PHYSICS OF PLASMAS 16, 060704 共2009兲
Tokamak profile prediction using direct gyrokinetic
and neoclassical simulation
J. Candy,1 C. Holland,2 R. E. Waltz,1 M. R. Fahey,3 and E. Belli1,a兲
1
General Atomics, San Diego, California 92186, USA
University of California–San Diego, La Jolla, California 92093, USA
3
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
2
共Received 11 May 2009; accepted 11 June 2009; published online 26 June 2009兲
Tokamak transport modeling scenarios, including ITER 关ITER Physics Basis Editors, Nucl. Fusion
39, 2137 共1999兲兴 performance predictions, are based exclusively on reduced models for core
thermal and particle transport. The reason for this is simple: computational cost. A typical modeling
scenario may require the evaluation of thousands of individual transport fluxes 共local transport
models calculate the energy and particle fluxes across a specified flux surface given fixed profiles兲.
Despite continuous advances in direct gyrokinetic simulation, the cost of an individual simulation
remains so high that direct gyrokinetic transport calculations have been avoided. By developing a
steady-state iteration scheme suitable for direct gyrokinetic and neoclassical simulations, we can
now compute steady-state temperature profiles for DIII-D 关J. L. Luxon, Nucl. Fusion 42, 614
共2002兲兴 plasmas given known plasma sources. The new code, TGYRO, encapsulates the GYRO 关J.
Candy and R. E. Waltz, J. Comput. Phys. 186, 545 共2003兲兴 code, for turbulent transport, and the NEO
关E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 50, 095010 共2008兲兴 code, for kinetic
neoclassical transport. Results for DIII-D L-mode discharge 128913 are given, with computational
and experimental results consistent in the region 0 ⱕ r / a ⱕ 0.8. © 2009 American Institute of
Physics. 关DOI: 10.1063/1.3167820兴
Temperature and density profiles in tokamaks are fundamentally limited by pressure-gradient-driven turbulence and
to a lesser extent by cross-field transport caused by collisions
共so-called neoclassical transport兲. A first-principles description of these transport processes can be obtained via direct
kinetic simulations. The computational cost of direct simulation has, to date, been far too high to be considered for modeling and performance-prediction purposes. A single gyrokinetic simulation computes the cross-field, or radial, transport
共outputs兲 for given temperature and density profiles 共inputs兲.
However, the modeler needs to solve a very different problem; that is, given plasma sources 共inputs兲 such as heating
power, the self-consistent profiles 共outputs兲 are to be determined. This represents a sort of inverse problem, which has
to date been far too expensive to study by direct kinetic
simulation. Instead, to solve the inverse problem, modelers
use reduced models for core thermal and particle transport
where a typical modeling scenario may require the evaluation of thousands of local transport fluxes. However, there is
serious concern that existing modeling tools cannot be reliably extrapolated to reactor-scale plasmas like ITER.1
By developing an iteration scheme suitable for solving
the inverse problem, we have been able to obtain steady-state
temperature profiles for DIII-D2 plasmas using the GYRO
code3 to repeatedly calculate turbulent particle and energy
fluxes. These so-called transport solutions also include firstprinciples neoclassical fluxes computed using NEO.4 A new
code, TGYRO, acts as a data manager, overseeing execution of
multiple simultaneous instances of both GYRO and NEO, with
a兲
URL: http://fusion.gat.com/theory/gyro.
1070-664X/2009/16共6兲/060704/4/$25.00
the ability to substitute calls to less-expensive analytic or
reduced transport models in place of calls to NEO or GYRO.
Results for DIII-D L-mode discharge 128913 共Refs. 5 and 6兲
are given, for which computational and experimental results
compare remarkably well over the simulation domain, with
the dominant discrepancy occurring in the electron temperature near the magnetic axis. This result complements the traditional approach,6,7 which focuses on flux rather than profile
comparison.
The transport equation for a plasma species a can be
derived from the Fokker–Planck equation for the distribution
function f a, which we write compactly as
df a
= Sa + 兺 Cab共f a, f b兲.
dt
b
共1兲
Explicit consideration of the source Sa 共which contains the
effect of plasma heating兲 and the nonlinear collision operator
Cab are essential for a proper description of plasma transport.
Starting from Eq. 共1兲, one can derive a comprehensive transport equation.8 In the present letter we do not attempt to
model all the effects described in the most general formulation 共for example, turbulent energy exchange9兲 but limit attention to the dominant terms only. We emphasize that the
only consistent formulation of the transport problem, even in
the relatively general circumstances considered in Ref. 8,
relies on the assumption of a formal separation between
equilibrium and fluctuations. Only in this case is the nonlinear collision operator annihilated to lowest order and only in
this case is the lowest-order distribution a local Maxwellian.
In order to formulate the transport problem in terms of direct
gyrokinetic and neoclassical simulations, we therefore resort
16, 060704-1
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060704-2
Phys. Plasmas 16, 060704 共2009兲
Candy et al.
to the perturbative expansion f a = f 0a + f a1 + ¯, where f na
⬃ 共␳i / a兲n. Here, ␳i = vti / ⍀ci is the ion gyroradius, vti
= 冑Ti / mi is the ion thermal velocity, and ⍀ci = zieB / 共mic兲 is
the ion-cyclotron frequency. We further define the ion sound
speed, cs = 冑Te / mi, and the ion sound gyroradius ␳s = cs / ⍀ci,
consistent with Ref. 4. We require the equilibrium to evolve
slowly, and the source to be “weak;” this is formally expressed as 共⳵t f 0a , Sa兲 ⬃ 共␳i / a兲2. Under these conditions, the
equilibrium f 0a is easily shown to be a local Maxwellian. If
the electron and ion temperatures are unequal, there are small
residual collision terms, for example, Cie共f 0i , f 0e兲, which are
O共me / mi兲 and give rise to classical exchange. For tokamak
parameters, these are comparable in magnitude to the
O共␳i / a兲2 source terms. Integrating over the phase volume
interior to the flux surface at r 共the half width of the flux
surface at the elevation of the midplane兲 gives transport
equations of the form
⳵ 具n0a典 1 ⳵
+
共V⬘⌫2a兲 =
⳵t
V⬘ ⳵ r
冓冕 冔
冓冕 冔
⳵ 具W0a典 1 ⳵
+
共V⬘Q2a兲 =
⳵t
V⬘ ⳵ r
d3vS2a ,
d 3v
m av 2
S2a ,
2
共2兲
共3兲
共4兲
We assume for modeling purposes that the density profile is
fixed, although this assumption will be relaxed in future
work. The integrated sources s2a, which have dimensions
power/volume, include contributions from auxiliary heating,
radiation loss, and thermonuclear reactions 共relevant for reactor studies兲. In addition, we lumped the classical energy
exchange described above, which is finite when T0e ⫽ T0i,
into s2a. In this limit, the perturbed distribution can be split
into three distinct parts: a part which depends on the gyroangle, a long-wavelength neoclassical part 共k⬜ = 0兲, and a
short-wavelength gyrokinetic part 共k⬜ ⬎ 0兲, which contains
the effects of turbulence. The second-order fluxes are then
exactly separable into a sum of neoclassical and turbulent
neo
turb
+ Q2a
.
contributions: Q2a = Q2a
Rather than solving Eq. 共4兲 directly, we prefer to solve
the volume-integrated form
Qa共r兲 =
1
V⬘共r兲
冕
r
dxV⬘共x兲sa共x兲 ⬟ QTa 共r兲,
冉冕
Ta共r兲 = Taⴱ exp
rⴱ
冊
dxza共x兲 .
r
共6兲
On a discrete grid 兵r j其, the temperature profile can be approximately determined using the trapezoidal rule
再冋
Ta共r j−1兲 = Ta共r j兲exp
册
共5兲
0
where, for brevity, we hereafter omit the ordering subscripts.
We refer to the quantity QTa as the target flux, since that is the
flux we will try to match by adjusting parameters in the
gyrokinetic and neoclassical simulations. Since the kinetic
冎
za共r j兲 + za共r j−1兲
关r j − r j−1兴 .
2
共7兲
To put the problem into discrete form, we define a vector of
gradients 共independent variables兲 za,j = za共r j兲, transport
fluxes, Qa,j = Qa共r j兲, and target fluxes QTa,j = QTa 共r j兲. Then, the
equations to be solved are
共8兲
Q̂a,j = Q̂Ta,j ,
where ⌫2a is the second-order particle flux, Q2a is the
second-order energy flux, n0a is the equilibrium number density, and W0a = 共3 / 2兲n0aT0a is the equilibrium energy density.
Also, V⬘ is the derivative of the flux surface volume, V共r兲,
with respect to r. We restrict attention in the present letter to
the steady-state transport problem for a pure plasma, such
that a = 共i , e兲, in which case the equations for energy transport take the form
1 ⳵
共V⬘Q2a兲 = s2a共r兲.
V⬘ ⳵ r
codes require the profile gradients as inputs, we define the
logarithmic gradients za ⬟ −共1 / Ta兲 ⳵ Ta / ⳵r. Then, if we
specify the temperature at an arbitrary matching radius rⴱ
共normally chosen in the vicinity of the pedestal兲 as
Taⴱ ⬟ Ta共rⴱ兲, then the gradients uniquely determine the profiles, Ta:
where a hat denotes a gyro-Bohm normalization; that is, Q̂
⬟ Q / QGB with QGB共r兲 = neTecs共␳s / a兲2. Note that Q̂a,j is extremely expensive to evaluate, whereas Q̂Ta,j is virtually free.
Thus, the goal is to apply Newton’s method in a way which
is as accurate as possible while still minimizing evaluations
of Q̂a,j. To construct an iteration scheme, we assume the
transport fluxes depend only upon the local gradients, which
is approximately true when quantities are normalized to QGB.
This assumption affects only the rate of convergence, not the
accuracy of the root. The Newton-iteration equation has the
form
Jaa⬘,jj⬘␦za⬘,j⬘ = − 关Q̂a,j共z0兲 − Q̂Ta,j共z0兲兴␩a,j ,
共9兲
where we used the shorthand z0 ⬟ 兵z0a,j其. We emphasize that
the part of the Jacobian,
Jaa⬘,jj⬘共z0兲 =
⳵ Q̂Ta,j
⳵ Q̂a,j
␦ jj⬘ −
,
⳵ za⬘,j
⳵ za⬘,j⬘
共10兲
associated with Q̂a,j is block diagonal. In Eq. 共9兲, we introduced an arbitrary relaxation parameter ␩a,j. The quality of a
root at a given stage of iteration is measured by the residual
R1a,j = 兩Q̂a,j共z1兲 − Q̂Ta,j共z1兲兩, where z1 = z0 + ␦z is the Newton update vector. If, after a Newton step, R1a,j ⬎ R0a,j occurs, the
values of the corresponding ␩a,j are reduced by a factor of 2,
the local gradients reset to z0aj, and the step is recomputed. In
practice, we observe that limiting the magnitude of the correction ␦za,j on any given step to a maximum value of ␦zmax
improves robustness. Finally, we approximate the derivatives
in the Jacobian matrix using a forward-difference approximation,
Q̂a,j共za⬘,j⬘ + ⌬z兲 − Q̂a,j共za⬘,j⬘兲
⳵ Q̂a,j
⯝
.
⳵ za⬘,j⬘
⌬z
共11兲
A desirable feature of this approximation is that the iteration
scheme, Eq. 共9兲, will usually converge to the exact root of
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Phys. Plasmas 16, 060704 共2009兲
Tokamak profile prediction…
TABLE I. Fixed shape and profile parameters used in
r/a
␳
q
␬
TGYRO
simulations.
␦
R/a
a␥E / cs
a / Lne
0.1
0.089
1.44
1.27
0.057
2.85
⫺0.0463
0.409
0.2
0.3
0.177
0.269
1.52
1.57
1.28
1.29
0.087
0.100
2.84
2.83
0.0572
0.0680
0.806
1.07
0.4
0.357
1.62
1.29
0.111
2.83
0.0671
1.22
0.5
0.449
1.72
1.30
0.133
2.82
0.0499
1.18
0.6
0.7
0.542
0.638
1.90
2.22
1.31
1.33
0.164
0.197
2.81
2.80
0.0408
0.0534
1.02
0.96
0.8
0.740
2.76
1.36
0.237
2.79
0.0746
1.08
hour兲. Nevertheless, planned improvements in the efficiency
of the iteration scheme will be crucial for runs at increased
resolution—in particular, for cases which resolve significantly higher wave numbers in order to capture the electronscale transport.
Figure 1 compares simulated with experimental temperature profiles 共black curves兲 for DIII-D discharge 128913.
Simulation results with matching condition at rⴱ / a = 0.7 共red
curves兲 give good agreement over the entire simulated range,
with some discrepancy near the magnetic axis for the elec-
3
Ti [keV]
the original equation even when ⌬z is relatively large. Computing the forward difference Jacobian requires 共Nz + 1兲
evaluations of Qa,j, where Nz is the number of profiles being
evolved 共2 in the present case兲. There is generally one additional call to Qa,j due to the relaxation strategy, which resets
␩a,j. Using a large value of ⌬z is generally necessary because
of significant intermittency in the time history of Qa and an
associated statistical variance in its time average. Thus, the
method contains the adjustable parameters ⌬z and ␦zmax. Optimal values for gyrokinetic transport simulation differ from
those suitable for transport simulation with traditional transport models. Although the method we described generalizes
to an arbitrary number of gradients and fluxes per grid point,
in the present letter we restrict attention to the case of two
fluxes, Qi and Qe.
To test the algorithm and establish a proof of principle,
we apply the algorithm at nine radii across a DIII-D discharge. This requires eight separate instances of the GYRO
and NEO codes at radii r j = j⌬r, where ⌬r = 0.1a and j
= 1 , . . . , 8, plus an additional grid point at r0 = 0 共the magnetic
axis兲, where Qa,0 = 0. We limit the simulation to 0 ⱕ r / a
ⱕ 0.8 because in another work it has been shown that GYRO
tends to underpredict the transport for r / a ⬎ 0.8.6 In Table I
we summarize various profile parameters fixed over the
simulation duration: ␳ is the square root of the normalized
toroidal flux, q is the safety factor, ␬ is the elongation, ␦ is
the triangularity, R / a is the aspect ratio, ␥E is the E ⫻ B
shearing rate, and Lne is the density gradient scale length. For
the iterations, we use ⌬z = 0.3/ a and ␦zmax = 0.4/ a. The GYRO
simulations retain the effects of plasma shape, equilibrium
radial electric field shear, kinetic electrons 共mi / me = 3672兲,
and electron collisions, but treat electrostatic fluctuations
only. Simulations were carried out using an 共Lx , Ly兲 / ␳s
= 共96, 80兲 domain covering the spectral range 0 ⱕ ky␳s
ⱕ 0.86 and 0 ⬍ kx␳s ⬍ 1.6, with a 128-point velocity-space
discretization and 10 orbit grid points along a field line. The
NEO simulations used a 170 spectral-element velocity-space
expansion, with 17 points along a field line. Each complete
TGYRO iteration requires four calls to GYRO/NEO 共three for
the Jacobian evaluation and one for the relaxation correction兲
at each of the eight radii, for a total of 32 calls per iteration.
The cases studied in this letter typically converge in about 12
iterations, or 384 total calls to GYRO/NEO. The total computing time 共Cray XT4兲 required to reach convergence is about
58 h on 1536 cores 共at a rate of 6.6 GYRO simulations per
Experiment
TGYRO r∗ /a = 0.5
TGYRO r∗ /a = 0.7
2
1
0
(a) 0
0.1
0.2
0.3
0.4 0.5
r/a
0.6
0.7
0.8
4
Experiment
TGYRO r∗ /a = 0.5
TGYRO r∗ /a = 0.7
3
Te [keV]
060704-3
2
1
0
(b) 0
0.1
0.2
0.3
0.4 0.5
r/a
0.6
0.7
0.8
FIG. 1. 共Color online兲 Steady-state TGYRO simulation results for ion 共a兲 and
electron temperature 共b兲. Black curves show experimental data from DIII-D
discharge 128913, red curves show simulation results with matching condition at rⴱ / a = 0.7, and blue curves show simulation results for matching at
rⴱ / a = 0.5.
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060704-4
Phys. Plasmas 16, 060704 共2009兲
Candy et al.
102
Qi = Qneo
+ Qturb
i
i
neo
Qi
Qi /QGB
101
QTi
100
10−1
10−2
(a) 0.1
0.2
0.3
102
Qe /QGB
0.6
0.7
0.8
0.7
0.8
turb
Qe = Qneo
e + Qe
Qneo
e
101
10
0.4 0.5
r/a
QTe
0
10−1
10−2
10−3
(b) 0.1
0.2
0.3
0.4 0.5
r/a
0.6
FIG. 2. 共Color online兲 Comparison of transport 共black curves兲 and target
共solid blue curves兲 fluxes for the TGYRO case with matching radius rⴱ / a
= 0.5, comparing both ion 共a兲 and electron 共b兲 channels. Also shown are the
neoclassical fluxes 共dashed lines兲, which become dominant for r / a ⬍ 0.1.
The QT effectively represent experimentally measured values.
tron temperature. The quality of agreement improves further
when the matching condition is moved inward to rⴱ / a = 0.5,
but the anomaly in Te near the magnetic axis persists. Additional insight into the character of the transport can be
gleaned by inspection of the various components of Eq. 共8兲.
Figure 2 indicates that the quality of the solution 共i.e., the
closeness of Qa to QTa 兲 in both channels is remarkably good
considering the statistical uncertainty involved in the gyrokinetic computation of Qturb
a / QGB. We can also identify four
distinct transport regions: 共1兲 a strong turbulence region,
r / a ⬎ 0.7, for which Qa / QGB ⬎ 10.0, 共2兲 a moderate turbulence region, 0.3⬍ r / a ⬍ 0.7, for which 1 ⬍ Qa / QGB ⬍ 10, 共3兲
a near-threshold region, 0.1⬍ r / a ⬍ 0.3, where turbulence is
very small, and difficult to estimate accurately, and 共4兲 a
turbulence-free region, r / a ⬍ 0.1, where the transport is neoclassical. Throughout regions 1 and 2, the ion-temperaturegradient 共ITG兲 instability is active, with drive from nonadiabatic trapped electrons becoming progressively stronger as
r / a increases. The significant region of nonmarginal 共i.e.,
Qa / QGB ⬎ 1兲 transport places serious limitations on discussion of turbulence near marginality.10
The radial electric field is taken as an input and is not
modified 共for example, by an equation for the evolution of
toroidal angular momentum兲. Moreover, the only effect of
flow shear retained in these simulations is the E ⫻ B shear
共the so-called parallel velocity shear and Coriolis drift effects
have been ignored兲. The E ⫻ B shear has been shown in the
past to significantly reduce the transport in GYRO simulations
of DIII-D plasmas3 and the present discharge is no exception.
The plasma performance is in fact significantly worse if E
⫻ B shear is ignored.
Figure 2 shows that the transport fluxes match the targets
almost perfectly. This result is significant in that it demonstrates that the temperate profiles in Fig. 1 are precise roots
of Eq. 共8兲. The small underprediction in the strong transport
region toward the edge is a feature of the discharge already
noted by Holland,6 where a series of high-resolution simulations were carried out at r / a ⬃ 0.8, with the conclusion that
the GYRO underprediction of transport is robust and not sensitive to simulation resolution. The tiny flux undershoot near
the core is less significant, since the transport is virtually
zero there. With regard to the overprediction of the electron
temperature, although we can say for certain that there is
missing electron-temperature-gradient mode transport at
electron scales that would act to lower the core electron temperature, the magnitude cannot be reliably estimated without
expensive multiscale ITG-ETG simulations. In previous
work11 we have shown that when there is significant ionscale instability, ETG modes account for only a small fraction 共roughly 10%兲 of the total electron transport. However,
this fraction can increase in cases where the ion-scale instabilities are partially or completely suppressed. This is indeed
the case deep in the core and work is presently underway to
quantify the amount of ETG-scale transport in the region
r / a ⬍ 0.4.
This work was supported by the U.S. DOE under Contract Nos. DE-FG03-95ER54309 and DE-FG02-07ER54917
as part of the FACETS SciDAC project and used the resources of the NCCS at ORNL under Contract No. DEAC05-00OR22725.
ITER Physics Basis Editors, Nucl. Fusion 39, 2175 共1999兲.
J. Luxon, Nucl. Fusion 42, 614 共2002兲.
3
J. Candy and R. Waltz, J. Comput. Phys. 186, 545 共2003兲.
4
E. Belli and J. Candy, Plasma Phys. Controlled Fusion 50, 095010 共2008兲.
5
A. White, L. Schmitz, G. McKee, C. Holland, W. Peebles, T. Carter, M.
Shafer, M. Austin, K. Burrell, J. Candy, J. C. DeBoo, E. J. Doyle, M. A.
Makowski, R. Prater, T. L. Rhodes, G. M. Staebler, G. R. Tynan, R. E.
Waltz, and G. Wang, Phys. Plasmas 15, 056116 共2008兲.
6
C. Holland, J. Candy, R. Waltz, A. White, G. McKee, M. Shafer, L.
Schmitz, and G. Tynan, J. Phys.: Conf. Series 125, 012043 共2008兲.
7
L. Lin, M. Porkolab, E. Edlund, J. Rost, C. Fiore, M. Greenwald, Y. Lin,
D. Mikkelsen, N. Tsujii, and S. J. Wukitch, Phys. Plasmas 16, 012502
共2009兲.
8
F. Hinton and R. Waltz, Phys. Plasmas 13, 102301 共2006兲.
9
R. Waltz and G. Staebler, Phys. Plasmas 15, 014505 共2008兲.
10
D. Newman, B. Carreras, P. Diamond, and T. Hahm, Phys. Plasmas 3,
1858 共1996兲.
11
J. Candy, R. Waltz, M. Fahey, and C. Holland, Plasma Phys. Controlled
Fusion 49, 1209 共2007兲.
1
2
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