Hall-coupling of Slow and Alfvén Waves at Low Frequencies in the Lower Solar Atmosphere

Abstract

The Hall effect due to weak ionization in the lower solar atmosphere is shown to produce significant coupling between slow magneto-acoustic and Alfvén waves, especially in highly inclined magnetic fields, and even at low frequencies (${\approx }\, 5$ mHz and above). Based on the exact magneto-acoustic linear wave solutions in a 2D isothermal model atmosphere, a perturbation approach is used to calculate the coupling to Alfvén waves polarized in the third dimension. First, a fast wave is injected at the bottom and is partially and often strongly reflected/converted to a down-going slow wave at the Alfvén-acoustic equipartition height, depending on magnetic field inclination, frequency, and wave number. This slow wave then couples strongly to the down-going Alfvén wave via the Hall effect for realistic Hall parameters. The coupling is strongest for horizontal wavenumbers oriented opposite to the field inclination, and magnetic fields around 100 G, for which large values of the Hall parameter are co-spatial with the region where slow and Alfvén waves have almost identical wave forms. Second, a slow wave is injected at the bottom, and found to couple even more strongly to up-going Alfvén waves in certain regions of the wavenumber–frequency plane where acoustic-gravity waves are evanescent. These results contrast with those for Hall-mediated fast-Alfvén coupling, which occurs higher in the atmosphere and is evident only at much higher frequencies.

Appendix: Recast of the Momentum Equation

In this appendix, we employ the momentum Equation 2 with the displacement vector written as $\boldsymbol{\xi}=\xi _{ \perp }\hat{\boldsymbol{e}}_{\perp }+\eta \,\hat{\boldsymbol{e}}_{y}+ \xi _{\parallel }\hat{\boldsymbol{e}}_{\parallel }$, where the ⊥ and ∥ directions are, respectively, perpendicular to and parallel to ${\boldsymbol{B}}_{0}$ in the $x$–$z$ plane. For each component we then find

$$\begin{aligned} &-i s^{2} \epsilon _{\text{H}} \cos \theta \bigl(-4 \bigl(-1 + \kappa ^{2}\bigr) \eta + s \bigl(5 \eta ' + s \eta ''\bigr) \bigr) \\ &\quad {}+ \frac{1}{\gamma } \bigl[-4 s^{2} \bigl(i \kappa \cos \theta + \gamma \bigl( \kappa ^{2} -\nu ^{2}\bigr) \sin \theta \bigr) \xi _{\parallel } + 2 i s^{2} \kappa \sin \theta \bigl(2 \xi _{\perp } + s \gamma \xi _{\perp }' \bigr) \\ &\quad {}+ \gamma \cos \theta \bigl(-4 \bigl(\kappa ^{2} \nu ^{2} + s^{2} \bigl(\kappa ^{2} -\nu ^{2}\bigr)\bigr) \xi _{\perp } - 2 i s^{3} \kappa \xi _{\parallel }' + s \nu ^{2} \bigl(\xi _{\perp }' + s \xi _{\perp }'' \bigr) \bigr) \bigr] = 0, \end{aligned}$$

(23)

$$\begin{aligned} &\nu ^{2} \bigl(4 \bigl(s^{2}-\kappa ^{2} \sin ^{2}\theta \bigr) \eta + s \cos \theta \bigl(( \cos \theta -4 i \kappa \sin \theta ) \eta ' + s \cos \theta \, \eta ''\bigr) \bigr) \\ &\quad {}+ \frac{s^{2} \epsilon _{\text{H}}}{\gamma \nu ^{2}} \bigl[-4 \bigl(\kappa ^{2}- \gamma \nu ^{2}\bigr) (i \cos \theta + \kappa \sin \theta ) (\sin \theta \, \xi _{\parallel } + \cos \theta \, \xi _{\perp }) \\ &\quad {}+ 4 \kappa \bigl(1-\gamma \nu ^{2}\bigr) (\cos \theta - i \kappa \sin \theta ) ( \cos \theta \, \xi _{\parallel } - \sin \theta \, \xi _{\perp }) \\ &\quad {}+ s \bigl(-i \bigl(2 \kappa ^{2} - 5 \gamma \nu ^{2}\bigr) \cos \theta + 2 \gamma \kappa \nu ^{2} \sin \theta \bigr) \bigl(\sin \theta \, \xi _{\parallel }' + \cos \theta \, \xi _{\perp }'\bigr) \\ &\quad {}+ s \kappa \bigl(\bigl(5 - 2 \gamma \nu ^{2}\bigr) \cos \theta - 2 i \kappa \sin \theta \bigr) \bigl(\cos \theta \, \xi _{\parallel }'-\sin \theta \, \xi _{\perp }' \bigr) \\ &\quad {}+ i s^{2} \gamma \nu ^{2} \cos \theta \bigl(\sin \theta \, \xi _{\parallel }'' + \cos \theta \, \xi _{\perp }''\bigr) \\ &\quad {}+ s^{2} \kappa \cos \theta \bigl(\cos \theta \, \xi _{\parallel }''-\sin \theta \, \xi _{\perp }''\bigr) \bigr] = 0, \end{aligned}$$

(24)

$$\begin{aligned} & i s^{2} \epsilon _{\text{H}} \sin \theta \bigl(-4 \bigl( \kappa ^{2}-1\bigr) \eta + s \bigl(5 \eta ' + s \eta ''\bigr) \bigr) \\ &\quad {}+ \frac{1}{\gamma } \bigl[s^{2} \cos \theta \bigl(4 \gamma \nu ^{2} \xi _{\parallel } - 4 i ( \gamma -1) \kappa \xi _{\perp } + s \gamma \bigl(3 \xi _{\parallel }' - 2 i \kappa \xi _{\perp }' + s \xi _{\parallel }'' \bigr) \bigr) \\ &\quad {}-\sin \theta \bigl(4 i s^{2} ( \gamma -1 ) \kappa \, \xi _{\parallel } + 4 \gamma \bigl(s^{2}-\kappa ^{2} \bigr) \nu ^{2} \xi _{\perp } \\ &\quad {}+ s \gamma \bigl(2 i s^{2} \kappa \, \xi _{\parallel }' + \bigl(3 s^{2} + \nu ^{2}\bigr)\xi _{\perp }' + s \bigl(s^{2} + \nu ^{2}\bigr) \xi _{\perp }'' \bigr) \bigr) \bigr] = 0. \end{aligned}$$

(25)

We note that it is only Equation 23 that is free from one of the second derivative terms, namely $\xi _{\parallel }''(s)$. By exploiting this leverage, we solve this equation for $\xi _{\parallel }(s)$ and eliminate all its derivatives entirely from Equation 25 to find an equation for $\xi _{\parallel }(s)$ only in terms of $\xi _{\perp }(s)$, $\eta (s)$ and their derivatives according to

$$\begin{aligned} \xi _{\parallel } =&\Delta ^{-1} \bigl\{ 2 i \gamma ^{2} \kappa \nu ^{2} s ^{3} \cos ^{2}\theta \, \xi _{\perp }^{(3)} \\ & {}+2 i \gamma \nu ^{2} s^{2} \cos \theta \bigl[(3 \gamma -2) \kappa \cos \theta -2 i \gamma \sin \theta \bigl(\kappa ^{2}+\nu ^{2} \bigr) \bigr] \xi _{\perp }'' \\ & {}+ i \gamma \nu ^{2} s \bigl[-2 i \gamma \sin 2 \theta \bigl( \kappa ^{2}+\nu ^{2} \bigr)-\kappa \bigl(\gamma \bigl(4 \kappa ^{2}-1 \bigr)+2 \bigr) \cos 2 \theta \\ & {}+\kappa \bigl(-4 \gamma \kappa ^{2}+\gamma +8 \gamma s^{2}-2 \bigr) \bigr]\xi _{\perp }' + 8 i\, \xi _{\perp } \bigl[2 \gamma \kappa \nu ^{2} \cos ^{2}\theta \bigl(\kappa ^{2}+(\gamma -1) s^{2} \bigr) \\ & {}+2 \sin \theta \bigl(\gamma \kappa \nu ^{2} s^{2} \sin \theta +i \cos \theta \bigl(\gamma ^{2} \kappa ^{2} \nu ^{2} \bigl(\kappa ^{2}+\nu ^{2} \bigr)+s ^{2} \bigl((\gamma -1) \kappa ^{2}-\gamma ^{2} \nu ^{4} \bigr) \bigr) \bigr) \bigr] \bigr\} \\ & {}- \epsilon _{\text{H}} R_{\parallel }, \end{aligned}$$

(26)

where

$$\begin{aligned} \Delta =& 8 s^{2} \bigl(\cos 2 \theta \bigl(\gamma ^{2} \nu ^{4}-( \gamma -1) \kappa ^{2} \bigr)-\gamma ^{2} \nu ^{4}-i (\gamma -2) \gamma \kappa \nu ^{2} \sin 2 \theta \\ & {}+\kappa ^{2} \bigl(\gamma \bigl(2 \gamma \nu ^{2}-1 \bigr)+1 \bigr) \bigr), \end{aligned}$$

(27)

and $R_{\parallel }$ is the Alfvén-to-acoustic driving term for displacements in the direction parallel to the unperturbed magnetic field, mediated by the Hall coupling, which describes producing an extra amount of acoustic wave powered by the Alfvén wave. Finally, introducing $\kappa _{0}$ and $\kappa _{z}$ defined in Equations 9a, 9b, Equation 26 yields the Hall-forced magneto-acoustic wave equation which takes the following form:

$$\begin{aligned} & 4 \bigl(s^{2} - \kappa ^{2} - 4 \kappa ^{4} - 4 \kappa ^{2} \kappa _{0} ^{2} + 4 \bigl(\kappa ^{4} + s^{2} \kappa _{z}^{2}\bigr) \sec ^{2}\theta - 4 i \kappa ^{3} \tan \theta + s^{2} \tan ^{2}\theta \bigr) \xi _{\perp } \\ &\quad {}+ 4 s \bigl(2 \kappa ^{2} + \kappa _{0}^{2} + \bigl(3 s^{2} - \kappa ^{2}\bigr) \sec ^{2} \theta + 4 i \kappa ^{3} \tan \theta \bigr) \xi _{\perp }' \\ &\quad {}+ 2 s^{2} \bigl(1 + 2 \kappa _{0}^{2} + 2 \bigl(s^{2} - \kappa ^{2}\bigr) \sec ^{2} \theta - 4 i \kappa \tan \theta \bigr) \xi _{\perp }'' \\ &\quad {}+ 4 s^{3} (1 - i \kappa \tan \theta )\xi _{\perp }^{(3)} +s^{4} \xi _{\perp }^{(4)} = \epsilon _{\text{H}} R_{\perp }, \end{aligned}$$

(28)

in which $R_{\perp }$ symbolizes the Alfvén-to-acoustic forcing term due to the Hall coupling for displacements in the direction perpendicular to the initial magnetic field, which is responsible for production of additional magneto-acoustic waves fueled by the Alfvén wave, and hence is a function of $\eta (s)$ only.