## 3. Two dimensional continuum equationsIn terms of the fluid displacement field , the linearized MHD equations can be cast in the following form: where , , and From Eqs. (2) and (3), it follows that and
. 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
where and
are the coordinates used to describe positions
within flux surfaces. We assume a time dependence
, where is the angular
frequency, and an azimuthal dependence , where
where is a 2-vector and is a 4-vector, and , ,
, and are rectangular
matrix operators. In the latter, and
appear as A formal way of solving Eqs. (20)-(22) is to invert Eq. (21), i.e., and to substitute the result in Eq. (20) to obtain a partial differential equation for that may be solved subject to appropriate boundary conditions. This formal solution breaks down on flux surfaces for which Eq. (25) supplemented with the appropriate longitudinal and azimuthal boundary conditions constitute a discrete eigenvalue problem on each flux surface . This equation determines the continuous spectrum. By varying , each discrete eigenvalue traces out a continuous set of eigenvalues , i.e., a continuum branch. The totality of branches constitute the continuous spectrum. The flux surfaces on which the formal solution method breaks down are singular surfaces in the sense that the corresponding continuum 'eigenfunctions' are singular at (Goedbloed 1975). The above equation can be reduced further to yield an explicit eigenvalue formulation. By elimination of and from Eq. (25), we obtain two coupled ordinary differential equations determining the continuous spectra, viz., where and Explicit forms of and appear in the appendix: Eqs. (A21) and (A22). Gravity only appears in the matrix and 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 are all real. When , the solutions represent stable oscillating continuum waves but when , 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 When 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 , which is the usual polarization perpendicular to the magnetic field. The slow magnetosonic wave solutions have and are polarized parallel to the magnetic field. From Eq. (26) and the explicit forms of the matrices and 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 is non-zero, the components and 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 , 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: The approximation signs indicate that the continuum ordering (Goedbloed 1975), i.e., is of the same order as and but much larger than , has been used to eliminate from the definitions of and . 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 and : where The variables , ,
where we introduced the gravitational potential
(). The term between
square brackets in the definition of The matrices and are both diagonal representing the uncoupled Alfvén and slow magnetosonic continuum waves. The matrix is an off-diagonal matrix with non-zero elements when both and , 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 conditionsUntil 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 with integer © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |