissible’ non-linear fits (on a logarithmic scale) to the dataset. In this context,
the word ‘admissible’ means that the RMSE
decreases significantly with respect to the best
fitting log-linear model (simple power law), andthat the model selection has been based on‘proper’ considerations, i.e. on relatively plausible physical arguments and/or simple model
extensions of existing first-order (log-linear)
Nuclear Fusion, Vol. 39, No. 12 (1999)

scalings, rather than on ‘artificially construed’
mathematical functions, directed towards especially high or especially low ITER confinement.
There are a number of related concepts (see [321])
that try to capture other interesting aspects. However, they are either rather difficult to carry out inpractice, or operational prescriptions, rather thanintrinsic definitions:(a) the discharges from a large number of ‘identical’ machines yield a distribution of confinement times, of which 95% should be situatedwithin the interval;(b) exchanging the two or three thermal energy
measurements available per machine (based on
Wdia , Wmhd , Wkin ), yields a large number ofdifferent predictions from which one can delete
2237

ITER Physics Basisthe 5% of extreme ones to obtain an interval
estimate;(c) one can construct jackknife type interval estimates by deleting one machine at a time fromthe database and looking at the variation in
the ITER prediction.
Alternatively, one can perform some cross validationby comparing the predicted confinement times of thetokamak deleted with the values actually observed.
For practical application we return toapproach (i). Following the argumentation in [321],
we will base the evaluation of the log-linear interval
estimate on the multiplication factorpc = cα σ̂/ Neff H,
withqcα = uα 4 Neff H/64,√
instead of on the classical value c = 2σ̂/ N . Here,
Neff denotes the ‘effective number’ of data points,
which means that multiple observations in all variables (within measurement error) are excluded. (Asusual, uα is defined by the property that the probability of a standard normal random variable toexceed uα equals α). The quantity
Neff H = −

N
tok
X

Neff ,j lb(Neff ,j /Neff )j=1

where Ntok is the number of tokamaks, lb denotes
the binary logarithm, and H is the Shannon uncertainty measure [444,445] reflecting the degree of nonuniformity of the distribution of the data points over
the tokamaks. For two tokamaks, each with 32 effec√
tive data points, c is equal to the familiar 2σ̂/ N
corresponding to a classical (two sided) 95% interval.
For other values of Ntok and Neff ,1 , . . . , Neff ,Ntok , anormalization to this situation has been made. Thisapproach provides at least some safeguard against
remaining moderate systematic influences that arealmost always present, but are not reduced as thesquare root of the sample size increases. (The absolute value of this calibration depends somewhat ona practical judgment of the specific situation.) The
estimated values of Neff ,1 , . . . , Neff ,Ntok can be foundin Table 13. In our case, for the DB3 ELMy dataset,
Neff = 867, H/Hmax = 0.785, Hmax = lb(11) =3.46, which makes cα = 4.8 and c = 1.5% instead of
the classical value c = 0.8%. The scaling of the ELM
free data, multiplied by a factor 0.85 to account fora roughly estimated confinement loss due to ELMs
2238

in ITER, supports, in general, the prediction basedon the ELMy dataset. Hence, this prediction is basedon more information than the ELMy dataset alone.
In fact, for the total (ELMy and ELM free) DB3
dataset, we have N = 2529, Neff = 1600, H/Hmax =0.795, Hmax = 3.46, cα = 5.6 and c = 1.27%. Alongthis line of thought, the additional information fromthe ELM free data reduces the width of the log-linear
interval from the ELMy dataset by some 15% toabout 1.6 times the width according to the classicalformulation.
From the second approach, a 95% confidence
interval for ITER confinement should include theinterval 4–7.5 s according to the three types of nonlinear scalings analysed in Section 2.6.4. Although,obviously, not all possible non-linear models havebeen investigated, the non-linear scalings discussed
stem from a range of analyses by different investigators and by using different approaches. By applying automatic optimization procedures towards highand low predictions for ITER, which disregards the‘admissibility’ criterion above, it is possible to arrive
at more extreme predictions. We will not pursue thisapproach any further here, but set the interval to
3.5–8 s.
Originated by Tukey [446], it has become a popular method to apply jackknife type estimators for thevariance of the estimator of an unknown parameterof interest, partly in view of the ease with which they
can be (electronically) calculated. Nevertheless, thestatistical basis of jackknife type arguments (‘leavingout one tokamak in turn’) is quite intricate. In [321] it
was found (by error propagation analysis) that theselead to some type of cross validated interval estimatespc = cα σ̂cr,ef f / Neff H
that are roughly a factor 2 larger than those usingpc = cα σ̂/ Neff H
if the assumption of a log-linear model is not abandoned. In [322] it was argued that by applying thesimple variance formula (coined ‘shotgun’ estimator in Efron’s fundamental paper [447]) applied to
the DB3 working dataset, the width of the interval estimate increases by an additional factor of2, which amounts to about six times the classicalwidth. As explained in more detail in [321], these two
approaches relax, in different degrees, the full classical log-linear model assumptions, albeit without
accurately specifying the class of alternative regression models assumed. The accuracy of the latter
Nuclear Fusion, Vol. 39, No. 12 (1999)

Chapter 2: Plasma confinement and transport
approach depends, notably, on the number of tokamaks (rather than on the effective number of timeslices) being sufficiently large to yield reasonableasymptotic approximations.
A further complication arises from the large variation in scaling exponents from different tokamak
operating periods, even with respect to the present
non-linear scalings. When modelled by a, somewhat
artificial, log-linear random coefficient model [321],
these lead to a different type of error propagation
than a constant times the classical error. One of
the basic problems seems to be that the shape ofthe true regression surface is not yet sufficiently wellknown, and the statistical methodology for fitting‘catastrophic type’ response functions [320] is still in
an initial stage of its development. As discussed in
Section 6.4, there are various additional issues associated with practical physical modelling of plasmatransport and with the question of hidden variables
that may have additional influence on confinement.
Further work in this area is being pursued.
As described in [320], on the basis of considerations of the type outlined here, while utilizing theinformation at the Naka-95 expert group meeting,a 95% (log-non-linear) interval estimate for ITER,
based on DB2, was set at 3.5–9 s, and a 95% (loglinear) interval estimate at 4.2–7.8 s, centred aroundthe point estimate of 6.0 s. (The latter interval corresponds, numerically, to roughly a 66% log-non-linear
interval.) The present analysis, based on the various
subsets of the extended database DB3, suggests a
95% log-non-linear interval estimate of 3.5–8 s, anda 23 log-non-linear (95% log-linear) interval of 4.4–
6.8 s, centred around the point estimate 5.5 s. This
result is graphically displayed in Fig. 13. The shift
with respect to the previous interval is partly due
to the new data and additional analysis and partly
to the change (from the IDR to the DDR values)of the reference operating point. The dependence ofthe log-linear interval width on the actual operating point (in the neighbourhood of a reference operating point) has been expressed analytically by asimple linearized formula in [321]. For constructing
the corresponding log-non-linear interval estimate,a more computer (and human intervention) intensive approach is still required, except for the simple
situation in which the actual operating point does
not deviate very much from the reference operating
point.
Nuclear Fusion, Vol. 39, No. 12 (1999)

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(Manuscript received 2 April 1998
Final manuscript accepted 23 April 1999)
E-mail address of V.S. Mukhovatov:mukhovv@itergps.naka.jaeri.go.jp
Subject classification: F1, Te; F1, Tt; F2, Te; F1, Tt

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