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Astron. Astrophys. 339, 215-224 (1998)

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4. Calculation of the absorption coefficient

To study the resonant absorption of magnetosonic waves, we derive an expression for the absorption coefficient [FORMULA] from the amplitudes of the incident and the reflected wave at [FORMULA].

The absorption coefficient is defined as

[EQUATION]

where [FORMULA] and [FORMULA] are the [FORMULA]components of the energy flux densities of the incident and the reflected wave respectively. [FORMULA] and [FORMULA] can be expressed in terms of the [FORMULA]components of the group velocities [FORMULA] and the total wave energy densities [FORMULA] as

[EQUATION]

After some algebra (ade, Csík, Erdelyi & Goossens, 1997) the expression (24) for the absorption coefficient reduces to

[EQUATION]

where [FORMULA] and [FORMULA] are amplitudes of total pressure perturbation induced by incident and reflected waves respectively.

The terms 'incident wave' and 'reflected wave' are determined by the sign of the [FORMULA]component of the group velocity in the sense that the incident waves have [FORMULA] while the reflected waves have [FORMULA].

According to inequalities (17), the sign of the z-component of the phase velocity and the z-component of the group velocity are the same for fast magnetosonic waves and opposite for slow magnetosonic waves. Fast magnetosonic waves, therefore, carry energy towards the nonuniform layer while slow magnetosonic waves carry it in the opposite direction. Going back to the solution (15), we see that the amplitudes [FORMULA] and [FORMULA] are related to waves propagating in the positive z-direction and in the negative z-direction respectively. This means that [FORMULA] and [FORMULA] are the amplitudes of the reflected and the incident wave respectively when the wave is fast magnetosonic wave and vice versa for slow waves.

The amplitudes of the Eulerian perturbation of total pressure [FORMULA] and [FORMULA] for the incident and the reflected wave are

[EQUATION]

Finally, applying the boundary conditions of continuity of P and [FORMULA] at [FORMULA] to solutions (15) and (16), we relate the amplitudes [FORMULA] and [FORMULA] to the calculated values [FORMULA] and [FORMULA] from the nonuniform layer as follows:

[EQUATION]

The absorption coefficient (26) is now completely determined by relations (27) and (28) for an incident wave of given frequency [FORMULA] and propagation angles [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998
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