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Astron. Astrophys. 354, 334-348 (2000)

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Appendix A: calculation of the WKB solutions

The phase mixing equation for the perturbed velocity in a gravitationally stratified atmosphere with a radially diverging background magnetic field is given by

[EQUATION]

where [FORMULA] and [FORMULA].

Now set [FORMULA] so that

[EQUATION]

Setting [FORMULA] with [FORMULA] and only taking into account second order derivatives in the damping term we get

[EQUATION]

To obtain a second order partial differential equation with constant coefficients, we eliminate the first order derivative by settings its coefficient equal to zero:

[EQUATION]

To leading order, the coefficient of X is constant in R if:

[EQUATION]

where we have chosen the positive square root to match the outgoing wave condition. We then find

[EQUATION]

Neglecting the second term in the coefficient of X which is of smaller order, Eq. (A.1) becomes:

[EQUATION]

To obtain a solution to the non-zero dissipation case, we set

[EQUATION]

where [FORMULA] and [FORMULA]. From Eq. (A.5) we then get

[EQUATION]

and with

[EQUATION]

we find,

[EQUATION]

The full solution for the perturbed velocity then becomes:

[EQUATION]

with [FORMULA]

and [FORMULA].

The solution for the perturbed magnetic field can be obtained in a similar manner.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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