Instabilities: Convective versus Absolute

Abstract

The response of a medium to a pulse disturbance is determined by the zeros of the dispersion function $D(k,\omega )$ in the complex domain of both the wave number $k$ and the frequency $\omega$, and the asymptotic nature of the response near the origin depends on the velocity $V$ of the frame of reference of the observer. In particular, a crucial feature for an unstable medium is the behavior, as a function of $V$, of the position of the saddle points of $\tilde{\omega }\mathrm{}\left(k\right)=\omega \mathrm{}\left(k\right)-kV$, defined by the requirements $D(k,\omega )=0, \frac{\partial D}{\partial k}+\frac{V\partial D}{\partial \omega }=0,$ since $\mathrm{Im}\tilde{\omega }$ for suitably defined saddle points gives the asymptotic growth in frame $V$. Thus, if $\frac{d^2\omega }{d{k}^2}$ is taken along the contour $D=0\mathrm{}$, the differential equations for the variation of a saddle point $\frac{\delta k}{\delta V}={\left(\frac{d^2\omega }{d{k}^2}\right)}^{-1\mathrm{}}, \frac{\delta \omega }{\delta V}=V{\left(\frac{d^2\omega }{d{k}^2}\right)}^{-1\mathrm{}}, \frac{\delta \tilde{\omega }}{\delta V}=-k$ (only two of which are independent) can be used to plot the asymptotic pulse shape. A new phenomenon of general interest to the theory of singularities of a function of two complex variables is described which, in the present context, is associated with the significance or insignificance of a given saddle point in contributing to instability, and with how this significance may change with $V$. This (mathematical) phenomenon may also be of relevance to elementary-particle theory through its inportance to the theory of the analytic structure of scattering amplitudes. The results of the theory are applied to certain examples of plasma physics, obtaining for these cases quantitative descriptions of the asymptotic shape of a pulse disturbance in unstable media.