Slow sound approximation

Examining the expression (138) for matrix element $E_{22}$, we find that the first term of $E_{22}$ can be written as

$$\frac{{\textmu}_0^{- 1} B_0^2 + \gamma p_0}{B_0^2} ={\textmu}_0^{- 1} \left( 1 + \frac{\gamma \beta}{2} \right),$$ (214)

where $\beta \equiv p_0 / (B_0^2 / 2{\textmu}_0)$, while the second term of $E_{22}$ can be written as

$${\textmu}_0^{- 1} \frac{\gamma p_0}{\omega^2 \rho_0} \mathbf{B}_0 \cdot \nabla \left( \frac{\mathbf{B}_0 \cdot \nabla}{B^2_0} \right) \approx {\textmu}_0^{- 1} \left( \frac{\gamma p_0}{\omega^2 \rho_0} k_{\parallel}^2 \right),$$

$$\approx {\textmu}_0^{- 1} \left( \frac{\gamma p_0}{V_A^2 \rho_0} \right)$$

$$= {\textmu}_0^{- 1} \left( \frac{\gamma p_0}{\frac{B_0^2}{{\textmu}_0 \rho_0} \rho_0} \right)$$

$$= {\textmu}_0^{- 1} \left( \frac{\gamma \beta}{2} \right) .$$ (215)

where $k_{\parallel }$ is the parallel wave vector and we have used the approximation $\omega / k_{\parallel} \approx V_A$. Using Eqs. (214) and (215), the ratio of the second term to the first term of $E_{22}$ is written as $(\gamma \beta / 2) / [1 + \gamma \beta / 2]$. For low $\beta$ ( $\beta \ll 1$) equilibrium, the ratio is small and therefore the second term of $E_{22}$ can be dropped. This approximation is called the slow sound approximation in the literature[6,7]. Numerical results indicate this approximation will remove all the sound continua while keeping the Alfven continua nearly unchanged.