MHD waves

For low-$\beta$ plasmas, with $\beta \ll 1$ (also referred to as cold plasmas) the stresses in the plasma are predominantly magnetic. We seek MHD wave solutions in a cold magnetized plasma. In treating small-amplitude waves, the MHD equations are linearized, keeping only terms linear in the amplitude of the wave ( ${\bf B}_{1}$, $\eta_{1}$, and ${\bf U}_{1}$). We seek plane wave solutions; i.e., solutions that vary in space and time as ${\rm exp}[-i(\omega t - k x)]$(assuming that the plane wave propagates in the x-direction, with ${\bf k} = k {\bf {\hat x}}$). Additional assumptions are that the background magnetic field ${\bf B}_{0}$and plasma density $\eta_{0}$ are uniform, that there are no background currents or electric fields, and that there is no bulk fluid motion. Our starting equations are:

$$\frac{\partial \eta}{\partial t} +{\mbox{\boldmath$\nabla \cdot$ }} (\eta {\bf U}) = 0 \, ,$$ (3.49)

$$\eta \frac{\partial {\bf U}} {\partial t} = \frac{1}{\mu_{0}} {\mbox{\boldmath$(\nabla$ }} {\bf\times B}) {\bf\times B} \, ,$$ (3.50)

$$\frac{\partial {\bf B}} {\partial t} = {\mbox{\boldmath$\nabla \times$ }} ({\bf U \times B}) \, .$$ (3.51)

After linearizing, and replacing the time and spatial derivatives by $\partial/\partial t \rightarrow - i \omega$and $\partial/\partial x \rightarrow i k$, these equations become

$$\omega \eta_{1} - k \eta_{0} U_{1x} = 0 \,$$ (3.52)

$$\omega \eta_{0} {\bf U}_{1} - k ( {\bf {\hat x}} ({\bf B_{1} \cdot B_{0}}) -B_{0x} {\bf B_{1}})/\mu_{0} = 0 \, ,$$ (3.53)

$$\omega {\bf B}_{1} + k(B_{0x} {\bf U}_{1} - U_{1x} {\bf B}_{0}) =0 \, .$$ (3.54)

Without loss of generality we assume that ${\bf B}_{0}$ lies in the x-z plane, with ${\bf B}_{0} = (B_{0} \cos \theta, 0, B_{0} \sin \theta)$, where $\theta$ is the angle between ${\bf B}_{0}$ and ${\bf k}$. After eliminating ${\bf B}_{1}$ from () and (), the three equations relating components of ${\bf U}_{1}$ are written in the following matrix form:

$$\left[ \begin{array}{ccc} \left( {\omega^{2}}/{k^{2}} - v_{A}... ... U_{1x} \\ U_{1y} \\ U_{1z} \end{array}\right] = {\bf0} \, ,$$ (3.55)

where the Alfvèn velocity v_A satisfies

$$v_{A}=\left(\frac{B_{0}^{2}}{\mu_{0} \eta_{0}}\right)^{1/2} \, .$$ (3.56)

A solution for ${\bf U}_{1}$ exists only if the determinant of this matrix vanishes. This yields two independent non-trivial solutions for $\omega$as a function of k (known as the dispersion relation):

$$\omega^{2} = k^{2} v_{A}^{2} \cos^{2} \theta \, , \,\,\,\,\,\,\, \omega^{2} = k^{2} v_{A}^{2} \, .$$ (3.57)

The first solution corresponds to Alfvèn waves. After substituting the dispersion relation back into the matrix equation (), we find that a solution for ${\bf U}_{1}$ is only possible if U_1x = U_1z = 0. Thus Alfvèn waves are shear waves that shift plasma in the direction perpendicular to the plane containing the wavevector ${\bf k}$ and the background magnetic field ${\bf B}_{0}$, and that propagate with a phase velocity $v_{\phi} = \omega/k= v_{A} \cos \theta$. The wave motion in an Alfvèn wave may be attributed to an interplay between magnetic tension and plasma inertia. When a fluid element is displaced relative to ${\bf B}_{0}$ the magnetic field is displaced with the fluid. The field line becomes locally curved, which generates a tension force tending to straighten out the field line. The inertia of the plasma causes it to overshoot, setting up an oscillatory motion. The density of the fluid is unaffected by the propagating Alfvèn wave [ $U_{1x}=0 \Rightarrow \eta_{1} = 0$in ()]. The group velocity (velocity at which information propagates and the direction for energy flow) for Alfvèn waves satisfies

$${\bf v}_{g} = \left(\frac{\partial \omega}{\partial k_{x}}, \... ...al \omega}{\partial k_{z}} \right) = v_{A} {\bf {\hat b}} \, ,$$ (3.58)

so that the flow of energy associated with Alfvèn waves is directed along the background magnetic field direction.

The dispersion relation $\omega^{2} = k^{2} v_{A}^{2}$ corresponds to the magnetoacoustic mode. Substituting the dispersion relation into () yields the requirement that U_1y=0 (for $\theta \ne 0$), so that the fluid motion is in the plane containing ${\bf k}$ and ${\bf B}_{0}$. Because U_1x is not required to be zero, () implies that $\eta_{1}$ is also nonzero; i.e., magnetoacoustic waves affect the plasma density and are thus called compressional waves. For magnetoacoustic waves,

$${\bf v}_{g} = v_{A} {\bf {\hat k}} \, ,$$ (3.59)

so that wave energy may flow at an arbitrary angle to ${\hat {\bf b}}$, as opposed to Alfvèn waves (with ${\bf v}_{g} \parallel {\hat {\bf b}}$).

In a warm plasma, when $\beta$ is no longer small relative to unity, the plasma pressure terms can no longer be ignored. The pressure gradient term is reinserted in () and the adiabatic equation of state () closes the set of equations. In this case a linear analysis yields a dispersion relation with three solutions:

$$\omega^{2} = k^{2} v_{A}^{2} \cos^{2} \theta \, , \,\,\,\,\,\... ...^{2}-4v_{A}^{2} c_{S}^{2} \cos^{2} \theta ]^{\frac{1}{2}} \, ,$$ (3.60)

with the sound speed $c_{S}=\sqrt{\Gamma p_{0}/\eta_{0}}$. These three solutions correspond to the Alfvèn mode, and the fast (+) and slow (-) magnetoacoustic modes, so named because the phase speeds satisfy

$$v_{\rm fast} \ge v_{A} \ge v_{\rm slow} \, .$$ (3.61)

In the limit of small background magnetic field strengths, fast mode waves become gas sound waves, with the dispersion relation $\omega^{2} = k^{2} c_{s}^{2}$. In the cold plasma limit, fast mode waves become magnetoacoustic waves. In the small-field limit, slow mode waves become magnetoacoustic-like, with the dispersion relation $\omega^{2} = k^{2} v_{A}^{2} \cos^{2} \theta$. These only have magnetoacoustic properties for small angles $\theta$. In the cold plasma limit, slow mode waves (along the field lines) become gas-sound-like, with $\omega^{2} = k^{2} c_{S}^{2} \cos^{2} \theta$.