Turbulence wave number spectrum and TCCF reconstruction

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Turbulence in fusion plasma

Magnetically confined fusion plasma is a more complex system than the neutral fluid. In plasmas there are at least two fluids, electrons and ions, which cause great number of instabilities. Microinstabilities cause fluctuations of electric and magnetic fields which in its turn cause fluctuations in velocities and particle positions therefore microinstabilities have an influence on transport. Turbulence is induced by incoherent motion appearing from instabilities. It is rather frequent phenomenon in plasma experiments. Observations show that plasma is a fluctuating medium in all its parameters such as density, magnetic field, potential and temperature. Various instabilities that cause turbulence present in various regions of plasma with different characteristics: SOL, edge and core.
Drift wave microturbulence is considered nowadays to be the main source of anomalous transport in tokamak which usually results in loss of heat much faster than it is predicted by neoclassical approach. In figure 1.9. a comparison between neoclassical and turbulence thermodiffusional coefficients is shown.

Bohm or Gyro‐Bohm (drift wave) scaling for turbulence

In absence of a fundamental, first‐principles turbulence theory, heuristic, mixing length rules are often utilized to estimate size scaling of turbulent transport [64]. This approach invokes a random walk type of picture for diffusive processes using the scale length of turbulent eddies as the step size and the linear growth time of the instability as the step time. It predicts that if the eddy size increases with device size, the transport scaling is Bohm‐like, i.e., local ion heat diffusivity is given as: B cT (9) eB.
On the other hand, if the eddy size is microscopic (on the order of the ion gyroradius), the transport scaling is gyro‐Bohm, i.e., local ion heat diffusivity is given as: GB*B (10).
where * i a is ion gyroradius i normalized by the tokamak minor radius a. There is a long history of confinement scaling studies that have correlated the thermal and/or particle confinement with either Bohm or drift wave scaling laws. The issue is still actively debated as to which transport scaling is to occur under given confinement conditions [65].

Theoretical description of the turbulence wave number spectrum

A better understanding of turbulence transport requires precise comparison between experimental observation and theory. Macroscopic effects give general information on turbulent motion. It is clear that only macroscopic parameters or characteristics without detailed investigation of wave number and frequency spectra and oscillation amplitude do not allow to determine the exact type of turbulent motion which is in charge of given microscopic phenomenon. The turbulence energy spectrum function n2 describing fluctuation energy repartition over different spatial scales contains information on characters of underlying instabilities and mechanisms involved in energy transfer between different scales. Energy transfer towards smaller scales is called the direct cascade, towards larger scales it is called the inverse cascade. The wave number spectrum is the one of the few quantities that can be measured in a tokamak and allows a highly detailed comparison between experiment and theory [66].
Several models describing turbulence spectral characteristics exist: the dressed test particle model of fluctuations in plasma near equilibrium, 2D fluid turbulence and 3D model [67]. In this work we consider the 2D model as soon as the simplest fluid model in the first approximation gives a good description of turbulence behavior in plasmas.
Well known 3D Kolmogorov’s theory of high Reynolds number turbulence (K41 theory) gives the spectrum scaling of the direct cascade 5 3 [68, 69]. However, the behavior of the spectrum is dimensionally dependent. In magnetically confined toroidal plasmas the magnetic field B has two components: a toroidal component Bt produced by toroidal field coils and a poloidal component B produced by a toroidal plasma current. At first approximation plasma turbulence moving perpendicular to the magnetic field can be considered as two‐dimensional in poloidal cross section of the tokamak supposing central symmetry. Experimentally, a 2D fluid is realized by a thin but wide layer where movements are mainly horizontal.
In this work Kraichnan‐Leith‐Batchelor (KLB) model of statistically stationary forced homogeneous isotropic 2D turbulence is considered [69]. This theory predicts existence of two inertial ranges: an energy inertial range with an energy spectrum scaling of 5 3 and an enstrophy inertial range with an energy spectrum scaling 3 . The existence of two conserved quantities complicates the construction of theory. Energy and enstrophy are injected into the flow by some external forcing at some intermediate wave number range min f max . The most of energy transfers towards low and forms the inverse cascade, the most of enstrophy transfers downscale towards high and is called the enstrophy cascade of direct cascade. Energy dissipates at large scale due to friction between the box size vortices and the boundary, the enstrophy dissipates at small scales due to molecular viscosity [71]. The inverse enstrophy and forward energy cascades are neglected however in reality there are small fractions of upscale enstrophy flux and downscale energy flux.

Cold plasma approximation

At present, most fusion experiments operate at plasma temperature below 5 keV. Waves traveling at phase velocities close to the speed of light are concerned. In this case the thermal velocity of electrons is much less than the phase velocity Te ph 1 . Cold plasma approximation is used to describe the propagation of most electromagnetic waves in tokamak plasma. The meaning of the approximation is that thermal motion of particles is neglected comparing to the motion caused by propagating electromagnetic wave [113, 114].
It is also assumed that there is no collisional damping (or Landau damping) on the time scale of plasma electrons as required for cold plasma approximation. Electrons are initially considered at rest, except for movement induced by wave fields.

High frequencies

Ion and neutral particle motion is neglected as well due the relation me mi 1 as soon as high frequency electromagnetic waves ci are studied. Only electrons contribute to the plasma dielectric tensor over the time of flight.

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We suppose the anisotropy is introduced only by external magnetic field B0 . In this work inhomogeneous anisotropic plasma is considered where refractive index depends on the propagation direction.

Propagationg waves

The electromagnetic wave propagating into plasma is supposed to be monochromatic. It could be described in usual way: E E 0 exp( i t) (30).
where2 f is the microwave angular frequency of the wave. The phase velocity, ph k gives the rate of propagation of a point of constant phase on the wave. If the wave frequency or amplitude is modulated the wave possesses the group velocity gr k . The dispersion relation ( k) contains information on phase and group velocities, propagation region, reflection points, resonance points, damping, wave growth. Another property of the electromagnetic wave is polarization which is defined by the orientation and phase of the electric field of the wave E . There are three types of polarization: linear, circular and elliptical.

Linear approximation

As soon as restrictions of small‐amplitude waves are imposed it is possible to apply linear theory of perturbations. All the perturbations f of the quantity f in this work are assumed to be small as well f f 1 . This permits to use linear relations to describe wave propagation in plasma knowing that the input power of reflectometer is not able to modify the background plasma parameters.

Propagation in homogeneous plasma

Taking into account approximations introduced in 2.1.1., we consider plane wave propagation in the uniform and homogeneous plasma in external magnetic field B0 . According to linear approximation we perform Fourier analysis of Maxwell equations. By taking the curl of the eq. (22) and combining it with the eq. (23) and transforming operators ik and t i we obtain the wave equation (Helmholtz equation): 2 k ( k E ) E 0 (31). where is the dielectric tensor related to the conductivity as follows: 4 1 (32) i Eq. (31) can be rewritten in a form: c NN N2 1 E 0 (33).

Table of contents :

I. Introduction
1.1. The world energy problem
1.2. Nuclear fusion: energy source for the future
1.3. The tokamak
1.3.1. Tokamaks in this work Tore Supra FT‐2 JET ITER Main parameters of machines mentioned in this work
1.4. Turbulence in fusion plasma
1.4.1. How fluctuations cause anomalous transport
1.4.2. Bohm or Gyro‐Bohm (drift wave) scaling for turbulence
1.4.3. Theoretical description of the turbulence wave number spectrum
1.4.4. Examples of turbulence wave number spectra
1.4.5. Turbulence suppression Radial electric field shear Zonal Flows
1.5. Turbulence diagnostics
1.6. Radial correlation reflectometry
1.7. Scope of this work
II. Theoretical background of radial correlation reflectometry
2.1. Propagation of electromagnetic waves in plasmas
2.1.1. Approximations and restrictions used Stationary plasma Cold plasma approximation High frequencies Anisotropy Propagationg waves Linear approximation
2.1.2. Propagation in homogeneous plasma Perpendicular propagation
2.1.3. Propagation in inhomogeneous plasma Wentzel – Kramers – Brillouin approximation
2.2. Plasma density fluctuations
2.3. Mechanism of back and forward Bragg scattering
2.4. Reflectometry principles
2.4.1. Standard reflectometry for plasma density profile masurements
2.4.2. Fluctuation reflectometry
2.5. Basic assumptions and equations in 1D analysis
2.5.1. Reciprocity theorem
2.6. Scattering signal in case of linear plasma density profile
2.6.1. Asymptotic forms of the characteristic integral Contribution of the pole Contribution of the branch point Contribution of the stationary phase points
2.6.2. Asymptotic forms of scattering signal
2.6.3. Numerical computation example
2.6.4. WKB representation of Airy function
2.6.5. Long wavelength limit
2.7. Scattering signal in case of arbitrary plasma density profile
2.7.1. Numerical computation example for parabolic plasma density profile
2.7.2. Short summary on validity domain of Helmholtz equation solutions
2.8. The RCR CCF
2.8.1. RCR CCF for linear plasma density profile
2.8.2. RCR CCF for arbitrary plasma density profile
2.9. Turbulence spectrum reconstruction from the RCR CCF
2.10. Direct transform formulae for RCR
2.10.1. Forward transformation kernel
2.10.2. Numerical simulation example of forward kernel usage
2.10.3. Inverse transformation kernel
2.11. Ideas for a combined diagnostic using reflectometry and other density fluctuation diagnostic
2.11.1. Forward and inverse transforms for ICF
2.12. Summary
III. Numerical modeling
3.1. Numerical model
3.1.1. Numerical solution of unperturbed Helmholtz equation
3.1.2. Reflectometry signal partial amplitude integral computation
3.1.3. Signal CCF computation
3.1.4. Turbulence wave number spectrum and TCCF reconstruction
3.2. O‐mode probing in case of linear plasma density profile
3.2.1. Reconstruction of turbulence spectrum and CCF for large machine CCF and spectrum reconstruction in conditions relevant to experiment
3.2.2. Reconstruction of the turbulence spectrum and CCF for small machine. Standard conditions of reconstruction at FT‐2 Optimized reconstruction in more realistic conditions
3.2.3. Amplitude CCF computation
3.2.4. Inhomogeneous turbulence
3.3. O‐mode probing in case of density profile close to experimental one
3.3.1. Tore Supra – like plasma density profile
3.3.2. Plasma density profile with a steep gradient
3.4. Synthetic X‐mode RCR experiment
3.5. Summary
IV. Applications to experiments
4.1. General remarks on data analysis
4.1.1. Reflectometer generic scheme
4.1.2. Quadrature phase detection
4.1.3. Probing range and step
4.1.4. Statistical analysis
4.2. Results of RCR experiment at Tore Supra
4.2.1. Reflectometers at Tore Supra
4.2.2. Phase calibration
4.2.3. Data analysis and interpretation Probing with equidistant spatial step Probing with exponentially growing spatial step
4.2.4. Summary
4.3. Experimental results obtained at FT‐2 tokamak
4.3.1. Radial correlation reflectometers at FT‐2
4.3.2. O‐mode probing from HFS
4.3.3. X‐mode probing from HFS
4.3.4. Summary
4.4. Results of experimental campaign at JET
4.4.1. RCR diagnostic at JET
4.4.2. Experimental results Shot #82671 data analysis Shot #82633 data analysis
4.4.3. Summary
Future plans
Appendix A. Stationary phase method
Appendix B. 4th order Numerov scheme


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