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

Abstract

In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (v), convectivity (V), and damping (s) in cylindrical geometry. For this purpose, the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on the heat equation in cylindrical geometry using the symmetry (Neumann) boundary condition at the plasma center. This means that the approximations derived here should be used only to estimate transport coefficients between the plasma center and the off-axis perturbative source. If the effect of cylindrical geometry is small, it is also possible to use semi-infinite domain approximations presented in Part I and Part II of this series. A number of new approximations are derived in this part, Part III, based upon continued fractions of the modified Bessel function of the first kind and the confluent hypergeometric function of the first kind. These approximations together with the approximations based on semi-infinite domains are compared for heat waves traveling towards the center. The relative error for the different derived approximations is presented for different values of the frequency, transport coefficients, and dimensionless radius. Moreover, it is shown how combinations of different explicit formulas can be used to estimate the transport coefficients over a large parameter range for cases without convection and damping, cases with damping only, and cases with convection and damping. The relative error between the approximation and its underlying model is below 2% for the case, where only diffusivity and damping are considered. If also convectivity is considered, the diffusivity can be estimated well in a large region, but there is also a large region in which no suitable approximation is found. This paper is the third part (Part III) of a series of three papers. In Part I, the semi-infinite slab approximations have been treated. In Part II, cylindrical approximations are treated for heat waves traveling towards the plasma edge assuming a semi-infinite domain.

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

“…The solution for the semi-infinite domain approximations of a third order polynomial, i.e., (20), is always given by (22). This is because the argument of z of the correct solution must be situated in [0, p/4].…”

“…In Part III, approximations to estimate v are derived and discussed for heat waves traveling towards the center (inwards) using either semi-infinite domain assumptions or a symmetry boundary condition. 22 The latter is the natural boundary condition for a cylindrical geometry.…”

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

“…Note that the coefficients of Eqs. ( 13) and (20) do not contain the effect of χ HP , V HP nor τ . Therefore, the prediction of the wavenumber becomes inaccurate when large values of χ HP , V HP or τ −1 exist.…”

“…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].…”