Alfvén waves
Alfvén waves, dispersion relations in ideal and diffusive medium; Hartmann boundary layer.
Alfvén waves are magnetohydrodynamic (MHD) waves with Lorentz force (or the magnetic tension) as restoring force. In this section I review some of the key aspects of this type of ways, including the equations, dispersion relations, phase and group velocities, etc.
Governing equations
I start by stating the governing equations of the system. Here it is already assumed that the medium is both homogeneous and incompressible, so that
\[\nabla\cdot \mathbf{u} = 0,\quad \rho \equiv Cst.\]which already filters out all acoustic waves and stratification / arbitrary heterogeneity of the setup. @ferraro_reflection_1954 has a short section on Alfvén waves in stratified atmosphere, where the wave solution takes the form of Bessel functions, and is shown to be damped as it enters stratification region. This might be important in stars, planetary atmosphere, and might also become important if similar stratification occurs in Earth’s core. When the incompressibility assumption is dropped, one obtains magneto-acoustic (magnetosonic) waves, which are hybrid between MHD waves and acoustic waves, with wave velocities in between Alfvén waves and acoustic waves. When both the fluctuation in density and the volumetric strain rate are comparatively negligible (), the assumptions should still work, and it should suffice in studying the reflection and transmission near the boundary.
I start from Navier-Stokes equation and induction equation in a homogeneous medium,
\[\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u} = - \nabla \frac{P}{\rho} + \frac{1}{\rho \mu_0} (\nabla\times \mathbf{B})\times \mathbf{B} + \nu \nabla^2 \mathbf{u},\]\(\frac{\partial \mathbf{B}}{\partial t} = \nabla\times (\mathbf{u}\times \mathbf{B}) + \eta \nabla^2 \mathbf{B}.\) The total fields (\(\mathbf{u}\), \(\mathbf{B}\)) are decomposed into background fields (\(\mathbf{U}_0\), \(\mathbf{B}_0\)) and perturbation fields (\(\mathbf{u}\), \(\mathbf{b}\)). We take the background velocity field to be zero (\(\mathbf{U}_0 = \mathbf{0}\)), so that no advection occurs. The magnetic background field \(\mathbf{B}_0\) is a constant in both space and time, representing a time-invariant uniform field. This approximation should hold as long as both the characteristic time scale and the length scale of \(\mathbf{B}_0\) are much larger than those of the perturbed fields (in fact, the ratio should be greater than \(|\mathbf{B}_0|/|\mathbf{b}|\)).
To linearize the set of equations, the perturbation magnetic field is considered much smaller than the background field (\(|\mathbf{b}|\ll |\mathbf{B}_0|\)), and the perturbation velocity field is at the same order of magnitude as magnetic field, in the sense that \(|\mathbf{u}|\sim |\mathbf{b}|/\sqrt{\rho \mu_0}\). Collecting only the first order terms, we obtain the linearized system
\[\begin{aligned} \frac{\partial \mathbf{u}}{\partial t} &= - \nabla \frac{P}{\rho} + \frac{1}{\rho \mu_0} (\nabla\times \mathbf{b})\times \mathbf{B}_0 + \nu \nabla^2 \mathbf{u}, \\ \frac{\partial \mathbf{b}}{\partial t} &= \nabla\times (\mathbf{u}\times \mathbf{B}_0) + \eta \nabla^2 \mathbf{b}. \end{aligned}\]which can be further rearranged into a more symmetric form using vector identities
\[\frac{\partial \mathbf{u}}{\partial t} = \frac{1}{\rho \mu_0} \mathbf{B}_0 \cdot \nabla \mathbf{b} + \nu \nabla^2 \mathbf{u} - \nabla P_\mathrm{eff}',\]\(\frac{\partial \mathbf{b}}{\partial t} = \mathbf{B}_0 \cdot \nabla \mathbf{u} + \eta \nabla^2 \mathbf{b}.\) where the effective pressure is a sum of mechanical pressure and the magnetic pressure, divided by the density of the medium
\[P'_\mathrm{eff} = \frac{P}{\rho} + \frac{\mathbf{B}_0\cdot \mathbf{b}}{\rho \mu_0}.\]The total magnetic pressure is of course \((\mathbf{B_0} + \mathbf{b})^2/2\rho \mu_0\), but given the previous assumption that \(\mathbf{B}_0\) is constant vector, and only the first-order terms are retained, using \((\mathbf{B}_0 + \mathbf{b})^2\) in the numerator would be equivalent to stating \(2\mathbf{B}_0\cdot \mathbf{b}\), which is what has been obtained directly from \((\nabla\times \mathbf{b})\times \mathbf{B}_0\).
Dispersion relation of the ideal system
Neglecting the pressure gradient, and neglecting the viscous as well as magnetic diffusion, the equations can be combined into a second-order wave equation, in the form
\[\frac{\partial^2 \mathbf{u}}{\partial t^2} = \frac{(\mathbf{B}_0\cdot \nabla)^2}{\rho \mu_0} \mathbf{u},\qquad \frac{\partial^2 \mathbf{b}}{\partial t^2} = \frac{(\mathbf{B}_0\cdot \nabla)^2}{\rho \mu_0} \mathbf{b},\]which, with a plane wave ansatz of any sort (equivalently converting the equation into frequency-wavenumber domain), immediately yields the dispersion relation for Alfvén waves in diffusionless medium
\[\omega^2 = \frac{1}{\rho \mu_0} \left(\mathbf{k}\cdot \mathbf{B}_0\right)^2 = \frac{B_0^2}{\rho \mu_0} \left(\mathbf{k}\cdot \widehat{\mathbf{B}}_0\right)^2 = V_A^2 \left(\mathbf{k}\cdot \widehat{\mathbf{B}}_0\right)^2.\]where \(\widehat{\mathbf{B}}_0\) is the unit vector in the direction of \(\mathbf{B}_0\), and \(V_A = B_0/\sqrt{\rho \mu_0}\) is the Alfvén wave velocity, as will become clearer in the next section. For the plane wave ansatz used in this article, please refer to eq.([eqn:wave-ansatz]{reference-type=”ref” reference=”eqn:wave-ansatz”}) and the related texts in the remark box. Since reversing the sign on \(\omega\) and \(\mathbf{k}\) simultaneously would yield the same physical solution (taking the real part of the complex wave yields the exact same expression, see remark box that follows), when describing the plane waves I shall take the convention that \(\omega > 0\). Under this convention the dispersion relation can be further written as
\[\omega = \pm V_A \left(\mathbf{k}\cdot \widehat{\mathbf{B}}_0\right) = \left\{\begin{aligned} V_A \left(\mathbf{k}\cdot \widehat{\mathbf{B}}_0\right),\quad \mathbf{k}\cdot \widehat{\mathbf{B}}_0 > 0 \\ - V_A \left(\mathbf{k}\cdot \widehat{\mathbf{B}}_0\right),\quad \mathbf{k}\cdot \widehat{\mathbf{B}}_0 < 0 \end{aligned}\right.\]The two solutions for \(\omega\) correspond to two propagation directions of the Alfvén wave. As will also become clear later, Alfvén waves can either propagate in or opposite the direction of the background magnetic field, which corresponds to the obtained two solutions.
It is already readily seen that Alfvén wave is an anisotropic wave. The isotropy is broken due to the fact that background magnetic field is the essential cornerstone for providing the magnetic tension, and the orientation of the magnetic field has a special status. It will also be seen that the orientation of the background magnetic field greatly complicates the reflection-refraction problem, compared to the isotropic waves, such as elastic waves, acoustic waves (seismic waves) and light waves in isotropic medium. The anisotropy dictates that, for a given temporal frequency \(\omega\),
\[\Big|\mathbf{k}\cdot \widehat{\mathbf{B}}_0\Big| = \frac{\omega}{V_A}\]meaning the wave vector has a fixed projection length on the background field.
::: mdframed In this article I shall use the following convention for plane wave
\[\mathbf{A}(\mathbf{r}, t) = \mathbf{A}_0 \exp\left\{i\left(\omega t - \mathbf{k}\cdot \mathbf{r}\right)\right\},\]and the conventions for respective Fourier transforms follow. Some intermediate steps and results in this article will be different by a sign compared to the alternative ansatz \(\exp\{i(\omega t + \mathbf{k}\cdot \mathbf{r})\}\), for instance the dispersion relation as shown in eq.([eqn:dispersion-ideal]{reference-type=”ref” reference=”eqn:dispersion-ideal”}). In the other convention, \(\mathbf{k}\) is opposite the direction in which the phase propagates, and \(\omega = V_A (\mathbf{k}\cdot \widehat{\mathbf{B}}_0)\) would represent a wave travelling in the opposite direction of \(\mathbf{B}_0\).
In the ideal case without diffusion, it can be easily shown that the perturbation velocity field \(\mathbf{u}\) and the magnetic field \(\mathbf{b}\) are proportional to one another. To this end, we can construct a plane wave solution for the perturbed fields
\[\mathbf{b} = \mathbf{b}_0 \exp\{i(\omega t - \mathbf{k}\cdot \mathbf{r})\}, \quad \mathbf{u} = \mathbf{u}_0 \exp\{i(\omega t - \mathbf{k}\cdot \mathbf{r})\}.\]The two waves share the same phase argument since their phases need to match in the coupled system of equations. Substituting the expression into the first-order ideal equation yields
\[\frac{\partial \mathbf{u}}{\partial t} = \frac{\mathbf{B}_0\cdot \nabla}{\rho \mu_0} \mathbf{b},\quad \Longrightarrow\quad i\omega \mathbf{u}_0 = - i \frac{\mathbf{B}_0\cdot \mathbf{k}}{\rho \mu_0} \mathbf{b}_0\quad \Longrightarrow\quad \mathbf{u}_0 = -\frac{V_A (\mathbf{k}\cdot \widehat{\mathbf{B}}_0)}{\omega\sqrt{\rho \mu_0}} \mathbf{b}_0.\]Taking into account the dispersion relation, we have
\[\mathbf{u}_0 = \left\{\begin{aligned} - \frac{\mathbf{b}_0}{\sqrt{\rho \mu_0}},\quad \mathbf{k}\cdot \widehat{\mathbf{B}}_0 > 0,\\ \frac{\mathbf{b}_0}{\sqrt{\rho \mu_0}},\quad \mathbf{k}\cdot \widehat{\mathbf{B}}_0 < 0. \end{aligned}\right.\]Therefore, the perturbed magnetic field and the velocity field are opposite one another (or have a \(\pi\) phase shift) when the wave is propagating in the same direction of the background magnetic field, and the perturbed fields are completely in-phase when the wave is propagating in the opposite direction of \(\mathbf{B}_0\). Either way, the magnetic field and the velocity field in Alfvén waves share the same polarization. :::
Dispersion relation of the diffusive system
With viscous and magnetic diffusion, the system of equations cannot be easily combined into one wave equation. Instead, the system of equations can be converted directly into frequency-wavenumber domain
\[\begin{aligned} i\omega \mathbf{u} &= -i\frac{\mathbf{B}_0\cdot \mathbf{k}}{\rho \mu_0} \mathbf{b} - \nu k^2 \mathbf{u}, \\ i\omega \mathbf{b} &= -i(\mathbf{B}_0\cdot \mathbf{k}) \mathbf{u} - \eta k^2 \mathbf{b}, \end{aligned}\]which can be rearranged into the linear system
\[\begin{aligned} (i\omega + \nu k^2) \mathbf{u} + i\frac{\mathbf{B}_0\cdot \mathbf{k}}{\rho \mu_0} \mathbf{b} = \mathbf{0}, \\ i(\mathbf{B}_0\cdot \mathbf{k}) \mathbf{u} + (i\omega + \eta k^2) \mathbf{b} = \mathbf{0}. \end{aligned}\]Naturally, for the system to have nontrivial solutions, the necessary condition is
\[\det \begin{pmatrix} i\omega + \nu k^2 & i \frac{\mathbf{B}_0 \cdot \mathbf{k}}{\rho \mu_0} \\ i (\mathbf{B}_0 \cdot \mathbf{k}) & i\omega + \eta k^2 \end{pmatrix} = - \omega^2 + i(\nu + \eta) k^2 \omega + \frac{(\mathbf{B}_0\cdot \mathbf{k})^2}{\rho \mu_0} + \nu \eta k^4 0\]which is the dispersion relation for the diffusive system. The relation can be rearranged as a biquadratic polynomial equation of \(k\),
\[\nu \eta k^4 + \left(V_A^2 \cos^2\gamma + i\omega(\nu + \eta)\right) k^2 - \omega^2 = 0\]where I have used \(\gamma = \langle \mathbf{B}_0, \mathbf{k} \rangle\), and the defined Alfvén wave velocity \(V_A = B_0 / \sqrt{\rho \mu_0}\). The roots of this equation give the spatial branch of the dispersion relations
\[k^2 = -\frac{V_A^2 \cos^2\gamma}{2\nu\eta} \left(1 + i\frac{\omega(\nu + \eta)}{V_A^2\cos^2\gamma}\right) \left[1 \pm \sqrt{1 + \frac{4\omega^2 \nu \eta}{V_A^4 \cos^4\gamma \left(1 + \frac{i\omega(\nu + \eta)}{V_A^2 \cos^2\gamma}\right)^2}}\right].\]The repetitive terms can be greatly simplified by introducing the notation
\[\mathrm{S}_\omega = \frac{2V_A^2}{\omega (\nu + \eta)}\]and the spatial branch of the dispersion relation can be rewritten as
\[k^2 = - \frac{V_A^2}{2\nu \eta} \left(\cos^2\gamma + i2\mathrm{S}_\omega^{-1}\right) \left[1 \pm \sqrt{1 + \frac{4\omega^2 \nu \eta}{V_A^4 \left(\cos^2\gamma + i2\mathrm{S}_\omega^{-1}\right)^2}}\right]\]The notation \(\mathrm{S}_\omega\) defined as such represents the ratio between the diffusion time scale and the Alfvén wave time scale, and is called the Lundquist number. In its general form, without specifying the length scale of interest, the Lundquist number is written as
\[\mathrm{S} = \frac{\tau_\alpha}{\tau_A} = \frac{L^2/\alpha}{L/V_A} = \frac{V_A L}{\alpha}.\]Choosing specific diffusion mechanism (specifying \(\alpha\)) and specific time scale (specifying \(L\)) yields a variety of variants of Lundquist number. In this case, we see that the diffusion mechanism of interest is the combined effect of viscous diffusion and magnetic diffusion; the length scale of interest is determined by the Alfvén wavelength at specified frequency, i.e.
\[\alpha \sim \frac{\nu + \eta}{2},\quad L \sim \frac{V_A}{\omega}\quad \Longrightarrow\quad \mathrm{S}_\omega = \frac{V_A^2/\omega}{(\nu + \eta)/2} = \frac{2V_A^2}{\omega(\nu + \eta)}.\]When Lundquist number \(\mathrm{S}\gg 1\), Alfvén wave time scale is much smaller than that of diffusion time scale; this means the damping of Alfvén waves at the specific length scale is small, and the propagation of Alfvén waves is allowed in the system. When \(\mathrm{S} \ll 1\), diffusion time scale is much smaller, and diffusion process dominates the system at given length scale, prohibiting the effective propagation of Alfvén waves. This can be readily seen from the dispersion relation, as follows.
At \(\mathrm{S}_\omega \ll 1\), the term \(\mathrm{S}_\omega^{-1}\) always dominates over other terms, reducing eq.([eqn:dispersion-spatial]{reference-type=”ref” reference=”eqn:dispersion-spatial”}) into the form
\[k^2 \approx - i \frac{V_A^2}{2\nu\eta} 2\mathrm{S}_\omega^{-1} \left(1 \pm \sqrt{1 - \frac{\omega^2 \nu \eta}{V_A^4}\mathrm{S}_\omega^2}\right) = -i \frac{\omega}{2\nu \eta} \left(\nu + \eta \pm |\nu - \eta|\right)\]which gives the ultimate solutions
\[k_1^2 \approx -i \frac{\omega}{\min(\nu, \eta)},\quad k_1 = \pm \frac{1 - i}{\sqrt{2}}\sqrt{\frac{\omega}{\min(\nu,\eta)}},\]\(k_2^2 \approx -i \frac{\omega}{\max(\nu, \eta)},\quad k_2 = \pm \frac{1 - i}{\sqrt{2}}\sqrt{\frac{\omega}{\max(\nu,\eta)}}.\) These solutions correspond to damped oscillations, which decays in the propagating direction. The characteristic decaying length is given by \(\bar{\lambda}_1 = \sqrt{2\min(\nu,\eta)/\omega}\) and \(\bar{\lambda}_2 = \sqrt{2\max(\nu,\eta)/\omega}\), respectively. The \(\sqrt{\omega/\alpha}\) scaling for \(k\), the characteristic wavelength scaling with \(\sqrt{\alpha/\omega}\) and the feature of damped oscillation reveal that these solutions correspond to Stokes-type boundary layers. The oscillation and damping is simply governed by the diffusion process, but not the magnetic tension.
At \(\mathrm{S}_\omega \gg 1\), or \(V_A^2 \gg \omega (\nu + \eta) \geq 2\omega \sqrt{\nu\eta}\), we expect to recover the regime where magnetic tension is important in the system, producing the propagation of Alfvén waves. Given some value of \(\gamma\) so that \(\cos\gamma \sim 1\) is at some finite magnitude, the inverse Lundquist number \(\mathrm{S}_\omega^{-1}\) will always be small compared to \(\cos^2\gamma\). To see the effect of diffusion in the system we can keep \(\mathrm{S}_\omega^{-1}\) in eq.([eqn:dispersion-spatial]{reference-type=”ref” reference=”eqn:dispersion-spatial”}) to its leading order,
\[\begin{aligned} k^2 &= - \frac{V_A^2\cos^2\gamma}{2\nu\eta} \left(1 + i\frac{2}{\mathrm{S}_\omega\cos^2\gamma}\right) \left[1 \pm \sqrt{1 + \frac{4\omega^2 \nu\eta}{V_A^4 \cos^4\gamma}\left(1 + i\frac{2}{\mathrm{S}_\omega \cos^2\gamma}\right)^{-2}}\right] \\ &\approx - \frac{V_A^2\cos^2\gamma}{2\nu\eta} \left(1 + i\frac{2}{\mathrm{S}_\omega\cos^2\gamma}\right) \left[1 \pm \left(1 + \frac{2\omega^2 \nu\eta}{V_A^4 \cos^4\gamma}\left(1 - i\frac{4}{\mathrm{S}_\omega \cos^2\gamma}\right)\right)\right] \end{aligned}\]which gives two ultimate solutions
\[k_1^2 \approx - \frac{V_A^2}{\nu\eta} \left(\cos^2\gamma + i 2 \mathrm{S}_\omega^{-1}\right),\quad k_1 \approx \pm \frac{V_A \cos\gamma}{\sqrt{\nu\eta}} \left(-\frac{1}{\mathrm{S}_\omega \cos^2\gamma} + i\right),\]\(k_2^2 \approx \frac{\omega^2}{V_A^2\cos^2\gamma} \left(1 - i \frac{2}{\mathrm{S}_\omega \cos^2\gamma}\right),\quad k_2 \approx \pm \frac{\omega}{V_A \cos\gamma} \left(1 - i \frac{1}{\mathrm{S}_\omega \cos^2\gamma}\right).\) The first solution \(k_1\) has a dominant imaginary part. In the case where \(\mathrm{S}_\omega^{-1} \ll 1\) is negligible this can be simply written as \(k_1\approx \pm i V_A \cos\gamma/\sqrt{\nu\eta} = \pm i/\delta\). This correspond to a Hartmann boundary layer. Properties of this layer is listed below.
::: mdframed The thickness, or the characteristic length scale over which the wave decays in a Hartmann layer, is given by
\[\delta_\mathrm{BL} = \frac{\sqrt{\nu\eta}}{V_A|\cos\gamma|} = \frac{\sqrt{\rho \mu_0 \nu \eta}}{B_0 |\cos\gamma|} = \frac{1}{B_\parallel}\sqrt{\frac{\rho \nu}{\sigma}}.\]Here \(B_\parallel = |{B}_0\cdot \hat{\mathbf{k}}|\) is the magnetic field strength in line with the wave vector. The boundary layer thickness scales with \(\sqrt{\nu\eta}/V_A\), and hence goes to zero when \(\nu,\eta \rightarrow 0\). Nevertheless, an infinitely small Hartmann layer might be able to accommodate finite velocity and magnetic field discontinuity at the boundary (), just like the free-slip boundary condition for inviscid fluid. The reason for that is the infinite conductivity assumption, which allows infinitely large current in the system, giving rise to magnetic field discontinuity. This boundary layer seems to play an important role in constructing solutions that satisfy continuity boundary conditions across the interface [@schaeffer_reflection_2012], even at high Lundquist numbers.
The thickness of the Hartmann boundary layer is another important length scale of the system. When variation occurs on a length scale comparable to or smaller than \(\delta_\mathrm{BL}\), the viscous and magnetic diffusion wins over, and promotes boundary layer behaviour; when the length scale of variation is much larger than \(\delta_\mathrm{BL}\), the magnetic tension is much more effective. This motivates the introduction of a dimensionless number, Hartmann number, which, supposedly, is the ratio of Lorentz force to viscous force. It can also be interpreted as the ratio of some characteristic length scale to the Hartmann layer thickness
\[\mathrm{Ha} = B L \sqrt{\frac{\sigma}{\rho\nu}} = \frac{L}{\delta_\mathrm{BL}}.\]For Hartmann layer, the amplitudes of \(\mathbf{u}\) and \(\mathbf{b}\) follow a relation different from the travelling wave solution. Recall at high Lundquist number, \(V_A^2 \gg \omega \eta\), the first-order equation gives
\[\mathbf{u} = \frac{i \mathbf{B}_0\cdot \mathbf{k}}{\rho \mu_0 (i\omega + \nu k^2)} \mathbf{b} \approx \frac{\mp B_0 \frac{V_A \cos^2\gamma}{\sqrt{\nu\eta}}}{\rho \mu_0 \frac{V_A^2}{\eta}\cos^2\gamma} \mathbf{b} = \mp \sqrt{\frac{\eta}{\nu}}\frac{\mathbf{b}}{\sqrt{\rho \mu_0}} = \mp \mathrm{Pm}^{-1/2} \frac{\mathbf{b}}{\sqrt{\rho\mu_0}}\]where \(\mathrm{Pm} = \frac{\nu}{\eta}\) is the magnetic Prandtl number. Not surprisingly, the amplitude in such boundary layer is skewed towards the field with smaller diffusion. :::
The second solution \(k_2\) reduces to the dispersion relation of Alfvén waves in diffusionless medium, i.e. \(k_2 = \pm \omega/V_A \cos\gamma\), when \(\mathrm{S}_\omega^{-1}\) is dropped from the multiplier (eq.[eqn:dispersion-ideal]{reference-type=”ref” reference=”eqn:dispersion-ideal”}). To first order, the role of non-negligible diffusion is to introduce damping with the coefficient \(1/\mathrm{S}_\omega \cos^2\gamma = \omega(\nu+\eta)/2V_A^2\cos^2\gamma\), which mildly damps the Alfvén wave as it propagates.
At the mildly diffusive limit \(\mathrm{S}_\omega \gg 1\), both solutions are anisotropic. For the Hartmann boundary layer solution, the thickness or spatial decay rate is constrained by the projection of magnetic field on the wave vector. For the travelling Alfvén wave solution, the wavenumber is also determined by the projection of the magnetic field on the wave vector. Conversely, it means the wave vectors in both solutions are only controlled in the direction of the background field.
Phase and group velocities
Phase velocity is the velocity at which the phase of a monochromatic wave travels. Given that a plane wave takes the form \(\exp(i(\omega t - \mathbf{k}\cdot \mathbf{r})) = \exp(i\mathbf{k}\cdot (\omega t \hat{\mathbf{k}} / k - \mathbf{r}))\), the phase velocity is given by
\[\mathbf{c}_p = \frac{\omega}{k}\hat{\mathbf{k}}.\]Apparently, the phase velocity always has the same direction as the wave vector. The wave vector for plane wave is exactly the indicator of phase propagation. When a collection of waves is present (in reality there is almost never standalone monochromatic plane wave, since that implies infinite energy), the velocity at which the wave packet near a frequency travels is different from the phase velocity. This is called the group velocity, and is given by
\[\mathbf{c}_g = \nabla_k \omega.\]Since wave packets and wave groups are the carrier of information and energy, group velocity is considered to be the velocity at which information and energy propagates. Although this seems to be true in many cases, I argue that this cannot replace the energy argument. Group velocity is a mathematical property, a property that arises due to the mathematical form of wave ansätze; energy flux is a physical property, which does not seem to be strictly linked to the wave ansatz a priori.
Both phase velocity and the group velocity can be derived from the dispersion relations. The dispersion relation with finite \(\mathrm{S}_\omega\) is complicated, and would give rise to dispersive waves. The Alfvén wave in ideal, diffusionless medium, however, is simple. The phase velocity is given by
\[\mathbf{c}_p = \frac{\omega}{k} \hat{\mathbf{k}} = \left\{\begin{aligned} V_A (\hat{\mathbf{k}}\cdot \widehat{\mathbf{B}}_0) \hat{\mathbf{k}} = V_A \cos\gamma \, \hat{\mathbf{k}} = \frac{\mathbf{B}_0\cdot \hat{\mathbf{k}}}{\sqrt{\rho \mu_0}} \hat{\mathbf{k}},\quad \mathbf{k} \cdot \widehat{\mathbf{B}}_0 > 0 \\ - V_A (\hat{\mathbf{k}}\cdot \widehat{\mathbf{B}}_0) \hat{\mathbf{k}} = -V_A \cos\gamma \, \hat{\mathbf{k}} = - \frac{\mathbf{B}_0\cdot \hat{\mathbf{k}}}{\sqrt{\rho \mu_0}} \hat{\mathbf{k}},\quad \mathbf{k} \cdot \widehat{\mathbf{B}}_0 < 0 \end{aligned}\right.\]which can be uniformly written as
\[\mathbf{c}_p = V_A |\cos\gamma| \, \hat{\mathbf{k}}.\]The magnitude of the phase velocity of Alfvén wave is fixed, as long as the orientation of the wave propagation is fixed. In this sense, Alfvén wave is diffusionless. Among all the direction of waves, the wave that propagates along the background field propagates the fastest.
For isotropic waves such as seismic waves and light waves in isotropic medium, not only are the waves dispresionless, but \(\mathbf{c}_p \parallel \mathbf{c}_g\). This is not the case for anisotropic waves as Alfvén wave. While the phase can travel in any direction except normal to the background field, the group velocity is always aligned with the background field
\[\mathbf{c}_g = \nabla_k \omega = \left\{\begin{aligned} V_A \widehat{\mathbf{B}}_0 = \frac{\mathbf{B}_0}{\sqrt{\rho \mu_0}},\quad \mathbf{k}\cdot \widehat{\mathbf{B}}_0 > 0, \\ - V_A \widehat{\mathbf{B}}_0 = - \frac{\mathbf{B}_0}{\sqrt{\rho \mu_0}},\quad \mathbf{k}\cdot \widehat{\mathbf{B}}_0 < 0. \end{aligned}\right.\]For an Alfvén wave propagating (in the sense of \(\mathbf{c}_p\) or \(\mathbf{k}\)) in the direction that forms an angle \(\gamma < \pi/2\) with the magnetic field \(\mathbf{B}_0\), i.e. downwind, the group velocity is in the direction of \(\mathbf{B}_0\). For an Alfvén wave propagating upwind, the group velocity is opposite the direction of \(\mathbf{B}_0\). Either way, the magnitude of group velocity is always given by the Alfvén wave velocity \(V_A\).