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Astron. Astrophys. 322, 995-1006 (1997)

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3. Two dimensional continuum equations

In terms of the fluid displacement field [FORMULA], the linearized MHD equations can be cast in the following form:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA], and P are the equilibrium density, magnetic field, and pressure, respectively, and [FORMULA], p, [FORMULA], and [FORMULA] are the perturbations of the magnetic field, pressure, density, and total pressure, respectively. This form of the linearized MHD equations shows that the derivatives of most of the perturbed quantities are in the direction parallel to the magnetic field (the [FORMULA] operators), whereas derivatives across the magnetic surfaces only occur through the terms [FORMULA] and [FORMULA].

From Eqs. (2) and (3), it follows that [FORMULA] and [FORMULA]. Hence, the magnetic field and current density lie on flux surfaces. Therefore, it is most natural to study the set of equations (15)-(19) in a flux coordinate system [FORMULA] where [FORMULA] and [FORMULA] are the coordinates used to describe positions within flux surfaces. We assume a time dependence [FORMULA], where [FORMULA] is the angular frequency, and an azimuthal dependence [FORMULA], where m is the azimuthal mode number. Single azimuthal Fourier modes can be considered because the equilibrium is axisymmetric and, as a consequence, these Fourier modes do not couple. In terms of generalized flux coordinates ([FORMULA], [FORMULA], [FORMULA]), the linearized MHD equations can then be written in an elegant form that stresses derivatives across the magnetic flux surfaces (Kieras & Tataronis 1982, Poedts & Goossens 1991):

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is a 2-vector and [FORMULA] is a 4-vector,

[EQUATION]

and [FORMULA], [FORMULA], [FORMULA], and [FORMULA] are rectangular matrix operators. In the latter, [FORMULA] and [FORMULA] appear as parameters and the derivatives of [FORMULA] and [FORMULA] only appear as linear combinations of first order derivatives. Both [FORMULA] and [FORMULA] are expressed in contravariant form, i.e., [FORMULA], and [FORMULA]. An explicit form of these equations is given in the appendix.

A formal way of solving Eqs. (20)-(22) is to invert Eq. (21), i.e.,

[EQUATION]

and to substitute the result in Eq. (20) to obtain a partial differential equation for [FORMULA] that may be solved subject to appropriate boundary conditions. This formal solution breaks down on flux surfaces [FORMULA] for which

[EQUATION]

Eq. (25) supplemented with the appropriate longitudinal and azimuthal boundary conditions constitute a discrete eigenvalue problem on each flux surface [FORMULA]. This equation determines the continuous spectrum. By varying [FORMULA], each discrete eigenvalue traces out a continuous set of eigenvalues [FORMULA], i.e., a continuum branch. The totality of branches constitute the continuous spectrum. The flux surfaces [FORMULA] on which the formal solution method breaks down are singular surfaces in the sense that the corresponding continuum 'eigenfunctions' are singular at [FORMULA] (Goedbloed 1975).

The above equation can be reduced further to yield an explicit eigenvalue formulation. By elimination of [FORMULA] and [FORMULA] from Eq. (25), we obtain two coupled ordinary differential equations determining the continuous spectra, viz.,

[EQUATION]

where

[EQUATION]

and

[EQUATION]

Explicit forms of [FORMULA] and [FORMULA] appear in the appendix: Eqs. (A21) and (A22). Gravity only appears in the matrix [FORMULA] and [FORMULA] is the third covariant component of the gravitational acceleration.

Since all matrices are Hermitian, Eq. (26) supplemented with the boundary conditions is a Hermitian eigenvalue problem. Hence, the eigenvalues [FORMULA] are all real. When [FORMULA], the solutions represent stable oscillating continuum waves but when [FORMULA], the solutions represent unstable continuum waves. The numerically calculated continua in this paper are all stable.

In Poedts & Goossens (1991), Eq. (26) was evaluated for an arcade model in a special geometry (constant Jacobian with the contravariant magnetic field components only depending on the flux coordinate) so that the H matrix vanished. Furthermore, gravity was not taken into account so that the matrix [FORMULA] vanished as well.

When [FORMULA] is zero, i.e., when the magnetic field is untwisted, the two differential equations (26) are no longer coupled and split into separate equations for the well-known Alfvén and slow magnetosonic continua. The Alfvén wave solutions have [FORMULA], which is the usual polarization perpendicular to the magnetic field. The slow magnetosonic wave solutions have [FORMULA] and are polarized parallel to the magnetic field. From Eq. (26) and the explicit forms of the matrices [FORMULA] and [FORMULA] in the appendix, it can easily be seen that gravity does not influence the Alfvén continua in the case of untwisted magnetic fields. This is due to the fact that the polarization of the Alfvén waves is perpendicular to the direction of the gravitational acceleration. However, the slow magnetosonic continua are influenced by gravity since the polarization of the corresponding waves has a non-zero component in the direction of gravity.

When [FORMULA] is non-zero, the components [FORMULA] and [FORMULA] no longer correspond to displacements in a flux surface that are perpendicular and parallel to the magnetic field. Although the notion of pure Alfvén and slow magnetosonic continuum waves must be abandoned when [FORMULA], it is convenient to project Eq. (26) onto directions parallel and perpendicular to the magnetic field and to rewrite the equations in terms of the parallel and perpendicular displacement components. The projected equations clearly show the coupling between Alfvén and slow magnetosonic continuum waves and are helpful in interpreting the results obtained in the next section. Within an astrophysical context these projected equations were first derived by Poedts et al. (1985). Within a plasma fusion context they were derived, without taking into account gravity effects, by Goedbloed (1975) and Pao (1975). To get a nice symmetrical form of the projected equations the following variables are used:

[EQUATION]

The approximation signs indicate that the continuum ordering (Goedbloed 1975), i.e., [FORMULA] is of the same order as [FORMULA] and [FORMULA] but much larger than [FORMULA], has been used to eliminate [FORMULA] from the definitions of [FORMULA] and [FORMULA]. Equality holds when the coordinate system is orthogonal. After some tedious algebra one then finds the following symmetric form of the continuum equations in terms of the components [FORMULA] and [FORMULA]:

[EQUATION]

where

[EQUATION]

and

[EQUATION]

[EQUATION]

The variables [FORMULA], [FORMULA], U, and V are defined by:

[EQUATION]

where we introduced the gravitational potential [FORMULA] ([FORMULA]). The term between square brackets in the definition of V is a modification of the Brünt-Väisälä frequency which now becomes a tensor. The eigenvalue equation (30) formally corresponds to Eq. (57) of Poedts et al. (1985), but the explicit expressions involving the matrices [FORMULA], [FORMULA], [FORMULA], and [FORMULA] have been simplified substantially here.

The matrices [FORMULA] and [FORMULA] are both diagonal representing the uncoupled Alfvén and slow magnetosonic continuum waves. The matrix [FORMULA] is an off-diagonal matrix with non-zero elements when both [FORMULA] and [FORMULA], i.e., the pressure and the geodesic curvature of the magnetic field should not vanish. Therefore, if one includes the chromospheric/photospheric boundary regions in a spectral study of coronal magnetic flux tubes, one can expect considerable coupling between Alfvén and slow magnetosonic waves. It should be noted that the coupling between these two types of continuum waves occurs if and only if the considered equilibrium is two-dimensional. For one-dimensional equilibria the geodesic curvature is zero and the component of the gravitational acceleration along the magnetic field necessarily vanishes.

3.1. Boundary conditions

Until now we have not specified what kind of boundary conditions we impose on the displacement vector field. By taking an azimuthal dependence of the form [FORMULA] with integer m we have already implicitly assumed periodicity in the angle [FORMULA]. The longitudinal boundary conditions need to be specified yet. The results shown in the next section are obtained with periodic longitudinal boundary conditions. Clearly, coronal magnetic flux tubes are not periodic structures. However, as a consequence of the huge density stratification, the amplitudes and derivatives at the bases are extremely small compared with the amplitudes and derivatives at the apex. Hence, we do not have to force the amplitude to be zero by means of the boundary conditions, i.e., we do not have to impose line-tied or flow-through boundary conditions, and we can just use periodic boundary conditions.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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