Build a Nachamkin plasmoid on purpose

Why the library suggests it

Jack Nachamkin, writing for the Air Force's Phillips Laboratory, works the spherically symmetric time-harmonic case and finds two things worth building against: a critical frequency above which electrons can carry the required current and small local disturbances raise the energy. At particular resonant sizes, its ideal spherical fields can match ordinary vacuum solutions at the surface, eliminating outgoing radiation in that model. This does not remove all losses; the report also discusses finite lifetimes. A stable vortical motion of the plasma exactly cancels the dominant electromechanical stress, and what is left falls away steeply as frequency rises (Force-Free Time-Harmonic Plasmoids, 1992).

Use the interactive to explore a simplified plasma-wave cutoff. Changing electron density changes that cutoff; above it, the circles show half-wavelength lengths. Nachamkin's spherical solutions also require fields to match at the surface.

Plasma waves and length scales

Electron density sets a cutoff for electromagnetic waves in this simplified cold, unmagnetized plasma model. Above it, the wavelength sets the lengths shown by the circles. Change the density or drive frequency to see those lengths move.

At this setting

Free-electron density

1E18

Drive frequency

2E10

Back to the opening setting

• Plasma-frequency cutoff
• Half-wavelength lengths

Plasma-frequency cutoff:

8.98E9 Hz

Refractive index:

0.893

First half-wavelength length:

8.41E-3 m

Half-wavelength lengths:

8.41E-3, 1.68E-2, 2.52E-2 m

Above the cutoff, this model has a propagating electromagnetic wave. The listed lengths are one, two and three half-wavelengths.

The report's additional conditions. Nachamkin's spherical solution also depends on matching fields at the surface and balancing electromagnetic forces with fluid motion.

Read the 1992 analysis

A separate line asks what holds many nanometre-scale plasmoid kernels together as one visible object, and finds a quantum exchange force between dipole-carrying ions that reaches a few per cent of the electromagnetic energy at liquid-water ion densities (Quantum exchange interaction of spherically symmetric plasmoids, 2012). The magnetic relaxation treatment in Magnetic Helicity, Spheromaks, Solar Corona Loops, and Astrophysical Jets (2017) concerns magnetically dominated plasmas, not the zero-magnetic-field radial oscillations in Dvornikov's model.

With approximately conserved magnetic helicity and a fixed boundary, magnetic energy can fall toward a Taylor state, with the magnetic field parallel to its curl. Physical confinement also requires force balance at the boundary.

The relaxation

In this model, magnetic energy decreases while helicity — the field's overall twist and linkage — stays fixed. The field is restricted to a sphere whose boundary it cannot cross. Move the control to concentrate the helicity in the lowest-energy field pattern allowed by that boundary.

At this setting

Energy released, at fixed helicity

0%

250 micrometres, the radius observed50 micrometres, the radius being aimed for

Let the tilt mode growBack to the tangled state

• Magnetic energy
• Magnetic helicity
• Magnetic axis

Magnetic energy:

3.823

Magnetic helicity:

1.000

Lowest energy this helicity allows:

2.247

Energy still above that floor:

70.2%

Force-free residual:

101.6%

Best-fitting eigenvalue:

8.425

Taylor eigenvalue at this radius:

1.8E4 m⁻¹

Magnetic axis:

0°

The field is tangled: several curl eigenmodes are present at once, so no single eigenvalue fits it, the Lorentz force does not vanish, and there is energy here that the linkage does not require.

The honest part: the relaxed state is not unconditionally stable. The tilt mode is the motion predicted to end these objects, and the minimisation cannot forbid it, because a rigid rotation leaves both quantities exactly where they were. What decides it is the boundary. The laboratory record is not one-sided either: at Princeton a field-reversed configuration has held plasmas more than ten thousand times longer than tilt theory predicted.

The two radii come from the dense-plasma-focus paper: 250 micrometres previously observed and 50 micrometres proposed as a target. The spherical interior field is an idealized calculation, not a field map measured in that experiment.

What to watch. A measurement of the field inside a laboratory plasmoid, compared against the eigenvalue the boundary fixes. That is the observation that would turn the relaxed state drawn here from a prediction into a reading.

Relaxation at fixed helicity is textbook magnetohydrodynamics, worked out for the spheromak, the coronal loop and the astrophysical jet alike.

Sources

• The relaxation argument, at book length
• The same principle, stated for the spheromak
• The force-free knot, applied to a fireball
• The two published plasmoid radii
• The tilt mode, and what the laboratory found

The experiment or build

First model how a source couples to the spherical fields. In Nachamkin's mode expansion, localized plane-wave or point sources do not supply the trapped fields; a current column along the plasmoid axis is discussed as a possible pumping route. Frequency and chamber dimensions alone are not a build specification.

An institutional plasma laboratory could use that coupling study to plan a discharge experiment, with high-voltage and radio-frequency safety controls reviewed before construction. Measure lifetime while varying drive frequency and geometry, and account for the drive and losses.

Compare any measured lifetime peaks or changes in stability with the report's frequency and spherical surface-matching conditions. These are proposed diagnostics. The report does not supply quantitative lifetime estimates.