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 theta = 0) these are referred to as the X, A and L modes, with the X and A modes having dispersion relation vertical bar z vertical bar = z(A) approximate to 1 - 1/2 beta(2)(A), where z = omega/k(parallel to)c is the phase speed and beta(A)c is the Alfven speed. The L mode dispersion relation is determined by a relativistic plasma dispersion function, z(2)W(z), which is negative for vertical bar z vertical bar < z(0) and has a sharp maximum at vertical bar z vertical bar = z(m), with 1 - z(m) < 1 - z(0) << 1. We give numerical estimates for the maximum of z(2)W(z) and for z(m) and z(0) for a one-dimensional Juttner distribution. The L and A modes reconnect, for z(A) > z(0), to form the O and Alfven modes for oblique propagation (theta not equal 0). For z(A) < z(0) the Alfven and O mode curves reconnect forming a new mode that exists only for tan(2) theta greater than or similar to z(0)(2) - z(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 theta approximate to 0. The resonance is also possible for the Alfven mode but only near a turnover frequency that has no counterpart for Alfven waves in a non-relativistic plasma.