Amplitude equations for electrostatic waves: Universal singular behavior in the limit of weak instability
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (United States)
An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the unstable manifold of the equilibrium {ital F}{sub 0}({ital v}). The mode eigenvalue arises from a simple zero of the dielectric {epsilon}{sub {ital k}}({ital z}); as the linear growth rate {gamma} vanishes, the eigenvalue merges with the continuous spectrum on the imaginary axis and disappears. The evolution of the mode amplitude {rho}({ital t}) is studied using an expansion in {rho}. As {gamma}{r_arrow}0{sup +}, the expansion coefficients diverge, but these singularities are absorbed by rescaling the amplitude: {rho}({ital t}){equivalent_to}{gamma}{sup 2}{ital r}({gamma}{ital t}). This renders the theory finite and also indicates that the electric field exhibits trapping scaling {ital E}{similar_to}{gamma}{sup 2}. These singularities and scalings are independent of the specific {ital F}{sub 0}({ital v}) considered. The asymptotic dynamics of {ital r}({tau}) can depend on {ital F}{sub 0} only through exp {ital i}{xi} where {ital d}{epsilon}{sub {ital k}}/{ital dz}={vert_bar}{epsilon}{sub {ital k}}{sup {prime}}{vert_bar}exp{minus}{ital i}{xi}/2. Similar results also hold for the electric field and distribution function. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.
- OSTI ID:
- 164964
- Journal Information:
- Physics of Plasmas, Vol. 2, Issue 1; Other Information: PBD: Jan 1995
- Country of Publication:
- United States
- Language:
- English
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