Aceptación de los trabajos 1 Controles

The first order toroidal corrections as a result of more accurate derivation of the vector potential of the perturbation field gives the following equation [2]:

Aϕ(r, θ, ϕ) =

X

m

m−1rAmn(r, θ) cos(mθ−nϕ). (3.14)

For large aspect ratio tokamaks, i.e. with R0/a1, the Fourier coefficients can be

presented in the asymptotical form [24]: Amn(r, θ)≈ Bcrcgm p 1 +rcosθ/R0 r rc m , (3.15)

where Bc = 2µ0Icn/θcrc = 0.23 T is a characteristic strength of the magnetic per-

turbation determined by the divertor current Ic = 15 kA, the minor radius of the

coils rc = 53.25 cm and the angular poloidal extension of the coils θc ≈2π/5. The

factor p1 +rcosθ/R0

−1

corresponds to the first order toroidal correction. In the toroidal approximation the poloidal spectrum is localized near the central mode m0 = 2πn/θc ≈20 instead of 12. The reason for the increased exponentm0 relative

to the cylinder result from the lower pitch angle of the equilibrium field at the high field side of the torus compared to the cylindrical case. If the perturbation coils would have been placed at the low field side, then the effective mode number would be smaller than the average value of 12. The total perturbation field shows a strong radial decay Aϕ ∝rm0−1.

The plasma pressure, described by βpol, influences the position of the resonance

surface via the Shafranov shift, which also changes the pitch angle of the field lines. The influence of the perturbation field on the field lines strongly depends on the angle between the field lines and the coils. As the pitch of the coils is fixed, the angle of the field lines is modified by changingβpol. The shift of the main resonance

44 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE

to lower m-numbers results again from the lowered pitch angle at the HFS for in- creased βpol-values.

The DED coils were laid out for a low value of βpol and for the high resonance

4 6 8 10 12 14 16 18 20 22 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 |H m |·1 0 -4 b_pol= 0.0 b_pol= 0.4 b_pol= 1.0 m-mode number

Figure 3.7: The absolute values of the perturbation spectrum for three different values of poloidal beta.

with the field lines at q= 3. From figure 3.7 one sees, that the calculations confirm the initial design. The real spectrum of the perturbation field is well approximated by the sin(m−m0)θc/(m−m0)θc- dependence obtained from the cylindrical model.

Atβpol = 0.0, the 12/4 mode is indeed close to the maximum of the curve. At higher

values ofβpol, the maximum shifts to lowerm-numbers and correspondingly to lower

values of q. Therefore, the resonances are deeper inside the plasma. Because of the strong radial decay of the perturbation field, this would result in a reduction of the island widths. However, one can counter-react by shifting the plasma such that the ”optimum” resonance remains at the same location.

A spectrum like given in figure 3.7 has to be calculated for all relevant reso- nant surfaces. From the value of Hm at that surface, the width of the individual

islands is derived. Figure 3.8 gives the island widths for conditions used in3.7. For higher beta poloidal the islands characterized by higher m-numbers are destroyed by overlapping with neighbors and do not exist anymore. The position of the island

3.3. THE PROGRAM “ATLAS” FOR THE TEXTOR-DED 45 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0 0.05 0.1 0.15 0.2 0.25 normalized flux

normalized island width

β_pol = 0.0 β_pol = 0.4 β_pol = 1.0 7 8 9 10 11 12 13 7 8 9 10 11 12 7 8 9 10 11

Figure 3.8: The width of the islands calculated for the same conditions as for figure

3.7. Numbers corresponds to the poloidal mode numbers.

chains is changing with βpol due to the Shafranov shift. As mentioned before, one

has many degrees of freedom to influence the island widths, e.g by plasma position or by Ip.

The superposition of the equilibrium and perturbation fields creates a three- dimensional topology of the magnetic field in the plasma edge. The field lines are deflected from their regular trajectories in the vicinity of the DED coils. In the figure 3.9 two field lines are traced for one poloidal turn. The calculations are done for the plasma current, Iplasma = 450kA and the poloidal beta βpol = 1.0. The ab-

scissa represents the poloidal angle and the ordinate – the minor radius. The lines visualize trajectories projected on the poloidal section, at toroidal angle ϕ = 0◦_.

The poloidal extension of the DED coils is marked as a yellow rectangle. The black dots represent the unperturbed trajectory of the field line (IDED = 0 kA) and the

red and blue, the field lines under the influence of the perturbation field (IDED = 15

kA). The change in the radial coordinates for the unperturbed field line is due to choice of the major radius of the plasma, which is shifted by one centimeter to the high field side (HFS) from the geometrical center of the torus. The same shift is

46 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE

Figure 3.9: Two field lines traced for one poloidal turn with Atlas codes for two cases: with and without the perturbation field. They are starting at the same radial and poloidal positions (r = 43.2 cm, θ = 0◦_{: (black dots) – no perturbation; (blue} dots) – Ic = 15 kA, ∆ϕ= 0◦; (red dots) – Ic = 15 kA, ∆ϕ= 5◦.

superimposed with the deflections coming from the DED field in the trajectories of the perturbed field lines. One sees that the trajectories of the perturbed field lines are deflected from their regular trajectories, only if their are in the vicinity of the DED coils, i.e. if the poloidal angle along the trajectory is in the range of the poloidal extension of the DED coils. If the field lines are outside of the poloidal extension of the DED coils, the field lines are unperturbed. In the figure 3.9 it is also shown, that the actual path of the field lines depends critically on the initial toroidal angle. Field lines with different starting points will undergo different orbits and reach different radial location. From this it follows that the structure of the ergodic and laminar zone becomes three dimensional. The deflection of the field line depicted as the red dots is completely different from the one depicted as the blue dots. The different starting toroidal angles in figure 3.9 are 0◦ _{and 5}◦_{. By}

3.4. DESCRIPTION OF THE VISUALIZATION METHODS 47