Parameterization of chaotic particle dynamics in a simple time‐dependent field reversal

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Abstract
We investigate single charged particle dynamics in the Earth's magnetotail using a simple, scale free magnetic field model which is explicitly time dependent, with a corresponding induction electric field. The time‐dependent Hamiltonian of particle motion in the simple model describes a system of two coupled oscillators which is driven. When the (time‐dependent) ratio of the oscillation frequencies is different from unity, the motion is regular, but if it approaches unity at some point on the trajectory, the motion becomes chaotic. A parameter α is found which characterizes the adiabaticity of the system, and the transition time for behavior in the system is given by at αt → 1. The condition at αt ≈ 1 is equivalent to κ ≈ 1 in the static parabolic field model discussed by previous authors (Buchner and Zelenyi, 1989). The explicit time dependence produces two possible classes of motion, ordered by α. If the reversal is thick and is folding slowly, so that α ≪ 1, the motion is a transition in behavior, from regular μ conserving to chaotic “cucumberlike” trajectories, when t ≈ 1/α. If on the other hand, the reversal is thin and folds quickly, so that α ≫ 1, the particles execute regular “ring type” trajectories once t > 1/α. Simple estimates of presubstorm magnetotail parameters indicate that electrons in a slowly thinning (15‐min time scale), thick (1 RE) sheet have α ≪ 1, whereas protons in a thin (several hundreds of kilometers) sheet which thins on a 5‐min time scale have α ≫ 1. Hence the behavior of the particles, and by implication, the field reversal which they support, will depend upon the adiabaticity of the system α as well as the “chaotization” parameter αt; this is shown only to be the case in a model which is explicitly time dependent and which includes the induction electric field.