In the magnetohydrodynamic description of a plasma,
we split the velocity , electric current ,
and electric field like so,
into a constant uniform equilibrium (subscript )
and a small unknown perturbation (subscript ):
Inserting this decomposition into the ideal form of the generalized Ohm’s law
and keeping only terms that are first-order in the perturbation, we get:
We do this for the momentum equation too,
assuming that (to be justified later).
Note that the temperature is set to zero, such that the pressure vanishes:
Where is the uniform equilibrium density.
We would like an equation for ,
which is provided by the magnetohydrodynamic form of Ampère’s law:
Substituting this into the above momentum equation,
and differentiating with respect to :
For which we can use Faraday’s law to rewrite ,
incorporating Ohm’s law too:
Inserting this back into the momentum equation for
thus yields its final form:
Suppose the magnetic field is pointing in -direction,
Then Faraday’s law justifies our earlier assumption that ,
and the equation can be written as:
Where we have defined the so-called Alfvén velocity to be given by:
Now, consider the following plane-wave ansatz for ,
with wavevector and frequency :
Inserting this into the above differential equation for leads to:
To evaluate this, we rotate our coordinate system around the -axis
such that ,
i.e. the wavevector’s -component is zero.
Calculating the cross products:
We rewrite this equation in matrix form,
using that :
This has the form of an eigenvalue problem for ,
meaning we must find non-trivial solutions,
where we cannot simply choose the components of to satisfy the equation.
To achieve this, we demand that the matrix’ determinant is zero:
This equation has three solutions for ,
one for each of its three factors being zero.
The simplest case is of no interest to us,
because we are looking for waves.
The first interesting case is ,
yielding the following dispersion relation:
The resulting waves are called shear Alfvén waves.
From the eigenvalue problem, we see that in this case
, meaning :
these waves are transverse.
The phase velocity and group velocity are as follows,
where is the angle between and :
The other interesting case is ,
which leads to so-called compressional Alfvén waves,
with the simple dispersion relation:
Looking at the eigenvalue problem reveals that ,
so these waves are not necessarily transverse, nor longitudinal (since is free).
The phase velocity and group velocity are given by:
The mechanism behind both of these oscillations is magnetic tension:
the waves are “ripples” in the field lines,
which get straightened out by Faraday’s law,
but the ions’ inertia causes them to overshoot and form ripples again.
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,