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Astron. Astrophys. 327, 377-387 (1997)

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3. Numerical procedure and boundary conditions

We solve Eq. (13) numerically subject to boundary conditions at the centre of the sheet, so that one has control over the parity of the solution (kink or sausage). At [FORMULA] we take [FORMULA] to be zero; the integration range is therefore [FORMULA]. Lengths and speeds are normalised against the half-width a of the sheet and the Alfvén speed [FORMULA] in the exterior; frequencies are measured in units of [FORMULA]. The second order ordinary differential equation (13) is written in terms of two first order equations in the new functions [FORMULA] and [FORMULA]:

[EQUATION]

such that

[EQUATION]

The boundary conditions at [FORMULA] for the kink mode are given by [FORMULA], [FORMULA]; for the sausage mode, [FORMULA] and [FORMULA] at [FORMULA]. The constant c is arbitrary. We impose that [FORMULA] is zero at [FORMULA] for both modes.

To obtain a solution of the Eq. (13) we use a fixed value of [FORMULA] and integrate the two first order equations (22) using the NAG routine D02BEF. The integration is performed between [FORMULA], where proper initial conditions for [FORMULA] and [FORMULA] are available, and [FORMULA], where the condition [FORMULA] must be satisfied. A Newton iteration, with NAG routine C05AXF, is done by changing [FORMULA] and integrating from [FORMULA] to [FORMULA] until the boundary condition at the second point is satisfied. The value of [FORMULA] is chosen to be large, typically [FORMULA], except for short wavelengths where it was found that a smaller [FORMULA] (e.g. 10a) worked better. For long (short) wavelengths the integration range is wide (narrow), in such a way that the frequencies and eigenfunctions are not affected by the boundaries. On locating a solution to Eq. (13), the velocity is examined in the exterior of the sheet. We consider only those solutions where the velocity decreases to zero outside the sheet, thus imposing an exponentially decreasing solution. We have made a careful comparison (see Smith 1997) of the results produced by this numerical procedure and those obtained analytically for the simpler problem of a magnetic slab (discussed by Edwin et al. 1986) and found excellent agreement, so we are confident that the procedure correctly determines the evanescent modes of a current sheet.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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