Abstract
A nonaxisymmetric stable magnetohydrodynamic(MHD) equilibrium within a prolate cylindrical conducting boundary has been produced experimentally at Swarthmore Spheromak Experiment (SSX) [M. R. Brown et al., Phys. Plasmas6, 1717 (1999)]. It has toroidal symmetry, helical distortion, and flat profile. Each of these observed characteristics are in agreement with the magnetically relaxed minimum magnetic energy Taylor state. The Taylor state is computed using the methods described by A. Bondeson et al. [Phys. Fluids24, 1682 (1981)] and by J. M. Finn et al. [Phys. Fluids24, 1336 (1981)] and is compared in detail to the measured internal magnetic structure. The lifetime of this nonaxisymmetric compact torus (CT) is comparable to or greater than that of the axisymmetric CTs produced at SSX; thus suggesting confinement is not degraded by its nonaxisymmetry. For both one and twospheromak initial state plasmas, this same equilibrium consistently emerges as the final state.
This research was supported by the U.S. Department of Energy grant No. DEFG0200ER54604 and Plasma Science and Innovation Center (PSICenter), the National Science Foundation Physics Frontier Center for Magnetic Self Organization (CMSO), and the Office of Naval Research.
I. INTRODUCTION
II. EXPERIMENT
III. TAYLOR STATES IN A CYLINDRICAL BOUNDARY
IV. RESULTS
V. DISCUSSION AND SUMMARY
Key Topics
 Spheromaks
 19.0
 Magnetohydrodynamics
 13.0
 Plasma diagnostics
 8.0
 Plasma temperature
 6.0
 Toroidal plasma confinement
 6.0
Figures
Sketch of the SSX device. Numbers indicate the following: (1) the cylindrical conducting boundary with ; (2) one of 12 linear magnetic probe arrays inserted radially into the device; and for one of the two plasma guns on each end of the device, the (3) inner and (4) outer coaxial electrodes linked by (5) magnetic flux from (6) an external coil (the ignitron switched capacitor banks are also indicated schematically). Each magnetic probe array makes eight measurements with radial spacing of 2.5 cm. The full set of twelve probes determines three right circular cross sections of the cylindrical plasma volume at , , and , each with four probes equally spaced azimuthally (i.e., at , 135°, 225°, and 315°), in addition to two planes of data. Only one plane of data is shown in this figure since at this early time (, where initiation of the discharge defines ) the two righthanded spheromaks have not interacted much and are largely axisymmetric.
Sketch of the SSX device. Numbers indicate the following: (1) the cylindrical conducting boundary with ; (2) one of 12 linear magnetic probe arrays inserted radially into the device; and for one of the two plasma guns on each end of the device, the (3) inner and (4) outer coaxial electrodes linked by (5) magnetic flux from (6) an external coil (the ignitron switched capacitor banks are also indicated schematically). Each magnetic probe array makes eight measurements with radial spacing of 2.5 cm. The full set of twelve probes determines three right circular cross sections of the cylindrical plasma volume at , , and , each with four probes equally spaced azimuthally (i.e., at , 135°, 225°, and 315°), in addition to two planes of data. Only one plane of data is shown in this figure since at this early time (, where initiation of the discharge defines ) the two righthanded spheromaks have not interacted much and are largely axisymmetric.
The measured magnetic structure of the relaxed state at , following the completion of the cohelicity merging process. The mostly axisymmetric initial state containing two spheromaks is shown in Fig. 1. Both planes of data are shown since this state is inherently nonaxisymmetric.
The measured magnetic structure of the relaxed state at , following the completion of the cohelicity merging process. The mostly axisymmetric initial state containing two spheromaks is shown in Fig. 1. Both planes of data are shown since this state is inherently nonaxisymmetric.
Time dependence of average , electron density , and carbon (impurity) ion temperature . Once the relaxed state forms the plasma has symmetry and selfsimilarly decays with about a folding time in and .
Time dependence of average , electron density , and carbon (impurity) ion temperature . Once the relaxed state forms the plasma has symmetry and selfsimilarly decays with about a folding time in and .
Measured magnetic structure for single plasma gun discharge. The initial spheromak state (a) is mostly axisymmetric, but the final state (b) is the same (except for an arbitrary global azimuthal rotation) as the nonaxisymmetric relaxed state seen in Fig. 2 for cohelicity merging.
Measured magnetic structure for single plasma gun discharge. The initial spheromak state (a) is mostly axisymmetric, but the final state (b) is the same (except for an arbitrary global azimuthal rotation) as the nonaxisymmetric relaxed state seen in Fig. 2 for cohelicity merging.
(a) The determinant of Eq. (4) evaluated for and . (b) dependence of the values of for and solutions. The minimum energy state (smallest ) is axisymmetric for and nonaxisymmetric for ). (c) The normalized axial wavenumber spectrum for the coefficients in the numerical solution [Eq. (3)] for show a narrowing peak around the value with increasing , indicating that the helical structure approaches that of the Taylor double helix solution. The peak value of each spectrum is normalized to one.
(a) The determinant of Eq. (4) evaluated for and . (b) dependence of the values of for and solutions. The minimum energy state (smallest ) is axisymmetric for and nonaxisymmetric for ). (c) The normalized axial wavenumber spectrum for the coefficients in the numerical solution [Eq. (3)] for show a narrowing peak around the value with increasing , indicating that the helical structure approaches that of the Taylor double helix solution. The peak value of each spectrum is normalized to one.
isosurfaces for the Taylor state (top) and two orthogonal views (middle and bottom) of a field line to illustrate the helical character of the structure and the simplified description of the structure as a twisted closed loop of flux. The dotted line is the cylindrical axis; the perpendicular line segments on the end walls (circles) indicate the rotation between the two views of the field lines.
isosurfaces for the Taylor state (top) and two orthogonal views (middle and bottom) of a field line to illustrate the helical character of the structure and the simplified description of the structure as a twisted closed loop of flux. The dotted line is the cylindrical axis; the perpendicular line segments on the end walls (circles) indicate the rotation between the two views of the field lines.
(a) isosurfaces for the Taylor state and two orthogonal views of a field line to illustrate the helical character of the structure. (b) Field lines drawn through the final state of an MHD simulation using the HiFi (Refs. 23) code initialized with the Bessel function doublespheromak state. The final state is consistent with the Taylor state structure. The cylindrical boundary is not shown in (b) but is in roughly the same horizontal orientation as the figures in (a).
(a) isosurfaces for the Taylor state and two orthogonal views of a field line to illustrate the helical character of the structure. (b) Field lines drawn through the final state of an MHD simulation using the HiFi (Refs. 23) code initialized with the Bessel function doublespheromak state. The final state is consistent with the Taylor state structure. The cylindrical boundary is not shown in (b) but is in roughly the same horizontal orientation as the figures in (a).
Measurements of for the cohelicity merging experiment [(a) and (c)] and for the single plasma gun discharge (single spheromak) experiment [(b) and (d)]. The radial profiles of [(a) and (b)] are consistent with the Taylor state value (lower dotted line at ); the next larger value giving an state (see Fig. 5) is indicated by the upper dotted line. The radial average of (excluding the two values at smallest ) stays very near the Taylor state value throughout the decay of the nonaxisymmetric selforganized state [(c) and (d)]. Figure adapted from Ref. 12.
Measurements of for the cohelicity merging experiment [(a) and (c)] and for the single plasma gun discharge (single spheromak) experiment [(b) and (d)]. The radial profiles of [(a) and (b)] are consistent with the Taylor state value (lower dotted line at ); the next larger value giving an state (see Fig. 5) is indicated by the upper dotted line. The radial average of (excluding the two values at smallest ) stays very near the Taylor state value throughout the decay of the nonaxisymmetric selforganized state [(c) and (d)]. Figure adapted from Ref. 12.
Contour plots of , , and determined experimentally (corresponding to the data of Fig. 2). Isosurfaces are separated by 150 G. Red/yellow colors (solid lines) are positive values; blue/green (dotted lines) are negative values. Figure adapted from Ref. 12.
Contour plots of , , and determined experimentally (corresponding to the data of Fig. 2). Isosurfaces are separated by 150 G. Red/yellow colors (solid lines) are positive values; blue/green (dotted lines) are negative values. Figure adapted from Ref. 12.
Contour plots of , , and for the calculated Taylor state with . The only free parameter is the total helicity, which was adjusted to give best agreement with the experimental results. Isosurfaces are separated by 150 G. Red/yellow colors (solid lines) are positive values; blue/green (dotted lines) are negative values. Figure adapted from Ref. 12.
Contour plots of , , and for the calculated Taylor state with . The only free parameter is the total helicity, which was adjusted to give best agreement with the experimental results. Isosurfaces are separated by 150 G. Red/yellow colors (solid lines) are positive values; blue/green (dotted lines) are negative values. Figure adapted from Ref. 12.
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