5. Calculation of the absorption coefficient
To study the resonant absorption of magnetosonic waves, we derive an expression for the absorption coefficient from the amplitudes of the incident and the reflected waves at .
The absorption coefficient is defined as
where and are the components of the energy flux densities of the incident and the reflected wave respectively. and can be expressed in terms of the components of the group velocities and the total wave energy densities as
The terms 'incident wave' and 'reflected wave' are therefore determined by the sign of the component of the group velocity in the sense that incident waves have while reflected waves have . and are equal in absolute value as they are related to the same location of the plasma. Consequently, they cancel out in the expressions (26 and (27) for calculating the absorption coefficient.
According to the inequalities (16) 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. It means that fast magnetosonic waves carry energy in the direction of the wave propagation, and slow magnetosonic waves carry energy in the opposite direction of the wave propagation.
In Eqs. (11) the amplitudes and are related to the waves propagating in the positive z -direction and the negative z -direction respectively. This means that and are the amplitudes of the reflected and the incident wave respectively when the wave is a fast magnetosonic wave. In the case of slow magnetosonic waves the situation is reversed. The amplitudes and are now related to the incident and the reflected wave respectively.
Therefore, we can write the total pressure amplitudes and for the incident and the reflected wave as
As we know the total pressure amplitudes of the incident and the reflected wave for both the slow and fast magnetosonic waves, we can calculate the related energy densities needed in (27) and the absorption coefficient (26).
The total wave energy density
is the sum of the kinetic , thermal and magnetic energy given as
The averaged values of perturbation squares in the above expressions are equal to one half of the related amplitude squares in the case of harmonic waves. Thus:
where and the superscripts indicate the reflected and the incident wave respectively, are omitted.
The perturbed quantities in (30) can all be expressed in terms of the Eulerian perturbation of total pressure for the region 2 with a uniform plasma as
When we substitute expressions (31) into (30), we can write
so that the total energy densities (29) for the incident and the reflected wave become:
Consequently, the absorption coefficient (26) can be reduced to the simple expression
with and given by relations (28).
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998