Turbulence-driven transport in the tokamak
In the tokamak plasma, fast motion of the particles along the magnetic field lines leads to the plasma parameters variation being small in that direction, which gives it the previously mentioned 2D character. Because of this, the plasma and in particular the plasma turbulence is often described in the poloidal plane roughly perpendicular to the external magnetic field.
In these 2D cross-sections of the tokamak, plasma parameters are not homogenous . Due to the localization of the heating and the radial transport of the particles and energy across the magnetic surfaces to the outside the plasma volume, the profiles of density and temperature are usually peaked at the plasma core (central area of the poloidal cross-section) with the pressure decreasing.
In the presence of the pressure gradient, appearance of the random periodic perturbation of the electric potential causes a periodic perturbation of electron density. Certain effects, for example collisions can cause the “lag” in phase between the potential and density perturbation, which leads to the further accumulation of energy by the perturbation, which results in the development drift-wave turbulence (more detailed information on the mechanism can be found for example in ). Developed drift-wave in turn causes the local elimination of the pressure gradients.
These small-scale instabilities via the inverse energy cascade cause the development of the plasma density and temperature fluctuations at larger scales (see Fig. 2.1) which in turn lead to anomalously high transport and significant degradation of plasma particle and energy confinement . The scale of the turbulence in the radial direction is however somewhat limited by the radial variation of the plasma perpendicular velocity , often called the velocity shear. The difference of velocity on the adjacent magnetic surfaces “breaks up” turbulent cells and causes their radial scales to dwindle. An illustration of this mechanism is given on the figure 2.1. The complete interplay between plasma flows and turbulence is, however, very complicated and is currently a topic of active research.
Turbulent processes cause fluctuations of multiple plasma parameters : density (δn), temperature of electrons (δTe) and ions (δTi), electric field potential (δφ) and magnetic field (δB).Doppler reflectometry technique, discussed in this thesis, however, focuses on measuring the density fluctuations δn, which is why they will be considered from here on out.
While plasma generally possesses toroidal geometry, if the size of the probing zone is much smaller than the minor and major radius of the plasma, to describe DR diagnostic analytically it is sufficient to consider so-called slab geometry, where Cartesian coordinate system is used for plasma parameters description. In such system x generally corresponds to radial component r of toroidal coordinate system, y corresponds to poloidal coordinate, which would normally be expressed as rφ through poloidal angle φ. Finally, often neglected tororidal direction is substituted by z coordinate. Such a simplification neglects curvature effects, but provides the clarity toroidal system sometimes lacks. Slab geometry will be mostly used within the analytical approach of this thesis as well as in the introductory chapters 2 and 3.
Due to the random nature of the turbulence and consequently δn, it is generally described by its statistical characteristics or averaged values. The turbulence spatial scales are conveniently described by its k power spectrum in the Fourier space. Due to the 2D nature of the tokamak plasma, the spectrum is described in the following by its dependency on radial (κ) and poloidal (q) wavenumbers. Such power spectrum within the area where turbulence can be considered statistically stationary and homogenous is linked to the turbulence two-point CCF by Wiener- Khinchin theorem: Here |δn(κ,q)|2 is the power spectrum of the density fluctuations, a statistical value rather than a random one. The measure of the random turbulence amplitude is usually given by the value of δn r.m.s.
Typical tokamak instabilities
While the exact power laws differ from the ones presented on the Fig 2.1 the general turbulence spectra measured experimentally do have the characteristic knee-shape , with two distinct areas, the transition between which is characterized by the inverse ion Larmor radius √ where Ti is ion temperature and mi is ion mass.
As it was mentioned before, the main cause of the anomalous transport in the tokamak is the drift-wave instabilities driven by the gradients on density and temperature in the plasma . The four main types of instabilities are ion temperature gradient (ITG) mode, electron temperature gradient mode (ETG), trapped electron mode (TEM) and trapped ion mode (TIM) (which is often negligible, but can play a role in some cases). The term trapped here refers to the particles being trapped mainly as the plasma LFS, as mentioned in section 1.1. Due the trapped particles having a rather peculiar banana shaped (in the poloidal cross-section) trajectory they can cause instabilities specific to toroidal magnetic devices . Such instabilities typically have a scale larger than the width of the “banana” in the radial direction and therefore the TIM has a much larger scale than TEM.
ETG mode  possesses the smallest scale of the order of ρce , which means that k⊥ρci >> 1. This mode is driven by the gradient of electron temperature and is also stabilized by the density gradient.
TEM  has the intermediate scale, which is wider that electron Larmor radius and can be comparable to ion one. As a result, for this mode k⊥ρci ≤ 1. This mode is driven by both density and electron temperature gradients, but is limited by collisions, which lead to trapped particles becoming passing ones. Although collisions can also drive it causing the dissipative TEM to manifest, it was shown  that strong enough collisions will still act as a damping mechanism. TIM  is the largest-scale mode, which possesses k⊥ρci <<1 and can lead to substantial transport, but is usually not the dominating instability in the tokamak discharge. Similarly to TEM, TIM is driven by ion temperature and density gradients and is limited by collisions.
An illustration of different instabilities’ respective scales as well as an example of growth rates for HT-7 device is presented on the figure 2.4. The FT-2 tokamak that is going to be studied in this work is dominated by TEM instability according to the linear analysis .
In plasma, in most of the cases density is treated as a function of magnetic surface due to a much faster transport along magnetic field compared to the one across it. The magnetic field is generally considered to only be dependent on the major radius. As a result, the cut-off (reflective) surfaces of the waves in plasma usually roughly repeat the shape of magnetic surfaces for O-mode. For X-mode, which possesses both cut-off and resonances dependent on magnetic field, the situation is a bit more complicated. The cut-off layer is no longer directly connected to the magnetic closed surfaces or density.
The two typical situations of X-mode probing are presented on Fig 3.1. The two pictures there present the color map of density for FT-2 tokamak poloidal cross-section according to the numerical modelling results, which will be explained in more detail in later chapters. Red lines correspond to probing frequency being equal to the right cut-off formula (3.9), while blue ones to it being equal to the UHR frequency from the same formula.
In the case of lower magnetic field (left figure), the cut-off covers the resonance making the situation suitable for the reflectometry experiment, while in the opposite case reflectometry can only be used from the LFS, otherwise the wave is absorbed in the UHR and while the scattering still can happen, it falls in the domain of enhanced scattering diagnostic, which differs from DR.
One thing that was not mentioned before is that while cut-off defines the coordinate where the total wavenumber goes to zero, reflection actually happens when the radial component of the probing beam wavenumber is zero. In the case where the wave has nonzero poloidal and toroidal wavenumber components, this point, called the turning point is located before the cut-off in the path of the wave.
Linear modelling tools
In some cases, there is no need for a model to include complicated nonlinear interactions of the probing wave with the plasma. Such a model could be relevant for example when the application is dedicated to analytical result validations, which were derived in linear approximation, as it will be done in the future sections of this thesis. Another possible application is the analysis of experimental results to check if a “linear” interpretation is applicable, as it was done in .
Such linear computations require only the evaluation of the electric field in the case of plasma unperturbed by the turbulence. This can often be done analytically, but in other cases will require numerical solution of the Helmholtz equation (3.14).
When the unperturbed field is obtained the Helmholtz equation for the next order can be solved. However an even simpler method is using the reciprocity theorem and formula (3.19) and calculating directly the signal received by the antenna. In the best case, when the unperturbed field can be found analytically, the task is simplified and only requires an evaluation of the 2D integral for each turbulence realization.
Moreover, for the analytical RCDR studies, instead of performing multiple calculations to properly describe random nature of the turbulence, expression (3.21) can be used in conjuncture with the Wiener-Khinchin theorem, to obtain result within one computation with the help of turbulence power spectrum. A detailed description and application of such an approach to validate analytical results can be found in . A schematic roadmap of the linear modelling approach is given at the figure 4.1: Within this thesis, such an approach will be used for the validation of analytical results produced by the study of the tilted turbulence effects on RCDR, and turbulence structures tilt angle measurement technique described in section 3.4.
The main advantage of the linear reciprocity theorem approach is an extremely fast numerical computation (although in the case of benchmarking with experiment, such as  the main limitation comes from the plasma modelling code rather than wave propagation code). The main disadvantage is, naturally, the nonlinear effects being neglected. Another potentially problematic peculiarity of this approach is that it requires the information about the density perturbations and unperturbed profile separately. The process of extracting the fluctuations out of the full density profile can, in principle, impose some additional limitations on the possibility of description of the low-frequency turbulence and should be performed carefully.
The limit of validity of the linear model is given by the conditions (3.24) and (3.25), except for the fact that in the case of X-mode probing an additional factor is added to it as dictated by  to account for the complicated dependence of X-mode wavenumber on the plasma parameters. However, as previously stated in the section 3.5, there is still room for exploration of the formula (3.25) within the rigorous wave theory and by the end of this thesis another condition will be obtained as a new nonlinearity criterion.
Nonlinear modelling tools
Since the end of last century, the state of art method of performing the wave propagation calculations is the so-called finite difference time domain (FDTD) computations. This approach is based on using finite difference method to estimate spatial derivatives of the quantities as the difference between the neighboring points of the spatial grid. The time derivatives then can be used to calculate the state of a wave at the next temporal step. This way, the system of Maxwell’s equations (3.1) can be turned into a linear algebra problem. It is still necessary to add some version of Ohm’s law to define current to complete the system, which corresponds to a linear response of the plasma. So, when we speak about nonlinearity, we are only considering the non-linear behavior of the probing wave amplitude. As a consequence, there is no way to access such nonlinear effects as soliton generation, due to the fact a second order of the plasma response is required . An approach where all the components of wave field are computed is called full-wave computation and is actively used in the modelling of waves in plasma.
The current equation is usually written in cold plasma approximation, neglecting the collisions and ultimately using the dielectric tensor given by (3.4). The spatial grid is introduced, and since the electric and magnetic fields are dependent on each other’s vorticity, “staggered” grids are usually used, meaning that the electric and magnetic fields are calculated on two different grids shifted by half a grid step apart. Similarly, an iterative process of calculation electric field in half a timestep then recalculating magnetic field based on it to then recalculate electric field is used. Such method was suggested by Yee in 1966 and its numerical stability was later demonstrated, although it is still a topic of active research . Further details on the specifics of the Yee method and FDTD calculations in general can be found in .
FDTD method, unlike the Helmholtz equation solvers performs the calculation in time domain, meaning that it is necessary to perform the computation long enough to describe the studied process. Combined with the fact that numerical stability requirement also places an upper bound on the timestep size , this means that FDTD methods are rather demanding resource-wise. Although, due to the fact that normally the equation system for each grid point only includes the neighboring points (due to finite difference method), the computations can be parallelized efficiently.
The code used in this thesis project is the full-wave IPF-FD3D code  created by Carsten Lechte. It is used within this thesis in its 2D configuration for both validation of analytical studies and the creation of the synthetic DR and RCDR diagnostics. The full-wave computation is performed under the cold plasma approximation. As an input the code receives the matrices of background density, density perturbations and external magnetic field. The main output used is the complex amplitude recorded by receiver after an amount of time-steps sufficient to exclude all the transient processes (propagation of incident and scattered wave, multiple scattering). The computations were performed for “frozen” turbulence, meaning that within a computation there was no temporal dependence of the density or magnetic field. This is justified by the fact that even when temporal behavior of the signal was studied, the timescale of density changes due to drift-wave turbulence (μs<) was much bigger than the timescale of wave propagation (~ns) in the studied cases. To obtain the temporal dependence a set of calculations on consecutive “frozen” snapshots of density was performed and this way the time dependence of complex amplitude was obtained.
To develop synthetic diagnostic capable of reproducing experimental results, aside from using the wave propagation code, it is imperative to have as precise density and magnetic field profiles as possible. To produce a density profile describing the perturbations of density numerical modelling is used.
The most complete description of plasma is given by the kinetic theory, where kinetic equation for particle distribution function is used. However, using kinetic approach is often challenging, and moments of the equation are calculated to produce a system of transport equations for particles, momentum and energy. Sometimes these momentum equations are transformed to obtain a system of magnetohydrodynamic (MHD) equations  describing plasma as a magnetized fluid.
Numerical modelling of the plasma is usually done within one of these analytical models. Oftentimes a set or transport equations is solved, with sources, sinks and transport coefficients either prescribed externally obtained by coupling with other codes. Examples of such approach is the code ASTRA  and B2SLOPS , often used for describing the core and the edge regions of the tokamak respectively. The MHD approximation, however, generally breaks down when the phase velocity and the scale of the instability becomes comparable to the velocity and the finite orbit width of the particle. That makes drift-wave instabilities unsuitable for the MHD studies and therefore a code with full kinetic description of particles rather than a fluid approach is required. Since the goal of the thesis was the creation of synthetic diagnostic measuring the parameters of drift-wave turbulence, it was necessary to use a kinetic code.
Most kinetic codes used for plasma modelling, are the gyrokinetic codes in which the kinetic equation is averaged over the gyromotion of particles around the magnetic field lines. Some of these codes solve kinetic equations directly (for example GENE code ), while others, including the ELMFIRE code  used in this thesis, use Monte Carlo method – creation of randomly generated virtual particles with statistical characteristics of the real plasma and tracking their movement to obtain the distribution functions.
Table of contents :
1.1. Magnetic confined fusion
1.2. Turbulence in magnetized plasma
1.3. Microwave diagnostics
1.4. Scope of this thesis
2. Turbulence and its characterization
2.1. Turbulence-driven transport in the tokamak
2.2. Turbulence characteristics
2.3. Typical tokamak instabilities
2.4. Measurement of turbulence parameters
3. Doppler reflectometry and radial correlation Doppler reflectometry techniques
3.1. Theoretical basis
3.2. Measured quantities
3.3. Validity of interpretation
3.4. Linear scattering effects
3.5. Nonlinear scattering effects
4. Numerical modelling
4.1. Linear modelling tools
4.2. Nonlinear modelling tools
4.3. Plasma modelling
5. FT-2 tokamak experiment
5.1. Studied discharge
5.2. DR and RCDR setup
6. Analytical study of cylindrical geometry effects
6.1. Basic equations
6.2 BS signal
6.2. FS signal
6.3. Total signal and CCF
7. RCDR analysis for tilted turbulent structures
7.1. Basic equations
7.2. CCF at suppressed small-angle scattering
7.3. CCF for small-angle scattering dominance
7.4. Validation with numerical modelling
8. Nonlinear regime transition study
8.1. Basic equations
8.2. Quadratic scattering signal
8.3. Nonlinear transition criterion
8.4. Analytical results discussion
8.5. Numerical validation
8.6. Conclusions and discussion
9. Synthetic DR and RCDR diagnostics
9.1. Experimental benchmarking of X-mode modelling
9.2. Different ELMFIRE cases used for modelling
9.3. Nonlinear scattering effects in X-mode modelling
9.4. O-mode synthetic diagnostic