Investigate the stability of a bent cross-tail current sheet
We investigated the magnetotail’s current sheet stability to the cross-tail transversal mode utilizing analytical, as well as 2.5D linear (Korovinskiy et al., 2018c) and 3D non-linear MHD simulations (Korovinksiy et al., 2019). It is found that in plane current sheets, stable and unstable branches of the solution coexist. With increasing tilt angle, the growth rate rises and for sufficiently large tilt angles (~0.5 fmax), the stable solution becomes unobservably small compared to the unstable mode (see Fig. 1). For the maximum possible value of f ( ~40°, consisting of maximum of 33° from dipole tilt angle and 8° from non-radial propagation of the solar wind), the growth rate is 2.25 times bigger compared to the growth rate in a plane current sheet. Furthermore, it was found that the so-called double gradient instability corresponds to the compressible ballooning mode developing in a strongly stretched tail region. With downtail distance the velocity perturbation vector is rotating from the horizontal to the vertical direction, indicating the transition from the conventional ballooning mode to the double gradient mode.
Investigate the interplay of kink and sausage modes in a bent current sheet
In a 2.5D numerical simulation (Korovinskiy et al., 2018c), it was found that the symmetry of the solution of MHD equations in a bent current sheet is lost (cf. Fig. 2). For a plane current sheet (f=0), perturbations can be either symmetric (i.e., kink) or anti-symmetric (i.e., sausage) with respect to the current sheet center. In a bent current sheet, the solutions are asymmetric, consisting of a symmetric kink part and an anti-symmetric sausage part. With growing tilt angle the ratio of amplitudes of these two modes tends to unity. It was found that the asymmetry is most pronounced for f=20° and that perturbations are localized in the magnetotail’s summer hemisphere (for negative tilt angles, as used in the simulation). With this, in a bent current sheet, both kink and sausage modes coexist.
Investigate the relation of bent current sheets to substorms
In a statistical analysis (Kubyshkina et al., 2018) it was found that substorms occur almost two times more frequent when the IMF and solar wind parameters Bx and vz have the same sign as Bz. Since the magnetospheric current sheet bends for non-zero Bx and vz, one can derive a relation of current sheet bending and substorm onset out of this finding.
In a 2.5D simulation (Korovinskiy et al., 2018c) the perturbation of the potential energy (δW) in plane and bent current sheet configurations is studied. Over the course of time, a concurrence of stable (δW>0) and unstable (δW<0) modes is found (see Fig. 3). For a plane current sheet the unstable mode dominates after ~ 1.5 to 2 hours, which is rather long compared to substorm timescales. However, in bent current sheets, the unstable mode dominates much faster – for the maximum possible tilt angle (~40°) it dominates after about 5 minutes. Hence, if bending is induced on a current sheet, it becomes fully unstable after a time period that is consistent with substorm onset time scales. The same situation is also found in 3D non-linear simulations (Korovinskiy et al., 2019).
Investigate the influence of reconnection on the evolution of instabilities
In a 3D nonlinear MHD simulation it was found that reconnection enhances the growth rate of the double gradient mode for a factor of about 2, but it does not shift the threshold of non-linear stabilization of the mode (see Fig. 4). With this, reconnection affects the growth rate but not the maximum amplitude of the perturbation.
Investigate the relation of instabilities with entropy
The field line entropy (S) was calculated in Korovinskiy et al. (2018c) for different angles of current sheet bending. S demonstrates a smooth monotonic profile along the current sheet center and increases tailward for any value of tilt angle f. Hence, the stability of a Kan-like current sheet to the transversal mode is not governed by the entropy criterion.
Generalization of the instability criterion to bent current sheets
Analogous to the characteristic flapping frequency (Equation (7) in Erkaev et al., 2007) we derived the necessary instability criterion for bent current sheets, reflected in Equation (41) in Korovinskiy et al., 2019. Under the simplifying assumptions of the Double Gradient Model, Equation (41) turns into the characteristic flapping frequency of that model. Since Equation (41) includes several previously neglected terms, it is applicable in the near-Earth region (|vx|>|vz|) and also for bent current sheets. With this, equation (41) allows a representation for a much broader range of situations. Furthermore, this generalization allowed us to understand that the instability is controlled by the second derivative of the total pressure after ∂x ∂z in the near Earth region (where|vx|>|vz|) and by the second derivative of the total pressure after ∂z² more tailward (where| vx|<|vz|). Because the spatial variation of the total pressure is larger in the near-Earth region, the overall instability is controlled mainly by the mixed derivative of the total pressure.
Energy budget of double gradient/ballooning instability
The temporal energy evolution (kinetic, internal, magnetic, total energy) was investigated by means of a 3D non-linear MHD simulation (Korovinskiy et al. 2019). Three stages could be found: An initial settling phase, followed by a phase of exponential growth and finally a phase of non-linear stabilization. It is found that the kinetic energy is growing during the linear stage at the expense of the internal energy (see Fig. 6). The increase in magnetic energy is small compared to the increase of the kinetic energy and can therefore be neglected. The energy conservation within the computational box allowed the application of the energy principle of Bernstein (1958) and the mode identification (compressible ballooning mode).
Applicability of the Kan model
In Korovinskiy et al. (2018b), a generalized Kan-like model was compared with the empirical T96 Tsyganenko model.
It was found that parameters in the analytical model can be adjusted to fit a wide range of averaged magnetotail configurations (see Fig. 8). The best agreement between analytical and empirical models is obtained for the midtail at distances beyond 10–15 RE at high levels of magnetospheric activity. The essential model parameters (current sheet scale, current density) are compared to Cluster data of magnetotail crossings. The best match of parameters is found for single-peaked current sheets with mediu
Field line curvature-related stability criterion for plane current sheets
In the course of studies of the influence of the local total pressure maximum on current sheet stability (Korovinskiy et al., 2018a), a new criterion – related to the field line curvature – was derived. The plane current sheet is stable with respect to the MHD flapping mode, if the magnetic field curvature radius is decreasing in tailward direction before the X-line and increasing behind it. This criterion does not contradict the Schindler-Birn criterion (Schindler and Birn, 2004). Instead, it has advantages over it since it provides the necessary and sufficient condition for the mode stability and is more local, since it requires calculations only along the sheet center and not within the entire domain.
Generalization of the double gradient model to oblique waves
In Korovinskiy and Kiehas (2016) The double-gradient model of magnetotail flapping oscillations/instability is generalized for the case of oblique propagation in the equatorial plane. The transversal direction Y (in GSM reference system) of the wave vector is found to be preferable, showing the highest growth rates of kink and sausage double-gradient unstable modes (see Fig. 8). Growth rates decrease with the wave vector rotating toward the X direction. It is found that neither waves nor instability with a wave vector pointing toward the Earth/magnetotail can develop. These findings explain why flapping waves are observed in the Y-direction.
Dispersion curve of flapping oscillations in plane current sheet
In the simple double gradient model, the phase velocity is monotonically decreasing with wavenumber. However, by solving the exact solutions of linearized MHD equations (Korovinskiy et al. 2018a), it was found in that the dispersion curve of flapping oscillations can have a local maximum and hence the phase velocity as function of wave number can have a local maximum as well (see Fig. 9). Such behavior was observationally confirmed by Rong et al. (2018).
Occurrence rate of fast flows observed by ARTEMIS
In Kiehas et al., 2018, a five year statistical ARTEMIS study was conducted to investigate the occurrence rate of earthward and tailward fast flows near lunar orbit. It was found that a significant fraction of fast flows is directed earthward, comprising 43% (vx >400 km/s) to 56% (vx >100 km/s) of all observed flows (see Fig. 10). This suggests that near-Earth and midtail reconnection are equally probable of occurring on either side of the ARTEMIS downtail distance. For fast convective flows (vx >400 km/s), this fraction of earthward flows is reduced to about 29%, which is in line with reconnection as source of these flows and a downtail decreasing Alfvén velocity.
Dawn-dusk asymmetry of fast flows observed with ARTEMIS
More than 60% of tailward convective flows occur in the dusk sector (as opposed to the dawn sector), while earthward convective flows are nearly symmetrically distributed between the two sectors for low AL (>−400 nT) and asymmetrically distributed toward the dusk sector for high AL (< −400 nT) (see Fig. 11). This indicates that the dawn-dusk asymmetry is more pronounced closer to Earth and moves farther downtail during high geomagnetic activity. This is consistent with similar observations pointing to the asymmetric nature of tail reconnection as the origin of the dawn-dusk asymmetry of flows and other related observables. We infer that near-Earth reconnection preferentially occurs at dusk, whereas midtail reconnection (X >−60 RE) likely occurs symmetric across the tail during weak substorms and asymmetric toward the dusk sector for strong substorms, as the dawn-dusk asymmetric nature of reconnection onset in the near-Earth region progresses downtail.