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
where and .
Now set so that
Setting with and only taking into account second order derivatives in the damping term we get
To obtain a second order partial differential equation with constant coefficients, we eliminate the first order derivative by settings its coefficient equal to zero:
To leading order, the coefficient of X is constant in R if:
where we have chosen the positive square root to match the outgoing wave condition. We then find
To obtain a solution to the non-zero dissipation case, we set
where and . From Eq. (A.5) we then get
The full solution for the perturbed velocity then becomes:
The solution for the perturbed magnetic field can be obtained in a similar manner.
© European Southern Observatory (ESO) 2000
Online publication: January 31, 2000