Modeling aperiodic magnetospheric oscillations

We present an analysis of a Hall-magnetohydrodynamics model of the magnetospheric plasma with finite Larmor radius effect. Through a bifurcation analysis of the resultant nonlinear system, we show that this nonlinear model does not possess a limit cycle, which rules out regular periodic oscillations with constant amplitude. However, it does result in a train of magnetosonic solitons, localized in space, with amplitudes increasing in time, which are largely in agreement with what is usually observed in the magnetopause region. We call these oscillations aperiodic magnetospheric oscillations. We emphasize that most of the train of solitary oscillations observed by the Cluster fleet and other spacecrafts do not have constant amplitudes: they either continuously increase or decrease. These train of solitons with nonconstant amplitudes is a primary solution of our model.

Figure 1

Oscillations in $b_z$ without the FLR effect. The initial points, respectively, for the largest to the smallest amplitudes are $(b_y,b_z)=(0,0.7),(0,0.9),$ and (0,1.2).

Figure 2

Nullclines (a), (b) of Eqs. (45) and (46) in the $(b_y,b_z)$ plane for different values of $\alpha$ showing the equilibrium points of the system (the intersections of $b_z$ with the $b_y=0$ line). (c) The position of the equilibrium points are shown with the stream plot of the flow in the $(b_y,b_z)$ plane for the case $\alpha ={82}^{\circ }$. The three intersections of the flows with the central horizontal line indicates the corresponding equilibrium points.

Figure 3

(a) The stream plot of Eqs. (45) and (46) in the $b_z\text{−}{b}_y$ plane is shown for a realistic parameter set. The solid line shows the overall flow pattern. (b) The progression of behaviors of the equilibrium points of Eqs. (45, 46) for the same set of parameters (see Sec. 5) in the $\left(\kappa \text{−}{b}_z\right)$ plane. The arrow indicates the Hopf bifurcation point.

Figure 4

Oscillations of the geomagnetic field $B$ as detected by Cluster spacecraft 1 on February 3, 2002 (a) between 08:30:00 and 09:15:00 hours (UT). The dashed lines indicate the divisions of groups of solitons, and the arrows indicate few individual solitons. (b) The plot of the corresponding power spectrum density $\mathcal{P}\left(f\right)$ of these oscillations, plotted against the sampling frequency $f$, which shows no detectable linear wave train.

Figure 5

(a) Alfvén velocity $V_A$, ion inertial length, ${\lambda }_i$ plasma $\beta$, and pressure anisotropic parameter $a_p$ as detected by Cluster spacecraft 1 corresponding to the event shown in Fig. 4. (b) The field variation after an MVA analysis.

Figure 6

Position of the Cluster spacecrafts between 08:00:00 and 09:00:00 hours (UT) on February 3, 2002, as shown by the red arrow (reproduced in scale with the help of NASA's 4D Orbit Viewer). The figure is drawn with earth at the center in the GSE coordinates.

Figure 7

A numerical solution of Eqs. (45) and (46) for the parameter regime shown in Fig. 5 and Table 1.

Figure 8

Domain of nonlinear oscillations of Eqs. (45) and (46) in the $M_A\text{−}cos\alpha$ plane for $\beta =20$. The plasma parameters are taken from the Cluster observations (see Table 1) $a_p=-0.2,{\beta }_e=0.3$ with $\gamma =1$ and $\kappa =0$. The value of $\left|k\right|$ is color coded from black (small $\left|k\right|$) to red (large $\left|k\right|$).