Poincar´e plot

The technique to visualize the mapping is the Poincar´e plot. The principle of the Poincar´e plots is explained in the section 2.2. For our application each point on the graph marks the intersection of a field line with the chosen poloidal section. The sketch illustrating these method is presented in figure 2.4. The tokamak torus is presented there with four poloidal sections, the distance between them being 90◦_{. Because of the toroidal symmetry of the 12/4 mode of the perturbation field,}

n = 4, one can use all those sections to mark the intersections of the field lines. This procedure reduces the computational time by factor of 4. The field line is traced either to maximum number of steps (e.g. 1000 steps) or until it intersects the divertor target plates at rdiv = 47.7 cm. This latter method ”simulates” the

neutralization of the particles hitting the wall. An example of the Poincar´e plot for the TEXTOR boundary with the DED in 12/4 mode is shown in figure 3.10. The parameters for the calculations are presented in table 3.2. All other calculations discussed in this subsection are performed with the same set of parameters. In figure 3.10a one sees the Poincar´e plot in polar representation. The angle variable is the poloidal angle of the plasma; the radius corresponds to the minor radius. The ergodic and laminar regions cover only small fraction of the plasma minor radius (typically 5-7 cm), therefore the unfolded representation is better for the analysis of the topology of the magnetic field in the plasma edge. The transition is made by cutting the polar graph at θ = 0◦ _{and subsequently unfolding. The result of the}

transformation is shown in figure 3.10b. The abscissa represents the geometrical poloidal angleθ from 0◦ _{to 360}◦ _{and the ordinate – the minor radius of the plasma,}

48 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE a) b) 7/4 25/12 17/8 9/4 19/8 8/4 Y q=2 10/4 Y q=2.5 1 /42 Y q=3 [m/n] fingers structure DED coils edge cor e

poloidal angle -

q

minorr

adius-

[cm]

r

Figure 3.10: A Poincar´e plot for the magnetic field lines in the plasma edge calculated with Atlas: a) in polar representation; b) in unfolded representation.

3.4. DESCRIPTION OF THE VISUALIZATION METHODS 49

plasma major radius of LCSF R0 = 1.74 m

plasma minor radius a = 46.7 cm

plasma current Iplasma = 450 kA

toroidal field at R0 Bϕ = 1.88 T

poloidal beta βpol = 0.0

DED current distribution 12/4

DED current per coil IDED = 15 kA

Table 3.2: Input parameters for Atlas used for creation of the Poincar´e plot presented in figure 3.10.

in the edge region expressed in centimeters. The poloidal extension of the DED coils is marked as a rectangular area in the top of the figure, which spans the angular range from θmin = (180−36)◦, toθmax = (180 + 36)◦. The minor radius of the

coils is rcoils = 53.25 cm. In this figure all characteristic features of the ergodized

edge of the TEXTOR-DED can be seen. The visible island chains are marked on the right-hand side ordinate with the corresponding safety factor value, e.g. the m/n= 10/4 mode contains 10 islands.

As discussed earlier, each island chain is generated by different modes. The number of the modes is related to the q-values of the resonant surfaces. Between the main island chains (q = 8/4) and (q = 9/4), the (q = 25/12) at r = 39.8 cm and (q = 17/8) at r = 40 cm islands exist. They are generated by higher order perturbation fields created by the modes with symmetry n = 8 and n = 12 respectively; however, their influence on global or local plasma parameters is insignificantly small. The island chains created at the q = 12/4 and q = 11/4 surface are completely destroyed, because they are very close to the DED coils. In the presented structure figure one can distinguish three different regions:

1. Good confinement zone –r .40 cm:

According to the KAM theorem (see section 2.2.3) the non-resonant surface, remains undestroyed for sufficiently small perturbation, and only slightly per- turbed. The surfaces remain closed as long as the island chains surrounding

50 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE

them do not overlap. The trajectories of the field lines are distorted in the vicinity of the DED coils, i.e. on the HFS, as it was presented in section3.3.1. The field lines remain always on the same flux surface, and after infinite num- ber of poloidal turns ”fill” the flux surface.

Between the non-resonant surfaces island chains exist, created on the q = 2 and q = 7/4 flux surfaces. It is assumed that the island chains for q 6 2 can be tolerated as long as they do not overlap. If they overlap, they might cause disruptions as was found in Tore Supra. The boundary of the good confine- ment zone is between the island chains at q= 25/12 andq = 17/8 at a minor radius of r≈40 cm.

2. Ergodic region – 40 cm .r .44 cm:

The typical picture of the ergodic zone in the TEXTOR-DED experiment is the volume with mixture of the areas with completely destroyed flux surfaces and some remnants of islands. The trajectories of the field lines in the ergodic region are irregular, i.e. it is practically impossible to predict the trajectory over long distances (long as compared to the Kolmogorov length). Initially neighboring field lines will deviate from each other substantially and in an unpredictable way. This is in contrast to field lines in the good confinement zone and in islands. There, the field line trajectories remain inside the island chain and are separated from the stochastic areas. They still keep a regular character. Because a limiting wall (divertor target plates) is placed inside the ergodic zone, the character of the field lines is changed to an ”open chaotic system”. Even though the field lines may remain for very long path inside the ergodic zone, they will finally hit the wall. It was found that the field lines leave the ergodic zone in a well ordered way, namely along the so called ”fingers”. They are rather thin (∆θ <5◦_{) and surround the laminar zone.}

3. Laminar zone –r &43.5 cm:

Close to the divertor coils, the effects of the near field from the divertor coils is so significant, that the field lines are very strongly deflected towards the wall; thus these field lines have short connection lengths. The laminar zone

3.4. DESCRIPTION OF THE VISUALIZATION METHODS 51 is visible in the Poincar´e plot as the white regions between and outside the fingers. The Poincar´e points of the previous (q = 11/4) surface and outwards have practically vanished in the laminar zone. In order to visualize a laminar zone the new imaging technique was developed, called a laminar plot [34]. The principles of the laminar plot are explained in the following section.