Planetary period modulations of Saturn's magnetotail current sheet: A simple illustrative mathematical model

We mathematically model the modulation effects on Saturn's equatorial magnetotail and magnetodisk current sheet produced by the combined magnetic field perturbations of the northern and southern planetary period oscillation (PPO) systems, specifically north-south displacements associated with the radial perturbation field and thickness modulations associated with the colatitudinal perturbation field. Since the phasing of the two PPO systems is taken to be related to the radial field perturbations, while the relative phasing of the colatitudinal perturbations is opposite for the two systems, the north-south oscillations reinforce when the two PPO systems are in phase, while the thickening-thinning effects reinforce when they are in antiphase. For intermediate relative phases we show that when the northern PPO system leads the southern the sheet is thicker when moving south to north than when moving north to south, while when the northern PPO system lags the southern the sheet is thicker when moving north to south than when moving south to north, thus leading to sawtooth profiles in the radial field for near-equatorial observers, of opposite senses in the two cases. Given empirically determined modulation amplitudes, the maximum sawtooth effect is found to be small when one system dominates the other, but becomes clear when the amplitude of one system lies within a factor of 2 of the other.