Ambipolar Potential and Pastukhov Confinement

The tandem mirror confines ions with an electrostatic potential set by charge neutrality; the Pastukhov relation ties confinement time to that potential.

Charge neutrality sets the potential

In the burner the end plugs raise the electron density locally, and quasineutrality forces an electrostatic potential Phi that confines the central-cell ions in a potential well. Solving for Phi is a nonlinear root problem: the potential must make ion and electron loss rates equal (ambipolarity) at every axial location. This potential is the burner's central control target, the analogue of the breeder's plasma shape.

Ambipolarity (equal particle loss rates):

  Gamma_i(Phi) = Gamma_e(Phi)   at each axial cell

Quasineutrality:
  sum_s q_s n_s(Phi, B, T_s) = 0

Boltzmann electrons (fast species):
  n_e(z) = n_e0 * exp( e (Phi(z) - Phi_0) / T_e )

Pastukhov confinement scaling

The confinement time of ions trapped by both the magnetic mirror and the electrostatic potential follows the Pastukhov relation: it grows roughly exponentially with the ratio of confining potential to ion temperature, modulated by the mirror ratio and a logarithmic factor. This is why building and holding the potential is everything for the burner.

Pastukhov-type ion confinement time:

  n tau_i  ~  ( R_m / log(R_m) ) * ( e*Phi_c / T_i ) *
              exp( e*Phi_c / T_i ) * (collisional factor)

  Phi_c : confining potential depth
  R_m   : mirror ratio (set by plug/throat fields)
  Strong (exponential) sensitivity to Phi_c / T_i

The solver and its honesty

The stack solves the coupled quasineutrality-ambipolarity system for Phi(z) as a nonlinear two-point problem, then uses Phi to predict confinement. But the operating regime is 166-830x beyond any device, so the Pastukhov extrapolation is un-post-dictable today; the He-3 fuel it presumes is ~400x current domestic supply. Both are reported as gates, not solved away.

# solve ambipolar potential profile (schematic)
def residual(Phi):
    n = quasineutral_densities(Phi, B_profile, T)
    return Gamma_i(Phi, n) - Gamma_e(Phi, n)   # ambipolarity
Phi = newton_krylov(residual, Phi_guess)
tau_i = pastukhov(Phi, R_m, T_i)
# confidence band dominated by 166-830x regime extrapolation

Because confinement depends exponentially on Phi_c/T_i, small modeling errors become large confinement errors - a mathematical reason the burner is treated as a design study and never as a proven operating point.