System Size Effects on Gyrokinetic Turbulence
B.F. McMillan, X. Lapillonne, S. Brunner, and L. Villard
Centre de Recherches en Physique des Plasmas, Association Euratom-Confe ´de ´ration Suisse,
Ecole Polytechnique Fe ´de ´rale de Lausanne, PPB, 1015 Lausanne, Switzerland
Japan Atomic Energy Agency, Higashi-Ueno-6-9-3, Taitou, Tokyo 110-0015, Japan
A. Bottino, T. Go ¨rler, and F. Jenko
Max Planck Institut fu ¨r Plasmaphysik, Boltzmannstrasse 2, D-85748 Garching, Germany
(Received 30 June 2010; published 4 October 2010)
The scaling of turbulence-driven heat transport with system size in magnetically confined plasmas is
reexamined using first-principles based numerical simulations. Two very different numerical methods are
applied to this problem, in order to resolve a long-standing quantitative disagreement, which may have
arisen due to inconsistencies in the geometrical approximation. System size effects are further explored by
modifying the width of the strong gradient region at fixed system size. The finite width of the strong
gradient region in gyroradius units, rather than the finite overall system size, is found to induce the
diffusivity reduction seen in global gyrokinetic simulations.
DOI: 10.1103/PhysRevLett.105.155001 PACS numbers: 52.30.Gz, 52.35.Ra, 52.65.Tt
The ultimate purpose of gyrokinetic simulations of to-
kamaks is to predict the transport properties of fusion
reactors. This is often restricted to the less ambitious
goal of predicting how the transport properties scale with
system size, so that the properties of a reactor can be
estimated using measurements of transport in smaller
Most of the scaling studies to date [1,2] have used the
Cyclone model equilibrium , based loosely on an ex-
periment on the DIII-D tokamak . The Cyclone parame-
ters have been used as a standard case for benchmarking
gyrokinetic codes, and as a baseline case for parameter
scans with both global and flux-tube codes, with a variety
of approximations taken for the background geometry and
magnetic field. The heat flux is quite sensitive to minor
model changes because the baseline parameters are rela-
tively close to marginal nonlinear stability. As a result,
both finite system size effects and geometrical assumptions
can make a relatively large difference to the predicted heat
diffusivity (up to a factor of 2), so that direct comparisons
between code results are exacting.
In this Letter we reexamine the effects of global system
size and temperature gradient profiles on the heat diffusiv-
ity. To improve the confidence in the final results we
perform simulations with two gyrokinetic codes with
very different numerical methods, which have been care-
fully benchmarked, both independently [5–7] and against
each other . The global-GENE code [6,7] is a general-
ization of the flux-tube Eulerian GENE  code to global
geometry. ORB5  is a global particle-in-cell code.
For the equilibrium description, we choose the circular
concentric model of Refs. [5,10], which is convenient for
numerical studies as it provides simple expressions for
geometrical and magnetic field quantities, although it is
not a true MHD equilibrium. A flux-tube gyrokinetic
analysis finds linear growth rates and a heat diffusivity
for this model which are quite close to those obtained using
an appropriate MHD equilibrium. This is not the case for
typical implementations of the s-? model, where the as-
sumption that the geometrical poloidal angle is a straight
field line coordinate leads to inconsistencies between geo-
metric quantities and the field strength which are first order
in the inverse aspect ratio .
Ion temperature gradient (ITG) turbulence is driven on
the scale of the ion gyroradius, ?i, but a related quantity,
the ion sound radius ?s, is typically used to parameterize
the turbulent length scale. The value of the parameter
1=??¼ a=?s, where a is the tokamak minor radius, is
thus a useful dimensionless measure of system size. In
order to evaluate how global effects influence turbulence
levels, a ??scan is carried out using the global-GENE and
ORB5 codes, for 1=??2 ½70;140;180;280;560?, and the
results are compared to a GENE flux-tube simulation. The
Cyclone parameters are T ¼ ZTe, a=R ¼ 0:36, ?n¼ 2:23,
?T¼ 6:96, with T and Tethe ion and electron tempera-
tures, respectively, R the major radius, and ?nand ?Tthe
logarithmic density and temperature gradient at midradius,
respectively.Electrons aremodeled asan adiabatic species,
and the simulations are performed in the electrostatic limit.
We choose a safety factor profile qðrÞ ¼ ð0:854 þ
2:184r2Þ=½1 ? ðra=RÞ2?1=2, where r is the distance from
the magnetic axis, normalized to the minor radius. For this
profile, qð0:5Þ ¼ 1:4 and sð0:5Þ ¼ ðr=qÞdq=dr ¼ 0:8 to
match the Cyclone parameters at midradius. Two very
PRL 105, 155001 (2010)
8 OCTOBER 2010
? 2010 The American Physical Society
similar functional forms of the initial temperature gradient
profile are used. For the first,
for r 2 ½r0? ?r=2;r0þ ?r=2?, and zero elsewhere, with
r?¼ r ? r0? ?r=2. The other functional form is
1 ? cosh?2
The ion density profiles are defined using the same func-
tional form, with the scale factor ?nreplacing ?T. Both
ORB5 and GENE were run with r0¼ 0:5, with ?r¼ 0:8 and
0.7 for ORB5 and GENE, respectively, so that R=LT¼
?RdlnðTÞ=dr is constant over most of the minor radius
and equal to ?T. In this section, GENE uses the first func-
tional form, and ORB5 the second. ORB5 uses a canonical
Maxwellian to define the background distribution, rather
than a local Maxwellian, so the reconstructed temperature
profiles differ slightly from nominal values (the bracketing
described below corrects for this effect).
For the GENE code, 33 toroidal modes were examined,
with ky2 ½0:048;1:6? for ??¼ 1=180, and similar ky
range for other ??. For the baseline case at ??¼ 1=180,
the number of grid points is 150, 64, and 16 in the radial,
binormal, and field line direction, respectively. For differ-
ent ??,thenumberof radialgridpointsis scaledtokeepthe
resolution constant in units of ion gyroradius. 64 grid
points are used in vk, and 16 in ?. The simulation domain
is restrictedtoaradialdomain0:1 < r < 0:9,and,for??¼
1=180, a 1=3 wedge of the full torus. The values of the
hyperdiffusion terms, added for stability reasons, can be
found in Ref. .
For ORB5, the ??¼ 1=180 simulations were performed
with a field grid of 128 ? 512 ? 256 points in the radial,
poloidal, and toroidal directions, respectively. 45 toroidal
modes were kept, and the simulation domain is a 1=2
wedge of the full torus from 0 < r < 1. 320 ? 106markers
were used, and a Krook operator with a coefficient
0:0035cs=a, & 2:5% of the peak linear growth rate, for
the range of ??considered (this is at least a factor of 4
smaller than suggested by convergence requirements ).
For different ??, the number of field grid points in each
direction is scaled proportional to 1=??, the fractional size
scaled according to 1=??, to keep the number of markers
per mode constant.
Dirichlet boundary conditions are used in GENE for the
electrostatic potential at the inner and outer radial points,
whereas ORB5 imposes Dirichlet conditions only at the
outer boundary. The boundary conditions on the distribu-
tion function are also slightly different, and are described
in detail in Refs. [5,6]. The restriction to a wedge of the
torus of roughly constant size in gyroradius units is moti-
vated by earlier convergence studies.
Both codes use a heat source proportional to the tem-
perature deviation ?T ¼ Tðr;tÞ ? Tðr;0Þ; as a conse-
quence, the steady state temperature gradient differs
somewhat from the initial temperature gradient, in a way
which cannot accurately be estimated prior to a simulation
run. In order to compute the heat diffusivity over this ??
scan at a desired temperature gradient, two simulations
were carried out for each ??value, with ?T¼ 7:1 and
?T¼ 7:5. This gives us a pair of values for the final
temperature gradients which closely bracket the target
temperature gradient R=LT¼ 7:0. The diffusivity ?icor-
responding to the target gradient is then computed using
linear interpolation. In Fig. 1, the resulting heat diffusiv-
ities ?iðR=LTi¼ 7Þ obtained with both global-GENE and
ORB5 are shown in gyro-Bohm units (?GB¼ ?2
function of ??, and the results are compared to a GENE flux-
The relative difference between the flux-tube and global
heat diffusivity is below 10% for 1=??* 280, confirming
the qualitative results of Refs. [2,12] that the global gyro-
kinetic codes reproduce the flux-tube diffusivities in the
limit ??! 0, where the gyro-Bohm scaling is prescribed.
Gyro-Bohm diffusivity scaling is expected in this limit
based on a mixing length argument, in which eddies on
turbulence scale, ??s, are able to transport energy across
an eddy width on the time scale of the driving instability,
?a=cs. The minor radius enters the time scale because, for
fixed geometry, the growth rate is dependent on the field
line curvature and inverse gradient lengths, which scale
The ORB5 and global-GENE results show a similar ??
dependence, as well as an excellent overall agreement,
within 10%, between the two codes. An estimate of the
scs=a) as a
0 100200 300
FIG. 1 (color online).
scs=a) obtained with global-GENE and ORB5 as function of
1=??. A radial window, r 2 ½0:4;0:6? is used, and a time average
over tcs=R 2 ½210;450?. The results are compared with the
corresponding GENE flux-tube simulation. The error bar repre-
sents simulation-to-simulation variability.
Ion heat diffusivity ?i=?GB (?GB¼
PRL 105, 155001 (2010)
8 OCTOBER 2010
simulation-to-simulation variability of the measured ?i
due to turbulent variation has been obtained forthe 1=??¼
180 case by carrying out three independent ORB5 simula-
tions with different initial conditions . The diffusivities
predicted by the two codes are essentially within the error
bars, despite the very different numerical methods and a
somewhat different set of approximations. In the ??! 0
limit, both diffusivity curves asymptote to ?i=?GB’
2:8–2:9, similar to the value found in flux-tube simulations
using the circular model equilibrium (?i=?GB¼ 3:0) or a
MHD equilibrium (?i=?GB¼ 3:3) .
Results of the GYRO code  using an s-? approxima-
tion, can be compared against the set of results of the GTC
code  with finite aspect ratio effects ‘‘removed’’
(although presumably some finite aspect ratio effects,
like magnetic trapping, were kept) to try to reproduce the
s-? results. These GTC results appear to asymptote to at
least ?i=?GB? 2:5 in the flux-tube limit. On the other
hand, the global GYRO results in Ref.  converge to-
wards a value of ?i=?GB’ 1:9 in the limit ??! 0.
The circular analytical equilibrium used for the GTC ??
scaling runs of Ref. , yields linear growth rates close to
those using the numerical MHD equilibria , and ap-
pears to be similar to the equilibrium used by GENE and
ORB5. Diffusivities from the GTC ??scans of Ref. 
converge towards ?i=?GB’ 3:4 in the limit ??! 0, which
is in relatively good agreement with the present results
?i=?GB’ 2:8–2:9. This agreement occurs despite notable
differences between these two studies, including the use of
shorter simulations in Ref. .
In the s-? approximation the geometry specification is
inconsistent withthe magnetic
However, in their manipulations of the gyrokinetic equa-
tions, code authors often assume that the magnetic field
and geometry are consistent. The final equations used
might, for example, include terms like the divergence of
the magnetic field, which is nonzero in the s-? model, or
these might have been explicitly set to zero by code
authors, given that physical fields are divergence free.
The results of using the s-? approximation will thus not
only be incorrect, but incorrect in a way which may vary
from code to code, so that results are not easily reproduc-
ible. Many flux-tube codes using the s-? model find a
diffusivity of ?i=?GB? 1:7  for the Cyclone test case,
substantially different from the value ?i=?GB? 3:3 found
using an MHD equilibrium, or 3.0 found using the circular
concentric model . The use of the s-? model may also
explain why the results of Ref.  differ from ours. We
suggest that future benchmarks be run using consistent
For a fixed local temperature gradient, the shape of the
temperature profile is known to modify turbulence inten-
sity . To quantify this effect, two extra pairs of simula-
tions were run at ??¼ 1=280, with the normalized width
?rof the strong temperature region set to ?0:4 and ?0:2,
approximately one-half and one-quarter of thevalues in the
previous section (see Fig. 2). We again use pairs of simu-
lations with slightly different nominal temperature gra-
dients, to bracket the desired final temperature gradient.
Here, both codes use Eq. (2) for temperature and density
gradient profiles, but parameters are otherwise identical to
the simulations described above. The mean flux levels of
these simulations, together with those of the previous
section, are plotted in Fig. 3, against the measured width
of the strong gradient region ??
gyroradius. Thewidth ?ris the full width at half maximum
of the initial reconstructed temperature gradient profile;
because the heating operator is quite effective at maintain-
ing these profiles, using late time values makes very little
eff¼ ??=?r, in units of
0 0.20.4 0.60.8
FIG. 2 (color online).
ted against radius for simulations with ??¼ 1=280, and ?r¼
0:2, 0.4, 0.8, in order of increasing width of the plotted peak. The
dashed lines show initial gradients, and the solid lines show the
late time average over the last half of the simulation time.
Logarithmic temperature gradients plot-
0 100 200 300 400500
ORB5, δ x scan
GENE, δ x scan
ORB5, ρ* scan
GENE, ρ* scan
FIG. 3 (color online).
½150;410?R=csversus the width of the strong gradient region,
eff, in gyroradius units.
Average flux levels r 2 ½0:4;0:6? for t 2
PRL 105, 155001 (2010)
8 OCTOBER 2010
difference. The good agreement between the data points in Download full-text
Fig. 3 suggests that the variation in diffusivity levels in the
??scan is a result of the changing width of the strong
gradient region (in units of gyroradius) rather than the
system size ??itself. Note that several of the data points
in Fig. 3havevery similar ??
eff, and levels of diffusivity, but
very different ??. Any influence due to the overall system
size ??is seen to be small compared to that of ??
diffusivity reduction at large ??
be important not only in small tokamaks, but also in
transport barriers, where strong gradients appear over a
small proportion of the minor radius.
The radial window used for flux averaging becomes
comparable to the width of the strong gradient region for
the extreme case ?r¼ 0:2. However, the measured diffu-
sivity is not very sensitive to the choice of the averaging
window: if the width of the flux averaging window is
reduced proportionately to ?r, the results are essentially
Because changing ??does not appear to substantially
modify the diffusivities, for fixed ??
responsible for the flux reduction should exist in the flux-
tube-like limit where ??! 0, but ??
This limit is different from the usual flux-tube limit only
through the choice of a modified initial temperatureprofile,
and the use of a heat source term to maintain this tempera-
ture gradient profile; the temperature, shear, and density
would be constant across the radial domain, as in a stan-
dard flux-tube simulation.
Profile shearing is a form of phase mixing which occurs
because the local mode frequency !ðrÞ (calculated for a
homogeneous plasma) has a nonzero radial derivative, so
that radial phase mixing occurs, and modes tend to secu-
larly become tilted with time. This flux reduction mecha-
nism is analogous to that due to sheared zonal flows. At
region, where the logarithmic temperature gradient is
nearly constant, depends on ??, but the results are clearly
unaffected by this variation: profile shearing cannot be
causing the diffusivity reduction at large ??
Some nonlocality is expected because turbulent eddies
have radial extent of ?20 gyroradii, so sense an interval of
the minor radius, but this is not enough to explain the
significant flux reduction seen at ??
edge of the strongly driven region is 60 gyroradii from the
center of the averaging window. Information appears to
propagate over several eddy lengths (or turbulence decor-
relation lengths). The diffusion of turbulent intensity, or
‘‘turbulence spreading’’ , is one possible way to ex-
plain strong nonlocal effects. The presence of larger scale
structures, like zonal flows, or avalanches  is also
expected to lead to substantial nonlocality.
effwould thus be expected to
eff, the mechanisms
effis held constant.
eff, the variation of !ðrÞ over the strong gradient
eff¼ 1=120, where the
To summarize, for the Cyclone test parameters, we
confirm that the diffusivities of finite plasmas approach
the flux-tube limit from below in the large system size
limit, in agreement with Ref. . Quantitatively, we find an
asymptotic diffusivity of ?i=?GB’ 2:8–2:9. This substan-
tially differs from the value found in Ref. , which used
an inconsistent s-? equilibrium model, but in rough agree-
ment with the results of Ref. , which used a more
realistic equilibrium. Moreover, we find that the reduction
of flux at finite system size is largely the result of the finite
width of the unstable region in gyroradius units, rather than
the overall tokamak size.
This work was supported in part by the Swiss National
Science Foundation, a grant from the Swiss National
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