2.1 Wave dispersion in two frames

Dispersion in any plasma may be described by its dielectric tensor $\unicode[STIX]{x1D612}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k})$ and the wave equation written in the form $\unicode[STIX]{x1D6EC}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k})e_{j}=0$ , with $\boldsymbol{e}$ the polarization vector and with wave equation tensor

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k})={\displaystyle \frac{c^{2}(k_{i}k_{j}-|\boldsymbol{k}|^{2}\unicode[STIX]{x1D6FF}_{ij})}{\unicode[STIX]{x1D714}^{2}}}+\unicode[STIX]{x1D612}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta. The dispersion equation is given by setting the determinant of $\unicode[STIX]{x1D6EC}_{ij}$ to zero. In appendix A we derive the non-zero components of $\unicode[STIX]{x1D6EC}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k})$ as given by (A 10), making the low-frequency approximation in § A.2.1 and the non-gyrotropic approximation in § A.2.2. The dispersion equation then may be written as, $\unicode[STIX]{x1D6EC}_{ij}(\unicode[STIX]{x1D714},\boldsymbol{k})\rightarrow \unicode[STIX]{x1D6EC}_{ij}$ ,

(2.2) $$\begin{eqnarray}\displaystyle \det \unicode[STIX]{x1D6EC}_{ij}=\unicode[STIX]{x1D6EC}_{22}(\unicode[STIX]{x1D6EC}_{11}\unicode[STIX]{x1D6EC}_{33}-\unicode[STIX]{x1D6EC}_{13}^{2})=0. & & \displaystyle\end{eqnarray}$$

The dispersion relation for any specific wave mode is a specific solution of the dispersion (2.2). Here we derive and discuss the wave properties in the rest frame of the plasma; we discuss the wave properties Lorentz transformed to the pulsar frame in Part 2. We argue for the use of a Jüttner distribution to describe a pulsar plasma (Arendt & Eilek Reference Arendt and Eilek2002), which is an even function of $\unicode[STIX]{x1D6FD}$ in the plasma rest frame. For a distribution that is an even function of $\unicode[STIX]{x1D6FD}$ the non-zero components of $\unicode[STIX]{x1D6EC}_{ij}$ are given by (A 11).

For parallel propagation, $\unicode[STIX]{x1D703}=0$ , we have $\unicode[STIX]{x1D6EC}_{13}=0$ so that the solutions of (2.2) are $\unicode[STIX]{x1D6EC}_{22}=0$ , $\unicode[STIX]{x1D6EC}_{11}=0$ and $\unicode[STIX]{x1D6EC}_{33}=0$ which give the dispersion equation for the parallel X mode, parallel Alfvén or A mode and the parallel longitudinal or L mode, respectively, with explicit expressions

(2.3a-c ) $$\begin{eqnarray}\displaystyle z^{2}=z_{\text{A}}^{2},\quad z^{2}=z_{\text{A}}^{2},\quad \unicode[STIX]{x1D714}^{2}=\unicode[STIX]{x1D714}_{\text{p}}^{2}z^{2}\text{Re}W(z)\equiv \unicode[STIX]{x1D714}_{\text{L}}^{2}(z), & & \displaystyle\end{eqnarray}$$

where $z_{\text{A}}$ is given in (A 11), $\text{Re}W(z)$ denotes the real component of $W(z)$ , with $W(z)$ defined in (A 8). For simplicity in writing we omit $\text{Re}$ in the definition of $\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)$ . The parallel X and A modes are identical and may also be expressed as $\unicode[STIX]{x1D714}^{2}=z_{\text{A}}^{2}c^{2}k_{\Vert }^{2}$ .

For oblique propagation, $\unicode[STIX]{x1D703}\neq 0$ , the solution $\unicode[STIX]{x1D6EC}_{22}=0$ gives the oblique X mode,

(2.4a,b ) $$\begin{eqnarray}\displaystyle z^{2}=z_{\text{A}}^{2}(1+\tan ^{2}\unicode[STIX]{x1D703}/b)\quad \text{or}\quad \unicode[STIX]{x1D714}^{2}=z_{\text{A}}^{2}(1+\tan ^{2}\unicode[STIX]{x1D703}/b)c^{2}k_{\Vert }^{2}, & & \displaystyle\end{eqnarray}$$

where $b$ is given in (A 11). The solution $\unicode[STIX]{x1D6EC}_{11}\unicode[STIX]{x1D6EC}_{33}-\unicode[STIX]{x1D6EC}_{13}^{2}=0$ gives the Alfvén and the O modes,

(2.5a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}={\displaystyle \frac{(z^{2}-z_{\text{A}}^{2})\,\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)}{z^{2}-z_{\text{A}}^{2}-b\tan ^{2}\unicode[STIX]{x1D703}}}\quad \text{or}\quad z^{2}=z_{\text{A}}^{2}+{\displaystyle \frac{\unicode[STIX]{x1D714}^{2}\,b\tan ^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\text{L}}(z)$ is given in (2.3). For $\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)>0$ , the O mode is given by (2.5) over $z_{\text{A}}^{2}+b\tan ^{2}\unicode[STIX]{x1D703}<z^{2}\leqslant \infty$ and the Alfvén mode is over $z_{0}^{2}\leqslant z^{2}<z_{\text{A}}^{2}$ with $z_{0}$ such that $\unicode[STIX]{x1D714}_{\text{L}}^{2}(z_{0})=0$ . The case $\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)<0$ is more subtle and is discussed below.

The polarization vector $\boldsymbol{e}$ corresponding to a solution of $\unicode[STIX]{x1D6EC}_{22}=0$ is along the 2-axis, which implies that the X mode is strictly transverse. Any solution of $\unicode[STIX]{x1D6EC}_{11}\unicode[STIX]{x1D6EC}_{33}-\unicode[STIX]{x1D6EC}_{13}^{2}=0$ corresponds to a polarization vector, $\boldsymbol{e}$ , in the 1-3 plane with

(2.6) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{e_{1}}{e_{3}}}=-{\displaystyle \frac{\unicode[STIX]{x1D6EC}_{33}}{\unicode[STIX]{x1D6EC}_{13}}}=-{\displaystyle \frac{\unicode[STIX]{x1D6EC}_{13}}{\unicode[STIX]{x1D6EC}_{11}}}. & & \displaystyle\end{eqnarray}$$

Longitudinal polarization corresponds to $\boldsymbol{e}=(\sin \unicode[STIX]{x1D703},0,\cos \unicode[STIX]{x1D703})$ , and the only strictly longitudinal waves are for parallel propagation, $\sin \unicode[STIX]{x1D703}=0$ , satisfying the dispersion equation $\unicode[STIX]{x1D6EC}_{33}=0$ , which implies that the L mode is strictly longitudinal.

There are three modes present for either parallel propagation (parallel X, A and L modes) or oblique propagation (oblique X, Alfvén and O modes). In the rest frame of the plasma, each mode has a forward-propagating component, $z>0$ , and a backward-propagating component, $z<0$ . The backward-propagating portion is a mirror image of the forward-propagating one about $z=0$ . We restrict our discussion to the forward-propagating solution from which the properties of the backward-propagating part can be readily inferred.

3.1 One-dimensional Jüttner distribution

Several different choices have been made for the distribution function of the electrons in a pulsar magnetosphere, including a power law (Kaplan & Tsytovich Reference Kaplan and Tsytovich1973, §17), a relativistically streaming Gaussian distribution (e.g. Egorenkov et al. Reference Egorenkov, Lominadze and Mamradze1983; Asseo & Melikidze Reference Asseo and Melikidze1998) and water-bag and bell distributions (Arons & Barnard Reference Arons and Barnard1986; Melrose & Gedalin Reference Melrose and Gedalin1999). Although the distribution function for the electrons and positrons in the rest frame of the plasma is not known, a relativistic thermal distribution is one approximate form suggested by numerical models for the cascade that generates the pair plasma (Hibschman & Arons Reference Hibschman and Arons2001; Arendt & Eilek Reference Arendt and Eilek2002). We choose a 1-D Jüttner distribution (Melrose & Gedalin Reference Melrose and Gedalin1999; Melrose et al. Reference Melrose, Gedalin, Kennett and Fletcher1999; Asseo & Riazuelo Reference Asseo and Riazuelo2000) and suggest that this should be the default choice, with a specific reason being required to justify any other choice. (In Part 2 a streaming distribution is modelled by Lorentz transforming a Jüttner distribution from its rest frame to the frame in which it is streaming.)

The combined distribution function, for electrons plus positrons, is then

(3.1) $$\begin{eqnarray}\displaystyle g(u)={\displaystyle \frac{n\,e^{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}}}{2K_{1}(\unicode[STIX]{x1D70C})}},\quad \text{with}~n=\int _{-\infty }^{\infty }\,\text{d}u\,g(u), & & \displaystyle\end{eqnarray}$$

where $u=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}$ and $n$ is the number density. The parameter $\unicode[STIX]{x1D70C}=mc^{2}/T$ is the ratio of the rest energy of the electron to the temperature in energy units, with $\unicode[STIX]{x1D70C}=1$ corresponding to $T=0.511~\text{MeV}\approx 0.6\times 10^{10}\,K$ , and $K_{1}(\unicode[STIX]{x1D70C})$ is the Macdonald function of order 1. One has $\langle \unicode[STIX]{x1D6FE}\rangle \approx 1/\unicode[STIX]{x1D70C}$ for a highly relativistic distribution where $\unicode[STIX]{x1D70C}\ll 1$ .

3.2 RPDF for a Jüttner distribution

The integrand of (A 8) defining the RPDF $W(z)$ is singular at $\unicode[STIX]{x1D6FD}=z$ for $|z|<1$ . The singularity is treated following the Landau prescription: $\unicode[STIX]{x1D714}\rightarrow \unicode[STIX]{x1D714}+i0$ (we assume real $k_{\Vert }$ ). With $z=\unicode[STIX]{x1D714}/k_{\Vert }c$ , this implies $z\rightarrow z+i0$ for $k_{\Vert }>0$ and to $z\rightarrow z-i0$ for $k_{\Vert }<0$ . For $\unicode[STIX]{x1D714}>0$ , the resonant denominator is then replaced by $i\unicode[STIX]{x03C0}\,\text{sgn}(k_{\Vert })\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FD}-z)$ where $\text{sgn}(k_{\Vert })=k_{\Vert }/|k_{\Vert }|$ is the sign of $k_{\Vert }$ . The RPDF may be expressed as

(3.2) $$\begin{eqnarray}\displaystyle W(z)=\left\{\begin{array}{@{}ll@{}}\left\langle {\displaystyle \frac{1}{\unicode[STIX]{x1D6FE}^{3}(\unicode[STIX]{x1D6FD}-z)^{2}}}\right\rangle ,\quad & \text{for}\,|z|>1,\\[12.0pt] \displaystyle {\displaystyle \frac{1}{n}}\left[\text{i}\unicode[STIX]{x03C0}\text{sgn}(k_{\Vert }){\displaystyle \frac{\text{d}g(u)}{\text{d}\unicode[STIX]{x1D6FD}}}|_{\unicode[STIX]{x1D6FD}=z}-2\unicode[STIX]{x1D6FE}^{2}g(u)|_{\unicode[STIX]{x1D6FD}=z}\right.\quad \\[12.0pt] \displaystyle \left.\quad -\,{\wp}\int _{-1}^{1}\,\text{d}\unicode[STIX]{x1D6FD}\,{\displaystyle \frac{g(u)|_{\unicode[STIX]{x1D6FD}=z}-g(u)}{(\unicode[STIX]{x1D6FD}-z)^{2}}}\right],\quad & \text{for}\,|z|\leqslant 1,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where ${\wp}$ denotes a Cauchy principal value integral, and $u=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}/(1-\unicode[STIX]{x1D6FD}^{2})^{1/2}$ . The expression for $|z|>1$ follows from a partial integration of (A 8) and that for $|z|\leqslant 1$ is derived in  appendix B. Note that the expression for $|z|>1$ can be misleading if applied to $|z|<1$ . Specifically, the form for $|z|>1$ is real and positive definite, whereas for $|z|<1$ the RPDF is complex and its real part is negative for $0<|z|\leqslant z_{0}$ , where $z_{0}$ is identified below. For a Jüttner distribution the imaginary part of $W(z)$ follows from (3.2) with (3.1) implying

(3.3a,b ) $$\begin{eqnarray}\displaystyle \text{Im}z^{2}W(z)=-\text{sgn}(k_{\Vert }){\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}z^{3}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}^{3}\exp (-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}})}{2K_{1}(\unicode[STIX]{x1D70C})}},\quad \unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}={\displaystyle \frac{1}{\sqrt{1-z^{2}}}}. & & \displaystyle\end{eqnarray}$$

The RPDF $W(z)$ for the distribution (3.1) can be expressed in terms of another RPDF,

(3.4a,b ) $$\begin{eqnarray}\displaystyle W(z)={\displaystyle \frac{1}{2K_{1}(\unicode[STIX]{x1D70C})}}{\displaystyle \frac{\unicode[STIX]{x2202}T(z,\unicode[STIX]{x1D70C})}{\unicode[STIX]{x2202}z}},\quad T(z,\unicode[STIX]{x1D70C})=\int _{-1}^{1}\,\text{d}\unicode[STIX]{x1D6FD}\,{\displaystyle \frac{\text{e}^{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D6FD}-z}}. & & \displaystyle\end{eqnarray}$$

The properties of the RPDF $T(z,\unicode[STIX]{x1D70C})$ were summarized by Godfrey, Newberger & Taggart (Reference Godfrey, Newberger and Taggart1975), cf. also Melrose (Reference Melrose2008). We note two alternative forms for $T(z,\unicode[STIX]{x1D70C})$ given by Godfrey et al. (Reference Godfrey, Newberger and Taggart1975),

(3.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}T(z,\unicode[STIX]{x1D70C})=\text{e}^{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}}\ln {\displaystyle \frac{1-z}{1+z}}+\displaystyle \int _{-1}^{1}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FD}}{\unicode[STIX]{x1D6FD}-z}}(\text{e}^{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}}-\text{e}^{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}}),\\[11.0pt] \displaystyle T(z,\unicode[STIX]{x1D70C})=-2\unicode[STIX]{x1D70C}\int _{0}^{z}{\displaystyle \frac{\text{d}x}{[(1-x^{2})(1-z^{2})]^{1/2}}}K_{1}\left[\left({\displaystyle \frac{1-x^{2}}{1-z^{2}}}\right)^{1/2}\unicode[STIX]{x1D70C}\right]+i\unicode[STIX]{x03C0}\text{e}^{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FE}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

In our detailed calculations we compared all three forms, and confirmed their equivalence.

Figure 1. The RPDF $z^{2}W(z)$ is plotted as a function of $z$ for 1-D Jüttner distributions. The thick curves correspond to the real part and the thin curves to the imaginary part of $z^{2}W(z)$ for $\unicode[STIX]{x1D70C}=50$ (dotted), $\unicode[STIX]{x1D70C}=10$ (dashed) and $\unicode[STIX]{x1D70C}=1$ (solid). The imaginary parts are identically zero for $z\geqslant 1$ and negative for $z<1$ . Note that $z$ increases from right to left to facilitate comparison with dispersion curves shown below.

Examples of $z^{2}W(z)$ for the distribution (3.1) are shown in figure 1 for three temperatures, ranging from a non-relativistic value, $\unicode[STIX]{x1D70C}=50\gg 1$ (dotted), to a value, $\unicode[STIX]{x1D70C}=1$ (solid), where relativistic effects are significant. A similar plot was presented by Melrose & Gedalin (Reference Melrose and Gedalin1999), and we make two notable changes in figure 1; we include the imaginary parts, shown by the thin curves, and we plot the curves such that the superluminal regime, $z>1$ is to the left of $z=1$ and the subluminal regime is to the right of $z=1$ . The peak in the RPDF evident in figure 1 becomes higher, narrower and closer to $z=1$ with decreasing $\unicode[STIX]{x1D70C}$ . We show this peak on a fine scale and on a very fine scale in figure 2 for $\unicode[STIX]{x1D70C}=0.1$ (dashed) and $\unicode[STIX]{x1D70C}=0.01$ (solid). Note that the shape of the peak near $z=1$ scales in a characteristic way with $\unicode[STIX]{x1D70C}$ . Numerical estimates based on the scaling apparent in these figures are given below.

For numerical results and plots we use, unless stated otherwise, pulsar period $P=1~\text{s}$ , period derivative ${\dot{P}}=10^{-15}$ , emission height at radius $r/r_{\text{L}}=0.1$ , where $r_{\text{L}}=Pc/2\unicode[STIX]{x03C0}$ is the light-cylinder radius, and multiplicity $\unicode[STIX]{x1D705}=10^{5}$ . For these parameter values $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 5.0\times 10^{3}$ & $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 3.6\times 10^{3}$ for $\unicode[STIX]{x1D70C}=0.1$ , and $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 1.6\times 10^{3}$ & $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 1.1\times 10^{3}$ for $\unicode[STIX]{x1D70C}=0.01$ , where $\unicode[STIX]{x1D6FE}_{\text{A}}$ is given by (A 12). In some calculations we vary $\unicode[STIX]{x1D6FD}_{\text{A}}$ which can be achieved by varying a number of pulsar parameters as

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{A}^{2}\approx 2.6\times 10^{7}\left({\displaystyle \frac{10}{\langle \unicode[STIX]{x1D6FE}\rangle }}\right)\left({\displaystyle \frac{10^{5}}{\unicode[STIX]{x1D705}}}\right)\left({\displaystyle \frac{\unicode[STIX]{x1D6FE}_{s}}{10^{3}}}\right)\left({\displaystyle \frac{{\dot{P}}/P^{3}}{10^{-15}}}\right)^{1/2}\left({\displaystyle \frac{r/r_{L}}{0.1}}\right)^{-3}. & & \displaystyle\end{eqnarray}$$

Figure 2. As for figure 1 but with $\unicode[STIX]{x1D70C}z^{2}W(z)$ plotted against $(1-z)/\unicode[STIX]{x1D70C}^{2}$ for $\unicode[STIX]{x1D70C}=0.1$ (dashed) and $\unicode[STIX]{x1D70C}=0.01$ (solid) around $z=1$ on a fine scale (a) and on a very fine scale (b). With these scalings of the vertical and horizontal axes, the plots for $\unicode[STIX]{x1D70C}=0.1$ and $\unicode[STIX]{x1D70C}=0.01$ are nearly indistinguishable.

3.3 Properties of the RPDF

Figure 3. A plot of $z^{2}\text{Re}W(z)$ (thick solid) and $z^{2}\text{Im}W(z)$ (thin solid) over $1>z>0$ illustrating specific values of $z$ : $z_{\text{m}}$ where $z^{2}\text{Re}W(z)$ is a maximum, $z_{0}$ where $z^{2}\text{Re}W(z)$ passes through zero, $z_{\text{min}}$ where $z^{2}\text{Re}W(z)$ is a minimum, $z_{\text{Imin}}$ where $z^{2}\text{Im}W(z)$ is a minimum, and $z_{\text{e1,2}}$ where $|z^{2}\text{Re}W(z)|=|z^{2}\text{Im}W(z)|$ .

In figure 3 we present a typical plot of the real (thick solid) and imaginary (thin solid) part of the RPDF $z^{2}W(z)$ over $1>z>0$ . We denote the critical $z$ values and the corresponding value of the real and imaginary parts of $z^{2}W(z)$ at these $z$ values. These critical values of $z$ are $z_{\text{m}}$ where $z^{2}\text{Re}W(z)$ is a maximum, $z_{0}$ where $z^{2}\text{Re}W(z)$ passes through zero, $z_{\text{min}}$ where $z^{2}\text{Re}W(z)$ is a minimum, $z_{\text{Imin}}$ where $z^{2}\text{Im}W(z)$ is a minimum and $z_{\text{e}1,2}$ where $|z^{2}\text{Re}W(z)|=|z^{2}\text{Im}W(z)|$ . We note that

(3.7) $$\begin{eqnarray}\displaystyle 0<z_{\text{e}2}<z_{\text{min}}<z_{0}<z_{\text{Imin}}<z_{\text{e}1}<z_{\text{m}}<1. & & \displaystyle\end{eqnarray}$$

In the rest frame of the plasma these critical points only depend on the value of $\unicode[STIX]{x1D70C}$ . For $\unicode[STIX]{x1D70C}\ll 1$ , the critical $z$ values vary as $z\approx 1-\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D70C}^{2}$ and the corresponding Lorentz factors as $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}\approx \unicode[STIX]{x1D6FC}_{2}/\unicode[STIX]{x1D70C}$ , and the real and imaginary components of $z^{2}W(z)$ vary as ${\approx}\unicode[STIX]{x1D6FC}_{3}/\unicode[STIX]{x1D70C}$ . We give approximate values of $\unicode[STIX]{x1D6FC}_{i}$ in table 1 for both $\unicode[STIX]{x1D70C}\ll 1$ and $\unicode[STIX]{x1D70C}=1$ . We plot these critical points in figure 4 where $\unicode[STIX]{x1D70C}$ varies between $10^{-2}$ and 50.

Table 1. Empirical values for parameters $\unicode[STIX]{x1D6FC}_{i}$ for $i=1,2,3$ .

Figure 4. (a) Plots of $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}$ against $\unicode[STIX]{x1D70C}$ with $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}$ evaluated at $z=z_{\text{m}}$ (thick solid), at $z=z_{0}$ (dashed), at $z=z_{\text{min}}$ (thin solid), at $z=z_{\text{Imin}}$ (dotted), at $z=z_{\text{e}1}$ (thick dash-dotted) and at $z=z_{\text{e}2}$ (thin dash-dotted). (b) Magnitude of $\text{Re}z^{2}W(z)$ at $z=z_{\text{m}}$ (thick solid), at $z=z_{\text{min}}$ (thin solid), at $z_{\text{e}1}$ (thick dash-dotted) and $z_{\text{e}2}$ (thin dash-dotted) where $|\text{Im}z^{2}W(z)|=|\text{Re}z^{2}W(z)|$ , and at $z=z_{\text{Imin}}$ where $|\text{Im}z^{2}W(z)|$ (dotted).

The average value of powers of $\unicode[STIX]{x1D6FE}$ over a Jüttner distribution are related to $\unicode[STIX]{x1D70C}$ as shown in table 2 (Melrose & Gedalin Reference Melrose and Gedalin1999). The approximations given for $\unicode[STIX]{x1D70C}\ll 1$ are particularly important and simple.

Table 2. Averages over a Jüttner distribution as given by Melrose & Gedalin (Reference Melrose and Gedalin1999). The functions $K_{i}(\unicode[STIX]{x1D70C})$ are modified Bessel functions of second type, and $Ki_{n}(\unicode[STIX]{x1D70C})$ are Bickley functions defined as the $n^{\text{th}}$ integral of $K_{0}(\unicode[STIX]{x1D70C})$ .

4.1 Cold-plasma limit

The cold-plasma limit corresponds to

(4.1a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\rightarrow \infty ,\quad \langle \unicode[STIX]{x1D6FE}\rangle \rightarrow 1,\quad \text{and}\quad z^{2}W(z)\rightarrow 1. & & \displaystyle\end{eqnarray}$$

The non-zero terms of $\unicode[STIX]{x1D6EC}_{ij}$ are then given by (A 11) with $a=1+1/\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$ , $b=1$ and $z^{2}W(z)=1$ . This limit may also be treated using the cold-plasma model, with the ions replaced by positrons. In cold-plasma theory, it is conventional to solve the dispersion equation for the square of the refractive index, $N^{2}=k^{2}c^{2}/\unicode[STIX]{x1D714}^{2}=1/z^{2}\cos ^{2}\unicode[STIX]{x1D703}$ , as a function of $\unicode[STIX]{x1D714}$ and the angle $\unicode[STIX]{x1D703}$ of wave propagation (Stix Reference Stix1962).

4.1.1 The X mode

The dispersion equation for the X-mode in the cold-plasma limit follows from (2.4) using (4.1) as

(4.2a,b ) $$\begin{eqnarray}\displaystyle z^{2}={\displaystyle \frac{z_{\text{A}}^{2}}{\cos ^{2}\unicode[STIX]{x1D703}}},\quad \text{giving}~N^{2}={\displaystyle \frac{1}{z_{\text{A}}^{2}}}=1+{\displaystyle \frac{1}{\unicode[STIX]{x1D6FD}_{\text{A}}^{2}}}. & & \displaystyle\end{eqnarray}$$

The X mode may be interpreted as a magnetoacoustic wave with $z_{\text{A}}c$ the magnetohydrodynamic (MHD) speed when the displacement current is included. The polarization vector for the X mode is along the 2-axis, that is, along the direction $\boldsymbol{k}\times \boldsymbol{B}$ .

4.1.2 Parallel A and L modes

With the cold-plasma assumption (4.1) one obtains from (2.3) the expression for A mode in a cold plasma as

(4.3a,b ) $$\begin{eqnarray}\displaystyle z^{2}=z_{\text{A}}^{2},\quad \text{implying}~N^{2}={\displaystyle \frac{1}{z_{\text{A}}^{2}}}. & & \displaystyle\end{eqnarray}$$

From (4.2) and (4.3) it is evident that the dispersion relations for the X and A modes are the same for parallel propagation. The polarization of the A mode is along the $1$ -axis.

The expression for the L mode in a cold plasma is obtained from (2.3) using (4.1) as

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}=\unicode[STIX]{x1D714}_{\text{p}}^{2}. & & \displaystyle\end{eqnarray}$$

The polarization is longitudinal, which is along $\boldsymbol{B}$ for parallel propagation.

4.1.3 Alfvén and O modes

The Alfvén and O modes (2.5) using the cold-plasma limit (4.1) reduce to

(4.5a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}={\displaystyle \frac{(z^{2}-z_{\text{A}}^{2})\,\unicode[STIX]{x1D714}_{\text{p}}^{2}}{z^{2}-z_{\text{A}}^{2}-\tan ^{2}\unicode[STIX]{x1D703}}}\quad \text{or}\quad z^{2}=z_{\text{A}}^{2}+{\displaystyle \frac{\unicode[STIX]{x1D714}^{2}\,\tan ^{2}\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{p}}^{2}}}. & & \displaystyle\end{eqnarray}$$

For $\unicode[STIX]{x1D6FD}_{\text{A}}^{2}\gg 1$ we have $z_{\text{A}}^{2}\approx 1$ which allows us to write (4.5) as

(4.6) $$\begin{eqnarray}\displaystyle N^{2}\approx {\displaystyle \frac{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{p}}^{2}}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{p}}^{2}\cos ^{2}\unicode[STIX]{x1D703}}}\approx 1-\frac{\unicode[STIX]{x1D714}_{\text{p}}^{2}}{\unicode[STIX]{x1D714}^{2}}\sin ^{2}\unicode[STIX]{x1D703}, & & \displaystyle\end{eqnarray}$$

where the final approximation applies for $\unicode[STIX]{x1D714}^{2}\gg \unicode[STIX]{x1D714}_{\text{p}}^{2}$ . Equation (4.6) implies propagating waves for $\unicode[STIX]{x1D714}^{2}>\unicode[STIX]{x1D714}_{\text{p}}^{2}$ and for $\unicode[STIX]{x1D714}^{2}<\unicode[STIX]{x1D714}_{\text{p}}^{2}\cos ^{2}\unicode[STIX]{x1D703}$ , with a stop band (evanescent waves) in the range $\unicode[STIX]{x1D714}_{\text{p}}^{2}\cos ^{2}\unicode[STIX]{x1D703}<\unicode[STIX]{x1D714}^{2}<\unicode[STIX]{x1D714}_{\text{p}}^{2}$ . The higher-frequency branch corresponds to the O mode and the lower-frequency branch to the Alfvén mode. For this cold-plasma case the reconnection of the L and A modes to form these two oblique modes is illustrated in figure 5, which is similar to a figure presented by Lyutikov (Reference Lyutikov1999).

In a relativistic plasma, an approximation to (2.5) that is similar to (4.6) may be obtained by regarding $\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)$ as a constant (over a small range of $z$ values) and assuming $b\approx 1$ , $z_{\text{A}}^{2}\approx 1$ . The dispersion relation reduces to

(4.7) $$\begin{eqnarray}\displaystyle N^{2}\approx {\displaystyle \frac{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)\cos ^{2}\unicode[STIX]{x1D703}}}. & & \displaystyle\end{eqnarray}$$

Analogous to the cold-plasma case, as $\unicode[STIX]{x1D703}$ increases the reconnected modes move apart, the O mode to higher $\unicode[STIX]{x1D714}$ and larger $z$ , and the Alfvén mode to lower $\unicode[STIX]{x1D714}$ .

Figure 5. Wave properties in the cold limit. The axes of the inset are $(\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{p}}-1)\times 10^{6}$ and $(ck_{\Vert }/\unicode[STIX]{x1D714}_{\text{p}}-1)\times 10^{6}$ . The dotted line in the inset is the light line, $z=1$ , and the dashed line is the X mode.

4.1.4 Cross-over frequency

The dispersion relations for the A and L modes for strictly parallel propagation cross each other at the cross-over frequency

(4.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{L}}(z_{\text{A}})\equiv \unicode[STIX]{x1D714}_{\text{co}},\quad z=z_{\text{A}}. & & \displaystyle\end{eqnarray}$$

A cross-over always occurs in the cold-plasma limit, for which the dispersion relations reduce to the horizontal line, $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}$ (L mode), and the oblique line, $z=z_{\text{A}}$ (A and X modes) that passes through the origin in the $\unicode[STIX]{x1D714}$ - $z$ plane.

4.2 Dispersion curves for $\unicode[STIX]{x1D70C}=20$

The transition from the cold-plasma limit, $\unicode[STIX]{x1D70C}\rightarrow \infty$ , to the highly relativistic limit, $\unicode[STIX]{x1D70C}\ll 1$ , leads to a dramatic change in the wave properties. To follow how this transition occurs, it is helpful to consider two intermediate cases: one where thermal effects are significant and relativistic effects are small, and another where relativistic effects are important. We choose the intermediate cases $\unicode[STIX]{x1D70C}=20$ and $\unicode[STIX]{x1D70C}=1$ .

4.2.1 L mode for $\unicode[STIX]{x1D70C}=20$

The dispersion curves are illustrated for $\unicode[STIX]{x1D70C}=20$ in figure 6. The dispersion relation for the L mode changes from the horizontal line $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}$ in a cold plasma to the solid black curve in figure 6, which may be regarded as a plot of $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{p}}$ versus $ck_{\Vert }/\unicode[STIX]{x1D714}_{\text{p}}=(\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{p}})/z$ . The cutoff frequency, $\unicode[STIX]{x1D714}_{x}=\unicode[STIX]{x1D714}_{\text{L}}(\infty )=\langle 1/\unicode[STIX]{x1D6FE}^{3}\rangle \unicode[STIX]{x1D714}_{\text{p}}$ , moves to just below $\unicode[STIX]{x1D714}_{\text{p}}$ due to a relativistic correction $\langle 1/\unicode[STIX]{x1D6FE}^{3}\rangle$ . The frequency increases with increasing $k_{\Vert }$ , crosses the light line at $\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{\text{L}}(1)=(2\langle \unicode[STIX]{x1D6FE}\rangle -\langle 1/\unicode[STIX]{x1D6FE}\rangle )\unicode[STIX]{x1D714}_{\text{p}}$ , $k_{\Vert }c=\unicode[STIX]{x1D714}_{1}$ and reaches its maximum at $z=z_{\text{m}}$ , corresponding to $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{p}}\approx 1.5$ and $ck_{\Vert }/\unicode[STIX]{x1D714}_{\text{p}}=2.6$ .

Figure 6. Dispersion curves for $\unicode[STIX]{x1D70C}=20$ . The solid black curves correspond to the L and A modes for $\unicode[STIX]{x1D703}=0$ , and the other nested curves are for the O and Alfvén modes with $\unicode[STIX]{x1D703}$ increasing in steps of 0.25 rad. The X mode (not shown) is degenerate with the A mode for $\unicode[STIX]{x1D703}=0$ .

The L mode is double valued, with a higher-frequency portion and a lower-frequency portion joining at what we refer to as a turnover. The turnover, as a function of $z$ , occurs at $z=z_{\text{m}}$ , where the RPDF has its maximum. The lower-frequency portion of the dispersion curve extends to $\unicode[STIX]{x1D714}=0$ , which is approached along the line $z=z_{0}$ (or $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{p}}=z_{0}ck_{\Vert }/\unicode[STIX]{x1D714}_{\text{p}}$ ), where the RPDF passes through zero. The higher-frequency portion may be interpreted as a counterpart of the parallel Langmuir mode in a non-relativistic, magnetized plasma; the L mode dispersion relation (2.3) may be approximated by $\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)=\unicode[STIX]{x1D714}_{\text{p}}^{2}+3k_{\Vert }^{2}V^{2}$ with $\unicode[STIX]{x1D70C}=c^{2}/V^{2}\gg 1$ . The lower-frequency portion of the dispersion curve is in a region of strong Landau damping, and we do not discuss this branch further.

4.2.3 Oblique O and Alfvén modes for $\unicode[STIX]{x1D70C}=20$

For $\tan \unicode[STIX]{x1D703}\neq 0$ the L and A modes reconnect to form the O mode, which moves to the upper left as $\unicode[STIX]{x1D703}$ increases, and the Alfvén mode, which moves to the lower right as $\unicode[STIX]{x1D703}$ increases. The turnover, corresponding to the peak in the RPDF, is in the Alfvén mode for $z_{\text{A}}>z_{\text{m}}$ . The reconnected Alfvén mode consists of two branches, and for sufficiently small $\unicode[STIX]{x1D703}$ , these are the familiar Alfvén mode with $z\approx z_{\text{A}}$ , $\unicode[STIX]{x1D714}<\unicode[STIX]{x1D714}_{\text{co}}$ , and a branch that follows the L mode for $z_{\text{A}}>z\gtrsim z_{\text{m}}$ . We refer to the latter as the turnover branch. The wave properties remain topologically similar as $\unicode[STIX]{x1D70C}$ decreases, but become increasingly distorted from the mildly relativistic case $\unicode[STIX]{x1D70C}=20$ as $\unicode[STIX]{x1D70C}$ decreases to ${\lesssim}1$ .

Figure 7. Dispersion curves for $\unicode[STIX]{x1D70C}=1$ in the same form as figure 6.

4.3 Dispersion curves for $\unicode[STIX]{x1D70C}=1$

For the case $\unicode[STIX]{x1D70C}=1$ shown in figure 7, relativistic effects are important. Comparing the dispersion curves in figure 7 with those in figure 6, an obvious difference is the cutoff frequency $\unicode[STIX]{x1D714}_{x}=\unicode[STIX]{x1D714}_{\text{L}}(\infty )$ , which is only marginally below $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}$ for $\unicode[STIX]{x1D70C}=20$ , and is significantly below $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}$ for $\unicode[STIX]{x1D70C}=1$ . Another notable difference is the frequency $\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{\text{L}}(1)$ at which the dispersion curve for the L mode crosses the light line: this is marginally above $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}$ for $\unicode[STIX]{x1D70C}=20$ , and significantly above $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}$ for $\unicode[STIX]{x1D70C}=1$ . A further difference is a narrowing of the dispersion curve for the (reconnected) Alfvén mode, with the higher-frequency and lower-frequency portions of the dispersion curves moving closer together and closer to the light line. These intrinsically relativistic features become increasingly important as $\unicode[STIX]{x1D70C}$ decreases to $\ll 1$ .

5.1 L mode for $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$

Approximate forms for the dispersion relation for the L mode, given by (2.3), for large $z$ and for $z\approx 1$ , were given in the early literature (Lominadze & Mikhailovskiǐ Reference Lominadze and Mikhailovskiǐ1979; Lominadze et al. Reference Lominadze, Mikhaǐlovskiǐ and Sagdeev1979). These approximations are derived by expanding $z^{2}W(z)$ in powers of $1/z\ll 1$ and in powers of $|1-z|\ll 1$ , respectively, and retaining only the lowest-order terms,

(5.1) $$\begin{eqnarray}\displaystyle z^{2}W(z)\approx \left\{\begin{array}{@{}ll@{}}\displaystyle \left\langle {\displaystyle \frac{1}{\unicode[STIX]{x1D6FE}^{3}}}\right\rangle \left[1+{\displaystyle \frac{3}{z^{2}}}\left(1-{\displaystyle \frac{\langle \unicode[STIX]{x1D6FE}^{-5}\rangle }{\langle \unicode[STIX]{x1D6FE}^{-3}\rangle }}\right)\right]\approx {\displaystyle \frac{1}{\langle \unicode[STIX]{x1D6FE}\rangle }}\left(1+{\displaystyle \frac{3}{z^{2}}}\right),\quad & \text{for}~|z|\gg 1,\\[11.0pt] 2\langle \unicode[STIX]{x1D6FE}\rangle +(z^{2}-1)4\langle \unicode[STIX]{x1D6FE}^{3}\rangle \approx 2\langle \unicode[STIX]{x1D6FE}\rangle [1+12(z^{2}-1)\langle \unicode[STIX]{x1D6FE}\rangle ^{2}],\quad & \text{for}~|1-z^{2}|\ll 1,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where the final forms apply for a Jüttner distribution with $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$ (Melrose & Gedalin Reference Melrose and Gedalin1999),

(5.2a,b ) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FE}^{n}\rangle =n!\langle \unicode[STIX]{x1D6FE}\rangle ^{n},\quad \langle \unicode[STIX]{x1D6FE}^{-1}\rangle =\langle \unicode[STIX]{x1D6FE}\rangle ^{-1}[\ln (2\langle \unicode[STIX]{x1D6FE}\rangle )-0.577], & & \displaystyle\end{eqnarray}$$

(5.2c,d ) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FE}^{-3}\rangle \approx \langle \unicode[STIX]{x1D6FE}\rangle ^{-1},\quad \langle \unicode[STIX]{x1D6FE}^{-5}\rangle \approx {\textstyle \frac{2}{3}}\langle \unicode[STIX]{x1D6FE}\rangle ^{-1}. & & \displaystyle\end{eqnarray}$$

We rederive the approximations (5.1) in  appendix D by expanding in powers of $1/z^{2}$ and $z-1$ , respectively. Although we find that the approximation for $z^{2}\gg 1$ is well justified, that for $z-1$ is based on an expansion that converges only for $z^{2}=1$ . We suggest that the approximation (5.1) for $|1-z^{2}|\ll 1$ should not be used, at least without further justification.

The cutoff frequency $\unicode[STIX]{x1D714}_{x}$ and the frequency $\unicode[STIX]{x1D714}_{1}$ are given by

(5.3a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{x}=\unicode[STIX]{x1D714}_{L}(\infty )=\langle 1/\unicode[STIX]{x1D6FE}^{3}\rangle \unicode[STIX]{x1D714}_{\text{p}},\quad \unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{L}(1)=(2\langle \unicode[STIX]{x1D6FE}\rangle -\langle 1/\unicode[STIX]{x1D6FE}\rangle )\unicode[STIX]{x1D714}_{\text{p}}, & & \displaystyle\end{eqnarray}$$

with $\langle 1/\unicode[STIX]{x1D6FE}^{3}\rangle \approx 1/\langle \unicode[STIX]{x1D6FE}\rangle$ for $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$ . For a cold plasma one obtains $\unicode[STIX]{x1D714}_{x}=\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{\text{p}}$ .

The dispersion relation, $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{L}}(z)$ , just above the cutoff frequency $\unicode[STIX]{x1D714}_{x}$ is then given by

(5.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\text{L}}^{2}\approx \unicode[STIX]{x1D714}_{x}^{2}+k_{\Vert }^{2}c^{2}, & & \displaystyle\end{eqnarray}$$

which reproduces a known result (Lominadze & Mikhailovskiǐ Reference Lominadze and Mikhailovskiǐ1979; Lominadze et al. Reference Lominadze, Mikhaǐlovskiǐ and Sagdeev1979; Melrose & Gedalin Reference Melrose and Gedalin1999). However, analogous approximate dispersion relations for $z^{2}\approx 1$ are questionable for the reason discussed above. Specifically, although one can evaluate $z^{2}W(z)$ and all its derivatives at $z^{2}=1$ , the Taylor series in $(z^{2}-1)$ involving these derivatives appears to have zero radius of convergence (appendix D).

5.2 O mode for $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$

The O mode has the same cutoff frequency, $\unicode[STIX]{x1D714}_{x}$ , as the L mode independent of $\unicode[STIX]{x1D703}$ . As $\unicode[STIX]{x1D703}$ increases the dispersion curve for the O mode deviates increasingly from that of the L mode frequency. One may rewrite (2.5) in the form

(5.5a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}(z,\unicode[STIX]{x1D703})={\displaystyle \frac{\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)}{1+a(z)\tan ^{2}\unicode[STIX]{x1D703}}},\quad a(z)={\displaystyle \frac{b}{z_{\text{A}}^{2}-z^{2}}}. & & \displaystyle\end{eqnarray}$$

One has $z^{2}>z_{\text{A}}^{2}+b\tan ^{2}\unicode[STIX]{x1D703}$ for the O mode, and hence $a(z)<0$ , which implies that the frequency is an increasing function of $\unicode[STIX]{x1D703}$ . Except for the small range $\unicode[STIX]{x1D703}^{2}\lesssim 2/\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$ the dispersion curve for the O mode is entirely in the superluminal range. The resonance condition $\unicode[STIX]{x1D6FD}_{\text{s}}=z$ , where $\unicode[STIX]{x1D6FD}_{\text{s}}c$ is the streaming speed, cannot be satisfied for the O mode except for this tiny range of angles. An approximate dispersion relation for the (superluminal) O mode follows by setting $z_{\text{A}}^{2}\rightarrow 1$ , $b\rightarrow 1$ in (5.5),

(5.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\text{O}}^{2}(z,\unicode[STIX]{x1D703})\approx \unicode[STIX]{x1D714}_{\text{L}}^{2}(z){\displaystyle \frac{(z^{2}-1)\cos ^{2}\unicode[STIX]{x1D703}}{z^{2}\cos ^{2}\unicode[STIX]{x1D703}-1}}, & & \displaystyle\end{eqnarray}$$

which applies for $z^{2}>1$ when $\unicode[STIX]{x1D6FD}_{\text{A}}^{2}\gg 1$ . The frequency, $\unicode[STIX]{x1D714}_{\text{O}}(1,\unicode[STIX]{x1D703})$ , at which the dispersion curve crosses the light line increases with increasing $\unicode[STIX]{x1D703}$ ,

(5.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\text{O}}^{2}(1,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D714}_{1}^{2}{\displaystyle \frac{1-z_{\text{A}}^{2}}{1-z_{\text{A}}^{2}-b\tan ^{2}\unicode[STIX]{x1D703}}}\approx {\displaystyle \frac{\unicode[STIX]{x1D714}_{1}^{2}}{1-\unicode[STIX]{x1D6FD}_{\text{A}}^{2}\unicode[STIX]{x1D703}^{2}/2}}, & & \displaystyle\end{eqnarray}$$

where the approximation applies for $\unicode[STIX]{x1D6FD}_{\text{A}}^{2}\gg 1$ , $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$ and $\unicode[STIX]{x1D703}^{2}\ll 1$ .

An approximate dispersion relation for the O mode for $z=1/N\cos \unicode[STIX]{x1D703}\gg 1$ and $\unicode[STIX]{x1D714}^{2}\gg \unicode[STIX]{x1D714}_{\text{L}}^{2}(z=1/N\cos \unicode[STIX]{x1D703})$ that follows from (4.7) using (5.1) is

(5.8) $$\begin{eqnarray}\displaystyle N_{\text{O}}^{2}\approx \frac{1-\unicode[STIX]{x1D714}_{\text{p}}^{2}\sin ^{2}\unicode[STIX]{x1D703}/\langle \unicode[STIX]{x1D6FE}\rangle \unicode[STIX]{x1D714}^{2}}{1+3\unicode[STIX]{x1D714}_{\text{p}}^{2}\sin ^{2}\unicode[STIX]{x1D703}\cos ^{2}\unicode[STIX]{x1D703}/\langle \unicode[STIX]{x1D6FE}\rangle \unicode[STIX]{x1D714}^{2}}\approx 1-{\displaystyle \frac{\unicode[STIX]{x1D714}_{\text{p}}^{2}}{\langle \unicode[STIX]{x1D6FE}\rangle \unicode[STIX]{x1D714}^{2}}}[\sin ^{2}\unicode[STIX]{x1D703}(1+3\cos ^{2}\unicode[STIX]{x1D703})], & & \displaystyle\end{eqnarray}$$

where the final expression applies for $\unicode[STIX]{x1D714}^{2}\gg (1+3\cos ^{2}\unicode[STIX]{x1D703})\unicode[STIX]{x1D714}_{\text{p}}^{2}\sin ^{2}\unicode[STIX]{x1D703}/\langle \unicode[STIX]{x1D6FE}\rangle$ .

5.3 Alfvén mode for $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$

The turnover branch of the Alfvén mode for $z_{\text{A}}>z_{\text{m}}$ and $\unicode[STIX]{x1D70C}\ll 1$ is dominated by the peak in $z^{2}W(z)$ . For the case $\unicode[STIX]{x1D70C}=1$ shown in figure 7, the Alfvén mode consists of a thin loop, with the higher-frequency and lower-frequency portions joining at a turnover that corresponds to the peak at $z=z_{\text{m}}$ in the RPDF. The maximum frequency of the Alfvén mode decreases with increasing $\unicode[STIX]{x1D703}$ approximately along the line $z=z_{\text{m}}$ . For $\unicode[STIX]{x1D70C}\approx 1/\langle \unicode[STIX]{x1D6FE}\rangle \ll 1$ the loop becomes narrower with decreasing $\unicode[STIX]{x1D70C}$ . The format adopted in figures 6 and 7 makes it increasingly difficult to illustrate the dispersive properties due to the curves being strongly concentrated near $z=1$ .

An alternative format is used in figure 8 to show the dispersion curves near $z=1$ : we plot $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\text{p}}$ on a log scale as a function of $(1-z)/\unicode[STIX]{x1D70C}^{2}$ for $\unicode[STIX]{x1D70C}=0.1$ , with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 1.2\times 10^{2}$ and $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 87$ , and $0.01$ , with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 1.2\times 10^{3}$ and $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 8.7\times 10^{2}$ , where $\unicode[STIX]{x1D6FE}_{\text{A}}$ is defined by (A 12). The solid curves correspond to the L mode for $\unicode[STIX]{x1D703}=0$ and the solid vertical line is the dispersion curve $z=z_{\text{A}}$ for the A mode. The curves to the left of $(1-z_{\text{A}})/\unicode[STIX]{x1D70C}^{2}$ correspond to the O mode and the curves to the right of $(1-z_{\text{A}})/\unicode[STIX]{x1D70C}^{2}$ correspond to the Alfvén mode. Each of $1-z_{\text{A}}\approx 1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$ , $1-z_{\text{m}}$ and $1-z_{0}$ is proportional to $\unicode[STIX]{x1D70C}^{2}\approx 1/\langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ as discussed in § 3.3, so that the locations of the left asymptote, the peak and the right asymptote of the Alfvén mode all scale with $\unicode[STIX]{x1D70C}^{2}$ and so coincide in the two figures.

Figure 8. Dispersion curves for $\unicode[STIX]{x1D703}=0$ (black solid), $0.25\unicode[STIX]{x1D70C}$  rad (black dashed), $0.5\unicode[STIX]{x1D70C}$  rad (black dotted), $0.75\unicode[STIX]{x1D70C}$  rad (red solid) and $0.1\unicode[STIX]{x1D70C}$  rad (red dashed): (a) $\unicode[STIX]{x1D70C}=0.1$ with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 1.2\times 10^{2}$ , (b) $\unicode[STIX]{x1D70C}=0.01$ with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 1.2\times 10^{3}$ . The black solid curve corresponds to the L mode, the solid vertical line at $z=z_{\text{A}}$ corresponds to the A mode with the O mode to its upper left and the Alfvén mode to its lower right. The Alfvén mode exists between $z=z_{\text{A}}$ , which is very close to zero in the figure with $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 87$ for $\unicode[STIX]{x1D70C}=0.1$ and $\unicode[STIX]{x1D6FE}_{\text{A}}=8.7\times 10^{2}$ for $\unicode[STIX]{x1D70C}=0.01$ , and $z=z_{0}$ . The maximum in the dispersion curve occurs near $z=z_{\text{m}}$ .

The Alfvén mode (for $\unicode[STIX]{x1D703}\neq 0$ ) exists only for $z<z_{\text{A}}$ or $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}<\unicode[STIX]{x1D6FE}_{\text{A}}\approx \unicode[STIX]{x1D6FD}_{\text{A}}$ . Its dispersion relation, given by (5.5) with $a(z)>0$ , becomes

(5.9a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\text{A}}^{2}(z,\unicode[STIX]{x1D703})={\displaystyle \frac{\unicode[STIX]{x1D714}_{\text{L}}^{2}(z)}{1+a(z)\unicode[STIX]{x1D703}^{2}}},\quad a(z)\approx {\displaystyle \frac{\unicode[STIX]{x1D6FD}_{\text{A}}^{2}\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}^{2}}{\unicode[STIX]{x1D6FD}_{\text{A}}^{2}-\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}^{2}}}. & & \displaystyle\end{eqnarray}$$

The Alfvén mode is in the range where Landau damping is non-zero. Assuming that the waves are weakly damped only for $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}^{2}\gtrsim \unicode[STIX]{x1D6FE}_{\text{m}}^{2}$ , the approximations $z_{\text{m}}\approx 1-0.013\unicode[STIX]{x1D70C}^{2}$ , given by equation table 1, and $\langle \unicode[STIX]{x1D6FE}\rangle \approx 1/\unicode[STIX]{x1D70C}$ imply that the waves are weakly damped for $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}\gtrsim 6\,\langle \unicode[STIX]{x1D6FE}\rangle$ .

5.4 Condition for reconnection of modes

The dispersion curves for parallel propagation, $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{L}}(z)$ and $z=z_{\text{A}}$ , do not necessarily cross in a relativistic plasma. There are three possibilities:

1. (i) $z_{\text{A}}>z_{\text{m}}$ : the line $z=z_{\text{A}}$ crosses the higher-frequency portion of the L mode dispersion curve $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{L}}(z)$ in the range $1<z<z_{\text{m}}$ ; reconnection in this case is similar to that in the cold-plasma limit. The peak in the dispersion curve is in the (reconnected) Alfvén mode.

2. (ii) $z_{\text{m}}>z_{\text{A}}>z_{0}$ : the line $z=z_{\text{A}}$ crosses the higher-frequency portion $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{L}}(z)$ in the range $z_{\text{m}}>z_{\text{A}}>z_{0}$ .

3. (iii) $z_{\text{A}}<z_{0}$ : the line $z=z_{\text{A}}$ does not cross $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{L}}(z)$ .

Simple estimates for a pulsar magnetosphere suggest very large values, $\unicode[STIX]{x1D6FD}_{\text{A}}\gg 1$ , but before concentrating on this case we comment on the second and third cases.

Figure 9. Similar to figure 8 but for $z_{\text{m}}>z_{\text{A}}>z_{0}$ and as a function of angle in steps of $0.0625\unicode[STIX]{x1D70C}$  rad: (a) $\unicode[STIX]{x1D70C}=0.1$ with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 36$ and $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 25$ , (b) $\unicode[STIX]{x1D70C}=0.01$ with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 3.6\times 10^{2}$ and $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 2.5\times 10^{2}$ .

5.4.1 Case $z_{\text{m}}>z_{\text{A}}>z_{0}$

Examples of the dispersion curves for $z_{\text{m}}>z_{\text{A}}>z_{0}$ are shown in figure 9 for $\unicode[STIX]{x1D70C}=0.1$ and $\unicode[STIX]{x1D70C}=0.01$ . For $\unicode[STIX]{x1D703}=0$ , the solid curve corresponds to the L mode, and the solid vertical line corresponds to the A mode at $z=z_{\text{A}}$ . The maximum in the solid curve is at $z=z_{\text{m}}$ , to the left of $z=z_{\text{A}}$ . This peak leads to a maximum and a minimum in the O mode dispersion curve for small $\unicode[STIX]{x1D703}$ . The maximum and minimum become smoothed out with increasing $\unicode[STIX]{x1D703}$ , and the dispersion curves are then qualitatively similar to those for $z_{\text{A}}>z_{\text{m}}$ when $z_{\text{A}}^{2}+b\tan ^{2}\unicode[STIX]{x1D703}>z_{\text{m}}^{2}$ . As $\unicode[STIX]{x1D703}$ increases the Alfvén mode moves downward, with asymptotes at $z=z_{\text{A}}$ and $z=z_{0}$ remaining fixed. Only the branch with $z\approx z_{\text{A}}$ is Alfvén like, and we refer to the branch that links the asymptotic forms as the turnover branch. Landau damping becomes important in the range $z_{\text{m}}>z>z_{0}$ , and this damping needs to be taken into account in a more detailed discussion of this case.

Figure 10. Similar to figure 8 but for $z_{\text{A}}<z_{0}$ and as a function of angle in steps of $0.45\unicode[STIX]{x1D70C}$  rad: (a) $\unicode[STIX]{x1D70C}=0.1$ with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 7.1$ and $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 5$ , (b) $\unicode[STIX]{x1D70C}=0.01$ with $\unicode[STIX]{x1D6FD}_{\text{A}}\approx 71$ and $\unicode[STIX]{x1D6FE}_{\text{A}}\approx 50$ .

6.1 Ratio of electrical to total energy

In general the energy in waves in a specific mode in a dispersive medium may be separated into electric, magnetic and kinetic contributions, with the kinetic contribution attributed to the perturbations in the motion of particles forced by the wave (e.g. Melrose & McPhedran Reference Melrose and McPhedran1991, chap. 15). The total energy, which is the sum of the three, is determined by the dispersion theory, allowing one to infer the kinetic energy contribution. The magnetic energy is zero in a longitudinal wave. The ratio of the electric to total energy, denoted as $R_{\text{L}}(z)$ for the L mode, is relevant because the rate at which work is done by any source term involves only the electric energy, but the energy that appears in waves is the total energy.

The ratio $R_{\text{L}}(z)$ is evaluated in terms of $W(z)$ in appendix C. Using the form (A 8) for the RPDF, this gives

(6.1) $$\begin{eqnarray}\displaystyle R_{\text{L}}(z)=-{\displaystyle \frac{W(z)}{z\,\text{d}W(z)/\text{d}z}}={\displaystyle \frac{1}{2z^{2}}}\left\langle {\displaystyle \frac{z^{2}+\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D6FE}^{3}(z^{2}-\unicode[STIX]{x1D6FD}^{2})^{2}}}\right\rangle \left\langle {\displaystyle \frac{z^{2}+3\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D6FE}^{3}(z^{2}-\unicode[STIX]{x1D6FD}^{2})^{3}}}\right\rangle ^{-1}, & & \displaystyle\end{eqnarray}$$

which is strictly valid only for superluminal waves, $z\geqslant 1$ . For $z\gg 1$ and $z\rightarrow 1$ (6.1) gives

(6.2a,b ) $$\begin{eqnarray}\displaystyle R_{\text{L}}(z)\approx {\displaystyle \frac{1}{2}}-{\displaystyle \frac{3z^{2}+1}{2z^{2}(z^{2}+3)}},\quad R_{\text{L}}(1)\approx {\displaystyle \frac{\langle \unicode[STIX]{x1D6FE}\rangle }{4\langle \unicode[STIX]{x1D6FE}^{3}\rangle }}\approx {\displaystyle \frac{1}{24\langle \unicode[STIX]{x1D6FE}\rangle ^{2}}}, & & \displaystyle\end{eqnarray}$$

respectively, where the final approximation applies for a Jüttner distribution, $\langle \unicode[STIX]{x1D6FE}^{n}\rangle =n!\langle \unicode[STIX]{x1D6FE}\rangle ^{n}$ for $\langle \unicode[STIX]{x1D6FE}\rangle \gg 1$ . One finds that $R_{\text{L}}(z)$ decreases with decreasing $z$ , from $1/2$ in the limit $z\rightarrow \infty$ , to $1/24\langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ at $z=1$ . For $z$ in the range $1\gtrsim z\geqslant z_{\text{m}}\approx 1-0.13\unicode[STIX]{x1D70C}^{2}$ it is convenient to plot the inverse, $1/R_{\text{L}}(z)$ , rather than $R_{\text{L}}(z)$ itself, as illustrated in figure 11. The form of $1/R_{\text{L}}(z)\propto 1/\unicode[STIX]{x1D70C}^{2}\approx \langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ scales with $(1-z)/\unicode[STIX]{x1D70C}^{2}$ , increasing with decreasing $z$ through its value ${\approx}24\langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ at $z=1$ to a maximum ${\approx}38\langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ at $z\approx 1-0.004\unicode[STIX]{x1D70C}^{2}$ , and then decreasing with decreasing $z$ , passing through zero at $z=z_{\text{m}}$ and becoming negative in the region $z<z_{\text{m}}$ ; as already remarked, we do not consider solutions in the region $z<z_{\text{m}}$ where Landau damping is strong.

6.2 Group speed

The velocity of energy propagation of waves in a dispersive medium may be identified as the ratio of the energy flux to the energy density, which is the group velocity, written here as $\unicode[STIX]{x1D6FD}_{\text{g}}c$ . It is shown in  appendix C that $\unicode[STIX]{x1D6FD}_{\text{g}}$ for the L mode is given by

(6.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\text{g}}={\displaystyle \frac{\text{d}[z^{2}W(z)]/\text{d}z}{z\,\text{d}W(z)/\text{d}z}}=z[1-2R_{\text{L}}(z)]. & & \displaystyle\end{eqnarray}$$

Figure 12. The function $\unicode[STIX]{x1D70C}^{2}/[1-\unicode[STIX]{x1D6FD}_{\text{g}}(z)]$ is plotted over $(1-z)/\unicode[STIX]{x1D70C}^{2}$ for $\unicode[STIX]{x1D70C}=0.1$ (solid) and 0.01 (dashed). The two curves are indistinguishable over the scale shown. The thin dotted vertical line indicates the light line.

Approximations for $z\gg 1$ and $z=1$ are

(6.4a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\text{g}}(z)\approx {\displaystyle \frac{1}{z}}{\displaystyle \frac{3z^{2}+1}{z^{2}+3}},\quad \unicode[STIX]{x1D6FD}_{\text{g}}(1)\approx 1-{\displaystyle \frac{1}{12\langle \unicode[STIX]{x1D6FE}\rangle ^{2}}}, & & \displaystyle\end{eqnarray}$$

respectively. Thus, the group speed is zero at the cutoff frequency $\unicode[STIX]{x1D714}_{x}$ (where $z\rightarrow \infty$ ), and it increases with decreasing $z>1$ , approaching unity, but remaining subluminal ( $\unicode[STIX]{x1D6FD}_{\text{g}}<1$ ) for $z\rightarrow 1$ . For $1>z\gg z_{\text{m}}$ the final expression in (6.3), and the form of $R_{\text{L}}(z)$ shown in figure 11, imply that $\unicode[STIX]{x1D6FD}_{\text{g}}$ gets closer to unity with decreasing $z$ , reaching its maximum value at the maximum of $R_{\text{L}}(z)$ ; it then decreases to zero at $z=z_{\text{m}}$ , as shown in figure 12. It follows from the scaling in the figure that $(1-\unicode[STIX]{x1D6FD}_{\text{g}})/(1-z)$ is approximately independent of $\unicode[STIX]{x1D70C}$ for $\unicode[STIX]{x1D70C}\ll 1$ .

We do not discuss the region $z<z_{\text{m}}$ in detail, but remark that (6.3) has properties that preclude the interpretation of $\unicode[STIX]{x1D6FD}_{\text{g}}$ as the speed of (wave) energy propagation for $z<z_{\text{m}}$ : it becomes negative for $z<z_{\text{m}}$ , singular at the zero of $\text{d}W(z)/\text{d}z$ , where it changes sign to very large (superluminal group speed) and positive for smaller $z$ .

6.3 Wave properties for $1\geqslant z\gg z_{\text{m}}$

The only waves that can satisfy the resonance condition for a beam-driven instability in a pulsar plasma have properties that are quite different from those of Langmuir waves that can be beam driven in a non-relativistic plasma. Here we comment briefly on three unusual properties of the parallel-propagating waves in the L mode: the dispersion relation, the ratio of electric to total energy and the group speed.

In is convenient to write the condition $z=\unicode[STIX]{x1D6FD}_{\text{b}}$ for resonance with a beam with speed $\unicode[STIX]{x1D6FD}_{\text{b}}$ in terms of the corresponding Lorentz factor $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D6FE}_{\text{b}}$ . For beam-driven growth to overcome Landau damping in the background plasma requires $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}\gg \unicode[STIX]{x1D6FE}_{\text{m}}$ , with $\unicode[STIX]{x1D6FE}_{\text{m}}\approx 6\langle \unicode[STIX]{x1D6FE}\rangle$ for $\unicode[STIX]{x1D70C}\lesssim 1$ . In the example shown in figure 8, this corresponds to a tiny range of the solid black dispersion curve: to the right of $(1-z)/\unicode[STIX]{x1D70C}^{2}=0$ and to the left of the peak in the curve at $z=z_{\text{m}}$ . Over this range $\unicode[STIX]{x1D714}_{\text{L}}(z)$ is a rapidly decreasing function of $z$ , with

(6.5) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{1}{\unicode[STIX]{x1D714}_{\text{L}}(z)}}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D714}_{\text{L}}(z)}{\text{d}z}}={\displaystyle \frac{1}{2z^{2}W(z)}}{\displaystyle \frac{\text{d}[z^{2}W(z)]}{\text{d}z}}={\displaystyle \frac{1}{z}}\left[1-{\displaystyle \frac{1}{2R_{\text{L}}(z)}}\right]\approx -{\displaystyle \frac{1}{2R_{\text{L}}(z)}}, & & \displaystyle\end{eqnarray}$$

where the approximation applies for $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D719}}\gg \unicode[STIX]{x1D6FE}_{\text{m}}$ . With $1/2R_{\text{L}}(z)$ ranging between $12\langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ and $19\langle \unicode[STIX]{x1D6FE}\rangle ^{2}$ for $z<1$ , cf. figure 11, it follows that the frequency $\unicode[STIX]{x1D714}_{\text{L}}(z)$ is a very strong function of $z$ over the tiny range of $z$ where resonant wave growth is possible. This is quite different from Langmuir waves, for which the frequency is only a weak function over a wide range of phase speeds. It is also quite different from ion (or electron) acoustic waves, for which the phase speed is approximately equal to the ion (or electron) sound speed.

A second feature that is not ‘Langmuir like’ is that the ratio of electric to total energy $R_{\text{L}}(z)$ is very small for L mode waves in the range $1>z\gg z_{\text{m}}$ . In contrast, for Langmuir waves the ratio is approximately $1/2$ , corresponding to approximate equipartition between electric energy and kinetic energy of forced motions in the wave. The very small value of $R_{\text{L}}(z)$ is unusual when compared with Langmuir waves, but not when compared with ion acoustic waves, which have $R_{\text{L}}\approx k^{2}\unicode[STIX]{x1D706}_{\text{De}}^{2}\ll 1$ , where $k$ is the wavenumber and $\unicode[STIX]{x1D706}_{\text{De}}$ is the electron Debye length.

The group speed for L waves is only marginally subluminal, $1-\unicode[STIX]{x1D6FD}_{\text{g}}(z)\ll 1$ , in the relevant range, $1>z\gg z_{\text{m}}$ . The small group speed for Langmuir waves implies that the wave energy that grows in a given region due to a beam-driven instability remains localized to that region as the beam propagates through it. However, $\unicode[STIX]{x1D6FD}_{\text{g}}(z)\approx 1$ for the relevant L mode waves implies that the wave energy propagates away at nearly the speed of light, impeding any wave growth.