Abstract
The linear stability of a class of forcefree equilibria in cylindrical geometry is investigated. The class consists of cylindrically symmetric forcefree equilibria for which the ratio between the parallel current density and the magnetic field is a step function of the radius. It is suggested that plasmas in reversed field pinches could be roughly represented by such equilibria as a consequence of a small departure from an initial forcefree state with constant , the latter being reached after a relaxation process according to the classical theory proposed by Taylor [Phys. Rev. Lett.33, 1139 (1974)]. A fully analytical derivation of the tearing stability parameter for such class of equilibria is given. It is then shown with one explicit example how the presence of a downward step of relatively small height can destabilize the innermost resonant mode, which would otherwise be stable if were constant. A possible implication of this mechanism for the formation of cyclic quasisingle helicity states observed in reversed field pinches is proposed. Considerations on the ideal stability of the class of equilibria under investigation are also given.
We would like to acknowledge discussions with several RFX colleagues, and in particular D. Bonfiglio, S. Cappello, D. F. Escande, P. Martin, S. Ortolani, and R. Paccagnella. This work was partly supported by the Istituto Gas Ionizzati (IGI) of the Consiglio Nazionale delle Ricerche in the framework a FIRB collaboration. This work was also partly supported by the Euratom Communities under the contract of association between EURATOM/ENEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
I. INTRODUCTION
II. FORCEFREE EQUILIBRIA WITH STEPPED PROFILE
III. LINEAR STABILITY
IV. CONCLUSIONS
Key Topics
 Toroidal plasma confinement
 14.0
 Reversed field pinch
 11.0
 Magnetic islands
 10.0
 Current density
 9.0
 Magnetic fields
 9.0
Figures
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 .
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 .
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.
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.
(a), (b) Values of and , respectively, as functions of during an evolution from an initial Taylor state with and .
(a), (b) Values of and , respectively, as functions of during an evolution from an initial Taylor state with and .
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 .
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 .
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).
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).
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.
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.
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 .
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 .
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.
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.
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 .
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 .
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.
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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