Requerimientos de Construcción 1 Calidad de los tubos y del material

In order to discuss the properties of the perturbation spectrum created by the cur- rents flowing in the DED coils we will use the simplest cylindrical model to explain the principles. Such a formulation allows us to discuss the properties of the spectrum without loosing generality, i.e. the conclusions are also valid for the perturbation spectrum derived (more accurate) for the toroidal case by [2]. In Atlas the latter one is used.

The perturbation, which is mainly determined by the axial component of the vector potential and is presented as a Fourier series

Az =

X

m

3.3. THE PROGRAM “ATLAS” FOR THE TEXTOR-DED 39

Calculation of the perturbation spectrum ( , ),

-profile, conversion coefficients ( , , A h q r r mn mn m m Jm)

Spectrum

Poincaré plot

Field line tracing, topological properties of flux tubes

Laminar plot

Trajectories

Figure 3.5: The scheme presenting the organization of the Atlas codes. The Fourier components of the perturbation spectrum (eq. 2.12), transformation coefficients (eq.

2.2) and values of safety factor are calculated and stored. These files are used in several applications, e.g to obtain a Poincar´e plot.

where θ is the poloidal angle. The amplitudes of the Fourier components are: Am =

Z π

−π

A∗_m(r)·sinmθsinm0θ·f(θ)dθ, (3.7)

where m0 = 12 is the central mode number and A∗m(r) is the radially dependent

part of the poloidal mode. The function f gives the poloidal localization of the perturbation field and it is defined as follows:

f =    1 ifθc 6θ6θc 0 ifθ < −θc and θ > θc (3.8)

40 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE

where −θc 6 θ 6 θc is the poloidal extension of the perturbation currents on the

surface of the cylinder. The solution of the integral 3.7 is

Am =A∗m(r)

sin(m−m0)θc

(m−m0)θc

. (3.9)

The poloidal spectrum of the perturbation Am is localized near the central mode

m0 = 12 and it consists of several modes: m∈ h10, . . . ,14i. In linear approximation

the interaction of the perturbation field with the resonance surfaces can be treated as the interaction of single modes with the corresponding flux surfaces; thus we can analyze the Fourier components of Az separately. To find the power of the radial

dependence one needs to solve the equation:

∇ ×(∇ ×Az) = 0, r 6=rcoil (3.10)

for the amplitude Am of the single poloidal mode. The radial and azimuthal com-

ponents of the vector ∇ ×Az are:

∇ ×Am|r = imAmeimθ (3.11)

∇ ×Am|θ = −

∂Am

∂r (3.12)

After including these components in equation3.10one obtains the linear differential equation for the radial part of the Fourier coefficients:

d2_A∗ m dr2 + 1 r dA∗ m dr − m2 r2 A ∗ m = 0. (3.13)

The second order differential equation has the solution: Az =A0rm+B0r−m. In or-

der to avoid singularity, A0 = 0 for r > rcoil and B0 = 0 for r < rcoil. The constants

A0 and B0 are derived from the boundary conditions such that δBr is continuous

at r = rcoil and δBϕ fulfils the jumping condition resulting from ∇ ×B~ =~j. The

radial decay of the poloidal modes is very rapid and what is important to notice the modes with higher m-number decay faster with the distance from the DED coils. The perturbation field is localized within the range of few centimeters on the high- field side of the tokamak.

3.3. THE PROGRAM “ATLAS” FOR THE TEXTOR-DED 41 a) 440 460 480 500 520 540 560 0.7 0.75 0.8 0.85 0.9 0.95 1

plasma current [kA]

r/r c m = 12 m = 10 b) 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 0.7 0.75 0.8 0.85 0.9 0.95 1

toroidal magnetic field [T]

r/r

c

m = 10 m = 12

Figure 3.6: The relative position of two resonant surfaces (m indicates poloidal mode number, n in both cases equals 4)as a function of: a) the plasma current with fixed Bϕ = 2.25 T and b) the toroidal magnetic field with fixedIp = 440 kA

42 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE

of the corresponding Fourier component of the perturbation field. Because of the very strong decay of the poloidal spectrum with the distance from the perturbation coils, the width of the islands centered at a given resonant flux surface strongly depends on the distance between the flux surface and the perturbation coils. This distance is given by the q-profile and the differential Shafranov shift. As it is dis- cussed in chapter1the latter quantity shifts the centers of the flux surfaces towards the low field side, thus increasing the distance of the resonant surfaces from the origin of the perturbation field. The q-profile (see eq. 1.3) depends on the toroidal component of the magnetic field and the plasma current. For fixed toroidal magnetic field (Bϕ) the radius of the resonance surface increases with the growing plasma cur-

rent (Ip); this dependence is presented in figure 3.6a. The relative positions of the

m = 12,10 surfaces changes almost linear with the plasma current. Figure 3.6b shows the dependence of the radial position of the rational surface as the function of the toroidal magnetic field (Bϕ) for fixedIp. As it can be expected from equation

1.3 the radius of the flux surface decreases with increasing Bϕ. Therefore, adjust-

ing the q-profile, one can locate the resonant surface closer to the DED coils and thus increase the island size. As we can see in figure 3.6 for the standard toroidal magnetic field in TEXTOR Bϕ(r= 0) = 2.2 T theq = 12/4 surface is close enough

to the DED coils to be ergodized for relatively high plasma currents Ip > 500 kA;

however typically a lower plasma current in the TEXTOR tokamak is used . Hence, for the TEXTOR-DED operation, the Bϕ on the tokamak axis is lowered to 1.9 T

in order to achieve bigger DED effect.

The ergodic region is created when the island from neighboring island chains overlap. The width of each island is proportional to the different mode of the per- turbation. In the typical case of the q-profile (see figure 1.3) the resonances with lower m number lie closer to the plasma center, thus the width of the island with lower m is smaller than the width of the one characterized by m+1. The distance between the overlapping island chains is defined by the magnetic shear. With the smaller shear the distance of the resonances is larger. If one fixes an outer resonant surface by an appropriate plasma shift to a given radius, then the separation of the other, inner resonances move away from the DED coils. Because of the strong radial

3.3. THE PROGRAM “ATLAS” FOR THE TEXTOR-DED 43 decay of the DED field these resonances react by smaller islands and therefore the ergodization becomes weaker. In toroidal geometry, the effect explains the weaker ergodization for plasmas with an increased value of the plasma pressure, indicated by βpol.