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Drifts in a tokamak plasma

If the effects of the micro turbulence are neglected, the theoretical description of plasma evolves in two directions. In the case of a slow plasma motion, the solution of the kinetic equation leads to the so-called two-fluid hydrodynamics, which in many cases reduces to one-fluid magnetohydrodynamics (MHD). In another extreme case, when the characteristic time is much smaller than the collisional time, the plasma is described by the Vlasov equation [21].
In the case of a strong magnetic field, the decomposition with a small parameter, the ratio of the Larmor radius to the characteristic scale, is used. This approach is simple and efficient for a tokamak plasma, where the magnetic and electric forces are dominant compared to the gravitational one. The equation of motion for a single particle can be written as: mα dv = qα(E + v × B), (2.1).
where mα and qα are the mass and the charge of a particle α (electron or ion). Supposing the magnetic field to be uniform and time independent B = Bez, where B = const, and the absence of an electric field, the motion of the particle can be easily described. The resulting particle orbit is a helical trajectory following the magnetic field lines. The rotation frequency, called also gyro or cyclotron frequency, is: Wc,α = |qα|B (2.2).
and the radius of the orbit (gyro- or Larmor radius) is: Wc,α |qα|B rL,α = v ⊥,α = mαv ⊥,α , (2.3). where v⊥,α is the velocity component perpendicular to the magnetic field. It is convenient to analyse the movement of the gyro-center called the guiding centre.

Wave propagation in a magnetised plasma

The propagation of an electromagnetic wave in plasma presents a special interest in the frame of tokamak plasma diagnostics with microwaves, particularly for the reflectometry application. This section summarises the basics of the microwave propagation in a magnetised plasma which will be used in the next chapters. The wave propagation equation is derived from the Maxwell’s equations and the electron equation of motion. As the results on the refractive index appear in many papers and books, only a quick overview will be made in this section. The assumptions used to obtain the refractive index will be discussed and the difference between O and X mode will be explained. We will consider phenomena which are faster than the typical time of collisional relaxation.

Dielectric permittivity tensor

The wave creates a small perturbation with wave vector k⊥||ex of the homogenous equilibrium perpendicular to a constant magnetic field B = B0ez. The background electric field is E0 = 0. One can find the dielectric permittivity as it is shown in [28] by calculating the current density response ji to the wave field E from Ohm’s law: ji = σikEk, ˜ (2.12).
where σ is the conductivity tensor, the index k appearing twice implies summation over all components x, y, z. Then the dielectric permittivity tensor has a simple form: ik = 1 − α ω2 ! (δik − bibk) + 1 ω2 ! bibk + i ω2 Wcα eikjbj, pα − α pα pα ω2 − W2 ω2 α ω(ω2 − W2 ) X cα X X cα (2.13). where ωpα = s Z n e2 (2.14) is the plasma frequency, Wcα is the cyclotron frequency, b = B/B, α indicates electrons or ions. This expression can be obtained from the MHD approach too, and stays correct in the approximation of a cold plasma or long wave-length perturbations (kv → 0) without spatial dispersion.

Radial resolution of reflectometry

Without going into the details of the reflectometry technique which will be described in the Chapter 5, the radial resolution of this method can be discussed based on the wave propagation principles. For a continuous medium it is impossible to talk about one point of reflection, however, in a tokamak plasma one still defines the cutoff layer as the positions where the refractive index for a given probing wave frequency goes to zero, because after this point the wave electrical field decreases exponentially. The cutoff layers are represented schematically for a tokamak cross-section in Fig. 2.4 for O-mode (Fig. 2.4a) and X-mode (Fig. 2.4b) waves. The surfaces of the constant poloidal flux are shown with dashed lines and labeled with the value of ρpol, while the cutoff surfaces are colour-coded for frequencies 40–110 GHz. The density profile is shown in Fig. 2.4c.
Generally speaking there are also back-scattered reflections before the cutoff layer (Fig. 2.5), which are neglected if the plasma density doesn’t have strong gradients [28] and should be taken into account in the turbulent tokamak plasma. We should distinguish therefore the main reflection from the cutoff layer that serves for density profile measurements and the reflections from the density fluctuations which follow the Bragg rule:
kf − kwave = kwave, (2.22).

MHD instabilities and ELMs

In the MHD model the plasma is considered as a quasineutral fluid of charged particles. Therefore magnetic fields induce currents, which in their turn locally polarize the plasma and modify the magnetic field itself. The equations that describe MHD are a combination of the Navier-Stokes and Maxwell’s equations. The MHD instabilities, associated to growing MHD modes, arise due to the current and/or pressure gradients.
The magnetic structure of a tokamak consists of the nested tori of the magnetic surfaces. The safety factor along a field line is defined by the ratio between the number of turns in the toroidal and in the poloidal directions before the field line closes in itself. The radial profile of the safety factor depends mainly on the current profile (the stronger is the current, the higher is q). From the Ampere’s law q(r) can be expressed as a function of the plasma current Ip(r) inside the flux surface: µ0RIp(r) q(r) = 2πr2Bφ(r = 0) , (3.1).
where r is the minor radius of the magnetic surface and Bφ(r = 0) is the toroidal magnetic field on the magnetic axis. The twisting of the field lines is different for each magnetic surface, this causes the magnetic shear: q dr s(r) = r dq . (3.2).
When q(r) is a rational number q(r) = m/n, the magnetic field lines close after m poloidal and n toroidal turns and this case fcilitates the instability development.
MHD modes can be divided into two types: ideal and resistive modes [35]. In ideal MHD the plasma is considered to be perfectly conductive. In this case, the magnetic flux is conserved within each magnetic surface. This approach is only applicable when the plasma collisionallity is low and the resistivity is small. The resistivity of a tokamak plasma can become important in the vicinity of the rational surfaces. Then the ideal MHD approximation is not valid and the resistive MHD is introduced. The enhanced resistivity can result in the formation of magnetic islands or magnetic turbulence. Thus the magnetic structure can be broken by magnetic reconnection, releasing the stored magnetic energy as waves, in particle acceleration or heat.
Similar to q(r) of the magnetic filed lines, an MHD mode can be characterized by its poloidal number m and its toroidal number n. A resonance is possible on the rational surfaces with the safety factor q = m/n. Some of the MHD modes are:
• Ideal kink modes m = 1, driven by the current density gradient.
• Ballooning mode (ideal or resistive), driven by the pressure gradient.
• Peeling mode, a sub-case of the kink mode with low toroidal mode numbers satisfying the condition m − nq(a) = 0, m > 1.
• Tearing mode (resistive), driven by the current density gradient.
The stability of the kink modes is defined by the Kruskal-Shafranov criterion [36] according to which q(r) at the minor plasma radius a should satisfy q(a) > 1. (3.3).
In reality an external kink mode is destabilised already at q = 2, in addition there is a tendency for disruptions to occur more often for q < 3 [37]. These stability limits typically lead to the safety factor profile in the order of 1 at ρpol = 0 and higher than 3 at the separatrix. For the ballooning modes the stability is given by the Troyon’s limit with respect to the normalised β [38]: βNlim = β aB = 3. (3.4).
Ballooning modes as pressure-driven instabilities impose the most strict upper limit on the plasma pressure for a given value of the magnetic field amplitude [39]. They tend to be localized on the low field side, in the region with unfavourable magnetic field line curvature.
During a tokamak discharge if enough energy is put into the plasma, the plasma reaches a state with a better confinement. This regime is called H-mode. Although the H-mode is the most favourable regime for a potential fusion reactor, it is associated with a confinement destructive phenomenon called edge localized modes or ELMs. ELMs lead to a periodic relaxation of the edge pressure profile [37]. Experimentally ELMs produce repetitive bursts in the Dα emission that indicates enhanced particle transport from the main plasma to the divertor. During an ELM the density and temperature gradients decrease abruptly (< ms) in the edge pointing out a confinement deterioration. After an ELM the plasma pressure recovers until the next ELM.

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Table of contents :

1 Introduction 
1.1 Fusion
1.2 The tokamak concept
1.3 H-mode
1.4 Thesis goals and outline
2 Basics of plasma physics 
2.1 Charged particle dynamics in a tokamak
2.1.1 Drifts in a tokamak plasma
2.1.2 Collisional transport
2.2 Wave propagation in a magnetised plasma
2.2.1 Dielectric permittivity tensor
2.2.2 O- and X-mode polarisation
2.2.3 Radial resolution of reflectometry
3 Instabilities and turbulence in fusion plasmas 
3.1 MHD instabilities and ELMs
3.2 Turbulence
3.2.1 Drift waves
3.2.2 Turbulent spectra
3.2.3 Turbulence suppression by shear flow
3.2.4 Limit-cycle oscillations
3.2.5 H-mode, turbulence and confinement
4 ASDEX Upgrade and diagnostics for temperature, density and electric field measurements 
4.1 ASDEX Upgrade tokamak
4.2 Equilibrium reconstruction at ASDEX Upgrade
4.3 Laser interferometer
4.4 Electron cyclotron emission radiometer
4.5 Thomson scattering
4.6 Lithium beam emission spectroscopy
4.7 Charge exchange recombination spectroscopy
4.8 Magnetic pick-up coils
4.9 Profile reflectometers
4.10 Doppler reflectometer
5 Ultra-fast swept reflectometry 
5.1 Ultra-fast swept reflectometer history and design
5.1.1 UFSR history
5.1.2 UFSR installation at ASDEX Upgrade
5.1.3 UFSR perfomance tests
5.2 Data analysis methodes
5.2.1 Density profile reconstruction
5.2.2 Step to the ASDEX database
5.2.3 Wavenumber power spectra and turbulence level
5.2.4 Frequency power spectra
5.2.5 Correlation length
6 Experimental results on the L-H transition
6.1 Dynamics and mechanisms of the L–H transition
6.2 Comparison of L- and H-mode turbulence characteristics
6.3 I-phase and limit-cycle oscillations
6.3.1 Radial electric field as main player during I-phase
6.3.2 Density and electric field dynamics during I-phase
6.3.3 Established I-phase as ELM-like phenomenon
6.3.4 Controversial early I-phase
6.4 Summary
7 Edge coherent modes 
7.1 Observation of modes with the UFSR
7.2 Edge coherent modes in the ELM free phase of the H-mode
7.3 Edge coherent modes in between ELMs
7.4 Edge coherent modes during I-phase
7.5 Summary
8 Conclusions 


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