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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.