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

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Appendix A: derivation of the 2D continuum equations

In this appendix, we derive the equations governing the two-dimensional continua for axisymmetric magnetic flux tubes. The equations are derived for curvilinear coordinates. Hence, extensive use of tensor calculus is made (Arfken, 1990).

A flux coordinate system [FORMULA] is used in which the coordinate [FORMULA] labels the flux surfaces. Since [FORMULA] is a flux coordinate, [FORMULA] and, hence, [FORMULA]. The covariant components of the metric tensor are denoted as [FORMULA]. The Jacobian of the coordinate transformation is denoted by J. A [FORMULA] with a single subscript is a covariant component of the gravitational acceleration [FORMULA]. The symbol [FORMULA] represents the parallel gradient operator [FORMULA],

[EQUATION]

A differential operator will work on all the factors to its right unless these factors are surrounded by parentheses in which case the differential operator only works on the factors between the parentheses. A last point of notation is the use of the Einstein convention for repeated indices.

The contravariant components of the perturbed magnetic field [FORMULA] defined in Eq. (16) are given by:

[EQUATION]

whereas Eqs. (17)-(18) for the perturbed pressure and the density become:

[EQUATION]

Using Eqs. (A2)-(A5) and the equilibrium force balance equation, we can obtain an expression for [FORMULA] in terms of the total perturbed pressure [FORMULA] and the contravariant displacement components, viz.,

[EQUATION]

[EQUATION]

From this equation it then follows that

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

On substitution of Eqs. (A7)-(A8) in the [FORMULA] -component of the momentum equation (15), we obtain an expression for [FORMULA]:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

After substitution of Eq. (A7), the covariant form of the [FORMULA] - and [FORMULA] -components of the momentum equation (15) read:

[EQUATION]

and

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In the same way, substitution of Eq. (A7) in Eq. (16) results in the following expressions for the [FORMULA] - and [FORMULA] -components of Eq. (16):

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The six equations (A8)-(A13) now have the required form expressed by Eqs. (20)-(21), where Eqs. (A8)-(A9) correspond to Eq. (20) while Eqs. (A10)-(A13) correspond to Eq. (21). The equations governing the continuum are found by equating the right hand sides of Eqs. (A10)-(A13) to zero:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Solving for [FORMULA] and [FORMULA] from Eqs. (A16)-(A17) and substituting the results in Eqs. (A14)-(A15) leads to the following set of coupled, ordinary second order differential equations in [FORMULA]:

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The elements of the matrices [FORMULA] and [FORMULA] are given by:

[EQUATION]

and

[EQUATION]

In the derivation of Eqs. (A8)-(A22) we made use of the facts that the equilibrium quantities do not depend on [FORMULA], that [FORMULA] is a flux function, and hence only depends on [FORMULA], and that [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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