Wave dispersion in pulsar plasma. Part 1. Plasma rest frame

Wave dispersion in a pulsar plasma (a one-dimensional, strongly magnetized, pair plasma streaming highly relativistically with a large spread in Lorentz factors in its rest frame) is discussed, motivated by interest in beam-driven wave turbulence and the pulsar radio emission mechanism. In the rest frame of the pulsar plasma there are three wave modes in the low-frequency, non-gyrotropic approximation. For parallel propagation (wave angle $\unicode[STIX]{x1D703}=0$ ) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation $|z|=z_{\text{A}}\approx 1-1/2\unicode[STIX]{x1D6FD}_{\text{A}}^{2}$ , where $z=\unicode[STIX]{x1D714}/k_{\Vert }c$ is the phase speed and $\unicode[STIX]{x1D6FD}_{\text{A}}c$ is the Alfvén speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function, $z^{2}W(z)$ , which is negative for $|z|<z_{0}$ and has a sharp maximum at $|z|=z_{\text{m}}$ , with $1-z_{\text{m}}<1-z_{0}\ll 1$ . We give numerical estimates for the maximum of $z^{2}W(z)$ and for $z_{\text{m}}$ and $z_{0}$ for a one-dimensional Jüttner distribution. The L and A modes reconnect, for $z_{\text{A}}>z_{0}$ , to form the O and Alfvén modes for oblique propagation ( $\unicode[STIX]{x1D703}\neq 0$ ). For $z_{\text{A}}<z_{0}$ the Alfvén and O mode curves reconnect forming a new mode that exists only for $\tan ^{2}\unicode[STIX]{x1D703}\gtrsim z_{0}^{2}-z_{\text{A}}^{2}$ . The L mode is the nearest counterpart to Langmuir waves in a non-relativistic plasma, but we argue that there are no ‘Langmuir-like’ waves in a pulsar plasma, identifying three features of the L mode (dispersion relation, ratio of electric to total energy and group speed) that are not Langmuir like. A beam-driven instability requires a beam speed equal to the phase speed of the wave. This resonance condition can be satisfied for the O mode, but only for an implausibly energetic beam and only for a tiny range of angles for the O mode around $\unicode[STIX]{x1D703}\approx 0$ . The resonance is also possible for the Alfvén mode but only near a turnover frequency that has no counterpart for Alfvén waves in a non-relativistic plasma.

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