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Astron. Astrophys. 331, 392-404 (1998)

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3. Equations and solutions

We simplify the problem by Fourier analyzing in [FORMULA], z and t. All perturbed quantities are assumed proportional to

[EQUATION]

where m is an integer.

In ideal MHD, the displacements in a compressible 1D cylindrical plasma can be described by a set of two first-order differential equations for the radial component of the Lagrangian displacement, [FORMULA] and the perturbed total pressure [FORMULA] (see, e.g., Appert et al. 1974):

[EQUATION]

[EQUATION]

where [FORMULA]. The other perturbed quantities ([FORMULA], [FORMULA],.....) can be computed once [FORMULA] and [FORMULA] are known. The sound speed and the Alfvén speed are defined as: [FORMULA] and [FORMULA] where the ratio of specific heats [FORMULA], as usual. The Alfvén and cusp frequency are defined by:

[EQUATION]

[EQUATION]

The coefficient functions [FORMULA], [FORMULA] and [FORMULA] depend on the equilibrium quantities and on the frequency [FORMULA]. When [FORMULA] (no twist in the magnetic field) then [FORMULA]. The coefficient [FORMULA] can be rewritten as [FORMULA] to obtain the (local) cut-off frequencies [FORMULA] and [FORMULA]:

[EQUATION]

and

[EQUATION]

To obtain the impedances at the tube boundary necessary to select eigenfrequencies and optimal driving frequencies, we have to solve the Eqs. (7)-(8) in both the internal and external region of the tube. The external region is homogeneous and non-magnetized, while the internal region is characterized by a (straight) magnetic field and (eventually) inhomogeneous equilibrium quantities.

3.1. Internal region

In a non-uniform plasma [FORMULA] and [FORMULA] are functions of position and they therefore determine an Alfvén and slow continuum. This gives possible singular points at [FORMULA] and [FORMULA] where:

[EQUATION]

or

[EQUATION]

Jump conditions can be obtained to cross the dissipative layer and avoid solving the dissipative equations. The exact results can be found in Sakurai et al. (1991), Goossens et al. (1995) and the numerical implementation in Stenuit et al. (1995).

A new variable [FORMULA] is thereby used. The treatment of the dissipative layer(s) [FORMULA] around the possible resonance(s) is based on an overlap region (Fig. 1) where the asymptotic dissipative MHD solutions (valid for [FORMULA]) and the simplified, ideal solutions (valid in the interval [FORMULA]) are matched. This matching leads to jump conditions over the resonant point. Analogue for slow resonances.

[FIGURE] Fig. 1. Schematic overview of the notations in the treatment of the dissipative layer

For the transmitted impedance, we have to use numerical integration to obtain the internal solution. We start with a power series for [FORMULA] and [FORMULA] around [FORMULA] as in Stenuit et al. (1995). The values of [FORMULA] and [FORMULA] at a value [FORMULA] are used to start the numerical integration of the ideal MHD wave equations by use of a Runge-Kutta scheme. This Runge-Kutta integration in combination with the jump conditions yields values for [FORMULA] and [FORMULA] at the boundary and therefore for the transmitted impedance. In this way we find a corresponding complex F - and G -value for each complex frequency, if the external solution is known. We subsequently iterate this procedure to locate the zeroes of these functions for a particular set of mode numbers m and [FORMULA].

3.2. External region

The waves in the uniform, non-magnetic region outside the flux tube are accurately described by the equations of ideal MHD. The solutions to these equations for [FORMULA] can be written in terms of Hankel functions as:

[EQUATION]

and

[EQUATION]

where [FORMULA] and [FORMULA] denote the Eulerian perturbation of the plasma pressure and the radial component of the Lagrangian displacement in the external region. [FORMULA] is the density outside the flux tube, and [FORMULA] and [FORMULA] are the radial wave number and frequency.

In these equations [FORMULA] and [FORMULA] are the Hankel functions of, respectively, the first and second order, and a prime on these symbols denote the derivative of the Hankel functions with respect to their argument.

The prescription to calculate the external horizontal wavenumber [FORMULA] is:

[EQUATION]

where we introduce a branch cut on the real axis as [FORMULA] in the complex [FORMULA] -plane and choose signs for [FORMULA] and [FORMULA] such that [FORMULA] (the asterisk denotes the complex conjugate) to remove the double-valuedness of the root (see Keppens 1996).

The complex impedance difference functions then become:

[EQUATION]

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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