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

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6. Results and conclusions

The results are obtained for the equilibrium model specified by (3). We have taken [FORMULA], [FORMULA], [FORMULA] and [FORMULA] which corresponds to the density ratio [FORMULA]. Lengths, velocities and time are expressed in terms of L, [FORMULA] and [FORMULA], respectively.

The absorption coefficient [FORMULA] is calculated for a given wave frequency [FORMULA] and a wave vector [FORMULA] that satisfy the equation (10) for propagating waves in the region 2 and the condition [FORMULA] for evanescent waves in the region 1. The wave vector components are further related to the wave propagation angles [FORMULA] and [FORMULA] according to expressions (17).

Since we do not intend to consider the Alfvén resonance alone, the frequencies and the wave vectors are chosen in such a way that they satisfy the condition [FORMULA]. This means that the driving frequency remains smaller than the maximal value of the cusp frequency as shown schematically in Fig. 2.

[FIGURE] Fig. 2. A schematical view of the considered wave propagation domains for slow and fast magnetosonic waves.

Thus we investigate how the properties of the wave absorption due to the resonant excitation of slow and Alfvén continua by slow and fast magnetosonic waves, depend on the wave frequency [FORMULA] and the propagation angles [FORMULA] and [FORMULA]. The angle [FORMULA] varies from [FORMULA] to [FORMULA] while the angle [FORMULA] ranges between [FORMULA] and [FORMULA]. In our model, however, the wave characteristics are symmetric with respect to [FORMULA] since [FORMULA] and [FORMULA] appear only through their squares. For this reason, [FORMULA] is taken in calculations.

In the case of a magnetosonic driving wave with an initially prescribed frequency [FORMULA], the location where the considered type of resonance occurs will depend on the [FORMULA] component of the wave vector i.e. on the propagation angles according to (17). However, for certain values of [FORMULA] and [FORMULA] the driving frequency can be very close to either the lower end points of the cusp ([FORMULA]) and the Alfvén continua ([FORMULA]) respectively, or to the upper end point of the cusp continuum ([FORMULA]).

In the first case, the cusp, resp. the Alfvén resonance, occurs very close to the boundary [FORMULA] ([FORMULA], [FORMULA]) meaning that the whole lower uniform layer is practically in resonance. This then results into computational difficulties. In the second case the cusp resonance occurs practically at [FORMULA] ([FORMULA]) i.e. at the limit when the region 1 becomes transparent for the considered waves, and computational difficulties arise again.

We first study how incoming slow magnetosonic waves are absorbed by coupling to local resonant slow waves. Results for the absorption rate are shown in Fig. 3a as a function of [FORMULA] and [FORMULA] for a typical dimensionless wave frequency [FORMULA].

[FIGURE] Fig. 3a and b. a The dependence of the absorption coefficient on the incident angle [FORMULA] and the azimuth angle [FORMULA] for slow MHD waves. b The top view of the surface [FORMULA] showing the boundary of the computational domain. The dimensionless wave frequency is [FORMULA].

In our model an incoming propagating slow magnetosonic wave enters into the nonuniform layer, and propagates up to the resonant point, where it is partially absorbed by the cusp resonance. After the resonance the wave becomes evanescent (see Fig. 2).

For values of angles [FORMULA] and [FORMULA] which produce the cusp resonance close to [FORMULA], the value of the absorption coefficient is artifically put to zero. In Fig. 4, a 3D plot shows the location of the cusp resonance [FORMULA] with [FORMULA] and for the parameter domain. The domain of angles [FORMULA] and [FORMULA] where [FORMULA], clearly coincides with the domain in which the absorption coefficient can not be calculated. Therefore a cut appears in Fig. 3a which separates the regions where the absorption can be and can not be calculated. The profile of the cut is shown in Fig. 3b showing the vertical view at the surface from Fig. 3a.

[FIGURE] Fig. 4. The location of the cusp resonance depending on angles [FORMULA] and [FORMULA] for slow MHD waves with [FORMULA]

The absorption rate reaches values close to [FORMULA] at [FORMULA] and [FORMULA] around [FORMULA]. For other values of [FORMULA] and [FORMULA] it smoothly decreases to well below [FORMULA] where the absorption surface is cut and ends abruptly.

The absorption coefficient is shown as a function of [FORMULA] (for fixed [FORMULA]) in Fig. 5a and as a function of [FORMULA] (for fixed [FORMULA]) in Fig. 5b. The curves end at angles where [FORMULA] as noted above.

[FIGURE] Fig. 5a and b. Absorption of slow MHD waves with the frequency [FORMULA]. a The dependence of the absorption coefficient [FORMULA] on the incident angle [FORMULA] at three indicated azimuthal angles [FORMULA]. b The dependence of the absorption coefficient [FORMULA] on the azimuthal angle [FORMULA] at three indicated incident angles [FORMULA].

The dependency of the absorption of slow waves on the wave frequency is presented in Fig. 6: a) for [FORMULA] and several values of [FORMULA] and b) for [FORMULA] and taking [FORMULA] as a parameter. These two fixed values where chosen as they yield the maximal value for [FORMULA] in Fig. 3. The plots now show that the absorption increases with the frequency to high values close to [FORMULA].

[FIGURE] Fig. 6a and b. The dependence of the absorption coefficient [FORMULA] on the wave frequency [FORMULA] for slow MHD waves for: a [FORMULA] and b [FORMULA] and other parameters as indicated.

Next we investigate the absorption of incoming fast magnetosonic waves by coupling to localised resonant slow waves. Since these waves have frequencies that are above the upper cutoff frequency [FORMULA] at [FORMULA], they reach first the Alfvén resonance [FORMULA] and then the cusp resonance [FORMULA] that is located within the layer so that [FORMULA]. Thus both resonances contribute to the absorption of fast waves provided the wave frequency [FORMULA] does not exceed the value of the cusp frequency at [FORMULA], i.e. for [FORMULA]. At higher frequencies when [FORMULA] the absorption occurs from the Alfvén resonance solely, because [FORMULA] is the maximal value of the cusp frequency.

The only way of avoiding the Alfvén resonance for fast waves is to consider waves propagating with [FORMULA]. Eqs. (17) show that for [FORMULA] the [FORMULA] component of the wave vector is zero. Hence, the Alfvén singularity disappears from Eqs (16). The absorption coefficient for three different frequencies is shown in Fig. 7a while Fig. 7b shows the absorption as a function of the frequency of the incoming wave for three different angles [FORMULA] in the case of [FORMULA]. A general conclusion is that the absorption of fast waves with [FORMULA] is less efficient when compared to the absorption of slow waves and does not exceed [FORMULA]. The curves end again where [FORMULA].

[FIGURE] Fig. 7a. Absorption of fast MHD waves with [FORMULA]. a The dependence of the absorption coefficient [FORMULA] on the incident angle [FORMULA] at three indicated wave frequencies [FORMULA]. b The dependence of the absorption coefficient [FORMULA] on the wave frequency [FORMULA] at three indicated angles of incidence [FORMULA].

Finally, if [FORMULA] the Alfvén resonance also occurs and the dependency of the absorption coefficient on [FORMULA] and [FORMULA] is shown in Fig. 8. In this case the wave dynamics is complex. An incoming propagating fast magnetosonic wave propagates up to the point, where its frequency equals to the cutoff frequency [FORMULA] for fast waves. From there on the wave tunnels until it hits the Alfvén resonance, where it is partially absorbed. It further tunnels up to the point where the wave frequency equals to the cutoff frequency [FORMULA] for slow waves. From this point on the wave propagates up to the cusp resonant point where it is partially absorbed. Subsequently the wave becomes evanescent (see Fig. 2).

For [FORMULA] and [FORMULA], the Alfvén resonance occurs very close to the lower boundary layer at [FORMULA], and computational difficulties arise, so we set the value of the absorption coefficien to zero. Outside of the domain of [FORMULA] and [FORMULA] where the absorption surface is present, the value of the absorption is set to zero again. The reason is that for those angles the driving frequency of the fast wave is higher than the maximal value of the cusp frequency [FORMULA], or it is close to [FORMULA] which causes the cusp resonance to be very close to [FORMULA], therefore we exclude these possibilities.

[FIGURE] Fig. 8 The dependence of the absorption coefficient on the incident angle and the azimuth angle for fast MHD waves.

[FIGURE] Fig. 9a and b. Absorption of fast MHD waves with the frequency [FORMULA]. a The dependence of the absorption coefficient [FORMULA] on the incident angle [FORMULA] at three indicated azimuthal angles [FORMULA]. b The dependence of the absorption coefficient [FORMULA] on the azimuthal angle [FORMULA] at three indicated incident angles [FORMULA].

The cross sectional plots in Fig. 9 clearly indicate that the absorption is again most efficient at incident angles [FORMULA] in the interval [FORMULA] and that it decreases with [FORMULA]. Fig. 10 shows that waves are more efficiently absorbed at lower frequencies.

[FIGURE] Fig. 10a and b. The dependence of the absorption coefficient [FORMULA] on the wave frequency [FORMULA] for fast MHD waves for: a [FORMULA] and b [FORMULA] and other parameters as indicated.

The results obtained so far are for an equilibrium with [FORMULA] and for [FORMULA]. They show that the highest values of the absorption coefficient occur for [FORMULA] and for [FORMULA]. The variation of the absorption coefficient as a function of [FORMULA] is shown in Fig. 11.

[FIGURE] Fig. 11. The dependence of the absorption coefficient [FORMULA] on the parameter [FORMULA] for fast and for slow MHD waves when: [FORMULA] and [FORMULA] and [FORMULA] as indicated.

The dependence of the absorption coefficient on the ratio [FORMULA] is given in Fig. 12 for three wave frequencies for slow magnetosonic wave only. The values of the remaining parameters are [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 12. The dependence of the absorption coefficient [FORMULA] on the ratio of the Alfvén speeds [FORMULA] for slow MHD waves when: [FORMULA], [FORMULA] and [FORMULA].

Up to now, we have studied general properties of absorption of magnetosonic waves by coupling to local resonant slow waves in a nonuniform plasma layer that separates two uniform regions. To put the present results in the solar context, we can take region 1 as the corona, region 2 as the photosphere, where the incoming magnetosonic waves are generated and the nonuniform layer as the chromosphere.

Let us take [FORMULA] and [FORMULA] for the sound speeds corresponding to temperatures [FORMULA] and [FORMULA] for the corona and the photosphere respectively and [FORMULA]. We then obtain for the related Alfvén speeds [FORMULA] and [FORMULA] or [FORMULA] for their ratio. Fig. 12 shows that a wave with a dimensionless frequency [FORMULA] can be absorbed close to [FORMULA] if the the directional angles are [FORMULA] and [FORMULA] at the Alfvén speed ratio of [FORMULA]. The value of [FORMULA] then corresponds to a period [FORMULA]:

[EQUATION]

if the thickness of the layer L is given in kilometers. Thus, if [FORMULA] which is a typical value for the thickness of the chromosphere, we obtain [FORMULA] for the wave period which does not differ significantly from the period of the observed solar p-oscillations. In other words, the solar p-oscillations can play a role in the process of chromospheric heating through the mechanism of enhanced absorption of MHD waves at the cusp resonance.

Another example of significant absorption of magnetosonic waves by coupling to local resonant waves occurs when the incoming waves are generated locally in the solar corona. To examine this possibility, we consider both uniform layers to be located in the corona and separated by a nonuniform layer with a thickness of L. Let the waves propagate through the region with a lower temperature of [FORMULA] and be evanescent in the other domain with the higher temperature of [FORMULA]. Taking [FORMULA], the related values for the sound and the Alfvén speeds are then: [FORMULA], [FORMULA], [FORMULA] and [FORMULA] respectively. The period [FORMULA] of the waves that are absorbed in the layer, is then [FORMULA], where [FORMULA] is the dimensionless frequency and L is given in kilometers. We can now take [FORMULA] and [FORMULA] which gives a period of [FORMULA] for the wave. Its absorption follows from Fig. 12 to be about [FORMULA] at [FORMULA].

To conclude, the resonant absorption of magnetosonic waves coupled to local resonant magnetosonoc waves under the considered conditions can be an efficient mechanism for the coronal heating, mostly by slow magnetosonic waves. Fast magnetosonic waves are less absorbed at this resonance.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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