Theory of Two Threshold Fields for Relativistic Runaway Electrons

This Letter presents a rigorous kinetic theory for relativistic runaway electrons in the near critical electric field in tokamaks. The theory provides a distribution function of the runaway electrons, reveals the presence of two different threshold electric fields, and describes a mechanism for hysteresis in the runaway electron avalanche. Two different threshold electric fields characterize a minimal field required for sustainment of the existing runaway population and a higher field required for the avalanche onset. The near-threshold regime for runaway electrons determines the time scale of toroidal current decay during runaway mitigation in tokamaks.

Figure 1

The flow velocity $U(p)$ defined by Eq. (7) for $Z=5$ and ${\overline{\tau }}_{\text{rad}}=70$. The values of the electric field $E$ are 1.8 for the solid curve, 1.7 for the dashed curve, and 1.65 for the dotted curve.

Figure 2

The contours of the sustainment field $E_0$ (solid curves) and the avalanche onset field $E_a$ (dashed curves).

Figure 3

Snapshot of the runaway electron distribution in momentum and pitch angle during the decay process. The pitch-angle parameter is $\lambda \equiv {\sin }^2\theta$.

Figure 4

Avalanche growth rate predicted by Eq. (11) (solid curve) in comparison with Eq. (18) from Ref. [6] (dashed curve) and the growth rate inferred from the dynamical model of Ref. [7] (dotted curve).