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

Žic, Tomislav; Vršnak, Bojan; Skender, Marina
The magnetic flux and self-inductivity of a thick toroidal current // Journal of Plasma Physics, 73 (2007), 5; 741-756 (međunarodna recenzija, članak, znanstveni)

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

Autori
Žic, Tomislav ; Vršnak, Bojan ; Skender, Marina

Izvornik
Journal of Plasma Physics (0022-3778) 73 (2007), 5; 741-756

Ključne riječi
Plasma; magnetohydrodynamics

Sažetak
We investigate numerically the magnetic flux and self-inductivity of a toroidal current $I$ of arbitrary aspect ratio ($R_0/r_0 = 1/\eta$, where $R_0$ and $r_0$ are the major and the minor torus radius, respectively). The total flux $\Psi$ is represented by the sum of the flux outside the torus envelope ($\Psi_o$) and the internal flux within the torus body ($\Psi_i$). Analogously, the total inductivity is expressed as $L = L_o + L_i$. The outside self-inductivity is determined directly from the magnetic flux $\Psi_o$, utilizing $\Psi_o = L_o I$. 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 $\Psi_o(\eta)$ and $L_o(\eta)$ depend only very weakly on the form of $j(r)$. On the other hand, $\Psi_i$ and $L_i$ do not depend on $\eta$, but depend on the form of $j(r)$. In the range $0.02 \lessapprox \eta \lessapprox 0.5$ the numerical values of $L_o$ can be very well fitted by the function of the form $L_o^{; ; ; ; ; fit1}; ; ; ; ; (\eta) = -A \log(\eta) - B$. Such a relation is analogous to that for a slender-torus, however, the coefficients are different. For $\eta\lesssim0.01$ the slender-torus approximation ($L_o^*$) matches the numerical results better than our function $L_o^{; ; ; ; ; fit1}; ; ; ; ;$, whereas for thicker tori, $L_o^{; ; ; ; ; fit1}; ; ; ; ;$ becomes more appropriate. It is shown that beyond $\eta\gtrapprox 0.1$, the departure of the slender-torus analytical expression from the numerical values becomes greater than 10\, \%, and the difference becomes larger than 100\, \% at $\eta\approx 0.55$. In the range $\eta\gtrapprox 0.5$ the numerical values of $L_o$ can be very well expressed by the function $L_o^{; ; ; ; ; fit2}; ; ; ; ; (\eta)=c_1(1-\eta)^{; ; ; ; ; c_2}; ; ; ; ;$. Furthermore, since the internal flux and inductivity become larger than that outside the envelope, $\Psi_i$ and $L_i$ become larger than $\Psi_o$ and $L_o$. The total inductivity $L_{; ; ; ; ; tot}; ; ; ; ; ^{; ; ; ; ; fit}; ; ; ; ; =L_o^{; ; ; ; ; fit}; ; ; ; ; +L_i$, calculated by appropriately employing our functions $L_o^{; ; ; ; ; fit1}; ; ; ; ;$ and $L_o^{; ; ; ; ; fit2}; ; ; ; ;$, never deviates by more than 1\, \% from the numerically determined values of $L_{; ; ; ; ; tot}; ; ; ; ;$.

Izvorni jezik
Engleski

Znanstvena područja
Fizika

Projekt / tema
007-0000000-1362 - Eruptivni procesi u Sunčevoj atmosferi (Bojan Vršnak, )
098-0982930-2864 - Fundamentalne interakcije u fizici elementarnih čestica i kozmologiji (Goran Duplančić, )

Ustanove
Geodetski fakultet, Zagreb

#### Časopis indeksira:

• Current Contents Connect (CCC)
• Web of Science Core Collection (WoSCC)
• Science Citation Index Expanded (SCI-EXP)
• SCI-EXP, SSCI i/ili A&HCI
• Scopus