Ballooning instability of a thin current sheet in the high-Lundquist-number magnetotail

Two-dimensional simulations of the magnetotail in the high-Lundquist-number regime indicate the slow growth of thin current sheets and an impulsive intensification of the cross-tail current density at near-Earth distances during a short interval just before the onset of the expansion phase, consistent with multisatellite observations. Such a two-dimensional magnetotail, symmetric along y and containing a thin current sheet, is found to be unstable to a symmetry-breaking, ideal compressible ballooning instability with high wave number along y. The linear instability is demonstrated by numerical solutions of the ideal ballooning eigenmode equation for a sequence of two-dimensional thin current sheet configurations in the impulsive growth phase. Line-tied boundary conditions at the ionosphere are imposed, and shown to play a crucial role in the stability analysis. It is suggested that the ideal ballooning instability, which has strong spatial variation along y, provides a possible mechanism for disrupting the crosstail current at onset. The problem of substorm onset remains one of the most intriguing and as yet, unresolved issues of magnetospheric physics. There is increasing appreciation among substorm researchers that the key to the problem of onset lies in the growth phase when the magnetotail is prepared for the violent relaxation dynamics that follows. During the growth phase, intensification of the cross-tail current [Kokubun and McPherron, 1981] leads to a tail-like reconfiguration of the nightside magnetic field at near-Earth distances (< 10 R_E) [Kaufmann, 1987]. Observations suggest the formation of a thin current sheet with width less than 1 R_E in the near-Earth region before the onset of the expansion phase [Mitchell et al., 1990; Sergeev et al., 1990; Lui et al., 1992]. The onset involves a sudden reduction or disruption of the cross-tail current in the near-Earth region (7-11 R_E) [Lui, 1978; Takahashi et al., 1987; Lopez et al., 1990; Jacquey et al., 1991; Lui et al., 1992; Ohtani et al., 1992]. As a result, the magnetic field configuration at low latitudes becomes more dipolar, and the plasma sheet expands. Multi-point observations of current disruption at near-Earth distances by Ohtani et al. [1992] provide some strong constraints on theoretical models of substorms. In these observations, after a period of sluggish growth (∼ 0.5-1.5hr), the cross-tail current density exhibits rapid, impulsive growth for a short time-interval (< 1 min) just before onset. Following the impulsive growth phase, the current disrupts on a very short time scale (< 10 s). These observations are of great interest because they show the presence of two distinct time scales in the growth phase, followed by a third time scale during which rapid disruption of the tail current occurs. Recently, Ma et al. [1995] have performed a two-dimensional (2D), resistive MHD simulation of the magnetotail in the high-Lundquist-number (S) regime, relevant for the weakly collisional magnetotail. The S-value realized in this simulation (including the effect of numerical diffusion) is approximately 10⁵. Ma et al. show that it is possible to account quantitatively for the slow as well as the impulsive enhancement of a thin current sheet in the growth phase within the framework of high-S MHD. However, the 2D simulations assume that y is an ignorable coordinate which imposes a stringent constraint on the dynamics of the thin current sheet. Ma et al. suggest that this may be why the 2D simulation fails to account for the observed fast time scale of current disruption. We are thus motivated to consider three-dimensional (3D) dynamics that break the assumed symmetry along y. In this paper, we demonstrate that the stretched magnetotail containing a thin current sheet in the impulsive, pre-onset phase is unstable to an ideal ballooning instability with very rapid spatial variation along y. This is the first time that a full ballooning eigenmode analysis has been carried out for realistic magnetotail configurations with thin current sheets, including line-tying boundary conditions at the ionosphere. We show that line-tying constraints have a significant effect in determining the type of ballooning modes that may be excited to trigger onset at near-Earth distances. Whereas incompressible ballooning modes are stabilized, compressible ballooning instabilities, with sub-Alfvénic growth rates, are shown to occur at near-Earth distances (≃ 10R_E) in the impulsive growth phase. Our calculation thus provides a plausible explanation of the three distinct temporal phases seen in the observations of Ohtani et al. [1992]: the slow growth, the impulsive pre-onset, and the sudden disruption of the cross-tail current at onset.