We consider a finite pressure plasma confined by a 2D curved magnetic field B(x, z) with a local curvature radius R
. 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-axis) the system is homogeneous.
Derivatives along the basis vectors are ∇
∙ ∇. The inhomogeneities of plasma and magnetic field are characterized by 3 local parameters: κP = P−1∇1P, κ
= B−1∇1B, and the field line curvature . The local equilibrium condition of plasma with scalar pressure P(x, z) can be written via these κ -parameters as follows:
Under equilibrium the plasma pressure P(x, z) is constant along a field line.
For the harmonic disturbance ~exp(− iωt) the linearized MHD equations are
whereρis unperturbed density, ξ is the plasma displacement, and b and p are disturbed magnetic field and pressure. We exclude b from the first equation (2) and proceed from the variables ξ3, p to the new variables: u = ∇ ∙ ξ and the normalized disturbance of the total pressure q = μ0B−2(p + Bb3/μ0). The variable u characterizes the plasma compression and is related to its field-aligned displacement by the relationship . As a result we get the linearized MHD equations, which coincide with equations for 2D case from (Cheng, 2002), but for different variables:
Here the following notations have been introduced: V
= B(μ0ρ)−1/2 is the Alfven velocity, is the Alfven wave number, V
= (γP/ρ)1/2 is the sound velocity, is the sound wave number, is the “cusp ” velocity, and . We introduce also the following operators:
Alfven poloidal ,
Alfven toroidal ,
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 asymptotic solution for the harmonics (ξ, u,q) ∝ exp(ik1x1 + ik2x2) of the system (2) for large transverse wave numbers may be searched in the form
(similar for u and q) where ε is a small parameter.
From the system of order ε−1 it follows that the azimuthal displacement component ξ2 and the perturbation of the total pressure q are small values of the order of . Further, in the zero-th order of ε the closed system of equations occurs
The system (4) is a special case of the system (71) from (Hameiri et al., 1991). It can be re-written in the following form (Klimushkin, 1998)
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
The spectral properties of the ballooning modes can be qualitatively understood with the use of local dispersion equation. Let us suppose that a disturbance has a small scale not in the transverse direction only, but also along the field line. In the geometrical optics approximation ȝ exp (ikǁx3) all operators turn into numerical factors , and . The linear system of differential equations (4) now becomes an algebraic system. The dispersion equation is obtained by setting the determinant of this system to zero, namely
where . Equation (6) is a quadratic in Ω2 with roots
where H = βκ
(2 + γβ)/2] sin2 α. The roots (7) are real for a real kǁ.
The relationship (7) describes two branches: fast (Ω+) mode, which transforms into an Alfven wave as β → 0, and slow (Ω −) mode. Their dispersion curves are shown in Fig. 1. The fast branch is stable for any kǁ. Only the slow mode can be unstable under the condition
This inequality is the generalization for oblique disturbance with k1 ≠ 0 of the ballooning instability condition (Liu, 1997). The maximal value of the growth rate as estimated from the dispersion relation (7) is
If κb ≪ κc, then, taking into account (1), (9) is reduced to .
The asymptotic formulas for disturbances with small scales in the field-aligned direction are obtained from (7) assuming kǁ ≫ κ
where . Here the parameter Λ = κ
has been introduced. Its absolute value ǀ Λ ǀ = R
/a is the ratio between the curvature radius R
and the plasma inhomogeneity scale . When Λ < 0 (outward pressure gradient) both branches are stable. When Λ > 0 (inward pressure gradient), the interval emerges according to (8), where , thus the slow mode branch turns out to be unstable.
2.3 The possibility of total reflection of poloidal Alfven waves
Analysis of the dispersion equation (7) shows that for real Ω there may occur regions where (see dispersion curves in Fig. 2). These regions are non-transparent (opaque) for poloidal Alfven waves. Their occurrence can be qualitatively shown with the use of the formula (7) for . Wave meets the turning points when its frequency Ω matches the cut-off frequency
The exact formula (12) differs somewhat from that in (Mager et al., 2009) obtained with the use of asymptotic relationship (11). In a disturbed nightside magne-tosphere with tailward extended field lines, the curvature ǀκ
(x3)ǀ increases sharply equatorward, therefore, in the near-equatorial region the condition may be fulfilled, and becomes negative in this area. This region with high β and locally curved field lines is non-transparent for poloidal Alfven waves.