3 Normal modes in a weak magnetic field

We now examine the asymptotic properties of waves and normal modes of a stratified atmosphere with a weak magnetic field (corresponding to the limit of small $\epsilon$, where $\epsilon = v_{\rm A,0}/c_{\rm S}$). The analytical results are used for the interpretation of the numerical solutions presented in Sect. 4. In order to get a physical picture of the solution, we consider the upward propagation of a wave, excited from below at z=0, in an isothermal atmosphere. It is well known that acoustic modes are easily reflected if the temperature of the medium changes with height (for a good discussion see Leibacher & Stein 1981). The slow mode can be reflected due to the increasing Alfvén speed with height from layers where $v_{\rm A}\sim c_{\rm S}$, through conversion into a fast mode (e.g. Zhugzhda et al. 1984). We implicitly assume that the properties of the atmosphere change abruptly at the top boundary, resulting in downward reflection of the waves. The lower boundary condition is chosen to simulate a forcing layer. This permits standing wave solutions. It should, however, be kept in mind that an isothermal atmosphere by itself does not trap modes, rather we use this assumption to understand the physical properties of the modes in a stratified atmosphere with a vertical field. Let us now derive approximate dispersion relations for various boundary conditions.

3.1 Rigid boundary condition

Let us first consider rigid boundary conditions, viz.

$$\xi_x = \xi_z = 0 ~~~{\rm at} ~~~~~z = 0 ~~~~{\rm and} ~~~~z = d,$$ (17)

here d is the height of the upper boundary, and D = d/His the dimensionless height.

Figure 4: Variation of the imaginary part of $\Omega$ with K for $\tilde\tau_{\rm R} = 0.5$. The different line styles corresponds to different $\epsilon$ values (which is a measure of magnetic field strength) as labelled.

The asymptotic properties of the solution in the purely adiabatic limit are presented in Hasan & Christensen-Dalsgaard (1992). Following the same line of treatment using Eqs. (12)-(16) and applying the boundary conditions given by Eq. (17) one can derive the following dispersion relation in the weak field limit

 

$$\left({\Omega^2\over\tilde\gamma}-K^2\right) \sin \tilde\theta \sin(K_zD) ~\hspace*{4cm}$$

$$=2\sqrt{\tilde\gamma} {\epsilon\over\Omega}{\rm e}^{D/4} \bigg\{ K_z K^2 \bigl[\cosh(D/4)\cos \tilde\theta \cos(K_zD) - 1\bigr]$$

$$+ \sinh(D/4)\cos\tilde\theta \sin(K_zD)\left[M \left({\Omega^2\over\tilde\gamma}-K^2\right) \right.$$

$$\left.\left. -K^2\left(\frac 1{\gamma\tilde\gamma} -\frac 12\right)\right]\right\} + O\left({\epsilon^2\over\Omega^2}\right),$$ (18)

where

$$\theta_0 =\theta(0) , \quad \theta_D=\theta(D) , \quad \quad\tilde\theta = 2(\theta_0 -\theta_D) ,$$ (19)

and K_z² is given by

$$K_z^2 = {\Omega^2\over\tilde\gamma} - K^2 \left(1 - {\tilde{\Omega}_{\rm BV}^2\over\Omega^2}\right) - {1\over 4} ,$$ (20)

and

$$M =K^2\frac {\tilde{\Omega}_{\rm BV}^2}{\Omega^2} -\frac 1{16} \cdot$$ (21)

For $\epsilon << \Omega$, the dispersion relation to lowest order in $\epsilon/\Omega$ becomes

$$\left({\Omega^2\over\tilde\gamma}-K^2\right) \sin \tilde\theta \sin(K_zD) = 0.$$ (22)

Equation (22) admits the following solutions

$$\sin(K_zD) = 0,$$ (23)

$$\sin \tilde\theta = 0,$$ (24)

$$\Omega = \sqrt{\tilde\gamma}~K .$$ (25)

We first consider the solution given by Eq. (23) which implies that $K_z D = n \pi$, where n is an integer and denotes the order of the mode. Using this condition Eq. (20) yields,

$${\Omega_i^4 \over \tilde\gamma} - \Omega_i^2 \left(K_{\rm t}^... ...\right) + K^2 \tilde{\Omega}_{\rm BV}^2 = 0 ~~~~~~~(i = p,g),$$ (26)

where K_t² = K_z² + K². Note that Eq. (26) looks very similar to the usual relation for p- and g-modes (see Banerjee et al. 1995) apart from the factor $\tilde{\gamma}$ and modified $\tilde{\Omega}_{\rm BV}$. Because of the presence of these two factors, the properties of these modes change drastically. Figures 3a, b give an overview of the behaviour of the g- and p-type modes respectively. Their properties are reflected in these two $K - \Omega$ diagrams produced by solving Eq. (26) for different values of the relaxation time $\tilde\tau_{\rm R}$. Figure 3a clearly shows that the g₁-mode has been pushed down to the low frequency part of the diagnostic diagram with decreasing value of $\tilde\tau_{\rm R}$. As $\tilde \tau_{\rm R} \rightarrow 0$, the g₁-mode tends to disappear, because $\tilde{\Omega}_{\rm BV} \rightarrow 0$. On the other hand, Fig. 3b shows that the inclusion of radiative exchange has not greatly changed the behaviour of the p₁-mode, apart from a decrease of the acoustic cutoff frequency.

The solution corresponding to Eq. (25) can be recognized as a modified Lamb mode (compare with $\Omega =~K$, for pure Lamb mode). Thus we expect a frequency shift of the adiabatic Lamb mode. Turning our attention to the solution given by Eq. (24), we find that these modes are the same magnetic modes present in the adiabatic conditions, which arise solely due to the presence of the magnetic field. The magnetic modes, hereafter referred to as m-modes, have frequencies

$$\Omega_{\rm m} = {\epsilon l \pi \over 2s},\qquad(l=1,2,\dots) ,$$ (27)

where $s= (1-{\rm e}^{-D/2})$. These modes are approximately transverse (it has be shown by Banerjee et al. (1995) that ${\xi_x^{(1,2)}/ \xi_z^{(1,2)}} \sim O(\theta)$). Physically, these modes can be interpreted as gravity-modified slow modes in a weak magnetic field. Thus these slow modes are not affected by the inclusion of radiative losses in the weak field limit. This result complements the result of Bogdan & Knölker (1989), where it was conjectured that the uniform magnetic field reduces the temperature perturbations associated with these waves and therefore suppresses the radiative damping of these disturbances. This aspect will be taken up later when we discuss the numerical solutions.

 

 

Table 1: Eigenfrequencies of different order p-modes for a model atmosphere with D = 10, B = 2kG, $\tilde\tau_{\rm R} = 0.5$ and K = 0.1.

	Adiabatic case			Radiative case			Isothermal case
	( $\tilde\tau_{\rm R} = 100$)			( $\tilde\tau_{\rm R} = 0.5$)			( $\tilde\tau_{\rm R} = 0.05$)
Mode	Re($\Omega$)	Im($\Omega$)	P(S)	Re($\Omega$)	Im($\Omega$)	P(S)	Re($\Omega$)	Im($\Omega$)	P(S)
p1	0.5903	0.0007	164	0.5203	0.0658	186	0.458	0.0125	211
p2	0.803	0.0009	120	0.7075	0.0895	136	0.624	0.017	155
p3	1.07	0.0013	90	0.94	0.119	103	0.828	0.0227	117

3.2 Zero-gradient boundary condition

If we use zero-gradient boundary conditions at the top and bottom of the layer,

$${{\d\xi_x} \over {\dz}} ={{\d\xi_z} \over {\dz}} =0 \quad \hbox{\rm at} \quad z=0 \quad \hbox{\rm and} \quad z=d .$$ (28)

In addition to the modes discussed previously, we find another wave mode, namely the gravity-Lamb mode. The dispersion relation for this mode is given by

$$K_z^2 + {1 \over 4} = 0.$$ (29)

Combining Eqs. (29) with (20) yields

$$\Omega^4 -\tilde\gamma \Omega^2 K^2 + \tilde\gamma K^2{\Omega}_{\rm BV}^2=0 .$$ (30)

This equation has the solution

$$\Omega ^2 = {\tilde\gamma K^2 \over 2} \left[ 1 \pm \left(1 ... ...\rm BV}^2 \over K^2 \tilde\gamma } \right)^{1/2} \right] \cdot$$ (31)

The solution resembles a modified gravity mode on the lower branch and a Lamb mode on the upper branch. In order to see this, consider the limit $K \rightarrow \infty$. The smaller solution in Eq. (31) has the limit $\Omega \simeq \tilde{\Omega}_{\rm BV}$, which is the dispersion relation for a modified g-mode for large K; the larger solution has the limit $\Omega \simeq \sqrt{\tilde\gamma} K$ for large K, which shows that the mode behaves like a modified Lamb wave.

Thus the separate modes have changed their behavior in the diagnostic diagram in the non-adiabatic case. It is important to know how these modified modes interact with one another in the presence of radiative losses. Mode coupling in the non adiabatic case will be different as compared to the adiabatic case studied by Banerjee et al. (1995) (the right hand side of Eq. (18) contributes to the coupling).