FAST-TO-ALFVÉN MODE CONVERSION MEDIATED BY THE HALL CURRENT. I. COLD PLASMA MODEL

Following Cally & Hansen (2011), we consider a cold MHD plasma with a uniform magnetic field ${{\boldsymbol{B}}}_{0}={B}_{0}(\mathrm{cos}\theta ,0,\mathrm{sin}\theta ).$ In the cold plasma (also called the zero-β) approximation, the sound speed is neglected in the perturbation equations compared to the Alfvén speed. This freezes out slow MHD waves, leaving only fast and Alfvén waves. The density stratification due to gravity (assumed to act in the negative x-direction) remains though, and produces an Alfvén speed a that increases (we shall assume) exponentially with x: $a(x)={a}_{0}\mathrm{exp}[x/2h],$ where h is the density scale height. See the appendix for details.

Hall current contributes an additional term to the electric field, which becomes

$$\begin{eqnarray}&&{\boldsymbol{E}}=-{\boldsymbol{v}}\times {\boldsymbol{B}}+\displaystyle \frac{{\boldsymbol{j}}\times {\boldsymbol{B}}}{{{en}}_{{\rm{e}}}},\end{eqnarray} \tag{ 1 }$$

where ${\boldsymbol{v}}$ is the mass-weighted combination of electron and ion fluid velocities, e is the elementary charge, and n_e is the electron number density (see Goedbloed et al. 2010). The current density ${\boldsymbol{j}}={\mu }_{0}^{-1}{\boldsymbol{\nabla }}\times {\boldsymbol{B}}$ (neglecting the displacement current as usual) and the Faraday equation $\partial {\boldsymbol{B}}/\partial t=-{\boldsymbol{\nabla }}\times {\boldsymbol{E}}$ may be combined with the momentum equation $\rho D{\boldsymbol{v}}/{Dt}={\boldsymbol{j}}\times {\boldsymbol{B}}$ to complete the description of the system. It is assumed that the plasma is collisionally dominated, so that the inertia of the neutrals plays a full role in the oscillations. Consequently, the full mass density ρ appears in the Alfvén speed $a=B/\sqrt{{\mu }_{0}\rho },$ and not just the ion density.

The linearized Hall-MHD equations may be combined into a single vector equation in the plasma displacement ${\boldsymbol{\xi }}$ (see the appendix). Fourier analyzing in time, ${\boldsymbol{\xi }}(x,y,z,t)={\boldsymbol{\xi }}(x,y,z)\;\mathrm{exp}(-i\omega t),$ and introducing the "Hall parameter"

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{\omega }{f\;{{\rm{\Omega }}}_{{\rm{i}}}},\end{eqnarray} \tag{ 2 }$$

where f is the ionization fraction and Ω_i is the mean ion gyrofrequency, these equations may be combined to yield

$$\begin{eqnarray}&&\left({\partial }_{\parallel }^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right){\boldsymbol{\xi }}=-{{\boldsymbol{\nabla }}}_{{\rm{p}}}\chi +i\left[{\boldsymbol{\nabla }}\tilde{\chi }\times {\hat{{\boldsymbol{e}}}}_{\parallel }-{{\rm{\nabla }}}^{2}\left(\tilde{{\boldsymbol{\xi }}}\times {\hat{{\boldsymbol{e}}}}_{\parallel }\right)\right],\end{eqnarray} \tag{ 3 }$$

where $\chi ={\boldsymbol{\nabla }}\;{\boldsymbol{\cdot }}\;{\boldsymbol{\xi }}$ is the dilatation. The Hall-parameter-scaled displacement $\tilde{{\boldsymbol{\xi }}}=\epsilon \;{\boldsymbol{\xi }}$ is also introduced, with $\tilde{\chi }={\boldsymbol{\nabla }}\;{\boldsymbol{\cdot }}\;\tilde{{\boldsymbol{\xi }}}.$ Furthermore, ${\hat{{\boldsymbol{e}}}}_{\parallel }={\hat{{\boldsymbol{B}}}}_{0}$ is the unit vector in the direction of the magnetic field, ${\partial }_{\parallel }={\hat{{\boldsymbol{B}}}}_{0}\;{\boldsymbol{\cdot }}\;{\boldsymbol{\nabla }}$ is the field-aligned directional derivative, and ${{\boldsymbol{\nabla }}}_{{\rm{p}}}={\boldsymbol{\nabla }}-{\hat{{\boldsymbol{B}}}}_{0}\;{\partial }_{\parallel }$ is the complementary perpendicular component of the gradient. The coordinates used are illustrated in Figure 1.

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The ionization fraction is $f={m}_{{\rm{i}}}{n}_{{\rm{i}}}/\rho ,$ with ${m}_{{\rm{i}}}$ and ${n}_{{\rm{i}}}$ as the mean ion mass and total number density, respectively. The mass of the electron is neglected relative to that of the ion. Charge neutrality ${n}_{{\rm{e}}}={{Zn}}_{{\rm{i}}}$ is assumed, where Z is the ion mean charge state. The ion gyrofrequency is denoted by ${{\rm{\Omega }}}_{{\rm{i}}}={{ZeB}}_{0}/{m}_{{\rm{i}}}.$ Equation (3) generalizes Equation (1) of Cally & Hansen (2011) by the addition of a term proportional to the ratio of the wave frequency to the ion gyrofrequency and inversely to the ionization fraction. For waves of helioseismic interest (frequencies of a few mHz), $\omega /{{\rm{\Omega }}}_{{\rm{i}}}$ is very small (the proton gyrofrequency is $15.2\;{B}_{0}\;{\rm{MHz}}$, for example, where B₀ is measured in Tesla), suggesting that very small ionization fractions and low field strengths are required for there to be any significant effect. This is discussed further in Section 7.

The plasma displacement is conveniently expressed in either the $(x,y,z)$ Cartesian coordinate system ${\boldsymbol{\xi }}(x)={\xi }_{x}{\hat{{\boldsymbol{e}}}}_{x}+{\xi }_{y}{\hat{{\boldsymbol{e}}}}_{y}+{\xi }_{z}{\hat{{\boldsymbol{e}}}}_{z},$ or in the magnetic flux system defined by the parallel direction (denoted by "$\parallel$"), the direction $(\mathrm{sin}\theta ,0,-\mathrm{cos}\theta )$ perpendicular to ${{\boldsymbol{B}}}_{0}$ but in the plane containing the field lines and the direction of stratification (denoted by "${\rm{\textquotedblleft }}\perp$"), and the y direction perpendicular to both: ${\boldsymbol{\xi }}(x)={\xi }_{\perp }{\hat{{\boldsymbol{e}}}}_{\perp }+{\xi }_{y}{\hat{{\boldsymbol{e}}}}_{y}.$ There is no component of displacement in the parallel direction because there is no restoring force that acts in that direction, having lost the gas pressure perturbation in the cold plasma approximation. We also use the subscript "p" to denote the plane perpendicular to ${{\boldsymbol{B}}}_{0},\;$ i.e., spanned by the unit basis vectors ${\hat{{\boldsymbol{e}}}}_{\perp }$ and ${\hat{{\boldsymbol{e}}}}_{y}.$ Decomposing Equation (3) into the two components, and Fourier analyzing in both time and the homogeneous spatial dimensions ${\boldsymbol{\xi }}(x,y,z,t)={\boldsymbol{\xi }}(x)\mathrm{exp}[i({k}_{y}y+{k}_{z}z-\omega t)],$

$$\begin{eqnarray}&&\left({\partial }_{\parallel }^{2}+{\partial }_{\perp }^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right){\xi }_{\perp }=-i{k}_{y}{\partial }_{\perp }{\xi }_{y}\\ &&\quad +\;i\left[i\;{k}_{y}{\partial }_{\perp }(\epsilon \;{\xi }_{\perp })-({\partial }_{\parallel }^{2}+{\partial }_{\perp }^{2})(\epsilon \;{\xi }_{y})\right]\end{eqnarray} \tag{ 4a }$$

and

$$\begin{eqnarray}&&\left({\partial }_{\parallel }^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}-{k}_{y}^{2}\right){\xi }_{y}=-i{k}_{y}{\partial }_{\perp }{\xi }_{\perp }\\ &&\quad +i\left[({\partial }_{\parallel }^{2}-{k}_{y}^{2})(\epsilon \;{\xi }_{\perp })-i\;{k}_{y}{\partial }_{\perp }(\epsilon \;{\xi }_{y})\right],\end{eqnarray} \tag{ 4b }$$

where ${\partial }_{\perp }={\hat{{\boldsymbol{e}}}}_{\perp }{\boldsymbol{\cdot }}{\boldsymbol{\nabla }},$ and of course ${\partial }_{\parallel }^{2}+{\partial }_{\perp }^{2}={\partial }_{x}^{2}-{k}_{z}^{2}$ is just the two-dimensional (2D) Laplacian in the $(\parallel ,\perp )$ or $(x,z)$ plane.

The derivatives ${\partial }_{\parallel }=\mathrm{cos}\theta \;{\partial }_{x}$ + $\mathrm{sin}\theta \;{\partial }_{z}$ = $\mathrm{cos}\theta \;{\partial }_{x}+i\;{k}_{z}\mathrm{sin}\theta$ and ${\partial }_{\perp }=\mathrm{sin}\theta \;{\partial }_{x}$ − $\mathrm{cos}\theta \;{\partial }_{z}=\mathrm{sin}\theta \;{\partial }_{x}$ − $i\;{k}_{z}\mathrm{cos}\theta$ cannot be completely written in terms of wavenumbers when the background depends on x, except in the WKB weakly inhomogeneous approximation (see Whitham 1974; Bender & Orszag 1978, for example), which we will use for expository purposes in Sections 2.2 and 2.3, and in part in Section 5. Otherwise though, the exact equations are retained.

As emphasized by Cally & Hansen (2011), cross-field wave propagation, ${k}_{y}\ne 0,$ couples the fast and Alfvén waves, respectively characterized by ${\xi }_{\perp }$ and ${\xi }_{y}$ when k_y is small. Equation (4) indicates that the Hall terms do this too, even when ${k}_{y}=0.$

Strictly speaking, pure Alfvén waves exist only in the 2D case ${k}_{y}=0,$ as only then is there a direction (${\hat{{\boldsymbol{e}}}}_{y}$) perpendicular to both the magnetic field and wave propagation direction in which the background medium is invariant. However, even with ${k}_{y}\ne 0,$ there are waves that are "asymptotically Alfvén" as $x\to \infty ,$ as shown by Frobenius expansion (see the appendix to Cally & Hansen 2011).

Assuming an exponentially decreasing density of scale height h, we may set the dimensionless quantity ${\omega }^{2}{h}^{2}/{a}^{2}={e}^{-x/h},$ thereby arbitrarily fixing the zero point of x. Note that this zero point is dependent on frequency as well as the background medium properties. The classical reflection point of the fast wave is at $\omega =a\;k,$ where ${k}^{2}={k}_{y}^{2}+{k}_{z}^{2},$ or equivalently $\omega h/a=\kappa ,$ where $\kappa ={kH}$ is the dimensionless wave number transverse to x. The significance of x = 0 then is that it is the point at which fast waves of transverse wavenumber $\kappa =1$ reflect. We shall mostly be concerned with $\kappa \ll 1$, for which transverse wavelength is much larger than the density scale height.

The local analysis of a cold plasma with Hall current is discussed in some detail by Damiano et al. (2009; Section III.B) for a fully ionized plasma. However, we reassess the equations here with a particular focus on a "precession" or oscillation between the fast and Alfvén waves that does not seem to have been fully appreciated up until now.

Neglecting all spatial variation in the background, or equivalently assuming that  all wavelengths are small compared to the density scale height, we may identify ${\partial }_{\parallel }=i\;{k}_{\parallel }$ and ${\partial }_{\perp }=i\;{k}_{\perp }.$ Then Equations (4a) and (4b) may be reduced to an algebraic matrix form ${\mathsf{D}}{\boldsymbol{\xi }}=0$ where

$$\begin{eqnarray}&&{\mathsf{D}}\;=\;\\ \left(\begin{array}{ll}\displaystyle \frac{{\omega }^{2}}{{a}^{2}}-{k}_{\parallel }^{2}-{k}_{\perp }^{2}+i\;\epsilon \;{k}_{y}{k}_{\perp } & -{k}_{y}{k}_{\perp }-i\;\epsilon ({k}_{\parallel }^{2}+{k}_{\perp }^{2})\\ -{k}_{y}{k}_{\perp }+i\;\epsilon ({k}_{\parallel }^{2}+{k}_{y}^{2}) & \displaystyle \frac{{\omega }^{2}}{{a}^{2}}-{k}_{\parallel }^{2}-{k}_{y}^{2}-i\;\epsilon \;{k}_{y}{k}_{\perp },\end{array}\right)\end{eqnarray} \tag{ 5 }$$

${\boldsymbol{\xi }}={({\xi }_{\perp },{\xi }_{y})}^{T}.$ The determinant of the coefficient matrix specifies the dispersion relation,

$$\begin{eqnarray}&&{\mathcal{D}}=\mathrm{det}{\mathsf{D}}=({\omega }^{2}-{a}^{2}{k}_{\parallel }^{2})({\omega }^{2}-{a}^{2}{k}^{2})-{\epsilon }^{2}{a}^{4}{k}_{\parallel }^{2}{k}^{2}=0,\end{eqnarray} \tag{ 6 }$$

where ${k}^{2}={| {\boldsymbol{k}}| }^{2}={k}_{\parallel }^{2}+{k}_{\perp }^{2}+{k}_{y}^{2},$ in accord with Equation (17) of Damiano et al. (2009; for f = 1). The Hall term couples the otherwise disjointed Alfvén wave, ${\omega }^{2}={a}^{2}{k}_{\parallel }^{2},$ and fast wave, ${\omega }^{2}={a}^{2}{k}^{2}.$ The eigenvectors, specifying the wave polarizations, are

$$\begin{eqnarray}&&{{\boldsymbol{\xi }}}_{\mathrm{fast}},{{\boldsymbol{\xi }}}_{{\rm{A}}}\;=\\ &&\quad \left[{k}_{\perp }^{2}-{k}_{y}^{2}-2i\;{k}_{\perp }{k}_{y}\epsilon \pm \sqrt{{({k}_{\perp }^{2}+{k}_{y}^{2})}^{2}+4{k}_{\parallel }^{2}{k}^{2}{\epsilon }^{2}}\;\right]{\hat{{\boldsymbol{e}}}}_{\perp }\\ &&\qquad +2\left[{k}_{\perp }{k}_{y}-i({k}_{\parallel }^{2}+{k}_{y}^{2})\epsilon \right]{\hat{{\boldsymbol{e}}}}_{y}\end{eqnarray} \tag{ 7 }$$

for the fast (+sign) and Alfvén (−sign) modes, respectively. Due to the imaginary terms proportional to , these describe elliptical polarization for $\epsilon \ne 0.$

In ideal MHD, the case ${k}_{\perp }={k}_{y}=0,\;$ $k={k}_{\parallel },$ is degenerate and the eigenvectors are arbitrary in the normal plane. The Hall term, however, splits the degeneracy, with eigenvalues satisfying

$$\begin{eqnarray}&&{({\omega }^{2}-{a}^{2}{k}^{2})}^{2}={\epsilon }^{2}{a}^{4}{k}^{4},\quad {\rm{i.e.}},\quad {\omega }^{2}={a}^{2}{k}^{2}\pm \epsilon \;{a}^{2}{k}^{2},\end{eqnarray} \tag{ 8 }$$

and eigenvectors $\pm i\;{\hat{{\boldsymbol{e}}}}_{\perp }+{\hat{{\boldsymbol{e}}}}_{y}.$ This dispersion relation accords with Equation (14.128) of Goedbloed et al. (2010; for f = 1). The small ${\mathcal{O}}(\epsilon )$ difference between the eigenfrequencies of the Alfvén and fast modes results in precession of the polarization of the combined "Alfvén wave" ${\omega }^{2}={a}^{2}{k}^{2},$ with precession frequency $\frac{1}{2}\left(\sqrt{1+\epsilon }-\sqrt{1-\epsilon }\right){ak}=\frac{1}{2}\epsilon {ak}+{\mathcal{O}}({\epsilon }^{3}),$ where $\epsilon ={ak}/{{\rm{\Omega }}}_{{\rm{i}}}f$ to leading order. This is the phenomenon of beating, and was noted by Cheung & Cameron (2012). It does not occur far away from the ${k}_{\perp }={k}_{y}=0$ degeneracy, or at least, the fast and Alfvén waves are more distinct there, so they would not be perceived in combination as a single precessing mode.

The analysis of Cheung & Cameron (2012; Section 3.1) returns the dispersion relation ${\omega }^{2}={a}^{2}{k}^{2}+{\sigma }^{2}$ (their Equation (16)), where $\sigma =(\epsilon /2){ak}$ in our notation, ignoring the ${\omega }^{2}={a}^{2}{k}^{2}-{\sigma }^{2}$ solution. This is at odds with our value from Equation (8) above, $\sqrt{\epsilon }\;{ak},$ and Equation (14.128) of Goedbloed et al. (2010; for f = 1). However, they then state that the precession frequency is (their) σ, i.e., $(\epsilon /2){ak},$ which is the correct formula to ${\mathcal{O}}({\epsilon }^{2}).$ A more correct statement of their formulation would have been $\omega ={ak}+\sigma ,$ so that ${\omega }^{2}={a}^{2}{k}^{2}+2\sigma {ak}+{\mathcal{O}}({\sigma }^{2}),$ or in other words ${\omega }^{2}={a}^{2}{k}^{2}+\epsilon \;{a}^{2}{k}^{2}$ to ${\mathcal{O}}(\epsilon ).$ Their analysis though was hampered by the assumption of an original ansatz that did not recognize the implicit coupling with the fast wave that makes the dispersion relation quartic in ω. The polarizations of single normal modes do not precess; they are just the unique eigenvectors.

Nevertheless, as depth increases in near-vertical field, with k_y and k_z fixed, the longitudinal Alfvén wavenumber ${k}_{\parallel }\approx {k}_{x}\approx \omega /a$ increases exponentially and the degenerate case is approached. We shall see that this is indeed the circumstance in which Hall-induced mode conversion occurs most strongly, mediated by precession of the joint polarization of the fast and Alfvén modes. This is discussed further in Section 2.3.

If we neglect the variation of a and with x, the basic Equation (3) may be recast in terms of $\chi ={\boldsymbol{\nabla }}\;{\boldsymbol{\cdot }}\;{\boldsymbol{\xi }}$ and $\zeta ={\hat{{\boldsymbol{e}}}}_{\parallel }\;{\boldsymbol{\cdot }}\;{\boldsymbol{\nabla }}\times {\boldsymbol{\xi }},$ which perfectly characterize the fast and Alfvén waves, respectively, at all wave orientations:

$$\begin{eqnarray}&&\left({{\rm{\nabla }}}^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right)\chi =-i\;\epsilon \;{{\rm{\nabla }}}^{2}\zeta \end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}&&\left({\partial }_{\parallel }^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right)\zeta =i\;\epsilon \;{\partial }_{\parallel }^{2}\chi .\end{eqnarray} \tag{ 9b }$$

Only the Hall term couples the two modes now.

Further neglecting the cross-field derivatives, ${\partial }_{\perp }={\partial }_{y}=0,$ the dispersion relation is transparently just as set out in Equation (8). The solution for χ and ζ contains linear combinations of trigonometric terms with rapid (Alfvénic) oscillations and slowly varying sinusoidal amplitudes.

For comparison with later numerical solutions, it is now convenient to solve for k with fixed wave frequency ω yielding four roots, $k=\pm \omega /(a\sqrt{1\pm \epsilon }),$ where the two ± signs are independent. The corresponding modes are therefore

$$\begin{eqnarray}&&\mathrm{exp}\left[\pm i\displaystyle \frac{\omega \;s/a}{\sqrt{1\pm \epsilon }}\right].\end{eqnarray} \tag{ 10 }$$

These combine to produce an Alfvénic wavenumber

$$\begin{eqnarray}&&{k}_{\mathrm{Alf}}=\displaystyle \frac{\omega }{2a}\left(\displaystyle \frac{1}{\sqrt{1-\epsilon }}+\displaystyle \frac{1}{\sqrt{1+\epsilon }}\right)=\displaystyle \frac{\omega }{a}+{\mathcal{O}}({\epsilon }^{2})\end{eqnarray} \tag{ 11 }$$

and an envelope wavenumber

$$\begin{eqnarray}&&{k}_{\mathrm{env}}=\displaystyle \frac{\omega }{2a}\left(\displaystyle \frac{1}{\sqrt{1-\epsilon }}-\displaystyle \frac{1}{\sqrt{1+\epsilon }}\right)=\displaystyle \frac{\epsilon \;\omega }{2a}+{\mathcal{O}}({\epsilon }^{3}).\end{eqnarray} \tag{ 12 }$$

The envelope is produced by the beating of the near-degenerate modes and describes an oscillatory transfer of energy between the compressive fast wave (χ) and incompressive Alfvén mode (ζ) on their journey through the Hall window. This will be confirmed numerically and generalized in propagation direction in Section 4. The spatial periodicity $4\pi a/\epsilon \omega$ (for small ) corresponds to a temporal periodicity $4\pi /\epsilon \omega ,\;$ i.e., circular frequency $\epsilon {ak}/2,$ which is just the "precession" frequency identified in Section 2.2.

Elementary analysis of the governing wave equations has already told us a lot. Both out-of-the-plane wave orientation k_y and Hall current couple fast and Alfvén waves, but in very different fashions. The former requires Alfvén speed stratification and operates locally near the fast wave reflection point. The latter applies everywhere that $\epsilon \ne 0,$ even in an unstratified plasma.

For a fixed Hall-effective window of thickness L, we must expect the fast-to-Alfvén conversion coefficient ${{\mathcal{A}}}^{+}$ (see Section 3.1) to display a $2\pi a/\omega L$ periodicity in when the wavevector is field-aligned. The quadratic dependence of ${{\mathcal{A}}}^{+}$ on ζ, which makes the periodicity $2\pi a/\omega L$ rather than $4\pi a/\omega L,$ has been taken into account. With increasing L or decreasing Alfvén speed a, the conversion coefficient oscillates ever more rapidly with . In a slowly varying atmosphere, the number of oscillations in passing from x₁ to x₂ would be

$$\begin{eqnarray}&&{N}_{\mathrm{osc}}={\displaystyle \int }_{{x}_{1}}^{{x}_{2}}\displaystyle \frac{\epsilon \omega }{2\pi a}\;{dx},\end{eqnarray} \tag{ 13 }$$

which shall be termed the "oscillation number." A half-integer value corresponds to total conversion, while a full integer yields zero net conversion. In practice, the Hall parameter and Hall window thickness L may be small enough or the Alfvén speed may be large enough that this periodic behavior is never seen for low-frequency waves.

For fixed frequency, the inverse quadratic dependence of the oscillation number on magnetic field strength is important (one factor of ${B}_{0}^{-1}$ through the ion gyrofrequency in , and the other through the Alfvén speed in the denominator). Ultimately, this means that Hall conversion can never be effective for mHz-frequency waves in sunspots (see Section 7).

We now turn to arbitrary magnetic field and wavevector orientations. In Section 3 we address the 2D case ${k}_{y}=0$ using a perturbation analysis. Both 2D and 3D cases are examined numerically in Section 4.

In the 2D case ${k}_{y}=0,$ the Hall terms proportional to $\omega /{{\rm{\Omega }}}_{{\rm{i}}}$ still couple ${\xi }_{\perp }$ and ${\xi }_{y}.$ To understand the nature of the coupling, it is useful to perform a perturbation analysis, expanding to first order in $\epsilon =\omega /{{\rm{\Omega }}}_{{\rm{i}}}f.$ Typically, $\epsilon \ll 1$ for waves of interest in the low solar atmosphere. At this stage, it is convenient to scale all lengths by h, or equivalently set h = 1.

We consider a (zeroth order) fast wave defined by

$$\begin{eqnarray}&&\left({\partial }_{\parallel }^{2}+{\partial }_{\perp }^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right){\xi }_{\perp 0}=\left({\partial }_{x}^{2}-{k}_{z}^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right){\xi }_{\perp 0}=0,\end{eqnarray} \tag{ 14 }$$

with ${\xi }_{y0}=0.$ With ${\omega }^{2}/{a}^{2}={e}^{-x},$ the appropriate solution is a Bessel function of the first kind,

$$\begin{eqnarray}&&{\xi }_{\perp 0}={J}_{2\kappa }\left(2{e}^{-x/2}\right),\end{eqnarray} \tag{ 15 }$$

representing the reflecting fast wave that is evanescent as $x\to \infty .$ Here $\kappa ={k}_{z}h$ is the dimensionless z-wavenumber.

The Alfvén wave driven through Hall coupling by this solution satisfies

$$\begin{eqnarray}&&\left({\partial }_{\parallel }^{2}+\displaystyle \frac{{\omega }^{2}}{{a}^{2}}\right){\xi }_{y1}=i\;\epsilon \;{R}_{H}(x),\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray}&&{R}_{H}=f\;{\partial }_{\parallel }^{2}\left({f}^{-1}{\xi }_{\perp 0}\right).\end{eqnarray} \tag{ 17 }$$

Equation (16) may be solved using the method of variation of parameters, or equivalently by constructing a Green's function. The solutions of the homogeneous equations are most conveniently expressed as Hankel functions ${e}^{-i\kappa x\mathrm{tan}\theta }{H}_{0}^{(1)}(2{e}^{-x/2}\mathrm{sec}\theta )$ and ${e}^{-i\kappa x\mathrm{tan}\theta }{H}_{0}^{(2)}(2{e}^{-x/2}\mathrm{sec}\theta ),$ respectively representing the leftward (downward) and rightward (upward) propagating Alfvén waves. Their Wronskian is $W=2\;i\;{\pi }^{-1}\mathrm{exp}[-2i\kappa \;x\mathrm{tan}\theta ].$ The formal driven inhomogeneous solution is then

$$\begin{eqnarray}{\xi }_{y1} & = & \epsilon \;\displaystyle \frac{\pi }{2}\;{e}^{i\kappa \;x\mathrm{tan}\theta }{\mathrm{sec}}^{2}\theta \left[{A}_{1}(x){H}_{0}^{(1)}(2{e}^{-x/2}\mathrm{sec}\theta )\right.\\ & & +\left.{A}_{2}(x){H}_{0}^{(2)}(2{e}^{-x/2}\mathrm{sec}\theta )\right],\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}{A}_{1}(x) & = & {\displaystyle \int }_{x}^{\infty }{e}^{i\;\kappa \;X\mathrm{tan}\theta }{H}_{0}^{(2)}(2{e}^{-X/2}\mathrm{sec}\theta )\;{R}_{H}(X)\;{dX}\\ & = & {\displaystyle \int }_{0}^{{e}^{-x}}{s}^{-i\;\kappa \mathrm{tan}\theta -1}{H}_{0}^{(2)}(2\sqrt{s}\mathrm{sec}\theta )\;{R}_{H}(-\mathrm{ln}s)\;{ds}\\ & = & {\displaystyle \int }_{{e}^{-x}}^{\infty }{a}_{1}(s)\;{ds},\end{eqnarray} \tag{ 19 }$$

and

$$\begin{eqnarray}{A}_{2}(x) & = & {\displaystyle \int }_{-\infty }^{x}{e}^{i\;\kappa \;X\mathrm{tan}\theta }{H}_{0}^{(1)}(2{e}^{-X/2}\mathrm{sec}\theta )\;{R}_{H}(X)\;{dX}\\ & = & {\displaystyle \int }_{{e}^{-x}}^{\infty }{s}^{-i\;\kappa \mathrm{tan}\theta -1}{H}_{0}^{(1)}(2\sqrt{s}\mathrm{sec}\theta )\;{R}_{H}(-\mathrm{ln}s)\;{ds}\\ & = & {\displaystyle \int }_{{e}^{-x}}^{\infty }{a}_{2}(s)\;{ds},\end{eqnarray} \tag{ 20 }$$

assuming that the integrals exist. The change of variable $s={e}^{-x}\in (0,\infty )$ has been introduced.

Conversion to the upgoing Alfvén wave is determined by the wave-energy conversion coefficient (see Cally & Hansen 2011)

$$\begin{eqnarray}&&{{\mathcal{A}}}^{+}={\epsilon }^{2}{\pi }^{2}{\mathrm{sec}}^{2}\theta {\left|{A}_{2}(\infty )\right|}^{2}={\epsilon }^{2}{{\mathcal{F}}}^{+},\end{eqnarray} \tag{ 21 }$$

defining ${{\mathcal{F}}}^{+}.$ Conversely, conversion to the downgoing Alfvén wave has the coefficient

$$\begin{eqnarray}&&{{\mathcal{A}}}^{-}={\epsilon }^{2}{\pi }^{2}{\mathrm{sec}}^{2}\theta {\left|{A}_{1}(-\infty )\right|}^{2}={\epsilon }^{2}{{\mathcal{F}}}^{-}.\end{eqnarray} \tag{ 22 }$$

${{\mathcal{A}}}^{\pm }=0$ indicates no conversion, and 1 means total conversion. The perturbation solution breaks down before either reach 1 as increases (see Figure 9 of Cally & Hansen 2011).

To gain an understanding of the Hall coupling process, we first explore the case of uniform ionization fraction f in a partially ionized gas, for which is uniform. Then

$$\begin{eqnarray}{R}_{H}(-\mathrm{ln}s) & = & {\partial }_{\parallel }^{2}{\xi }_{\perp 0}\\ & = & {s}^{\kappa +1}\left[{}_{0}{\tilde{F}}_{1}\left(;2\kappa +1;-s\right)\left(\displaystyle \frac{{\kappa }^{2}}{s}{e}^{2i\theta }-{\mathrm{cos}}^{2}\theta \right)\right.\\ & & -\left.\displaystyle \frac{i\;\kappa \mathrm{sin}2\theta }{s}{\;}_{0}{\tilde{F}}_{1}\left(;2\kappa ;-s\right)\right],\end{eqnarray} \tag{ 23 }$$

where ${}_{0}{\tilde{F}}_{1}(;b;z)={}_{0}{F}_{1}(;b;z)/{\rm{\Gamma }}(b)$ is a regularized ${}_{0}{F}_{1}$ hypergeometric function (Wolfram Research 2015; DLMF 2014, chapter 16). Specifically,

$$\begin{eqnarray*}&&{}_{0}{\tilde{F}}_{1}(;b;z)=\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{z}^{k}}{{\rm{\Gamma }}(b+k)\;k!}.\end{eqnarray*}$$

The evaluation of the integrals (21) and (22) for the A-coefficients presents numerical difficulties, as a₁ and a₂ are highly oscillatory. This is illustrated by the asymptotic behavior as $s\to \infty$ ($x\to -\infty$) in $-\pi \lt \mathrm{arg}\;s\leqslant \pi$ of the integrand ${a}_{2}(s)$ in A₂ for the case (23),

$$\begin{eqnarray}&&{a}_{2}(s)\sim \displaystyle \frac{{e}^{3i\pi /4}\;{e}^{2i\sqrt{s}\mathrm{sec}\theta }{s}^{-i\kappa \mathrm{tan}\theta }}{\pi {\mathrm{sec}}^{5/2}\theta }\\ &&\quad \left[\left({s}^{-1/2}+{\mathcal{O}}\left({s}^{-1}\right)\right)\mathrm{cos}\left(2\sqrt{s}-\pi \left(\kappa +1/4\right)\right)\right.\\ &&\qquad +\left.{\mathcal{O}}\left({s}^{-1}\right)\mathrm{sin}\left(2\sqrt{s}-\pi \left(\kappa +1/4\right)\right)\right].\end{eqnarray} \tag{ 24 }$$

Note that $\int {s}^{-1/2}$ exp  $(2i\alpha \sqrt{s})\mathrm{cos}2\sqrt{s}\;{ds}$ = $i{(1-{\alpha }^{2})}^{-1}$ exp $(2i\alpha \sqrt{s})\;[\alpha \mathrm{cos}(2\sqrt{s}-b)-i\mathrm{sin}(2\sqrt{s}-b)]$ for $\alpha \ne \pm 1,$ but $\int {s}^{-1/2}\mathrm{exp}(2i\sqrt{s})\mathrm{cos}2\sqrt{s}\;{ds}$ = $\left(i\sqrt{s}-\displaystyle \frac{1}{4}{e}^{4i\sqrt{s}}\right)\mathrm{sin}b$ + $\left(\sqrt{s}-\displaystyle \frac{1}{4}{{ie}}^{4i\sqrt{s}}\right)\mathrm{cos}b.$ This indicates that $| {A}_{2}(x)|$ diverges as $s\to \infty$ in the resonant case $\mathrm{sec}\theta =\pm 1,$ but that it converges otherwise, even though ${A}_{2}(x)$ itself is oscillatory with finite amplitude. The oscillatory behavior can be suppressed in ${A}_{2}(\infty )$ by integrating along a straight line path $0\lt | s| \lt \infty ,$ $0\lt \mathrm{arg}\;s\leqslant \pi$ rather than directly along the positive real s-axis. Doing so renders the numerical evaluation fast and accurate. Similarly, $| {A}_{1}(-\infty )|$ is best evaluated using a radius in the lower half complex s-plane.

Figure 2 shows ${{\mathcal{F}}}^{+}$ and ${{\mathcal{F}}}^{-}$ as functions of θ and κ. Several features are immediately apparent.

• 1.  
  ${{\mathcal{F}}}^{+}$ and ${{\mathcal{F}}}^{-}$ are the mirror images of each other in θ. This is to be expected, as ${A}_{1}(-\infty )$ and ${A}_{2}(\infty )$, with opposite signs in θ, are complex conjugates of each other.
• 2.  
  Both ${{\mathcal{F}}}^{+}$ and ${{\mathcal{F}}}^{-}$ are singular at $\theta =0,$ because of the perfect resonant coupling of the Hankel function solutions of the free Alfvén waves, with the fast wave driving term R_H. This is particularly apparent in Equation (24), where the fore-factor $\mathrm{exp}[2i\sqrt{s}]$ (when $\theta =0$) exactly matches with the later cosine term. This renders the A₁ and A₂ integrals divergent.
• 3.  
  There is only weak dependence on κ.

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Of course, the conversion coefficients cannot be infinite. Indeed, they cannot exceed 1. Multiplying ${{\mathcal{F}}}^{+}$ and ${{\mathcal{F}}}^{-}$ by the typically very small ${\epsilon }^{2}$ to get the actual conversion coefficients ${{\mathcal{A}}}^{\pm }$ restricts the coupling to small θ (near vertical field), but does not remove the singularity.

There are two reasons for our unphysical results for small θ. First, we have used a perturbation expansion. This clearly breaks down when the coupling is too strong.

Second, we have assumed an infinite region of coupling. Unlike the ideal MHD coupling by k_y investigated by Cally & Hansen (2011), which occurs locally in the neighborhood of the fast wave reflection point, Hall coupling in our simple model occurs throughout the fast wave's domain. In the Sun and stars though, Hall coupling is associated only with finite regions of very low ionization fraction. As we move deeper into the interior, ionization becomes near-total, and the effect is suppressed. This is illustrated in Figure 4, where the contribution to A₂ is (arbitrarily) restricted to $x\gt -5$ (left panel), that is, to the region extending upward from five scale heights beneath the $\kappa =1$ reflection point. Again we see the general preference for small θ, but the amplitudes are reduced by orders of magnitude compared to the full-domain case of Figure 2. The singularity at $\theta =0$ has of course disappeared. Restriction of the contribution to $x\gt -1$ greatly diminishes the coupling further. The interaction integral A₂ is displayed as a function of x or s in Figure 3 for two field inclinations. It is seen that it is highly oscillatory, but that the oscillatory behavior is pushed ever deeper as the field becomes more vertical. This limits the conversion in practice for fields of small inclination if the Hall interaction region is of limited extent.

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Another feature of Figure 4 is the shifting of the maximum conversion from vertical field $\theta =0^\circ$ as κ increases to ${\mathcal{O}}(1)$ and above. This is due to the overall wavevector tilting away from the vertical; the maximum is essentially in the field-aligned propagation direction.

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Discussion of the application of these insights to the weakly ionized temperature minimum region of the solar atmosphere is deferred until Section 7.

We now turn to full numerical solutions in 2D and 3D.

First, consider cases with ${k}_{y}=0,$ where Hall current is the only mechanism coupling the fast and Alfvén waves. Figure 5 shows the conversion coefficient for a range of parameters as a function of magnetic field inclination θ.

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The results are entirely consistent with the perturbation results above. Specific points of interest include the following.

• 1.  
  For small κ, conversion is most favored by a vertical or moderately inclined magnetic field, and is very weak for a highly inclined field.
• 2.  
  The dependence on dimensionless wavenumber $\kappa \ll 1$ is very weak, since the wave is essentially vertically propagating in any case.
• 3.  
  Restriction of the Hall window to $(-5,0,1)$ makes almost no difference compared to $(-5,5,1),$ indicating that the mode conversion is occurring overwhelmingly below x = 0. For comparison, with $\kappa =0.1,$ the fast wave reflection point is ${x}_{\mathrm{ref}}=-\mathrm{ln}{\kappa }^{2}=\mathrm{ln}100=4.6.$ This is very different from k_y-mediated conversion, which occurs near ${x}_{\mathrm{ref}}.$
• 4.  
  Restricting the Hall window to $(-2,2,1)$ greatly reduces the mode conversion, indicating that it is chiefly happening below $x=-2.$
• 5.  
  ${{\mathcal{A}}}^{+}$ is roughly symmetric about $\theta =0^\circ .$
• 6.  
  Conversion to the downward propagating Alfvén wave (Figure 6) is also roughly symmetric for small ${\epsilon }_{0}.$ However, at a large enough ${\epsilon }_{0},$ where ${{\mathcal{A}}}^{+}$ approaches 1, it moves to two lobes on either side of the central ${{\mathcal{A}}}^{+}$ peak. Of course, ${{\mathcal{A}}}^{+}+{{\mathcal{A}}}^{-}\leqslant 1,$ so they cannot both be 1 in the same case.
• 7.  
  Figure 7 illustrates the periodic behavior of ${{\mathcal{A}}}^{+}$ with ${\epsilon }_{0}$ at a small field inclination that was anticipated in Section 2.3. It is seen that the period increases with increasing height in the atmosphere (increasing Alfvén speed), and that it depends on a throughout the window. The periodicity will be discussed again in Section 4.2.

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Figure 8 displays $\zeta ={\hat{{\boldsymbol{e}}}}_{\parallel }\;{\boldsymbol{\cdot }}\;{\boldsymbol{\nabla }}\times {\boldsymbol{\xi }}$ for 2D strongly ($\theta =5^\circ$) and weakly ($\theta =60^\circ$) Hall-coupled cases. The quantity ζ preferentially selects the Alfvén wave. The effect of the lower part of the Hall window is particularly apparent in the strong coupling case, with its greatly enhanced ζ-amplitude.

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In three dimensions (${k}_{y}\ne 0$), the fast and Alfvén modes couple, preferentially near the fast wave reflection height, as discussed by Cally & Hansen (2011). The question now is, to what extent does the Hall current modify or add to this conversion?

We introduce the orientation angle ϕ, defined by ${\kappa }_{y}=\kappa \mathrm{sin}\phi$ and ${\kappa }_{z}=\kappa \mathrm{cos}\phi .$ Figure 9 shows how the upward Alfvén conversion coefficient ${{\mathcal{A}}}^{+}$ varies with ϕ in several cases, compared with the non-Hall case ${\epsilon }_{0}=0.$ As in 2D, the largest effects are at small θ, reducing to almost nothing at large θ. Specifically:

• 1.  
  The Hall parameter ${\epsilon }_{0}$ essentially shifts the conversion curve uniformly to the right at small θ. This produces a periodic dependence on that was foreshadowed in Section 2.3 and seen previously in Figure 7. Effectively, the Hall term adds linearly to the orientation ϕ, but with a magnitude that increases with the depth of the Hall window.
• 2.  
  At intermediate θ (45°), it enhances the conversion at negative ϕ but diminishes it at positive ϕ.
• 3.  
  There is practically no effect at $\theta =80^\circ .$

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Taking ion number density data from the quiet Sun Model C of Vernazza et al. (1981, Tables 12 and 17–24), based on a mixture of neutral and singly ionized H, He, C, Mg, Al, Si, and Fe, the mean atomic weight of ions at the temperature minimum is ${m}_{{\rm{i}}}=38.7$ amu, the total ion number density is ${n}_{{\rm{i}}}=2.29\times {10}^{17}\;{{\rm{m}}}^{-3},$ the mean ion gyrofrequency is ${{\rm{\Omega }}}_{{\rm{i}}}=2.47\times {10}^{6}{B}_{0}\;{\rm{rad}}\;{{\rm{s}}}^{-1},$ where B₀ is measured in Tesla, and the ionization fraction ${m}_{{\rm{i}}}{n}_{{\rm{i}}}/\rho$ is $f=3\times {10}^{-3}.$ The electron number density ${n}_{{\rm{e}}}={n}_{{\rm{i}}}$ is slightly below the value $2.495\times {10}^{17}{{\rm{m}}}^{-3}$ listed in Table 12, presumably because of the presence of other minor ions, but is close enough for our purposes. For a 5 mHz wave, this gives $\epsilon =4.2\times {10}^{-6}\;{B}_{0}^{-1}.$ The Alfvén speed is $a=400\;{B}_{0}\;\mathrm{km}\;{{\rm{s}}}^{-1}.$ For a Hall window of width L, measured in km, the oscillation number is therefore ${N}_{\mathrm{osc}}\approx 5\times {10}^{-11}{B}_{0}^{-2}L.$ At ${B}_{0}={10}^{-4}$ T (1 G) with $L\approx 600$ km, we have oscillation number ${N}_{\mathrm{osc}}\approx 3,$ but this diminishes quadratically with increasing field strength. It is already negligible for 10 G.

One way of enhancing Hall conversion is for the magnetic field to be highly inclined, so that the effective L is greatly increased, and the wave propagation direction is angled to match. This requires $\kappa \gg 1,$ but we do not expect much power at such wavenumbers in the p-mode spectrum. However, other locally excited waves may exhibit such behavior.

Ionization fractions are lower in sunspot umbrae (Maltby et al. 1986), but their much greater magnetic field strengths more than compensate, resulting in even smaller oscillation numbers.

Although Hall-mediated conversion of low-frequency waves is only effective for the very weak fields of the quiet Sun, the overall amount of converted energy can still be significant since the quiet Sun occupies most of the solar volume. Indeed, recent high-resolution measurements using Hanle and Zeeman effects reveal that the quiet Sun is full of weak magnetic fields of magnitudes below 10–100 G (Trujillo Bueno et al. 2004; Sánchez Almeida & Martínez González 2011). Therefore, the process identified here can contribute to chromospheric and coronal heating by facilitating the propagation of Alfvén waves into the upper atmosphere in quiet solar areas.

The Hall effect has already been identified in simulations as being crucial to the process of fast reconnection in collisionless plasmas by allowing inflow close to the Alfvén speed (Birn et al. 2001; Rogers et al. 2001; Shay et al. 2001). If, as commonly believed, small-scale reconnection is ubiquitous through the solar atmosphere, spanning both collisional and collisionless regimes, fast/Alfvén oscillation may play a significant role in light of the radically higher frequencies involved.

Escaping waves originating from these events may also be affected. It was suggested by Axford & McKenzie (1992) that high frequency waves with periods of the order of or less than one second can be released in "microflare" reconnections. By increasing by between two and three orders of magnitude, all other things being equal, reconnection waves would indeed be most susceptible to Hall-mediated conversion.

Turning now to star-forming regions, we take some typical values from the simulations of J. Wurster et al. (2015, in preparation), who find that Hall current aids collapse and disk formation if the magnetic field is anti-aligned with the rotation axis. Specifically, we set ${B}_{0}=4\times {10}^{-8}$ T, ${\rho }_{0}=7.5\times {10}^{-15}\;\mathrm{kg}\;{{\rm{m}}}^{-3},$ a mean ion mass of 24.3 amu, and an initial cloud size  $L=0.013$ parsec. The Alfvén speed is then about 400 ${\rm{m}}\;{{\rm{s}}}^{-1},$ and the ion gyrofrequency is 0.16 ${{\rm{s}}}^{-1}.$ Crucially, the ionization fraction is $f={10}^{-10}$ or lower. We take as the dynamical timescale the free-fall time of $T=2.4\times {10}^{4}\;{\rm{years}}$ and consider an Alfvén period of that order: $\omega =2\pi /T=8.3\times {10}^{-12}\;{{\rm{s}}}^{-1}.$ These give $\epsilon \sim 0.5,$ and ${N}_{\mathrm{osc}}\sim 0.7.$

These numbers are all very rough, and the result could vary by large amounts either way, but it is interesting that the oscillation number is at least plausibly of ${\mathcal{O}}(1).$ We might hypothesize that the process could even contribute to star formation by taking energy from incompressive Alfvén waves and converting it to compressive magnetoacoustic waves. Any increase in density could help precipitate gravitational collapse.