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Magneto-hydrodynamically stable axisymmetric mirrorsa)
a)Paper CT3 1, Bull. Am. Phys. Soc. 55, 61 (2010).
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Image of FIG. 1.
FIG. 1.

Closed (a) and open (b) magnetic configurations. In the second case, the zones of a high magnetic field near the ends of the device are called “mirrors.” Sometimes, the word “mirror” is used to describe the whole device of this type.

Image of FIG. 2.
FIG. 2.

The quadrupole mirror system of the MFTF-B facility: (a) the magnet system; (b) one of the flux surfaces. Zone 1 is the MHD-stable “anchor,” and zone 3 is an almost axisymmetric ambipolar “plug,” whereas zone 2 is a transition region; the central solenoid begins at the right upper corner and is terminated at the opposite end by the same complex system reversed left to right and rotated by 90o around the magnetic axis.

Image of FIG. 3.
FIG. 3.

(Color online) Cross-sections of the mirror device: (a) meridional cross-section; (b) equatorial cross section. A thick line represents a flux tube moved radially to a new location; ξ0 is the radial displacement in the equatorial plane The maximum MHD growth rate corresponds to flux-tubes having ribbon-like shape (narrow in the azimuthal direction, panel b).

Image of FIG. 4.
FIG. 4.

(Color online) Flaring of the magnetic field lines in the end tank. The mirror throat is at z = 0, the confinement zone (not shown) is at z < 0. Note the favorable curvature of the magnetic field lines. Highlighted is a limiting flux surface a(z).

Image of FIG. 5.
FIG. 5.

(Color online) Stabilization with sloshing ions. Curve 1 shows the plasma boundary a(z), curve 2 is a pressure p ||  + p⊥ of the sloshing ions with a narrow angular spread leading to a strong pressure peaking near the turning points; curve 3 is the product . All curves are in arbitrary units. The zones of favorable curvature are highlighted on curve 1.

Image of FIG. 6.
FIG. 6.

Kinetic stabilizer. (a) Upper line: outline of coils in a “double-conical” expander; lower line: one of the field lines. (b) Pressure distribution of the stabilizing beam, with a sharp maximum in the optimum location. There is a need in the presence of the highly conducting plasma, connecting the stabilizer with the central part of the facility. Courtesy R.F. Post, Ref. 32.

Image of FIG. 7.
FIG. 7.

Schematic of the end section of a tandem mirror, with gas injected in the expander where the field lines have a large favorable curvature. Courtesy R.F. Post, Ref. 32.

Image of FIG. 8.
FIG. 8.

A “short-fat” mirror as a stabilizer of a global mode. The plasma boundary has to be situated well within the separatrix passing through the null points. This ensures the conservation of the adiabatic invariant over the whole confinement region. The area where the boundary can be situated is shaded. If the boundary is closer to the axis than the shaded area, the non-paraxiality effects become weak and the system becomes unstable. A long confinement cell is attached to the stabilizer through one of the mirrors. The second stabilizer is situated at the opposite end of the confinement zone.

Image of FIG. 9.
FIG. 9.

A cusp stabilizer. This particular configuration is obtained by superposing a uniform solenoidal field and the field of a single coil with the current in a direction opposite to the current in the solenoid (solenoid is not shown). The solenoid continues to the left and to the right where it is ended by strong mirrors. The distance of a flux surface from the axis in the region of a uniform field is r, the distance of the separatrix is r 0.

Image of FIG. 10.
FIG. 10.

Normalized critical pressure profiles (82) for the simple divertor of Fig. 9 (bold line) and the snowflake divertor of Fig. 11 (thin line), A.U. The profiles correspond to the same pressure at r = r 0 /2. The abscissa axis is r/r 0 . The right panel shows vicinity of the separatrix.

Image of FIG. 11.
FIG. 11.

A snowflake divertor. Both the magnetic field and its first derivatives are zero in the singular point. To get such configuration, one has to split the coil of Fig. 9 into two coils and properly adjust the distance between them.

Image of FIG. 12.
FIG. 12.

Magnetic field lines and coils of the Large Axisymmetric Mirror Experiment (LAMEX). The field lines shown are for the configuration Bz = 200 G and mirror ratio 20. Courtesy J.R. Ferron, Ref. 104.

Image of FIG. 13.
FIG. 13.

The cross-section of a cylindrical conducting shell and a plasma with m = 5 and m = 1 perturbations. The magnetic field is directed along the axis of the cylinder. In a purely cylindrical case, for linear perturbations, the magnetic field outside the plasma is not perturbed (remains uniform and parallel to the axis), and no restoring force appears.

Image of FIG. 14.
FIG. 14.

(Color online) Ballooning perturbation in the system stabilized at one end only. At zero beta the system is stable but it becomes unstable if beta exceeds the limit determined by Eq. (87).


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Table I.

Characteristic parameters of mirror devices.a

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Table II.

FLR effect for devices listed in Table I.a

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Table III.

Effects of the end-tank stabilization for devices listed in Table I.a

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Table IV.

The effect of the partial line-tying on the plasma stability.a


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Magneto-hydrodynamically stable axisymmetric mirrorsa)