dorsal/arxiv
View SchemaWeakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves
| Authors | S. Phibanchon, M. A. Allen, G. Rowlands |
|---|---|
| Categories | |
| ArXiv ID | physics/0701287 |
| URL | https://arxiv.org/abs/physics/0701287 |
| DOI | 10.1017/S002237780700640X |
| Journal | Journal of Plasma Physics 73(6), 933-946 (2007) |
Abstract
We determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel-Korteweg-de Vries-Zakharov-Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio $b$ increases as the electron distribution becomes increasingly flat-topped. As $b$ and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle $\theta$ at which the perturbation is applied. Solutions whose minimum value is zero and travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of $\theta$ for which the first-order growth rate is not zero.
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"abstract": "We determine the growth rate of linear instabilities resulting from\nlong-wavelength transverse perturbations applied to periodic nonlinear wave\nsolutions to the Schamel-Korteweg-de Vries-Zakharov-Kuznetsov (SKdVZK) equation\nwhich governs weakly nonlinear waves in a strongly magnetized cold-ion plasma\nwhose electron distribution is given by two Maxwellians at slightly different\ntemperatures. To obtain the growth rate it is necessary to evaluate non-trivial\nintegrals whose number is kept to minimum by using recursion relations. It is\nshown that a key instance of one such relation cannot be used for classes of\nsolution whose minimum value is zero, and an additional integral must be\nevaluated explicitly instead. The SKdVZK equation contains two nonlinear terms\nwhose ratio $b$ increases as the electron distribution becomes increasingly\nflat-topped. As $b$ and hence the deviation from electron isothermality\nincreases, it is found that for cnoidal wave solutions that travel faster than\nlong-wavelength linear waves, there is a more pronounced variation of the\ngrowth rate with the angle $\\theta$ at which the perturbation is applied.\nSolutions whose minimum value is zero and travel slower than long-wavelength\nlinear waves are found, at first order, to be stable to perpendicular\nperturbations and have a relatively narrow range of $\\theta$ for which the\nfirst-order growth rate is not zero.",
"arxiv_id": "physics/0701287",
"authors": [
"S. Phibanchon",
"M. A. Allen",
"G. Rowlands"
],
"categories": [
"physics.plasm-ph",
"nlin.PS"
],
"doi": "10.1017/S002237780700640X",
"journal_ref": "Journal of Plasma Physics 73(6), 933-946 (2007)",
"title": "Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves",
"url": "https://arxiv.org/abs/physics/0701287"
},
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