3-D MHD Stability of Magnetotail Configurations With a Bz Hump

Abstract

Using three-dimensional ideal magnetohydrodynamics (MHD) simulations, we explore the stability of magnetotail configurations that include a local enhancement of $B_z$ (a “$B_z$ hump”). We focus on configurations that were previously found to be unstable in 2-D (neglecting cross-tail, $y$, variations as well as a cross-tail magnetic field component $B_y$) but approached final 2-D stable states. Not surprisingly, the 2-D unstable configurations were also found unstable in 3-D, developing 3-D structure after an initial rise as in 2-D. This is consistent with the fact that the selected $B_z$ hump configurations are characterized also by a local tailward decrease of field line entropy, which has been found to govern ballooning/interchange (B/I) instability (Schindler & Birn, 2004, https://10.1029/2004JA010537). The evolution of the electric field perturbation of the unstable 3-D modes showed an early exponential growth, which was only modestly faster than the 2-D mode. This might suggest that the unstable ballooning regime extends from $k_y\to \infty$ (proper ballooning modes) over finite $k_y$ even to $k_y\to 0$, at least for some equilibria. When the 3-D modes had evolved they grew significantly faster than the 2-D modes. This was associated with an evolution into several narrow beams. The 3-D modes did not approach stable final configurations. The final 2-D stable configurations approached from the unstable ones were found to be unstable in 3-D. These findings are again consistent with the fact that the 2-D evolution, governed by ideal MHD, did not alter the characteristics of the entropy function.