Regular and chaotic charged particle motion in magnetotaillike field reversals: 1. Basic theory of trapped motion

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Abstract
We give a systematic theoretical analysis of trapped nonadiabatic charged particle motion in two‐dimensional taillike magnetic field reversals. Particle dynamics is shown to be controlled by the curvature parameter κ, i.e., the ratio κ² = Rmin/ρmax between the minimum radius of curvature of the magnetic field and the maximum Larmor radius in it for a particle of given energy. κ≫1 corresponds to the usual adiabatic case with the magnetic moment μ as a first‐order invariant of motion. As κ decreases toward unity, the particle motion becomes stochastic due to deterministic chaos, caused by the overlapping of nonlinear resonances between the bounce‐ and the gyro‐motion. We determine the threshold of deterministic chaos and derive the related pitch angle diffusion coefficient which describes statistically the particle behavior in the limit κ → 1. Such behavior, which for κ ≅ 1 becomes strongly chaotic, applies, e.g., to thermal electrons in Earth's magnetotail and makes its collisionless tearing mode instability possible. We also show that in sharply curved field reversals, i.e., for κ1. Both types of trapped particle motion in sharply curved magnetic field reversals κ<1 are closely connected with fast oscillations perpendicular to the reversal plane. However, the trajectories are adiabatic only in the case that they permanently remain crossing the reversal plane. The adiabatic are of a ring type, i.e., they resemble rings in phase space and also in real physical space. For ring‐type orbits the action integral over the fast oscillations is an adiabatic invariant in the usual sense. On the other hand, the most common particle trajectories in a sharply curved field reversal with κ<1 are essentially of a cucumberlike quasi‐adiabatic type. For quasi‐adiabatic cucumberlike orbits the action integral over the fast oscillations is an adiabatic invariant only in a piecemeal way between successive traversals in the phase space of the fast motion of a separatrix between orbits which do and those, which do not cross the reversal plane. Due to the effect of separatrix traversals the slow motion shifts between different cucumber orbits with a conservation of the action integral on average but with its chaotic phase space diffusion even for very small perturbation parameters κ. The case κ<1 is applicable, e.g., to thermal ions and high‐energy electrons in Earth's magnetotail. Our findings lead to a systematic interpretation of particle observations in Earth's magnetotail and of numerous numerical calculations, carried out in the past. They also explain rather well, e.g., the pitch angle diffusion of plasma sheet particles, the isotropization of the plasma sheet electron distribution immediately before a substorm and provide with the transition to chaos a mechanism for the onset of a large‐scale tail instability and the explosion of isolated substorms. Further implications for magnetotail physics, such as acceleration processes and the influence of the particle escape from the field reversal will be discussed in a second related paper.