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Astron. Astrophys. 326, 1241-1251 (1997)

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5. 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 waves 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]

The terms 'incident wave' and 'reflected wave' are therefore determined by the sign of the [FORMULA] component of the group velocity in the sense that incident waves have [FORMULA] while reflected waves have [FORMULA]. [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA] are related to the waves propagating in the positive z -direction and 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 a fast magnetosonic wave. In the case of slow magnetosonic waves the situation is reversed. The amplitudes [FORMULA] and [FORMULA] are now related to the incident and the reflected wave respectively.

Therefore, we can write the total pressure amplitudes [FORMULA] and [FORMULA] for the incident and the reflected wave as

[EQUATION]

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

[EQUATION]

is the sum of the kinetic [FORMULA], thermal [FORMULA] and magnetic energy [FORMULA] given as

[EQUATION]

[EQUATION]

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:

[EQUATION]

where [FORMULA] and the superscripts [FORMULA] 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

[EQUATION]

When we substitute expressions (31) into (30), we can write

[EQUATION]

[EQUATION]

[EQUATION]

so that the total energy densities (29) for the incident and the reflected wave become:

[EQUATION]

Consequently, the absorption coefficient (26) can be reduced to the simple expression

[EQUATION]

with [FORMULA] and [FORMULA] given by relations (28).

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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