dorsal/arxiv
View SchemaAmplitude equations for coupled electrostatic waves in the limit of weak instability
| Authors | John David Crawford, Edgar Knobloch |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9708004 |
| URL | https://arxiv.org/abs/patt-sol/9708004 |
| DOI | 10.1017/S0022377898006540 |
Abstract
We consider the simplest instabilities involving multiple unstable electrostatic plasma waves corresponding to four-dimensional systems of mode amplitude equations. In each case the coupled amplitude equations are derived up to third order terms. The nonlinear coefficients are singular in the limit in which the linear growth rates vanish together. These singularities are analyzed using techniques developed in previous studies of a single unstable wave. In addition to the singularities familiar from the one mode problem, there are new singularities in coefficients coupling the modes. The new singularities are most severe when the two waves have the same linear phase velocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result the short wave mode saturates at a dramatically smaller amplitude than that predicted for the weak growth rate regime on the basis of single mode theory. In contrast the long wave mode retains the single mode scaling. If these resonance conditions are not satisfied both modes retain their single mode scaling and saturate at comparable amplitudes.
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"abstract": "We consider the simplest instabilities involving multiple unstable\nelectrostatic plasma waves corresponding to four-dimensional systems of mode\namplitude equations. In each case the coupled amplitude equations are derived\nup to third order terms. The nonlinear coefficients are singular in the limit\nin which the linear growth rates vanish together. These singularities are\nanalyzed using techniques developed in previous studies of a single unstable\nwave. In addition to the singularities familiar from the one mode problem,\nthere are new singularities in coefficients coupling the modes. The new\nsingularities are most severe when the two waves have the same linear phase\nvelocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result\nthe short wave mode saturates at a dramatically smaller amplitude than that\npredicted for the weak growth rate regime on the basis of single mode theory.\nIn contrast the long wave mode retains the single mode scaling. If these\nresonance conditions are not satisfied both modes retain their single mode\nscaling and saturate at comparable amplitudes.",
"arxiv_id": "patt-sol/9708004",
"authors": [
"John David Crawford",
"Edgar Knobloch"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1017/S0022377898006540",
"title": "Amplitude equations for coupled electrostatic waves in the limit of weak instability",
"url": "https://arxiv.org/abs/patt-sol/9708004"
},
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