The bump-in-tail instability

Many microinstabilities have both reactive and kinetic forms. From a mathematical viewpoint one treats the reactive form by ignoring the imaginary part of the dielectric tensor and solving a real dispersion equation to find complex solutions, and one treats the kinetic form by assuming real frequencies to a first approximation and then including weak (negative) damping. To be more specific, in the weak-beam instability (§3.3) thermal motions are neglected and the correction to the real part of the dielectric tensor due to the presence of trie beam leads to a cubic equation (3.11) for the frequency shift of the Langmuif waves; this cubic equation has a real solution and a pair of complex conjugate solutions in the regime of interest. The kinetic version of this instability is known as the bump-in-tail instability. It is treated by first finding the imaginary contribution of the beam to the dielectric tensor and using this to evaluate the imaginary part of the frequency shift.

It is apparent from the foregoing discussion that the reactive and kinetic versions should be limiting cases of a single instability. This may be shown by finding both the real and imaginary parts of the frequency shift simultaneously. The real and imaginary parts of the frequency can be found as a complex solution for ω as a function of real k to the complex dispersion equation KL(ω, k) = 0, where both real and imaginary parts of KL are retained.