Virial Theory of Direct Langmuir Collapse

Abstract
A general virial theorem is proved without any assumption of special symmetries and used to calculate the threshold and time for direct collapse of a two-dimensional Langmuir wave packet. The analysis is shown to apply to "bump-on-tail" instabilities, when the "bump" is broad, low, and at high velocity. The example of the type-III solar radioburst problem is treated both numerically and analytically.