Drift Shells and the Second and Third Adiabatic Invariants

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This chapter defines the fundamental concept of trapped particle drift shells. Two additional field-geometric concepts are introduced and added to the concept of guiding center: the guiding field line and the guiding drift shell. All this leads to the definition of the second adiabatic invariant and several related simplified expressions thereof valid for some special cases. The conservation of the second invariant is demonstrated (in Sect. A.2). Using this theorem, it is possible to trace drift shells in general fields and some practical recipes are given for numerical methods to accomplish this. The concepts of shell splitting and pseudo-trapping in azimuthally asymmetric fields are discussed, and consequences for particle trapping and diffusion are mentioned. Several examples are analyzed in detail and analytical expressions for near-equatorial particles are given. A special look is taken at the dipole field as a first approximation of the geomagnetic field. Old but still much used quantities are introduced and discussed, such as the L-value and the system of invariant coordinates. We examine the situation of near-equatorial particles (pitch angles near 90∘), for whose drift orbits analytical relationships can be written down for first-order magnetospheric field approximations. The last part of this chapter deals with slowly time-varying fields and the resulting effects on drift shells. We introduce the third adiabatic invariant, a purely field-geometric quantity (the magnetic flux enclosed by a drift shell), and demonstrate (in Sect. A.3) its constancy under adiabatic conditions. The process during the adiabatic change is examined “under a microscope”, showing that adiabatic constancy is not absolute but valid only when averaged over a drift period: under time-dependent conditions, identical particles on the same drift shell share a common drift shell only at times that are integer multiples of their drift period. Without proof, we mention that this really is also true for the other two adiabatic invariants and their related periodicities. A generalized L-parameter, or “L-star” is introduced, and a general method for its calculation is presented.