Flapping oscillations of the bent current sheet

We study the dependence of the flapping oscillations on the magnetotail current sheet bending, which is caused by the dipole tilt. Observations show that flapping waves propagate from the center of the current sheet to its flanks with a velocity one order of magnitude less than typical Alfvén speed. For our analysis we use the double gradient model (Erkaev et al., 2009) of the flapping oscillations, which predicts a small minimum of the total pressure (gas plus magnetic) across the current layer. It is the depth of the potential well in the total pressure which defines the period and the speed of the flapping waves. Using the extension of the Kan/Manankova equilibriums for the non-zero dipole tilt we investigate the depth of the potential well with respect to the current sheet bending rate. We show that with the growth of the dipole tilt angle the depth of the potential well becomes smaller, the period of the flapping oscillations increases, and oscillations become nonlinear. There exists the critical tilt angle, where the potential well disappears and flapping regime changes from oscillations to instability.