Experimental and theoretical studies of runaway electrons in toroidal devices are reviewed here, with particular reference to tokamaks. The complex phenomenology of runaway effects, which have been the subject of research for the past twenty years, is organized within the framework of a number of physical models. The mechanisms and rates for runaway production are discussed first, followed by sections on runaway-driven kinetic relaxation processes and runaway orbit confinement. Next, the equilibrium and stability of runaway-dominated discharges are reviewed. Models for runaway production at early times in the discharge and the scaling of runaway phenomena to larger devices are also discussed. Finally, detection techniques and possible applications of runaways are mentioned.

https://iopscience.iop.org/article/10.1088/0029-5515/19/6/008/pdf

Runaway electrons in toroidal discharges

The possibility of using intense relativistic electron beams (REBs) for heating plasmas in open systems is discussed. Within this context the following three sets of problems are discussed: REB transport in a vacuum with a strong magnetic field; beam equilibrium, stability and critical currents in a vacuum. Beam transport in a plasma; charge and current neutralization of the beam; reverse-current heating of the plasma, and macroscopic REB instabilities in the plasma. The theory of collective relaxation of REBs in a plasma, including quasi-linear and non-linear relaxation models; the role of plasma non-uniformity, and macroscopic effects during REB relaxation.

https://iopscience.iop.org/article/10.1088/0029-5515/14/6/012/pdf

Powerful relativistic electron beams in a plasma and in a vacuum (theory)

A relativistic electron beam propagating through an ummagnetized, underdense plasma exhibits a transverse instability due to the coupling of the beam centroid to plasma electrons at the ``ion-channel'' edge. The transverse wake field corresponding to this ``electron-hose'' effect is calculated in the ``frozen-field'' approximation for a low-current, cylindrical beam in a radially infinite plasma. The asymptotic growth of beam-centroid oscillations is computed, and the growth length is found to be very rapid, indeed much less than the betatron period of the beam. Results for a radially finite plasma and for a slab beam are noted. Damping and saturation mechanisms are discussed.

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Electron-hose instability in the ion-focused regime.

All the basic and physical aspects of ion-driven confinement fusion are timely reviewed. Both light- and heavy-ion beams are stressed out. The fundamental components : accelerating structure, beam, target and reactor vessel are thoroughly presented. A particular emphasis is given to the three crucial interfaces : beam-reactor, beam-target and target-reactor. Ion-plasma experiments of closely related concern are also investigated at length.

Inertial confinement fusion driven by intense ion beams

The resistive hose instability of a self‐pinched relativistic beam is examined with emphasis placed on the important case of the Bennett current profile JB(r) ∝ (1+r2/a2)−2. Previously known results applicable to a general profile are recovered and extended in several directions. An essential new feature in this study is the use of distributed particle mass to model orbital phase‐mixing effects produced by the anharmonic pinch field. Resonant growth is considerably reduced, and the instability when viewed in the beam reference frame is shown to be convective rather than absolute. The peak amplitude of a disturbance wave packet moves from the point of its inception in the beam pulse toward the pulse tail. The disturbance subsequently damps if the pulse length is finite; thus, propagation over distances that are long compared with the particle betatron wavelength is possible. The predicted growth rate and group velocity of the mode are shown to be in fair agreement with the results of numerical simulation.

Resistive hose instability of a beam with the Bennett profile

A Fokker–Planck equation is derived to study the evolution of a stable, low‐current beam propagating in a gas‐plasma medium. Small‐angle scattering of the beam particles by the medium causes diffusion in the phase space projected transverse to the direction of propagation. The projected components of dynamical friction vanish. As a result, there is a continued input of energy into the transverse particle motions, which is taken up in expansion against the pinch field. A quasi‐static Bennett equilibrium, with isothermal distribution of transverse momenta, is shown to be a similarity solution of the Fokker–Planck equation with scale radius increasing in accord with Nordsieck’s formula. An H theorem is proved and the Bennett distribution is shown to minimize both H and −dH/dt; hence, it is the time‐dependent asymptotic state. The predicted current profile and radius are shown to be in fair agreement with experiment.

Kinetic theory of a relativistic beam

The observed disruption of a self‐focused, relativistic electron beam propagating through a gas is shown to result from the growth of m=1 (’’kink’’ or ’’hose’’) perturbations. Measurements of the frequency dependence of the spatial amplification rate are presented. An upper cutoff to the frequency range for hose amplification is observed, in agreement with a theoretical model that includes the damping effects of a spread in the particle betatron frequency.

Measurements of hose instability of a relativistic electron beam

A heavy‐ion beam driver for inertial confinement fusion is subject to filamentation instability over a broad range of beam and plasma background conditions. The case of a beam injected into a gas‐filled reactor vessel, where finite pulse length and propagation distance play an important role in limiting mode growth, is analyzed. The effects of transverse thermal spread, spherical convergence to the pellet, and finite magnetic decay rate of eddy currents are included in this treatment. It is concluded that a cold beam will be severly disrupted unless the product of magnetic plasma frequency and propagation time is not large compared with unity. If this condition is not met, mode growth may still be limited to about six e folds by adding transverse velocity spread such that the pulse tail is in a state of pinch equilibrium. However, this approach causes much of the pulse to be lost by thermal expansion.

Filamentation of a heavy‐ion beam in a reactor vessel

An analysis of the stability of the kink mode of a relativistic charged‐particle beam propagating in a toroidally confined plasma is presented. The essential feature is the treatment of the beam as a distinct component of the system having finite velocity and inertia in addition to its current. To simplify the analysis, the plasma background is assumed to be a pressureless, uniform fluid obeying the laws of perfect magnetohydrodynamics while the beam is treated as a cold, rigid body. To simulate toroidal geometry, periodic boundary conditions are imposed in the axial direction of a straight, cylindrical volume. The walls of the cylinder have perfect conductivity. A strong solenoidal field is externally imposed, but the beam is the only source of the poloidal field. It is found that a modification of the stability condition of Kruskal and Shafranov applies; the onset of instability corresponds to the appearance of closed particle orbits rather than the more severe condition of closed field lines. The maxim...

Edward P. Lee

Papers

Resistive hose instability of a beam with the Bennett profile

Kinetic theory of a relativistic beam

Measurements of hose instability of a relativistic electron beam

Filamentation of a heavy‐ion beam in a reactor vessel

Low‐frequency hydromagnetic kink mode of a relativistic beam