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

M. Van Berkel, N. Tamura, G. M.D. Hogeweij, H. J. Zwart, S. Inagaki, M. R. De Baar, K. Ida

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) 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.

Original languageEnglish
Article number4901311
JournalPhysics of Plasmas
Volume21
Issue number11
DOIs
Publication statusPublished - Nov 1 2014

Fingerprint

traveling waves
diffusivity
damping
heat
estimates
approximation
transport properties
convection
geometry
hypergeometric functions
Bessel functions
slabs
boundary conditions
harmonics
thermodynamics
radii

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics

Cite this

Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. III. Cylindrical approximations for heat waves traveling inwards. / Van Berkel, M.; Tamura, N.; Hogeweij, G. M.D.; Zwart, H. J.; Inagaki, S.; De Baar, M. R.; Ida, K.

In: Physics of Plasmas, Vol. 21, No. 11, 4901311, 01.11.2014.

Research output: Contribution to journalArticle

@article{d0c0148eefe64338a4cc74e636254bbc,
title = "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 (χ), convectivity (V), and damping (τ) 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.",
author = "{Van Berkel}, M. and N. Tamura and Hogeweij, {G. M.D.} and Zwart, {H. J.} and S. Inagaki and {De Baar}, {M. R.} and K. Ida",
year = "2014",
month = "11",
day = "1",
doi = "10.1063/1.4901311",
language = "English",
volume = "21",
journal = "Physics of Plasmas",
issn = "1070-664X",
publisher = "American Institute of Physics Publising LLC",
number = "11",

}

TY - JOUR

T1 - Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. III. Cylindrical approximations for heat waves traveling inwards

AU - Van Berkel, M.

AU - Tamura, N.

AU - Hogeweij, G. M.D.

AU - Zwart, H. J.

AU - Inagaki, S.

AU - De Baar, M. R.

AU - Ida, K.

PY - 2014/11/1

Y1 - 2014/11/1

N2 - In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) 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.

AB - In this paper, a number of new explicit approximations are introduced to estimate the perturbative diffusivity (χ), convectivity (V), and damping (τ) 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.

UR - http://www.scopus.com/inward/record.url?scp=84912570015&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84912570015&partnerID=8YFLogxK

U2 - 10.1063/1.4901311

DO - 10.1063/1.4901311

M3 - Article

AN - SCOPUS:84912570015

VL - 21

JO - Physics of Plasmas

JF - Physics of Plasmas

SN - 1070-664X

IS - 11

M1 - 4901311

ER -