ACERO ESTRUCTURAL FY=4200 KG/CM2.

Let us assume a sheared magnetic field in the plasma edge with the shear value

of s = −r2(_qdq/dr2_R0 ). The shear guaranties the existence of more than one resonant

surface (i.e. with rational q-value). For a given set of and a magnetic equilibrium the radial magnetic perturbation field consists of a poloidal and toroidal spectrum (typically one toroidal and few poloidal modes), which is equivalent to the spatial Fourier transform of e.g. the radial magnetic field component. To assure proper performance of the ergodic divertor the applied perturbation field has to fulfill several requirements:

• the toroidal n an poloidal m spectrum of the perturbation field has to be in resonance with the flux surfaces at the plasma boundary (typicallym/n&2); • it must not perturb theq = 2 surface, because the system flips from stabilized MHD (2,1) mode to destabilized (2,1) mode [36,20]. It can lead to disruption, therefore it should be avoided;

• the radial extent of the stochastic domain has to be wide enough in order to decouple the divertor region, which determines the plasma wall interactions, from the core (i.e. the width should be of order of ionization scale).

In order to describe, the effect of the external perturbation field on the magnetic field topology let us consider growing perturbation field from zero:

1. If the spectrum of the perturbation is sufficiently broad and the amplitudes of the modes are high enough, initially few flux surfaces are destroyed by the resonant radial perturbation. According to the Poincar´e – Birkhoff theorem (see Sec2.2.3 ) magnetic field lines on the flux surfaces with q-value resonant to the perturbation spectrum (e.g. q = mi

n ,where i = 1, . . . , k) create island

chains. The field lines belonging to the islands always stay in the volume defined by the island chain boundaries. The flux surfaces, which have safety factors non resonant to the perturbing magnetic field remain closed.

3.2. THE BASICS OF THE ERGODIZATION 35 2. If the amplitude of the spectrum grows then the islands width grows. At some point the last closed flux surface between two neighboring island chains is destroyed and islands start to overlap. The volume, where this happens is called “stochastic” or “ergodic”.

The natural measure of the ergodicity is the Chirikov parameter, defined as: σChir =

∆m+1,n+ ∆m,n

|rm+1,n−rm,n|

, (3.1)

where ∆ is the half width of islands andr is a minor radius of the island chain. The width of the islands is a function of the resonant component in the perturbation spectrum: ∆m,n = 4 hmn(ψ) dq−1_/dΨ 1/2 , (3.2)

where ψ represents the resonant flux surface and hmn is an amplitude of the

Fourier component in the perturbation spectrum defined in equation (2.16). A transition to stochasticity occurs, whenσChir &1, i.e. the islands from neighboring

chains start to overlap. The flux surfaces between the overlapping islands are de- stroyed and the ergodic layer is created.

According to the ergodic hypothesis 1 _{field lines will “fill” all the available vol-}

ume. The charged particles follow the field lines, hence transport will be different in the stochastic domain. One should expect an enhancement of the effective radial heat and particle diffusion coefficients, which – in a quasi-linear approximation – are proportional to the product DF Lvth, where DF L is the field line diffusion coefficient

and vth – the thermal velocity. In the highly ergodized case (withσChir(m, n)>1)

DF L can be defined as:

DF L = X resonant(m,n) πqR0 hm,n Bϕ0 2 , (3.3)

where Bϕ0 is the toroidal magnetic field at the axis [42].

1_{If the dynamical system is ergodic than a trajectory of any point in a phase space after a}

36 CHAPTER 3. FIELD LINES IN THE ERGODIZED EDGE

At the outermost boundary, where the near field effects are significant, magnetic field lines have short wall-to-wall connection lengths –Lc. In chaos theory a charac-

teristic length for the separation of the neighboring orbits is the Kolmogorov length [36]: LK =πqR0 πσ_Chir 2 (3.4) In general it is proven, that stochasticity prevails if LC(r, θ, ϕ) LK(r). If the

LC(r, θ, ϕ).LK(r) the connection to wall is a dominant feature.

unperturbed plasma core

ergodic region

laminar region

perturbation coils

divertor target plates

tokamak vessel

reference sections

Figure 3.4: The sketch of the different regions created by the superposition of the tokamak equilibrium and the DED perturbation field presented in the poloidal cross- section

In the latter case the transport of energy and particles is not of diffusive character like in the ergodic zone; it is of convective/conductive character. In contrast to the proper ergodic zone, the connection lengths of the magnetic field lines have smooth and continuous properties, however, with some sharp boundaries. This region has been termed as a laminar volume [16]. The structure of the different magnetic regions created by the DED is presented in figure 3.4.

The goal of the DED in the static case is to redistribute the heat fluxes to larger areas. The particle and heat distribution pattern is strongly dependent on

3.3. THE PROGRAM “ATLAS” FOR THE TEXTOR-DED 37