2.1.1 Solenoidal constraint

Let us first consider the solenoidal constraint (2.5). We rewrite it in terms of $Q$-variables, through the expression of $\boldsymbol {B}$ in terms of $\boldsymbol {Q}^\pm$

(2.7)\begin{equation} \boldsymbol{B}=\frac{1}{2\alpha}(\boldsymbol{Q}^+-\boldsymbol{Q}^-). \end{equation}

Inserting that into (2.5) allows us to write

(2.8)\begin{equation} 0=\frac{1}{2\alpha}\boldsymbol{\nabla}\boldsymbol{\cdot}(\boldsymbol{Q}^+-\boldsymbol{Q}^-)-\frac{1}{2\alpha} (\boldsymbol{Q}^+-\boldsymbol{Q}^-)\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\alpha}, \end{equation}

or, after simplification,

(2.9)\begin{equation} 0=\boldsymbol{\nabla}\boldsymbol{\cdot}(\boldsymbol{Q}^+-\boldsymbol{Q}^-)- (\boldsymbol{Q}^+-\boldsymbol{Q}^-)\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\alpha}. \end{equation}

Considering the limiting case of $\alpha ^2=1/\mu \rho, \boldsymbol {Q}^\pm =\boldsymbol {Z}^\pm, \boldsymbol {Z}^+-\boldsymbol {Z}^-=\boldsymbol {V}_{A}$, we recover equation (10) of Marsch & Mangeney (Reference Marsch and Mangeney1987).

2.1.2 Conservation of mass

Next, we rewrite the conservation of mass (2.3). We use it for finding an expression for $({{\rm D}^\pm }/{{\rm D} t})(\ln {\rho })$, where ${\rm D}^\pm /{\rm D} t=\partial /\partial t+\boldsymbol {Q}^\pm \boldsymbol {\cdot }\boldsymbol {\nabla }$ is the derivative co-moving with the wave, in the so-called waveframe. We find

(2.10)\begin{align} \frac{{\rm D}^\pm}{{\rm D} t} (\ln{\rho}) & = \frac{\partial\ln{\rho}}{\partial t}+ \boldsymbol{Q}^\pm\boldsymbol{\cdot}\boldsymbol{\nabla} \ln{\rho}, \end{align}

(2.11)\begin{align} & = \frac{\partial\ln{\rho}}{\partial t} + \boldsymbol{V} \boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\rho} \pm \alpha\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\rho}, \end{align}

(2.12)\begin{align} & =-\nabla\boldsymbol{\cdot}\boldsymbol{V}\pm \alpha\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\rho}, \end{align}

where the continuity equation (2.3) was used in the last equation. To this last equation, we add on the right-hand side ($\mp \times$ (2.9)) to find

(2.13)\begin{align} \frac{{\rm D}^\pm}{{\rm D} t} (\ln{\rho}) & =-\frac{1}{2} \boldsymbol{\nabla}\boldsymbol{\cdot}(\boldsymbol{Q}^++\boldsymbol{Q}^-)\mp \boldsymbol{\nabla}\boldsymbol{\cdot}(\boldsymbol{Q}^+-\boldsymbol{Q}^-)\pm \frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\rho\alpha^2}, \end{align}

(2.14)\begin{align} & =-\frac{1}{2}\boldsymbol{\nabla}\boldsymbol{\cdot}(3\boldsymbol{Q}^\pm-\boldsymbol{Q}^\mp)\pm \frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\rho\alpha^2}, \end{align}

where we have used the expressions for $\boldsymbol {V}$ in terms of $\boldsymbol {Q}^\pm$ in the equations

(2.15)\begin{equation} \boldsymbol{V}=\tfrac{1}{2}(\boldsymbol{Q}^++\boldsymbol{Q}^-). \end{equation}

When the limit of $\alpha ^2\to 1/\sqrt {\mu \rho }$ is considered, the last term of (2.14) cancels out and (16) of Marsch & Mangeney (Reference Marsch and Mangeney1987) is readily recovered.

2.1.3 Momentum equation

Now we turn to the momentum equation and the induction equation, (2.2) and (2.4), which form the key equation (17) of Marsch & Mangeney (Reference Marsch and Mangeney1987). Following their lead, we add (2.2)$\pm \alpha$(2.4). In the first step, we use the expansion of $\boldsymbol {Q}^\mp \boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {Q}^\pm$ as

(2.16)\begin{equation} \boldsymbol{Q}^\mp\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{Q}^\pm= \boldsymbol{V}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{V}\mp \alpha \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{V}\pm \alpha \boldsymbol{V}\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{B} -\alpha^2\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{B}\pm \boldsymbol{B}\boldsymbol{Q}^\mp \boldsymbol{\cdot}\boldsymbol{\nabla}\alpha, \end{equation}

where we have used the vector identity $\boldsymbol {C}\boldsymbol {\cdot }\boldsymbol {\nabla }(\,f\boldsymbol {D})=f\boldsymbol {C}\boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {D}+\boldsymbol {D}(\boldsymbol {C}\boldsymbol {\cdot }\boldsymbol {\nabla } f)$ for any vector fields $\boldsymbol {C}$ and $\boldsymbol {D}$ and scalar field $f$. We also define the parameter

(2.17)\begin{equation} \Delta \alpha^2=\alpha^2-\frac{1}{\mu\rho}. \end{equation}

The $\Delta \alpha ^2$ parameter expresses how far a wave's phase speed is from the Alfvén speed. Since a wave can be slower or faster than the Alfvén speed, the $\Delta \alpha ^2$ parameter may be positive or negative, despite the square! The square in the notation is kept for dimensional purposes to keep $\Delta \alpha$ in the same units as $\alpha$. In the limit of $\alpha ={1}/{\sqrt {\mu \rho }}$, the parameter $\Delta \alpha ^2$ will turn to 0: $\Delta \alpha ^2=0$ and $\boldsymbol {Q}^\pm =\boldsymbol {Z}^\pm$ turns into the classical Elsässer variable. Here, it is also useful to point out that it will be convenient to use expressions with $\rho \alpha ^2$, which are constant in this limit.

With the above expressions, we obtain from (2.2)$\pm \alpha$(2.4) the result

(2.18)\begin{equation} \frac{\partial \boldsymbol{Q}^\pm}{\partial t}\mp \boldsymbol{B}\frac{\partial \alpha}{\partial t}=-\boldsymbol{Q}^\mp\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{Q}^\pm \pm \boldsymbol{B}\boldsymbol{Q}^\mp \boldsymbol{\cdot}\boldsymbol{\nabla}\alpha-\Delta \alpha^2\boldsymbol{B} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{B}-\frac{1}{\rho}\boldsymbol{\nabla} P_{T}\mp \alpha \boldsymbol{B}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}. \end{equation}

The first two terms on the right-hand side group with the left-hand side to form the co-moving derivative

(2.19)\begin{equation} \frac{{\rm D}^\mp }{{\rm D} t}\boldsymbol{Q}^\pm \mp \boldsymbol{B}\frac{{\rm D}^\mp}{{\rm D} t} \alpha=-\frac{1}{\rho}\boldsymbol{\nabla} P_{T} -\Delta \alpha^2\boldsymbol{B} \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{B} \mp\alpha \boldsymbol{B}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{V}. \end{equation}

We now find an expression for the terms on the right-hand side. For the total pressure term, we find

(2.20)\begin{align} \frac{1}{\rho}\boldsymbol{\nabla} P_{T}& =\frac{1}{\rho}\boldsymbol{\nabla} \left(\,p+\frac{B^2}{2\mu}\right)\!,\\ & = v_{s}^2 \boldsymbol{\nabla} \ln{\rho} + \frac{1}{8\alpha^2} (\alpha^2-\Delta \alpha^2)\boldsymbol{\nabla} (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2 \nonumber\end{align}

(2.21)\begin{align} & \quad +\frac{1}{8}(\alpha^2-\Delta \alpha^2)(\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2\boldsymbol{\nabla}\left(\frac{1}{\alpha^2}\right)\!, \end{align}

(2.22)\begin{align} & = v_{s}^2 \boldsymbol{\nabla} \ln{\rho} + \frac{1}{8} \left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) \boldsymbol{\nabla} (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2 \nonumber\\ & \quad -\frac{1}{4}\left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2\boldsymbol{\nabla} \ln{\alpha}, \end{align}

where we have used the adiabatic relationship of $p(\rho )$ which introduces the expression for the sound speed $v_{s}=\sqrt {{\gamma p}/{\rho }}$.

The second term on the right-hand side of (2.19) can be rewritten with the expression for $\boldsymbol {B}$ in terms of $\boldsymbol {Q}^\pm$ as

(2.23)\begin{align} -\Delta \alpha^2\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{B} & =-\frac{1}{4} \frac{\Delta \alpha^2}{\alpha^2} (\boldsymbol{Q}^+-\boldsymbol{Q}^-)\boldsymbol{\cdot} \boldsymbol{\nabla}(\boldsymbol{Q}^+-\boldsymbol{Q}^-) \nonumber\\ & \quad +\frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) ((\boldsymbol{Q}^+-\boldsymbol{Q}^-)\boldsymbol{\cdot}\boldsymbol{\nabla} \ln{\alpha}). \end{align}

The third term on the right-hand side of (2.19) should be handled through the modified version of the continuity relation (2.13). From that equation, we have that

(2.24)\begin{align} \mp\alpha \boldsymbol{B}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{V} & ={\pm} \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \frac{{\rm D}^\mp}{{\rm D} t} (\ln{\rho}) -\left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \boldsymbol{\nabla}\boldsymbol{\cdot} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) \nonumber\\ & \quad + \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \left(\left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \boldsymbol{\cdot}\boldsymbol{\nabla} \ln{\rho\alpha^2}\right)\!. \end{align}

Substituting everything in (2.19), we now have

(2.25)\begin{align} \frac{{\rm D}^\mp }{{\rm D} t}\boldsymbol{Q}^\pm & \mp \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right)\frac{{\rm D}^\mp }{{\rm D} t} \ln{\alpha}=-v_{s}^2 \boldsymbol{\nabla} \ln{\rho} - \frac{1}{8} \left(1-\frac{\Delta\alpha^2}{\alpha^2}\right)\boldsymbol{\nabla} (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2 \nonumber\\ & \quad + \frac{1}{4}\left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2\boldsymbol{\nabla} \ln{\alpha} -\frac{1}{4}\frac{\Delta\alpha^2}{\alpha^2} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) \boldsymbol{\cdot}\boldsymbol{\nabla} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) \nonumber\\ & \quad +\frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) ((\boldsymbol{Q}^+-\boldsymbol{Q}^-)\boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\alpha}) \pm \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right)\frac{{\rm D}^\mp}{{\rm D} t} (\ln{\rho}) \nonumber\\ & \quad - \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \boldsymbol{\nabla}\boldsymbol{\cdot} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) + \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \left(\left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \boldsymbol{\cdot}\boldsymbol{\nabla} \ln{\rho\alpha^2}\right)\!. \end{align}

After moving the right-hand side convective derivative to the left-hand side and subsequently adding (${\pm } ({(\boldsymbol {Q}^+-\boldsymbol {Q}^-)}/{4})\times$ (2.14)) and using (2.9), we obtain the final result

(2.26)\begin{align} \frac{{\rm D}^\mp }{{\rm D} t}\boldsymbol{Q}^\pm & \mp \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{4}\right) \frac{{\rm D}^\mp }{{\rm D} t}\ln{\rho\alpha^2}=-v_{s}^2 \boldsymbol{\nabla} \ln{\rho} - \frac{1}{8} \left(1-\frac{\Delta\alpha^2}{\alpha^2}\right)\boldsymbol{\nabla} (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2 \nonumber\\ & \quad +\frac{1}{4}\left(1-\frac{\Delta\alpha^2}{\alpha^2}\right) (\boldsymbol{Q}^+-\boldsymbol{Q}^-)^2\boldsymbol{\nabla} \ln{\alpha} \nonumber\\ & \quad -\frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) \boldsymbol{\cdot}\boldsymbol{\nabla} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) + \frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (\boldsymbol{Q}^+-\boldsymbol{Q}^-) \boldsymbol{\nabla}\boldsymbol{\cdot}(\boldsymbol{Q}^+-\boldsymbol{Q}^-) \nonumber\\ & \mp \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{8}\right) \boldsymbol{\nabla}\boldsymbol{\cdot}(3\boldsymbol{Q}^\pm-\boldsymbol{Q}^\mp) + \left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{4}\right) \left(\left(\frac{\boldsymbol{Q}^+-\boldsymbol{Q}^-}{2}\right) \boldsymbol{\cdot}\boldsymbol{\nabla}\ln{\rho\alpha^2}\right)\!. \end{align}

Taking the limit of $\alpha =\sqrt {1/\mu \rho }$ allows us to confirm that this equation converges in that case to equation (17) of Marsch & Mangeney (Reference Marsch and Mangeney1987). The last term on the left-hand side and the last three terms on the right-hand side are terms parallel to the magnetic field. We remind the reader that these equations are valid for any choice of $\alpha$ (satisfying basic dimensional arguments).

2.2.1 Alfvén waves

As expected, the $y$-component (2.35) is separated from the other equations. This equation is rewritten in the following system:

(2.37)\begin{gather} (\omega - \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{Q}^-_0)\delta Q^+_y = \frac{1}{2} \frac{\Delta \alpha^2}{\alpha^2} \alpha B_0k_z (\delta Q^+_y-\delta Q^-_y), \end{gather}

(2.38)\begin{gather}(\omega - \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{Q}^+_0)\delta Q^-_y = \frac{1}{2} \frac{\Delta \alpha^2}{\alpha^2} \alpha B_0k_z (\delta Q^+_y-\delta Q^-_y), \end{gather}

resulting in a dispersion relation

(2.39)\begin{equation} \omega^2 - \boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{Q}_0^++\boldsymbol{Q}_0^-)\omega + (\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{Q}^+_0)(\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{Q}^-_0) + \frac{1}{2} \boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{Q}_0^+-\boldsymbol{Q}_0^-)\frac{\Delta \alpha^2}{\alpha}B_0k_z=0, \end{equation}

with solutions

(2.40)\begin{align} \omega & = \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{V}_0 \pm \sqrt{( \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{V}_0)^2 - (\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{Q}^+_0)(\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{Q}^-_0) - \Delta \alpha^2 k_z^2 B_0^2 } \end{align}

(2.41)\begin{align} & = \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{V}_0 \pm \sqrt{\left(\frac{\boldsymbol{k}}{2}\boldsymbol{\cdot}(\boldsymbol{Q}_0^+-\boldsymbol{Q}_0^-)\right)^2 - \Delta \alpha^2 k_z^2 B_0^2} \end{align}

(2.42)\begin{align} & = \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{V}_0 \pm k_zB_0 \sqrt{\alpha^2-\Delta \alpha^2} \end{align}

(2.43)\begin{align} & = \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{V}_0 \pm \frac{k_zB_0}{\sqrt{\mu\rho_0}}, \end{align}

which nicely converges to the well-known Alfvén wave solution $\omega =\boldsymbol {k}\boldsymbol {\cdot }(\boldsymbol {V}_0\pm \boldsymbol {V}_{A})=\boldsymbol {k}\boldsymbol {\cdot } \boldsymbol {Z}^\pm _0$.

This subsection also points us in the direction of the meaning and importance of the $\alpha$ parameter. If we would change variables to the co-moving frame (co-moving with $\boldsymbol {Q}^\pm _0$), then that frame would require that either $\omega ^\pm =0$ separately. Implementing these conditions in (2.37) and (2.38), leads to the (single) condition

(2.44)\begin{equation} \frac{1}{2} \frac{\Delta \alpha^2}{\alpha^2} \alpha B_0k_z (\delta Q^+_y-\delta Q^-_y)=0. \end{equation}

From this condition, we obtain that $(\delta Q^+_y-\delta Q^-_y)\equiv 0$ or that $\Delta \alpha ^2\equiv 0$. The former condition would lead to $\delta Q^\pm _y\equiv 0$ through the companion equation (e.g. (2.38) for $\omega ^-=0$), which tells us that there is no physical solution with non-zero amplitude. The latter condition $\Delta \alpha ^2=0$ leads to the well-known solution $\alpha ^2=1/\mu \rho _0$, which is equivalent to the limit where the $Q$-variables coincide with the Elsässer variables. This thus shows that the Elsässer variables are the only co-propagating waveframe variables in which the Alfvén waves have a non-zero amplitude. It shows that $\alpha$ should be chosen according to the phase speed, through the solution of $\omega ^\pm =0$

(2.45)\begin{equation} 0=\omega^\pm= \omega - \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{Q}^\pm_0 = \omega - \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{V}_0 \mp \alpha \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{B}_0, \end{equation}

resulting in an expression for $\alpha$

(2.46)\begin{equation} \alpha ={\pm} \frac{\omega - \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{V}_0}{\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{B}_0}. \end{equation}

The reader is cautioned to be careful with this expression, given that the expression diverges if $k\to 0$ or perpendicular $\boldsymbol {k}$ and $\boldsymbol {B}_0$.

2.2.2 Magnetoacoustic waves

Let us now investigate magnetoacoustic waves as they appear in terms of $Q$-variables. For the specific geometry chosen without loss of generality in § 2.2, linear magnetoacoustic modes perturb the $Q$-variables in the $x$–$z$ plane, and density. The system of equations to be solved for magnetoacoustic modes is composed of (2.33)–(2.36), except 2.35, which were treated in the previous subsection, yielding Alfvén waves. The dispersion relation is given by the determinant of this system of 5 equations for 5 unknowns. However, it turns out that, in this system, there are only 4 independent equations, (2.33) being linearly dependent on the other equations. Instead, we use the linearised solenoidal constraint (2.32) as a fifth equation

(2.47)\begin{equation} k_x (\delta Q_x^+- \delta Q_x^-) + k_z (\delta Q_z^+- \delta Q_z^-) = 0. \end{equation}

Next, we use the standard dispersion relation of magnetosonic waves. Assuming that $|k| = 1$, so that $k_z = \cos (\theta )$ and $k_x = \sin (\theta )$, with $\theta$ being the angle between the background magnetic field $B_0\boldsymbol {e}_z$ and the wavevector $\boldsymbol {k}$, and that there are no background flows $V_0 = 0$, the dispersion relation is

(2.48)\begin{equation} \alpha B_0 \cos(\theta)(V_{A0}^2 v_{s0}^2 \cos^2(\theta) - \omega^2(V_{A0}^2 + v_{s0}^2) + \omega^4) = 0.\end{equation}

Note that we have not yet assumed any form for $\alpha$, which is not needed for isolating the magnetoacoustic solutions. If we assume a form for $\alpha$ like in (2.46), we recover the fifth, trivial, solution of the dispersion relation, $\omega = 0$, the entropy wave, which represents non-propagating perturbations of plasma density and temperature. The other four solutions are the up- and downward-propagating (with respect to $\boldsymbol {e}_z$) fast and slow magnetoacoustic modes, as found also elsewhere through e.g. the velocity representation of MHD (Goedbloed & Poedts Reference Goedbloed and Poedts2004)

(2.49)\begin{equation} \omega_{s,f} = \pm_{{\rm ud}} \sqrt{\frac{1}{2} (V_{A0}^2+ v_{s0}^2)} \sqrt{1\pm_{{\rm sf}} \sqrt{1-\frac{\cos ^2(\theta ) (4 V_{A0}^2 v_{s0}^2)}{(V_{A0}^2+ v_{s0}^2)^2}}}. \end{equation}

Here, we have 4 solutions with the symbol $\pm _{{\rm ud}}$ differentiating between upward- and downward-propagating waves, and the symbol $\pm _{{\rm sf}}$ is the usual differentiation between the slow and fast magnetoacoustic waves. Recovering the magnetoacoustic solutions demonstrates the validity of the formulation of compressible MHD equations in terms of the $Q$-variables.

Using the eigenvalues in terms of $\omega$ (2.49), the eigenfunctions for $\boldsymbol {Q^\pm }$ can be determined for fast and slow waves from (2.30)–(2.32). The $Q$-variables can also be computed directly from the velocity and magnetic field eigenfunctions, if we assume a form for $\alpha$. Note that the definition of $\alpha$ from (2.46) diverges for purely perpendicularly propagating ($k_z = 0$) fast waves, thus this definition is not suitable for fast waves. This uncovers a curious property of (2.33)–(2.36), in that advection (in the form of the co-moving advective derivative) is only explicitly present along the magnetic field, leaving the definition of the phase speed in $\alpha$ only in terms of $k_z$. A straightforward remedy is then to use the full magnitude of the wavevector instead of only the $k_z$ component in the definition of $\alpha = \omega k^{-1} B_0^{-1}$.

In figure 1 we represent the parallel and perpendicular eigenfunctions of the $Q$-variables for fast and slow waves. From this figure, it is clear that only the perpendicular components are separated as a function of propagation direction with respect to the background magnetic field. In other words, $Q^+_{s,f \perp }$ is non-zero only when $\boldsymbol {k}\boldsymbol {\cdot }\boldsymbol {B_0} > 0$, and $Q^-_{s,f \perp }$ is non-zero for $\boldsymbol {k}\boldsymbol {\cdot }\boldsymbol {B_0} < 0$. The parallel components $Q^\pm _{s,f \parallel }$ are generally both perturbed, thus, based on the present form of the $Q$-variables, parallel perturbations cannot be separated into parallel- and anti-parallel-propagating components. The parallel components $Q^\pm _{s,f \parallel }$ vanish only for purely parallel-propagating fast waves, which are just Alfvén waves polarised in plane. In the next section (§ 2.2.3), we show that this is because of the connection of $Q_\parallel$ to the magnetic pressure.

Figure 1. Polar plots representing the $\theta$-dependence of the magnitude of the parallel (a) and perpendicular (b) $Q^\pm$-variables, for both fast and slow waves, indicated with the subscripts $f$ and $s$, respectively. The magnitudes are normalised by multiplying with the phase speed $\omega _{s,f}/|k|$ where $|k| = 1.$ Here, the plasma-$\beta$ is set to 0.2.

We conjecture that the full separation of waves, including the component parallel to the background field, is possible by constructing a waveframe variable which includes the total pressure or density perturbation as well in its formulation, but this will solely work in a homogeneous plasma where such a neat separation is possible.

2.2.3 Kink waves

In order to model kink waves, we start from (2.30)–(2.31), written out in components. Once again, we use the same frame of reference: the magnetic field $\boldsymbol {B}_0$ is pointing in the $z$-direction, and we also take the flow in the $z$-direction $\boldsymbol {V}_0=V_0\boldsymbol {e_z}$. Additionally, we take the assumption of a pressureless plasma $v_{s}=0$ and we take a density step function at $x=0$, with a constant density $\rho _{L}$ ($\rho _{R}$) on the left (right) side of the interface. In each half-space, the waves may be Fourier analysed in $y$, $z$ and $t$, putting every quantity proportional to $\exp {({\rm i}k_z z-{\rm i}\omega t)}$, where we have once again considered $k_y\equiv 0$ as in § 2.2. The resulting equations will be just like (2.33)–(2.36), except that the terms with $k_x$ will be replaced by a derivative ${\rm d}/{{\rm d}x}$. In what follows, we ignore (2.35), because we will not concentrate on the Alfvén waves, but rather on the kink waves, which are solely polarised in the $x,z$-directions for $k_y=0$.

Following the earlier strategy, we take (e.g.) $\omega ^+_{L,R}=0$ to find the upward-propagating kink waves. This immediately implies a connection

(2.50)\begin{equation} V_{L}+\alpha_{L}B_{L}=V_{R}+\alpha_{R}B_{R} ,\end{equation}

between $\alpha _{R,L}$. Each quantity in this equation is the corresponding background quantity in the left half-space or right half-space, respectively, for subscripts $L$ and $R$. With this assumption, we then have the following set of equations:

(2.51)\begin{gather} 0 =-\frac{1}{2} \frac{{\rm d}}{{\rm d}x} (3 \delta Q^+_x-\delta Q^-_x)-\frac{1}{2} k_z (3\delta Q^+_z-\delta Q^-_z) + \alpha B_0k_z \delta R, \end{gather}

(2.52)\begin{gather} \begin{aligned} 0 & =- \frac{1}{4}\left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) (Q_0^+-Q_0^-) \frac{{\rm d}}{{\rm d}x} (\delta Q^+_z-\delta Q^-_z) \nonumber\\ & \quad -\frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (Q_0^+-Q_0^-){\rm i}k_z (\delta Q^+_x-\delta Q^-_x), \end{aligned}\end{gather}

(2.53)\begin{gather} \begin{aligned} -\, {\rm i}k_z(Q_0^+-Q_0^-)\delta Q^+_x & =- \frac{1}{4}\left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) (Q_0^+-Q_0^-) \frac{{\rm d}}{{\rm d}x} (\delta Q^+_z-\delta Q^-_z) \nonumber\\ & \quad -\frac{1}{4} \frac{\Delta \alpha^2}{\alpha^2} (Q_0^+-Q_0^-){\rm i}k_z (\delta Q^+_x-\delta Q^-_x), \end{aligned}\end{gather}

(2.54)\begin{gather} \begin{aligned} & -\frac{Q_0^+-Q_0^-}{4}\left[\frac{{\rm d}}{{\rm d}x}(\delta Q_x^++\delta Q_x^-) +{\rm i}k_z(\delta Q_z^++\delta Q_z^-)\right] \nonumber\\ & \quad =-\frac{1}{4}\left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) (Q_0^+-Q_0^-) {\rm i}k_z (\delta Q^+_z-\delta Q^-_z) \nonumber\\ & \qquad + \frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (Q_0^+-Q_0^-) \frac{{\rm d}}{{\rm d}x} (\delta Q^+_x-\delta Q^-_x), \end{aligned} \end{gather}

(2.55)\begin{gather} \begin{aligned} & - {\rm i}k_z(Q_0^+-Q_0^-)\delta Q^+_z +\frac{Q_0^+-Q_0^-}{4} \left[\frac{{\rm d}}{{\rm d}x}(\delta Q_x^++\delta Q_x^-) +{\rm i}k_z(\delta Q_z^++\delta Q_z^-)\right] \nonumber\\ & \quad =- \frac{1}{4}\left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) (Q_0^+-Q_0^-) {\rm i}k_z (\delta Q^+_z-\delta Q^-_z) \nonumber\\ & \qquad +\frac{1}{4}\frac{\Delta \alpha^2}{\alpha^2} (Q_0^+-Q_0^-) \frac{{\rm d}}{{\rm d}x} (\delta Q^+_x-\delta Q^-_x), \end{aligned} \end{gather}

in which all quantities are subscripted with $R$ and $L$, respectively, for each half-space. Combining (2.53) and (2.52) isolates $\delta Q_x^+$ as

(2.56)\begin{equation} {\rm i}k_z(Q_0^+-Q_0^-)\delta Q^+_x=0, \end{equation}

showing that the kink wave is uniquely described by $\delta Q_x^-$ only, because $\delta Q_x^+$ is 0 if $k_z\neq 0$ and $B_0\neq 0$. If we find a value for $\alpha _{R,L}$ and $\omega$, then the kink wave is written with only one of $\delta Q_x^\pm$, as was the intention of the $Q$-variables for separating upward- and downward-propagating waves. Similarly, from the combination of (2.54) and (2.55) (and using $\delta Q_x^+=0$), we obtain

(2.57)\begin{equation} \frac{{\rm d}}{{\rm d}x} \delta Q_x^-- {\rm i}k_z (\delta Q^+_z-\delta Q^-_z)=0. \end{equation}

Thus, we obtain a set of equations describing the kink waves (or any other wave under these assumptions) from (2.52) and (2.57)

(2.58)\begin{gather} 0=\frac{{\rm d}}{{\rm d}x} \delta Q_x^-- {\rm i}k_z (\delta Q^+_z-\delta Q^-_z), \end{gather}

(2.59)\begin{gather}0 = \left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) \frac{{\rm d}}{{\rm d}x} (\delta Q^+_z-\delta Q^-_z) - \frac{\Delta \alpha^2}{\alpha^2} {\rm i}k_z \delta Q^-_x. \end{gather}

Introducing a new variable $\varPi =\delta Q^+_z-\delta Q^-_z$, we obtain the set

(2.60)\begin{gather} 0=\frac{{\rm d}}{{\rm d}x} \delta Q_x^-- {\rm i}k_z \varPi, \end{gather}

(2.61)\begin{gather}0 = \left(1-\frac{\Delta \alpha^2}{\alpha^2}\right) \frac{{\rm d}\varPi}{{\rm d}x} - \frac{\Delta \alpha^2}{\alpha^2} {\rm i}k_z \delta Q^-_x. \end{gather}

This set is reminiscent of the coupled differential equations between the perturbed total pressure and displacement that other works have found for the description of kink waves (Appert, Gruber & Vaclavik Reference Appert, Gruber and Vaclavik1974; Goossens, Hollweg & Sakurai Reference Goossens, Hollweg and Sakurai1992; Ismayilli et al. Reference Ismayilli, Van Doorsselaere, Goossens and Magyar2022), which have a strong correspondence to the currently modelled surface Alfvén waves (Goossens et al. Reference Goossens, Andries, Soler, Van Doorsselaere, Arregui and Terradas2012).

Since each quantity is constant in each half-space, we can substitute one of the equations in the other. Then, we obtain a single second-order differential equation

(2.62)\begin{equation} \frac{{\rm d}^2}{{{\rm d}x}^2}\delta Q^-_x +k_z^2 \left(\frac{\Delta \alpha^2}{\alpha^2-\Delta \alpha^2}\right) \delta Q^-_x=0. \end{equation}

In the left and right half-spaces, we consider respectively the solution

(2.63)\begin{equation} \delta Q^-_{x,{L}}=A_{L} \exp{(\kappa_{L} x)},\quad \delta Q^-_{x,{R}}=A_{R} \exp{(-\kappa_{R} x)}, \end{equation}

where

(2.64)\begin{equation} \kappa^2=k_z^2\left\vert \frac{\Delta \alpha^2}{\Delta \alpha^2-\alpha^2}\right\vert. \end{equation}

The solution for $\varPi$ can be calculated from (2.60).

Next, we need to apply boundary conditions at $x=0$. Namely, we take (as usual)

(2.65)\begin{gather} 0=[v_x], \end{gather}

(2.66)\begin{gather}0=[P'], \end{gather}

where $P'=B_0b_z/\mu$ is the perturbed total pressure and the square brackets are differences between the left and the right of the interface. Note that the extra term in the Lagrangian pressure perturbation is 0 in the linear regime, since the magnetic pressure is uniform in each half-space. Translated to our variables, these boundary conditions are

(2.67)\begin{gather} 0=[Q_x^-], \end{gather}

(2.68)\begin{gather}0=\left[\frac{B_0\varPi}{\alpha}\right]\!. \end{gather}

The first condition states that $A_{L}=A_{R}$, while the second condition results in the dispersion relation

(2.69)\begin{equation} {-}\frac{\kappa_{L}B_{L}}{\alpha_{L}}=\frac{\kappa_{R}B_{R}}{\alpha_{R}}. \end{equation}

Squaring this relation, and inserting the expression for $\kappa ^2=k_z^2|\mu \rho \alpha ^2-1|$, we obtain

(2.70)\begin{equation} \left(\mu\rho_{L}-\frac{1}{\alpha_L^2}\right)B_{L}^2=-\left(\mu\rho_{R}-\frac{1}{\alpha_R^2}\right)B_{R}^2, \end{equation}

where we have used the fact that the absolute values in $\kappa ^2$ take a different sign on either side of the interface. Solving this equation in conjunction with (2.50) (and considering $V_0=0$ for simplicity), we obtain finally the allowed values for $\alpha$

(2.71)\begin{equation} \alpha B_0=\alpha_{L}B_{L}=\alpha_{R}B_{R}=\sqrt{\frac{B_{R}^4+B_{L}^4}{\mu (\rho_{R}B_{R}^2+ \rho_{L}B_{L}^2)}},\quad \omega=k_z\sqrt{\frac{B_{R}^4+B_{L}^4}{\mu(\rho_{R}B_{R}^2+\rho_{L}B_{L}^2)}}. \end{equation}

Given that $B_{L}=B_{R}=B_0$ for a pressureless plasma, these equations reduce to

(2.72)\begin{equation} \alpha =\alpha_{L}=\alpha_{R}=\sqrt{\frac{2}{\mu(\rho_{R}+\rho_{L})}},\quad \omega=k_z\sqrt{\frac{2B_0^2}{\mu(\rho_{R}+\rho_{L})}}, \end{equation}

as is well known from other works.

2.2.4 General waves in field-aligned flows

Now we will prove explicitly that the proper choice of $\alpha$ splits the $Q$-variable between wave modes of propagation directions. We follow the derivation of Magyar, Van Doorsselaere & Goossens (Reference Magyar, Van Doorsselaere and Goossens2019b) and their equation (19). In this subsection, we consider the general configuration with a magnetic field pointing in the $z$-direction, but still dependent on $x$ and $y$. Moreover, we also take the background flow along the magnetic field

(2.73)\begin{equation} \boldsymbol{B}_0=B_0(x,y)\boldsymbol{e}_z,\quad \boldsymbol{V}_0=V_0(x,y)\boldsymbol{e}_z. \end{equation}

Let us now consider the linearised induction equation

(2.74)\begin{equation} \frac{\partial \boldsymbol{b}}{\partial t}=\boldsymbol{\nabla}\times((\boldsymbol{V_0}+\boldsymbol{v})\times \boldsymbol{B}_0), \end{equation}

of which we will only consider the perpendicular component. We can reduce this induction equation with vector identities to

(2.75)\begin{equation} \frac{\partial \boldsymbol{b}_\perp}{\partial t}=\boldsymbol{B}_0 \boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{v}_\perp- \boldsymbol{V}_0\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{b}_\perp. \end{equation}

Here, we have naturally used that $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {B}_0=\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {b}=0$, but we have also used $\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {V}_0=0$ because $\boldsymbol {V}_0$ only has a $z$-component that does not depend on $z$. Using Fourier analysis for the ignorable coordinates $z$ and $t$, we then have

(2.76)\begin{equation} {-}{\rm i}\omega \boldsymbol{b}_\perp= {\rm i}k_z(B_0 \boldsymbol{v}_\perp- V_0 \boldsymbol{b}_\perp). \end{equation}

Using (2.7) and (2.15), we then have

(2.77)\begin{equation} (\omega-k_zV_0+k_z\alpha B_0) \boldsymbol{\delta Q}^+_\perp= (\omega - k_z V_0-k_z\alpha B_0) \boldsymbol{\delta Q}^-_\perp. \end{equation}

This equation shows that the correct choice of $\alpha$ indeed splits a wave mode with a specific $\omega$ and $k_z$ between different $\boldsymbol {\delta Q}_\perp$ components. Using the $Q$-variable terminology, the equation is more elegantly written as

(2.78)\begin{equation} (\omega - \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{Q}_0^-)\boldsymbol{\delta Q}^+_\perp= (\omega - \boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{Q}_0^+)\boldsymbol{\delta Q}^-_\perp. \end{equation}

This equation states that a wave with phase speed $\boldsymbol {Q}_0^\pm$ has the associated wave only present in $\boldsymbol {\delta Q}_\perp ^\mp$, with the other $Q$-variable $\boldsymbol {\delta Q}_\perp ^\pm =0$.