Introduction
Particle Simulation
Discussion
Full orbit simulations of impurity transport in
spherical tokamaks with strongly-sheared electric
fields
C. G. Wrench1, E. Verwichte1 , K. G. McClements2
1
Centre for Fusion, Space and Astrophysics
University of Warwick
Coventry, CV4 7AL, UK
c.g.wrench@warwick.ac.uk
2
EURATOM/CCFE Fusion Association
Culham Science Centre
Abingdon, Oxfordshire, OX14 3DB, UK
TTG Conference, Córdoba
7th September 2010
O2.09 TTG 2010
1/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Introduction
• Advanced tokamak operating scenarios
use strongly sheared electric fields to
suppress turbulence (Chen 2001)
• In spherical tokamaks (STs), such as
MAST, radial electric fields associated
with edge transport barriers can vary
on length scales of the order of the ion
Larmor radius (Meyer, 2009; Shaing,
1998)
• Standard neoclassical theory not valid
• We use full-orbit test-particle approach
to study collisional impurity transport
in vicinity of transport barriers in STs
O2.09 TTG 2010
2/17
Radial electric field as
measured in the edge of
MAST (Meyer, 2009)
C. G. Wrench
Introduction
Particle Simulation
Discussion
Radial E-field: effect on collisionless orbits
• Impact of a sheared radial
1
electric field on collisionless
orbits is to:
fraction
• Reduce orbit width of
trapped ions
• See: Shaing (1998) and Tau
(1993).
• Orbit squeezing shown by
Shaing et al. (1998) to affect
neoclassical heat conductivity
• Collisional particle transport
also affected
O2.09 TTG 2010
3/17
0.6
Vertical Position [m]
• Increase trapped particle
0.8
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0.2 0.4 0.6 0.8 1 1.2
Major Radius [m]
Demonstration of orbit squeezing
for an Ar12+ ion.
C. G. Wrench
Introduction
Particle Simulation
Discussion
Test Particle Simulation
• CUEBIT: Full orbit test particle simulation code
• Solves the Lorentz-Langevin equation:
mz
dv
mz
= Ze (E + v × B) −
(v − uf ) + mz a(t)
|
{z
} |τ
dt
{z
}
Lorentz
Langevin
(1)
using an implicit finite difference method
• Solves test particle (Z 2 nz ni ) trajectories for a prescribed
MAST-like equilibrium
• All fields and bulk ion profiles prescribed and assumed static
• Electrostatic potential prescribed and of the form:
Φ = Φ0 arctan
Ψ − Ψ1
∆Ψ
(2)
• We assume that the dominant terms in the bulk ion force
balance at the transport barrier are ∇p and ZeEr
O2.09 TTG 2010
4/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Simulation Results: α-particles
4
(a)
1
x 10
(b)
2
1.5
1
(d)
1
0.5
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0
4
2.5
−0.5
−1
0.2 0.4 0.6 0.8 1 1.2
Major radius [m]
Particle density [m−3]
Height [m]
0.5
Particle density [m−3]
2.5
x 10
(c)
2
−0.5
1.5
−1
1
0.2 0.4 0.6 0.8 1 1.2
Major radius [m]
0.5
0
0
0.1
0.2
0.3
0.4
Minor radius [m]
0.5
0.6
Simulation of 104 α-particles without, (a) and (b), and with, (c) and (d), a
sheared radial electric field.
O2.09 TTG 2010
5/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Simulation Results: Ne10+ particles
4
(a)
1
x 10
(b)
4
1
(d)
2
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0
4
6
−0.5
−1
0.2 0.4 0.6 0.8 1 1.2
Major radius [m]
Particle density [m−3]
Height [m]
0.5
Particle density [m−3]
6
x 10
(c)
−0.5
4
−1
2
0
0
0.2 0.4 0.6 0.8 1 1.2
Major radius [m]
0.1
0.2
0.3
0.4
Minor radius [m]
0.5
0.6
Simulation of 104 Ne10+ -particles without, (a) and (b), and with, (c) and (d),
a sheared radial electric field.
O2.09 TTG 2010
6/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Simulation Results: W20+ particles
4
(a)
1
x 10
(b)
4
3
1
(d)
2
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0
4
5
−0.5
−1
0.2 0.4 0.6 0.8 1 1.2
Major radius [m]
Particle density [m−3]
Height [m]
0.5
Particle density [m−3]
5
x 10
(c)
4
−0.5
3
−1
2
0.2 0.4 0.6 0.8 1 1.2
Major radius [m]
1
0
0
0.1
0.2
0.3
0.4
Minor radius [m]
0.5
0.6
Simulation of 104 W20+ -particles without, (a) and (b), and with, (c) and (d), a
sheared radial electric field.
O2.09 TTG 2010
7/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Simulation Results
10+
2+
Vertical position [m]
20+
Ne
He
W
1
1
1
0.5
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
0.5
1
1.5
Major radius [m]
0.5
1
1.5
Major radius [m]
0.5
1
1.5
Major radius [m]
Poloidal distribution of 104 impurity ions after 100 ms. Red line indicates
location of peak radial electric field.
O2.09 TTG 2010
8/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Scaling with particle parameters
Consider collisional equation of motion (Langevin equation):
mz
dv
= Ze (E + v × B) −
(v − uf ) + mz a(t)
mz
(3)
dt
τ
To make analytics tractable, we assume Cartesian geometry and
E = Ex = const. Solve
Ze
dvx
+ νvx =
Ex + Ωvy
(4)
dt
mz
dvy
+ νvy = −Ωvx
(5)
dt
to find:
Ex ν
1
(6)
vx = v⊥ e −νt sin Ωt +
B Ω 1 + ν 2 /Ω2
1
Ex
(7)
vy = v⊥ e −νt cos Ωt −
B 1 + ν 2 /Ω2
Note: x-direction is a proxy for the radial direction
O2.09 TTG 2010
9/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Scaling with particle parameters: Drag force drift
Drag force drift in the same direction as the electric field:
vx = v⊥ e −νt sin Ωt +
1
Ex ν
B Ω 1 + ν 2 /Ω2
(8)
This implies that there is:
• an inward-pinch in the presence of a negative radial electric
field (Ex < 0) and
• an outward-pinch in the presence of a positive electric field.
In tokamaks we have that ν 2 Ω2 , therefore, drag force drift is
1/2
vdrag
O2.09 TTG 2010
10/17
Ex m e 3 ni ln Λ
'Z √ i
3/2
6 2π 3/2 20 Ti B 2
(9)
C. G. Wrench
Introduction
Particle Simulation
Discussion
Vertical position [m]
Scaling with particle mass
1
1
1
1
0.5
0.5
0.5
0.5
0
0
0
0
−0.5
−0.5
−0.5
−0.5
−1
−1
0.5
1
Major radius [m]
1.5
−1
0.5
1
Major radius [m]
1.5
−1
0.5
1
Major radius [m]
1.5
0.5
1
Major radius [m]
1.5
4
4
x 10
Ar10+
Ne10+
Mo10+
W10+
3.5
Particle density [m−3]
3
Above: Poloidal distributions for (left
to right) Ne10+ , Ar10+ , Mo10+ and
W10+ .
Left: Density profiles in minor radius
for impurities with varying masses but
the same charge state (same species as
above).
2.5
2
1.5
1
0.5
0
0
O2.09 TTG 2010
0.1
0.2
0.3
0.4
0.5
0.6
Averaged minor radial coordinate [m]
11/17
0.7
C. G. Wrench
Introduction
Particle Simulation
Discussion
Characterising transport: Diffusion coefficients
Diffusivities are estimated from simulated particle flux and density
gradient:
Γz (10)
Deff = −
dnz /dx E =Emax
Assuming particle flux is composed of usual diffusion and pinch
terms, we may relate drag force drift to transport:
Γz = −D
dnz
− vnz
dx
(11)
D
+1
(12)
implies
Deff =
nz v
Γz
where we take υ to be drag force drift velocity at peak of electric
field. Therefore:
Deff
1
=
(13)
D
1 + αZ
O2.09 TTG 2010
12/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Scaling of effective diffusivities with particle mass and
charge
2.5
1
CUEBIT data
Linear fit
2
0.8
0.7
0.6
D0/Deff − 1
Normalised effective diffusion coefficient
0.9
0.5
1.5
1
0.4
0.3
0.5
0.2
0.1
0
0
20
40
60
80
100
120
Particle mass number
140
160
180
200
0
1
2
3
4
5
6
7
Particle charge number
8
9
10
11
Normalised effective diffusion
Normalised effective diffusion
coefficient for various charge states of
coefficient for various impurities
neon.
species, all with Z=10+ .
Diffusion coefficients are normalised to measured coefficient in the absence of
an electric field
O2.09 TTG 2010
13/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
Scaling with electric field parameters
Variation of effective diffusion
coefficient with change in electric field
width and strength for C6+ ions.
Scaling is of the form:
b1 (E −E0 )
∆r
Deff = D0
∆r0
0.121E
∆r
m2 s−1
= 9.436
0.159ρL
0.08
Electric field width [m]
0.07
0.06
0.6
0.05
0.4
0.2
0.04
0.8
0.03
0.02
0.3
0.01
0
0
0.7
0.9
−1
O2.09 TTG 2010
0.5
−2
−3
−4
−5
−6
Peak electric field strength [kVm−1]
14/17
−7
Here ∆r is the electric barrier width in
meters and electric field strength, E , is
measured in kVm−1
C. G. Wrench
Introduction
Particle Simulation
Discussion
Comparison of Carbon Profiles with Experiment
18
6
x 10
Scaled CUEBIT C6+ density
Particle density [m−3]
5
4
3
2
1
0
0
Carbon density profile in MAST as
measured by the Charge Exchange
Diagnostic. Image courtesy J. McCone.
O2.09 TTG 2010
15/17
0.1
0.2
0.3
0.4
Minor radius [m]
0.5
0.6
Scaled C6+ density profile as computed
by CUEBIT.
C. G. Wrench
Introduction
Particle Simulation
Discussion
Concluding remarks
• A strongly sheared electric field can strongly modify collisional
transport
• We have used full-orbit test-particle approach to study
collisional impurity transport in spherical tokamaks with
strongly-sheared electric fields
• Collisional drag in presence of radial E < 0 produces local
inward pinch, which increases with Z but is independent of
impurity mass
• Negative implication of this is the retention of highly ionised
species, e.g. tungsten
• Effective diffusion scales with the electric field width with a
power proportional to the electric field strength
O2.09 TTG 2010
16/17
C. G. Wrench
Introduction
Particle Simulation
Discussion
References
• Chen, H., Hawkes, N. C., Ingesson, L. C., Peacock, N. J. and
Haines, M. G., Nuclear Fusion 41 pp. 31, 2001.
• Hinton, F. L. and Kim, Y.-B., Physics of Plasmas 2, pp. 159,
1995.
• Meyer et al., Nuclear Fusion 49 104017, 2009.
• Shaing, K. C. and Hazeltine, R. D., Physics of Fluids 4, pp.
2547, 1992.
• Shaing, K. C., Aydemir, A. Y. and Hazeltine, R. D., Physics
of Plasmas 5, pp. 3680, 1998.
• Tau, Y. -Q., et al., Phys. Fluids B 5, pp. 344, 1993.
O2.09 TTG 2010
17/17
C. G. Wrench