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).

Modelling Ripple Transport in Two DimensionsZ. Rao1, M. Hole1, K.G. McClement2, M. Fitzgerald1

1Res. School Phys. Sci. and Eng., Australian National University, Canberra ACT 02002EURATOM/CCFE Fusion Association, Culham Science Centre, UK.

I. Introduction

III. CUEBIT2

• 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.

• 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.

V. Chaotic Particle Orbits

VI. Chaos and Magnetic Moment

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.

Figure 1: the MAST tokamak (CCFE)

• 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.

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 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

•Ripple amplitude along x direction (as a function of z):

• The finite width of the trajectories comes from the finite Larmor radius. The guiding centre motions of these particles are following similar orbits.

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.

• 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 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.

• 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.

Figure 6: Initial conditions: as indicated in fig. 5. vII varies from 20000m/s to 42000m/s.

• In the given time, the particles may resume the regular guiding centre motion. However the Larmor radius is different.

Figure 7: Initial vII :upper left: 20000m/s, upper right: 30000m/s, lower left: 40000m/s, lower right: 42000m/s.

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|.

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.

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.

Figure 9: Initial conditions: as above. Left: normalised Larmor radius versus x position. Right: magnetic moment versus x position;

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).

Particle released at:

• 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:

• Onset of chaotic behaviour is observed in a 2D mock-up 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.

• Individual plot of four of the trajectories in Fig.6:

VII. Conclusion

• 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

A particle undergoing chaotic orbit experiences sharp changes in the normalised Larmor radius and magnetic moment as it reaches the field minimum.