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Linear stability analysis of force-free equilibria close to Taylor relaxed states
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View: Figures


Image of FIG. 1.
FIG. 1.

Safety profile for three stepped- equilibria corresponding to (dashed line), (solid line), and (thick solid line). The three profiles belong to the same family of equilibria originated by an initial Taylor state characterized by and .

Image of FIG. 2.
FIG. 2.

Comparison between current density profiles of a stepped- equilibrium (solid line) with and of the corresponding initial Taylor state (dashed line) characterized by and . (a) refers to the poloidal component of , whereas (b) refers to the toroidal component.

Image of FIG. 3.
FIG. 3.

(a), (b) Values of and , respectively, as functions of during an evolution from an initial Taylor state with and .

Image of FIG. 4.
FIG. 4.

The figure shows the relations, as given by the formula (16), between the resonance radii and the values of for the modes (1,6), (1,7), and (1,8), assuming . The aspect ratio is equal to 4.34. The intersections of the curves for and with determine the minimum and maximum admissible values of , respectively, in order to have a resonance with the mode (1,7) and to exclude the resonance with modes with and .

Image of FIG. 5.
FIG. 5.

The plot shows an example of safety profile for a stepped- equilibrium, near marginal stability, with , , , and . This equilibrium resonates to the right of the step with the mode (1,8), inside the step with the mode (1,7) and does not resonate with the mode (1,6).

Image of FIG. 6.
FIG. 6.

Plot comparing the stability parameter as a function of for a Taylor (dashed line) and stepped- (solid line) equilibrium for modes with . The presence of the step destabilizes the modes resonating to the left of the step while keeping the other modes stable. In particular, for the aspect ratio under consideration, i.e., , the mode (1,7) resonates at and the corresponding value of is equal to 0.12, i.e., just above the marginal stability condition. The values of the parameters are those of Fig. 5.

Image of FIG. 7.
FIG. 7.

The plot shows the dependence of the stability parameter on for (solid line) and (dashed line). The mode (, ) in both cases resonates at . Increasing the distance of the step from results in a higher value of required in order to make the mode (, ) unstable. In both cases the value of goes to infinity for some critical value of .

Image of FIG. 8.
FIG. 8.

The plot shows the stability parameter as a function of for modes with (solid), (dashed), and (thick). In all cases is negative over almost the entire domain, thus implying stability. The only region of potential instability is located just to the left of the step. For the given aspect ratio, however, none of the modes under consideration resonates in that region. Parameters are as in Fig. 5.

Image of FIG. 9.
FIG. 9.

The plot refers to an example of ideally unstable equilibrium. The eigenfunction has a zero between 0 and 1 and the eigenfunction satisfies the boundary condition at . and have been plotted with a solid line for and for , respectively, and with a dashed line elsewhere. One can then see from the slopes of the solid lines at (value of chosen for this example) that the difference in the logarithmic derivative is positive, which indicates instability. The plot refers to (, ) perturbations of a Taylor equilibrium with .

Image of FIG. 10.
FIG. 10.

Plot showing the quantity for the stepped- equilibrium characterized by the values of the parameters given in Fig. 5. The quantity is negative over the entire range of values of nonresonant wave numbers , thus indicating ideal stability of the equilibrium.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Linear stability analysis of force-free equilibria close to Taylor relaxed states