Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T14:37:09.240Z Has data issue: false hasContentIssue false
  • English
  • Français

The magnetic flux and self-inductivity of a thick toroidal current

Published online by Cambridge University Press:  01 October 2007

TOMISLAV ŽIC ,
BOJAN VRŠNAK  and
MARINA SKENDER
TOMISLAV ŽIC
Affiliation:
Hvar Observatory, Faculty of Geodesy, Kačićeva 26, HR-10000 Zagreb, Croatia ([email protected])
BOJAN VRŠNAK
Affiliation:
Hvar Observatory, Faculty of Geodesy, Kačićeva 26, HR-10000 Zagreb, Croatia ([email protected])
MARINA SKENDER
Affiliation:
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany Rudjer Bošković Institute, Bijenička 54, HR-10001 Zagreb, Croatia
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate numerically the magnetic flux and self-inductivity of a toroidal current I of arbitrary aspect ratio (R0/r0 = 1/η, where R0 and r0 are the major and the minor torus radii, respectively). The total flux Ψ is represented by the sum of the flux outside the torus envelope (Ψo) and the internal flux within the torus body (Ψi). Analogously, the total inductivity is expressed as L = Lo + Li. The outside self-inductivity is determined directly from the magnetic flux Ψo, utilizing Ψo = LoI. On the other hand, the internal inductivity is evaluated as the magnetic energy contained in the poloidal field. The calculations are performed for three different radial profiles of the current density, j(r).

It is found that Ψo(η) and Lo (η) depend only very weakly on the form of j(r). On the other hand, Ψi and Li do not depend on η, but depend on the form of j(r). In the range 0.02 ≲ η ≲ 0.5, the numerical values of Lo can be very well fitted by the function of the form Lofit1(η) = −A log(η) − B. Such a relation is analogous to that for a slender torus, although the coefficients are different. For η ≲ 0.01 the slender-torus approximation (Lo*) matches the numerical results better than our function Lofit1, whereas for thicker tori, Lofit1 becomes more appropriate. It is shown that, beyond η ≳ 0.1, the departure of the slender-torus analytical expression from the numerical values becomes greater than 10%, and the difference becomes larger than 100% at η 0.55. In the range η 0.5, the numerical values of Lo can be very well expressed by the function Lofit2(η)=c1 (1 − η)c2. Furthermore, since the internal flux and inductivity become larger than that outside the envelope, Ψi and Li become larger than Ψo and Lo. The total inductivity Ltotfit = Lofit + Li, calculated by appropriately employing our functions Lofit1 and Lofit2, never deviates by more than 1% from the numerically determined values of Ltot.

Type
Papers
Information
Journal of Plasma Physics , Volume 73 , Issue 5 , October 2007 , pp. 741 - 756
Copyright
Copyright © Cambridge University Press 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bellan, P. M., Yee, J. and Hansen, J. F. 2001 Spheromaks, solar prominences, and Alfvén instability of current sheets. Earth, Planets, and Space 53, 495–499.CrossRefGoogle Scholar
[2]Braithwaite, J. 2006 The stability of toroidal fields in stars. Astron. Astrophys. 453, 687.CrossRefGoogle Scholar
[3]Lovelace, R. V. E., Romanova, M. M. and Biermann, P. L. 1998 Magnetically supported tori in active galactic nuclei. Astron. Astrophys. 338, 856–862.Google Scholar
[4]Chen, J. 1989 Effects of toroidal forces in current loops embedded in a background plasma. Astrophys. J. 338, 453–470.CrossRefGoogle Scholar
[5]Chen, J. and Krall, J. 2003 Acceleration of coronal mass ejections. J. Geophys. Res. 108, 1410.Google Scholar
[6]Titov, V. S. and Démoulin, P. 1999 Basic topology of twisted magnetic configurations in solar flares. Astron. Astrophys. 351, 707.Google Scholar
[7]Vršnak, B. 1990 Eruptive instability of cylindrical prominences. Solar Phys. 129, 295.CrossRefGoogle Scholar
[8]Vršnak, B., Ruždjak, V. and Rompolt, B. 1991 Stability of prominences exposing helical-like patterns. Solar Phys. 136, 151–167.CrossRefGoogle Scholar
[9]Williams, D. R., Török, T., Démoulin, P., Driel-Gesztelyi, L. van and Kliem, B. 2005 Eruption of a kink-unstable filament in NOAA active region 10696. Astrophys. J. 628, L163–L166.CrossRefGoogle Scholar
[10]Birn, J., Forbes, T. G., and Hesse, M. 2006 Stability and dynamic evolution of three-dimensional flux ropes. Astrophys. J. 645, 732.CrossRefGoogle Scholar
[11]Garren, D. A. and Chen, J. 1994 Lorentz self-forces on curved current loops. Phys. Plasmas 1, 3425.CrossRefGoogle Scholar
[12]Hansen, J. F. and Bellan, P. M. 2001 Experimental demonstration of how strapping fields can inhibit solar prominence eruptions. Astrophys. J. 563, L183.CrossRefGoogle Scholar
[13]Kliem, B. and Török, T. 2006 The torus instability. Phys. Rev. Lett. 96, 255 002.CrossRefGoogle ScholarPubMed
[14]Shafranov, V. D. 1966 Plasma equilibrium in a magnetic field. Rev. Plasma Phys. 2, 103.Google Scholar
[15]Priest, E. R. 1982 Solar Magneto-hydrodynamics. Dordrecht: Reidel.CrossRefGoogle Scholar
[16]Burlaga, L. F. 1998 Magnetic clouds and force-free fields with constant alpha. J. Geophys. Res. 93, 7217.CrossRefGoogle Scholar
[17]Batygin, V. V. and Toptygin, I. N. 1962 Problems in Electrodynamics. New York: Academic Press.Google Scholar
[18]Lin, J., Forbes, T. G., Isenberg, P. A. and Démoulin, P. 1998 The effect of curvature on flux-rope models of coronal mass ejections. Astrophys. J. 504, 1006–1019.CrossRefGoogle Scholar
[19]Chen, J. 1996 Theory of prominence eruption and propagation: interplanetary consequences. J. Geophys. Res. 101, 27499.CrossRefGoogle Scholar
[20]Bothmer, V. and Schwenn, R. 1998 The structure and origin of magnetic clouds in the solar wind. Ann. Geophysicae 16, 1–24.CrossRefGoogle Scholar