Ballooning modes and their stability in a near-Earth plasma
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 13 February 2012
Accepted: 22 July 2012
Published: 10 June 2013
As a possible trigger of the substorm onset, the ballooning instability has been often suggested. The ballooning disturbances in a finite-pressure plasma immersed into a curved magnetic field are described with the system of coupled equations for the Alfven and slow magnetosonic modes. The spectral properties of ballooning disturbances and instabilities can be characterized by the local dispersion equation. The basic system of equations can be reduced to the dispersion equation for the small-scale in transverse direction disturbances. From this relationship the dispersion, instability threshold, and stop-bands of the Alfvenic and slow magnetosonic modes have been determined. The field-aligned structure of unstable mode is described with the solution of the eigenvalue problem in the Voigt model. We have also analyzed in a cylindrical geometry an eigenvalue problem for the stability of ballooning disturbances with a finite scale along the plasma inhomogeneity. The account of a finite scale in the radial direction raises the instability threshold as compared with that in the WKB approximation.
1. Introduction: Ballooning Instability of Near-Earth Plasma as a Substorm Trigger
The key dilemma of the physics of terrestrial space environment is related to the identification of the substorm onset mechanism: does it occur in the magnetotail owing to magnetic field reconnection, or in a closed field line region as a result of some still unidentified instability? Tamao and his colleagues were the first who suggested that the ballooning instability could be a possible trigger of the substorm explosive phase (Miura et al., 1989; Ohtani and Tamao, 1993). This instability can be imagined as a distortion of radial gradient of hot plasma pressure by locally outward expansion and inward intrusion due to the azimuthally oscillating mode. The analysis of satellite observational data led Roux et al. (1991) to the suggestion that an instability driven by the plasma pressure gradient is responsible for the field-aligned current generation during the substorm onset. Later the idea about the ballooning instability as an onset trigger has been extensively elaborated (e.g., Lee and Wolf, 1992; Cheng et al., 1994; Liu, 1997; Cheremnykh et al., 2004; Agapitov et al., 2007).
In realistic magnetosphere the mechanisms of the ballooning instability in the near-Earth tail and reconnection in a distant magnetotail are probably coupled. The computer experiments with advanced models of the magnetosphere showed that the substorm onset was caused by violation of the balance between the thermal plasma pressure and Ampere’s force, resulting into the plasma expulsion and field lines extension into the magnetotail (Raeder et al., 2010). As a result, decrease of the magnetic component normal to the current sheet destabilized the tearing instability and stimulated the magnetic field reconnection. Thus, though the main substorm power is released via the reconnection, the substorm onset trigger could be the ballooning instability. To understand better the physical mechanisms of the processes involved in the substorm development, the results of the numerical modeling and in-situ satellite observations are to be compared with simplified, but more explicit, theoretical models.
A theoretical approach to the study of the ballooning instability is based on a complicated system of coupled equations for the poloidal Alfven waves and slow magnetosonic (SMS) modes in a finite-pressure plasma, immersed in a curved magnetic field B (e.g., Southwood and Saunders, 1985; Walker, 1987; Hameiri et al., 1991; Klimushkin and Mager, 2008). Favorable conditions for the instability growth emerge at a steep plasma pressure drop held by curved field lines. Such a condition may occur before sub-storm onset on strongly extended field lines.
The easiest way to comprehend qualitatively the basic features of the unstable modes and instability condition is the analysis of the local dispersion equation. The dispersion equation, obtained using a local analysis of this system, is widely used for geophysical applications both for the examination of plasma stability, and for the description of spectral properties of ULF wave phenomena in the nighside auroral magnetosphere (Safargaleev and Maltsev, 1986; Ohtani and Tamao, 1993; Liu, 1997; Golovchanskaya and Kullen, 2005).
However, the exact form of the dispersion equation used by different authors happens to be somewhat different, and the obtained results differ, too. Thus, it is necessary to check the derivation of the dispersion equation from basic MHD equations and the transfer to various limiting cases, which will be done in this paper.
In order to evaluate a field-aligned scale of ballooning-unstable modes, we consider the eigenvalue problem in the self-consistent analytical model of magnetic field and finite-pressure plasma (Voigt, 1986). However, an estimation of the instability criterion in the local approximation for a particular magnetic shell has necessarily a qualitative character because of the WKB approximation in the radial direction used during the solution of relevant equations. In a realistic situation an unstable mode near a steep gradient of the plasma pressure has a finite scale across the magnetic shells, which cannot be described by the WKB approximation.
Therefore, in this paper we consider the global stability with the use of very simplified magnetic field geometry (a cylindrical field with constant curvature field lines), which has enabled us to obtain some analytical results.
2. MHD Plasma Equilibrium and Linearized Dynamic Equations
We consider a finite pressure plasma confined by a 2D curved magnetic field B(x, z) with a local curvature radius R c . We introduce a local orthogonal basis, organized by the B field geometry: e3 = B/B is in field-aligned direction, and e1 = e2 × e3 corresponds to the radial direction across magnetic shell. Along e2 = e y (Y-axis) the system is homogeneous.
The typical time scale of the substorm explosive phase (~1−2 min) is much less than the Alfven transit time along the extended field lines from the magnetotail to the ionosphere (~10 min). Therefore, influence of the ionospheric boundaries on the ballooning mode properties can be neglected, assuming that the growing disturbances are localized in the near-equatorial region of the nightside magne-tosphere. The influence of the ionospheric boundary conditions on the ballooning modes was considered in many papers (e.g., Cheremnykh and Parnowski, 2006).
2.1 Asymptotic theory of transverse small-scale disturbances
For small-scale in the transverse (across B) direction disturbances the linearized MHD equations of a finite pressure plasma can be simplified and reduced to a system of ordinary differential equations for coupled Alfven and SMS modes. General approach to the 3D case with account for the gravity and plasma rotation effects was outlined by Hameiri et al. (1991). Our analysis of 2D configuration is given in another form.
The relationship between different disturbance components is given by formulas , and . Both Alfven and SMS modes convey the field-aligned current . For the ballooning disturbances (k2 ≪ k1) the component b2 → 0, hence .
2.2 Local dispersion equation
2.3 The possibility of total reflection of poloidal Alfven waves
3. Field-Aligned Structure of Unstable Modes
4. Global MHD Stability of the Ballooning Mode in a Cylindrical Geometry
In contrast to the local analysis of the plasma stability in Section 2, here we perform a global analysis, but in the framework of a simple cylindrical model. The cylinder axis is along coordinate y, which corresponds to the azimuthal direction in the magnetosphere. Magnetic field lines are circles with radius r. This simplified model possesses all the typical features necessary for occurrence of the ballooning instability: field line curvature and a plasma pressure gradient. This model can be considered as an element of a more general configuration, specifically in the region of strongly disturbed field lines. The structure of the disturbance harmonic along a field line is assumed to be ∝ exp(− iωt + ivθ), where the parameter v determines the field-aligned wave number as kǁ = v/r.
5. The Boundary Problem for the Growth Rate Determination
We consider the boundary problem for the system (15) in a semi-infinite interval with the boundary condition ξ(R e ) = 0 and the requirement of bounded ξ (r ) as r → ∞. The system parameters for which instability is possible are determined by the occurrence of discrete eigenvalues . The values , when this problem has a solution, determine the growth rates Γ = Im ω > 0 for the corresponding eigenfunctions.
The dependence of eigenvalues on the azimuthal wave number k y for a given v = 0.5 is shown in Fig. 5 (right-hand panel). The model used enables us to consider the instability pattern for an arbitrary k y . This consideration shows that the instability is possible even for azimuthally large-scale modes, . The growth rate increases rapidly with the increase of k y and at gradually reaches the saturation.
The description of the ballooning instability by (Ohtani and Tamao, 1993) owing to a mathematical error (Liu, 1997) resulted in an incorrect conclusion about the possible instability of both wave branches. Miura et al. (1989) predicted a possible instability of Alfven-type disturbances with the growth rate . However, their assumption u = 0 (or ξǁ = 0) used for isolating the Alfven mode turns out to be inconsistent with the second equation from the basic system (4). Analysis of the dispersion relationship (6) in the poloidal limit k1 = 0 shows that no branch can intersect the line . The relation from (Miura et al., 1989) contradicts this condition. Thus, the Alfven-type branch is always stable, .
The favorable conditions for the balloon instability growth may occur under strongly extended into the magne-totail field line before the substorm onset (Zhu et al., 2009). At the linear stage of the ballooning instability a disturbance grows exponentially, though drift effects may produce oscillatory growth and azimuthal drift with the velocity of about the Larmor drift velocity (Miura et al., 1989).
Using the energy principle, the ideal MHD stability of ballooning-type perturbations for a “hard ” ionospheric boundary condition was considered by Lee and Wolf (1992). For self-consistent analytical model of the magnetosphere, they found that if a magnetotail configuration was stable to interchange, it was also stable against symmetric ballooning. We suppose that in a realistic magnetosphere the ballooning modes are to be localized near the mag-netospheric equator, where the parameter β and curvature rapidly increase. Indeed, the field-aligned structure of unstable ballooning modes described within the Voigt model is strongly localized in the vicinity of the top of field lines.
In a radial direction the growing disturbances are localized in the region of pressure gradient, whereas the scale of fundamental mode is about the pressure inhomogeneity scale. The radial fundamental mode has a lowest threshold in respect to the parameter β0, whereas the threshold value β0 is higher than it follows from the local criterion. The az-imuthally small-scale disturbances with k y a ≫ 1 have the growth rate of the fundamental mode Γ ≃ (V A /a)ImΩ ≃ 0.03 c−1 for V a = 500 km/s and a = 0.8R e . This value agrees with the estimate following from the local approximation (9). The characteristic growth time of the instability ~ Γ−1 ≃ 30 s is about the typical time scale of the substorm explosive phase.
The cylindrical model has enabled us to consider the stability for an arbitrary k y , which showed that even an azimu-tally large-scale mode, k y a ≪ 1, can be destabilized. Such mode, named KY0 mode, indeed was found unexpectedly during numerical modeling (Raeder et al., 2010). The instability growth rate increases with the increase of k y in the range , and reaches the saturation for the azimuthal wave numbers . This behavior fits the results of the numerical modeling (Zhu et al., 2004).
For the considered model the source of free energy for the ballooning instability is the excess of the hot plasma pressure in the radial direction. However, our model does not take into account few factors which might be significant for the instability development: pressure anisotropy (Cheng and Qian, 1994), drift effect (Miura et al., 1989), kinetic effects (Klimushkin and Mager, 2008), and the azimuthal pressure gradient related to a background field-aligned current (Ivanov et al., 1992; Golovchanskaya and Maltsev, 2003). The ballooning instability can be excited nonlin-early by an external trigger (Hurricane et al., 1999). For such “MHD detonation ” the magnetospheric plasma must be near the threshold level, determined by the linear theory of instability, and an external trigger must have a magnitude sufficient to transfer the system into the nonlinear explosive growth phase.
We have discussed the dispersion relationship, instability threshold, and stop-bands for the Alfvenic-type and SMS-type modes which can be used for space applications. The Alfven-type branch is always stable. For poloidal Alfven waves with frequencies less than the cut-off frequency (12) a non-transparent region may occur, which makes the wave propagation along the whole field line impossible.
In the region with a steep drop of the finite pressure plasma, held by curved magnetic field lines, an instability of the SMS-type mode may become feasible. When the ballooning instability threshold is exceeded the hot plasma expands locally outward. The disturbance grows aperiodi-cally, though drift effects (neglected here) may produce an oscillatory growth. Unstable disturbance is strongly localized in the region of a field line with high β and strong curvature.
In the radial direction unstable disturbances are non-propagating modes localized near the plasma gradient. The fundamental mode has the lowest threshold, whereas the threshold per se is somewhat higher than what follows from the local criterion.
This study is supported by grant 13-05-90436 from RFBR and Program 22 of Russian Academy of Sciences. We appreciate constructive suggestions of both reviewers.
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