(Color online) Examples of numerically computed co-interchange displacements. The axis (axial direction) is indicated with an arrow. The unperturbed pressure isosurface is shown in (a). The results of axial mode are illustrated in (b), and an radial mode in (c).
(Color online) Side view (a) and end-on view (b) of the MRX device.
(Color) The computed midplane magnetic perturbations for co-interchange modes. The perturbation for the radial mode is illustrated in (a), and the and perturbations for the axial mode is shown in (b) and (c).
(Color) The measured (a) and (b) perturbations in MRX, for .
(Color) The time evolution of three discharge sets is illustrated in (a)–(l). Colors represent toroidal field, arrow represent poloidal field, and a few streamlines are shown in green to guide the eye. The horizontal black line in (e)–(l) indicate the presence of the center column. The bottom section illustrates (m) the poloidal flux evolution, (n) the , evolution, and (o) the , evolution. The colors of the traces correspond to the colors of the letters above. Note the different scales in (n) and (o). See text for further details.
Magnitude of (a) the , (b) the , and (c) the perturbations to (axially polarized mode) at in the midplane, for helium discharges. The open symbols are for cases without the center column, and closed symbols for cases with the center column. Some representative error bars are shown.
Radial profiles of the , 2, and 3 amplitudes, as a function of minor radius. This case is for a discharge without a center column; the tilt mode has a radially more broad perturbation field than the and 3 axial modes.
Magnitude of (a) the , (b) the , and (c) the perturbations to (corresponding to radially polarized modes) at in the midplane, for helium discharges. The open symbols are for cases without the center column, and closed symbols for cases with the center column. Some representative error bars are shown.
Plasma decay-time normalized to the Alfvén time, plotted against : (a) the , (b) , and (c) magnetic perturbations, in helium. The points in (a) with are not included in (b). Open symbols are for cases without the center column, and closed symbols for cases with the center column.
Plasma decay time normalized to the Alfvén time, plotted against (a) the , (b) , and (c) perturbations to , for helium plasmas. Open symbols are for cases without the center column, and closed symbols for cases with the center column.
(Color online) Time evolution of a discharge with a significant radial shift. The frames (a)–(e) illustrate the evolution of the shift at the toroidal angle of the 90-channel probe. A large outward shift of the plasma is visible at . The time evolution of the , and 2 modes is illustrated in (f), and the poloidal flux in (g). Vertical lines indicate the times of the 2D plots. The plasma “bounces” off the high-field region outside the shaping coils; the shift amplitude decreases before the poloidal flux decays away.
(Color online) Three example equilibria computed with the MRXFIT code. These equilibria are for the plasmas whose evolution is illustrated in Fig. 5. The case in (a) has an external field mirror ratio of 2.4, but no center column. The case in (b) is limited on the inboard side by the center column. The case in (c) displays the result when the shaping-field coils are used to pull the plasma out, leading to a plasma with a very oblate shape.
Plasma shape parameters as derived from the MRXFIT code, The different frames show (a) the minor radius (a), (b) the aspect ratio , (c) the elongation , and (d) the triangularity . These geometric quantities are defined in Eq. (11). Open/closed symbols correspond to discharges without/with the center column.
Parameters from the rigid-body model for tilting/shifting. The tilting torque and shifting force are illustrated in (a) and (b), while the tilt and shift growth rates are illustrated in (c) and (d). Open/closed symbols are for cases without/with central conductor.
The five separatrix shapes used in the HYM calculations. The mirror ratio varies from 2.3 to 4 among these shapes. All plasmas are limited by the center column, which is modeled as a perfectly conducting boundary in the simulation.
The growth rate of the axially (a) and radially (b) polarized co-interchange instabilities, as a function of toroidal mode number (), for the configurations in Fig. 15.
(Color online) Time evolution plots from nonlinear HYM simulations: (a) Normalized axial shift of the magnetic axis (red) and relative radial displacement of plasma boundary at the midplane (blue). The dashed line shows value of radial velocity at the plasma boundary. (b) The magnetic energy for modes normalized to the total magnetic field energy. Field reversal is lost at .
Algorithm of the MRXFIT code.
Measured and predicted magnetic fields: (a) and (b) for one of the coil triplets of the 90-channel probe. The measured magnetic field is shown in the black line, the prediction with only the PF windings is draws with a dot-dashed line (---), and the prediction with the PF windings and flux conserver is shown with a dashed line (---), The prediction using the flux conserving vessel accurately matches the measurement.
(Color online) Reconstruction of an FRC in MRX, for a case with a mirror ratio of 2.9. The reconstructed poloidal flux contours are illustrated in (a) and the pressure profile is plotted in (b). The measured and fit midplane magnetic profiles are illustrated in (c).
The growth rate of “local modes,” as a function of mirror ratio, based on Eq. (B4). The growth rate is reduced as the plasma shape becomes more oblate.
Parameters for the helium FRCs discussed in this paper.
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