MHD Waves

When the time variations are much slower than the speed of light, we can neglect the electromagnetic waves discussed above. However, slower waves are allowed in a magnetised plasma. Unlike the pure sound discussed earlier, the addition of a background magnetic field introduces a preferred direction and the waves are now anisotropic.

Consider a perturbation to a uniform state at rest, with magnetic field and density denoted respectively by

$${\bf B}_{0} \hbox{ and }\rho_{0},$$

both uniform, that generates a small flow and a perturbed magnetic field so that

$${\bf v} = {\bf v}_{1} \hbox{ and } {\bf B} = {\bf B}_{0} + {\bf B}_{1}.$$

We linearise the MHD equations

$$\mu_{0}\rho {D{\bf v}\over Dt} = \left (\nabla \times {\bf B}\right )\times {\bf B},$$

$${\partial {\bf B}\over \partial t} = \nabla \times ({\bf v}\times {\bf B}),$$

and, assuming no pressure gradient term $\nabla p = 0$ and incompressible perturbations $\nabla \cdot {\bf v}_{1} = 0$, we obtain

$$\mu_{0}\rho_{0}{\partial {\bf v}_{1}\over \partial t} = \left (\nabla \times {\bf B}_{1}\right )\times {\bf B}_{0},$$ (6.21)

$${\partial {\bf B}_{1}\over \partial t} = \nabla \times ({\bf v}_{1}\times {\bf B}_{0}),$$ (6.22)

where

$$\nabla \cdot {\bf B}_{1} = 0.$$ (6.23)

Since the coefficients are again constant we look for solutions of the form

$$A e^{i({\bf k}\cdot {\bf r} - \omega t)} = Ae^{i(kx + ly + mz - \omega t)},$$

for a constant amplitude $A$ (each variable has its own constant amplitude), so that

$$-\mu_{0}\rho_{0}\omega {\bf v}_{1} = ({\bf k}\times {\bf B}... ...bf B}_{0}){\bf B}_{1} - ({\bf B}_{0}\cdot {\bf B}_{1}){\bf k},$$ (6.24)

and

$$-\omega {\bf B}_{1} = {\bf k}\times ({\bf v}_{1}\times {\bf... ...bf B}_{0}){\bf v}_{1} - ({\bf k}\cdot {\bf v}_{1}){\bf B}_{0}.$$ (6.25)

Now we make use of the equations

$$\nabla \cdot {\bf v}_{1} = 0, \quad \Rightarrow \quad {\bf k}\cdot {\bf v}_{1} = 0,$$

$$\nabla \cdot {\bf B}_{1} = 0, \quad \Rightarrow \quad {\bf k}\cdot {\bf B}_{1} = 0.$$

Now we take the scalar product of (6.24) with ${\bf k}$. Thus, we obtain

$${\bf B}_{0}\cdot {\bf B}_{1} = 0,$$

and so we have

$$-\mu_{0}\rho_{0}\omega {\bf v}_{1} = ({\bf k}\cdot {\bf B}_{0}){\bf B}_{1},$$

$$-\omega {\bf B}_{1} = ({\bf k}\cdot {\bf B}_{0}){\bf v}_{1}.$$

These two homogeneous equations only have a solution if $\omega$ and ${\bf k}$ are related through the dispersion relation

$$\omega^{2} = {({\bf k}\cdot {\bf B}_{0})^{2}\over \mu_{0}\rho_{0}},$$ (6.26)

or

$$\omega^{2} = \vert{\bf k}\vert^{2}v_{A}^{2}\cos^{2}\theta,$$ (6.27)

where $\theta$ is the angle between the equilibrium magnetic field direction and the direction of propagation of the wave and

$$v_{A}^{2} = {B_{0}^{2}\over \mu_{0} \rho_{0}}.$$ (6.28)

$B_{0}^{2}$ is the field strength squared and $v_{A}$ is the Alfvén speed. These magnetic waves propagate at the Alfvén speed that depends on the magnetic field strength and the density.

Thus, a magnetised plasma allows the propagation of Alfvén waves which propagate at a speed $\omega /\vert{\bf k}\vert$ equal to the Alfvén speed when directly along fieldlines but at a slower speed when propagating at an angle to the field.