4 Numerical results

The behavior of the MAG waves is reflected in their properties in the $K - \Omega$ diagram namely the variation of the real and imaginary part of the complex frequency with the horizontal wave number K. The solutions were obtained by solving Eq. (11) numerically, using a complex version of the Newton-Raphson-Kantorovich scheme (Cash & Moore 1980) subject to a different sets of boundary conditions. Banerjee et al. (1997) presented the numerical solutions for the weak field case subjected to rigid boundary conditions. In this paper we would like to compare our theoretical results with some observational results so we concentrate here on higher magnetic field strengths.

First we show the effect of the strength of the magnetic field on the damping of these waves. To delineate the influence of the magnetic field, we choose the m₂-mode, which is predominantly magnetic in nature. Figure 4 shows the variation of the imaginary part of the frequency (which is a measure of the damping) with K for fixed $\tilde\tau_{\rm R} = 0.5$, D=1, and $\gamma =5/3$ (for rigid boundary conditions). The different line styles correspond to different $\epsilon$ values. It clearly reveals that as we increase the value of $\epsilon$ (increasing magnetic field strength) the imaginary part reduces, indicating less damping of these wave modes. This result is in agreement with the conclusions drawn by Bogdan & Knölker (1989), that the magnetic field suppresses radiative damping. For horizontal magnetic field Bünte & Bogdan (1994) also reported a "stiffening'' of the atmosphere with increasing $\epsilon$ values.

Figure 5: Region in the diagnostic diagram for moderate field strength ( $\epsilon = 0.1$) and $\tilde\tau_{\rm R} = 0.5$, where the modified Lamb mode, magnetic modes and the gravity-Lamb mode are present. These results are for zero gradient boundary conditions.

Figure 6: Variation of the imaginary part of $\Omega$ with K for the same set of parameters as in Fig. 5. Panel  a) shows the m1-mode and panel  b) the m2-mode.

Let us now consider a moderate magnetic field strength. Figure 5 shows a region in the diagnostic diagram for $\epsilon = 0.1$ ( $B \sim 240$ G) and $\tilde\tau_{\rm R} = 0.5$, subject to the zero-gradient boundary conditions. The mode coupling in this case is much more complicated because we have three mode interaction regions as indicated. As K increases, the m₁-mode begins to acquire the character of a modified Lamb mode. Figure 6a, which shows the variation of imaginary part of the frequency of the m₁-mode with K also reveals that, there is an enhancement as it approaches an avoided crossing (near K=0.8) followed by a suppression due to mode transformation. Up to K=0.8 this mode behaves as a magnetic Lamb mode and after the mode transformation it becomes a magnetic type. This process is repeated at higher frequency (around K=2). Note the large drop due to magnetic field suppression. Figure 6b shows the variation of the imaginary part of the frequency for the m₂-mode (Fig. 5) with K. The two peaks correspond to modified Lamb and m-mode coupling and modified gravity-Lamb (gL-) and m-mode coupling respectively. Note the steep rise of the imaginary part after the avoided crossing which indicates the effect of the gravity mode. Figure 5 also shows the lower branch of the modified gravity-Lamb mode (indicated as $\tilde\Omega_{gL}$) which was absent in the purely adiabatic case.