This week is the start of a several columns devoted to studying wave propagation in plasmas. The wave behavior in a plasma are very rich due to the collective behavior of the plasma that results from its desire to keep charge neutrality. The presentation here is strongly influenced by and closely follows the approach presented in Introduction to Plasma Physics with Space and Laboratory Applications by Gurnett and Bhattacharjee.
Waves in a plasma represent the propagation over long distances of a complicated patterns of interaction between the charge particles that make up the plasma and the electromagnetic fields present. Thus the starting point is the Lorentz force law
\[ m_s \frac{d \vec v_s}{d t} = q_s (\vec E + \vec v_s \times \vec B ) \; , \]
where $$s$$ can be thought of as a multi-index labeling both particle and species, and
Maxwell’s equations.
The particular form of Maxwell’s equations – microscopic versus macroscopic – that is best to use is a bit of a thorny question. On one hand, the electrons and ions that comprise a plasma are free to move about over very large distances. On the other, they act together to maintain charge neutrality. Gurnett and Bhattacharjee seem to split the difference by analyzing both the microscopic set
\[ \begin{array}{ccc} \nabla \cdot \vec E & = & \rho/\epsilon_0 = (\rho_{free}+\rho_{pol})/\epsilon_0 \\ \nabla \cdot \vec B & = & 0 \\ \nabla \times \vec E & = & – \partial_t \vec B \\ \nabla \times \vec B & = & \mu_0 \vec J + \mu_0 \epsilon_0 \partial_t \vec E \end{array} \]
and the macroscopic set
\[ \begin{array}{ccc} \nabla \cdot \vec D & = & \rho_{free} \\ \nabla \cdot \vec B & = & 0 \\ \nabla \times \vec E & = & – \partial_t \vec B \\ \nabla \times \vec H & = & \vec J_{free} + \partial_t \vec D \end{array} \; \]
and then enforcing a sort of consistency relation between them. The central notion is that all of the charges are to be counted as bound charges so that the usual linear relation between the displacement vector, the electric field, and the polarization – $$\vec D = \epsilon_0 \vec E + \vec P$$ holds. Gurnett and Bhattacharjee are not particularly clear on how to treat the magnetic constitutive relation other than to say
This half-and-half approach seems to be related to the points raised in the discussion in Section 3.2 of Chen’s book, where he points out that the dependence of the magnetic moment of the individual particles is given by
\[ \mu = \frac{m v_{\perp}^2}{2 B} \; , \]
which makes the magnetization, $$\vec M$$, nonlinear in $$|\vec B|$$, thus complicating the purely-macroscopic approach.
In any event, the Fourier transform is the usual tool used to tackle wave phenomena, since it provides a transform framework that handles both spatial and time variations in a convenient way. The Fourier transform replaces all time derivatives with multiplications by frequency and replaces all spatial derivatives with multiplication by a wave number:
\[ \partial_t \rightarrow – i \omega \]
\[ \nabla \cdot \rightarrow \vec k \cdot \]
\[ \nabla \times \rightarrow \vec k \times \]
Using this translations, the microscopic Maxwell equations become
\[ \begin{array}{ccc} i \hat k \cdot \vec E & = & \rho/\epsilon_0 \\ i \hat k \cdot \vec B & = & 0 \\ i \hat k \times \vec E & = & i \omega \vec B \\ i \hat k \times \vec B & = & \mu_0 \vec J – i \omega \mu_0 \epsilon_0 \vec E \end{array} \; ,\]
while the macroscopic equations become
\[ \begin{array}{ccc} i \hat k \cdot \vec E & = & 0 \\ i \hat k \cdot \vec B & = & 0 \\ i \hat k \times \vec E & = & i \omega \vec B \\ i \hat k \times \vec B & = & – i \omega \mu_0 \vec D \end{array} \; .\]
It is understood that the vector quantities in both set of equations are the Fourier transforms and, thus, depend on frequency and wave number and not time and position. Also note that the assumptions that a plasma has only bound charges and no magnetization results in no free density or current terms in the macroscopic set.
To close the set of equations, two constitutive relations need to be adopted. The first is the generalized Ohm’s law that relates the current to the electric field
\[ \vec J = \overleftrightarrow{\sigma} \cdot \vec E \; \]
through the conductivity tensor $$\overleftrightarrow{\sigma}$$.
The second one is a generalized dielectric relation between the displacement vector and the electric field
\[ \vec D = \epsilon_0 \overleftrightarrow{K} \cdot \vec E \]
through the dielectric tensor $$\overleftrightarrow{K}$$.
A formal connection between the conductivity and dielectric tensors can be made by equating the right-hand sides of Ampere’s law in microscopic and macroscopic form to yield
\[ \mu_0 \overleftrightarrow{\sigma} \cdot \vec E – i \omega \epsilon_0 \mu_0 \vec E = – i \omega \mu_0 \epsilon_0 \overleftrightarrow{K} \cdot \vec E \; .\]
A simple re-arrangement of terms results in the dielectric tensor express as
\[ \overleftrightarrow{K} = \overleftrightarrow{1} – \frac{ \overleftrightarrow{\sigma} }{i \omega \epsilon_0 } \; .\]
With the connection of the conductivity and dielectric tensors made, the microscopic equation now take a back seat to the macroscopic ones. The wave equation results when substituting Ampere’s equation into Faraday’s. Doing so yields
\[ \vec k (\vec k \times \vec E) + \frac{\omega^2}{c^2} \overleftrightarrow{K} \cdot \vec E = 0 \; .\]
Defining an index of refraction vector as
\[ \vec n = \frac{c}{\omega} \vec k\]
and using the fundamental relation
\[ \mu_0 \epsilon_0 = \frac{1}{c^2} \]
gives the form
\[ \vec n \times (\vec n \times \vec E) + \overleftrightarrow{K} \cdot \vec E = 0 \; .\]
The only way for this equation to be satisfied with a non-trivial electric field is for the operator that acts on it to be singular. If the equation can be written as
\[ \overleftrightarrow{D}(\vec n, \omega) \cdot \vec E = \vec 0 \]
then the determinant of $$\overleftrightarrow{D}(\vec n, \omega)$$ must be zero.
Unfortunately, $$\overleftrightarrow{D}(\vec n, \omega)$$ hides its structure with the complicated division into two pieces, one of which involves a double cross-product. Fortunately, there are convenient tools from mechanics that can tame the first term and join that which is split.
The first step is to recognize that a vector cross product can be represented in terms of matrix multiplication
\[ \vec n \times \doteq \left[ \begin{array}{ccc} 0 & -n_z & n_y \\n_z & 0 & -n_x \\ -n_y & n_x & 0 \end{array} \right] \; ,\]
which is often denoted as
\[ \vec n \times \equiv \vec n^{\times} \; .\]
This step is very convenient since it puts the cross product into matrix form and it rids the need for the parentheses. The double product immediately results from a second application of the matrix $$\vec n^{\times}$$ giving
\[ \vec n \times (\hat n \times \vec E) \doteq \left[ \begin{array}{ccc} -n_y^2 – n_z^2 & n_x n_y & n_x n_z \\n_x n_y & -n_x^2 – n_z^2 & n_y n_z \\n_x n_z & n_y n_z & -n_x^2 -n_y^2 \end{array} \right] \; .\]
Of course, this same result falls out from the expansion of $$\vec n \times (\vec n \times \vec E) = \vec n (\vec n \cdot \vec E) – \vec E (\vec n \cdot \vec n) $$ using BAC-CAB.
The operator $$\overleftrightarrow{D}(\vec n, \omega)$$, which I will call the dispersion tensor, is then recognized as the sum
\[ \overleftrightarrow{D}(\vec n, \omega) = (\vec n^{\times})^2 + \overleftrightarrow{K} = (\vec n^{\times})^2 + \overleftrightarrow{1} – \frac{\overleftrightarrow{\sigma}}{i \omega \epsilon_0} \; .\]
Much of the relevant behavior of the waves come from mining relationships that come from the requirement that the determinant of the dispersion tensor is zero. This is the coupled-equation form of the requirement that the wave operator $$\nabla^2 – c^{-2} \partial_t^2$$ is zero when operating on the a valid waveform (e.g. $$f(x-ct) + g(x+ct)$$).
The strategy for finding the dispersion tensor is as follows. First compute the conductivity tensor using some model of the plasma. The possible ways of tackling this seem to be single particle motion from the Lorentz law, magnetohydrodynamic methods, and kinetic theory. Next get the dielectric tensor by plugging the conductivity tensor into the relation above. Finally, add this piece to $$({\vec n^{\times}})^2$$.
Next week I’ll examine the implementation of this strategy for a cold plasma.