Influence of the Magnetic Field Inclination on Magneto-Acoustic-Gravity Waves in the Solar Atmosphere

Abstract

Observations indicate that the magnetic field inclination plays an important role in wave propagation in sunspots and active regions. We explore the properties of these waves using the exact solution of MHD equations in a plasma permeated by an inclined magnetic field. The obtained $K-\Omega$ (wavenumber–frequency) diagrams indicate a decrease in the cut-off frequency with increasing magnetic field inclination. Typical 5-min waves are allowed to propagate upwards to the chromosphere if the magnetic field is sufficiently tilted. The ratio of Alfvén speed to sound speed plays a key role in determining the cut-off frequency. We find very low cut-off frequencies even when this ratio is small at the bottom of the model atmosphere. Wave-associated energy is investigated over the whole $K-\Omega$ domain under consideration. As the magnetic field inclination increases, we find that modes become progressively magneto-acoustic through a large portion of the $K-\Omega$ domain. Gravity-related energy becomes increasingly secondary. This effect depends on location in the $K-\Omega$ diagram and the inclination angle of the magnetic field. At small $K$ and $\Omega$, we find that the cut-off frequency is low even though the gravity-associated energy density is not small.

Appendix A: Alternative Derivation of the Exact Solutions of MAG Wave Equations

We use the Frobenius method to obtain solutions of the differential Equations 3 and 4. Starting with the corresponding series:

$$\begin{aligned} u_{x}=\sum _{n=0}^{\infty} X_{n} \zeta ^{n+\nu}\, , \end{aligned}$$

(14)

and

$$\begin{aligned} u_{z}=\sum _{n=0}^{\infty} Z_{n} \zeta ^{n+\nu}\, . \end{aligned}$$

(15)

We substitute these series into Equations 3 and 4 and gather terms of the same order “n” in $\zeta$. This gives the recurrence relations:

$$\begin{aligned} &\left ( \Omega ^{2} -K^{2}\right ) X_{n}+\left [ - 4\Omega ^{2} K^{2} +\Omega ^{2}(n+2+\nu )^{2}\right ] X_{n+2} \\ & = - iK\left ( \frac{1}{\gamma}+\frac{n+\nu}{2}\right ) Z_{n} + \left [ -4\tau \Omega ^{2} K^{2} + \tau \Omega ^{2}(n+2+\nu )^{2} \right ] Z_{n+2}\, , \end{aligned}$$

(16)

$$\begin{aligned} &iK\left ( \frac{\gamma - 1}{\gamma}+\frac{n+\nu}{2}\right ) X_{n}+ \left [ 4\tau \Omega ^{2} K^{2} -\tau \Omega ^{2}(n+2+\nu )^{2} \right ] X_{n+2} = \left ( -\Omega ^{2}\right . \\ & \left . -\frac{(n+\nu )(n+2+\nu )}{4}\right ) Z_{n} + \left [ 4 \tau ^{2} \Omega ^{2} K^{2} - \tau ^{2}\Omega ^{2}(n+2+\nu )^{2} \right ] Z_{n+2} \, . \end{aligned}$$

(17)

We multiply the recurrence Equation 16 by $\tau$ and add it to Equation 17. This leads to:

for $n\geqslant 0$

$$\begin{aligned} Z_{n}=- \frac{2iK(\nu + n)+4ik(\frac{\gamma - 1}{\gamma})+4\tau (\Omega ^{2}-K^{2})}{(\nu + n)[\nu + n +2(1+iK\tau )]+4(\frac{iK\tau}{\gamma}+\Omega ^{2})} X_{n} \, . \end{aligned}$$

(18)

If we take $Z_{n}$ and $Z_{n+2}$ from the recurrence Equation 18 and substitute them into Equation 16 or 17 (they give the same results), we have:

for $n\geqslant 2$

$$\begin{aligned} X_{n}=- \frac{[(\nu + n)(\nu + n -2)+4(\Omega ^{2}-K^{2})+4\frac{K^{2}}{\gamma \Omega ^{2}}(\frac{\gamma - 1}{\gamma})]}{[(\nu + n)^{2}- 4 K^{2}]} \frac{r}{q} X_{n-2}\, , \end{aligned}$$

(19)

where

$$\begin{aligned} q=[(\nu + n)^{2}+2(\nu + n)(1+2iK\tau )+4\Omega ^{2}(1+\tau ^{2})+4iK \tau (1+iK\tau )]\, , \end{aligned}$$

and

$$\begin{aligned} r= \frac{[(\nu +n)(\nu + n + 2(1+iK\tau ))+4(\frac{iK\tau}{\gamma}+\Omega ^{2})]}{[(\nu +n -2)(\nu + n + 2iK\tau ))+4(\frac{iK\tau}{\gamma}+\Omega ^{2})]} \, . \end{aligned}$$

These results are identical to the ones in Schwartz and Bel (1984). They use, however, a different method (a matrix method) including two other equations derived from Equations 3 and 4. Schwartz and Bel (1984) derive a solution of a system of four equations. The solutions obtained in this Appendix (using only the two original Equations 3 and 4) allow us to cross check with Schwartz and Bel’s solutions and help validate both.