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34th EPS Conference on Plasma Phys. Warsaw, 2 - 6 July 2007 ECA Vol.31F, P-4.041 (2007)
Variance reduction in computations of neoclassical transport
in stellarators using a δ f method ∗
K. Allmaier1 , S. V. Kasilov2,1 , W. Kernbichler1 , G. O. Leitold1 , V. V. Nemov2,1
1 Association
EURATOM-ÖAW, Institut für Theoretische Physik - Computational Physics,
Technische Universität Graz, Petersgasse 16, A–8010 Graz, Austria
2 Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and
Technology”, Akademicheskaya Str. 1, 61108 Kharkov, Ukraine
In absence of radial electric field, mono-energetic transport coefficients are determined by
the steady state solution of the linearized drift kinetic equation for the normalized perturbation
of the distribution function fˆ (marker),
∂
+ vk h · ∇ − LC fˆ = ψ̇ ≡ Vg · ∇ψ,
LD fˆ ≡
(1)
∂t
where vk , h, LC and ψ̇ are parallel velocity, unit vector along the magnetic field, Lorentz
collision operator and co-variant ψ-component of the drift velocity, respectively. Here, ψ is a
flux surface label, and a marker is defined through the local Maxwellian distribution function
fM and the total distribution function f via f = fM − fˆ∂ fM /∂ ψ. The mono-energetic radial
diffusion coefficient and the normalized bootstrap coefficient, respectively, are given by
+
* Z1
* Z1
+
3
1
1
1
Dmono = −
dλ fˆψ̇ ,
dλ fˆλ B ,
λbb = −
(2)
2
h|∇ψ|i
B
2
ρ
2
L 0
h|∇ψ|i
−1
−1
where λ = vk /v is the pitch parameter, ρL is the Larmor radius in the reference magnetic field
B0 , B is the magnetic field module and h. . . i denotes the average over the volume between
neighboring flux surfaces. The quantity λbb is linked to the equilibrium (bootstrap) current
density jk and gradient of the pressure p by λbb = − jk B (c h|∇ψ|i dp/dψ)−1 . In the follow√
ing Dmono is normalized by the plateau diffusion coefficient Dplateau = πvρL2 (8 2ιR)−1 where
ι and R are the rotational transform and major radius, respectively.
In order to introduce the Monte Carlo operator it is convenient to re-write (1) in the integral
form using a Green’s function G defined by
LD G(t, z, z0 ) = 0,
G(0, z, z0 ) = (g(z0 ))−1/2 δ (z − z0 ),
(3)
where z = (ϑ , ϕ, λ ) and g is a metric determinant of flux coordinates (ψ, ϑ , ϕ). This Green’s
R
function is normalized to 1, d3 z (g(z))1/2 G(t, z, z0 ) = 1. Thus, a formal solution to Eq. (1) is


fˆ(t, z) =
∗ This
Z
1/2 
d3 z0 (g(z0 ))
G(t − t0 , z, z0 ) fˆ(t0 , z0 ) +
Zt
t0
dt ′ G(t − t ′ , z, z0 )ψ̇(z0 ) .
(4)
work, supported by the European Communities under the contract of Association between EURATOM
and the Austrian Academy of Sciences, was carried out within the framework of the European Fusion Development Agreement. The views and opinions expressed herein do not necessarily reflect those of the European
Commission. Additional funding is provided by the Austrian Science Foundation, FWF, under contract number
P16797-N08.
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If a steady state solution is looked for, fˆ(t, z) = fˆ(z), Eq. (4) becomes an integral equation for
F(z) = (g(z))1/2 fˆ(z) given below also in the operator form,
F(z) =
Z
d3 z0 K(z, z0 )F(z0 ) + Q(z) ≡ K F + Q,
(5)
where K(z, z0 ) = (g(z))1/2 G(∆t, z, z0 ), ∆t is the integration time step and
Q(z) =
Z
1/2
3
d z0 (g(z)g(z0 ))
Z∆t
0
dt ′ G(t ′ , z, z0 )ψ̇(z0 ) ≈ ∆t (g(z))1/2 ψ̇(z).
(6)
Introducing the Monte Carlo operator, Z(∆t, z0 ), being the random position of a test particle
starting at z0 after a single time step modeled in a standard way by the random change of λ
in accordance with LC and integration of particle drift equations, the kernel of the integral
equation is given by an expectation value K(z, z0 ) = δ (z − Z(∆t, z0 )). The solution of (5) by
direct iterations can be presented as an expectation value of an integral along the stochastic
orbit,
F=
∞
∞
k=0
k=0
∑ K k Q = C0 ∑ w0 δ (z − zk ),
zk = Z(∆t, zk−1 ),
w0 = ∆t ψ̇(z0 ),
(7)
where C0 = d3 z (g(z))1/2 and the random starting point z0 is chosen with the probability
density δ (z − z0 ) = C0−1 (g(z))1/2 . The averages (2) are given by expectation values as
R
Dmono = −
1
∞
∑ w0ψ̇(zk ),
h|∇ψ|i2 k=0
λbb = −
∞
3
∑ w0λk B(zk ).
ρL B0 h|∇ψ|i k=0
(8)
When k∆t exceeds a few collision times, the correlation between zk and w0 is lost and, therefore, such terms in (8) tend to zero, e.g. w0 ψ̇(zk ) → w0 ψ̇(zk ) = 0 because the expectation
R
value w0 = C0−1 ∆t d3 z (g(z))1/2 ψ̇(z) = 0 due to Liouville’s theorem. Thus, a finite sum over
k is sufficient in (8). The method described by (8) has rather low variance in computations
of Dmono , however, for λbb variance has a very unfavorable scaling with collisionality. Indeed,
only the orbits originating in the boundary layer in velocity space of the width ∆λ ∼ (Lc /lc )1/2
where Lc = 2πR/ι and lc = vτc are the connection length and mean free path, respectively, contribute to λbb . In addition, the contribution of a particle from the boundary layer is ∆λ times
smaller than of a normal passing particle because of a higher trapping probability. Therefore,
the variance in λbb scales for this method as (lc /Lc )2 in the long mean free path regime.
It should be noted that test particles at each step keep the same equilibrium distribution,
δ (z − zk ) = C0−1 (g(z))1/2 . Therefore w0 λk− j B(zk− j ) = w j λk B(zk ) where w j = ∆t ψ̇(z j ), and
∞
1 K
∑ Wk λk B(zk ),
K→∞ K
k=0
∑ w0λk B(zk ) = lim Wk λk B(zk ) = lim
k=0
k→∞
k
Wk =
∑ w j.
(9)
j=0
The procedure in (9) corresponds to a standard δ f method [1] where the test particle weight
Wk is an integral of ψ̇ along a stochastic orbit. In a tokamak, the variance in λbb does not
scale with the collisionality, and the required CPU time for this method scales linearly with
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lc /Lc . However, in stellarators variance in λbb again recovers the scaling (lc /Lc )2 because
due to the non-zero bounce-averaged drift of trapped particles large contributions to Wk are
acquired, which scale as ψ̇τc and which are weakly correlated with the value of λk after they are
detrapped. To overcome this problem, in Ref. [2] trapped particles with large Wk are replaced
with particles with Wk = 0 which formally introduces some bias in the result.
For a formally “unbiased” method it is convenient to split the source in (5) into “passing”
and “trapped” sources Q p = χQ and Qt = Q − Q p using χ = tanh ((λ − λt−p )/∆λ ) where
λt−p is a trapped-passing boundary. Results for transport coefficients for these two sources are
added up at the end. The problem with Q p is solved with the standard method (9) because
accumulation of large weights is avoided there. For the problem with Qt the formal solution
to (5) is presented as F = FM + ∆FM where FM satisfies a similar equation with a different
source term, FM = K FM + QM , and ∆FM is known,
1 M−1 k
QM =
∑ K Q,
M k=0
∆FM =
M−1 ∑
k=0
k+1
1−
K k Q.
M
(10)
∆FM is used only in averages (2) while QM is computed by scoring weights on the 3D grid. On
the grid weights are partly annihilated. After the first such iteration, the module of the weight
is fixed and then iterations are repeated formally putting F = FM and Q = QM and applying
the same procedure to the resulting integral equation. Due to annihilation and fixed module
of the weight, the number of test particles needed for sampling the source term from the grid
is decreasing and iterations are stopped when this number is below 1. The number of steps
M for a single iteration is chosen to be much smaller than collision time and large enough
in order to fill the grid using a limited number of test particles. Since source terms generated
in this way are small in the passing and boundary region, particles are generated there with
smaller weights and particles which enter the boundary layer from the trapped side are split
in such a way that the number of particles in the passing and trapped regions is of the same
order. As a result, variance of this method is reduced to the scaling lc /Lc . In addition, due to
the decay of test particle number with iterations, the CPU time is also reduced and scales as
(lc /Lc )3/2 for given accuracy which is much better than scaling (lc /Lc )3 of the standard δ f
method. Results of testing of the method for a few toroidal devices stay in good agreement with
results of NEO-2, a field line tracing code which computes transport coefficients in arbitrary
collisionality regimes [3], as shown in Figs. 1-3.
References
[1] M. Sasinowski and A.H. Boozer, Phys. Plasmas 2, 610 (1995)
[2] M.Yu. Isaev, S. Brunner, W.A. Cooper, et al., Fusion Science & Technology 50, 440
(2006)
[3] W. Kernbichler, S. V. Kasilov, G. O. Leitold, V. V. Nemov and K. Allmaier, 33rd EPS
Conference on Plasma Physics, Rome, 19–23 June 2006, ECA Vol. 30I P-2.189 (2006)
34th EPS 2007; K.Allmaier et al. : Variance reduction in computations of neoclassical transport in stellarator...
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15
0
10
−2
10
λbb
Dmono/Dplateau
NEO2
MC
NEO2
MC
10
−4
5
10
−6
10
0
−8
−7
10
−6
10
10
−5
10
−4
−3
10
Lc/ lc
10
−2
10
−1
−8
0
10
−7
10
10
−6
10
−5
10
10
−4
−3
10
Lc/ lc
10
−2
10
−1
0
10
10
Figure 1: Normalized diffusion coefficient Dmono /Dplateau (left) and bootstrap coefficient
λbb (right) as functions of collisionality parameter Lc /lc for a tokamak. The computation
methods are indicated in the legend.
4
8
10
NEO2
MC
6
3
10
5
λbb
Dmono/Dplateau
NEO2
MC
7
2
10
4
3
2
1
10
1
0
0
10
−4
−3
10
10
−2
Lc/ lc
10
−1
10
−1
0
−4
−3
10
10
10
−2
Lc/ lc
10
−1
10
0
10
Figure 2: The same as in Fig. 1 for LHD configuration with R=375 at half toroidal flux.
1
10
NEO2
MC
NEO2
MC
1
0.6
0
10
λbb
Dmono/Dplateau
0.8
0.4
0.2
0
−1
10
−0.2
−4
10
−3
10
−2
10
Lc/ lc
−1
10
0
10
−4
10
−3
10
−2
10
Lc/ lc
−1
10
0
10
Figure 3: The same as in Fig. 1 for the standard configuration of W7-X at half toroidal
flux.