MHD Waves

Let us investigate the small amplitude waves which propagate through a spatially uniform MHD plasma. We start by combining Eqs. (681)-(684) and (687)-(688) to form a closed set of equations:

$$\frac{d\rho}{dt} + \rho\,\nabla\cdot{\bf V} = 0,$$ (702)

$$\rho\,\frac{d{\bf V}}{dt} + \nabla p - \frac{(\nabla\times{\bf B})\times {\bf B}}{\mu_0} = {\bf0},$$ (703)

$$-\frac{\partial{\bf B}}{\partial t}+ \nabla\times( {\bf V} \times {\bf B}) = {\bf0},$$ (704)

$$\frac{d}{dt}\!\left(\frac{p}{\rho^{\Gamma}}\right) = 0.$$ (705)

Next, we linearize these equations (assuming, for the sake of simplicity, that the equilibrium flow velocity and equilibrium plasma current are both zero) to give

$$\frac{\partial\rho}{\partial t} + \rho_0\,\nabla\cdot{\bf V} = 0,$$ (706)

$$\rho_0\,\frac{\partial {\bf V}}{\partial t} + \nabla p - \frac{(\nabla\times{\bf B})\times {\bf B}_0}{\mu_0} = {\bf0},$$ (707)

$$-\frac{\partial{\bf B}}{\partial t}+ \nabla\times( {\bf V} \times {\bf B}_0) = {\bf0},$$ (708)

$$\frac{\partial}{\partial t}\!\left(\frac{p}{p_0} - \frac{{\Gamma}\,\rho}{\rho_0}\right) = 0.$$ (709)

Here, the subscript 0 denotes an equilibrium quantity. Perturbed quantities are written without subscripts. Of course, $\rho_0$, $p_0$, and ${\bf B}_0$ are constants in a spatially uniform plasma.

Let us search for wave-like solutions of Eqs. (706)-(709) in which perturbed quantities vary like $\exp[\,{\rm i}\,({\bf k}\!\cdot\!{\bf r} - \omega t)]$. It follows that

$$-\omega\,\rho + \rho_0\,{\bf k}\!\cdot\!{\bf V} = 0,$$ (710)

$$-\omega\,\rho_0\,{\bf V} + {\bf k}\,p - \frac{ ({\bf k}\times{\bf B})\times {\bf B}_0}{\mu_0} = {\bf0},$$ (711)

$$\omega\,{\bf B} + {\bf k}\times({\bf V}\times{\bf B}_0) = {\bf0},$$ (712)

$$-\omega\left(\frac{p}{p_0} - \frac{{\Gamma}\,\rho}{\rho_0}\right) = 0.$$ (713)

Assuming that $\omega\neq 0$, the above equations yield

$$\rho = \rho_0\,\frac{{\bf k}\!\cdot\!{\bf V}}{\omega},$$ (714)

$$p = {\Gamma}\,p_0\,\frac{{\bf k}\!\cdot\!{\bf V}}{\omega},$$ (715)

$${\bf B} = \frac{ ({\bf k}\!\cdot\!{\bf V})\,{\bf B}_0 - ({\bf k}\!\cdot\!{\bf B}_0)\,{\bf V}}{\omega}.$$ (716)

Substitution of these expressions into the linearized equation of motion, Eq. (711), gives

$$\left[ \omega^2 - \frac{({\bf k}\!\cdot\!{\bf B}_0)^2}{\mu_0\,\rho_0} \right] {\bf V} = \left\{ \left[ \frac{{\Gamma}\,p_0}{\rho_0} + \frac{B_0^{~2}}{\m... ...\cdot\!{\bf B}_0)}{\mu_0\,\rho_0}\,{\bf B}_0 \right\} ({\bf k}\!\cdot\!{\bf V})$$

$$- \frac{({\bf k}\!\cdot\!{\bf B}_0)\, ({\bf V}\!\cdot\!{\bf B}_0)}{\mu_0\,\rho_0} \,{\bf k}.$$ (717)

We can assume, without loss of generality, that the equilibrium magnetic field ${\bf B}_0$ is directed along the $z$-axis, and that the wave-vector ${\bf k}$ lies in the $x$-$z$ plane. Let $\theta$ be the angle subtended between ${\bf B}_0$ and ${\bf k}$. Equation (717) reduces to the eigenvalue equation

$$\left( \begin{array}{ccc} {\scriptstyle \omega^2 - k^2\,V_A^... ...{c}V_x\\ [0.5ex] V_y\\ [0.5ex] V_z\end{array}\right) = {\bf0}.$$ (718)

Here,

$$V_A = \sqrt{\frac{B_0^{~2}}{\mu_0\,\rho_0}}$$ (719)

is the Alfvén speed, and

$$V_S = \sqrt{\frac{{\Gamma}\,p_0}{\rho_0}}$$ (720)

is the sound speed. The solubility condition for Eq. (718) is that the determinant of the square matrix is zero. This yields the dispersion relation

$$(\omega^2 - k^2\,V_A^{~2}\,\cos^2\theta)\left[ \omega^4 - \o... ...V_S^{~2}) + k^4\,V_A^{~2}\,V_S^{~2}\,\cos^2\theta \right] = 0.$$ (721)

There are three independent roots of the above dispersion relation, corresponding to the three different types of wave that can propagate through an MHD plasma. The first, and most obvious, root is

$$\omega = k\,V_A\,\cos\theta,$$ (722)

which has the associated eigenvector $(0,V_y, 0)$. This root is characterized by both ${\bf k}\cdot{\bf V} =0$ and ${\bf V}\cdot {\bf B}_0=0$. It immediately follows from Eqs. (714) and (715) that there is zero perturbation of the plasma density or pressure associated with this root. In fact, this root can easily be identified as the shear-Alfvén wave, which was introduced in Sect. 4.8. Note that the properties of the shear-Alfvén wave in a warm (i.e., non-zero pressure) plasma are unchanged from those we found earlier in a cold plasma. Note, finally, that since the shear-Alfvén wave only involves plasma motion perpendicular to the magnetic field, we can expect the dispersion relation (722) to hold good in a collisionless, as well as a collisional, plasma.

The remaining two roots of the dispersion relation (721) are written

$$\omega = k\,V_+,$$ (723)

and

$$\omega = k\,V_-,$$ (724)

respectively. Here,

$$V_\pm = \left\{\frac{1}{2}\left[V_A^{~2} + V_S^{~2} \pm \sqr... ...- 4\, V_A^{~2}\,V_S^{~2}\,\cos^2\theta} \right]\right\}^{1/2}.$$ (725)

Note that $V_+\geq V_-$. The first root is generally termed the fast magnetosonic wave, or fast wave, for short, whereas the second root is usually called the slow magnetosonic wave, or slow wave. The eigenvectors for these waves are $(V_x,0,V_z)$. It follows that ${\bf k}\cdot{\bf V} \neq 0$ and ${\bf V}\cdot {\bf B}_0\neq0$. Hence, these waves are associated with non-zero perturbations in the plasma density and pressure, and also involve plasma motion parallel, as well as perpendicular, to the magnetic field. The latter observation suggests that the dispersion relations (723) and (724) are likely to undergo significant modification in collisionless plasmas.

In order to better understand the nature of the fast and slow waves, let us consider the cold-plasma limit, which is obtained by letting the sound speed $V_S$ tend to zero. In this limit, the slow wave ceases to exist (in fact, its phase velocity tends to zero) whereas the dispersion relation for the fast wave reduces to

$$\omega = k\,V_A.$$ (726)

This can be identified as the dispersion relation for the compressional-Alfvén wave, which was introduced in Sect. 4.8. Thus, we can identify the fast wave as the compressional-Alfvén wave modified by a non-zero plasma pressure.

In the limit $V_A\gg V_S$, which is appropriate to low-$\beta$ plasmas (see Sect. 3.13), the dispersion relation for the slow wave reduces to

$$\omega \simeq k\,V_S\,\cos\theta.$$ (727)

This is actually the dispersion relation of a sound wave propagating along magnetic field-lines. Thus, in low-$\beta$ plasmas the slow wave is a sound wave modified by the presence of the magnetic field.

The distinction between the fast and slow waves can be further understood by comparing the signs of the wave induced fluctuations in the plasma and magnetic pressures: $p$ and ${\bf B}_0\!\cdot\! {\bf B}/\mu_0$, respectively. It follows from Eq. (716) that

$$\frac{{\bf B}_0\!\cdot\!{\bf B}}{\mu_0} = \frac{{\bf k}\!\cd... ...dot\!{\bf B}_0)\,({\bf B}_0\!\cdot\!{\bf V})} {\mu_0\,\omega}.$$ (728)

Now, the $z$- component of Eq. (711) yields

$$\omega\,\rho_0\,V_z = k\,\cos\theta\,p.$$ (729)

Combining Eqs. (715), (719), (720), (728), and (729), we obtain

$$\frac{{\bf B}_0\!\cdot\!{\bf B}}{\mu_0} =\frac{V_A^{~2}}{V_... ...\left(1-\frac{k^2\,V_S^{~2}\,\cos^2\theta} {\omega^2}\right)p.$$ (730)

Hence, $p$ and ${\bf B}_0\!\cdot\! {\bf B}/\mu_0$ have the same sign if $V>V_S \,\cos\theta$, and the opposite sign if $V<V_S\,\cos\theta$. Here, $V=\omega/k$ is the phase velocity. It is straightforward to show that $V_+> V_S\,\cos\theta$, and $V_-<V_S\,\cos\theta$. Thus, we conclude that in the fast magnetosonic wave the pressure and magnetic energy fluctuations reinforce one another, whereas the fluctuations oppose one another in the slow magnetosonic wave.

Figure 16: Phase velocities of the three MHD waves in the $x$-$z$ plane.

Figure 16 shows the phase velocities of the three MHD waves plotted in the $x$-$z$ plane for a low-$\beta$ plasma in which $V_S<V_A$. It can be seen that the slow wave always has a smaller phase velocity than the shear-Alfvén wave, which, in turn, always has a smaller phase velocity than the fast wave.