Nonadiabaticity in mirror machines

Abstract

An analytic technique for calculating magnetic-moment jumps Δμ of particles in magnetic traps, previously derived for particular two-dimensional vacuum fields, is generalized to nonvacuum fields of arbitrary complexity and applied to high-β mirror machines. The size of a jump depends on the behavior of the magnetic-field strength B (s) near the singularities of B in the complex s plane, where real s measures position along a field line. It is demonstrated that an intrinsic complication of mirror-machine magnetic configurations is the presence of multiple singularities of B, which become closely spaced for field lines near the axis. An expansion in r2 is used to determine Δμ in the closely spaced regime. The analytic theory is compared with results from a particle-orbit code for several axisymmetric nonvacuum fields, and is found to be in excellent agreement in both the well separated and closely spaced singularity regimes. Finite-β effects are examined using axisymmetric model fields derived from the long, thin equilibrium approximation, with parameters representative of present (2XIIB) and future (MFTF) mirror experiments. It is shown that increasing β appreciably lowers the maximum energy at which particles are confined. The effect is particularly significant if the plasma length is small compared with the vacuum field axial scale length and if the gyroradius is small compared with the plasma radius. The degradation is substantially reduced when the gyroradius at the energy limit is comparable to the plasma radius.

Butterfly pitch angle distribution of relativistic electrons in the outer radiation belt: Evidence of nonadiabatic scattering

“…For example, at > 3∕2, we do not expect to observe any pitch angle diffusion for equatorial electrons [see Delcourt et al, 1995;Young et al, 2008], while at < 3∕2 this diffusion is important (see section 3 and Shibahara et al [2010]). Additionally, in the vicinity of the loss cone (at small pitch angles), the nonadiabatic scattering is nondiffusive (i.e., the averaged jump is not equal to zero and is always positive, see Delcourt et al [1994Delcourt et al [ , 1995), while analytical estimates give only Δ ∼ sin 0 with zero average value for a random uniform distribution of 0 [Howard, 1971;Cohen et al, 1978;Birmingham, 1984;Chirikov, 1987;Varma, 2003]. Moreover, as we discuss in Appendix B, most analytical expressions for Δ should be tested by numerical calculations because these expressions were derived using not well justified methods.…”

“…In case of magnetic field configuration with the cusp region (i.e., X line), Howard [1971] derived the expression for Δ ∼ exp(− 2F ( 0 )) for an arbitrary order of singularity H 0 ∕ ∼ ( − 0 ) p (here p > 0, 0 is the equatorial pitch angle, i.e., the factorF does not depend on the particle energy). Using the same approach, Cohen et al [1978] obtained expressions for jumps of the magnetic moment in various configurations of magnetic field typical for mirror machines (magnetic traps). Several other configurations of magnetic field were considered in papers by Chen et al [1985], Tagare [1986], Basu and Rowlands [1986], and Yavorskij et al[2002] (see also reviews by Chirikov [1987] and Varma [2003]).…”

“…To derive equation (B5) Birmingham[1984] performed the integration in equation (B2) along the unperturbed trajectory, i.e., the conservation of was assumed for the integral calculation (the similar approach can be found in many other publications, e.g., Howard [1971], Cohen et al [1978], Chen et al [1985], Tagare [1986], Basu and Rowlands [1986], and Yavorskij et al [2002]). This procedure cannot be fully justified [see Neishtadt, 1981].…”

Charged-particle orbits near a magnetic null point

“…The jumps are such that ∆µ`exp(k1\ε). For a specific calculation of ∆µ for a wide range of magnetic field configurations, see for example Cohen et al (1978). In this case, the long-time behaviour of particles can be understood in terms of a map relating the values (µ n θ n ) before a jump to the values (µ n+" , θ n+" ) after a jump.…”

“…Thus, for Q 1, the particle is excluded from the origin, the position of the zero of the magnetic field. For Q 1, adiabatic theory applies, and the value of the jump ∆µ was given some time ago by Howard (1971). A typical orbit is shown in Fig.…”