RESUMEN
In this Letter we investigate factors that influence the effective critical electric field for runaway-electron generation in plasmas. We present numerical solutions of the kinetic equation and discuss the implications for the threshold electric field. We show that the effective electric field necessary for significant runaway-electron formation often is higher than previously calculated due to both (1) extremely strong dependence of primary generation on temperature and (2) synchrotron radiation losses. We also address the effective critical field in the context of a transition from runaway growth to decay. We find agreement with recent experiments, but show that the observation of an elevated effective critical field can mainly be attributed to changes in the momentum-space distribution of runaways, and only to a lesser extent to a de facto change in the critical field.
RESUMEN
M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether's theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal's theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether's theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.