2 The wave equation with Newtonian cooling

We confine our attention to an isothermal atmosphere with a vertical magnetic field which is unbounded in the horizontal direction. Using a fluid description and assuming an ideal plasma, we write the energy equation as,

$$\partial_{\rm t} {\delta T\over T} + \delta v_{z} {1\over T} ... ...ot \delta \vec{v} = - {1\over\tau_{\rm R}} {\delta T\over T} ,$$ (1)

where $\delta \vec{v} = \partial _{\rm t} {\bf\vec\xi}$ is the velocity perturbation, T is the temperature of the gas and $\delta T$the temperature perturbation. We assume the Lagrangian displacement ${\vec\xi}$ varies as $\sim$ ${\rm e}^{{\rm i}(\omega t-kx)}$, where $\omega$ is the angular frequency and k is the horizontal wave number. We allow for radiative losses, approximating them by Newton's law of cooling (e.g. Spiegel 1957; Mihalas & Mihalas 1984), which assumes that the temperature fluctuations are damped radiatively on a time scale $\tau_{\rm R}$, given by

$$\tau_{\rm R}= {\rho c_{\rm v} \over 16\chi \sigma T^{3}} ,$$ (2)

where $\chi$ is the mean absorption coefficient per unit length, $c_{\rm v}$ the specific heat per unit volume, and $\sigma$ the Stefan-Boltzmann constant. For simplicity, we assume that $\tau_{\rm R}$ is constant over the atmosphere. If the vertical dimension of the perturbation is small compared to the local scale height, the relation between the Lagrangian perturbations in pressure p and density $\rho$is then approximately given by $\delta p/p \simeq \gamma^\ast \delta \rho/\rho$, where

$$\gamma^{\ast} (\omega) = { 1 + \i \omega \tau_{\rm R}\gamma \over 1 + \i \omega \tau_{\rm R}} \cdot$$ (3)

With these assumptions, the linearized equations for MAG waves are given by a system of two coupled differential equations (Banerjee et al. 1997),

$$\left[v_{\rm A}^2 {\d^2\over \d z^2}\ - (\tilde{\gamma} c_{\r... ...(\tilde{\gamma} c_{\rm S}^2{\d\over \d z\ }-g\right)\xi_z =0 ,$$ (4)

$$\left[\tilde{\gamma} c_{\rm S}^2 {\d^2\over \d z^2} - \tilde{... ...2{\d\over \d z }-(\tilde{\gamma}\gamma -1)g\right] \xi_x =0 ,$$ (5)

where $\xi_z$ and $\xi_x$ are the amplitudes of the vertical and horizontal components of the displacement, g is the acceleration due to gravity, and $\tilde{\gamma} = \gamma^{\ast}/\gamma$. The adiabatic sound speed and the Alfvén speed are given, respectively, by

$$c_{\rm S}=\ \sqrt{\gamma p \over \rho} \quad {\rm and} \quad v_{\rm A}= \frac B{\sqrt{4\pi\rho}} \cdot$$ (6)

We should point out here that the equation governing the propagation of the purely transverse Alfvén waves is decoupled from Eqs. (4) and (5) and will not be considered in the present investigation. We have implicitly assumed that the propagation and the motions of the MAG modes are confined to the x-z plane. This involves no loss of generality. Equations (4) and (5) have the same structure as the linearized wave equation for adiabatic perturbations (see Eqs. (1), (2) of Hasan & Christensen-Dalsgaard 1992), apart from the appearance of the parameter $\tilde{\gamma}$, which incorporates radiative cooling. In non-dimensional form Eq. (3) can be written as

$$\gamma^{\ast} = { 1 + \i \Omega \tilde \tau_{\rm R}\gamma \over 1 + \i \Omega \tilde \tau_{\rm R}} ,$$ (7)

where the dimensionless relaxation time scale is given by $\tilde \tau_{\rm R}= (c_{\rm S}/ H) \tau_{\rm R}$, where $H = p/\rho g$ is the scale height of the atmosphere, which is constant for an isothermal medium and the dimensionless frequency, $\Omega$, is given by expression (9). As $\Omega\tilde\tau_{\rm R}$ varies from 0 to $\infty$, $\gamma ^\ast$ describes a semi-circle in the complex plane (see Bünte & Bogdan 1994).

Figure 2: The variation of the real part (solid line) and the imaginary part (dashed line) of the effective Brunt-Väisälä frequency $\tilde{\Omega}_{\rm BV}$, (in dimensionless units) as a function of $\Omega\tilde\tau_{\rm R}$.

Figures 1a, b show the variation of the real and imaginary parts of $\gamma ^\ast$ respectively as a function of $\Omega\tilde\tau_{\rm R}$. We find that for $\Omega\tilde\tau_{\rm R} < 0.1$, $\gamma ^\ast$ approaches the isothermal limit, and $\gamma^{\ast} = 1$. For $\Omega\tilde\tau_{\rm R} > 10,$Re( $\gamma^{\ast}) = 5/3 = \gamma$. Thus in the limit ${\tau_{\rm R}}, \rightarrow \infty$ i.e. in the limit of adiabatic perturbations, $\gamma^{\ast} = \gamma$ and $\tilde{\gamma} =1$. Letting $\tilde{\gamma} =1$ in Eqs. (4) and (5) we recover the linearized equations given by Hasan & Christensen-Dalsgaard (1992). The imaginary contribution to $\gamma ^\ast$ is maximal for $\Omega\tilde\tau_{\rm R} \sim 1$. Thus to study the maximal effect of radiative heat exchange we choose our parametric values such that $\Omega\tilde\tau_{\rm R}$ becomes close to 1. In order to obtain a dimensionless wave equation we introduce three dimensionless parameters,

 

K =kH , (8)

$$\Omega ={\omega H \over c_{\rm S}} ,$$ (9)

and the dimensionless vertical coordinate

$$\theta = {\omega H\over v_{\rm A}} = {c_{\rm S} \over v_{\rm A,0}} ~ \Omega ~ {\rm e}^{-z/(2H)} ,$$ (10)

where v_A,0 is the Alfvén speed at z=0. In terms of the variables defined by Eqs. (8)-(10), Eqs. (4) and (5) can be combined into a fourth-order differential equation for $\xi_x$,

 

$$\left\{ \theta^4 {\d^4\over \d\theta ^4} + 4\theta ^3 {\d ^3\over... ...gamma}} -K^2\right)+4\theta ^2 \right] \theta^2 {\d ^2\over \d\theta^2} \right.$$

$$-\left[ 1-4\left({\Omega^2 \over \tilde{\gamma}} +K^2\right)-12\theta ^2 \right] \theta {\d \over \d\theta }$$

$$+\left.16\left[\left({\Omega^2 \over \tilde{\gamma}} +K^2\left({\... ...ight) \theta ^2 - {\Omega^2K^2 \over \tilde{\gamma}} \right] \right\} \xi_x=0 ,$$ (11)

where $\tilde{\Omega}_{\rm BV}^2 =({\gamma^{\ast}} -1)/{\gamma}{\gamma^{\ast}}$ is the square of the effective Brunt-Väisälä frequency (in dimensionless units). This Brunt-Väisälä frequency is a function of frequency in the non-adiabatic case.

Figure 2 shows the dependence of $\tilde{\Omega}_{\rm BV}$ on the radiative relaxation time $\tilde\tau_{\rm R}$. The solid line depicts the real part whereas the dashed line represents the imaginary part of $\tilde{\Omega}_{\rm BV}$. Note that for $\Omega\tilde\tau_{\rm R} > 10$, $\tilde{\Omega}_{\rm BV}$ reaches a constant value of 0.5 (corresponding to the adiabatic limit). On the other hand, as $\Omega\tilde\tau_{\rm R} \rightarrow 0$, Re( $\tilde{\Omega}_{\rm BV}) < 0.1$. Thus, in the isothermal limit, $\tilde{\Omega}_{\rm BV}$, which is the higher cutoff frequency for the g-modes, is very low. The consequences of this will be taken up again when we discuss the properties of g-modes in detail. Figure 2 also reveals that the imaginary part of $\tilde{\Omega}_{\rm BV}$ is significant only for $\Omega\tilde\tau_{\rm R} \sim 1$.

The general solution of Eq. (11) can be expressed in terms of Meijer functions (Zhugzhda 1979) as follows:

$$\xi_x^{\rm (h)}=G^{1 2}_{2,4}\left (\mu_{{h}}, \begin{array}{... ...cdots ,& \mu_4 \end{array}\left. \right\vert\theta^2 \right) ,$$ (12)

where $(i,h=1,\dots , 4;\ i\ne h)$ and

$$\mu_{1,2} ={(1\pm \i\alpha)\over 2}, \qquad \mu_{3,4}=\pm K ,$$ (13)

$$a_{1,2} ={(1\pm\phi)\over 2} ,$$ (14)

$$\alpha =\sqrt{4{\Omega^2 \over \tilde\gamma} - 1} ,$$ (15)

$$\phi = \sqrt{-\alpha^2 +4K^2(1-\tilde{\Omega}_{\rm BV}^2/\Omega^2)} .$$ (16)

These solutions are very similar in nature to the purely adiabatic case (see Banerjee et al. 1995). Once $\xi_x^{\rm (h)}$ is known, it is fairly straightforward to determine the corresponding solutions $\xi_z^{\rm (h)}$ from either of Eqs. (4) or (5). The complete solutions satisfying the required boundary conditions can be constructed as linear combinations of $\xi_x^{\rm (h)}$and $\xi_z^{\rm (h)}$.

Figure 3: Diagnostic diagram for non adiabatic modes. Panel  a) for g1-mode and  b) for p1-mode. Different line styles correspond to different radiative relaxation time $\tilde\tau_{\rm R}$ as labelled.