Transcript Modelling ripple transport in two dimensions

Modelling Ripple Transport in Two
Dimensions
Z. Rao1, M. Hole1, K.G. McClement2, M. Fitzgerald1
1Res.
School Phys. Sci. and Eng., Australian National University, Canberra ACT 0200
2EURATOM/CCFE Fusion Association, Culham Science Centre, UK.
I. Introduction
• Finite number of external coils in a tokamak introduces
a ripple in the toroidal magnetic field.
• The ripple strength increases as the major radius
increases. Particle transport in the radial direction can
happen, which leads to particle loss. This transport
increases with the increase of the ripple strength.
Figure 1: the MAST
tokamak (CCFE)
• Motivation: to understand the effects of toroidal field
ripple on particle confinement in a 2D model.
• The code CUEBIT(CUlham Energy-conserving orBIT)1
is used to solve the full orbit trajectories for single
particles.
• Individual plot of four of the trajectories in Fig.6:
Particle
released
at:
Figure 7: Initial vII :
upper left: 20000m/s,
upper right: 30000m/s,
lower left: 40000m/s,
lower right: 42000m/s.
Figure 3: Initial conditions: x=0.4m, y=0m, z=0.37m.
v⊥=60000m/s. vII varies from 30000m/s, 35000m/s, 40000m/s to
45000m/s (inner to outer).
• A spread of the region sampled by the phase space
can be seen at the maxima and minima of vII (when
the particle approaches the field minimum (kx =
(2n+1)π)).
• The trajectories of the above particles in (x,z) plane:
• In the given time, the particles may resume the regular
guiding centre motion. However the Larmor radius is
different.
VI. Chaos and Magnetic Moment
Selecting two particles with the same initial v⊥, vII and
the same initial y and z, but one with x=0.4m and the
other x=0.45. The z excursion of the latter one is higher.
II. Ripple Field in 2D
• 2D model: the guiding toroidal field is in the x direction
and the poloidal field is in the y direction. The
unperturbed equilibrium field is
with <<1.
• A ripple-type perturbation is put in the x(toroidal)
direction. Expressions for such a field satisfying a
current-free equilibrium (
) are:
where B1<<B0.
• A vector potential whose curl yields this field is:
Figure 4: Initial conditions: as indicated in fig. 3. Red: vII=30000m/s,
green: vII=35000m/s, black: vII=40000m/s, blue: vII=45000m/s.
• The finite width of the trajectories comes from the
finite Larmor radius. The guiding centre motions of
these particles are following similar orbits.
V. Chaotic Particle Orbits
• Chaotic behaviour of the particle is observed as it
approaches the field minimum, if initial x position is
moved to x=0.45m rather than x=0.4m.
Figure 2: Contour plot of Ay
on (x,z) plane, indicating
the magnetic field direction
and the field strength, with
k=10,B0=1T, B1=0.01T
Figure 5: Phase plot in (x, vII) space. Initial conditions: x=0.45m,
y=0m, z=0.37m. v⊥=60000m/s. vII varies from 20000m/s to
42000m/s (inner to outer). Chaotic change in vII can be seen in both
trapped or passing particles.
•Ripple amplitude along x direction (as a function of z):
III.
• Fig. 6 shows the trajectories of the above particles in
(x,z) plane on the same plot. Up to the onset of
chaotic motion, the guiding centre motions of these
particles follow similar orbits.
2
CUEBIT
• CUEBIT solves the Lorentz force equation
by iteration on the following set of equations
where E=0 since the field is a current-free equilibrium.
IV. Particle Orbits
• Particle trapping along the x direction can happen
because of the presence of the ripple.
• Cases of trapped and passing particles in Fig.3:
phase plots in the (x,vII) space of deuterium particles
released at the same position (red line), ripple
amplitude about 40%, with the same initial v⊥ (same
magnetic moment) and different initial vII (different
pitch angle and energy).
Figure 6: Initial conditions: as indicated in fig. 5. vII varies
from 20000m/s to 42000m/s.
• At the field minimum,
the z excursion of the
particle is at maximum,
where the ripple
amplitude is the largest.
Here, as the particle
approaches the field
minimum, δB→B0.
• This suggests that
the onset of chaotic
behaviour may occur
at region with large
δB/B0
Figure 8: Initial conditions: red: x=0.4m, blue: x=0.45m; y=0m,
z=0.37m. v⊥=60000m/s, vII=30000m/s. Hence the blue one is
released at an initially higher |B| position.
Hall3 showed that when a particle passes through the
field minimum, there is a nonadiabatic change in
magnetic moment μ. The change in μ is related to the
local Larmor radius of the particle, and the local
magnetic field scale length defined by |B|/grad|B|.
Figure 9: Initial conditions: as above. Left: normalised Larmor
radius versus x position. Right: magnetic moment versus x
position;
A particle undergoing chaotic orbit experiences sharp
changes in the normalised Larmor radius and magnetic
moment as it reaches the field minimum.
VII. Conclusion
• Onset of chaotic behaviour is observed in a 2D mockup of field ripple in a tokamak.
• This is related to earlier work in astrophysics work.
The breaking of the adiabatic invariant of magnetic
moment at field minimum is confirmed.
• Future work could be determining the quantitative
change in μ within the particle parameter space.
Particle loss due to chaotic behaviour in tokamaks
could be estimated.
VIII. References
[1] McClements, K.G.(2005), Phys. Plasmas 12 072510.
[2] Hamilton, B., McClements, K.G., Fletcher, L. and Thyagaraja, A. (2003), Solar
Phys. 214: 339-352.
[3] Hall, A.N., (1980), Astron. Astrophys. 84 40-43.