Abstract
This chapter formalizes the definition of drift velocity in general terms for arbitrary field configurations; some specific second order drifts are discussed. We show why guiding center particles follow curved magnetic field lines (a non-trivial fact) and discuss the conditions for that to happen. Since drifts are, by definition, perpendicular to the local magnetic field vector, we then examine the parallel motion of the guiding center, introducing the all-important concepts of particle trapping, mirror points and periodic bounce motion. In discussing the concept of bounce period we show that, like a pendulum at rest, even non-bouncing equatorial (90∘-pitch angle) particles have an intrinsic bounce period (with which external perturbations can resonate). In the second part of this chapter we take a close look at the guiding center particle’s parallel acceleration and total kinetic energy change along a magnetic field line in the presence of a field-aligned electric field, deriving the energy equation and the so-called betatron and Fermi accelerations. The chapter ends with a thorough discussion of the effects of given potential (conservative) parallel electrostatic fields on a particle’s bounce motion, identifying distinct regions of behavior in a parallel/perpendicular velocity map—a subject of importance in auroral physics.
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Notes
- 1.
It is advisable to revisit the above derivation process starting with Fig. 2.1 and relation (2.3). That process really developed in stages: what the figure intended to show implicitly was a “pre-GCS” in which the particle was circling free of external forces—i.e., a system which was moving with a transverse drift velocity, sum of \({\boldsymbol U}\) (1.34) and \({\boldsymbol V }_{F}\) (1.31) and in which the corresponding motion-induced field force balanced the external field forces. In such a system the particle gyrates with constant speed v ⊥ ∗ (example of Sect. 1.3). However, due to any inhomogeneity of the magnetic field, there was another resultant force, the term (2.6), which leads to an additional drift and the “final GCS”. Now the particle’s transverse velocity v ⊥ is no longer independent of \(\varphi\) (see expression (1.41)), but its cyclotron average is equal to the (constant) velocity in the “pre-GCS” (see expression (1.42)). It is precisely this average transverse velocity that enters in the definition of the magnetic moment (1.46). Confused again? Unfortunately, this detail is conceptually important, especially for the fundaments of plasma physics.
- 2.
For the relativistic version, use (1.29) as the relation between the relativistic magnetic moment and kinetic energy.
- 3.
When the local time derivative ∂ B∕∂ t is entirely due to changes in the external current intensities, but not their configuration in space, one usually calls this process a betatron acceleration; if the current intensities are constant, but their position or distribution changes in space, one calls it a Fermi acceleration. This distinction is made mainly in astrophysics. However, the particle doesn’t care about what causes the local field to change!
- 4.
This justifies the entire discussion in Sect. 1.6 of equatorial particles: even if their pitch angles deviate a bit from 90∘, during their drift they will always be tied to an equilibrium position on the minimum-B surface.
References
Y.T. Chiu, M. Schulz, Self-consistent particle and parallel electrostatic field distributions in the magnetospheric-ionospheric auroral region. J. Geophys. Res. 83, 629–642 (1978)
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Roederer, J., Zhang, H. (2014). Higher Order Drifts and the Parallel Equation of Motion. In: Dynamics of Magnetically Trapped Particles. Astrophysics and Space Science Library, vol 403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41530-2_2
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DOI: https://doi.org/10.1007/978-3-642-41530-2_2
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