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Torus Instability and Coronal Mass Ejections

Stability of various forms of a toroidal magnetic configuration is an important issue in many astrophysical and laboratory phenomena. The central quantity in analyzing MHD instabilities of astrophysical and laboratory toroidal magnetic structures is the self-inductivity $L$, of the longitudinal current $I$, since it provides calculation of the hoop force. In stability analyzes, the analytical expression for $L$, obtained by applying the slender-torus approximation, is usually used. Since in real astrophysical and laboratory situations the slender-torus approximation is not really appropriate, we investigate numerically the self- inductivity of a torus 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). We evaluate $L$ by calculating the magnetic flux $\Psi$ associated with the current, and utilizing $\Psi=LI$. The total flux is presented as a sum of the flux outside the torus envelope ($\Psi_o$) and the internal flux within the torus body ($\Psi_i$), so the total inductivity is also expressed as $L=L_o+L_i$. The procedure is performed for three different radial profiles of the current density, $j(r)$. It is found that the inductivity $L_o$ only very weakly depends on the form of $j(r)$. On the other hand, $L_i$ does not depend on $\eta$, but depends 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 of 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 flux inside the torus envelope becomes larger than that outside the envelope, $L_i$ becomes larger than $L_o$. The values of total inductivity $L_o+L_i$, calculated by appropriately employing our functions $L_o^{;fit1};$ and $L_o^{;fit2};$, never departure for more than 1\, \% from the numerically determined values of $L$.

MHD Instability ; CMEs ; Solar Physics

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