5. Results and discussion
Numerical calculations are performed with physical quantities normalized as follows: velocities are scaled to , the lengths are are in units of L while the density and the temperature are normalized to and respectively.
We consider four different models of the equilibrium state, model Ia, Ib and IIa, IIb. In models I the value of the plasma beta is , while in models II we take which is a better approximation of the higher layers in the solar atmosphere. Each of the models I and II consist of two variants a and b corresponding to the temperature ratios and respectively. The dimensionless speed of the mass flow is in the interval meaning that we consider flows which are subalfvenic in Region 1.
The aim of this paper is to show, how an equilibrium flow changes the properties of waves with a given frequency and with a given propagation direction. Therefore, we restrict our analysis to waves that enter the nonuniform layer at a prescribed angle of incidence equal to and at the azimuthal angle . This case assumes which allows both for the the cusp and the Alfvén resonance to appear. These particular values for and were chosen because they yield the largest resonant absorption in a static model (ade, Csík, Erdélyi and Goossens, 1997). The results presented in this paper are all for incoming waves with dimensionless frequency .
Knowing the basic state parameters (5) and the characteristics of the wave, we calculate the absorption coefficient (26) for both slow and fast incoming waves. As said before, we study the absorption of those waves which propagate only through region 2 while they are evanescent in region 1. The absorption coefficient is not calculated if the waves can also propagate in region 1. These cases occur when the shifted frequency falls within one of the following intervals: or as seen in Fig. 2.
5.1. Results in models I
Let us first consider the results coming from the interaction of an incoming magnetosonic wave with the nonuniform layer described by model I. In model I which can be relevant in the lower parts of the solar atmosphere, like for example the low chromosphere. Figs. 3 and 4 show the absorption coefficient (26) as a function of for slow and for fast MHD waves respectively.
Figs. 3 and 4 make it very clear that even a slow equilibrium flow can have a drastic effect on the wave properties. In absence of a flow, i.e. in a static equilibrium, the energy of the wave is partially absorbed by resonant absorption. A small change in the equilibrium flow can remove the positive absorption present in the static equilibrium and even replace it by negative absorption. The negative absorption means that the amplitude of the reflected wave is larger than the amplitude of the incident wave. This phenomenon of over-reflection occurs for both types of waves if the shifted frequency becomes negative and falls within one of intervals 6-10 in Fig. 2. Over-reflection thus exists at positive values of if the Doppler shifted frequency (downwards, in Fig. 1) matches the negative value of the local cusp frequency (intervals 6-10 in Fig. 2):
This requires flows with .
The second relation in (29) can be understood as the energy equation in quantum mechanics indicating the flow as the energy source for over-reflection. We point out, that the considered over-reflection and normal wave absorption cannot occur in absence of dissipative losses at resonances.
If takes positive values, which occurs for , the absorption coefficient becomes positive and the incoming wave is partially absorbed. The resonant wave absorption thus occurs for all negative values of and for positive values not exceeding :
In Fig. 3a the results are shown for the temperature ratio . There are four different velocity intervals where the absorption coefficient is nonzero. Going from the left to the right these intervals correspond to the absorption at the Alfvén resonance, absorption at the cusp resonance, over-reflection at the cusp resonance, and over-reflection at the Alfvén resonance, which coincide with domains 1, (4, 5), (6, 7) and 10 in Fig. 2, respectively. Domains 2, 3, 8 and 9 are not present in our calculations since in model I and II the relation is always hold, which means that the cusp and the Alfvén resonances can not occur together for the same value of the flow speed . The labels C and A correspond to the pure cusp and Alfvén resonance, respectively. Absorption and over-reflection occur over a wider interval of velocity for the Alfvén resonance than for the cusp resonance.
Comparison of Fig. 3b with Fig. 3a reveals that the shape of the absorption and the over-reflection curves at the cusp resonance do not change significantly with the temperature ratio. The main difference is that the smaller temperature ratio results smaller peak values for the absorption (from 70 % in Fig. 3a to 50 % in Fig. 3b) and for the over reflection (250 % in Fig. 3a to 100 % in Fig. 3b). In Fig. 3b the absorption and the over-reflection at the Alfvén resonance shows an oscillatory behaviour, and the peak values are larger than in the previous case.
Turning to the fast incoming magnetosonic waves, Figs. 4a and 4b shows the absorption coefficient as a function of the flow speed for temperature ratios and , respectively. The main conclusion from these figures is that the fast waves are less sensitive to the variation of the flow speed. After a smooth increase the absorption approximately stays on its maximal value and then suddenly drops to zero. Over-reflection is also present, but less prominent than for the slow waves. The oscillatory behaviour is not present in Fig. 4b, just in contrast to Fig. 3b. The change of the temperature ratio does not have a strong influence on the peak values in Figs. 4a and 4b. The main difference between the two figures is that the absorption and the over-reflection take place in different intervals of the flow speed , coming from the fact that the variation of the temperature ratio results in a variation in the width of the cusp and the Alfvén continua. For example, in the case of the temperature ratio the given incoming wave even can not be over reflected in the subalfvenic regime since is valid for .
5.2. Results in models II
Model II has , which is a more realistic value for the plasma beta in the higher solar atmosphere. In Figs. 5a and 5b we present the variation of the absorption coefficient as a function of the flow speed for the temperature ratios and , respectively.
Comparison of Figs. 5-6 to Figs. 3-4 of model I, leads us to the following conclusions. For slow incoming waves the reduction of results in a reduction of the peak values for both temperature ratios while its effect on the fast MHD waves is less pronounced. The oscillatory behaviour of the curves at the pure Alfvén resonance is more pronounced in Fig. 5b than in Fig. 3b.
The results for fast incoming waves are shown in Figs. 6a and 6b. The variation of the plasma beta does not have a strong influence on the absortion curves for both temperature ratios. The main difference between Figs. 4ab and Figs. 6ab comes from the slight change of the absorption and over-reflection at the cusp resonance.
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998