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## δf equations in Fourier space

The nonlinear gyrokinetics is the domain ofmassively parallel high performance numerical simulations through the computation of nonlinear dynamics within a description, which includes self-consistent multi-scale interactions. It usually requires millions of CPU hours and makes anything beyond medium nonlinear runs dominated by a single type of instability over a small range of scales, impractical. As a result of the complexity of numerical implementation, sometimes it is hard to understand and isolate important physical mechanisms. In this regard, reduced models appear as intermediate tools, that are necessary for isolating important physical mechanisms, to provide guidance to large scale gyrokinetic simulation efforts as well as comparison with experiments. Based on these ambitions, the following reduced model, which stands on the simplification of the nonlinear dynamics, without losing any essential features, has been worked out for the trapped particle turbulence model. In this thesis the simulation is implemented in Fourier space, which means: fs ¡ α,ψ,μ,E ¢ = X k f s k ¡ μ,E ¢ eikαα+ikψψ .

### Description of nonlinear interactions

In theory, if the medium is infinite, the wave vector k is a continuum quantity, so in the Fourier plane there are infinite number of waves and in the absence of quantization due to boundary conditions, the number of triads defined by k+p+q = 0 is also infinite,which results in an infinite number of nonlinear interactions. The nonlinearity is the most complicated part in turbulence and analytic solution for such a problem is probably impossible.

However in numerical simulation, the wave vector should be discretized in a cartesian or polar coordinate, linearly or logarithmically, and the number of waves is not infinite. For a given range of discretized waves, the number of interacting triads is fixed, which can be found out easily by a program based on the conditions fixed by the wave vector constraint: k+p+q = 0. The triad information then can be written in a table as a coupling card. Based on this coupling card, it is possible to give an analytic expression of the nonlinear terms. Of course even with an analytic expression, the system is still very complicated and it is difficult to find a solution by theoretical analysis, due to the average operator (Bessel function) and the integration over the kinetic coordinates, etc, but it can be resolved numerically.

In order to give an analytic expression of the nonlinear terms, it is necessary to start fromthe coupling information defined by the wave vector constraint: k+p+q = 0.

#### Different approaches and connections to previous models

In the previous section, a description of the nonlinear terms (i.e. Fourier transform of the Poisson bracket) is worked out in the wave number plane. In simulation, the wave numbers k,p,q can be linear, logarithmic or even random, which is not a limit in this description. However in practice, one must consider the capacity (i.e. memory, threads,etc) of the machine and the intention of the research. For example if the nonlinear physics in small scale is of interest, it may be necessary to use logarithmic discretization of the wave number magnitudes, to describe a large range of scales easily. The number of the couplings in logarithmic discretization is much less than that in linear discretization, which will take less time and consume less computer resources to run the simulation. This was the main motivation for the development of ”shell models”. Logarithmic discretization means that the wave number k is discretized as: k = kn = k0g n .

**Threshold of the temperature gradient driven instability**

The instability threshold is defined as the point where the imaginary part of the roots of the dispersion relation becomes exactly equal to zero: ℑ(ω) = 0, and therefore it allows to distinguish between stable (γ < 0) and unstable (γ > 0) modes. In the multispecies model of interest here, the two branches can coexist at the threshold, corresponding to TIM or TEM. Depending on the sign of the real frequency, the TIM or TEMthresholds can be defined by setting ℑ(ω) = 0 in (3.5), (3.6) and (3.7) in the dispersion relation. Since the frequency is exactly real at the threshold, the plasma dispersion function can be seen as two parts, which are the real part given by ǫ(ω) and the imaginary part given by g (ω). Note that this is only valid for the frequency at the threshold (i.e, ℑ(ω) = 0, ω is purely real). In order that the plasma dielectric function vanish, the real and imaginary parts should both vanish, that is.

**Threshold of κT and ion to electron temperature ratio τ**

Even though the TIM linear threshold is found higher than the TEM linear threshold for the set of parameters given in figure 3.2, this is in fact only valid for values such that: τ ≥ 1.0. The effect of the ion to electron temperature ratio is investigated in figure 3.2 in order to fully understand the relationship between TIM and TEM linear thresholds. Note that the interval τ ∈ [0.5; 2] is clearly sufficient to cover most of the temperature ratios observed experimentally [7]. Figure 3.2 shows the TIM and TEM critical temperature gradients as functions of the ion to electron temperature ratio. Four regions appear, where TIM and TEM become alternatively stable or unstable. The TEMare found easier to destabilize while increasing τ, on the other hand the TIM instability is harder to destabilize for increasing values of τ.

**Table of contents :**

**1 Introduction: turbulence and transport in Tokamaks **

1.1 Nuclear fusion

1.1.1 Nuclear fusion reaction

1.1.2 Lawson criterion

1.2 Kinetic description of plasma turbulence

1.2.1 Vlasov equation

1.2.2 Quasi-neutrality equation

1.2.3 Difficulty of nonlinear simulation

1.3 Reduced model

1.3.1 Trapped particle model

1.3.2 Simplifications in fluid turbulence

**2 Bounce averaged gyrokinetics δf equations in Fourier space. **

2.1 Introduction

2.2 Bounce averaged gyrokinetics

2.2.1 Model equations

2.2.2 Scale separation

2.2.3 Normalization

2.3 δf equations in Fourier space

2.4 Description of nonlinear interactions

2.4.1 Description of k+p+q = 0

2.4.2 Description of the nonlinear terms

2.4.3 Different approaches and connections to previous models

2.4.4 Conserved quantities.

2.5 Conclusion

**3 Linear Phase **

3.1 Linear dispersion relation

3.1.1 Plasma dielectric function: ǫ(ω)

3.1.2 Singularity and residue

3.2 Threshold of the temperature gradient driven instability

3.2.1 Threshold of κT and the wave number k

3.2.2 Threshold of κT and ion to electron temperature ratio τ

3.2.3 Threshold of κT and the trapped particle ratio ft

3.3 Linear instability

3.3.1 Numerical method: argument principle

3.3.2 Linear instability for isotropy system: γ(k) with k = kα = kψ

3.3.3 Linear instability for anisotropic system: γ ¡ kψ, kα ¢ .

3.4 Conclusion

**4 Isotropic model: Sabra and GOY **

4.1 Model equations

4.1.1 Phase approximation

4.1.2 Model equations

4.2 Nonlinear simulation of the Sabra model

4.2.1 Code verification

4.2.2 k spectra of the electrostatic potential energy Eφ

4.2.3 k spectra of the entropy Ef and kinetic effect

4.2.4 The effect of free parameter α

4.3 Influence of phase information: GOY vs Sabra

4.3.1 Oscillatory dynamics of the GOYModel

4.3.2 Comparison of entropy

4.3.3 Oscillation of the k-spectra

4.4 Conclusion

**5 Anisotropic model: LDM **

5.1 LDMof bounce averaged gyrokinetics

5.1.1 LDMGrid

5.1.2 Vlasov-Poisson equation

5.1.3 Numerical scheme

5.1.4 Definition of zonal flow and dissipation

5.2 Kinetic ions

5.2.1 Temporal spectrumof Eφ and Efi

5.2.2 Spectrum of electrostatic potential Eφ in wave number plane

5.2.3 k spectra of Eφ and Efi

5.3 Kinetic electrons

5.3.1 Temporal spectrumof Eφ and Efe

5.3.2 Predator-prey dynamics between zonal flow and turbulence modes

5.3.3 Spectrum of Eφ in wave number plane

5.3.4 k spectra of electrostatic potential energy Eφ and entropy Efe

5.4 Fully kinetic system

5.4.1 Temporal spectrumof Eφ, Efi and Efe

5.4.2 Spectrum of Eφ in wave number plane

5.4.3 k spectra of Eφ, Efi and Efe

5.4.4 Comparison of the k-spectra: adiabaticity

5.5 Conclusion

**6 Conclusion and perspective **

**A Calculating the coefficients of GOY model **

**B The linear dispersion relation solver **

B.1 Linear dispersion relation ǫ(ω)

B.2 Eigenvalue solver

**C Formulation of gyro correction **

**D Comparison ofJ0s and its approximations **

**E Electrostatic potential energy Eφ and entropy Efs **

**F Method of numerical integration **