Turbulence and geodesic acoustic modes in fusion plasmas 

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Energy flow and interactions

This section aims to give an overview of the various energy sources, sinks and energy transfer mechanisms that can be encountered in a magnetically confined fusion experiment, which are schematically shown in Fig. 3.5.
Figure 3.5: Schematic overview of the energy flow in a turbulent system. Interactions between different components of the system and dissipation mechanisms are also shown. From Ref. [23]. External energy is introduced to the system via plasma heating, which is described in more detail in Sec. 4.2. This leads to the formation of radially varying density and temperature profiles. The main source of energy input into the various linear instabilities which can occur in the system are the gradients in density, temperature and pressure. Section 3.2 will introduce the most important of these instabilities. The saturated instabilities drive broadband turbulence through nonlinear coupling. As described in Sec. 3.1.3, dual cascades will lead both to viscous dissipation of energy at small scales, but also to energy transfer into larger scales such as zonal flows or geodesic acoustic modes (GAMs), which are the subject of this work. When the scales reach the order of the plasma size, they can affect both the growth rate of the linear instabilities, and the overall plasma equilibrium that is responsible for the plasma profiles and gradients. Via the Reynolds stress, the turbulence can also lead to the formation of radially localised zonal flows. Both static zonal flows and the oscillating geodesic acoustic modes are very important turbulence phenomena, as they can take energy out of the broadband turbulence, and can be dissipated through collisional and Landau damping. They can also reguChapter late the turbulence through velocity shearing. A more detailed description of zonal flows and geodesic acoustic modes is given in Sec. 3.4.

Linear instabilities

This section describes the mechanism of drift wave turbulence, which is responsible for turbulent transport of heat and particles, as well as a number of related linear instabilities. These are exponentially growing modes as predicted by the linearised two-fluid equations. As outlined in the previous section, these modes can interact with other components of a turbulent systems, such as zonal flows.

Core turbulence instabilities

While the drift wave is driven by strong density gradients in the edge, turbulence further towards the plasma core is more often influenced by the temperature gradients. The ion temperature gradient mode (ITG mode) is an instability that is believed to be a contributor to turbulent transport [25, 26]. It is illustrated in Fig. 3.7. The ITG mode can be compared to the Rayleigh-Taylor instability, which is encountered in neutral fluids with a heavy fluid layer on top of a lighter fluid. In the case of a plasma, these layers are replaced by higher and lower ion temperature regions. The ion (and electron) pressure profile in a tokamak is typically peaked in the centre and decreases gradually towards the edge. In the plasma core, temperature gradients are usually stronger than density gradients. Due to the curvature and causes the local curvature drift velocity of ions to change. This results in density and potential perturbations causing radial E ×B drifts. On the tokamak high-field side (left) these drifts are oriented in the opposite direction from the initial temperature perturbation and stabilise the plasma, however on the tokamak low-field side (right) the perturbation grows further. The overall mode is unstable once a certain critical gradient is exceeded. ∇B drifts, ions and electrons move in opposite directions (particle charge qp = ±e), normally along lines of equal pressure (i.e. temperature, if the density is assumed to be locally constant) [3]: vC,∇B = 􀀀 2Wk +W⊥ Rc × B qpR2 cB2 . (3.4).

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Shear flows and turbulence suppression

Now that some of the important turbulent instabilities have been introduced, it is important to understand the mechanisms by which turbulence can be suppressed, and thus plasma confinement can be increased.
It is believed that radially sheared plasma velocity, due to E × B flows driven by a sheared radial electric field E′r = ∂Er/∂r, is an important contributor to nonlinear stabilisation and the suppression of turbulence. The radial electric field in the edge region of tokamak plasmas shows strong variation, as shown in Fig. 3.8. Inside of the last closed flux surface (see Sec. 4.4), Er is negative and determined by the radial pressure gradient ∂p/∂r: Eedge r ≈ 1 en ∂p ∂r . (3.5).
In the scrape-off layer (SOL) outside of the last closed flux surface, parallel currents to the divertor target plates affect the radial electric field so that it changes sign and is determined by the plasma potential gradient, which is proportional to the temperature gradient [37]: ESOL r = − ∂ ∂r ∝ − ∂T ∂r . (3.6).

Table of contents :

1 Introduction 
2 Nuclear fusion and basic plasma physics 
2.1 Introduction to nuclear fusion
2.2 Magnetic confinement fusion using tokamaks
3 Turbulence and geodesic acoustic modes in fusion plasmas 
3.1 Turbulence
3.1.1 Introduction to turbulence
3.1.2 Turbulence in tokamaks
3.1.3 Turbulence spectra
3.1.4 Energy flow and interactions
3.2 Linear instabilities
3.2.1 Drift waves
3.2.2 Core turbulence instabilities
3.3 Shear flows and turbulence suppression
3.4 Zonal flows and geodesic acoustic modes
3.4.1 Zonal flows
3.4.2 Geodesic acoustic mode
3.4.3 GAM properties and state of GAM research
4 ASDEX Upgrade tokamak 
4.1 Details and operation
4.2 Heating systems
4.3 Important diagnostics
4.4 Plasma configuration
5 Doppler reflectometry 
5.1 Wave propagation and reflectometry
5.2 ASDEX Upgrade V-band Doppler reflectometers
6 Data analysis 
6.1 Determining Doppler shift and amplitude
6.1.1 Density profile reconstruction
6.1.2 Beam tracing code torbeam
6.1.3 Doppler shift measurements
6.2 GAM detection
6.3 Comparison of sliding window FFT and MUSIC techniques
6.4 Cross-correlation techniques
6.4.1 Spatio-temporal cross-correlation
6.4.2 Two-point correlation method
7 GAM frequency scaling 
7.1 Description of experiments
7.2 Comparison to existing models
7.2.1 Comparison with Winsor’s basic scaling
7.2.2 Comparison with Conway’s heuristic model
7.2.3 Comparison with Angelino’s fluid model
7.2.4 Comparison with Sugama and Watanabe’s gyrokinetic model .
7.2.5 Comparison with Gao’s gyrokinetic model
7.3 Further influences on the GAM frequency
7.4 Discussion
8 GAM amplitude scaling 
8.1 GAM amplitude scaling with κb and q
8.2 GAM drive
8.3 GAM damping
8.3.1 Collisionless damping for negligible kr
8.3.2 Collisionless damping with finite orbit width effects
8.3.3 Comparison of the collisionless damping rates
8.3.4 Collisional damping
8.4 Discussion
9 GAM structure and propagation 
9.1 Description of experiments
9.2 GAM structure
9.2.1 Multiple eigenmode GAMs
9.2.2 Continuum GAM
9.2.3 Single eigenmode GAM
9.2.4 Width of GAM region
9.3 GAM propagation
9.3.1 Cross-correlation tilt and cross-phase derivative
9.3.2 Two-point correlation method
9.4 Magnetic GAM signature and structure
9.5 Discussion
10 Summary and conclusion 
Bibliography 

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