Cold plasma approximation

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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
1.3.1.1. Tore Supra
1.3.1.2. FT‐2
1.3.1.3. JET
1.3.1.4. ITER
1.3.1.5. 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
1.4.5.1. Radial electric field shear
1.4.5.2. 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
2.1.1.1. Stationary plasma
2.1.1.2. Cold plasma approximation
2.1.1.3. High frequencies
2.1.1.4. Anisotropy
2.1.1.5. Propagationg waves
2.1.1.6. Linear approximation
2.1.2. Propagation in homogeneous plasma
2.1.2.1. Perpendicular propagation
2.1.3. Propagation in inhomogeneous plasma
2.1.3.1. 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
2.6.1.1. Contribution of the pole
2.6.1.2. Contribution of the branch point
2.6.1.3. 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
3.2.1.2. CCF and spectrum reconstruction in conditions relevant to experiment
3.2.2. Reconstruction of the turbulence spectrum and CCF for small machine.
3.2.2.1. Standard conditions of reconstruction at FT‐2
3.2.2.2. 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
4.2.3.1. Probing with equidistant spatial step
4.2.3.2. 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
4.4.2.1. Shot #82671 data analysis
4.4.2.2. Shot #82633 data analysis
4.4.3. Summary
Conclusion
Future plans
Appendix
Appendix A. Stationary phase method
Appendix B. 4th order Numerov scheme
References