Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. II. Semi-infinite cylindrical approximations

Abstract

In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (v), convectivity (V), and damping (s) in a cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based upon the heat equation in a semi-infinite cylindrical domain. The approximations are based upon continued fractions, asymptotic expansions, and multiple harmonics. The relative error for the different derived approximations is presented for different values of frequency, transport coefficients, and dimensionless radius. Moreover, it is shown how combinations of different explicit formulas can yield good approximations over a wide parameter space for different cases, such as no convection and damping, only damping, and both convection and damping. This paper is the second part (Part II) of a series of three papers. In Part I, the semi-infinite slab approximations have been treated. In Part III, cylindrical approximations are treated for heat waves traveling towards the center of the plasma.

Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. III. Cylindrical approximations for heat waves traveling inwards

“…If V ¼ 0, (2) can be simplified in terms of modified Bessel functions I and K of order ¼ 0 resulting in Refs. 6 and 9…”

“…From the literature, it is well known that these ratios of transcendental functions can be approximated by truncation of their continued fraction representation. 2,11,12 Based on this concept, a number of new approximations are derived, which are summarized in three tables in Sec. IV, and their derivations can be found in Appendix.…”

Method for Estimating Harmonic Frequency Dependence of Diffusion Coefficient and Convective Velocity in Heat Pulse Propagation Experiment

“…When the radius of interest for the analysis is not far enough from the boundaries, the obtained transport coefficients can also suffer the boundary effect. The effective distance to the boundaries may depend on the wavelength of the heat pulse, and thus inversely depend on the frequency [19]. Therefore, the estimation error is expected to decrease at the higher harmonic frequency.…”

“…There are several models to estimate the transport coefficients from the radial amplitude decay and the phase difference of the temperature perturbation [17][18][19][20].…”