3.18, lie within the extent of two different pair of stripes, which are separated by another one. It means that the Lc = 2 flux tube interconnects different modes of
the perturbed field.
The individual flux tubes are quite similar to a normal scrape-off layer of the poloidal divertor. However, at the boundaries large gradients in the velocity may occur, as the flux tubes can carry the particles with different thermal velocities. The structure of the laminar zone is dominated by the single-turn flux tubes, but also other, in particular Lc = 2 are present. The one-turn flux tubes connect either
two parts of the same footprints stripe or two parts of different stripes. The field lines with longer then one connection lengths have more complex topology. The field lines forming this flux penetrate deeper into the ergodic region, therefore they should bring hotter particles to the wall, thus the wall temperature should be higher in the area, where it is hit by those particles.
3.7
The variation of the ergodization with plasma
current and poloidal beta
The perturbation spectrum plays a key role in determining the topology of the magnetic field. As shown in section 3.3.1 the main quantities, which vary the per- turbation spectrum (assuming a constant plasma position) are the plasma current, which changes the q-profile, and the poloidal beta, which changes the position of the resonances and the pitch angle of the field lines. The Atlas code is well suited to study the effects of Ip and βpol variation on the structure of the ergodic and the
laminar zone. In figures 3.20 to 3.23 several graphs show the Poincar´e plots (fig.
3.20), laminar plots at the low-field side (fig. 3.21) and the high-field side fig. 3.21) for varying Ip and βpol. In figure 3.23 the series of footprints plots is shown. In all
the figures plasma current is varied from 350 kA to 500 kA, and poloidal beta from 0.0. to 1.0. The other input parameters are the same as in table 3.2.
64 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE
3.7.1
Variation of the poloidal beta
The first case considered is a change in topology of the ergodic and the laminar zone with changing poloidal beta. The strong influence of βpol results from the
increasing separation of the flux surfaces with increasing plasma pressure (differential Shafranov shift). As already discussed in section3.3.2this leads to a decrease of the ergodization. In the figures3.20–3.23this is presented by the variation from column to column. One can see that a lower current (e.g. Ip = 350 kA, top row) the laminar
zone almost does not exist for the highβpol. The level of ergodization is so low, that
all the island chains still remain, some of them overlapping. The finger structure is not very well developed and in the picture of magnetic footprints the very short field lines are dominating (LC < 0.5) (top right graph in figure 3.23). With decreasing
βpol the level of ergodization is increasing (this effect is discussed in section 3.3.1).
For βpol = 0.5 the ergodic region is thickest, while for βpol = 0.0 the laminar zone
plays the key role. In figure3.19the Chirikov parameter is plotted versus the plasma minor radius. Each point in the plot is positioned in between two interacting island chains, the number by the points denote corresponding poloidal mode numbers. Due to the differential Shafranov shift points calculated for βpol = 1.0 lie closer to the
plasma center (black curve) than those calculated for βpol = 0.0 (blue curve). The
value of the Chirikov parameter calculated for the overlapping island chains with q = 14/4 and q = 15/4 is about 3 times higher for βpol = 0.0 than for βpol = 1.0.
For the case of βpol = 0.0 the Chirikov parameter exceeds the critical value σCh = 1
at r >42cm. In the case of βpol = 1.0 it never exceeds this value. If we look at the
corresponding laminar plots (i.e. top left and top right graph in figures 3.22 and
3.22), we can see that a well developed laminar zone is connected to a high value of the Chirikov parameter at the plasma edge.
The pattern with four-stripes appears in all graphs in figure 3.23 and as ex- pected, the stripes are broader for a higher level of ergodization. With increasing the ergodization level they split into two stripes. As was shown in section 3.6 each stripe consists of two parts, which at a lower level of ergodization are connected, forming one structure and separate with increasing the ergodization level. According
3.7. THE VARIATION OF THE ERGODIZATION 65 32 34 36 38 40 42 44 46 48 0 0.5 1 1.5 2 2.5 3 radius [cm] Chirikov parameter σ Chir m=7,8 m=8,9 m=9,10 m=10,11 m=11,12 m=12,13 m=13,14 m=14,15 m=15,16 m=7,8 m=8,9 m=9,10 m=10,11 m=11,12 m=12,13 m=13,14 m=14,15 m=15,16 βpol = 0.0 βpol = 1.0
Figure 3.19: The Chirikov parameter versus the plasma minor radius calculated with the Atlas code for two different values of beta poloidal. The plasma current is 350 kA, the rest of the input parameters are the same as in table 3.2. The numbers denote poloidal mode numbers of the overlapping island chains.
to model calculations [13, 26] the particle and heat flux density deposition patterns follow the structure of the laminar zone, forming patterns similar to the magnetic footprints. Thus the effect of broadening of the footprint stripes should be measur- able, which will be demonstrated in chapter 4.
In the case of Ip = 500 kA (last row in figure3.23) we can see that for all values
of poloidal beta the footprints stripes have split into pairs which are narrower and sharper at a high degree of ergodization. The gap between two divided parts belong- ing to one stripe shrinks with increasing value ofβpol. The size of the laminar zone
is almost the same for all values of poloidal beta, but the topology of the laminar zone becomes simpler for smallerβpol, i.e. the single- and double-turn flux tubes are
66 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE
3.7.2
Variation of the plasma current
The plasma current is the second major parameter; it influences the distance of the resonant flux surfaces from the DED coils, thus the equilibrium part of the Hamiltonian (see eq. 2.12). With increasing plasma current the flux surfaces are getting closer to the perturbation coils, thus the amplitudes of the perturbation modes are increasing as well. Plasma current has also indirect influence on the perturbation spectrum: change of the q-profile changes the pitch angle of the field lines. If we consider columns in figure 3.20, we can notice that at the lower plasma currents the ergodic region is dominating, at higher values – the laminar zone. In figure 3.7 the maximum of the perturbing spectrum for βpol = 1.0 lies at 9/4 6
m/n610/4. For the given conditions (see table3.2) the flux surfaces corresponding to these numbers remain in the plasma volume for 350 kA < Ip < 500 kA. At the
low currents they are too far from the ergodizing coils, and therefore the ergodic region is created only at the flux surfaces with higher q-values, e.g. q > 14/4. If the plasma current grows, the q= 9/4 and q= 10/4 flux surfaces are getting closer to the perturbation field and the amplitudes of the Fourier components 9/4 and 10/4 are increasing; they reach their maximum at Ip = 500 kA. The field lines
are deflected so strongly to the DED coils, that they start to form the laminar zone. Thus, by changing the plasma current one can define – to some degree – the ratio between the ergodic and the laminar region. For other values of beta poloidal the strongest perturbation modes are characterized by higher values of m (i.e. m = 11/4 for βpol = 0.5 and m = 12/4 for βpol = 0.0). Thus the maximum
of the ergodization is reached at different values of the plasma current. By the maximum of the ergodization we understand the situation, where the size of the laminar zone is maximized.