Mirror Interchange Stability and Minimum-B

Open-field mirrors are prone to flute/interchange instability; stability requires a magnetic well (minimum-B), and the burner's ability to build one is a live physics gate.

The interchange drive

A simple magnetic mirror has unfavorable field-line curvature: plasma can interchange with vacuum and lower its energy, driving the flute (interchange) instability - the mirror analogue of the breeder's ballooning drive. Stability requires that field-line curvature be favorable on average, i.e. the field magnitude increases away from the plasma: a minimum-B or magnetic-well configuration.

Interchange stability criterion (average minimum-B):

  Integral  dl / B  * ( flux-tube volume ) must satisfy
     d/dpsi [ Integral dl / B ]  < 0   (well condition)

Equivalently the average curvature is favorable:
  < kappa . grad p >  < 0   (stabilizing)

  A magnetic well means B increases off-axis in all directions.

Building a well in a tandem mirror

Minimum-B is achieved with non-axisymmetric coils (historically Ioffe bars / baseball / yin-yang geometry) that raise B off-axis. The tension is fundamental: the high-field plug (26.49 T) and throat (17 T) that provide confinement must coexist with a field shape that keeps the well. Whether a feasible-stress coil set can deliver both simultaneously is precisely where a burner gate bites.

# average magnetic-well figure of merit along a flux tube
def well_depth(B_of_l, dl):
    U = sum(dl / B_of_l)                 # specific flux-tube volume
    return -dU_dpsi(U)                   # >0 -> magnetic well (stable)
# stability requires well > 0 at the operating beta

Why this is a gate, not a result

At the design bore the plug coil is overstressed ~3-3.9x, so the specified field-and-well combination is infeasible as drawn. That is a structural-mechanics fact upstream of the plasma physics: without a feasible coil, the stabilizing minimum-B cannot be assumed. The stack therefore represents burner interchange stability with the coil-stress constraint bound to it, never reporting a stable well without also reporting the ~3-3.9x overstress.

• Interchange/flute mode is the dominant open-field MHD threat.
• Minimum-B (magnetic well) is the stabilizing requirement.
• Coil overstress ~3-3.9x means the well-providing geometry is not yet feasible.

These are simulation findings for a machine that does not exist; they are stated plainly because they define what must be solved, not hidden to make the concept look finished.