We introduce the concept of the local wavenumber and frequency spectral density, which can be estimated using spatially fixed, point data sources (’’fixed probe pairs’’), and discuss the relationship of this spectral density to the conventional wavenumber and frequency spectral density and the cross‐power spectral density. The local wavenumber and frequency spectral density is shown to be equivalent to the conventional wavenumber and frequency spectral density when the fluctuation is stationary and homogeneous and consists of a superposition of wave packets; such a fluctuation is the basic model used in many turbulence theories. A digital method for estimating the local wavenumber spectrum is described and applied to the study of drift‐wave turbulence in an rf‐excited discharge. The statistical dispersion relation and wavenumber spectral width, computed from the local wavenumber and frequency spectrum of the drift‐wave turbulence, are compared with the conventional spectral moments computed using the correlation method of Iwama and Tsukishima; good agreement is found over a wide range of frequency. A frequency‐integrated wavenumber spectrum is computed; both frequency and wavenumber spectral indices are found independently. The local wavenumber and frequency spectrum is a completely new approach to the use of fixed‐probe data, and we believe it can greatly extend the quantity of information available from fixed probes, which are the principle tools in many, if not most, fluctuation experiments.

Estimation of wavenumber and frequency spectra using fixed probe pairs

A four‐grid electrostatic energy analyzer for measurements of the ion velocity distribution and the emission of secondary electrons on the electrodes of low‐pressure radio frequency glow‐discharge systems has been conceived. Problems arising from poor analyzer design are discussed and the performance of the presented analyzer is shown for measurements of the ion velocity distribution in pure hydrogen, helium, and argon discharges. Moreover, the secondary electron yields on aluminium, stainless steel, copper, and amorphous silicon exposed to radio frequency argon, helium, and hydrogen plasmas are determined in situ, for the first time to our knowledge. In parallel‐plate radio frequency discharges secondary electron emission involves the contributions of ions, fast neutrals, metastables, and photons impinging on the electrode surfaces. Therefore, secondary electron emission must be considered as a global phenomenon. Global secondary electron emission can be up to ten times larger than only ion‐induced secon...

Retarding‐field analyzer for measurements of ion energy distributions and secondary electron emission coefficients in low‐pressure radio frequency discharges

https://www.acert.cornell.edu/PDFs/JMagnReson177_184_2005.pdf

Maximum entropy: A complement to Tikhonov regularization for determination of pair distance distributions by pulsed ESR

A method is discussed for reconstruction of the emissivity profile from projections. The emissivity profile is expanded into Fourier–Bessel expansions, of which coefficients are determined by the use of least‐square‐fitting method with the help of the Akaike information criterion. Tomographic reconstruction of the m=1 mode structure is obtained in JIPP T‐II tokamak during the precursor oscillation of the sawtooth crash. The m=1 mode structure fits Wesson model rather than the Kadomtsev model.

Tomography of m=1 mode structure in tokamak plasma using least‐square‐fitting method and Fourier–Bessel expansions

This chapter reviews multichannel broadband measurement of the soft-X-ray radiation and total radiation in magnetically confined fusion plasma experiments Common detector types used (including bolometers), details of their application, and interpretation of their measurements are described An introduction is given to the application of computed tomography methods in the mathematical reconstruction of emission profiles from multiple (approximately) line-integral measurements, taking into account the specific circumstances common in magnetically confined fusion plasma experiments Although the emphasis is on two-dimensional tomography of poloidal cross sections, the applications of Abel inversion, three-dimensional tomography, vector tomography, and other specific methods are briefly discussed Several examples of the application and the plasma parameters that can be derived are given

Chapter 7: Tomography Diagnostics: Bolometry and Soft-X-Ray Detection

The use of the Phillips–Tikhonov regularization is proposed for numerically stabilizing the ill‐conditioned plasma image reconstructions. An objective function to be minimized leads to a linear estimator of the image intensity distribution and, with the aid of the singular value decomposition, makes it possible to use the generalized cross validation for optimizing a regularization parameter. An excellent behavior of the estimator with computational facility is obtained on the Hα emission computerized tomography of a toroidal plasma.

Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation

Spectrum parameter estimation concerning a stationary and homogeneous random field is examined in terms of quantities obtainable from a pair of fixed probes. The estimators of the mean wave number and the wave‐number spectral width are derived by relating wave‐number spectral moments to spatial derivatives of the complex covariance of the field, the effect of finite probe size being taken into account. The estimate biases associated with the probe separation can be greatly reduced by employing an algorithm based on the polar‐form representation of the complex covariance, while the variances of the estimates have a tendency to be enhanced with the decrease of the probe separation. An optimum range of the probe separation is theoretically determined.

Estimation of wave‐number spectrum parameters from fixed probe‐pair data

The covariance function of a stationary and homogeneous random field, represented by an analytic signal, is related to its wave‐number spectrum moments. By using a pair of fixed probes with an appropriately small separation, one can estimate the mean wave number and the wave‐number spectral spread. This method may be useful for the experimental study of fluctuation phenomena in plasmas, fluids, etc.

A correlation method for estimating wave‐number spectrum moments of fluctuations

A study of computerized tomography is made for the purpose of reconstructing the ion velocity distribution function of plasma in the three‐dimensional velocity space. A conventional type of ion energy analyzer with retarding grids is rotated in a plasma in order to acquire the data to be processed. Experimental results are presented on an argon plasma with a low‐energy ion beam, which is generated in a double‐plasma machine.

Three‐dimensional reconstruction of the ion velocity distribution function of plasma with a retarding‐grid analyzer

To measure the electron temperature profile of tokamak plasma in a high confinement mode with edge localized mode (ELMy H-mode), a method of data analysis is presented on the fast-scanning Fourier transform spectrometer (FTS), and results in the JT-60U tokamak are described. In the ELMy H-mode, where the FTS interferogram of electron cyclotron emission (ECE) becomes incomplete due to nonthermal emission pulses of ELM, the spectrum of ECE is estimated by using the maximum entropy method (MEM) of nonparametric type. The optimization of the regularization parameter is studied in terms of adopting the generalized cross validation (GCV) from the linear regularization of Tikhonov–Phillips. A study is also performed on a method to remove the nonthermal emission pulses from interferogram data before the MEM analysis, and on an extension of minimum cross entropy in order to meet with the missing of interferogram values in the region of zero path difference. From the application to experimental data of many shots, it is found that the electron temperature profiles can be automatically measured using the FTS during the ELMy H-mode operation of the tokamak.

Naofumi Iwama

Papers

Phillips–Tikhonov regularization of plasma image reconstruction with the generalized cross validation

Estimation of wave‐number spectrum parameters from fixed probe‐pair data

A correlation method for estimating wave‐number spectrum moments of fluctuations

Three‐dimensional reconstruction of the ion velocity distribution function of plasma with a retarding‐grid analyzer

Maximum Entropy Estimation of Electron Cyclotron Emission Spectra from Incomplete Interferograms in ELMy H-mode Tokamak Experiment