dorsal/arxiv
View SchemaUniversal trapping scaling on the unstable manifold for a collisionless electrostatic mode
| Authors | John David Crawford |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9407001 |
| URL | https://arxiv.org/abs/patt-sol/9407001 |
| DOI | 10.1103/PhysRevLett.73.656 |
| Journal | Phys. Rev. Lett. 73, 656 (1994) |
Abstract
An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the two-dimensional unstable manifold of the equilibrium. The mode amplitude $\rho(t)$ decouples from the phase due to the spatial homogeneity of the equilibrium, and the resulting one-dimensional dynamics is analyzed using an expansion in $\rho$. As the linear growth rate $\gamma$ vanishes, the expansion coefficients diverge; a rescaling $\rho(t)\equiv\gamma^2\,r(\gamma t)$ of the mode amplitude absorbs these singularities and reveals that the mode electric field exhibits trapping scaling $|E_1|\sim\gamma^2$ as $\gamma\rightarrow0$. The dynamics for $r(\tau)$ depends only on the phase $e^{i\xi}$ where $d\epsilon_{{k}} /dz=|{\epsilon_{{k}}}|e^{-i\xi/2}$ is the derivative of the dielectric as $\gamma\rightarrow0$.
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"abstract": "An amplitude equation for an unstable mode in a collisionless plasma is\nderived from the dynamics on the two-dimensional unstable manifold of the\nequilibrium. The mode amplitude $\\rho(t)$ decouples from the phase due to the\nspatial homogeneity of the equilibrium, and the resulting one-dimensional\ndynamics is analyzed using an expansion in $\\rho$. As the linear growth rate\n$\\gamma$ vanishes, the expansion coefficients diverge; a rescaling\n$\\rho(t)\\equiv\\gamma^2\\,r(\\gamma t)$ of the mode amplitude absorbs these\nsingularities and reveals that the mode electric field exhibits trapping\nscaling $|E_1|\\sim\\gamma^2$ as $\\gamma\\rightarrow0$. The dynamics for $r(\\tau)$\ndepends only on the phase $e^{i\\xi}$ where $d\\epsilon_{{k}}\n/dz=|{\\epsilon_{{k}}}|e^{-i\\xi/2}$ is the derivative of the dielectric as\n$\\gamma\\rightarrow0$.",
"arxiv_id": "patt-sol/9407001",
"authors": [
"John David Crawford"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevLett.73.656",
"journal_ref": "Phys. Rev. Lett. 73, 656 (1994)",
"title": "Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode",
"url": "https://arxiv.org/abs/patt-sol/9407001"
},
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